tag:blogger.com,1999:blog-2516188730140164076Fri, 24 Jun 2016 15:14:10 +0000mathematical thinkingCommon CoreMOOCmathematics educationFibonacciLeonardo of Pisaalgebraonline educationCourseraDragonBoxJo BoalerKhan AcademyMOOCsPisaSteve Jobseducational video gamesgame-based learningproblem solvingAlpine Road TrailApple MacintoshJordan EllenbergKeith DevlinMultiplicationPi DayStanford Universityapplications of mathematicsarithmeticconceptual understandingeducational startupsinfinitymountain bikingpiproofsquantitative literacyscience of patternstransition courseAndrew HackerAndrew NgArea of a circle formulaAronszajn treeAsa ButterfieldAspen Ideas FestivalBedtime MathCreateSpaceDan MeyerDaphne KollerDavid LeonhardtDerek MullerETSEdge questionEinstein’s energy equationEric MazurErlwangerEuler’s IdentityEuler’s equationFinnish educationGOFMGeorg CantorGeorge BooleGregory’s SeriesH-STARHilbert's HotelHindu-arabic arithmeticHypercubesIMOITSInquiry-Based LearningIs algebra necessaryJames Paul GeeJohn BrockmanK-12 educationKarim AniKen RobinsonKönig’s LemmaLaura OverdeckLearnLabLiber abbaciLiping MaM.C. EscherMTT2KMarilyn BurnsMath and musicMath moviesMonument Valley gameMoore MethodNSAOECDPIAACPaul CohenPeer InstructionPhil DaroPolyaPythagoras’ equationR L MooreRiccardiana LibrarySERPSanfordSebastian ThrunSergio CorreaStandard algorithmSteven StrogatzSugata MitraTIMSSTeachingTest FlightUdacityUniversity of ChicagoUri TreismanVeritasiumWikipedia contributionWorld Science FestivalWuzzit TroubleYouCubedabstractional-Khwarizmiamygdalaan approach that sucksanswer gettingappreciation of mathematicsbeauty in mathematicsboolean algebraboolean logiccalculuscollaborative workingcomputer science educationcountable setcyclingdistance learningdot diagramseducation researcheducational appseducational strandseducational tsunamieducational videoselementary arithmeticequityexceptional performancefailureforcinggeometrygooglehigher educationinstructioninstructional videointelligence analysislarge cardinalslearning outcomeslearning scienceliteracymachine gradingmath and dancemath appsmath competitionsmath learningmath learning appmath learning video gamemath on TVmath video gamesmathematical mindsetmathematical modelmathematical modelingmathematical proofsmathematicsmathematics and societymathematics transition coursememesmultiple-choice questionsmusicnature of mathematicsnumber sensenumeracyonline learningpeer evaluationpop quizzespovertyprocedural masteryproof techniquespublish-on-demandquantitative reasoning requirementrepeated additionrepeated addition (not)representations of mathematicsscience videoself-publishingset theorysoft mathematicstalent searchtextbookstreesuncountable setDevlin's AngleDevlin's Angle is a monthly column sponsored by the Mathematical Association of America. Mathematician Keith Devlin is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guyon NPR's Weekend Edition.http://devlinsangle.blogspot.com/noreply@blogger.com (Mathematical Association of America)Blogger59125tag:blogger.com,1999:blog-2516188730140164076.post-5470449280026104921Tue, 07 Jun 2016 20:45:00 +00002016-06-08T09:12:33.013-04:00Aronszajn treecountable setHilbert's HotelinfinityKönig’s Lemmatreesuncountable setInfinity and IntuitionOn May 30, Gary Antonick’s always interesting Numberplay section in the New York Times featured a<a href="http://wordplay.blogs.nytimes.com/2016/05/30/frenkel-cantor/?_r=0"> contribution</a> by Berkeley mathematician Ed Frenkel on the difficulties the human mind can encounter when trying to come to grips with infinity. If you have not yet read it, you should.<br /><br />Infinity offers many results that are at first counter-intuitive. A classic example is Hilbert's Hotel, which has infinitely many rooms, each one labeled by a natural number printed on the door: Room 1, Room 2, Room 3, etc., all the way through the natural numbers. One night, a traveler arrives at the front desk only to be told be the clerk that the hotel is full. "But don't worry, sir," says the clerk, "I just took a mathematics course at my local college, and so I know how to find you a room. Just give me a minute to make some phone calls." And a short while later, the traveler has his room for the night. What the clerk did was ask every guest to move to the room with the room number the next integer. Thus, the occupant of Room 1 moved into Room 2, the occupant of Room 2 into Room 3, etc. Everyone moved room, no one was ejected from the hotel, and Room 1 became vacant for the newly arrived guest.<br /><br />This example is well known, and I expect all regular readers of <i>MAA Online</i> will be familiar with it. But I expect many of you will not know what happens when you step up one level of infinity. No sooner have you started to get the hang of the countable infinity (cardinality aleph-0), and you encounter the first uncountable infinity (cardinality aleph-1) and you find there are more surprises in store.<br /><br />One result that surprised me when I first came across it concerns trees. Not the kind the grow in the forest, but the mathematical kind, although there are obvious similarities, reflected in the terminology mathematicians use when studying mathematical trees.<br /><br />A tree is a partially ordered set (T,<) such that for every member x of T, the set {y in T : y < x} of elements below x in the tree is well ordered. This means that the tree has a preferred direction of growth (often represented as upwards in diagrams), and branching occurs only in the upward direction. It is generally assumed that a tree has a unique minimum element, called the root. (If you encounter a tree without such a root, you can simply add one, without altering the structure of the remainder of the tree.)<br /><br />Since each element of a tree lies at the top of a unique well ordered set of predecessors, it has a well defined height in the tree - the ordinal number of the set of predecessors. For each ordinal number k, we can denote by T_k the set of all elements of the tree of height k. T_k is called the k'th level of T. T_0 consists of the root of the tree, T_1 is the set of all immediate successors of the root, etc.<br /><br />Thus, the lower part of a tree might look something like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-V6yTO9KD2dA/V1czATl5u3I/AAAAAAAAKhc/2rbB4C0uGcADNvAltnT6YoEyFDrqqN3OQCLcB/s1600/tree.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-V6yTO9KD2dA/V1czATl5u3I/AAAAAAAAKhc/2rbB4C0uGcADNvAltnT6YoEyFDrqqN3OQCLcB/s1600/tree.jpg" /></a></div><br /><br />(It could be different. There is no restriction on how many elements there are on each level, or how many successors each member has.)<br /><br />A classical result of set theory, sometimes called König's Lemma, says that if T is an infinite tree, and if each level T_n, for n a natural number, is finite, then T has an infinite branch, i.e., an infinite linearly ordered subset.<br /><br />It's easy to prove this result. You define a branch {x_n : n a natural number} by recursion. To start, you take x_0 to be the root of the tree. Since the tree is infinite, but T_1 is finite, there is at least one member of T_1 that has infinitely many elements above it. Let x_1 be one such element of T_1. Since x_1 has infinitely many elements above it and yet only finitely many successors on T_2, there is at least one successor of x_1 on T_2 that has infinitely many elements above it. Let x_2 be such an element of T_2. Now define x_3 in T_3 analogously so it has infinitely many elements of the tree above it, and so on. This simple process clearly defines an infinite branch {x_n : n a natural number}.<br /><br />Having seen why König's Lemma holds, it's tempting to argue by analogy that if you have an uncountable tree T (i.e., a tree whose cardinality is at least aleph-1) and if every level T_k, for k a countable ordinal, is countable, then T has an uncountable branch, i.e., a linearly ordered subset that meets level T_k for every countable ordinal k.<br /><br />But it turns out that this cannot be proved. It is possible to construct an uncountable tree, all of whose levels T_k, for k a countable ordinal, are countable, for which there is no uncountable branch. Such trees are called Aronszajn trees, after the Russian mathematician who first constructed one.<br /><br />Here is how to construct an Aronszajn tree. The members of the tree are strictly increasing (finite and countably transfinite), bounded sequences of rational numbers. The tree ordering is sequence extension. It is immediate that such a tree could not have an uncountable branch, since its limit (more precisely, its set-theoretic union) would be an uncountable strictly increasing sequence of rationals, contrary to the fact that the rationals form a countable set.<br /><br />You build the tree by recursion on the levels. T_0 consists of the empty sequence. After T_k has been constructed, you get T_(k+1) by taking each sequence s in T_k and adding in every possible extension of s to a strictly increasing (k+1)-sequence of rationals. That is, for each s in T_k and for each rational number q greater than or equal to the supremum of s, you put into T_(k+1) the result of appending q to s. Being the countable union of countably many sets, T_(k+1) will itself be countable, as required.<br /><br />In the case of regular recursion on the natural numbers, that would be all there is to the definition, but with a recursion that goes all the way up through the countable ordinals, you also have to handle limit ordinals - ordinals that are not an immediate successor of any smaller ordinal.<br /><br />To facilitate the definition of the limit levels of the tree, you construct the tree so as to satisfy the following property, which I'll call the Aronszajn property: for every pair of levels T_k and T_m, where k < m, and for every s in T_k and every rational number q that exceeds the supremum of s, there is a sequence t in T_m which extends s and whose supremum is less than q.<br /><br />The definition of T_(k+1) from T_k that I just gave clearly preserves this property, since I threw in EVERY possible sequence extension of every member of T_k.<br /><br />Suppose now that m is a limit ordinal and we have defined T_k for every k < m. Given any member s of some level T_k for k < m, and any rational number q greater than the supremum of s, we define, by integer recursion, a path (s_i : i a natural number) through the portion of the tree already constructed, such that its limit (as a rational sequence) has supremum q.<br /><br />You first pick some strictly increasing sequence of rationals (q_i : i a natural number) such that q_0 exceeds the supremum of s and whose limit is q.<br /><br />You also pick some strictly increasing sequence (m_i : i a natural number) of ordinals less than m that has limit m and such that s lies below level m_0 in the tree.<br /><br />You can then use the Aronszajn property to construct the sequence (s_i : i a natural number) so that s_i is on level m_i and the supremum of s_i is less than q_i.<br /><br />Construct one such path (s_i : i a natural number) for every such pair s, q, and let T_k consist of the limit (as a sequence of rationals) of every sequence so constructed. Notice that T_k so defined is countable.<br /><br />It is clear that this definition preserves the Aronszajn property, and hence the construction may be continued.<br /><br />And that's it.<br /><br />NOTE: <i>The above article first appeared in Devlin’s Angle in January 2006. Seeing Frenkel’s Numberplay article prompted me to revive it and give it another airing.</i>http://devlinsangle.blogspot.com/2016/06/infinity-and-intuition.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4125738874907900223Wed, 04 May 2016 19:32:00 +00002016-05-04T15:33:15.004-04:00algebraCommon CoreDragonBoxITSLearnLabmathematical mindsetmathematics educationrepresentations of mathematicsAlgebraic Roots – Part 2What does it mean to “do algebra”? In Part 1, published here last month, I described how algebra (from the Arabic <i>al-Jabr</i>) began in 9th Century Baghdad as a way to approach arithmetical problems in a systematic way that scales. It was a way of thinking, using logical reasoning rather than (strictly speaking, in addition to) arithmetical calculation, and the first textbook on the subject explained how to solve problems that way using ordinary language, not symbolic expressions. Symbolic algebra was introduced later, in 16th Century France.<br /><br />Just as the formal algorithms of Hindu-Arabic arithmetic make it possible to do arithmetic in a purely procedural, rule-following way (without the need for any thought), so too symbolic algebra made it possible to solve algebraic problems by manipulating symbolic expressions using formal rules, again without the need for any thought.<br /><br />Over the ensuing centuries, schools focused more and more exclusively on the formal, procedural rules of arithmetic and symbolic algebra, driven in part by the needs of industry and commerce to have large numbers of people who could carry out computations for them, and in part for the convenience of the school system.<br /><br />Today, however, we have digital devices that carry out arithmetical and algebraic procedural calculations for us, faster and with greater accuracy, shifting society’s needs back to <i>arithmetical</i> and <i>algebraic thinking</i>. This is why you see the frequent use of those terms in educational circles these days, along with <i>number sense</i>. (All three terms are so common that definitions of each are easily found on the Web by searching on the name, as is also the case for the more general term <i>mathematical thinking</i>.)<br /><br />As more (and hopefully better) technological aids are developed, the nature of the activity involved in solving an arithmetical or algebraic problem changes, both for learning and for application. The fluent and effective use of arithmetical calculators, graphing calculators (such as <i>Desmos</i>), spreadsheets, computer algebra systems (such as <i>Mathematica</i> or <i>Maple</i>), and <i>Wolfram Alpha</i>, are now marketable skills and important educational goals. Each of these tools, and others, provides a different representation of numbers, numerical problems, and algebraic problems.<br /><br />One consequence of this shift that seemed to take an entire generation of parents off guard, is that mastery of the “traditional algorithms” for solving arithmetic and algebraic problems, which were developed to optimize human computations and at the same time create an audit trail, and which used to be the staple of school mathematics instruction, became a much less important educational goal. Instead, it is evidently far more valuable for today’s students to spend their time working with algorithms optimized to develop good arithmetical and algebraic thinking skills, that will (among other things) support fluent and effective use of the new technologies.<br /><br />I said “evidently” above, since to those of us in the education business, it was just that. With hindsight, however, it seems clear that in rolling out the Common Core State Standards, those in charge should have put much more effort into providing that important background context <i>that was evident to them</i> but, clearly, <i>not</i> evident to many people <i>not</i> working in mathematics education.<br /><br />I was not involved in the CCSS initiative, by the way, but I doubt I would have done any better. I still find it hard to wrap my mind round the fact that the “evident” (to me) need to modify mathematics education to today’s world is actually not at all evident to many of my fellow citizens—even though we all live and work in the same digital world. I guess it is a matter of the <i>educational perspective</i> those of us in the math ed business bring to the issues.<br /><br />But even those of us in the education business can sometimes overlook just how much, and how fast, things have changed. The most recent example comes from a highly respected learning research center, LearnLab in Pittsburgh (formerly called the Pittsburgh Science of Learning Center), funded by the National Science Foundation.<br /><br />The tweet shown below caught my eye a few weeks ago.<br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-EDt6JE0oojk/Vyoqv7wKDgI/AAAAAAAAKgs/YqCv_omwGB8uqA_STta-h6h8Ty0cCjbTwCLcB/s1600/Dragonbox.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="333" src="https://4.bp.blogspot.com/-EDt6JE0oojk/Vyoqv7wKDgI/AAAAAAAAKgs/YqCv_omwGB8uqA_STta-h6h8Ty0cCjbTwCLcB/s400/Dragonbox.PNG" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br />The tweet got my attention because I am familiar with DragonBox, and include it in the (very small) category of math learning apps I usually recommend. (I also know the creator, and have given occasional voluntary feedback on their development work, but I have no other connection to the company.)<br /><br />“Ineffective”? “#dragonboxfail”? Those are the words used in the tweet. <i>But neither can possibly be true</i>. DragonBox provides <i>an alternative representation</i> for linear equations in one unknown. Anyone who completes the game (for want of a better term) has demonstrated mastery of algebraic thinking for single variable linear problems. Period. (There is a separate issue of the representation that I will come to later.)<br /><br />Indeed, since the mechanics in DragonBox are essentially isomorphic to the rules of classical symbolic algebra (as taught in schools for the last four hundred years), completing the game demonstrates mastery of those mechanics too. From a logical perspective then, the tweet made no sense. All very odd for an official tweet from a respected, federally-funded research institute. Suspecting what must be going on, I looked further.<br /><br />The tweet was in response to a <a href="https://www.edsurge.com/news/2016-03-13-enter-the-dragonbox-can-a-game-really-teach-third-graders-algebra">review</a> of DragonBox, published by EdSurge. I recognized the name of the reviewer, Brady Fukumoto, a former game developer I had meet a few times. It was a well analyzed review. Overall, I agreed with everything Brady said. In particular, he spent some time comparing “doing algebra in the DragonBox representation” to “doing algebra using the traditional symbolic equations representation”, pointing out how much richer is the latter—but noting too that the former can result in higher levels of student engagement. Hardly the “promote” of a product that LearnLab accused him of. Indeed, Brady correctly summarized, and referenced (with a link) the Carnegie Mellon University <a href="http://www.cs.cmu.edu/~ylong/papers/Long&Aleven_ITS2014.pdf">study</a> the LearnLab tweet implicitly referred to.<br /><br />I recommend you read Brady’s review. It gets at many aspects of the “what does it mean to do algebra?” issue. As does playing DragonBox itself, which toward the end gradually replaces its initial “game representation” with the standard symbolic equation representation on a touch screen (a process often referred to as deconcretization).<br /><br />Unlike the tweet, the CMU paper was careful in stating its conclusion. The authors say, and Brady quotes, that they found DragonBox to be “ineffective in helping students acquire skills in solving algebra equations, <i>as measured by a typical test of equation solving</i>.” (The emphasis is mine.)<br /><br />Now we are at the root of that odd tweet. (One should not make too much of a tweet, of course. Twitter is an instant medium. But, rightly or wrongly, tweets in the name of an organization or a public figure are generally viewed as PR, presenting an authoritative, public stance.) The folks at LearnLab, their knowledge of educational technology notwithstanding, are assuming a perspective in which one particular representation of algebra is privileged; namely, the traditional symbolic one. (Which is the representation they adopt in developing their own algebra instruction app, an Intelligent Tutoring System called <i>Lynnette</i>.) But as I pointed out last month, that representation became the dominant one entirely by virtue of what was at that time the best available distribution technology: the printing press.<br /><br />With newer technologies, in particular the tablet computer (“printed paper on steroids”), other representations are possible, some better suited to learning, others to applications. To be sure, there are learning benefits to be gained from mastering symbolic algebra, perhaps even from doing so using paper-and-pencil, as Brady points out in his review. But at this stage in the representational technology development, we should adopt a perspective of all bets being off when it comes to how to best represent algebra in different contexts. I think it highly unlikely that we will ever again view algebra as something you learn or do exclusively by using a pen to pour symbols onto a page.<br /><br />Indeed, with his background in video game design, Brady ends his review by rating DragonBox according to three metrics:<br /><br /><b>Fun Factor – A:</b> I collected all 1,366 stars available in DragonBox 1 and 2 and had a great time.<br /><br /><b>Academic Value – B:</b> I worry that many will underestimate the effort needed to transfer DragonBox skills to practical algebra proficiency.<br /><br /><b>Educational Value – A+:</b> Anytime a kid leaves a game with thoughts like, “algebra is fun!” or “hey, I’m really good at math!” that is a huge win.<br /><br />The LearnLab researchers are locked into the second perspective: what he calls Academic Value. (So too is Brady, to some extent, with his use of the phrase “practical algebra proficiency” to mean “symbolic algebra proficiency.”)<br /><br />Make no mistake about it, transfer from mastery in an interactive engagement on a tablet to paper-and-pencil math is not automatic, as both Brady and the CMU researchers observe. To modify the old horse aphorism, DragonBox takes its players right to the water’s edge and dips their feet in, but still the players have difficulty drinking. (My best guess is that, for most learners it takes a good teacher to facilitate transfer.)<br /><br />I note in passing that initially I had difficulty playing DragonBox. My problem was, classical, symbolic algebra is a second language to me that I have been fluent in since childhood and use every day. I found it difficult mastering the corresponding actions in DragonBox. Transfer is difficult in both directions.<br /><br />At the present moment in time, those of us in education (or learning research) should absolutely not assume any one representation is privileged. Particularly so when it comes to learning. In that respect, Brady is right to note that DragonBox’s success in terms of his third metric (essentially, attitude and engagement) is indeed “a huge win.”<br /><br />In the world in which our students will live their lives, arithmetic, algebra, and many other parts of mathematics, should be learned, and will surely be applied, in multimedia environments. All the evidence available today suggests that mastery of the traditional symbolic representation will be a crucial ingredient in becoming proficient at arithmetic and algebra. But the more effective practitioners are likely to operate with the aid of various technological tools. Indeed, for some future practitioners, mastery of the traditional symbolic representation (which is, remember, just a user interface to a certain kind of thinking) may turn out to be primarily just a key step in the cognitive process of achieving conceptual understanding, not used directly in applications, which may all be by way of mathematical reasoning tools.<br /><br />Exactly when, in the initial learning process, it is best to introduce the classical symbolic representation is as yet unclear. What the evidence of countless generations of students-turned-parents makes abundantly clear, however, is that teaching only the classical symbolic approach is a miserable failure. That much is affirmed every time a parent posts on social media that they are unable to understand a Common Core math question that requires nothing more than understanding the place-value representation of integers. (Which is true of most of the ones I have seen posted.)<br /><br />There is some evidence (see for example Jo Boaler’s <a href="http://www.amazon.com/Mathematical-Mindsets-Unleashing-Potential-Innovative/dp/0470894520/ref=sr_1_1?ie=UTF8&qid=1462196234&sr=8-1&keywords=mathematical+mindsets">new book</a>) that a more productive approach is to use learning technologies to develop and sustain student engagement and develop a growth mindset, and provide learning environments for safe, productive failure, with the goal of developing number sense and general forms of creative problem solving (mathematical thinking), bringing in symbolic representations and specific techniques as and when required.<br /><br />**Full declaration: I should note that my own work in this area, some of it through my startup company <a href="http://www.brainquake.com/">BrainQuake</a>, adopts this philosophy. The significant <a href="http://documents.brainquake.com/backed-by-science/Stanford-Pope_Mangram_SUMMARY.pdf">learning gains </a> obtained with our <a href="http://wuzzittrouble.com/">first app</a> were in number sense and creative problem solving for a novel, complex performance task. Acquisition of traditional “basic skills” with our app comes about (intentionally, by design) as a valuable by-product. The improvement we see in the basic skills category is much more modest, and may well be better achieved by a tool such as LearnLab’s ITS. In a world where we have multiple representations, it is wise to make effective use of them all, according to context. It is not a case of an interface “fail”; to say that (with or without a hashtag) is to remain locked in past thinking. Easy to do, even for experts. Rather, in an era when algebra is being forced to return to its roots of being a way of thinking to help us solve practical problems, using all available representations in unison can provide us with a major win.http://devlinsangle.blogspot.com/2016/05/what-does-it-mean-to-do-algebra-in-part.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-7982488435981996827Mon, 04 Apr 2016 19:33:00 +00002016-04-04T15:54:32.484-04:00al-KhwarizmialgebraCommon Coremathematics educationAlgebraic roots – Part 1<div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/--sdywWmDWlA/VwKtXmK9fKI/AAAAAAAAKfo/XdGmL94Fr2Y9_nPd-OzAqsnUkrz2BEt2Q/s1600/Fig1.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="228" src="https://3.bp.blogspot.com/--sdywWmDWlA/VwKtXmK9fKI/AAAAAAAAKfo/XdGmL94Fr2Y9_nPd-OzAqsnUkrz2BEt2Q/s400/Fig1.jpeg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig. 1: A problem from the first ever algebra textbook.</td></tr></tbody></table>The first ever algebra text book was written in Baghdad around 830CE, by the Persian mathematician Muhammad ibn Musa al-Khwarizmi, our modern word “algebra” coming from the Arabic term al-Jabr, a technique for balancing an equation, described in the book. If you were a student – or a teacher – back then, the problem shown above (Figure 1) is the kind of thing you would be faced with in your math class. It is a direct translation from the original Arabic of a problem in al-Khwarizmi’s book.<br /><br />Most modern readers, on seeing this for the first time and being told it is an algebra problem, are surprised that there are no symbols. Yet it is clearly not an “algebra word problem” in the usual sense. It’s just about numbers. It is, in fact, a quadratic equation problem. Figure 2 below is the same problem as we would present it in an algebra textbook today.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-p5thdoqU-9I/VwKtnx4kZ5I/AAAAAAAAKfs/qp0hTRwi6GQn6qM852TwXSa60iPd0srtg/s1600/Fig2.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="196" src="https://4.bp.blogspot.com/-p5thdoqU-9I/VwKtnx4kZ5I/AAAAAAAAKfs/qp0hTRwi6GQn6qM852TwXSa60iPd0srtg/s320/Fig2.jpeg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig. 2: Al-Khwarizmi's quadratic equation in modern notation.</td></tr></tbody></table><br /><i>Symbolic </i>algebra, as we understand it today, was not introduced until the Sixteenth Century, when the French mathematician François Viète took what until then had been a discipline presented in prose, and turned it into the symbolic process we are familiar with today.<br /><br />This is not to say that mathematicians back in Ninth Century Persia did not use symbolic expressions in their work. They surely did. The issue is how they presented it in textbook form. In the days when books were handwritten and duplicated by hand-copying, the author of a mathematics book was faced with a problem that other writers did not have to worry about: faithful copying. Copying of manuscripts was largely done by monks in monasteries. While masters of the written word – they did, after all, “live by a book” — few monks mastered mathematics, and hence could not be relied upon to create an accurate copy of anything other than prose. Aware of this issue, authors of mathematics books wrote everything in prose.<br /><br />With the introduction of the printing press in the Fifteenth Century, however, everything changed. Indeed, one of the first printed books published after Gutenberg printed his famous edition of the Bible was an Italian book on practical arithmetic. True, to handle a symbolic textbook, you have first to master the linguistic rules for reading, writing, and manipulating symbolic expressions, but once you do, algebra becomes a whole lot easier to do, as a line-by-line comparison of Figures 1 and 2 makes abundantly clear. (Actually, it’s lines-by-line!)<br /><br />Notice, however, that the two presentations of the quadratic problem specify the same problem, and both solutions are, from a logical deduction point of view, the same. To some extent, the al-Khwarizmi’s prose version describes what goes on in your head when you solve the problem. At least — <i>and this is where I am going with this</i> — it does if you solve the problem by thinking about it. <br /><br />With the symbolic presentation, it is possible to reduce the solution of an algebra problem to the <i>mindless </i>(literally), algorithmically-specified manipulations of symbols. Ever since the invention of the printing press, generations of students quickly discovered that you can pass an algebra test by mastering a collection of symbolic-manipulation rules.<i> No understanding necessary</i>. Moreover, when taught this way, the teacher’s job became immeasurably easier. It is easier to teach rules to be followed than to develop thinking skills, and it is easy to evaluate a student’s work if the goal is simply to check that it accords with the rules and arrives at the correct answer. (Indeed, teachers soon realized that the quickest way to grade a student’s work was to first see if the answer is correct, and only if it is not look at the symbolic working.)<br /><br />While the student in Ninth Century Baghdad solved (linear and quadratic) equations by performing essentially the same steps as a student would today, with the problem presented in words, and the solution written out (presumably) in words, it can’t be carried out in a mindless fashion. The human mind can learn to follow rules for manipulating symbols, without knowing what they mean, but words are so much an integral part of human thinking that we cannot use them without their having meaning (albeit possibly a meaning other than the one intended by the author of an algebra book).<br /><br />There is, then, a potential loss in taking algebra from a prose presentation to a symbolic one: namely, the student can lose the appreciation that algebra is a <i>powerful way of thinking</i> with countless uses in the everyday world. Instead of algebra being a codification of human logical thinking that emerges from within, it becomes a set of externally imposed, and often arbitrary-seeming rules to be mastered by repetitive practice. The natural, relevant, and empowering becomes the artificial, pointless, and tedious. (Those of us who like symbolic algebra see beyond the rules.)<br /><br />“When will I ever use algebra?” today’s student justifiably asks. In terms of rule-based, symbol manipulation, the answer is, for most people (not all – and this is educationally significant), “Never.” But in terms of algebra, that codified way of thinking that has evolved and developed considerably since al-Khwarizmi’s day, the answer is, “All the time.” (Whenever you use a spreadsheet, for example.)<br /><br />In the introduction to his algebra book, al-Khwarizmi declared that he was presenting<br /><br /><i> “... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.”</i><br /><br />This was cutting edge stuff back then. It doesn’t get much more practical than that! <br /><br />As al-Khwarizmi explains, he was asked to write his book by the Caliph, who recognized the importance — for trade and engineering in particular (both of which were crucial to the regional society at the time) — of making those new methods of calculation widely available. The Caliph’s reasoning was as sound and significant then as it would be today. When a society reaches a state of development where trade and commercial and financial activity go beyond two people engaging in one-off transactions, it needs a more efficient tool than basic arithmetic. What is required is <i>arithmetic-at-scale</i>. When you boil it down to its essence, that is what algebra is. Al-Khwarizmi’s book codifies and formalizes the numerical reasoning that people use in their daily personal and professional lives in a fashion that enables them to operate at scale.<br /><br />In the years since the printing press made it possible to produce algebra textbooks that used symbolic representations, the focus in the algebra class has gradually shifted from being about sophisticated reasoning about numbers to an often mindless game of symbol manipulation. For several centuries that could be justified on the grounds that the only effective way for society to be able to handle the arithmetic-at-scale required to advance was to train lots of people to carry out the necessary calculations. And for that, the most efficient way is to use rule-based, symbolic manipulation. The people carrying out those calculations no more had to understand what they were doing than the electronic calculator on your iPhone has to understand what it is doing. All that matters it that it – the human symbolic-algebraist or the calculator app — gets the right answer.<br /><br />But now that those of us in more advanced societies (and in a great many less advanced societies, for that matter) do have ready access to those powerful calculating devices, devices that in addition to performing numerical calculations can also solve algebraic problems (arithmetic-at-scale, aka the electronic spreadsheet), the once-important societal need for many human symbolic calculators has gone away. What is required today is that people can make effective use of those new tools. That has shifted the emphasis back from symbolic-rule-mastery to the kind of formalized, rigorous thinking about quantitative matters that, thanks to al-Khwarizmi, we call algebra. Only now, we are back to the realm, not of symbol manipulation, but codified, logical, rigorous thinking about issues in our lives and in the world we inhabit.<br /><br />To be sure, symbolic algebra is not going away. It is way too powerful to ignore. But whereas it used to be possible to provide a rationale for teaching algebra as pure, rule-based symbolic manipulation (albeit a societal rationale that views people as fodder for industry), it makes no sense to teach it that way today.<br /><br />Which is why the Common Core now directs the focus not on the symbolic rules that dominated math instruction for centuries past, but on sophisticated mathematical thinking skills that develop and require a deeper understanding of numbers. This is why there is now so much talk of “number sense” and why Mary and Johnny are coming home from school with homework questions that their parents often find strange and occasionally incomprehensible. <br /><br />In other words, algebra has returned to its roots. (Pun intended.)<br /><br /><b>END OF PART 1</b><br />In Part 2 of this commentary, to be published here next month, I will look at how those same digital technologies that have rendered obsolete much of what used to constitute K-12 algebra education, have provided new ways to teach the subject that are ideally suited to the way we use — and will increasingly use even more — algebra. After all, if the printing press turned algebra from prose to symbolic expressions, what will algebra look like now that the digital computer, and in particular the tablet device, has largely replaced the printing press?<br /><br /><b>NOTE</b>: I realize that there is little in this month’s post that is new to MAA members. But as I know from emails and comments I receive, <i>Devlin’s Angle</i> posts find their way to a wide variety of readers, occasionally onto the desks of governors, education administrators, and others who play a role in the nation’s education system. With so much media attention currently being given to a mathematics education proposal being made by an individual having little knowledge of mathematics or current mathematics education (see last month’s <a href="http://devlinsangle.blogspot.com/2016/03/the-math-myth-that-permeates-math-myth.html">column</a>), I thought it timely to bring us back to an appreciation of algebra (i.e., algebraic thinking) that was apparent to a Ninth Century Caliph in Baghdad, and which is even more relevant to our lives today than it was back then. <br /><br />I cannot avoid ending by observing that 2016 will surely go down as the year when the US media devoted more media space and time to individuals pontificating on topics they knew almost nothing about, than they did to experts, of which the United States has large numbers with global reputations. I think many editors would benefit from a (good) course in algebraic thinking.http://devlinsangle.blogspot.com/2016/04/algebraic-roots-part-1.htmlnoreply@blogger.com (Mathematical Association of America)3tag:blogger.com,1999:blog-2516188730140164076.post-5688277157532213592Tue, 01 Mar 2016 15:11:00 +00002016-03-02T10:53:47.183-05:00algebraAndrew HackerCommon CoreIs algebra necessarymathematics educationquantitative literacyThe Math Myth that permeates “The Math Myth”March 1 saw the publication of the book <i>The Math Myth: And Other STEM Delusions</i>, by Andrew Hacker. MAA members are likely to recognize the author’s name from an opinion piece he published in the New York Times in 2012, with the arresting headline "<a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html">Is Algebra Necessary?</a>"<br /><br />Yes, I thought you’d remember it! It’s almost up there with John Lennon’s murder in terms of knowing where you were at the time you first heard of it. But just to be sure we are all on the same page, let me recap that, in that essay, Hacker, a retired college professor of political science who over the years had taught some non-majors math courses, laid out a case for dropping algebra as a required course in K-12 and college.<br /><br />Before I dive into Hacker’s new book, you would be advised to refresh your memory of the case he presented in that article, since his book is essentially an extension of what he said then, expanded to cover the entire Common Core Mathematical Standards. Prior to writing this review, I wrote an <a href="http://www.huffingtonpost.com/dr-keith-devlin/andrew-hacker-and-the-cas_b_9339554.html">article for the Huffington Post</a> in which I summarized, with my commentary, his 2012 article, together with a <a href="http://chronicle.com/article/The-Case-Against-Mandating/235500">recent interview</a> he gave to the Chronicle of Higher Education.<br /><br />In my article, I noted that Hacker has no idea what algebra really is. His focus is entirely on school algebra as it is very often taught, as a collection of rules for manipulating symbolic expressions. What his argument actually establishes, with sound arguments and good examples, are two conclusions I would agree with:<br /><ol><li>Algebra as typically taught in the school system is presented as a meaningless game with arbitrary rules that does more harm than good.</li><li>There are strong arguments for teaching algebra as it was originally developed and how professional mathematicians today view it.</li></ol>I’ll leave you to read my HuffPost piece for more of the gory details. For the benefit of lay readers who may come to this site, I should though repeat here the brief summary I gave in that article of the difference between algebra (as mathematicians understand and practice it) and the rule-based-manipulation-of-symbolic-expressions that so often passes for algebra in our schools.<br /><br />First codified by the Persian mathematician al-Khwarizmi in his book <i>The Compendious Book on Calculation by Completion and Balancing</i> (balancing = <i>al-Jabr</i>), written in Bhagdad around CE 820, algebra is a powerful method for solving numerical problems more efficiently than by arithmetic. It does so by introducing two new ways of handling numerical problems.<br /><br />First, algebra provides methods for handling entire classes of numbers, rather than specific ones. (That’s where those x’s, y’s, and z’s come in, but that’s just an implementation detail introduced in France several centuries later.)<br /><br />Second, it provides a way to find numerical answers not by computing, which is often very difficult, but by reasoning logically to hone-in on the answer, using whatever information is available. Thus, whereas in arithmetic you work forwards, starting with numbers and computing with them to arrive at an answer, in algebra you work backwards, starting by postulating an answer and reasoning logically to figure out what it is. True, this powerful application of human logical reasoning capacity frequently gets boiled down to mastering various symbolic procedures to “Solve for x,” but again that’s just a particular implementation. Numerical forensics would be a sexier, and more descriptive, term for the real thing.<br /><br />The familiar symbolic expressions calculus usually taught in schools as “algebra” was a particular implementation of al-Khwarizmi’s ways of thinking introduced by the French mathematician François Viète in the 16th Century (700 years after algebra first began) to streamline paper-and-pencil problem solving. A more recent implementation of algebra is the computer spreadsheet.<br /><br />Since his new book follows the same line of attack as his 2012 opinion piece, but with his sights widened from school algebra to the Common Core, instead of crafting another analytic essay, I will do what Hacker himself does, and list a number of examples to make my case. More precisely, I’ll select some of the 20 instances (in a book of just over 200 pages) where I found a claim that is either plain wrong, wildly misleading, or otherwise problematic, and ask where he went wrong. In marking 20 pages, it’s likely I missed some. There were so many wild and inaccurate claims, I frequently found myself skimming through.<br /><br />First though, I should repeat what I said in my HuffPost article about his algebra piece. Just as his essay actually amounted to a strong argument in favor of teaching algebra to all students (albeit not the rule-based manipulations of formulas so often presented in place of algebra), so too his book includes a strong argument in favor of Common Core Math. In the same way that Hacker mischaracterized algebra in 2012, so too his portrayal of the CCSSM (Common Core State Standards for Mathematics) is totally at odds with the real thing—though not quite so far off if you turn your attention from the Standards themselves to some <i>implementations</i> of the CC.<br /><br />One of the book’s flaws is that Mr Hacker seems to get carried away with the flow of his rhetoric, since for the most part his argument consists of the erection of a series of straw men which he then, in time-honored tradition, proceeds to attack.<br /><br />“It’s a waste of time forcing kids to master azimuths and asymptotes,” he cries [not an exact quote] as early as page 2.<br /><br />I had to look up the word azimuth, since in my entire career as a mathematician and mathematics educator, I had never come across it. According to Wikipedia, azimuth is a “concept used in navigation, astronomy, engineering, mapping, mining and artillery.” I ran a search for the word on the entire, 93-page CCSSM document and, as I expected, it did not turn up. Straw man.<br /><br />Asymptotes are a different matter, of course, since a general sense of asymptotic behavior of functions is useful in many walks of life. The word is mentioned, but just once, in the CCSSM, in the section on Interpreting Functions (F-IF), where it says:<br /><br /><i>Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.</i><br /><br />That’s it. One mention, buried towards the end of the document, in the section that says the student should:<br /><ul><li>Understand the concept of a function and use function notation</li><li>Interpret functions that arise in applications in terms of the context</li><li>Analyze functions using different representations</li></ul>From the overall thrust of Hacker’s argument, I think it’s clear he believes this kind of knowledge is indeed important for everyone to have. But it’s also clear it is not a central pillar of the CC, to be used on page 2 to set the scene for what his book is about.<br /><br />Unfortunately, this example is indeed a good characterization of his overall argument: to knock down straw men.<br /><br />“<i>We’re told that if our nation is to stay competitive, on a given morning all four million of our fifteen-year-olds will be studying azimuths and asymptotes</i>,” he writes. (I am still on page 2, with over 200 more pages to go.) He provides no citation regarding who, exactly, is making this proclamation for the nation’s future. It’s not just disingenuously misleading, it’s about as far from reality as you could imagine, and not because of those azimuths. (See momentarily for the real story.)<br /><br />He continues, “<i>Then, to graduate from high school, they will face tests on radical notations and elliptical equations</i>.”<br /><br />To be sure, you will find mention of the word radical in the CCSSM, in the context of “Work with radicals and integer exponents” in the Section on Expressions and Equations (8.EE), which provides the helpful illustration,<br /><br />“<i>For example, estimate the population of the United States as 3 × 10<sup>8</sup> and the population of the world as 7 × 10<sup>9</sup>, and determine that the world population is more than 20 times larger</i>.”<br /><br />Again, this is exactly the kind of thing Hacker says (towards the end of his book) students should be able to do! And it is entirely reasonable that they be asked to demonstrate that ability on a test.<br /><br />“Elliptical equations” is another straw man.<br /><br />The point is, what Hacker keeps attacking are straw men. The CCSS is just what its name implies, a set of standards. It is not a curriculum, nor does it specify anything remotely like a daily, or even weekly timetable. How and when teachers across the land cover the various standards is for them, or perhaps their school district, to decide. As far as the CCSS are concerned, teachers can operate fluidly, depending on how their class progresses. (And no one will even suggest that they mention azimuths, let alone force the class to master them.)<br /><br />I would hazard a guess that Hacker has never looked at the CCSS document. Nor sat in on many math classes, as I have, and observed what actually goes on in today’s schools.<br /><br />Caveat: I get to see classes to which, for one reason or another, I have been invited to visit. Likely they are some of the best, since their teachers invite me along so their students can talk for a while with someone who has devoted a career to mathematical research. I hear enough stories to be prepared to believe things are often a lot worse. Perhaps even as bad as Hacker says. But his book is purported to be about educational policy, not what you can actually find in good or bad classrooms.<br /><br />Not only does Hacker give no indication he is familiar with the Common Core—the real one, not the azimuth-strewn, straw-man version he creates—he gives every indication he does not understand mathematics as it is practiced today. (He also does not know that pi is irrational, but I’ll come to that later.)<br /><br />Certainly, the examples he selects to illustrate the irrelevancy (in today’s world) of some of the test problems students are asked to solve simply demonstrate that he is lacking the basic, every-day, number sense he is arguing for. Let me give just three examples.<br /><br />On page 48, Hacker presents a question he took from an MCAT paper. It provides some technical data and asks what happens to the ratio of two inverse-square law forces between charges of given masses when the distance between them is halved. The context Hacker provides for this question is that medical professionals needs to be able to read and understand the mathematics used in technical papers. His claim is that this requirement does not extend to the physics of electrical and gravitational forces. In that, he is surely correct. But anyone with a grain of number sense will recognize at once that the setting is totally irrelevant. It’s a simple question about what happens to a ratio when the underlying scale is changed. The answer, of course, is nothing happens. It’s a ratio. The changes to the numerator and denominator cancel out. The ratio remains the same.<br /><br />What this question is asking for is, <i>Do you understand what a ratio is</i>? Surely that is something that any medical professional who will have to read and understand journal articles would need to know. Hacker completely misses this simple observation, and presents the question as an example of baroque mathematical testing run amok.<br /><br />On page 70, he presents a question from an admissions test for selective high schools. A player throws two dice and the same number comes up on both. The question asks the student to choose the probability that the two dice sum to 9 from the list 0, 1/6, 2/9, 1/2, 1/3. Hacker’s problem is that the student is supposed to answer this in 90 seconds. Now, I share Hacker’s disdain for time-limited questions, but in this case the answer can only be 0. It’s not a probability question at all, and no computation is required. It just requires you to recognize that you can never get a sum of 9 when two dice show the same number. As with the MCAT question, the question is simply asking, <i>Do you understand numbers</i>? In this case, do you recognize that the sum of two equal numbers can never be odd.<br /><br />Finally, on page 101, Hacker presents a list of mathematics requirements high school students must meet in order to study at Harvard and similar universities. The list includes the names of various kinds of analytic functions. As usual, Hacker seems overwhelmed by the technical terms, or worries that the students will be, but all the list is asking for is that students can read graphs and charts and know what they represent in terms of growth and change. An essential skill, surely, for anyone in today’s information-rich world, not just students at elite universities.<br /><br />You get the pattern surely? Hacker’s problem is he is unable to see through the surface gloss of a problem and recognize that in many cases it is just asking the student if she or he has a very basic grasp of number, quantity, and relationships. Yet these are precisely the kinds of abilities he argues elsewhere in the book are crucial in today’s world. He is, I suspect, a victim of the very kind of math teaching he rightly decries—one that concentrates on learning rules and mastering formal manipulations, with little attention to understanding.<br /><br />This, surely, explains why he would write (page 96), “<i>Reasoning mathematically may be a nice skill, but it is not relevant to most of life. We reason about many things: parenting, marriage, careers. Do we learn how to reason about these things by learning algebra?</i>”<br /><br />If he had asked instead if we learn such reasoning in a typical school algebra class, I would agree with his implied answer of “No.” But algebra arose by codifying the everyday reasoning people carried out—and still carry out today—about the numerical or quantity aspects of any human activity that involves them. (Trade, commerce, and civil engineering were the original applications.)<br /><br />From that historical perspective, it is absolutely clear that learning algebra can help us master such reasoning. It helps by providing an opportunity to carry out that kind of reasoning free of the complexities a problem generally brings with it when it arises in a real world context.<br /><br />The tragedy of <i>The Math Myth</i> is that Hacker is actually arguing for exactly the kind of life-relevant mathematics education that I and many of my colleagues have been arguing for all our careers. (Our late colleague Lynn Steen comes to mind.) Unfortunately, and I suspect because Hacker himself did not have the benefit of a good math education, his understanding of mathematics is so far off base, he does not recognize that the examples he holds up as illustrations of bad education only seem so to him, because he misunderstands them.<br /><br />The real myth in <i>The Math Myth</i> is the portrayal of mathematics that forms the basis of his analysis. It’s the same myth you see propagated in Facebook posts from frustrated parents about Common Core math homework their children bring home from school.<br /><br />In the interests of their overall cardiovascular health, I have to recommend that math educators do not read <i>The Math Myth</i>. But if you do, perhaps you should start with the final chapter, titled “Numeracy 101.” Here, at least, you will find things you are likely to agree with, as he lays out what he believes would be a good quantitative literacy course for college students.<br /><br />But even there, where all seems warm and friendly and positive, you will be jolted by Hacker’s fundamental lack of knowledge of mathematics. He writes,<br /><br />“<i>Along with phenomena like earthquakes and cyclones, nature also has some numbers that control or explain how the world works. One of them is pi, whose</i> 3.14159<i> goes on indefinitely, at least as far as we know</i>.”<br /><br />Yes, you read that last part correctly.<br /><br />“Few people writing today … can make more sense of numbers” proclaims the <i>Wall Street Journal</i> on the cover of Hacker’s book. Well, if that’s the view of the newspaper that purports to have the expertise to cover the nation’s financial markets, it is only a matter of time before we have another financial meltdown.http://devlinsangle.blogspot.com/2016/03/the-math-myth-that-permeates-math-myth.htmlnoreply@blogger.com (Mathematical Association of America)9tag:blogger.com,1999:blog-2516188730140164076.post-7574208066658684828Thu, 11 Feb 2016 15:49:00 +00002016-02-11T16:40:24.816-05:00exceptional performanceHypercubesmathematical modelingTheorem: You are exceptional“Everyone excels at something.” We hear it all the time, usually said to console someone who is miserable after underperforming at something. Parents, in particular, often fall back on it with their children. What few people realize, though, is that the statement can be mathematically verified. You need only consider a collection of 200 essentially independent human performance characteristics for 98% of people to measure as exceptional in at least one of them, where exceptional is defined as being in the top or bottom 1%. (The mathematics gives extremal values; if you want to effectively guarantee being in the top 1%, you need more characteristics. The phenomenon is asymptotic.) <br /><br />This result is a consequence of a rather surprising, but little known, observation about high-order hypercubes: as the dimension increases, the proportion of points in the interior (i.e., not on the bounding shell) decreases without limit.<br /><br />Here is how you can prove to your child, spouse, student, best friend, or whoever, that they—or you, as the circumstances may require—can or will excel at something.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-5aVnO2xvjz0/VrokChM7TsI/AAAAAAAAKc0/8EX50jzeub8/s1600/Fig1.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="118" src="https://2.bp.blogspot.com/-5aVnO2xvjz0/VrokChM7TsI/AAAAAAAAKc0/8EX50jzeub8/s200/Fig1.jpg" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 1. The bell curve (normal distribution)</span></td></tr></tbody></table><br />Everyone is familiar with the bell curve (normal distribution) showing the typical distribution of performance measures of a single characteristic across a sufficiently large population. This graph captures the fact that the scores for the majority of the population cluster around an “average”, middling value, with only a few individuals at either end (exceptionally poor or exceptionally good).<br /><br />For the purposes of the multi-dimensional computation, we can start with a geometrically simpler model, namely the closed interval [0,100], as in Figure 2. We define the <i>exceptional</i> points to be those in the unit intervals at each end. In this model, for a single characteristic, only 2% of the population are exceptional. The remaining 98% are “normal."<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-tCymcfRBOkU/VryqRi4oAHI/AAAAAAAAKds/DJQjtHJYhWw/s1600/Devlin%2Bnew%2Bfig.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://2.bp.blogspot.com/-tCymcfRBOkU/VryqRi4oAHI/AAAAAAAAKds/DJQjtHJYhWw/s1600/Devlin%2Bnew%2Bfig.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 2. A simple model of exceptionality in one characteristic</span><br /><div><br /></div></td></tr></tbody></table>Now consider two characteristics, X and Y (assumed to be independent). The distribution then is represented by a 100x100 square with an inner 98x98 cube, as in Figure 2.<br /><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-lYwIYcaoWWY/VryqmDLkSZI/AAAAAAAAKdw/94ISWc-PVoA/s1600/Devlin%2Bnew%2Bfigure.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://3.bp.blogspot.com/-lYwIYcaoWWY/VryqmDLkSZI/AAAAAAAAKdw/94ISWc-PVoA/s1600/Devlin%2Bnew%2Bfigure.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 3. A simple model of exceptionality in two characteristics</span></td></tr></tbody></table><br />An individual’s X measure is shown by the x-coordinate, their Y measure by the y-coordinate. The ordinary individuals are represented by points in the inner square; the exceptional individuals by points in the outer perimeter region.<br /><br />The total number of points is 100x100. The number of normal points is 98x98. So the number of exceptional points is 10,000 – 9,604 = 396.<br /><br />The proportion of exceptional points is thus 396/10,000 = 0.0396, i.e., 3.96%. Thus, more individuals are classified as exceptional when you consider two characteristics (3.96% as opposed to 2%).<br /><br />Going to three characteristics, X, Y, and Z, the model will be a 100x100x100 cube with an inner 98x98x98 cube, as in Figure 4.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-sNY53gvcS9M/Vrok_bk8b6I/AAAAAAAAKdc/Xu-qb4bYA_U/s1600/Devlin%2Bfig4.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://2.bp.blogspot.com/-sNY53gvcS9M/Vrok_bk8b6I/AAAAAAAAKdc/Xu-qb4bYA_U/s1600/Devlin%2Bfig4.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 4. A simple model of exceptionality in three characteristics</span></td></tr></tbody></table><br />The volume of the outer cube (representing the total population) is 1,000,000. The volume of the inner cube (representing the normal individuals) is 941,192. Thus the volume of the perimeter-region (representing the exceptional individuals) = 1,000,000 – 941,192 = 58,808. Hence, the proportion of exceptional individuals = 58,808/1,000,000 = 0.0588, i.e. 5.88%.<br /><br />So far, everything seems fairly straightforward and reasonable. Going beyond three characteristics, the model is a hypercube of four or more dimensions, and we can no longer provide meaningful illustrations. But by now we have grown familiar with the idea that the model represents exceptional individuals by points in the outer 1% shell. To see what this entails, let’s jump to 10 characteristics, X<sub>1</sub>,…,X<sub>10</sub>. In that case, our model will represent the situation as a 100<sup>10</sup> hypercube with an inscribed 98<sup>10</sup> hypercube.<br /><br />The volume of the outer hypercube (~ total population) = 100<sup>10</sup>. The volume of the inner hypercube (~ normal individuals) = 98<sup>10</sup>. Thus, the volume of the perimeter-region (~ exceptional individuals) = 100<sup>10</sup> – 98<sup>10</sup>, and the proportion of exceptional individuals = (100<sup>10</sup> – 98<sup>10</sup>)/100<sup>10</sup>. At this point, it’s time to bring in Wolfram Alpha to do the calculation. This gives the result that, with 10 characteristics, 18.29% of the population is exceptional.<br /><br />With 100 characteristics, X<sub>1</sub>,…,X<sub>100</sub>, our model gives: Volume of hypercube (~ total population) = 100<sup>100</sup>. Volume of inner hypercube (~ normal individuals) = 98<sup>100</sup>. Volume of perimeter- region (~ exceptional individuals) = 100<sup>100</sup> – 98<sup>100</sup>. Proportion of exceptional individuals = (100<sup>100</sup> – 98<sup>100</sup>)/100<sup>100</sup>. Calling on Wolfram Alpha again, we compute that with 100 characteristics, 86.74% of the population is exceptional.<br /><br />With 200 characteristics, X<sub>1</sub>,…,X<sub>200</sub>, our model gives: Volume of hypercube (~ total population) = 100<sup>200</sup>. Volume of inner hypercube (~ normal individuals) = 98<sup>100</sup>. Volume of perimeter-region (~ exceptional individuals) = 100<sup>200</sup> – 98<sup>200</sup>. Proportion of exceptional individuals = (100<sup>200</sup> – 98<sup>200</sup>)/100<sup>200</sup>. So with 200 characteristics, 98.24% of the population is exceptional. (Once again calling on the services of the normally unflappable Wolfram Alpha.)<br /><br />And there’s our result.<br /><br />Of course, we have been working with a model. As always, that entails making various assumptions and simplifications. If the final result surprises you, you have two choices. Either go back and modify your initial assumptions and generate another model. Or accept the result and modify the prejudices that led to your surprise. <br /><br />In this case, we have to accept that in higher dimensions, almost all the material in an equal-sided, rectangular, <i>solid</i> (!) box is on the outer shell. The (solid) inside is almost empty.<br /><br />When we consider more dimensions to a situation, the math can sometimes lead us to a counter-intuitive—but correct—conclusion we did not expect. Not everyone can accept that.<br /><br />Yes, in this US Election Season, this is a story with a moral. <a href="https://1.bp.blogspot.com/-vUhjWsgB9yc/Vryrb7op4BI/AAAAAAAAKeE/Kmjb2tYa-f4/s1600/simely.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://1.bp.blogspot.com/-vUhjWsgB9yc/Vryrb7op4BI/AAAAAAAAKeE/Kmjb2tYa-f4/s1600/simely.jpg" /></a><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br />http://devlinsangle.blogspot.com/2016/02/theorem-you-are-exceptional.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-3603082345084297799Fri, 01 Jan 2016 13:30:00 +00002016-01-28T11:01:05.113-05:00an approach that sucksBedtime MathLaura Overdeckmath appsmath learningStanford UniversityUniversity of ChicagoWuzzit TroubleDo your kids find learning math hard? There may be an app for that!<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-CV53K7QV8Es/VnHsJZHXpwI/AAAAAAAAKcE/U9gPiFFjWH4/s1600/BedtimeMath.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-CV53K7QV8Es/VnHsJZHXpwI/AAAAAAAAKcE/U9gPiFFjWH4/s320/BedtimeMath.png" width="213" /></a></div>If you are like me, you probably sigh and switch off when you read an article with a title claiming kids’ math scores show significant improvement after using some great new app for a few minutes each day. <br /><br />In which case, you may have paid little attention when a<a href="http://news.sciencemag.org/brain-behavior/2015/10/bedtime-problems-boost-kids-math-performance"> news article</a> came out in <i>Science</i> magazine recently, reporting a new study showing that after just one year of parents using a bedtime-story-telling app called<i> <a href="https://itunes.apple.com/us/app/bedtime-math/id637910701?mt=8">Bedtime Math</a></i> with their young children, those kids were three months ahead of fellow students whose parents were using an app to provide non-mathematical stories. In fact, children of math-anxious parents showed even greater improvement, ending up six months ahead. <br /><br />If you happened to see the article, you likely assumed it was essentially a piece of marketing, where a bogus “study” was carried out to produce the “results” the marketing folks wanted. After all, there are no magic bullets in the math ed world, right? <br /><br />My reaction was very different. I immediately wanted to know more. The reason being, as I recounted in <a href="http://devlinsangle.blogspot.com/2015/12/life-inside-impossible-escher-figure.html">last month’s column</a>, the same thing had happened a few months earlier with a math learning app I had created, a mobile game called <i><a href="https://itunes.apple.com/us/app/wuzzit-trouble/id600190128?mt=8">Wuzzit Trouble</a></i>. (Actually, I should have written “we” there. Paul McCartney may have sung that he had “got by with a little help from my friends,” but the reality was it took a lot of work by all four Beatles to make them a global phenomenon, and in the apps business it usually takes a whole team of highly talented people to create a great product. My team are listed <a href="http://www.brainquake.com/core/">here</a>.) A Stanford classroom study led by Prof Jo Boaler had found significant math learning after just two hours play of <i>Wuzzit Trouble</i> spread over four weeks.<br /><br />The <i>Bedtime Math</i> study was unlikely to have been a bogus marketing “study”, I felt, since it had been carried out at the University of Chicago, which is a great university. True, as the <i>Science</i> article noted, the study was funded by an entity called the Overdeck Family Foundation, whose chair, Laura Overdeck, a former astrophysics major at Princeton, established the nonprofit Bedtime Math Foundation, which created and supports the app. Some might read that and smell a rat – as some did when they first read of the Stanford <i>Wuzzit Trouble</i> study.<br /><br />But to my mind such a reaction says more about the reader than the researchers. We are talking about an educational app made and distributed for free by a nonprofit organization. Why would anyone want to fake data? Really!<br /><br />In fact, even if the app were for sale – say for a mind boggling $4.99 – the idea that seven researchers at a major university would fake a study about a small children’s app is simply not credible. As a number of news articles have made clear, the price for a university researcher faking a scientific study is dismissal from the university and the end of their career. When it does happen, the motivation is invariably massive career prestige and fame, or a huge follow-up research grant (or both). Neither of which are likely to result from an at best encouraging, small scale classroom study of <i>Bedtime Math</i>, <i>Wuzzit Trouble</i>, or any other kids’ app.<br /><br />If a foundation or a company wanted to run a fake study for marketing purposes, they could simply do it themselves, or else farm it out to an unscrupulous, individual researcher. Such people are to be found, sometimes associated with universities. (Google “intelligent design” or “climate change denial” for examples.) <br /><br />Certainly, James Stigler, a well known educational psychologist at UCLA, is not skeptical. <i>Science </i>quotes him as saying, "I think it's a fantastic study. But it is just the beginning." <br /><br />Another respected scholar, Andee Rubin, a mathematician and computer scientist at the nonprofit education R&D company TERC in Cambridge, Mass, has a similar reaction. <i>Science</i> quotes him as observing, "I'm interested in teasing it apart and seeing what makes this effective." <br /><br />Those are pretty much the same as the reaction I had to Stanford’s <i>Wuzzit Trouble</i> results, which prompted me to draw up the list of possible explanatory factors I published in my last column.<br /><br />With both the full paper and a cover article available in <i>Science</i>, all I will do here is provide a brief summary of the cover article.<br /><br />The Chicago team recruited 587 first-graders from 22 schools in the Chicago metropolitan area. The parents of each child were given a tablet computer with which to read to the child at bedtime. 420 families were told to use it to work through word problems related to counting, shapes, arithmetic, fractions, and probability using <i>Bedtime Math</i>. Another 167 families were instructed to use a reading app. With a standardized test, the researchers assessed all the subjects' mathematics performance at the beginning and end of the school year.<br /><br />As you would expect, use of the reading app made little difference to the children's math performance. In contrast, children who used the math app two or more times per week outpaced peers whose family rarely used it, ending up three months ahead.<br /><br />Perhaps most important, use of the app brought students whose parents said they were anxious about math up to par with those whose parents were at ease with the subject. Among children whose family rarely used the math app, those with math-phobic parents made only half as much progress as the children of parents comfortable with math. <br /><br />The researchers make some suggestions as to what may be going on. My own best guess, based on several months reflections on the Stanford and (subsequent) Finnish studies of <i>Wuzzit Trouble</i>, are consistent with what they think. Namely:<br /><br />We have created a system where learning is walled off from everyday life. Particularly in math. The “math classroom” operates according to its own rules. Even with a truly great teacher – and I have met many – there are many restrictions on what can be done. Not least because of an incessant rhythm of performance testing. <br /><br />Go into most math classrooms and what you see will most likely bring to mind a room full of clerks in the pre-computer age when companies employed large numbers of numerically-able people to crunch their numbers. (Young people will have to rely on old photographs or depictions in movies.) Which was, of course, what the system was set up to provide.<br /><br />The classroom certainly does not look remotely like a room full of professional mathematicians at work. The first words that might come to mind if you were to walk into such a room would be “fun”, “engagement”, “argument”, “passion”, “social interaction”. <br /><br />Nor does it look like the human activity that hundreds of thousands of years of natural selection have inbred into us to maximize learning in the young: play. (Some wise person once said that “play is the work of the child.” I agree.) <br /><br />And there is something else that evolution hard-wired into is: our love of stories. Effective political speeches are usually laden with stories of individual people, and for very good reason. Because they are powerful.<br /><br />At which point, it’s probably a good idea to do what we math instructors tell our student to do: look for patterns. Well, what do we see when we look at professional mathematicians at work and kids in their ideal learning activity?<br /><br />Fun, engagement, argument, passion, social interaction, play, stories. <br /><br />Those are all, I would argue, <b><i>essential</i></b> ingredients for good learning. Yet you would have difficulty finding <b><i>any</i></b> in many math classrooms. <br /><br />Indeed, society seems to have gotten into a mindset that these items are distractions that you have to eliminate to achieve good math learning. Even when good teachers do their best to inject some of those valuable features into their classes, they have to operate within a system that disapproves. And everyone knows that, most of all the kids.<br /><br />No wonder then, that when you have a well designed, engaging app – a game, a puzzle, a family-supporting bedtime story provider, or whatever – you will likely get good results. Because apps, if properly designed, create their own environment.<br /><br />Though I labored long and hard to create <i>Wuzzit Trouble</i>, and I am sure Laura Overdeck worked equally hard on <i>Bedtime Math</i>, all to good effect for sure, my strong suspicion is our apps work as well as they do <b><i>in such a short time</i></b> primarily because of <b><i>what they are not</i></b>. Namely they are not the typical school classroom approach to mathematics education.<br /><br />How else can you explain dramatic results after a few hours engaging with an app, other than it unlocking what had hitherto been shackled by the chains of an Industrial Revolution conception of mathematics learning? Learning something that is genuinely new takes time and a lot of effort. Freeing something that is already there – if only in embryonic form – is much quicker. If so, then this means that our received wisdom that it takes a lot of hard work, repetitive practice, frustration, tears, and pathological levels of anxiety to achieve competency in mathematics may simply be very strong evidence that our approach sucks.http://devlinsangle.blogspot.com/2016/01/do-your-kids-find-learning-math-hard.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-883903209776583603Fri, 04 Dec 2015 19:54:00 +00002015-12-16T17:40:34.303-05:00game-based learningJames Paul GeeJo BoalerM.C. EscherMonument Valley gameLife inside an impossible Escher figure<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-JomIwhHa-6o/VmHfWQuhX5I/AAAAAAAAKb0/8aQUlnZ9rkc/s1600/MonumentValley2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="175" src="http://3.bp.blogspot.com/-JomIwhHa-6o/VmHfWQuhX5I/AAAAAAAAKb0/8aQUlnZ9rkc/s400/MonumentValley2.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><span id="goog_2140169023"></span><span id="goog_2140169024"></span><br />When the M.C. Escher inspired puzzle video-game<i> <a href="http://www.monumentvalleygame.com/" target="_blank">Monument Valley</a> </i>came out last year, I knew I had to check it out. The more so when it started getting rave reviews and winning awards. But with so many other things to hold my attention, I never managed to get round to it. The recent decision of the creators to make a version available for free prompted me to finally take a quick look. Not that it had been expensive. Rather, a tweet about the new free version happened to come when I had an hour or so of free time on my hands. <br /><br />That free hour got immediately swallowed up, as did many more hours after that. I was hooked from the getgo. <br /><br />Anyone who is intrigued by those impossible figures that Escher made famous, even those of you with little or no interest in puzzle (video-) games, will surely be captivated by <i>Monument Valley</i>, where the solution to many of the puzzles involves orienting the figure to create an illusion of a continuous object. For when the player views the object as continuous, characters in the game that traverse the figures can move along it. Impossible chasms that prevented a character’s progress suddenly disappear as you rotate the entire figure just the right way.<br /><br />It’s not a learning game. I don’t see a player learning any new mathematics. But what it does is provide a rich, immersive experience of complex geometrical spaces <i>from the inside</i>. As a player, your task is to assist the princess on her quest, which involves finding her way through a fantasy world of Escher-like structures, the geometry of which you can sometimes change as you progress. By projecting yourself into the princess, you get a sense of what it would be like to live in such a world. <br /><br />And a beautiful world it is. The creators, based in the UK and operating under the name UsTwo, have crafted a series of truly gorgeous fantasy worlds, which you encounter one after another. It is not so much a game as a collection of interactive pieces of art where you play with, and experience, geometric shapes.<br /><br />In fact, it is the artistic creation that the developers bring to the work. The idea of taking Escher worlds and turning them into a puzzle game goes back to a 2007 video game called <a href="https://www.youtube.com/watch?v=QfICeBtVv8U" target="_blank">Echodrome</a>, designed for the Sony Play Station 3 by the Japanese designer Jun Fujiki. By all accounts it was fiendishly difficult, and never broke out beyond a small group of hard-core puzzle aficionados. <br /><br /><i>Monument Valley</i> shows the huge difference presentation can make. If you want to hold people’s attention, you often need to think carefully about the medium. The message on its own may not be enough. That holds in the math class or the math lecture hall as much as in a video game.<br /><br />Regular readers of this MAA blog or my other blog profkeithdevlin.org will know that I have a long-standing interest in video games, particularly so as an educational medium, where I am professionally active as a player, a learning researcher, and an entrepreneur. <br /><br />In fact, much of my career has involved looking for ways to use different media to make mathematics accessible to as many people as possible. I have authored many “<a href="http://www.amazon.com/Keith-Devlin/e/B000APRPC6/ref=ntt_athr_dp_pel_1" target="_blank">popular mathematics books</a>”, written for newspapers (MAA compilation of some of my articles<a href="http://www.amazon.com/All-Math-thats-Fit-Print/dp/0883855151" target="_blank"> here</a>), worked on television programs (including <a href="https://vimeo.com/127338218" target="_blank"><i>A Mathematical Mystery Tour</i></a>, BBC-tv 1984; <a href="http://www.amazon.com/Life-Numbers-Danny-Glover/dp/B000FVQM10" target="_blank"><i>Life by the Numbers</i></a>, PBS 1998; and <i><a href="http://www.amazon.com/Numb3rs-The-Complete-First-Season/dp/B000ERVJKE" target="_blank">NUMB3RS</a></i>, CBS, 2005-2008), and of course there is my regular <a href="http://profkeithdevlin.com/MathGuy.html" target="_blank"><i>Math Guy</i></a> radio gig for NPR, which started in 1994. More recently, in 2012, I launched the first ever <a href="https://www.coursera.org/course/maththink" target="_blank">math MOOC</a> on Coursera (the seventh session just ended). I even made a foray into using music, song, and dance, with the 2007 show <a href="http://profkeithdevlin.com/HE.html" target="_blank"><i>Harmonius Equations</i></a>. To me, video games are one more medium to carry mathematical content.<br /><br />In fact, when it comes to K-12 mathematics, video games are in many ways the most effective medium we currently have to provide good math learning, as I tried to articulate in a <a href="http://www.amazon.com/Mathematics-Education-New-Era-Learning/dp/1568814313/ref=sr_1_1?ie=UTF8&s=books&qid=1299701150&sr=8-1" target="_blank">book</a> I wrote in 2011, and a presentation I gave at the big Teaching and Learning 2014 conference in Washington D.C. last year, a 20-min video summary of which is available <a href="https://vimeo.com/89583763" target="_blank">here</a>.<br /><br />Until recently, there was relatively little research available to put any flesh onto educators’ beliefs/hopes/suspicions that video games could yield good math learning outcomes. That is starting to change. (Reports from two classroom studies, one of I was involved in, are due to be published in the <i><a href="http://journal.seriousgamessociety.org/" target="_blank">International Journal of Serious Games</a></i> this month. Preprints are available<a href="http://documents.brainquake.com/backed-by-science/Stanford-Pope-Mangram.pdf" target="_blank"> here</a> and <a href="http://profkeithdevlin.com/Papers/Kiili-Devlin_2015.pdf" target="_blank">here</a>.)<br /><br />Certainly, the results obtained in those two papers raise more questions than they answer. (Moreover, pending further, and substantially larger, studies, the results themselves have to be viewed as tentative.) What we are seeing is that, for mathematics in the K-8 range, significant learning outcomes can be observed after a video-game intervention of as little as two hours play spread over a month or so. (Some measures show an increase of 20% over a comparison group.)<br /><br />I’d seen reports earlier that made similar claims, and dismissed them as product- marketing masquerading as research. It was only when the first of the two particular studies I cited above came out in late 2014, carried out by Prof <a href="https://www.youcubed.org/" target="_blank">Jo Boaler</a>’s research group at Stanford University’s Graduate School of Education, using my own math learning video game <i><a href="http://www.brainquake.com/our-platform/" target="_blank">Wuzzit Trouble</a></i> as the intervention, that I sat up and really took notice. <br /><br />In fact, I did more than that. Together with a research colleague from Tampere University in Finland, Prof Kristian Kiili, who, like me, has founded a math-learning video game company, and who was spending the year at Stanford, we carried out our own study. (The second of the two papers I cited.) Kiili was developing a fractions learning game, <a href="http://kristiankiili.com/semideus/" target="_blank"><i>Semideus</i></a>, and wanted to see how well it could serve as an evaluation tool. So we repeated essentially the same study Boaler’s team had done, with <i>Wuzzit Trouble</i> as the intervention, but instead of a written pre- and post-test (which the Boaler team used), we used <i>Semideus</i>. The results were very similar to those obtained in the previous study. (With the added twist that this time we found transfer — in a game context — from the whole number arithmetic of <i>Wuzzit Trouble</i> to the fractional reasoning of <i>Semideus</i>.)<br /><br />Something is going on, that’s for sure. But what? It did not take long to come up with a fairly long list of possible factors. Among the many things that a (well designed) math learning game can offer, all which are known to have a positive impact on learning, are:<br /><br /><ul><li><a href="http://www.teachinglearning2014.org/media/blog/breaking-the-symbol-barrier" target="_blank">Breaking the Symbol Barrier</a> – human-friendly representation (not the traditional abstract symbols of math textbooks).</li><li>Focus on developing number sense and problem solving ability.</li><li>High level of engagement.</li><li>Instant feedback (both positive and negative).</li><li>Steady flow of dopamine – known to have positive impact on memory formation and consolidation.</li><li>Learning through failure – in a playful, safe environment.</li><li>“Failure” treated – and regarded – as “not yet succeeded”.</li><li>Constant sense of “I can do this on the next try.”</li><li>Lots of repetition – but at the demand of the student/player.</li><li>Student/player is in control.</li><li>Student/player has ownership.</li><li>Growth Mindset – good games encourage and develop this. (This is the important notion <a href="http://mindsetonline.com/whatisit/about/" target="_blank">Carol Dweck</a> is famous for.)</li><li>Fluid intelligence (Gf) – games require and develop this. (Loosely speaking, this is the ability to hold several pieces of information in the mind at the same time and reason fluidly with them.)</li><br />I have written about many (not all) of these factors in my series of video game learning articles in my blog <a href="http://profkeithdevlin.org/" target="_blank">profkeithdevlin</a>. (See also the many writings and videos on games and learning by <a href="http://www.jamespaulgee.com/" target="_blank">Prof James Paul Gee</a>.)<br /><br />My current guess is that all of these factors, and likely others, are at play in those dramatic learning outcomes. The only way to find out for sure, of course, is to do more research. A lot more. Prof Kiili, now back in Finland, is already hard at work on that, as am I and some of my colleagues at Stanford. And we are by no means alone. The field is wide open. Stay tuned. (Even better, get involved.) Truly, it’s an exciting time to be involved in mathematics education.<br /><br />Meanwhile, I have to sign off. <i>Monument Valley</i> is calling.<br /></ul>http://devlinsangle.blogspot.com/2015/12/life-inside-impossible-escher-figure.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5710359148963460476Mon, 02 Nov 2015 14:42:00 +00002015-11-02T10:08:08.328-05:00boolean algebraboolean logicGeorge BooleToday is George Boole’s 200th Birthday<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-8ujtPLDmUDk/Vjd1_NUXN-I/AAAAAAAAKaQ/KHAsRw9lO4I/s1600/Google.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="233" src="http://1.bp.blogspot.com/-8ujtPLDmUDk/Vjd1_NUXN-I/AAAAAAAAKaQ/KHAsRw9lO4I/s400/Google.png" width="400" /></a></div><br />Today, November 2, 2015, marks the 200th anniversary of the birth of George Boole, one of the most influential mathematicians of all time – though it would be long after his death that his influence would manifest itself, when the growth of the modern digital age made significant aspects of our lives boolean. (To the degree that adjectival use of his name is no longer capitalized nor in need of italicization.)<br /><br />Born in England, Boole spent the major part of his mathematical career as a professor at Queen’s College Cork, and the Irish mathematical community has been actively celebrating Boole’s life, work, and legacy throughout this year. Of particular note, is an Irish ballad, “The Mathematician - The Bould Georgie Boole”, specially written for the occasion and performed by the Arthur Céilí Band, which you can hear, with visual biographic accompaniment about Boole, on YouTube and Vimeo:<br /><a href="https://www.youtube.com/watch?v=05IMBfkpn_Mhttps://vimeo.com/143768018" target="_blank">https://www.youtube.com/watch?v=05IMBfkpn_M</a><br /><a href="https://vimeo.com/143768018">https://vimeo.com/143768018</a>. <br /><br />The lyric and a download link to the song are available at:<br /><a href="https://arthurceiliband.bandcamp.com/releases">https://arthurceiliband.bandcamp.com/releases</a>. <br /><br />In a more academic vein, University College Cork has created a video biography available at:<br /><a href="https://www.youtube.com/watch?v=y-eav8-EEY4">https://www.youtube.com/watch?v=y-eav8-EEY4</a>. <br /><br />And US-based Irish mathematician Colm Mulcahy has a <a href="http://blogs.scientificamerican.com/guest-blog/the-bicentennial-of-george-boole-the-man-who-laid-the-foundations-of-the-digital-age/" target="_blank">celebratory article</a> in Scientific American.<br /><br />There is a lot more available on the Boole Bicentennial that digital search technology (part of Boole’s legacy) makes easy to find, so I’ll keep this post short and let you explore on your own.<br /><br />Be sure to log on to Google today. The company logo for the day is an active demonstration of boolean algebra using colors.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-_oR0H_8RErY/Vjd2Gp6TmiI/AAAAAAAAKaY/Y1hNI0T5mHw/s1600/Google2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="263" src="http://4.bp.blogspot.com/-_oR0H_8RErY/Vjd2Gp6TmiI/AAAAAAAAKaY/Y1hNI0T5mHw/s400/Google2.png" width="400" /></a></div><br />http://devlinsangle.blogspot.com/2015/11/today-is-george-booles-200th-birthday.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-7730381965233577730Fri, 02 Oct 2015 15:14:00 +00002015-10-02T12:10:00.474-04:00appreciation of mathematicsbeauty in mathematicscalculusLetter to a calculus student – The Sequel<br />Devlin’s Angle for July 2006 was titled <a href="https://www.maa.org/external_archive/devlin/devlin_06_06.html" target="_blank">Letter to a calculus student</a>. In it, I tried to describe, as briefly but as effectively as I could, the deep beauty there is in calculus, a beauty that arises from the depth of human brilliance that it took for the human mind to find a way to tame the infinite, and bend it to our use, a beauty made the more so by the enormous impact calculus has had on life on Earth.<br /><br />In my essay, I acknowledged that there was little chance any calculus student would be able to understand what I was trying to convey. I wrote:<br /><br /><i>“Those techniques [of calculus] are so different from anything you have previously encountered in mathematics, that it will take you every bit of effort and concentration simply to learn and follow the rules. Understanding those rules and knowing why they hold can come only later, if at all. Appreciation of the inner beauty of the subject comes later still. Again, if at all.</i><br /><br /><i>I fear, then, that at this stage in your career there is little chance that you will be able to truly see the beauty in the subject. Beauty - true, deep beauty, not superficial gloss - comes only with experience and familiarity. To see and appreciate true beauty in music we have to listen to a lot of music - even better we learn to play an instrument. To see the deep underlying beauty in art we must first look at a great many paintings, and ideally try our own hands at putting paint onto canvas. It is only by consuming a great deal of wine - over many years I should stress - that we acquire the taste to discern a great wine. And it is only after we have watched many hours of football or baseball, or any other sport, that we can truly appreciate the great artistry of its master practitioners. Reading descriptions about the beauty in the activities or creations of experts can never do more than hint at what the writer is trying to convey.</i><br /><br /><i>My hope then is not that you will read my words and say, "Yes, I get it. Boy this guy Devlin is right. Calculus is beautiful. Awesome!" What I do hope is that I can at least convince you that I (and my fellow mathematicians) can see the great beauty in our subject (including calculus). And maybe one day, many years from now, if you continue to study and use mathematics, you will remember reading these words, and at that stage you will nod your head knowingly and think, "Yes, now I can see what he was getting at. Now I too can see the beauty."</i><br /><br />I then proceeded to describe, as articulately as I could, the beauty there is to be seen in calculus, or at least the beauty I see in it, taking the reader on a guided tour of the standard definition of the derivative, but from the perspective of how it takes advantage of what the human brain can do, while circumventing what it cannot.<br /><br />I ended my essay by quoting poet William Blake’s <i>Auguries of Innocence</i>, saying:<br /><br /><i>That's what [the derivative limit formula] asks you to do: to hold infinity in the palm of your hand. To see an infinite (and hence unending) process as a single, completed thing. Did any work of art, any other piece of human creativity, ever demand more of the observer? And to such enormous consequence for Humankind? If ever any painting, novel, poem, or statue can be thought of as having a beauty that goes beneath the surface, then the definition of the derivative may justly claim to have more beauty by far. </i><br /><br />As I noted above, I was really writing for my fellow mathematicians. I knew then, as I still acknowledge today, that what I had written was true: it is impossible to experience the beauty in many human creations until one has sufficient experience.<br /><br />It was then, with great pleasure, that I received the following email a few weeks ago (on August 17), which I reproduce in its entirety, unedited, with the permission of the sender. I hope you enjoy it to. And, if you are a math instructor at a college or university, maybe print off this blog post and pin it somewhere on a corridor in the department as a little seed waiting to germinate.<br /><br /><div style="text-align: center;">* * * * * *</div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-oSGU6TMzAos/Vg6MkQyb53I/AAAAAAAAKZo/9Yen-npdpbM/s1600/WWU.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="237" src="http://3.bp.blogspot.com/-oSGU6TMzAos/Vg6MkQyb53I/AAAAAAAAKZo/9Yen-npdpbM/s400/WWU.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Western Washington University, in Bellingham, WA</td></tr></tbody></table><br />Hi Dr. Devlin,<br /><br />My name is Murray Pendergrass. I am a math student at Western Washington University, a small public liberal arts college in the Pacific Northwest where I am pursuing a BS in Mathematics.<br /><br />Sometime around 2006 you authored a post on Devlin's Angle titled "Letter to a calculus student" and I suppose someone in the math department at my school enjoyed it because it has been tacked to a bulletin board on the math floor for quite sometime. I would have only been going into the 8th grade when it was originally posted, with absolutely no idea that I would ever become interested in mathematics. I did take a calculus course my junior year of high school, but I don't think I could even briefly explain what a derivative was by the time the course was over (time well spent, obviously).<br /><br />I must have first seen your article either my sophomore or junior year of college, 2014 most likely. I would have either been in precalculus or calculus I (differential calculus), and still completely unaware that I would end up declaring a math major. At that time I would have still been a member of the business school. I was probably waiting outside a professor's office for office hours when the title caught my eye,<br /><br />" 'Letter to a calculus student' … Hmm, maybe I should read this."<br /><br />However being the impatient person that I am, I believe I started in and thought "ok this is boring, I'll check the next page and see if it gets better,<br /><br />"Nope, second page is boring too. Oh well."<br /><br />And I have to admit, it was not until last night that I actually read the whole thing through for the first time.<br /><br />But not long after that first initial and brief encounter with the letter my passion for mathematics truly began to develop and I realized that you can actually major in math without being a child prodigy (yes I actually thought this for quite sometime). It would have been shortly after this time, less than a year ago, that I realized I wanted to major in math. Since changing majors, very few hours have been spent not working on math.<br /><br />I was studying at school late last evening when I decided to take a break and cruise up and down the hallway when for the second time in my life I noticed the letter tacked to the bulletin board. I must walk past it every single day but it was not until last night that it caught my eye again and I thought "I've seen this before! Oh wow I should give it a shot now that I am passionate about math."<br /><br />In the very first sentence you open with a quote by Bertrand Russell, someone I have taken great interest in over the last year since mathematical logic has become a particular interest of mine. I immediately knew this was going to be a whole different experience reading this letter, and I was right.<br /><br />What provoked me to feel the need to write you this letter was that I feel I am a precise example of the reader you are mentioning when you say,<br /><br />"And maybe one day, many years from now, if you continue to study and use mathematics, you will remember reading these words, and at that stage you will nod your head knowingly and think, 'Yes, now I can see what he was getting at. Now I too can see the beauty.' "<br /><br />Just as you predicted, the first time I made an attempt to read the letter "there was little chance I could see the true beauty in math", a statement so true that I could not only fail to see the beauty in math but I could not even read a letter about someone else promising me that even though I couldn't see the beauty, it was there.<br /><br />It was quite a shock to me to read the letter last night and realize what a strange coincidental experience it was to randomly come across it a year after diving head first into the world of mathematics. It felt like a testament to myself of the progress I have made in math over the last year, a type of progress that cannot be explained or noticed through grades or high marks but by reading and truly relating to a mathematicians admiration for the beauty in math.<br /><br />Before college I lived a bit of a bumpy life, it was a long and interesting road getting to where I am now. I will spare you the details as this letter has already turned out to be longer than I expected but I can truly say that finding math has been the best thing that has ever happened to me. In a lot of ways it has set me free. I am very grateful to have the opportunity to study math at a university, to study something I am passionate about, and to reflect on how my relationship with math has evolved. I also must note that I hope I don't sound naive! I know I have only been doing math for a little over a year, which might sound like child's play to a Doctor of Philosophy in Mathematics. I am ecstatic that I have reached the point where I can appreciate mathematical beauty and I am also confident that math will continue to fascinate me and reveal its beauty for many years to come. Like most things, math is a journey not a destination.<br /><br />Overall, I just felt the need to write to you because I thought you might enjoy knowing that even 9 years after you wrote it there are still students thinking for the first time:<br /><br />"Yes, now I can see what he was getting at. Now I too can see the beauty".<br /><br />Thank you,<br /><br />Murray Pendergrass<br /><br />http://devlinsangle.blogspot.com/2015/10/letter-to-calculus-student-sequel.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5955268167176696663Tue, 01 Sep 2015 13:52:00 +00002015-09-09T09:37:29.298-04:00Asa ButterfieldIMOmath competitionsMath moviesA Brilliant Young Mind: The IMO goes to the movies<div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-csyetufhC1k/VeS5Nvk3eXI/AAAAAAAAKYU/RpPAwQiiByo/s1600/ABYM6.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="266" src="http://3.bp.blogspot.com/-csyetufhC1k/VeS5Nvk3eXI/AAAAAAAAKYU/RpPAwQiiByo/s400/ABYM6.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Jo Yang as Zhang Mei and Asa Butterfield as Nathan Ellis in <i>A Brilliant Young Mind</i>. <i>Credit: Samuel Goldwyn Films</i></td></tr></tbody></table><br />Mainstream movies about mathematicians used to be a rarity, but are now fairly common. <i>Good Will Hunting</i>, <i>Proof</i>, <i>A Beautiful Mind</i>, <i>Pi</i>, <i>Cube</i>, <i>The Bank</i>, <i>Travelling Salesman</i>, <i>The Imitation Game</i> come immediately to mind. So too does <i>Stand and Deliver</i> if you include mathematics teaching. <br /><br />The titles I just listed are such good movies, there is now a high bar to success in this growing genre. In particular, the movie has to have a good story, a strong cast, and it needs to get the math right – and moreover do so in a way that intrigues the audience but does not detract from the pace of the story. The new movie <i>A Brilliant Young Mind</i>, by British director Morgan Matthews, meets that standard. <br /><br />Due to be released in the US on September 11, <i>A Brilliant Young Mind</i>, starring the hugely talented young British Actor Asa Butterfield (who starred as Ender in <i>Ender’s Game</i>, opposite Harrison Ford) was first screened in the UK last year, originally under the title “X + Y”, on which more later. <br /><br />The film focuses on the <a href="https://www.imo-official.org/">International Mathematical Olympiad</a> (IMO), the competition held annually around the world, where national teams of six pre-collegiate students compete for individual and team medals. The movie follows one particular British student as he goes through the grueling process of preparing for and taking the test to qualify for team pre-selection in the British National Mathematical Competition, going off to a training camp in Taiwan, where the final team of six is selected in a mock IMO competition, and then heading to Cambridge, England, for the international competition itself.<br /><br />Both the mathematics and the mathematics competitions are handled well. (More later.) Mathematicians will not be disappointed on that score. <br /><br />I have to admit that, on first viewing, I felt that the romantic thread between the Asa Butterfield character (Nathan Ellis) and the young female Chinese math whiz he meets at the training camp, played by Jo Yang, was a crude injection to create a movie with mainstream audience appeal. In particular, I thought the dramatic ending (you’ll have to watch the movie to find out if it is a happy or sad ending) was way over the top. <br /><br />But then I watched the original BBC documentary that <i>A Brilliant Young Mind</i> director Matthews made back in 2006, on which he based the movie, and guess what? The story in <i>A Brilliant Young Mind</i> stays pretty close to real life! Right down to what at first viewing of the movie I thought were syrupy shots included purely for cinematic romantic effect. (Cue the rainbow in the background as the British and Chinese math whizzes travel by train through the British countryside. Taken right out of real life!) <br /><br />So if my jaded-by-Hollywood mathematical colleagues find themselves, like me, lamenting to themselves, “Why do movie makers spoil the real story with all that romantic mush?” you should suppress that reaction at once. What the movie gives you is a dramatic (and dramatized) recreation of real life.<br /><br />That, by the way, is why I like the movie’s original title in the UK: “X + Y”, a nicely succinct way to link the mathematics problem solving and the romantic engagement. Still the title <i>A Brilliant Young Mind</i> does convey the idea of a young version of the brilliant (but, like Butterworth’s Nathan Ellis, mentally troubled) John Nash in the movie <i>A Beautiful Mind</i>, and Matthews himself pulled on the same association with the title of his earlier documentary.<br /><br />Despite taking his basic storyline straight out of real life, Matthews does (of course) take plenty of dramatic license in order to give us a watchable movie. He is, after all, telling a fictional story, albeit one based (unusually closely) on real life. Few in the audience will have much interest in the mathematics, or even math competitions (besides, perhaps, being surprised that such things exist), but everyone likes a good story about people. And that is what <i>A Brilliant Young Mind</i> delivers.<br /><br />In particular, I suspect many of my fellow mathematicians will also balk at the portrayal of several of the British IMO team selectees as exhibiting various forms of autism. (In real life, good mathematicians of all ages run the full spectrum of human characteristics, with <i>the vast majority</i> of math whizzes being just like everyone else.) But that aspect too is what you will find in the documentary. (But see my postscript comments at the end.)<br /><br />The IMO team members who the Butterfield character interacts with in the movie are also clearly based on real-life counterparts in the documentary. In particular, the student who has made his way onto the team by learning a lot of mathematical facts and procedures that he can regurgitate and apply at speed, but falters when it comes to having to apply original thinking. (Both the US and the UK math education systems encourage and reward that approach, which is why they do so poorly in the international PISA tests, which look for original thought. My Stanford colleague <a href="http://joboaler.com/">Jo Boaler</a> has a new book on that misguided, and sad, state of affairs, <i>Mathematical Mindsets</i>, coming out in the Fall.)<br /><br />The only complaint you could make about Matthews is the choices he made in selecting the footage he shot for the original documentary. But that is true for any documentary film. Matthews followed the UK 2006 IMO team through the entire competition process, and then told a story based on what he had captured.<br /><br />Interestingly enough, another documentary film maker, George Csicery, followed the US IMO team at the same time. You can compare the two documentaries. Matthews’ BBC documentary is <a href="http://topdocumentaryfilms.com/beautiful-young-minds/">available online</a> for free streaming. Csicery’s film <i>Hard Problems</i>, is <a href="http://www.hardproblemsmovie.com/">available for purchase</a> from the MAA ($19.95 to members).<br /><br />Enough of all these words. We’re talking about a movie, after all. Time to watch some movie clips.<br /><br />You can watch the official trailer for <i>A Brilliant Young Mind</i> <a href="http://www.rottentomatoes.com/m/a_brilliant_young_mind/">here</a>.<br /><br />Mathematicians will particularly like the one short sequence where the movie shows a brilliant mathematical mind in action, solving a problem, which you can see in isolation in this officially sanctioned YouTube <a href="https://www.youtube.com/watch?v=mYAahN1G8Y8">video</a>. It is a superbly chosen (and acted) example. No fake numerical mumbo jumbo here. Genuine mathematical thinking in action. And good mathematical thinking to boot. Any math instructor would surely love to have a student produce such a solution for the class.<br /><br />During the year the two documentaries were made, the IMO was held in Ljubljana, Slovenia (not Cambridge, England, as in the movie). You can see the actual problems the competitors faced <a href="https://www.imo-official.org/problems/IMO2006SL.pdf">here</a>. (With sample solutions.)<br /><br />Finally, for long lists of scenes in movies that feature a mathematician or a math problem, see <a href="http://www.math.harvard.edu/~knill/mathmovies/" target="_blank">here</a> and <a href="http://www.qedcat.com/moviemath/" target="_blank">here</a>.<br /><br />Enjoy the film!<br /><br /><b>EDITORIAL POSTSCRIPT</b><br /><br />Both the movie and the BBC documentary raise some issues that concern me as a mathematician. The main danger of any documentary or movie is if viewers (and for a film like <i>A Brilliant Young Mind</i>, the audience may well include young kids showing an early interest in mathematics) get the impression that what they see is representative of the field. This of course, is true for pretty well any movie, be the topic crime detection, politics, business, law, the military, sports, or whatever. I think it is particularly worrying in mathematics, because most people have a very impoverished, and often completely erroneous perception of mathematics. Both of Matthews’ films trouble me on that count.<br /><br />These thoughts are in no way a criticism of the movie I am reviewing. It is what it is. I think it is a good movie and I like it. (Though as I noted, in my case much of my positive evaluation comes from knowing that the things in the movie that initially turned me off as being unrealistic and contrived turned out to be true!) Rather, the issues I raise are general ones about the public perception of mathematics. In making both his documentary and the movie, Morgan Matthews set out to make good films. His goal was not to improve the public understanding of mathematics. That, on the other hand, is something I have spent a great deal of my career focusing on. Hence this postscript, separate from my review.<br /><br />First, it has to be said that competition mathematics is in many respects a very different activity than the professional mathematics that most of us in the business pursue. For one thing, competition math requires speed, whereas many good mathematicians are slow thinkers. (I certainly am.) <br /><br />There is also something very unusual about the kinds of problems that the IMO presents. Of necessity, they have to be solvable in at most an hour, and in many cases, the way to go about solving them is to be very conscious of that time limitation. They have to depend on seeing a particular insight or trick. Some people are naturally good at that kind of problem solving, but it can also be to some extent taught – which is what goes on at those IMO training camps. On the other hand, most mathematics problems that the pros grapple with are very different. In many cases, no one has any idea if there is a solution at all, or how long it may be.<br /><br />Moreover, the connection between being good at competition math and having the aptitude to succeed in professional mathematics is not at all clear-cut. Certainly, some IMO medal winners have gone on to pursue mathematics at university level, but not all of them have gone on to lead a successful career in mathematics. (Some have.) And many of the best mathematicians in the world have never in their lives had any interest in competition math. Though the two domains do have abstract mathematics in common, they are in many ways very different activities.<br /><br />So to a child or the parent of a child who shows aptitude toward mathematics, I would say this. If you fancy the idea of competition math, give it a try. If you do well, enjoy the experience. It will certainly show that you have some abilities that could help you succeed in a mathematical career. But if you find you do not enjoy it, or if you like it but do not do well, that in no way means you could not grow up to be a top rank mathematician.<br /><br />Another unintended message that math competitions tend to convey is that you have to have a special talent for mathematics (a “math gene” if you like). This notion that mathematics is something for the “gifted and talented” is pervasive in many cultures, and it is total BS. The two most important factors in achieving success in mathematics are wanting to do math and growing up in a supportive, educationally rich, sociocultural environment. Not only is the world of mathematics replete with examples of world class mathematicians who will tell you flat out how many hours of effort it took them to get to that point, and how others helped them on their way, there is also a growing body of evidence from nueroscience studies to support the hypothesis that mathematics, like pretty well any other human endeavor, is 5% inspiration and 95% perspiration.<br /><br />Society would do well to banish that term “gifted and talented” once and for all, and replace it with something more accurate. “Motivated and bloody hard working” is my nomination. (Individual mileages do, of course, vary.)<br /><br />My final editorial remark is something I touched upon in my review. The movie, focuses on a small group of mathematics students, and one of them in particular, who exhibit various forms and degrees of autism. True, the same was true of the young students in the director’s earlier documentary, but that clearly reflects the particular perceptual lens the director brought to the project. That lens is made dramatically clear by the opening scenes in the documentary where we meet one of the competitors, Jos. Matthews set out to portray IMO participants as being unusual and different. And he found some.<br /><br />Cleary, getting to represent their country in an international competition in of itself makes them different. But presenting them as intrinsically different is a definite editorial decision.<br /><br />Contrast Matthews’ documentary with Csicery’s. In the latter, focusing on the US team at the same IMO, the director sets out to convey the very ordinariness of the participants, highlighting not what is different about them (they love math being the main thing) but how much they are just like any other kids of that age. <br /><br /><a href="http://www.zalafilms.com/about.html">Csicery</a>, as many readers of this column will know, makes documentary films about mathematicians and mathematics as a profession, and he makes them primarily for the mathematics and mathematics education professions. So he strives to inform an audience who are interested in mathematics. This, of course, is very different from <a href="http://www.imdb.com/name/nm2051728/">Matthews</a>, who sets out to make movies that intrigue viewers who do not necessarily have any interest in the topic, whatever it is. Csicery succeeds with his audience by being as accurate and representative as he can, while also managing to tell a story. Matthews has to tell a good human interest story that hangs on some strong characters, with everything else revealed during the film. They are doing different things.<br /><br />I found Matthews’ documentary fascinating and highly engaging, and I’m really glad it inspired him to turn it into a movie. You should watch both. <br /><br />But if what you want is to get a good overall sense of the world of competitive math, you should watch Csicery’s documentary. The two documentaries provide very different perceptions of the same IMO competition.<br /><br />http://devlinsangle.blogspot.com/2015/09/a-brilliant-young-mind-imo-goes-to.htmlnoreply@blogger.com (Mathematical Association of America)1tag:blogger.com,1999:blog-2516188730140164076.post-8022703553387304016Sat, 01 Aug 2015 04:03:00 +00002015-08-01T00:03:00.121-04:00game-based learningmath video gamesHard fun – video games creep into the math classroom<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-mzOCEgJUkRM/Vbuc763Q00I/AAAAAAAAKX4/NVqaMj4z-qk/s1600/video%2Bgames.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-mzOCEgJUkRM/Vbuc763Q00I/AAAAAAAAKX4/NVqaMj4z-qk/s320/video%2Bgames.jpg" width="209" /></a></div>This month’s musings were inspired by the appearance of Greg Toppo’s excellent new book <a href="http://www.amazon.com/Game-Believes-You-Digital-Smarter/dp/1137279575/ref=sr_1_1?ie=UTF8&qid=1438312811&sr=8-1&keywords=greg+toppo" target="_blank"><i>The Game Believes in You: How Digital Play Can Make Our Kids Smarter</i></a>. In it, Toppo, who is USA Today's national K-12 education writer, does an excellent job of not only surveying the current scene in educational video games, he also exhibits a deep understanding of, and appreciation for, the educational potential of well designed video games. I have gone on record as saying it will likely turn out to be the most influential book on the role of video games in education since James Paul Gee’s 2003 classic <a href="http://www.amazon.com/Video-Games-Learning-Literacy-Second-ebook/dp/B00OFL6RDE/ref=sr_1_1?s=books&ie=UTF8&qid=1438313160&sr=1-1&keywords=video+games+gee" target="_blank"><i>What Video Games Have to Teach Us About Learning and Literacy</i></a>.<br /><br />Like it or loath it, video games are slowly finding their way into the nation’s math classes, as teachers and parents increasingly see video games as a valuable educational resource. For instance, according to a recently published survey designed by the Joan Ganz Cooney Center, 55% of teachers report having their students play video games at least once a week, with 47% of teachers saying low-performing students benefited most from the use of games. (<i>Games and Learning</i>, 2015)<br /><br />Well-designed educational video games offer meaningful learning experiences based on principles of situated learning, exploration, immediate feedback, and collaboration. The power of experiential learning in engaging contexts that have meaning for learners has been demonstrated in several studies (e.g. Lave, 1988; Nunes et al, 1993, Shute & Ventura 2013).<br /><br />But when it comes to education, not all games are equal. Of the many mathematics education video games (or gamified apps) available today (Apple’s App Store lists over 20,000), the majority focus on traditional drill to develop mastery of basic skills, particularly automatic recall of fundamental facts such as the multiplication tables. They require repetition under time pressure. Such games make no attempt to teach mathematics, to explore mathematical concepts, or to help students learn how to use mathematical thinking to solve real world problems. Their purpose is purely to make repetitive drill more palatable to students.<br /><br />The proliferation of such games is in large part a consequence of the mathematics education many Americans have experienced: teacher and textbook instruction emphasizing isolated facts, procedures, memorization, and speed.<br /><br />So widespread is this educational model in the US, that many American parents, teachers, and game developers think that this is the nature of mathematics, a perception that can result in underdeveloped <i>mathematical proficiency</i>. (See, for example, Boaler 2002; Boaler 2008; or Fosnot & Dolk 2001.)<br /><br />While command of basic computation skills was a valuable asset to previous generations, in an era where fast, accurate computation is cheaply and readily available (in our pockets and briefcases, and on our desks), the crucial ability has shifted to what is often called mathematical proficiency: the ability to solve a novel problem that requires creative, multi-step reasoning, making appropriate use of computational technology as and when required.<br /><br />The National Research Council’s recognized this significant change in the nation’s mathematical needs in its seminal 2001 recommendations for the future of US K-12 mathematics education, which laid out the case for the promotion of mathematical problem solving ability, built on <i>number sense</i>. Number sense involves being mathematically proficient with numbers and computations. It moves beyond the basics to developing a deep understanding about properties of numbers, and thinking flexibly about operations with numbers.<br /><br />The last few years have seen the emergence of a tiny handful of video games designed to meet the educational requirements laid out by the National Research Council. Games such as Mind Research Institute’s K-5 focused <i>Jiji</i> games, <i>Motion Math</i>, <i>DragonBox</i>, <i>Refraction</i>, <i>Slice Fractions</i>, and my own <i>Wuzzit Trouble</i>. These games represent mathematics in a fashion native to the game medium (not the “symbolic” representations developed for the static page). They present the player with conceptually deep, complex problem solving tasks wrapped up in a game mechanic.<br /><br />As such, these games leverage the representational power of personal computers and touch-screen devices to provide students with a means to interact with mathematical concepts in a direct fashion, not mediated through a symbolic representation, thereby facilitating exploration and learning through interactive problem solving.<br /><br />In this context, it is worth reminding ourselves that the dominance of the printed symbol in the systemic mathematics education world is itself a product of the then-available technology, namely the invention of printing press in the 15th Century. Modern devices allow us to greatly expand on the symbolic interface, which for many people is a <i>known barrier</i> to mathematics learning (Nunes et al 1993, Devlin 2011).<br /><br /><b>References</b><br />Boaler, <i>Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning</i>, Revised and Expanded Edition. Mahwah, N.J. : L. Erlbaum, 2002.<br /><br />Boaler, “Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed‐ability approach,” <i>British Educational Research Journal</i>, vol. 34, no. 2, pp. 167–194, Apr. 2008.<br /><br />Fosnot & Dolk, <i>Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction.</i> Portsmouth, NH: Heinemann, 2001.<br /><br />Devlin, <i>Mathematics education for a new era: video games as a medium for learning.</i> CRC Press, 2011.<br /><br /><i>Games and Learning</i> report, 2015. <a href="http://www.gamesandlearning.org/2014/06/09/teachers-on-using-games-in-class/#" target="_blank">http://www.gamesandlearning.org/2014/06/09/teachers-on-using-games-in-class/#</a><br /><br />Lave, 1988. <i>Cognition in Practice: Mind, Mathematics and Culture in Everyday Life (Learning in Doing)</i>, Cambridge University Press.<br /><br />National Research Council, <i>Adding It Up: Helping Children Learn Mathematics</i>. Washington, DC: National Academies Press: National Academy Press, 2001, pp. 1–462.<br /><br />Nunes, Carraher, & Schliemann, 1993. <i>Street Mathematics and School Mathematics</i>, Cambridge University Press.<br /><br />Pope, Boaler, & Milgram 2015. “Wuzzit Trouble: The Influence of a Digital Math Game on Student Number Sense”, submitted to International Journal of Serious Games.<br /><br />Shute & Ventura, 2013. <i>Stealth Assessment: Measuring and Supporting Learning in Video Games</i>, MIT Press.<br /><br />http://devlinsangle.blogspot.com/2015/08/hard-fun-video-games-creep-into-math.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-7432777528338159738Wed, 08 Jul 2015 17:04:00 +00002015-07-08T13:04:00.854-04:00Aspen Ideas FestivalDavid LeonhardtJo BoalerJordan EllenbergKhan AcademySteven StrogatzIs Math Important?This month’s column is short on words, because I want to give you time to watch a <a href="http://video.pbs.org/video/2365521689/" target="_blank">great video</a> (1 hr 18 min in length) from the recent <a href="http://www.aspenideas.org/" target="_blank">Aspen Ideas Festival</a>. It’s a panel discussion (actually, two discussions, back-to-back) hosted by New York Times journalist David Leonhardt. The topic is the question that I have chosen as the title for this post: Is math important? What makes this particularly worth watching is the selection of speakers and the views they express.<br /><br />From the mathematical world there are Steven Strogatz of Cornell University and Jordan Ellenberg of the University of Wisconsin, and from mathematics education research there is Jo Boaler of Stanford University. They are joined by David Coleman, President of the College Board, education writer Elizabeth Green, author of the recent book <i>Building a Better Teacher</i>, Pamela Fox, a computer scientist working with Khan Academy, and financier Steve Rattner.<br /><br />The conversation is lively and informative, and moves along at a brisk, engaging pace, with each speaker given time to provide in-depth answers (a refreshing antidote to the idiotic “received wisdom” that today’s viewers are not capable of watching a video longer than two-and-a-half minutes, a Big Data statistic that almost certainly says more about the abysmal engagement quality of most videos than about audience attention span).<br /><br />That’s it from me. <a href="http://video.pbs.org/video/2365521689/" target="_blank">Here</a> is the video.<br /><br />http://devlinsangle.blogspot.com/2015/07/is-math-important.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-6147616842470664912Tue, 02 Jun 2015 17:40:00 +00002015-06-02T13:40:37.771-04:00ETSliteracynumeracyOECDPIAACPisaproblem solvingPIACC – PISA for grown-ups<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-GHZ76RaEwJc/VW3evpfCGzI/AAAAAAAAKWg/0oGkSfnN4hU/s1600/ETS_cover.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-GHZ76RaEwJc/VW3evpfCGzI/AAAAAAAAKWg/0oGkSfnN4hU/s320/ETS_cover.png" width="247" /></a></div>Greetings from 37,000 feet. As I write these words, I am on my way from San Francisco, California, to Boston, Massachusetts, to participate in a two-day workshop at Harvard, sponsored by the OECD (Organisation for Economic Co- operation and Development), to look at what should go into the math tests that will be administered to children around the world for PISA 2021.<br /><br />PISA, the Programme for International Student Assessment, gets such extensive press coverage each time one of its reports is published, that it really needs no introduction. Americans have grown used to the depressing fact that US school children invariably perform dismally, ranked near the bottom of the international league tables, with countries like Japan and Finland jostling around at the top.<br /><br />But chances are you have not heard of PIACC – the Programme for the International Assessment of Adult Competencies. The OECD introduced this new program a few years ago to investigate the nation-based adult skillsets that are most significant to national prosperity in a modern society: literacy, numeracy, and problem solving in a technology-rich environment (PS-TRE).<br /><br />Whereas the PISA surveys focus on specific age-groups of school students, PIAAC studied adults across the entire age range 16 to 65.<br /><br />The first report based on the PIAAC study was published in fall 2013: <a href="http://dx.doi.org/10.1787/9789264204256-en" target="_blank"><i>OECD Skills Outlook 2013: First Results from the Survey of Adult Skills</i></a>.<br /><br />A subsequent OECD report focused on PIACC data for US adults. The report’s title, <i><a href="http://dx.doi.org/10.1787/9789264204904-en" target="_blank">Time for the U.S. to Reskill</a></i>, gives the depressing-for-Americans headline that warns you of its contents. The skill levels of American adults compared to those of 21 other participating OECD countries were found to be dismal right across the board. The authors summarized US performance as “weak on literacy, very poor on numeracy,” and slightly below average on PS-TRE.<br /><br />“Broadly speaking, the weakness affects the entire skills distribution, so that the US has proportionately more people with weak skills than some other countries and fewer people with strong skills,” the report concluded.<br /><br />I have not read either OECD report. As happened when I never was able to watch the movie Schindler’s List, it is one of those things I feel I ought to read but cannot face the depression it would inevitably lead to. Rather, for airplane reading on my flight from Stanford to Harvard, I took with me a recently released (January 2015) report from the Princeton, NJ-based Educational Testing Service (ETS), titled <a href="http://www.ets.org/millennials" target="_blank">AMERICA’S SKILLS CHALLENGE: Millennials and the Future</a>.<br /><br />The ETS report disaggregates the PIAAC US data for millennials—the generation born after 1980, who were 16–34 years of age at the time of the assessment.<br /><br />The millennial generation has attained more years of schooling than any previous cohort in American history. Moreover, America spends more per student on primary through tertiary education than any other OECD nation. Surely then, this report would not depress me? I would find things to celebrate. <br /><br />Did I? Read on.<br /><br />This month’s column is distilled from the notes I made as I read through the ETS report. (These are summarizing notes. I did not bother to quote exactly, or even to use quotation marks when lifting a passage straight from the report. The originals of all the reports cited here are all freely available on the Web, so please go to the source documents to see what was originally written.)<br /><br />A central message emerging from the ETS report is that, despite all the costly and extensive education, US millennials on average demonstrate relatively weak skills in literacy, numeracy, and problem solving in technology-rich environments, compared to their international peers. Sigh.<br /><br />And this is not just true for millennials overall, it also holds for our best performing and most educated young adults, for those who are native born, and for those from the highest socioeconomic background. Moreover, the report’s findings indicate a decrease in literacy and numeracy skills for US adults when compared with results from previous adult surveys.<br /><br />Some of the data highlights: <br /><ul><li>In literacy, US millennials scored lower than 15 of the 22 participating countries.</li><li>In numeracy, US millennials ranked last.</li><li>In PS-TRE, US millennials also ranked last.</li><li>The youngest segment of the US millennial cohort (16- to 24-year-olds), who will be in the labor force for the next 50 years, ranked last in numeracy and among the bottom countries in PS-TRE.</li></ul>Even worse for those of us in higher education, this dismal picture holds for those with higher education:<br /><ul><li>US millennials with a four-year bachelor’s degree scored third from bottom in numeracy.</li><li>US millennials with a master’s or research degree were fourth from bottom.</li></ul>All very depressing. I fear that this state of affairs will continue all the time US education continues to be treated as a political football, with our nation’s children and their teachers treated as pawns while various groups fight political battles, and make decisions, based not on learning research (of which there is now a copious amount, much of it generated in US universities) but on uninformed beliefs and political ideology. [You were surely waiting for me to throw in my two cents worth of opinion. There it is.]<br /><br />To finish on a high note, we Americans famously like winners. So let’s raise a glass to the nations that came out on top in the rankings (in order, top first):<br /><br />Literacy: Japan, Finland, Netherlands<br /><br />Numeracy: Japan, Finland, Belgium<br /><br />PS-TRE: Japan, Finland, Australia<br /><br />In their own way, each of these countries seems to be doing education better than we are.<br /><br />Yet here’s the fascinating thing. I’ve spent time in all of those countries. They each have a lot to offer, and I like them all. I also was born and grew up in the UK, moving to the US as an adult in 1987. I am a lifelong educator. But for all its faults (and its education system is just one of a legion of things America does poorly) I’d rather live where I do now, in the USA, with Italy in second place. But that’s another story. A complicated story. (If you think California is a separate nation, and in many ways it is, then my preference statement needs further parsing.) Doing well on global tests of educational attainment is just one factor that we can use to measure quality of life.http://devlinsangle.blogspot.com/2015/06/piacc-pisa-for-grown-ups.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-2021852599804282078Thu, 07 May 2015 16:30:00 +00002015-05-07T12:31:54.889-04:00Jo BoalerK-12 educationYouCubed Time to re-read (or read) What’s Math Got To Do With It?Back in June 2010, I wrote <a href="https://www.maa.org/external_archive/devlin/devlin_06_10.html" target="_blank">a post</a> to this blog in which I summarized a new book on K-12 mathematics teaching by a former Stanford colleague of mine, Prof Jo Boaler. At the time, though I had met Jo a few times, I did not really know her; rather I was just one of many mathematical educators who simply admired her work, some of which she described in the book <a href="http://www.amazon.com/Whats-Math-Got-Do-Teachers/dp/0143115715/ref=sr_1_1?ie=UTF8&s=books&qid=1272973115&sr=1-1" target="_blank"><i>What's Math Got To Do With It?</i></a>, parts of which were the primary focus of my post.<br /><br />Not long after my post appeared, Jo returned to Stanford from the UK, and over time we got to know each other better. When I formed my mathematics educational technology company <a href="http://www.brainquake.com/" target="_blank">BrainQuake</a> in 2012, I asked her to be a founding member of its Board of Academic Advisors, all of whom are listed <a href="http://www.brainquake.com/our-team/" target="_blank">here</a>. When she was putting the final touches to the <a href="http://www.amazon.com/Whats-Math-Got-Transform-Mathematics/dp/0143128299/ref=pd_sim_b_4?ie=UTF8&refRID=1NSW50TCVF156P7KCJBJ" target="_blank">new edition</a> of her book, just published, she asked me to write a cover-quote, which I was pleased to do.<br /><br />I say all of this by way of disclosure.* For my primary aim in writing this month’s column is to persuade you to read (or re-read) my <a href="https://www.maa.org/external_archive/devlin/devlin_06_10.html" target="_blank">earlier post</a>, and ideally Jo’s book. The research findings she describes in the book highlight the lasting damage done to generations of K-12 students (and possibly consequent damage to the US economy when that generation of students enters the workforce) by continuing adherence to a classroom mathematics pedagogy that portrays math as a rule-based process of answer-getting, rather than a creative enterprise of understanding and problem solving.<br /><br />The woefully ill-informed “debate” about the benefits of the US Common Core State Mathematics Standards that has been fostered in between the appearances of the two editions of Boaler’s book, make her message even more important than it was when the first edition came out in 2009. While CCSS opponents espouse opinions, Boaler presents evidence – lots of it – that supports the approach to K-12 mathematics learning the CCSS promotes.<br /><br />If you want to see more of Prof Boaler’s efforts to improve K-12 mathematics education, see her teachers’ resource site <a href="http://www.youcubed.org/" target="_blank">YouCubed</a>, or sign up for her online course <a href="http://online.stanford.edu/course/how-to-learn-math-for-teachers-and-parents-s14" target="_blank">How to Learn Math: for Teachers and Parents</a>, which starts on June 16.<br /><br />Also, check out her latest post in <a href="http://hechingerreport.org/memorizers-are-the-lowest-achievers-and-other-common-core-math-surprises/" target="_blank">The Hechinger Report</a> where she presents some recent data about the problems caused by a lot of old-style rule-memorization math instruction.<br /><br />* NOTE: Prof Boaler’s Stanford research team also recently completed a small <a href="http://www.brainquake.com/backed-by-science/" target="_blank">pilot study</a> of BrainQuake’s mathematics learning (free-) app <a href="https://itunes.apple.com/us/app/wuzzit-trouble/id600190128?ls=1&mt=8" target="_blank">Wuzzit Trouble</a>, first reported by education technology journalist Jordan Shapiro in an April 27, 2015 <a href="http://www.forbes.com/sites/jordanshapiro/2015/04/27/stanford-study-shows-dramatic-math-improvement-from-playing-video-games-just-10-minutes-per-day/" target="_blank">article in Forbes Magazine</a>. (Prof Boaler is an academic advisor to BrainQuake but does so as a volunteer, and has no financial stake in the company.)http://devlinsangle.blogspot.com/2015/05/time-to-re-read-or-read-whats-math-got.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4160571695042815149Wed, 01 Apr 2015 20:02:00 +00002015-04-01T16:02:48.947-04:00abstractioncomputer science educationThe Importance of Mathematics Courses in Computer Science EducationThe confluence of two events recently reminded me of an article I wrote back in 2003 about the role of mathematics courses in university computer science education. [<i>Why universities require computer science students to take math</i>, Communications of the Association for Computing Machinery, Vol 46, No 9, Sept 2003, pp.36-39.]<br /><br />The first event was a request for me to be an advisor on a research project to develop K-12 computer science programs. The second was a forum discussion in my <a href="https://www.coursera.org/course/maththink" target="_blank">Mathematical Thinking MOOC</a>, currently in the middle of its sixth session.<br /><br />My MOOC attracts a lot of mid-career computer professionals, who bring a different perspective to some of the issues the course considers. The forum thread in question focused on what is meant by a statement of the form “Let x be such that P(x).“ In mathematics, use of this statement requires that there exists an object satisfying P. If the existence is not known, you should express the statement counterfactually, as “Let x be an object such that P(x), assuming such an object exists.”<br /><br />Some of the computer scientists, however, instinctively interpreted the statement “Let x be such that P(x)” as a variable declaration. This led them to give an “incorrect” answer to a question that asked then to identify exactly where the logic of a particular mathematical argument broke down. The <i>logic</i> failed with the selection of an object x that was not known to exist. In contrast, those computer scientists felt that things went wrong when the argument <i>subsequently</i> tried to do something with that x. That, they observed in the discussion, was where the program would fail.<br /><br />It was a good discussion, that highlighted the distinction between the currently accepted view of mathematics as primarily about properties and relations, and the pre-nineteenth century view that it was at heart procedural. As such, it served as a reminder of the value of mathematics courses in computer science education, and vice versa.<br /><br />The remainder of this post is what I wrote in the <i>CACM</i> back in 2003 (very lightly edited). I still agree with what I wrote then. (That is by no means always the case when I look at things I wrote more than a decade earlier.) I suspect that now, as then, some will not agree with me. (I actually received some ferociously angry responses to my piece.) Here goes.<br /><br />Some years ago, I gave a lecture to the Computer Science Department at the University of Leeds in England. Knowing my background in mathematics — in particular, mathematical logic — the audience expected that my talk would be fairly mathematical, and on that particular occasion they were right. As I glanced at the announcement of my talk posted outside the lecture room, I noticed that someone had added some rather telling graffiti. In front of the familiar header “Abstract” above the description of my talk, the individual had scrawled the word “Very”.<br /><br />It was a cute addition. But it struck me then, and does still, many years later, that it spoke volumes about the way many CS students view the subject. To the graffiti writer, operating systems, computer programs, and databases were (I assumed) not abstract, they were real. Mathematical objects, in contrast, so the graffiti-writer likely believed — and I have talked to many students who feel this way — are truly abstract, and reasoning about them is an abstract mental pursuit. Which goes to show just how good we humans are (perhaps also how effective university professors are) at convincing ourselves (and our students) that certain abstractions are somehow real.<br /><br />For the truth is, of course, that computer science is entirely about abstractions. The devices we call computers don’t, in of themselves, compute. As electrical devices, if they can be said to do anything, it’s physics. It is only by virtue of the way we design those electrical circuits that, when the current flows, obeying the laws of physics, we human observers can pretend that they are doing reasoning (following the laws of logic), performing a numerical calculation (following the laws of arithmetic), or searching for information. True, it’s a highly effective pretence. But just because it’s useful does not make it any less a pretence.<br /><br />Once you realize that computing is all about constructing, manipulating, and reasoning about abstractions, it becomes clear that an important prerequisite for writing (good) computer programs is an ability to handle abstractions in a precise manner. Now that, as it happens, is something that we humans have been doing successfully for over three thousand years. We call it mathematics.<br /><br />This suggests that learning and doing mathematics might play an important role in educating future computer professionals. But if so, then what mathematics? From an educational point of view, in order to develop the ability to reason about formal abstractions, it is largely irrelevant exactly what abstractions are used. Our minds, which evolved over tens of thousands of years to reason (largely imprecisely) about the physical world, and more recently the social world, find it extremely difficult accepting formal abstractions. But once we have learned how to reason precisely about one set of abstractions, it takes relatively little extra effort to reason about any other.<br /><br />But surely, you might say, even if I’m right, when it comes to training computer scientists, it makes sense to design educational courses around the abstractions the computer scientists will actually use when they graduate and go out to work in the technology field. Maybe so (in fact no, but I’ll leave that argument to another time), but who can say what the dominant programming paradigms and languages will be four years into the future? Computing is a rapidly shifting sand. Mathematics, in contrast, has a long history. It is stable and well tested.<br /><br />Sure, there is a good argument to be made for computer science students to study discrete mathematics rather than calculus. But, while agreeing with that viewpoint, I believe it is often overplayed. Here’s why I think this.<br /><br />A common view of education is that it is about acquiring knowledge — learning facts. After all, for the most part that is how we measure the effectiveness of education: by testing the students’ knowledge. But that’s simply not right. It might be the aim of certain courses, but it’s definitely not the purpose of education. The real goal of education is to improve minds — to enable them to acquire abilities and skills to do things they could not do previously. As William Butler Yeats put it, “Education is not about filling a bucket; it’s lighting a fire.” Books and USB memory sticks store many more facts than people do — they are excellent buckets — but that doesn’t make them smart. Being smart is about doing, not knowing.<br /><br />Numerous studies have shown that if you test university students just a few months after they have completed a course, they will have forgotten most of the facts they had learned, even if they passed the final exam with flying colors. But that doesn’t mean the course wasn’t a success. The human brain adapts to intellectual challenges by forging and strengthening new neural pathways, and those new pathways remain long after the “facts” used to develop them have faded away. The facts fade, but the abilities remain.<br /><br />If you want to prepare people to design, build, and reason about formal abstractions, including computer software, the best approach surely is to look for the most challenging mental exercises that force the brain to master abstract entities — entities that are purely abstract, and which cause the brain the maximum difficulty to handle. And where do you find this excellent mental training ground? In mathematics.<br /><br />Software engineers may well never apply any of the specific theorems or techniques they were forced to learn as students (though some surely will, given the way mathematics connects into most walks of life in one way or another). But that doesn’t mean that those math courses were not important. On the contrary. The main benefit of learning and doing mathematics, I would argue, is not the specific content; rather it’s the fact that it develops the ability to reason precisely and analytically about formally defined abstract structures.http://devlinsangle.blogspot.com/2015/04/the-importance-of-mathematics-courses.htmlnoreply@blogger.com (Mathematical Association of America)1tag:blogger.com,1999:blog-2516188730140164076.post-4247914760283743603Mon, 09 Mar 2015 04:00:00 +00002015-03-16T11:17:38.214-04:00cyclingEuler’s IdentityGregory’s SeriesMultiplicationpiPi Dayrepeated addition (not)Pi Day, Cyclical Motion, and a Great Video Explanation of Multiplication<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-wgUFZVC_bxk/VP8JvCHQ0ZI/AAAAAAAAKUQ/ZzLh7eYR4wI/s1600/Leibniz.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-wgUFZVC_bxk/VP8JvCHQ0ZI/AAAAAAAAKUQ/ZzLh7eYR4wI/s1600/Leibniz.jpg" height="240" width="320" /></a></div><br />March 14 is Pi Day, the day in the year when we celebrate the world’s most famous mathematical constant.<br /><br />Back in 1988, on March 14, a physicist called Larry Shaw organized the first Pi Day celebration at the Exploratorium in San Francisco, where he worked. It was meant to be just a one-off, fun event to get kids interested in math. Children were invited in to march around one of the Exploratorium’s circular rooms and end up eating fruit pies. But the idea took off, and ever since, March 14 has been Pi Day. Not just at the Exploratorium, but with celebratory events organized all across the United States, and in other parts of the world.<br /><br />In case you haven’t twigged it, we celebrate Pi Day on March 14 because, in American date format, that day is 3.14, which is pi to two decimal places.<br /><br />This year is a particularly special, once-in-a-century Pi Day, since the American format date this year is 3.14.15, pi to four decimal places. If you want more pi-accuracy, drink a toast to pi at time 9:26:53 (AM or PM), to get the first nine places 3.141592653.<br /><br />That degree of accuracy, by the way, is more than enough for practical purposes. If you use that value to calculate the circumference of the Earth, the answer will be accurate to within 1/4 inch.<br /><br />Though we have known since the 18<sup>th</sup> Century that pi is irrational (indeed, transcendental, thereby demonstrating that you cannot square a circle), calculating approximate values of pi has a long history. In ancient times, Babylonians, Egyptians, Greeks, Indians, and Chinese mathematicians calculated the first three or four places, and found fraction approximations like 22/7 and 355/113.<br /><br />In the 16<sup>th</sup> century, a German who presumably had a lot of time on his hands spent most of his life computing pi to 36 places, and a 19<sup>th</sup> century American went all the way to 707 places, but he mad a mistake after 527 places, so the last part of his answer was wrong.<br /><br />In more recent times, computers have been used to compute pi to well over a trillion places, in part for sport, but also to test the accuracy of high speed supercomputers.<br /><br />Of course, PI Day isn’t really just about pi, it’s an excuse to celebrate all of mathematics, and in particular stimulate interest in mathematics among children and young adults. You will find Pi Day events in schools and colleges, at science museums, and other venues. Teachers, instructors, and students organize all kinds of math-related events and competitions. The value of pi simply sets the date.<br /><br />With this year’s special Century edition, some large organizations are putting on celebratory events, among them the Museum of Mathematics in New York City (details of the event <a href="https://in.momath.org/civicrm/event/info?reset=1&id=386" target="_blank">here</a>), the Computer History Museum just south of San Francisco (details <a href="http://www.computerhistory.org/events/upcoming/#pi-day-celebration" target="_blank">here</a>), and the NASA Space Center in Houston (see <a href="http://spacecenter.org/cosmic-spring-break/pi-day/" target="_blank">here</a>). And at the big <a href="http://teachingandlearning2015.org/" target="_blank">Teaching and Learning Conference</a> in Washington D.C. this week, I’m hosting a Pi Celebration at 8:00AM on Saturday morning.<br /><br />There are many other celebrations. Check to see what is going on in your area. If there is a large science or technology organization nearby, they may well be putting on a Pi Day event.<br /><br />The media have been getting in on the act too. NPR will air one of my short <a href="http://web.stanford.edu/~kdevlin/MathGuy.html" target="_blank">Math Guy</a> conversations with Weekend Edition host Scott Simon this Saturday morning, and today’s New York Times ran a <a href="http://wordplay.blogs.nytimes.com/2015/03/09/%CF%80/?_r=0" target="_blank">substantial article</a> about pi by their regular Numberplay contributor Garry Antonick.<br /><br />Antonick led off with a short pi-related problem I provided him with, and in honor of the Pi Day of the Century, in place of the traditional photo of me at a blackboard, he picked an action shot of me cresting a mountain on a bicycle (pi motion if ever there were) in a Century (100 mile) ride back in 2013.<br /><br />He could not resist bringing in the famous Euler Identity, linking the five most significant constants of mathematics, <i>pi</i>, <i>e</i>, <i>i</i>, 0, and 1. This has always been my favorite mathematical identity, and Antonick quotes from a magazine article I wrote about it a few years ago.<br /><br />But truth be told, it is not my favorite pi fact. For the simple reason, it’s not really about pi, rather it is about multiplication and exponentiation. Pi gets in because both operations involve the number.<br /><br />My favorite pi fact, ever since I first came across it as a teenager (one of several eye-opening moments that motivated me to become a mathematician), is Leibniz’s series (sometimes called Gregory’s series), which dates from the 17<sup>th</sup> century. You write down an endless addition sum that starts out 1/1, minus 1/3, plus 1/5, minus 1/7, plus 1/9, etc. All the reciprocals of the odd numbers, with alternating signs.<br /><br />Since this sum goes on for ever, you can’t actually add it up term by term, but you can use mathematical techniques to determine the answer a different way. And that answer is pi/4.<br /><br />What does pi have to do with adding the reciprocals of the odd numbers? As with Euler’s Identity, Leibniz’s series provides a glimpse of the deep structure of numbers and arithmetic that lies just beneath the surface.<br /><br />Talking of which, I caused a huge stir a few years ago when I ran a series of Devlin’s Angle posts trying to rid people (in particular, math teachers) of their false (and educationally dangerous) belief that multiplication is repeated addition (and exponentiation is repeated multiplication).<br /><br />The initial series ran in <a href="http://www.maa.org/external_archive/devlin/devlin_06_08.html" target="_blank">June</a>, <a href="http://www.maa.org/external_archive/devlin/devlin_0708_08.html" target="_blank">July-August</a>, and <a href="http://www.maa.org/external_archive/devlin/devlin_09_08.html" target="_blank">September</a> 2008. When the barrage of facts I referenced in the third of those posts failed to stem the flood of disbelieving reactions of readers, I ran a lengthy post in <a href="http://www.maa.org/external_archive/devlin/devlin_01_11.html" target="_blank">January 2011</a> trying to convey the truly deep (and powerful) structure of multiplication.<br /><br />Still to little avail. Put repeated addition in the same bin as evolution by natural selection, climate change, and the <a href="https://www.maa.org/external_archive/devlin/devlin_05_07.html" target="_blank">Golden Ratio</a>. For many people, no amount of facts can overturn a long held and cherished belief. It’s a common human trait – fortunately not a universal one, else we’d still be living in caves and mud huts. (A politician who says “I am not a scientist” is effectively saying “I don’t understand the difference between building my mansion and a mud hut.”)<br /><br />Unfortunately, as a wordsmith, I did not, and do not, possess the skill to provide a really good explanation of multiplication. I had to resort to spinning a multi-faceted story based on scaling. Someone who does have what it takes to tell the story properly, using video, is Stanford mathematics and computer science senior undergraduate <a href="https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw/about" target="_blank">Grant Sanderson</a>. His recent <a href="https://www.youtube.com/watch?v=F_0yfvm0UoU" target="_blank">video on Euler’s Identity</a> is the best explanation of addition and multiplication I have ever seen. Period. Antonick embeds it in his New York Times piece. It deserves widespread circulation.<br /><br />The video actually goes on to discuss the exponential function, and then the Euler Identity, but I suspect many viewers will get lost at that point. The exponential function is pretty sophisticated. Much more so than addition and multiplication. In contrast, all it takes to understand those two staples of modern numerical life is to get beyond the ultimately misleading concepts many of us form in the first few years of our lives. Do that, and Sanderson’s video provides the rest.<br /><br />As is so often said, a picture can be worth a thousand words. Sanderson demonstrates that a motion picture can be worth a hundred thousand.<br /><br />NOTE: I did try song a few years ago, collaborating with a Santa Cruz choral group called Zambra. The result can be found <a href="http://web.stanford.edu/~kdevlin/HE.html" target="_blank">here</a>. There’s lots of pi stuff in those compositions. But it’s primarily musical interpretation of mathematics, not explanation. (For instance, check out our rendering of <a href="http://web.stanford.edu/~kdevlin/HE_QTmovies/Part_5.mp4" target="_blank">Leibniz’s series</a>.)<br /><br />Finally, I often get asked why we use the Greek letter pi to denote the ratio of the circumference of a circle to its <i>perimeter</i> of a circle to its diameter.. This convention goes back to the 18<sup>th</sup> Century.<br /><br />http://devlinsangle.blogspot.com/2015/03/pi-day-cyclical-motion-and-great-video.htmlnoreply@blogger.com (Mathematical Association of America)1tag:blogger.com,1999:blog-2516188730140164076.post-5667154965858089610Sun, 01 Feb 2015 05:00:00 +00002015-02-17T13:24:45.679-05:00The Greatest Math Teacher Ever?<em>Last month I wrote about the kind of mathematic learning experiences we need to design to prepare young people for life in the Twenty-First Century. I cited the hugely successful, pioneering educational work of the late Professor R L Moore of the University of Texas. This follow up article about Moore and his teaching method is a combination of two earlier Devlin’s Angle posts, from <a href="http://www.maa.org/external_archive/devlin/devlin_5_99.html" target="_blank">May 1999</a> and <a href="http://www.maa.org/external_archive/devlin/devlin_6_99.html" target="_blank">June 1999</a>. Other than adding a short paragraph at the end leading to further information about Moore, the only changes to my original text are minor updates to adjust for the passage of time.</em><br /><br /><b>The set-up</b><br /><br />Each year, the MAA recognizes great university teachers of mathematics. Anybody who has been involved in selecting a colleague for a teaching award will know that it is an extremely difficult task. There is no universally agreed upon, ability-based linear ordering of mathematics teachers, even in a single state in a given year. How much more difficult it might be then to choose a “greatest ever university teacher of mathematics.” But if you base your decision on which university teacher of mathematics has had (through his or her teaching) the greatest impact on the field of mathematics, then there does seem to be an obvious winner. Who is it?<br /><br />Most of us who have been in mathematics for over thirty years probably know the answer. Let me give you some clues as to the individual who, I am suggesting, could justifiably be described as the American university mathematician who, through his (it’s a man) teaching, has had the greatest impact on the field of mathematics.<br /><br />He died in 1974 at the age of 91. <br /><br />He was cut in the classic mold of the larger-than-life American hero: strong, athletic, fit, strikingly good looking, and married to a beautiful wife (in the familiar Hollywood sense). <br /><br />He was well able to handle himself in a fist fight, an expert shot with a pistol, a lover of fast cars, self-confident and self-reliant, and fiercely independent. <br /><br />Opinionated and fiercely strong-willed, he was forever embroiled in controversy. <br /><br />He was extremely polite; for example, he would always stand up when a lady entered the room. <br /><br />He was a pioneer in one of the most important branches of mathematics in the twentieth century. <br /><br />He was a elected to membership of the National Academy of Science, as were three of his students. <br /><br />The method of teaching he developed is now named after him. <br /><br />If you measure teaching quality in terms of the product - the successful students - our man has no competition for the title of the greatest ever math teacher. During his 64 year career as a professor of mathematics, he supervised fifty successful doctoral students. Of those fifty Ph.D.’s, three went on to become presidents of the AMS - a position our man himself held at one point - and three others vice-presidents, and five became presidents of the MAA. Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own and helping shape the development of American mathematics as it rose to its present-day position of world dominance.<br /><br />In the first half of the Twentieth Century, fully 25% of the time the president of the MAA was either a student or a grandstudent of this man. <br /><br />Other students and grandstudents of our mystery mathematician served as secretary, treasurer, or executive director of one of the two mathematical organizations and were editors of leading mathematical journals. <br /><br />After reaching the age of 70, the official retirement age for professors at his university, in 1952, he not only continued to teach at the university, he continued to teach a maximum lecture load of five courses a year - more than most full-time junior faculty took on. He did so despite receiving only a half-time salary, the maximum that state regulations allowed for someone past retirement age who for special reasons was kept on in a “part-time” capacity. <br /><br />He maintained that punishing schedule for a further eighteen years, producing 24 of his 50 successful Ph.D. students during that period. <br /><br />He only gave up in 1969, when, after a long and bitter battle, he was quite literally forced to retire. <br /><br />In 1967, the <em>American Mathematical Monthly</em> published the results of a national survey giving the average number of publications of doctorates in mathematics who graduated between 1950 and 1959. The three highest figures were 6.3 publications per doctorate from Tulane University, 5.44 from Harvard, and 4.96 from the University of Chicago. During that same period, our mathematician’s students averaged 7.1. What makes this figure the more remarkable is that our man had reached the official retirement age just after to the start of the period in question.<br /><br />Who was he? <br /><br /><strong>The answer</strong><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-8eVQb6UIN0o/VONKKMYZxOI/AAAAAAAAKTI/bbvlekI4yzE/s1600/Feb15_RLM.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://1.bp.blogspot.com/-8eVQb6UIN0o/VONKKMYZxOI/AAAAAAAAKTI/bbvlekI4yzE/s1600/Feb15_RLM.jpg" /></a></div>His name is Robert Lee Moore. Born in Dallas in 1882, R. L. Moore (as he was generally known) stamped his imprint on American mathematics to an extent that only in his later years could be fully appreciated. <br /><br />As I noted earlier, during 64 year career, the last 49 of them at the University of Texas, Moore supervised fifty successful doctoral students. Three of them went on to become presidents of the AMS (R. L. Wilder, G. T. Whyburn, R H Bing) -- a position Moore also held -- and three others vice-presidents (E. E. Moise, R. D. Anderson, M. E. Rudin), and five became presidents of the MAA (R. D. Anderson, E. Moise, G. S. Young, R H Bing, R. L. Wilder). Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own -- mathematical grandchildren of Moore -- and helping shape the development of American mathematics as it rose to its present-day dominant position. That’s quite a record!<br /><br />In 1931 Moore was elected to membership of the National Academy of Science. Three of his students were also so honored: G. T. Whyburn in 1951, R. L. Wilder in 1963, and R H Bing in 1965. <br /><br />In 1973, his school, the University of Texas at Austin, named its new, two-winged, seventeen-story mathematics, physics, and astronomy building after him: Robert Lee Moore Hall. This honor came just over a year before his death. The Center for American History at the University of Texas has established an entire collection devoted to the writings of Moore and his students. <br /><br /><strong>Discovery learning</strong><br /><br />Moore was one of the founders of modern point set topology (or general topology), and his research work alone puts him among the very best of American mathematicians. But his real fame lies in his achievements as a teacher, particularly of graduate mathematics. He developed a method of teaching that became widely known as “the Moore Method”. Its present-day derivative is often referred to as “Discovery Learning” or “Inquiry-Based Learning” (IBL).<br /><br />One of the first things that would have struck you if you had walked into one of Moore’s graduate classes was that there were no textbooks. On each student’s desk you would see the student’s own notebook and nothing else! To be accepted into Moore’s class, you had to commit not only not to buy a textbook, but also not so much as glance at any book, article, or note that might be relevant to the course. The only material you could consult were the notes you yourself made, either in class or when working on your own. And Moore meant alone! The students in Moore’s classes were forbidden from talking about anything in the course to one another -- or to anybody else -- outside of class. Moore’s idea was that the students should discover most of the material in the course themselves. The teacher’s job was to guide the student through the discovery process in a modern-day, mathematical version of the Socratic dialogue. (Of course, the students did learn from one another during the class sessions. But then Moore guided the entire process. Moreover, each student was expected to have attempted to prove all of the results that others might present.)<br /><br />Moore’s method uses the axiomatic method as an instructional device. Moore would give the students the axioms a few at a time and let them deduce consequences. A typical Moore class might begin like this. Moore would ask one student to step up to the board to prove a result stated in the previous class or to give a counterexample to some earlier conjecture -- and very occasionally to formulate a new axiom to meet a previously identified need. Moore would generally begin by asking the weakest student to make the first attempt -- or at least the student who had hitherto contributed least to the class. The other students would be charged with pointing out any errors in the first student’s presentation.<br /><br />Very often, the first student would be unable to provide a satisfactory answer -- or even any answer at all, and so Moore might ask for volunteers or else call upon the next weakest, then the next, and so on. Sometimes, no one would be able to provide a satisfactory answer. If that were the case, Moore might provide a hint or a suggestion, but nothing that would form a constitutive part of the eventual answer. Then again, he might simply dismiss the group and tell them to go away and think some more about the problem. <br /><br />Moore’s discovery method was not designed for -- and probably will not work in -- a mathematics course which should survey a broad area or cover a large body of facts. And it would obviously need modification in an area of mathematics where the student needs a substantial background knowledge in order to begin. But there are areas of mathematics where, in the hands of the right teacher -- and possibly the right students -- Moore’s procedure can work just fine. Moore’s own area of general topology is just such an area. You can find elements of the Moore method being used in mathematics classes at many institutions today, particularly in graduate courses and in classes for upper-level undergraduate mathematics majors, but few instructors ever take the process to the lengths that Moore did, and when they try, they rarely meet with the same degree of success.<br /><br />Part of the secret to Moore’s success with his method lay in the close attention he paid to his students. Former Moore student William Mahavier (now deceased) addressed this point:<br /><br />“Moore treated different students differently and his classes varied depending on the caliber of his students. . . . Moore helped his students a lot but did it in such a way that they did not feel that the help detracted from the satisfaction they received from having solved a problem. He was a master at saying the right thing to the right student at the right time. Most of us would not consider devoting the time that Moore did to his classes. This is probably why so many people claim to have tried the Moore method without success.”<br /><br />Another famous (now deceased) mathematician who advocated -- and has successfully used -- (a modern version of) the Moore method was Paul Halmos. He wrote: <br /><br />“Can one learn mathematics by reading it? I am inclined to say no. Reading has an edge over listening because reading is more active -- but not much. Reading with pencil and paper on the side is very much better -- it is a big step in the right direction. The very best way to read a book, however, with, to be sure, pencil and paper on the side, is to keep the pencil busy on the paper and throw the book away.”<br /><br />Of course, as Halmos went on to admit, such an extreme approach would be a recipe for disaster in today’s over-populated lecture halls. The educational environment in which we find ourselves these days would not allow another R. L. Moore to operate, even if such a person were to exist. Besides the much greater student numbers, the tenure and promotion system adopted in many (most?) present-day colleges and universities encourages a gentle and entertaining presentation of mush, and often punishes harshly (by expulsion from the profession) the individual who seeks to challenge and provoke -- an observation that is made problematic by its potential for invocation as a defense for genuinely poor teaching. But as numerous mathematics instructors have demonstrated, when adapted to today’s classroom, the Moore method -- discovery learning -- has a lot to offer.<br /><br />If you want to learn more about R. L. Moore and his teaching method, check out the web site: <a href="http://legacyrlmoore.org/">http://legacyrlmoore.org/</a>. <br /><br />But before you give the method a try, take heed of the advice from those who have used it: Plan well in advance and be prepared to really get to know your students. Halmos put it this way: <br /><br />“If you are a teacher and a possible convert to the Moore method ... don’t think that you’ll do less work that way. It takes me a couple of months of hard work to prepare for a Moore course. ... I have to chop the material into bite-sized pieces, I have to arrange it so that it becomes accessible, and I must visualize the course as a whole -- what can I hope that they will have learned when it’s over? As the course goes along, I must keep preparing for each meeting: to stay on top of what goes on in class. I myself must be able to prove everything. In class I must stay on my toes every second. ... I am convinced that the Moore method is the best way to teach there is -- but if you try it, don’t be surprised if it takes a lot out of you.”<br /><br /><strong>There is a caveat</strong><br /><br />When I wrote those two Devlin’s Angle posts back in 1999, I debated with myself whether to address a side to the Moore story that, particularly from a late Twentieth Century perspective, does not stand to his credit. The issue is race. <br /><br />Moore’s racial attitude was nothing unusual for a white person who was born and lived most of his life in Texas in the late Nineteenth Century and the first three quarters of the Twentieth. When the Civil Rights Act was passed in 1964 (yes, that recently!), making racial discrimination illegal, Moore was already long past retiring age, and just five years short of actually vacating his university office. Moreover, no one who regularly reads a newspaper would believe that racial discrimination in America is a thing of the past. Moore’s racial views are still not unusual in Texas and elsewhere. <br /><br />Were Moore not such a towering figure, his position on race (at least as demonstrated by his actions) would not merit attention. But like all great people, all aspects of his life become matters of scrutiny. Moore could have acted differently when it came to race, even back then, in Texas, but he did not. And from today’s perspective, that inevitably leaves an uncomfortable stain on his legacy. <br /><br />In writing my two 1999 columns, I chose to focus on Moore the university teacher, in particular to raise awareness of discovery learning in mathematics. The focus of Devlin’s Angle is, after all, mathematics and mathematics teaching. I did not want to distract from that goal with what is clearly a side issue, particularly such an explosive one. Moore’s larger-than-life character was clearly a significant part of his success. His racism (or at least racist behavior) was not a part of that success story – if for no other reason than because he never accepted any Black students. So I did not raise the issue. <br /><br />For the same reason, I have left this side of the Moore story to the end here. We can learn from Moore when it comes to designing good mathematics learning experiences, and even admire him as a highly gifted teacher, without condoning other aspects of his life, just as we can enjoy Wagner’s music without endorsing Nazism. I can however leave you with a pointer to an <a href="http://www.math.buffalo.edu/mad/special/RLMoore-racist-math.html" target="_blank">article</a> posted online by Mathematics Professor Scott Williams on 5/28/99, about the same time my articles appeared (and possibly in response to the first of them). Like it or not, Williams’ post shines light on another side to the Moore story. We can learn things from great people in ways other than taking a class from them, and we can perhaps learn things they were not trying to teach us. <br /><br /><strong>References</strong><br /><br />Paul R. Halmos (1975), The Problem of Learning to Teach: The Teaching of Problem Solving, <em>American Mathematical Monthly</em>, Volume 82, pp.466-470. <br />Paul R. Halmos (1985), <em>I Want to Be a Mathematician</em>, Chapter 12 (How to Teach), New York: Springer-Verlag, pp.253-265. <br />William S. Mahavier (1998), What is the Moore Method?, The Legacy of R. L. Moore Project, Austin, Texas: The University of Texas archives. <br /><br />http://devlinsangle.blogspot.com/2015/02/the-greatest-math-teacher-ever.htmlnoreply@blogger.com (Mathematical Association of America)1tag:blogger.com,1999:blog-2516188730140164076.post-7328069128642870018Thu, 01 Jan 2015 05:00:00 +00002015-01-03T12:18:12.901-05:00Common CoreEric MazurInquiry-Based LearningKen RobinsonPeer InstructionR L MooreSergio CorreaSugata MitraYour Father’s Mathematics Teaching No Longer Works<i>Gender-challenged title courtesy of this famous 1988 Oldsmobile TV commercial:</i><br /><br /><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/bAJ3-mbP1pY" width="560"></iframe></div><br />The start of a New Year is traditionally a time when we resolve to make changes. Change is particularly imperative in US mathematics education, which is built on a (Nineteenth Century) pedagogic model that long since passed its expiry date.<br /><br />In a nutshell, the school system we all grew up with was essentially developed in Nineteenth Century Britain to provided a global infrastructure to run the British Empire. In modern terms, the British Imperialists created an “Internet” and an “Internet of Things” using the best computational and manufacturing resources available at the time: people.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-RJnMC765zcQ/VKRBDS08HjI/AAAAAAAAKSY/sqPwde7hN_4/s1600/BritishEmpire1922.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-RJnMC765zcQ/VKRBDS08HjI/AAAAAAAAKSY/sqPwde7hN_4/s1600/BritishEmpire1922.jpg" height="301" width="500" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: x-small;">Controlled by an “Internet of human computers”: The British Empire in 1922.<br />Map from <a href="http://trivto.deviantart.com/" target="_blank">trivto.deviantart.com</a>.</span></td></tr></tbody></table>While few of us in K-16 education today see it as merely a process to prepare young people for work, we inherited a system built to do just that. <br /><br />Now we have an electronic, digital Internet, does it make sense to continue to use the old system? <br /><br />What do Twenty-First Century citizens need from their education? <br /><br />While not the only thing—not even close—equipping young people for work is still an important educational goal, both for the individual and for society as a whole. Accordingly, it makes sense for those of us in systemic education to be constantly aware of the skills that are actually required in the workplace. If those skills change, so should the education we provide. <br /><br />A good place to start is by asking the leaders of the leading companies what they look for when hiring new employees. The table below shows us what skills the Fortune 500 companies were asking for in 1970, then again thirty years later in 1999. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-66atIwXiVZo/VKRBkPopK0I/AAAAAAAAKSg/KVNnVihH4-o/s1600/Fortune500.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-66atIwXiVZo/VKRBkPopK0I/AAAAAAAAKSg/KVNnVihH4-o/s1600/Fortune500.png" height="220" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: x-small;">Fortune 500 most valued skills, cited in<br />Linda Darling-Hammond et al, <a href="http://www.testpublishers.org/assets/criteria-higher-quality-assessment_1.pdf" target="_blank">Criteria for High-Quality Assessment</a> (2013)</span></td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"></div><br />It makes for dramatic reading. While the required skills ranked from 4 through 9 remained unaltered, the top three changed completely. The most important skill in 1970, writing, dropped to number 10, while skills two and three, computation and reading, respectively, dropped off the top ten list entirely.<br /><br />The most important skill in the workplace at the start of the Twenty-First Century, according to those leading companies, is teamwork, which in a single generation had leapt up from number 10. The other two skills at the top, Problem Solving and Interpersonal Skills, were not even listed back in 1970. <br /><br />Clearly, the world (of work) has changed, at least for those of us living in advanced societies. Unfortunately, for those of us in the United States, and many other parts of the world, our education system has failed to keep up. <br /><br />In large part, this is because of the hard-to-avoid inertia that so often comes with national (or statewide) education systems. By and large, many politicians and bureaucrats are far less aware of rapidly changing workforce requirements than those in business, and politicians frequently pander to the often woefully uninformed beliefs of voters, who tend to resist change–especially change that will affect their children.<br /><br />In the US, we see this dramatically illustrated by the widespread resistance to the Common Core State Standards. In the case of mathematics, just look at how closely the eight basic <a href="http://www.corestandards.org/Math/Practice/" target="”_blank”">Mathematics Principles</a> of the CCSS align to that Fortune 500 list of required Twenty-First Century skills:<br /><ol><li>Make sense of problems and persevere in solving them.</li><li>Reason abstractly and quantitatively.</li><li>Construct viable arguments and critique the reasoning of others.</li><li>Model with mathematics.</li><li>Use appropriate tools strategically.</li><li>Attend to precision.</li><li>Look for and make use of structure.</li><li>Look for and express regularity in repeated reasoning.</li></ol>The fact is, any parent who opposes adoption of the CCSS is, in effect, saying, “I do not want my child prepared for life in the Twenty-First Century.” They really are. Not out of lack of concern for their children, to be sure. Quite the contrary. Rather, what leads them astray is that they are not truly aware of how the huge shifts that have taken place in society over the last thirty years have impacted educational needs.<br /><br />Having lived through those changes, parents have (for the most part) been able to build on their own education and cope with new demands. “What worked for me will work for my children,” they say. (They say that even when it patently did not work for them!) <br /><br />But the situation is very different for their children. They are being thrust straight into that new world. To prepare them for that, you need a very different kind of education: one based on understanding rather then procedural mastery, and on exploration rather than instruction.<br /><br />One of the best summaries of this societal change, and the resulting need for educational shift, that I know is the 22 minutes TED Talk <a href="http://www.ted.com/talks/sugata_mitra_build_a_school_in_the_cloud?language=en" target="_blank">Build a School in the Cloud</a>, given by the educational researcher Sugata Mitra, winner of the 2013 TED Prize. <br /><br />In his talk, not only does Mitra explain why we need to make radical changes to education, he provides examples, backed by solid evidence, of how a “Fortune 500 oriented,” team-based, exploratory approach works. In the late 1990s and throughout the 2000s, Mitra conducted experiments in which he gave children in India access to computers. Without any instruction, they were able to teach themselves a variety of things, from English to DNA replication. <br /><br />[See also Mitra’s <a href="http://www.ted.com/talks/sugata_mitra_the_child_driven_education" target="_blank">earlier talk</a> from 2010.] <br /><br /><a href="http://www.ted.com/talks/ken_robinson_changing_education_paradigms" target="_blank">Another good account</a> of this need for educational change is provided by Sir Ken Robinson, also in a TED Talk (11 minutes). <br /><br />These ideas are not new. Indeed, they are mainstream in educational research circles. They just have not permeated society at large. <br /><br />For instance, Harvard physicist Eric Mazur has been teaching by Inquiry-Based Learning (IBL), to use one of several names for this general approach, for over twenty-five years, since he first noticed that instructional lectures simply do not work. He describes his approach, and the reasons for adopting it, in his 2009 talk <a href="https://www.youtube.com/watch?v=rvw68sLlfF8" target="_blank">Confessions of a Converted Lecturer</a> (18 minutes, abridged version). <br /><br />In mathematics, the IBL approach goes way back to the 1920s. I wrote about the best known proponent in two Devlin’s Angle posts back in 1999: <a href="http://www.maa.org/external_archive/devlin/devlin_5_99.html" target="_blank">May</a>, <a href="http://www.maa.org/external_archive/devlin/devlin_6_99.html" target="_blank">June</a>. <br /><br />Moore’s ideas have been adapted and used successfully in present-day mathematics classrooms, as shown in the promotional video <a _blank="" href="https://www.youtube.com/watch?v=f6t6WiWYdgY" target="_blank">Creativity in Mathematics: Inquiry-Based Learning and the Moore Method</a> (20 minutes). <br /><br />Lest my account of R L Moore and that last video portrayal leaves you with the impression that IBL math is for bright college students, see also this <a href="http://www.wired.com/2013/10/free-thinkers/all/" target="_blank">WIRED magazine account</a> of the success Mexican teacher Sergio Juárez Correa had when he took a Mitra inspired approach into a poor school in Matamoros, a city of half-a-million known more for its drug trade than for being at the forefront of Twenty-First Century mathematics education. <br /><br />Remember, Bob Dylan sang this in 1964: <br /><br /><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/e7qQ6_RV4VQ" width="420"></iframe></div><br /><br />AS OF TODAY, THAT’S OVER FIFTY YEARS AGO!<br /><br />It’s long past time for the education system to catch up with the world outside the classroom. That should be our resolution for 2015. <br /><br />http://devlinsangle.blogspot.com/2015/01/your-fathers-mathematics-teaching-no.htmlnoreply@blogger.com (Mathematical Association of America)24tag:blogger.com,1999:blog-2516188730140164076.post-2309051667875538595Wed, 10 Dec 2014 15:00:00 +00002014-12-10T14:26:08.140-05:00Common Coregame-based learningmath learning appmath learning video gamemathematics educationnumber senseStandard algorithmHow do you find good math learning apps?There are approximately 20,000 math learning apps available on the App Store (classified as such by their creators). Google Play does not provide the corresponding figure for Android apps, but presumably there are a lot there as well.<br /><br />Most of those apps do little more than provide repetitive practice of very basic skills, primarily about numbers. They are essentially just animated flash cards. <br /><br />How can a parent, or a teacher, decide which apps are likely to benefit their child, or their students? I’ll come back to that later. <br /><br />First, let me say that there is not necessarily anything wrong with an app that is essentially just an animated flash card – unless parents buy them (or just download them, as the majority are free) thinking that putting them on their children’s iPad or whatever <em>is all they need to do</em> to improve their performance in math. <br /><br />In the days when the gateway to mathematics, and indeed much of everyday life, lay in mastering the multiplication tables and memorizing a few formulas for calculating areas and volumes, mastery of the basic number facts was indeed enough to start with. So it’s a pity those fun learning apps were not available back then. They would have made the acquisition of those fundamental facts and skills so much easier and far more enjoyable. <br /><br />Unfortunately, the very digital technologies that have put those learning apps into eager young hands have also provided tools that have rendered procedural mastery of those basic skills all but irrelevant. <br /><br />In today’s world, we use cheap, ubiquitous devices to do our calculations. It’s no longer important that all members of society have procedural mastery of basic arithmetic. What is required is the ability to make effective use of those digital devices, and what that depends upon is a good understanding of number – what is often referred to as <em>number sense</em>. <br /><br />Roughly speaking, having number sense means being proficient with quantities and operations with numbers. A person with number sense is able to represent number concepts with models, words and diagrams, to communicate numerical ideas, and solve problems involving numbers. She or he can flexibly compose and decompose numbers for computation and solving problems. They can evaluate the reasonableness of solutions to numerical problems, and make connections between multiple solution methods. They can communicate their number sense verbally and in writing. They notice and explore number patterns, make connections and conjectures, and communicate their thinking to others. Number sense goes beyond solving word problems and memorizing basic facts and procedures. It involves engaging in numbers and operations in ways that develop a deep understanding of the content, which provides a firm foundation for mathematical success. In particular, a strong background in number sense sets the stage for later success in algebra and other parts of mathematics.<br /><br />If that last paragraph sounds like something that emerged from a committee of mathematics education experts, it is because in essence it did. You find language like that in the National Research Council’s 2000 report <a href="http://www.nap.edu/catalog/9822/adding-it-up-helping-children-learn-mathematics" target="_blank">Adding it Up</a>, (which you can download for free from the National Academies Press) and in the preamble to the <a href="http://www.corestandards.org/Math/" target="_blank">Common Core State Standards for Mathematics</a>, which emphasize the development of number sense in young children. <br /><br />For sure, you cannot have number sense without being able to solve an arithmetic problem and get the right answer. What has changed is that it is no longer important to solve that problem by the fastest method, or by a standard method that leaves a paper audit trail that others can check. Our calculating devices do those for us. <br /><br />Much more important in today’s world is to be able to reason about the numbers in a problem from first principles, in a way that embodies the internal structure of the numbers. For as humans, we need to be able to operate when and where that calculator cannot: namely, when we are faced with a novel problem the real world has thrown up at us. <br /><br />It was a lack of recognition that the world has changed fundamentally that led the consequently-Internet-famous “Jack’s Dad” to pen his satirical “letter to his son” that went viral on social media earlier this year. (See the next link below.) <br /><br />Actually, Jack’s dad is an electronics engineer, so he was certainly aware of how much today’s world was different from the days of his own childhood. Unfortunately, as someone outside the world of education, he had just not connected the dots to understand what changes in education were required in order to properly prepare today’s kids to live, not just in our present world, but in the world they will help shape from it. <br /><br />One of the best summaries of the issues behind that social media firestorm that I came across was the <a href="https://christopherdanielson.wordpress.com/2014/04/06/5-reasons-not-to-share-that-common-core-worksheet-on-facebook/" target="_blank">April 6 response</a> to Jack’s Dad written by the math education blogger Christopher Danielson. <br /><br />Danielson’s observations about different kinds of expertise rang very true to me. Having devoted the first part of my mathematics career to mathematical research, it was my appointment to serve on the Mathematical Sciences Education Board in 2000, and the close contact with leading experts in mathematics teaching that resulted, that brought home to me just how little I knew about how people learn mathematics, and how (consequently) we should teach it. <br /><br />Put plainly, having a PhD in mathematics and a string of published research is absolutely nothing like enough background to speak with authority about K-12 mathematics learning. People like me can provide good advice on mathematical content; but not on mathematics teaching. That requires different knowledge and expertise. <br /><br />My own university, Stanford, famous for its very high standards in research, apparently recognizes this when it comes to hiring new faculty in Education. While I cannot speak with authority for the School of Education’s policies, I have observed that no one gets appointed to the faculty who has not spent several years in K-12 teaching. (<em>In addition</em> to having done and published first class research!) Whether or not K-12 experience is official hiring policy, it certainly plays out that way, and it seems to me to be a sensible criterion to demand. <br /><br />Going back to the standard algorithms and Jack’s Dad, a few months after his first post, on October 8, Danielson posted <a href="https://christopherdanielson.wordpress.com/category/talking-math-with-your-kids/" target="_blank">another excellent blog</a> on the degree to which the position occupied by the standard arithmetic algorithms (in actual fact, there are many variations, so there is no such thing as “<em>the</em> standard algorithms) has changed in the educational landscape – from being the main focus as a method for daily use, to an interesting and historically important example of a set of highly efficient paper-and-pencil algorithms that quite literally changed the world. Their significance was a consequence of the dominant information storage and communication technology of the time: flat, static writing surfaces such as parchment, blackboards, and paper. (I describe that story in my book <a href="http://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&qid=1306990867&sr=8-1" target="_blank">The Man of Numbers</a>.) <br /><br />I will note, in passing, that Danielson’s October post indicates that some math learning apps may in fact do harm to a child’s mathematics learning, an observation that should be coupled with my earlier remarks about choosing basic skills educational apps. <br /><br />What put these thoughts onto my front burner recently were some discussions I was having with members of the Scientific Advisory Board for my educational technology startup company <a href="http://www.brainquake.com/" target="_blank">BrainQuake</a>. <br /><br />If you check out our company’s <a href="http://www.brainquake.com/our-team/" target="_blank">Team page</a>, you will find we have recruited a number of world renown experts in mathematics education. Now you may think they are just there for marketing purposes – website name dropping. But you would be wrong. Each one is there because they bring very valuable, very specific expertise to the table. <br /><br />To someone not an expert in mathematics learning, the arithmetic puzzles in our launch app, <a href="http://wuzzittrouble.com/" target="_blank">Wuzzit Trouble</a>, may look as though they are just a series of problems we generated in an essentially random fashion, following the simple rule that the numbers should get "harder" the further a player goes in the game. But that is not the case. In a mathematics learning game, the mathematics ramp is just as critical as the level design of the game, and both require a lot of expertise to get it right. <br /><br />(Interestingly, another name on our website, John Romero, is a world expert in level design – the ramping in game-play – but he joined forces with us only after we had brought out <em>Wuzzit Trouble</em>, so you will only see the results of his genius in future products we bring out.) <br /><br />Which brings me back to my promise to provide advice on how to select good learning apps. It’s probably not a foolproof method, but a quick and easy way is to check out the website of the creators, and see who they have advising them on the learning side. <br /><br />There is always the danger that some of the names are there for little more than window dressing, but the majority of education experts (indeed, experts in any domain) are not likely to lend their name to an enterprise they do not believe in. So the presence of names of distinguished mathematics educators should give you a lot of confidence in the product. <br /><br />More to the point, the absence of such names should be taken as a serious warning. Quite frankly, it is not possible to design and build an educationally sound and effective learning app without a lot of expert input. <br /><br />And I mean a lot of expert input. I bring years of my own expertise to BrainQuake, but <em>Wuzzit Trouble</em> would not have been anything like as <a href="http://wuzzittrouble.com/press.html" target="_blank">educationally successful</a> as it has, if it had just been me on the mathematics side. <br /><br />There is your quick-and-easy quality check. If you use it, you will find that list of 20,000 apps suddenly shrinks down to a significantly smaller number. Fortunately, that number is not zero. There are some great math learning apps out there. You just have to choose wisely.<br /><br />http://devlinsangle.blogspot.com/2014/12/how-do-you-find-good-math-learning-apps.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-1100851642263199172Wed, 05 Nov 2014 19:25:00 +00002014-11-05T14:45:09.662-05:00answer gettingCommon Coreconceptual understandingmathematical thinkingPhil DaroproofsSERP Against Answer Getting<blockquote class="tr_bq">"Correct answers are essential... but they're part of the process, they're not the product. The product is the math the kids walk away with in their heads." —Phil Daro</blockquote><div class="MsoNormal">If you have not already watched Phil Daro's 17-minute video <a href="https://vimeo.com/79916037" target="_blank">Against Answer Getting</a>, you should do so right away. (I'll keep this post short to give you enough time to watch it in its entirety.)<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><a href="http://serpinstitute.org/index.php/people/staff/phil-daro/" target="_blank">Daro</a>, a longtime mathematics educator and leading figure in the national mathematics education community, is currently director of the San Francisco field site of <a href="http://serpinstitute.org/index.php/about/about-2/">SERP</a>, the Strategic Education Research Partnership. He was one of the mathematics educators who played a leading role in the formulation of the mathematics Common Core State Standards. (You know, one of those knowledgeable experts the StopCommonCore brigade keep claiming were not involved in CCSS development.)<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The video is full of powerful insights that the mathematics education community has accumulated over many years of research. My opening quote sums up the focus of the video. Here is another one I like:</div><blockquote class="tr_bq">"Mathematics does not break down into lesson-sized pieces." <span style="text-align: right;">—</span>Phil Daro</blockquote><div class="MsoNormal">This particular quote resonates with me. I adopted the same principle in the design of my MOOC <a href="https://www.coursera.org/course/maththink" target="_blank">Introduction to Mathematical Thinking</a>, currently about halfway through its fifth run.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Daro's focus, both in the video and in his work in general, is K-12 mathematics education. But it is very relevant to those of us in college-level mathematics education. When students come to college with a perception that mathematics is about "answer getting," we face the very uphill task of ridding them of that misleading mindset. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">True, for hundreds of years, getting answers was a key component of learning and doing mathematics. But these days, if we want answers in mathematics, we generally use one of a number of digital technologies. The job of today's mathematician (or typical user of mathematics) is problem solving. The part that requires a human mind is when the problem has a novel aspect. It was precisely to put the focus on the thinking part that I named my MOOC the way I did.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The principle requirement for being able to solve a novel problem is conceptual understanding. That is why the issues Daro raises in that video are so central to the mathematics education of the citizens of tomorrow.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The outdated mindset about the purpose of mathematics that many students bring with them when they transition from school to college is not the only problem many have to overcome. A parallel issue manifests itself when they start to learn about mathematical proofs (if they follow the mathematics path). <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">My MOOC students are currently right in the middle of that part of the course (proofs), and many are having a very hard time coming to understand what role proofs play and what (therefore) constitutes a good proof.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The dominant perception is that proofs are what mathematicians produce in order to determine mathematical truth. That, of course, is true (at least in an idealistic sense that guides mathematical progress), but as with arithmetic answer getting, it is only part of the story. And in terms of actual mathematical practice, a very small part of the story. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">As with answer getting in K-12 math, achieving a logically correct proof is a binary target (right or wrong), which make both very easy to evaluate for correctness and assign a numerical grade. (Ka-ching!)<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">But let's pause and ask ourselves how proofs work in practice. If you want to know if Fermat's Last Theorem is true, you consult a reliable source. Today, any moderately knowledgeable mathematician will tell you the answer: "Yes." Now you know. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">But what if you want to know <i>why</i>it is true. That's when you need to look at a proof. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">In terms of mathematical practice, proofs are about understanding. They are communicative devices we construct to convince ourselves and to convince others.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">In my MOOC, because I cannot assume the students have access to individualized, expert feedback on their work, I do not ask them to construct proofs. But I do present them with a range of purported proofs, some correct, others not, and ask them to evaluate them. The evaluation is in terms both of logical correctness and communicative effectiveness. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">To do this, I ask them to look at each purported proof in terms of five different factors: one logical correctness, the others focusing on communicative issues. Though the five factors are not independent variables, I ask them to treat them as such when evaluating a proof.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">This is the part of the course where those students who have had some exposure to proofs in the K-12 system tend to do worse than those who are new to proofs. They are simply not able to approach a proof other than in the "answer getting" mode of "Is it logically correct?"<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">This shows up dramatically with extremal cases. When I present them with a carefully constructed argument that is logically correct but provides no explanation, they will give it high marks across the board. But faced with an argument that is superbly articulated but has a logical flaw, they are psychologically unable to evaluate the structure of the argument. "It's wrong," they keep saying. End of story (for them).<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Of course, extremal examples are atypical, and often difficult to wrap our minds around. That's what makes them so valuable as learning devices. It's when the classroom rubber hits the road and we find ourselves using mathematical thinking in our lives or careers that it becomes important to have good communication skills.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Pick up a more advanced level mathematics book or research article and the chances are high that the arguments presented will contain errors. (Actually, the book does not have to be advanced. Euclid's <i>Elements</i> is littered with "proofs" that are not logically sound.) But if the arguments are well laid out, with adequate explanations, a suitably skilled reader can fix them as they go along<span style="text-align: right;">—</span>possibly with help from someone else. (That's definitely the case with <i>Elements</i>, though it took two thousand years before David Hilbert noticed that Euclid's own arguments left a lot of work to be done to make them genuine "proofs.")<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">It's the same in software engineering. Any useful program will have bugs. But if the code is well structured, and adequately annotated, someone else can dive in and fix it whenever a flaw manifests. A <i>good</i> computer programmer is not someone who writes error-free, working code; it is someone who writes working code that can easily be fixed or modified.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">I'll leave it as an exercise for the reader to identify the analogous issue in the natural sciences.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">If those of us in the education business want to do the best we can to prepare our students for life in the 21st century, we need to recognize that in an era when technologies provide instant answers (facts), the one ability they will need above anything else is (creative, reflective) thinking. <o:p></o:p></div>http://devlinsangle.blogspot.com/2014/11/against-answer-getting.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-458131571795950019Tue, 07 Oct 2014 18:21:00 +00002014-10-07T14:21:59.639-04:00Common Coreconceptual understandinglearning scienceprocedural masteryThe Straw TeacherWhen people argue for a position they hold because of political bias or some deep-rooted sense of conviction (as opposed to one arrived at by a process of reflection, weighing all sides of the issue), they often resort to straw-man tactics. This is particularly common in the U.S Math Wars, which these days are largely focused on the Common Core State Standards for mathematics.<br /><br />A particularly popular straw man – more precisely, a "straw teacher" (a term that nicely gets us out of gender issues) – is a math teacher who spends class time exclusively discussing mathematics concepts (whatever that means) and pays no attention to helping the students master any procedures.<br /><br /> I guess there may be such a teacher, somewhere, but I have to confess I have yet to meet one. Ditto for the straw teacher who says getting the right answer (if there is one) is not important. Teachers just don't do either of those.<br /><br />My colleagues who work in classroom teacher preparation do tell me that many math teachers do little else than drill on procedures (in some cases because they never set out to teach math, and don't really understand the concepts themselves), but in my walk of life I never meet them. I see the ones who became math teachers because they love mathematics and want to teach, and attend mathematics teacher conferences to exchange ideas and to learn more – which is where I meet them.<br /><br />Anyone who has a working knowledge of (1) what mathematics (really) is and (2) how the brain works knows that learning math in a useful way requires both mastery of a set of basic procedures and conceptual understanding of the mathematical notions those procedures are built on.<br /><br /> In practical terms, you need to master basic procedures in order to develop conceptual understanding, and you need conceptual understanding in order to avoid any procedural mastery being brittle and short-lived.<br /><br />So good math teaching involves both. And, for the record (yet again), both are called for in the CCSS.<br /><br />Absent the CC connection, I've written about this issue on a number of occasions before in this column. For instance:<br /><br />March 2006: <a href="http://www.maa.org/external_archive/devlin/devlin_03_06.html" target="_blank">How do we learn math?</a><br />September 2007: <a href="http://www.maa.org/external_archive/devlin/devlin_09_07.html" target="_blank">What is conceptual understanding?</a><br /><br />Both articles were written long before the Common Core was developed. They were also written when I was just starting to become more actively involved in K-12 education issues. (And before I inadvertently ignited the "repeated addition" firestorm in the summer of 2008.) But having just re-read them for the first time in many years, I still stand by what I wrote. So I won't repeat myself here.http://devlinsangle.blogspot.com/2014/10/the-straw-teacher.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4256371680968970147Tue, 02 Sep 2014 13:31:00 +00002014-09-02T10:45:37.588-04:00applications of mathematicsgeometrymathematical modelnature of mathematicsWill the Real Geometry of Nature Please Stand Up?Is fractal geometry “<strong><em>the</em></strong> geometry of nature”? I was asked this question recently in an email from someone who had watched the PBS video <a href="http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html" target="_blank"><em>Hunting the Hidden Dimension</em></a> that I worked on, and appeared in, a few years ago.<br /><br />It would have been easy to simply reply “Yes,” and for many audiences I would (and have) done just that—for this was by no means the first time I had been asked that question, or others very much like it. But the context in which this recent questioner raised the issue merited a less superficial response. So I wrote back to say that there is no such thing as <strong><em>the</em></strong> geometry of nature, or more generally, <strong><em>the</em></strong> mathematics of W, where W is some real world domain.<br /><br />The strongest claim that can be made is something along the lines of “Mathematical theory T is the best mathematical description (or model) we currently have of the real world domain (or phenomenon) W.” But even then, this statement is less definitive than it might first appear: In particular, what do we mean by “best”?<br /><br />Best in terms of understanding? (If so, then understanding by whom?)<br /><br />Best in terms of building something in W? (If so, then building out of what, using what tools, and for what use?)<br /><br />Best in terms of teaching someone about W? (If so, then teaching what kind of person in terms of age, background, education, motivation, etc.?)<br /><br />Slightly edited and extended, the next few paragraphs are what I wrote back to my correspondent:<br /><br />Nature is just what it is. Mathematics provides various ways to model our perception and experience of reality. Different parts of mathematics provide different models, some better than others. Fractal geometry provides one model that seems to accord with our observations, measurements, and experiences. But so too do the cellular automata models on which Steve Wolfram bases his “<a href="https://www.wolframscience.com/" target="_blank">New Kind of Science</a>.”<br /><br />Many of us think fractal geometry does a better job than cellular automata in helping us understand the natural world by virtue of its nature, but that reflects an assumed patterns/relationship conception of what constitutes science.<br /><br />I would prefer to call Wolfram’s framework a <strong><em>computational theory</em></strong> (of the world), rather than science. But the distinction is, I think, purely one of the meaning we attach to the relevant words (particularly “science”).<br /><br />Both approaches can be said to begin by looking at how nature works, but the moment you start to create a model, you leave nature and are into the realm of human theorizing. From then on, the only available metrics are (1) degree of fit to observations and measurements, (2) degree of utility, and (3) degree to which we find the model’s assumptions reasonable.<br /><br />There is lots of slack here.<br /><br />In (1), what are we observing and measuring? (They are often entities created by those very mathematical theories, e.g. mass, length, volume, velocity, momentum, temperature, etc.)<br /><br />In (2), how do we define utility? Doing stuff, building stuff, understanding stuff, teaching stuff, or something else? (Each with the various audience/use/purpose caveats I raised earlier.)<br /><br />Then there is (3). Unless we make some initial assumptions, we cannot get a theory off the ground. And make no mistake about it, we do begin with assumptions. Not arbitrary ones, to be sure—not even close to being arbitrary. For the resulting theory to be fully accepted (as a plausible explanation or model), it has to accord to any and all the available facts, and it has to be falsifiable—it should make claims or imply conclusions that we can attempt to prove wrong.<br /><br />For instance, a mathematical theory that implied 3 = 4 (as an identity of integers) would be immediately rejected.<br /><br />What about a theory that implies 0.999… = 1.0, where those three dots indicate that the decimal series continues for ever? According to the widely accepted, standard definitions that mathematicians use to provide meaning to the concept of an infinite sequence of decimal digits, this identity is correct. Indeed, it can be proved to be correct, starting from the reasonable, plausible, and accepted basic principles (axioms) for the real number system.<br /><br />Most university math students learn about the framework within which 0.999… is indeed equal to 1.0. (Though many of the popular “proofs” you come across are not rigorous.) As a result, many mathematically educated people will state, as if it were an absolute fact of the world, that 0.999 = 1.0. But that is not true. The identity holds because we have made some assumptions about how to handle infinity. It’s easy to overlook that fact. So let me provide a further example where it may be less easy to miss an underlying assumption.<br /><br />Graduate students of mathematics are introduced to further assumptions (about handling the infinite, and various other issues), equally reasonable and useful, and in accord both with our everyday intuitions (insofar as they are relevant) and with the rest of mainstream mathematics. And on the basis of those assumptions, you can prove that<br /><br />1 + 2 + 3 + … = –1/12.<br /><br />That’s right, the sum of all the natural numbers equals –1/12.<br /><br />This result is so much in-your-face, that people whose mathematics education stopped at the undergraduate level (if they got that far) typically say it is wrong. It’s not. Just as with the 0.999… example, where we had to construct a proper meaning for an infinite decimal expansion before we could determine what its value is, so to we have to define what that infinite sum means.<br /><br />It turns out that there is an entire branch of mathematics, called analytic continuation theory, that provides us with a “natural” meaning for (in particular) that sum. And when we calculate the value using that meaning, we arrive at the answer -1/12. See <a href="http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" target="_blank">this Wikipedia article</a> for a brief account.<br /><br />Incidentally, just as with the 0.999… example, you will find purported “intuitive proofs” floating around, among them <a href="https://www.youtube.com/watch?v=w-I6XTVZXww" target="_blank">this video</a> that went viral earlier this year, but those arguments too are not rigorous.<br /><br />Both frameworks, the one that yields a value for 0.999… and the one that produces a value for 1 + 2 + 3 + … , satisfy all the requirements of being reasonable, plausible, consistent with the rest of mainstream mathematics, and useful (in studies of real world phenomena, including physics). If you accept one, you really cannot reasonably deny the other. Rather, you have to accept the implications they yield, even if they at first seem counter to your expectations.<br /><br />True, neither identity accords with our experiences in the physical world, since those experiences do not involve any infinite quantities or processes. (So there is nothing to accord with!)<br /><br />One of the things surprising examples involving infinity remind us of is that mathematics is not “the true theory of the real world” (whatever that might mean). Rather, mathematical theories are mental frameworks we construct to help us make sense of the world. They survive or wither according to the degree to which they continue to accord with our real world experiences and to prove useful to us in conducting our individual and collective lives.<br /><br />To return to geometry. For most people, throughout human history the geometry of the world experienced was planar Euclidean geometry, which accords extremely well with our everyday experiences.<br /><br />But for the global air traveler (such as long distance airplane pilots), and for the astronauts in the International Space Station, spherical geometry is “<strong><em>the</em></strong> geometry.” In still other circumstances (for the most part, physics and cosmology), hyperbolic and elliptic geometries are the best frameworks.<br /><br />For the artist trying to represent three dimensions on a two-dimensional canvas (or the movie or video-game animator trying to represent three dimensions on a screen), projective geometry is the best framework.<br /><br />Picking up on my opening example, when you adopt a geometric perspective to try to understand growth in the natural world, you find that fractal geometry is the most appropriate one to hand.<br /><br />And, finally, when you adopt a geometric perspective to try to make sense of social life in today’s multi-cultural societies, you may find that higher dimensional Euclidean geometries seem to work best, as I explain in <a href="https://vimeo.com/90522211" target="_blank">this video</a> (30 minutes) taken from a talk I gave at a conference in New Mexico earlier this year. (The relevant segment starts at 3:20 and ends at 11:00.)<br /><br />The fact is, there is not just one geometry, and there is no such thing as “<strong><em>the</em></strong> geometry of W,” where W is a real world phenomenon or domain.<br /><br />Likewise for other branches of mathematics we develop and use to understand our world and to do things in our world.<br /><br />This means that, whereas, <strong><em>within mathematics</em></strong> there are “right answers,” when you apply mathematics to the world, that certainty and accuracy is only as good as the fit between the mathematics (as a conceptual framework) and the world.<br /><br />And now we are back, more or less, at the topic of my <a href="http://devlinsangle.blogspot.com/2014/08/most-math-problems-do-not-have-unique.html" target="_blank">previous <em>Devlin’s Angle</em> post</a>. It merits a second look. Given the nature of the modern world, with mathematical models playing such a major role, with major consequences (in banking, information storage, communication, transportation, national security, etc.), we should not lose track of the fact that mathematics is not the truth.<br /><br />Rather, it provides us with useful <strong><em>models</em></strong> of the world. As a result, it is a powerful and useful way of making sense of the world, and doing things in the world.<br /><br />This distinction was not particularly significant for anyone growing up in the 20th century and earlier. Back then, there was usually no danger in viewing mathematics as if it were the truth. But it is an absolutely critical distinction to keep in mind for those coming of age today.<br /><br />That New Mexico talk video I referred to a moment ago was in fact from a conference on middle school mathematics education, and was an attempt to raise awareness among middle school math teachers of the need to make their students aware of the way mathematics is used in the world they will live in and help shape, emphasizing not only mathematics’ strengths but also its limitations.<br /><br />When you think about what is at stake here, much of the current debate (largely uninformed on the opposition side) about the Common Core State Standards resembles nothing more than two elderly bald men arguing over ownership of a comb.<br /><br />In the case of the UK’s Falkland’s War of 1983, where <a href="http://en.wikiquote.org/wiki/Falklands_War" target="_blank">this analogy originated</a>, both sides appeared equally stupid. The sad aspect to the CCSS debate is that the level of ignorance (or malicious intent) on the “Stop” side forces many well-informed teachers and mathematics learning experts to devote time to the debate, lest ignorance prevail and our kids find themselves unable to survive in the world they inherit. (What the debate should focus on is how to properly implement the Standards. There be dragons, and someone needs to slay them.)<br /><br />WORTH LISTENING TO: American RadioWorks has just aired an excellent <a href="http://www.wnyc.org/story/greater-expectations-challenge-common-core/" target="_blank">radio documentary</a> about the Common Core, in which we hear from real teachers who have been using it, both in states where it has been implemented according to plan and others where the implementation has been modified.http://devlinsangle.blogspot.com/2014/09/will-real-geometry-of-nature-please.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4353549706709988001Fri, 01 Aug 2014 08:00:00 +00002014-08-01T04:00:02.236-04:00Common CoreDragonBoxGOFMJordan Ellenbergmathematical thinkingMost Math Problems Do Not Have a Unique Right AnswerOne of the most widely held misconceptions about mathematics is that a math problem has a unique correct answer.<br /><br />(Some of those who hold that view also think that there is just one correct way to get that answer. A far smaller group, to be sure, but still a worryingly large number. Still, my focus here is on the first false belief.)<br /><br />Having earned my living as a mathematician for over 40 years, I can assure you that the belief is false. In addition to my university research, I have done mathematical work for the U. S. Intelligence Community, the U.S. Army, private defense contractors, and a number of for-profit companies. In not one of those projects was I paid to find "the right answer." No one thought for one moment that there could be such a thing.<br /><br />So what is the origin of those false beliefs? It's hardly a mystery. People form that misconception because of their experience at school. In school mathematics, students are only exposed to problems that (a) are well defined, (b) have a unique correct answer, and (c) whose answer can be obtained with a few lines of calculation.<br /><br />But the only career in which a high school graduate can expect to continue to work on such problems is academic research in pure mathematics—and even then (and again speaking from many years of personal experience), cleanly specified problems that have (obtainable) "right answers" are not as common as you might think.<br /><br />Since the vast majority of students who go through school math classes do not end up as university research mathematicians, whereas many do find themselves in careers that require some mathematical ability, it's reasonable to ask why their entire school mathematics education focuses exclusively on one tiny fraction of all possible mathematics problems.<br /><br />The answer can be found by looking at the history of mathematics. Starting with the invention of numbers around 10,000 years ago, people developed mathematical methods to solve problems they faced in the world: arithmetic and algebra to use in trade and engineering, geometry and trigonometry for building and navigation, calculus for scientific research, and so forth. <br /><br />While some of that mathematics was required only by specialists (e.g. calculus), arithmetic and parts of algebra in particular were essential for everyday living. As a consequence, mathematicians wrote books from which ordinary people could learn how to calculate. From the very earliest textbooks (Babylonian tablets, Indian manuscripts, etc.), two kinds of problems were presented: algorithm ("recipes") problems that showed the steps to be carried out to do a particular kind of computation, presented without any context, and word problems, designed to help people learn how to apply a particular algorithm to solve a real world problem. Ancient and medieval textbooks had many hundreds of such problems, so that a trader (say) could find a problem almost identical in form to the one he (and back then use of mathematics was primarily a male activity) actually wanted to solve in his business. If he were lucky, all he would have to do is substitute his own numbers for those in the book's worked word problem. In other cases, the book might not provide an exact match, but by working through five or six problems that were close in form, the individual could learn how to solve his real problem.<br /><br />For the majority of people, that was enough. Life simply did not require anything more. The problems they faced in their everyday activities for which mathematics was needed were simple and routine. The mathematical word problems that today seem so unrealistic were by and large remarkably similar to the problems ordinary citizens faced every day. <br /><br />"When do I need to leave home in order to catch that train?" There wasn't an app to tell you the answer; you had to calculate it yourself. That word problem about trains leaving stations in your math class showed you how.<br /><br />Arithmetic, in particular, was an essential, basic life skill that remained so until the development of devices that automated the process in the 1960s. I am a member of the last generation for whom the question "What do I need arithmetic for?" simply did not arise. (We asked it about other parts of mathematics.)<br /><br />But that computer technology that eliminated the need for people to be good calculators led to a world in which there is a huge demand for higher order mathematical skills, starting with algebra. I wrote about this change in this column back in 1998, in a piece titled "<a href="https://www.maa.org/external_archive/devlin/devlin_3_98.html" target="_blank">Forget 'Back to Basics.' It's Time for 'Forward to (the New) Basics.</a><u>'"</u> Looking back at what I wrote then, I am amazed at just how much things have changed in the intervening 16 years. In September of that year, Google was founded, and the Web became a dominant force in our lives and our work.<br /><br />Today, we have instant access to vast amounts of information and to unlimited computing power. Both are now utilities, much like water and electricity. And that has led to a revolution in the mathematics ordinary citizens need in order to lead a fulfilling, productive life. In a world where procedural (i.e., algorithmic) mathematics is available at the push of a button, the need has shifted to what I and others have been calling mathematical thinking. <br /><br />I wrote about this in my <a href="http://devlinsangle.blogspot.com/2012/08/what-is-mathematical-thinking.html" target="_blank">September 2012 <i>Devlin's Angle</i></a>. Broadly speaking, mathematical thinking is a way of approaching problems that is based on classical mathematics, but takes account of the fact that computation (both numeric and symbolic) can be readily done by machines.<br /><br />In practical terms, what this means is that people can now focus all their attention on real-world problems in the form they are encountered. Knowing how to solve an equation is no longer a valuable human ability; what matters now is formulating the equation to solve that problem in the first place, and then taking the result of the machine solution to the equation and making use of it.<br /><br />In the 1960s, we got used to the fact that the arithmetic part of solving a mathematical problem could be done by machines. Now we are in a world where almost all the procedural mathematics can be done by machines.<br /><br />Of course, this does not mean we should stop teaching procedural mathematics to the next generation, any more than the introduction of pocket calculators meant we should stop teaching arithmetic. But in both cases, the reason for teaching changes, and with it the way we should teach it. The purpose shifts from mastering procedures—something that was necessary only when there were no machines to do that part—to understanding the concepts sufficiently well to make good use of those machines.<br /><br />Though this change in emphasis has been underway for some years now, it did not garner much attention in the United States until the rollout of the Common Core State Standards, which are very much geared towards the mathematical thinking needs of the 21st century. The degree to which many parents were shortsighted by the shift was made clear when some of them took to social media to complain about the kinds of homework questions their children were being asked to do. While some of those questions were truly, truly awful, others garnering a lot of critical SM comments were actually extremely good.<br /><br />What was particularly ironic was that many parents, faced with being unable to assist their child with elementary grade arithmetic homework, did not draw the obvious conclusion: "Gee, if I cannot understand something as basic as integer arithmetic—however it is done—there must have been something really lacking in my own education." Instead, they jumped to the totally off-the-wall conclusion that the current educational system must be wrong.<br /><br />That's like waking up in the morning to find your car won't start and saying, "Oh dear, the laws of physics don't work." The smart person says, "I need to replace the battery."<br /><br />I'll tell you something. I was taught math the "old-fashioned way" too, and some of those student arithmetic worksheets were new to me when I first saw them. But regardless of any views I might have as to how it is best taught in today's world, it didn't take a lot of effort to figure out what those kids were doing on those worksheets posted on Facebook. It was just whole number arithmetic for heavens sake! Anyone who understands the basic ideas of whole number arithmetic can figure it out.<br /><br />It was not my training as a professional mathematician that helped me here. It was the simple fact that I understand whole number arithmetic, something that goes back to my early childhood, when I did not even know there was such a thing as a professional mathematician, let alone aspire to be one. Unfortunately, many Americans were never taught to understand arithmetic, they were just trained to execute procedures. It's not their kids who are being short-changed. They—the parents—were!<br /><br />Breezing into this fray is University of Wisconsin mathematics professor Jordan Ellenberg, with his new book <a href="http://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/1594205221/ref=sr_1_1?ie=UTF8&qid=1406591086&sr=8-1&keywords=how+not+to+be+wrong" target="_blank"><i>How Not To Be Wrong</i></a>. I knew I would find a kindred spirit when I read the book's subtitle: “The Power of Mathematical Thinking.” With a <a href="https://www.coursera.org/course/maththink">Stanford MOOC</a> and an associated <a href="http://www.amazon.com/Introduction-Mathematical-Thinking-Keith-Devlin/dp/0615653634/ref=sr_1_5?ie=UTF8&qid=1342652878&sr=8-5&keywords=devlin+mathematical+thinking" target="_blank">textbook</a> both called <i>Introduction to Mathematical Thinking</i>, how could I not?<br /><br />Ellenberg's title is superb. In one fell swoop, it casts aside that old misconception that mathematics provides "right answers," replacing it with the far more accurate description that it is a great way to stop you being wrong. For, like me, he focuses not on the internal activities of pure mathematics, rather on how mathematics is used in today's real world.<br /><br />To be sure, also like me, Ellenberg has devoted a lot of his career to working in pure mathematics, so he loves searching for those "right answers," and he enjoys the subject in its own terms. We both know that there are eternal truths within mathematics (a better term would be "tautologies") and have experienced the thrill of going after them. But we both realize that what we do as pure mathematicians is a very specialist pursuit. The society that supports us when we do that does so largely because of the payoff in terms of the benefits that emerge when mathematical thinking is applied to real world problems.<br /><br />Ellenberg's book is chock full of examples of those benefits, from many walks of life, presented with a delightfully light touch. He grabs the reader's attention with his very first example, taken from the Second World War. The U. S. military chiefs wanted to reduce the number of warplanes that were being shot down. The obvious solution was to add more armor to protect them. But armor adds weight, which limits the distances that can be flown and the duration of the mission, as well as increasing the production cost. So the question was, where is the most effective place to put that extra protection? <br /><br />To answer this question, the chiefs brought in a team of mathematicians to analyze the evidence and determine what parts of the aircraft were most likely to be hit. They examined the fuselages of all the damaged planes that had flown back after being hit to see where the most damage was. It turned out that the engines had an average of 1.11 bullet holes per square foot, the fuel system had 1.55, the fuselages 1.73, and the rest of the plane 1.8. <br /><br />So where was the optimal place to add extra armor? According to the data, the fuselages took a lot of hits, while engines suffered the least damage. So an obvious suggestion was to add armor to the fuselages. But that was not what the mathematicians suggested. Their solution was to add the armor to the engines, the part that had fewer hits when the planes got back.<br /><br />And they were right. I'll leave you to figure out why that is the best solution. It's a great example of mathematical thinking. After you have convinced yourself why adding armor to the engines was the best strategy, you should buy a copy of Ellenberg's book and gain some understanding of just what mathematical thinking is, and why it is a crucial ability in today's world. <br /><br />(My own book on mathematical thinking is more of a "how to" guide, as is my MOOC. Another, excellent book on mathematical thinking, that is somewhere between Ellenberg's and mine, is Burger and Starbird's <a href="http://www.amazon.com/5-Elements-Effective-Thinking-ebook/dp/B008JUVDUE/ref=sr_1_3?ie=UTF8&qid=1406632051&sr=8-3&keywords=starbird+michael" target="_blank"><i>The 5 Elements of Effective Thinking</i></a>.)<br /><br />Finally, and to some extent switching gears (and definitely switching media), I want to draw your attention to a new video game, <a href="http://wewanttoknow.com/elements/" target="_blank">DragonBox Elements</a>, by the Norwegian-based educational technology company WeWantToKnow. The company made a splash with its first game, <a href="http://wewanttoknow.com/algebra/" target="_blank">DragonBox (Algebra)</a> a couple of years ago.<br /><br />Unlike my own work in educational videogames, through my company <a href="http://brainquake.com/" target="_blank">BrainQuake</a>, which is very strongly focused on real-world mathematical thinking, the DragonBox folks are seeking to enhance and strengthen school mathematics.<br /><br />When I first played the new Elements game, I was initially confused, since I approached it with a <a href="http://www.keycurriculum.com/" target="_blank">Geometer's Sketchpad</a> expectation. But Elements is not a geometry construction/exploration tool. The focus is on the importance of providing justification for steps in a proof. Knowing why something is true. And that is not only a key feature of GOFM (“Good Old Fashioned Math”), as was taught for two thousand years, it's one of the aspects of mathematics that is characteristic of mathematical thinking (as used in the real world). Euclid, the author of the first <i>Elements</i> (the book), would surely have approved.<br /><br />The modern world has not made GOFM redundant. What has changed, and drastically, is the way GOFM fits in with the rest of human activities. Unless you are going to make a career for yourself in pure mathematics research, GOFM today is simply an amazingly powerful tool for acquiring one of the most important cognitive capacities in the 21st century: mathematical thinking.<br /><br />In today's world, most of the important problems are complex and multi-faceted. There are few right answers. As Ellenberg demonstrates, mathematical thinking can help you choose better answers—and avoid being wrong.http://devlinsangle.blogspot.com/2014/08/most-math-problems-do-not-have-unique.htmlnoreply@blogger.com (Mathematical Association of America)3tag:blogger.com,1999:blog-2516188730140164076.post-4167422961418068667Tue, 01 Jul 2014 16:31:00 +00002014-07-01T13:04:50.095-04:00Common Coredot diagramsmathematical thinkingThe Power of Dots<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-tsiADZYB4hs/U7LfEeFMotI/AAAAAAAAJ14/DgOWU4KVhC8/s1600/Morass.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-tsiADZYB4hs/U7LfEeFMotI/AAAAAAAAJ14/DgOWU4KVhC8/s1600/Morass.jpg" height="240" width="320" /></a></div><div><br /></div>On June 29, the <i>New York Times</i> ran <a href="http://www.nytimes.com/2014/06/30/us/math-under-common-core-has-even-parents-stumbling.html?emc=edit_tnt_20140629&eml_thmb=1&nlid=4832777&tntemail0=y&_r=1" target="_blank">a story</a> about the Common Core Mathematics Standards. If ever you wanted proof of the dismal mathematics education most Americans have been provided, you will find it in the story’s “human interest lede,” which described one mother’s response to seeing her daughter’s homework. By taking the daughter out of school to teach her herself “the old fashioned way” she herself had been subjected to, this well-meaning parent was ensuring that, as had clearly been the case for the mother, the daughter too would not be exposed to real mathematical thinking—the kind that in today’s world is a key to the most attractive jobs. Instead she would be subjected to the same, dreary, rote-skills-drills inflicted on previous generations—a process designed to train people for routine work in the pre-computer era, but so hopelessly inadequate for the 21st century that parents are un-equipped to figure out for themselves the simple (albeit unfamiliar) math homework their children are assigned.<br /><br />Surely, if mathematics education should achieve one thing, it is develop the ability to figure things out for yourself. We’re not talking the Riemann Hypothesis here; the focus is basic school arithmetic, for heaven’s sake.<br /><br />To continue with the <i>Times</i> article, arrays of dots seemed to figure large in this parent’s dislike of the Common Core. She felt it was pointless to spend time drawing and staring at arrays of dots.<br /><br />True, it would be possible—and I am sure it happens—to generate tedious, and largely pointless, “busywork” exercises involving drawing arrays of dots. But the image of a Common Core math worksheet the <i>Times</i> chose to illustrate its story showed a very sensible, and deep use of dot diagrams, to understand structure in arithmetic. Much like the (extremely deep) dot array at the top of this article, which I’ll come to in a moment.<br /><br />To the girl’s parent, mathematics is about numbers, but that’s just a surface feature. It’s really about structure. And throughout the ages, mathematicians have used the most simple symbols possible to bring out and understand that structure: namely, dots and lines.<br /><br />The <i>Times</i>’ parent, so dismissive of time spent drawing and reflecting on dot diagrams, would, I am sure, think it a waste of time to devote any effort trying to make sense of the dot diagram I used to open this post. She would, I have no doubt, find it incomprehensible that an individual with a freshly-minted Ph.D. in mathematics would spend many months—at taxpayers’ expense—staring day-after-day at either that one diagram, or seemingly minor variations he would start each day by sketching out on a sheet of paper in front of him.<br /><br />Well, I am that mathematician. That diagram helped me understand the framework that would be required to specify an infinite mathematical object of the third order of infinitude (aleph-2) by means of a family of infinite mathematical objects of the first order of infinitude (aleph-0). The top line of dots represents an increasing tower of objects that come together to form the desired aleph-2 object, and each of the lower lines of dots represent shorter towers of aleph-0 objects. In the 1970s, a number of us used those dot diagrams to solve mathematical problems that just a few years earlier had seemed impossible.<br /><br />That particular kind of dot diagram was invented by a close senior colleague (and mentor) of mine, Professor Ronald Jensen, who called it a “morass.” He chose the name wisely, since the structure represented by those dots was extremely complex and intricate. <br /><br />In contrast, the simple, rectangular array implicitly referred to in the <i>New York Times</i> article is used to help learners understand the much simpler (but still deep, and far more important to society) structure of numbers and the basic operations of arithmetic, as was well explained in a subsequent <a href="http://talkingmathwithkids.com/2014/06/30/dots/" target="_blank">blog post</a> by mathematics education specialist Christopher Danielson. The fact is, dot diagrams are powerful, for learners and world experts alike. <br /><br />The problem facing parents (and many teachers) today, is that the present student generation is the one that, for the first time in history, is having to learn the mathematics the professionals use—what I and many other pros have started to call “mathematical thinking” in order to distinguish it from the procedural skills so important in past times.<br /><br />The reason for that is that in the world today’s students will graduate into, computation is as plentiful as water or electricity. The smartphone we carry around with us is much faster, and more accurate, in carrying out mathematical procedures than any human. <br /><br />In a single generation, society’s need for mathematical mastery has gone from procedural computation, to being able to make effective and reliable use of an effectively unlimited amount of automated computation. To put it bluntly, mastery of computational skills is no longer a marketable asset. The ability to make good use of computational power is where it’s at in math today.<br /><br />For almost all the three thousand years of mathematical development, the focus in mathematics was calculation (numerical, symbolic, or geometric). Learning mathematics meant learning how to perform those calculations, which boiled down to achieving mastery of various procedures. Mastery of any one procedure could be achieved by rote learning—doing many examples, all essentially the same—leaving the only truly creative mental task that of recognition of which procedure to apply to solve which problem.<br /><br />Numerical and symbolic calculation (arithmetic and algebra) are so simple and routine that we can program computers to do it for us. That is possible because calculation is essentially trivial. Perceiving and understanding structure, on the other hand, is something that (at least at the present time) requires human insight. It is not trivial and it is difficult. Dot diagrams can help us come to terms with that difficulty.<br /><br />When movie director Gus Van Sant was faced with introducing the lead character, Will Hunting (played by Matt Damon) in the hit 1997 film <i>Good Will Hunting</i>, establishing in one shot that the hero was an uneducated (actually, self-educated) mathematical genius, the first encounter we had with Will showed him drawing a dot diagram on a blackboard in an MIT corridor.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-yJFoRYfemaY/U7LfMQqs1TI/AAAAAAAAJ2E/Y7B_fUGac38/s1600/Good_Will_Hunting.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-yJFoRYfemaY/U7LfMQqs1TI/AAAAAAAAJ2E/Y7B_fUGac38/s1600/Good_Will_Hunting.jpg" height="180" width="320" /></a></div><br />You can be sure that when an experienced movie director like Gus Van Sant selects an establishing shot for the lead character, he does so with considerable care, on the advice of an expert. By showing Will writing a network of dots on a blackboard, Van Sant was right on the button in terms of portraying the kind of thing that professional mathematicians do all the time. <br /><br />The one bit of license Van Sant took was that the diagram we saw Matt Damon writing was not the solution to a problem that had taken an MIT math professor two years to solve. (Unless MIT math professors are a lot less smart than we are led to believe!) It was a real solution to a real math problem, all right. I am pretty sure it was chosen because it fitted nicely on one blackboard and looked good on the screen. It absolutely conveyed the kind of (dotty) activity that mathematicians do all the time—the kind of (dotty) thing I did in my early post-Ph.D. years when I was working with Prof Jensen’s morasses.<br /><br />But it’s actually a problem that anyone who has learned how to think mathematically should be able to solve in at most a few hours. Numberphile has an excellent <a href="http://www.youtube.com/watch?v=iW_LkYiuTKE" target="_blank">video</a> explaining the problem.<br /><br />So, <i>New York Times</i> story parent, I hope you reconsider your decision to take your daughter out of school to teach her the way you were taught. The kind of mathematics you were taught was indeed required in times past. But not any more. The world has changed dramatically as far as mathematics is concerned. As with many other aspects of our lives, we have built machines to handle the more routine, procedural stuff, thereby putting a premium on the one thing where humans vastly outperform computers: creative thinking. <br /><br />Those dot diagrams are all about creative thinking. A computer can understand numbers, and process millions of them faster than a human can write just one. But it cannot make sense of those dot diagrams. Because it does not know what any particular array of dots means! And it has no way to figure it out. (Unless a human tells it.)<br /><br />Next month I’ll look further into the distinction between old-style procedural mathematics and the 21st-century need for mathematical thinking. In particular, I’ll look at an excellent recent <a href="http://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/1594205221/ref=sr_1_1?ie=UTF8&qid=1404226390&sr=8-1&keywords=how+not+to+be+wrong" target="_blank">book</a>, Jordan Ellenberg’s <i>How Not to be Wrong</i>. <br /><br />The book’s title is significant, since it recognizes that the vast majority of real-world mathematical problems do not have a unique right answer, and that the real power of mathematical thinking is making sure you are not wrong. (The book’s subtitle is “The power of mathematical thinking.”)<br /><br />I’ll also look at a new mathematics video game that also focuses on mathematical thinking, this time, school-room Euclidean geometry. It’s called <a href="http://wewanttoknow.com/elements/" target="_blank">DragonBox Elements</a>.<br /><br />You might want to check out both.http://devlinsangle.blogspot.com/2014/07/the-power-of-dots.htmlnoreply@blogger.com (Mathematical Association of America)9tag:blogger.com,1999:blog-2516188730140164076.post-5809859714363089543Sun, 01 Jun 2014 09:00:00 +00002014-06-02T07:38:35.596-04:00Apple MacintosharithmeticFibonacciLeonardo of PisaSteve JobsDéjà vu all over again: Fibonacci and Steve Jobs — Part 2This month’s column is the second of a two-part video presentation of a public address I gave recently at Princeton, where I have been spending this semester as a Visiting Professor.<br /><br />The talk was based on my 2011 e-book <a href="http://www.amazon.com/Leonardo-Steve-Genius-Market-ebook/dp/B005BRR2TY/ref=sr_1_1?s=digital-text&ie=UTF8&qid=1310688753&sr=1-1" target="_blank"><i>Leonardo and Steve</i></a>, which itself was a supplement to my print book <a href="http://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&qid=1306990867&sr=8-1" target="_blank"><i>The Man of Numbers</i></a>, published the same year.<br /><br />Both the e-book and my presentation show how Jobs’s introduction of the Macintosh computer in 1984 was an almost exact replay of Leonardo of Pisa’s (Fibonacci) 13th Century introduction to Europe of Hindu-Arabic arithmetic.<br /><br /><a href="http://devlinsangle.blogspot.com/2014/05/deja-vu-all-over-again-fibonacci-and.html" target="_blank">Part 1 appeared last month</a>.<br /><div><div style="text-align: center;"><br /></div></div><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="281" mozallowfullscreen="" src="//player.vimeo.com/video/93532834" webkitallowfullscreen="" width="500"></iframe></div>http://devlinsangle.blogspot.com/2014/06/deja-vu-all-over-again-fibonacci-and.htmlnoreply@blogger.com (Mathematical Association of America)0