Is math really beautiful?

The above tweet caught my eye recently. The author is a National Board Certified mathematics teacher in New York City who has an active social media presence. Is his claim correct? Not surprisingly, a number of other mathematics educators responded, and in the course of the exchange, the author modified his claim to include the word “just”, as in “It isn’t just about beauty …” In which case, I think he is absolutely correct.

Like many mathematicians who engage in public outreach, I have frequently discussed the inherent elegance and beauty of mathematics, the wonder of its purity, and the power of its abstraction. And as a body of human knowledge, I maintain (as do pretty well all other mathematicians) that such descriptions of the subject known as pure mathematics are totally justified. (Cue: for the standard quotation, Google “Bertrand Russell mathematical beauty”.) Anyone who is unable to recognize it as such surely has not (yet) understood what (pure) mathematics is truly about.

In contrast, the activity of doing mathematics is indeed “messy,” as Pershan claims. That is the case not only for the activity of using mathematics to solve problems in the real world, but also the activity of engaging in pure mathematics research. The former activity is messy because the world is. The latter is messy because the logical elegance and beauty of (many) mathematical theories and proofs are characteristics of the finished product, not the process of development.

And there, surely, we have the motivation for Pershan’s comment. When we teach mathematics to beginners, we don’t do them any service by making claims about beauty and elegance if what they are experiencing is anything but. With good teaching of a well-designed curriculum, we can ensure that they are exposed to the beauty, of course, and perhaps experience the elegance. But it’s surely better to let them know that the messiness, the uncertainty, the repeated stumbles, and the blind allies they are encountering are part of the package of doing mathematics that the pros experience all the time, whether the doing is trying to prove a theorem or using mathematics to solve a real-world problem.

By chance, the same day I read that tweet, I came across an excellent online article on Medium about the huge demand for mathematical thinking in today’s data-rich and data-driven world. Like me, the author is a pure mathematician who, later in his career, became involved in using mathematics and mathematical thinking in working on complex real-world problems. I strongly recommend it. Not only does it convey the inherent messiness of real-world problems, it convincingly makes the case that without at least one good mathematical thinker on the team, management decisions based on numerical data can go badly astray. As the author states in a final footnote, he takes pleasure in the process of applying the rigor of mathematics to the complex messiness of real-world problems.

To my mind, therein lies another kind of mathematical beauty: the beauty of making productive use of the interplay between the abstract purity of formal rigor and the messy stuff of everyday life.

Calculation was the price we used to have to pay to do mathematics

By Keith Devlin

You can follow me on Twitter @profkeithdevlin

Ever since mathematics got properly underway around 3,000 years ago, there was only one way to achieve access to the field. You had to spend many years developing a fairly extensive calculation skillset. In the first instance, to pass the graduation and entrance examinations to gain initial access to the field. Then, once accepted into the world of mathematics, calculation of one kind or another was what all mathematicians spent the bulk of their mathematical time doing. Arguably, for most of mathematics history, the subject really was, to a large extent, primarily about calculation of one form or another. Newton, Leibniz, Bernoulli (any of them), Fermat, Euler, Riemann, Gauss, and the other greats of times past, were all superb masters of calculation. (We should also include Boole, since his famous Boolean algebra is also a calculation system.)

But whereas most laypersons seem to think that calculation is all there is to mathematics, surely none of the greats did. Calculation was an important tool (more accurately, a set of tools) you needed to do mathematics, they must have realized, but the essence of mathematics is much more, a plateau of knowledge that transcends all the calculation techniques.

In the 19th Century, that somewhat tacit understanding became explicit. The increasing complexity of the problems mathematicians tackled led to a series of results that defied the human intuition. (Several of them were referred to as “paradoxes”.) This led to an intense period of mathematical introspection, where the primary focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships. In other words, a shift in emphasis from doing to understanding. What had previously been implicit, became full-on explicit.

Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving something was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts. Mathematics was reconceptualized as “thinking in concepts” (Denken in Begriffen).

This was, in every sense, a mathematical revolution, with the primary revolutionaries being
leading mathematicians such as Lejeune Dirichlet, Richard Dedekind, Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann.

To give just one instance of the shift, prior to the nineteenth century, mathematicians were used to the fact that a formula such as  y = x² + 3x – 5  specifies a function that produces a new number y from any given number x. Then the revolutionary Dirichlet came along and said, forget the formula and concentrate on what the function does in terms of input-output behavior. A function, according to Dirichlet, is any rule that produces new numbers from old. The rule does not have to be specified by an algebraic formula. In fact, there's no reason to restrict your attention to numbers. A function can be any rule that takes objects of one kind and produces new objects from them.

This definition legitimized functions such as the one defined on real numbers by the rule:
If x is rational, set f(x) = 0; if x is irrational, set f(x) = 1.

Of course, you cannot draw a graph of such a monster. Instead, mathematicians began to study the properties of abstract functions, specified not by some formula but by their behavior. For example, you can investigate questions such as, is the function one-one, injective, surjective, continuous, differentiable, etc.?

For any mathematician alive today, mathematics is a subject that studies formally-defined concepts, with a focus on the establishment of truth (based on accepted axioms), with various forms of calculation (numerical, algebraic, set-theoretic, logical, etc.) being tools developed and used in the pursuit of those goals. That’s the only kind of mathematics we have known.

Except, that is, when we were at school. By and large, the 19th Century revolution in mathematics did not permeate the world’s school systems, which remained firmly in the “mathematics is about calculation” mindset. The one attempt to bring the school system into the modern age (in the US, the UK, and a few other countries), was the 1960s “New Math”. Though well-intentioned, its rollout was disastrous, in large part because very few teachers understood what it was about – and hence could not teach it well. The confusion caused to parents (other than mathematician parents) was nicely encapsulated by the satirical songwriter and singer Tom Lehrer (who taught mathematics at Harvard, and did understand New Math), in his hilarious, and pointedly accurate, song New Math.

As a result of the initial chaos, the initiative was quickly dropped, and school math remained largely unchanged while real-world uses of mathematics kept steadily changing, leaving the schools increasingly separated from the way people did math in their jobs. Eventually, the separation blew up into a full-fledged divorce. That occurred in the late 1980s. The divorce was finalized on June 23, 1988. That was the date when Steve Wolfram released his mammoth software package Mathematica. Within a few short years of that release, if not on the release-date, Mathematica (and a similar package released a few months later in Canada, Maple) could answer pretty well any school or university math exam question with at least a grade B+, and very often an A.

The days when calculation (of pretty well any kind, not just numerical) was the price humans had to pay to do mathematics were over.

Given that thirty years have passed since that initial epochal moment, and most of the world has still not woken up to the fact that the entire mathematical world has changed dramatically and forever, let me repeat the core of that statement in caps.

THE DAYS WHEN CALCULATION WAS THE PRICE HUMANS HAD TO PAY TO DO MATHEMATICS ARE OVER.

To be sure, after that symbolic 1988 date, it took a few years for the change to percolate through the system, gain momentum, and eventually reach critical mass. Three further developments were also hugely significant: the birth of the World Wide Web in 1989 and the browser in 1993, and the launch of Wolfram Alpha in 2009. (Others might want to add other factors. I’m being selective here.)

Talking about being selective, I’ve mentioned Wolfram products twice now. Though I was a member of Wolfram’s Mathematica Advisory Board in the first few years, I have no stake in or involvement with the company. While both Mathematica and Alpha were indeed major players in changing the way mathematics is done – particularly in applied settings – I am citing those particular products largely as icons, using two specific products to represent a range of new digital tools that were being developed around the world at that time. While Wolfram’s systems were ones I myself made early use of in my work, other mathematicians were also active in that digital mathematical revolution, using different systems. Still, Mathematica was the system that caught the public attention.

Since the turn of the new Millennium, I doubt if anyone making professional use of mathematics in their job, or indeed any adult using mathematics in their everyday lives, has taken out paper-and-pencil and followed a classical algorithm to add, subtract, multiply or divide numbers in an array of real-life size, or perform complex algebraic reasoning to solve systems of equations, or solve problems using calculus, or any other established mathematical procedure. Not only would it now be a waste of valuable human time and energy doing something a cheap machine can do in far less time with no possibility of error, but many of the problems that people encounter in their careers and lives have simply too much data for the human mind to handle. Those same digital tools that have made the execution of mathematical procedures unnecessary have also come to dominate and drive our world, so many of the problems that require mathematics in their solution are now simply beyond human capacity. That’s why Amazon Web Services has become such a behemoth for data storage and processing.

But that does not mean humans no longer need to have some mathematical skills. On the contrary, they are as crucial as ever – unless you are willing to be totally reliant on others, but personally, I have never felt comfortable doing that with things that are part of my life every day. What has changed are the specific mathematical skills required today. There are plenty of things computers cannot do or do poorly. Genuinely creative thinking and analogical reasoning are two obvious ones – though with today’s massive cloud computing resources we can use systems that provide an approximation often adequate for the purpose, and on occasion can be better than humans.

Mostly, however, where you need humans is going from a real-world challenge situation to formulating one or mathematical tasks that can help you make progress. Sometimes, progress means solving a real-world problem in the sense of getting a specific answer (say, a number), but much more commonly it’s about finding a better method, where “better” can mean faster, cheaper, safer, or whatever other criterion is important, and where the change may involve developing a new method or improving an existing one.

This way of using mathematics was the focus of that mini-course I gave at a California school (Nueva School) in January of this year, that I wrote about in the February, March, and April posts to this column.

Though several mathematicians and mathematics education scholars expressed agreement with what I wrote, my articles brought some critiques from teachers and parents. The critiques all made reference to my asides about the Common Core State Standards in the first two of the posts. Since “Devlin’s Angle” no longer seems to be a target for the CCSS social media trolls (likely because the yield of issues to react to relative to the length and substance of most of my posts makes it less rewarding to them), I made some efforts to find out what exactly it is about the CCSSs that they objected to. As far as I could ascertain, the issue was inevitably (and predictably) to do with particular implementations of the Standards in specific curricula or (and this seems to be the most common occurrence) claims that a particular homework exercise was a “Common Core exercise”, which of course it cannot be since the CCSS are, as the name indicates, purely a set of standards to attain, not in any way a curriculum or curriculum content.

More generally, in fact, pretty well all critiques of the CCSS are due to a complete misunderstanding of what they are, why they are, and what they say. The issue was nicely dealt with in this 2014 article in the Hechinger Report.

My reason for bringing the Common Core into my series of posts was to point out that the standards were developed precisely to help guide school districts, schools, and teachers in the tricky task of updating K-12 mathematics education to adequately prepare tomorrow’s citizens for life and work in a world where calculation is no longer a central pillar of mathematics.

Having said that, I should point out that the above statement in no way implies that we should drop the teaching of basic arithmetic and algebra from the school system. As I discussed in some length in the third of my Nueva-inspired articles, the change that is required in K-12 math education is not so much in the mathematical topics but the reason they are now being taught and, in consequence, the way they should be taught.

Teaching for execution is no longer the primary driver, since no one using mathematics in the real world does that anymore. What is now of crucial importance is teaching for understanding. Digital systems outperform humans to an insane degree when it comes to execution. But they don’t understand; people have to supply that.

I leave you with an image I pulled from one of those Common Core social media rants some time ago. (I no longer remember the exact source.)

Typical social media posts about Common Core mathematics.

I have three comments about the post on the left. First, the mathematics in the bottom left is not some fancy new algorithm, it is what a child wrote down in reasoning (sensibly) about a particular arithmetic problem. Second, if you are unable to follow what the child is doing, you would have trouble making effective use of mathematics in today’s world. It’s pretty basic. (Your kid just did it, right?) Third, if you are a parent and you don’t see why it is important that today’s school students acquire those math reasoning skills, please don’t communicate your skepticism to your children. Doing so would be a great disservice, to your child, to your child’s math teacher, and to society. The mathematical world has changed significantly. That occurred over twenty years ago. It is not going to change back. Sit back, relax, be encouraging, and let the kids take over. They do just fine with it.

REFERENCE: During the period when the computer revolutionized how mathematics is done, I edited the American Mathematical Society’s “Computers and Mathematics” section of their monthly notices publication, sent to all members. I wrote about the column and that period in general in a paper that I submitted to the Proceedings of the Jon Borwein Commemorative Conference, held in 2017. Borwein, who died tragically young in 2016, was a leading pioneer in bringing digital technologies into mathematics. You can access a preprint of the paper HERE.

The Power of Simple Representations

The great mathematician Karl Freidrich Gauss is frequently quoted as saying “What we need are notions, not notations.” [In “About the proof of Wilson's theorem,” Disquisitiones Arithmeticae (1801), Article 76.]

While most mathematicians would agree that Gauss was correct in pointing out that concepts, not symbol manipulation, are at the heart of mathematics, his words do have to be properly interpreted. While a notation does not matter, a representation can make a huge difference. The distinction is that developing or selecting a representation for a particular mathematical concept (or notion) involves deciding which features of the concept to capture.

For example, the form of the ten digits 0, 1, … , 9 does not matter (as long as they are readily distinguishable), but the usefulness of the Hindu-Arabic number system is that it embodies base- 10 place-value representation of whole numbers. Moreover, it does so in a way that makes both learning and using Hindu-Arabic arithmetic efficient.

Likewise, the choice of 10 as the base is optimal for a species that has highly manipulable hands with ten digits. Although the base-10 arithmetic eventually became the standard, other systems were used in different societies, but they too evolved from the use of the hands and sometimes the feet for counting: base-12 (where finger-counting used the three segments of each of the four fingers) and base-20 where both fingers and toes were used. Base-12 arithmetic and base-20 arithmetic both remained in regular use in the monetary system in the UK when I was a child growing up there, with 12 pennies giving one shilling and 20 shillings one pound. And several languages continue to carry reminders of earlier use of both bases — English uses phrases such as “three score and ten” to mean 70 (= 3x20 + 10) and French articulates 85 as “quatre-vingt cinq (4x20 + 5).

Another number system we continue to use today is base-60, used in measuring time (seconds and minutes) and in circular measurement (degrees in a circle). Presumably the use of 60 as a base came from combining the finger and toes bases 10, 12, and 20, allowing for all three to be used as most convenient.

These different base-number representation systems all capture features that make them useful to humans. Analogously, digital computers are designed to use binary arithmetic (base 2), because that aligns naturally with the two states of an electronic gate (open or closed, on or off).

In contrast, the shapes of the Hindu-Arabic numerals is an example of a superfluous feature of the representation. The fact that it is possible to draw the numerals in a fashion whereby each digit has the corresponding number of angles, like this

may be a historical echo of the evolution of the symbols, but whether or not that is the case (and frankly I find it fanciful), it is of no significance in terms of their use—the form of the numerals is very much in Gauss’s “unimportant notations” bucket.

On the other hand, the huge difference a representation system can make in mathematics is indicated by the revolutionary change in human life that was brought about by the switch from Roman numerals and abacus-board calculation to Hindu-Arabic arithmetic in Thirteenth Century Europe, as I described in my 2011 book The Man of Numbers.

Of course, there is a sense in which representations do not matter to mathematics. There is a legitimate way to understand Gauss’s remark as a complete dismissal of how we represent mathematics on a page. The notations we use provide mental gateways to the abstract notions of mathematics that live in our minds. The notions themselves transcend any notations we use to denote them. That may, in fact, have been how Gauss intended his reply to be taken, given the circumstances.

But when we shift our attention from mathematics as a body of eternal, abstract structure occupying a Platonic realm, to an activity carried out by people, then it is clear that notations (i.e., a representation system) are important. In the early days of Category Theory, some mathematicians dismissed it as “abstract nonsense” or “mere diagram chasing”, but as most of us discovered when we made a serious attempt to get into the subject, “tracing the arrows” in a commutative diagram can be a powerful way to approach and understand a complex structure. [Google “the snake lemma”. Even better, watch actress Jill Clayburgh explain it to a graduate math class in an early scene from the 1980s movie It’s My Turn.]

A well-developed mathematical diagram can also be particularly powerful in trying to understand complex real-world phenomena. In fact, I would argue that the use of mathematical representations as a tool for highlighting hidden abstract structure to help us understand and operate in our world is one of mathematics most significant roles in society, a use that tends to get overlooked, given our present day focus on mathematics as a tool for “getting answers.” Getting an answer is frequently the end of a process of thought; gaining new insight and understanding is the start of a new mental journey.

A particularly well known example of such use are the Feynmann Diagrams, simple visualizations to help physicists understand the complex behavior of subatomic particles, introduced by the American physicist Richard Feynmann in 1948.

A more recent example that has proved useful in linguistics, philosophy, and the social sciences is the “completion diagram” developed by the American mathematician Jon Barwise in collaboration with his philosopher collaborator John Perry in the early 1980s, initially to understand information flow.

A discussion of one use of this diagram can be found in a survey article I wrote in the volume Handbook of the History of Logic, Volume 7, edited by Dov Gabbay and John Woods (Elsevier, 2008, pp.601-664), a manuscript version of which can be found on my Stanford homepage. That particular application is essentially the original one for which the diagram was introduced, but the diagram itself turned out be to be applicable in many domains, including improving workplace productivity, intelligence analysis, battlefield command, and mathematics education. (I worked on some of those applications myself; some links to publications are on my homepage.)

To be particularly effective, a representation needs to be simple and easy to master. In the case of a representational diagram, like the Commutative Diagrams of Category Theory, the Feynmann Diagram in physics, and the Completion Diagram in social science and information systems development, the representation itself is frequently so simple that it is easy for domain experts to dismiss them as little more than decoration. (For instance, the main critics of Category Theory in its early days were world famous algebraists.) But the mental clarity such diagrams can bring to a complex domain can be highly significant, both for the expert and the learner.

In the case of the Completion Diagram, I was a member of the team at Stanford that led the efforts to develop an understanding of information that could be fruitful in the development of information technologies. We had many long discussions about the most effective way to view the domain. That simple looking diagram emerged from a number of attempts (over a great many months) as being the most effective.

Given that personal involvement, you would have thought I would be careful not to dismiss a novel representation I thought was too simple and obvious to be important. But no. When you understand something deeply, and have done so for many years, you easily forget how hard it can be for a beginning learner. That’s why, when the MAA’s own James Tanton told me about his “Exploding Dots” idea some months ago, my initial reaction was “That sounds cute," but I did not stop and reflect on what it might mean for early (and not so early) mathematics education.

To me, and I assume to any professional mathematician, it sounds like the method simply adds a visual element on paper (or a board) to the mental image of abstract number concepts we already have in our minds. In fact, that is exactly what it does. But that’s the point! “Exploding Dots” does nothing for the expert. But for the learner, it can be huge. It does nothing for the expert because it represents on a page what the expert has in their mind. But that is why it can be so effective in assisting a learner arrive at that level of understanding! All it took to convince me was to watch Tanton’s lecture video on Vimeo. Like Tanton, and I suspect almost all other mathematicians, it took me many years of struggle to go beyond the formal symbol manipulation of the classical algorithms of arithmetic (developed to enable people to carry our calculations efficiently and accurately in the days before we had machines to do it for us) until I had created the mental representation that the exploding dots process capture so brilliantly. Many learners subjected to the classical teaching approach never reach that level of understanding; for them, basic arithmetic remains forever a collection of incomprehensible symbolic incantations.

Yes, I was right in my original assumption that there is nothing new in exploding dots. But I was also wrong in concluding that there was nothing new. There is no contradiction here. Mathematically, there is nothing new; it’s stuff that goes back to the first centuries of the First Millennium—the underlying idea for place-value arithmetic. Educationally, however, it’s a big deal. A very big deal. Educationally explosive, in fact. Check it out!

The Math Gift Myth

My May post is more than a little late. The initial delay was caused by a mountain of other deadlines. When I did finally start to come up for air, there just did not seem to be any suitable math stories floating around to riff off, but I did not have enough time to dig around for one. That this has happened so rarely in the twenty years I have been writing Devlin’s Angle (and various other outlets going back to the early 1980s in the UK), that it speaks volumes against the claim you sometimes hear that nothing much happens in the world of mathematics. There is always stuff going on.

Be that as it may, when I woke up this morning and went online, two fascinating stories were waiting for me. What’s more, they are connected – at least, that’s how I saw them.

First, my Stanford colleague Professor Jo Boaler sent out a group email pointing to a New York Times article that quoted her, and which, she noted, she helped the author to write. Titled "No Such Thing as a Math Person," it summarizes the consensus among informed math educators that mathematical ability is a spectrum. Just like any other human ability. What is more, the basic math of the K-8 system is well within the capacity of the vast majority of people. Not easy to master, to be sure; but definitely within most people’s ability. It may be defensible to apply terms such as “gifted and talented” to higher mathematics (though I will come back to that momentarily), but basic math is almost entirely a matter of wanting to master it and being willing to put in the effort. People who say otherwise are either (1) education suppliers trying to sell products, (2) children who for whatever reason simply do not want to learn and find it reassuring to convince themselves they just don’t have the gift, or (3) mums and dads who want to use the term as a parental boast or an excuse.

Unfortunately, the belief that mathematical ability is a “gift” (that you either have or you don’t) is so well established it is hard to get rid of. Part of the problem is the way it is often taught, as a collection of rules and procedures, rather than a way of thinking (and a very simplistic one at that). Today, this is compounded by the rapid changes in society over the past few decades, that have revolutionized the way mathematics needs to be taught to prepare the new generation for life in today’s – and tomorrow’s – world. (See my January 1 article in The Huffington Post, "All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime," and its follow up article (same date), "Number Sense: the most important mathematical concept in 21st Century K-12 education.")

With many parents, and not a few teachers, having convinced themselves of the “Math Gift Myth,” attempts over the past several decades to change that mindset have met with considerable resistance. If you have such a mindset, it is easy to see what happens in the educational world around you as confirming it. For instance, one teacher commented on The New York Times article:

“Excuse me? I'm a teacher and I refute your assertion. I have seen countless individuals who have problems with math – and some never get it. The same goes for English. But, unless you've spent years in the classroom, it takes years to fully accept that observation. The article's writer is a doctor, not a teacher; accomplishment in one field does not necessarily translate readily to another.”

Others were quick to push back against that comment, with one pointing out that her final remark surely argues in favor of everyone in the education world keeping up with the latest scientific research in learning. We are all liable to seek confirmation of our initial biases. And both teachers and parents are in powerful positions to pass on those biases to a new generation of math learners.

In her most recent book, Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching, Prof Boaler lays out some of the considerable evidence against the Math Gift Myth, and provides pointers to how to overcome it in the classroom. The sellout audiences Boaler draws for her talks at teachers conferences around the world indicates the hunger there is to provide math learning that does not produce the math-averse, and even math-phobic, citizens we have grown accustomed to.

And so to that second story I came across. Hemant Mehta is a former National Board Certified high school math teacher in the suburbs of Chicago, where he taught for seven years, who is arguably best known for his blog The Friendly Atheist. His post on May 22 was titled "Years Later, the Mother Who 'Audited' an Evolution Exhibit Reflects on the Viral Response." Knowing Mehta’s work (for the record, I have also been interviewed by him on his education-related podcast), that title hooked me at first glance. I could not resist diving in.

As with The New York Times article I led off with, Mehta’s post is brief and to the point, so I won’t attempt to summarize it here. Like Mehta, as an experienced educator I know that it requires real effort, and courage, to take apart ones beliefs and assumptions, when faced with contrary evidence, and then to reason oneself to a new understanding. So I side with him in not in any way trying to diminish the individual who made the two videos he comments on. What we can do, is use her videos to observe how difficult it can be to make that leap from interpreting seemingly nonsensical and mutually contradictory evidence from within our (current!) belief system, to seeing it from a new viewpoint from which it all makes perfect sense – to rise above the trees to view the forest, if you will. The video lady cannot do that, and assumes no one else can either.

Finally, what about my claim that post K-12 mathematics may be beyond the reach of many individuals’ innate capacity for progression along that spectrum I referred to? Of course, it depends on what you mean by “many”. Leaving that aside, however, if someone, for whatever reason, develops a passionate interest in mathematics, how far can they go? I don’t know. Based on a sample size of one, me, we can go further than we think. I look at the achievement of mathematicians such as Andrew Wiles or Terrence Tao and experience the same degree of their being from a different species as the keen-amateur- cyclist-me feels when I see the likes of Tour de France winner Chris Froome or World Champion Peter Sagan climb mountains at twice the speed I can sustain.

Yet, on a number of occasions where I failed to solve a mathematics problem I had been working on for months and sometimes years, when someone else did solve it, my first reaction was, “Oh no, I was so close. If only I had tried just a tiny bit harder!” Not always, to be sure. Not infrequently, I was convinced I would never have found the solution. But I got within a hairsbreadth on enough occasions to realize that with more effort I could have done better than I did. (I have the same experience with cycling, but there I do not have a particular desire to aim for the top.)

In other words, all my experience in mathematics tells me I do not have an absolute ability limit. Nor, I am sure, do you. Mathematical proficiency is indeed a spectrum. We can all do better – if we want to. That, surely is the message we educators should be telling our students, be they in the K-8 classroom or the postgraduate seminar room.

Gifted and talented? Time to recognize that as an educational equivalent of the Flat Earth Belief. Sure, we are surrounded by seemingly overwhelming daily experience that the world is flat. But it isn’t. And once you accept that, guess what? From a new perspective, you start to see supporting evidence for the Earth being spherical.

Algebraic Roots – Part 2

What does it mean to “do algebra”? In Part 1, published here last month, I described how algebra (from the Arabic al-Jabr) 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.

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.

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.

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 arithmetical and algebraic thinking. This is why you see the frequent use of those terms in educational circles these days, along with number sense. (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 mathematical thinking.)

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 Desmos), spreadsheets, computer algebra systems (such as Mathematica or Maple), and Wolfram Alpha, 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.

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.

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 that was evident to them but, clearly, not evident to many people not working in mathematics education.

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 educational perspective those of us in the math ed business bring to the issues.

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.

The tweet shown below caught my eye a few weeks ago.

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.)

“Ineffective”? “#dragonboxfail”? Those are the words used in the tweet. But neither can possibly be true. DragonBox provides an alternative representation 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.)

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.

The tweet was in response to a review 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 study the LearnLab tweet implicitly referred to.

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).

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, as measured by a typical test of equation solving.” (The emphasis is mine.)

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 Lynnette.) 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.

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.

Indeed, with his background in video game design, Brady ends his review by rating DragonBox according to three metrics:

Fun Factor – A: I collected all 1,366 stars available in DragonBox 1 and 2 and had a great time.

Academic Value – B: I worry that many will underestimate the effort needed to transfer DragonBox skills to practical algebra proficiency.

Educational Value – A+: 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.

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.”)

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.)

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.

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.”

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.

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.)

There is some evidence (see for example Jo Boaler’s new book) 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.

**Full declaration: I should note that my own work in this area, some of it through my startup company BrainQuake, adopts this philosophy. The significant learning gains obtained with our first app 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.

Algebraic roots – Part 1

Fig. 1: A problem from the first ever algebra textbook.

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.

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.

Fig. 2: Al-Khwarizmi's quadratic equation in modern notation.

Symbolic 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.

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.

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!)

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 — and this is where I am going with this — it does if you solve the problem by thinking about it.

With the symbolic presentation, it is possible to reduce the solution of an algebra problem to the mindless (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. No understanding necessary. 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.)

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).

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 powerful way of thinking 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.)

“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.)

In the introduction to his algebra book, al-Khwarizmi declared that he was presenting

“... 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.”

This was cutting edge stuff back then. It doesn’t get much more practical than that!

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 arithmetic-at-scale. 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.

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.

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.

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.

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.

In other words, algebra has returned to its roots. (Pun intended.)

END OF PART 1
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?

NOTE: 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, Devlin’s Angle 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 column), 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.

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.

The Math Myth that permeates “The Math Myth”

March 1 saw the publication of the book The Math Myth: And Other STEM Delusions, 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 "Is Algebra Necessary?"

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.

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 article for the Huffington Post in which I summarized, with my commentary, his 2012 article, together with a recent interview he gave to the Chronicle of Higher Education.

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:

1. Algebra as typically taught in the school system is presented as a meaningless game with arbitrary rules that does more harm than good.
2. There are strong arguments for teaching algebra as it was originally developed and how professional mathematicians today view it.

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.

First codified by the Persian mathematician al-Khwarizmi in his book The Compendious Book on Calculation by Completion and Balancing (balancing = al-Jabr), 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.

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.)

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.

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.

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.

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 implementations of the CC.

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.

“It’s a waste of time forcing kids to master azimuths and asymptotes,” he cries [not an exact quote] as early as page 2.

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.

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:

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

That’s it. One mention, buried towards the end of the document, in the section that says the student should:

• Understand the concept of a function and use function notation
• Interpret functions that arise in applications in terms of the context
• Analyze functions using different representations

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.

Unfortunately, this example is indeed a good characterization of his overall argument: to knock down straw men.

“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,” 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.)

He continues, “Then, to graduate from high school, they will face tests on radical notations and elliptical equations.”

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,

“For example, estimate the population of the United States as 3 × 10⁸ and the population of the world as 7 × 10⁹, and determine that the world population is more than 20 times larger.”

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.

“Elliptical equations” is another straw man.

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.)

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.

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.

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.)

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.

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.

What this question is asking for is, Do you understand what a ratio is? 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.

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, Do you understand numbers? In this case, do you recognize that the sum of two equal numbers can never be odd.

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.

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.

This, surely, explains why he would write (page 96), “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?”

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.)

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.

The tragedy of The Math Myth 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.

The real myth in The Math Myth 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.

In the interests of their overall cardiovascular health, I have to recommend that math educators do not read The Math Myth. 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.

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,

“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 3.14159 goes on indefinitely, at least as far as we know.”

Yes, you read that last part correctly.

“Few people writing today … can make more sense of numbers” proclaims the Wall Street Journal 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.

How 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.

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.

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.

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 is all they need to do to improve their performance in math.

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.

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.

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 number sense.

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.

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 Adding it Up, (which you can download for free from the National Academies Press) and in the preamble to the Common Core State Standards for Mathematics, which emphasize the development of number sense in young children.

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.

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.

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.)

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.

One of the best summaries of the issues behind that social media firestorm that I came across was the April 6 response to Jack’s Dad written by the math education blogger Christopher Danielson.

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.

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.

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. (In addition 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.

Going back to the standard algorithms and Jack’s Dad, a few months after his first post, on October 8, Danielson posted another excellent blog 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 “the 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 The Man of Numbers.)

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.

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 BrainQuake.

If you check out our company’s Team page, 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.

To someone not an expert in mathematics learning, the arithmetic puzzles in our launch app, Wuzzit Trouble, 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.

(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 Wuzzit Trouble, so you will only see the results of his genius in future products we bring out.)

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.

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.

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.

And I mean a lot of expert input. I bring years of my own expertise to BrainQuake, but Wuzzit Trouble would not have been anything like as educationally successful as it has, if it had just been me on the mathematics side.

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.