What is mathematical thinking?

What is mathematical thinking, is it the same as doing mathematics, if it is not, is it important, and if it is different from doing math and important, then why is it important? The answers are, in order, (1) I’ll tell you, (2) no, (3) yes, and (4) I’ll give you an example that concerns the safety of the nation.

If you had any difficulty following that first paragraph (only two sentences, each of pretty average length), then you are not a good mathematical thinker. If you had absolutely no difficulty understanding the paragraph, then either you are already a good mathematical thinker or you could acquire that ability pretty quickly. (In the former case, you most likely pictured a decision tree in your mind. Doing that kind of thing automatically is part of what it means to be a mathematical thinker.)

Okay, I had my tongue firmly in my cheek when I wrote those opening paragraphs, but there is such a thing as mathematical thinking, it can be developed, and it is not the same as doing mathematics.*

In my last column, I talked about my decision to self-publish a really cheap textbook to accompany my upcoming MOOC (massively open online course) on Mathematical Thinking. At the time of writing this column, just shy of 40,000 students have registered – and there are over two more weeks before the class starts.

As a result of sending out a number of tweets, chronicling my experiences in developing my MOOC in a blog MOOCtalk.org, and posting some videos about the upcoming course on YouTube, I’ve already received a fair number of emails asking for details about the course. (At one point, so many so I had to temporarily shut off comments on MOOCtalk.org, lest WordPress closed me down under the assumption that with so much traffic it must be a porn site.)

In this column, I’ll answer one question that came up a number of times: What is mathematical thinking? In fact, I’ll do more, I’ll answer the four questions I opened with.

To people whose experience of mathematics does not extend far, if at all, beyond the high school math class, I think it’s actually close to impossible for them to really grasp what mathematical thinking is. I used to try to convey the distinction with an analogy. “K-12 mathematics is like a series of courses in digging trenches, pouring concrete, bricklaying, carpentry, plumbing, electrical wiring, roofing, and glazing,” I would say. And then, after a brief pause, I would continue,  “Mathematical thinking is the equivalent of architecting. You need all of those individual house-building skills to build a house. But putting those skills together and making use of them requires a higher-order form of thinking. You need someone who can design the building and oversee its construction.”

It is a great analogy. I felt sure it would convey the essence of mathematical thinking. But many conversations and email exchanges over the years eventually convinced me it was not working. Saying A is to B as C is to D works fine when the recipient has good understanding of A, B, and C and some understanding of D. But if they have not even a clue about D, or even worse, if they believe that D actually is C, then the analogy simply does not work. It’s one of those analogies that is brilliant if you are sufficiently familiar with all four components, but hopeless as a way to explain one in terms of the other three.

Once I realized that, I set out to find a better way to describe it. It took me most of a whole book to do it. Not the ultra-cheap textbook I mentioned above. That has a different purpose. Rather, my book on using video games in mathematics education.

Below, in about 850 words, is the nub of what I say in that book in about 75 pages. (Yes, that’s quite a compression ratio. Clearly, it’s lossy compression!) After the quote, I’ll give you a specific example of mathematical thinking from my own past involvement in national security research. (Don’t worry, my part was not classified. You can read it without me having to kill you.)

BEGIN QUOTE [pp.59–61]:

[Mathematical thinking is more than being able to do arithmetic or solve algebra problems. In fact, it is possible to think like a mathematician and do fairly poorly when it comes to balancing your checkbook. Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns. Moreover, it involves adopting the identity of a mathematical thinker.]

[For instance] like most people, when I am doing something routine, I rarely reflect on my actions. But if I’m do ing mathematics and I step back for a moment and think about it, I see myself [not just as someone who can do math, but] as a mathematician.

“Well, duh!” I hear you saying. “You are a mathematician.” By which I assume you mean that I have credentials in the field and am paid to do math. But I have a similar feeling when I am riding my bicycle. I’m a fairly serious cyclist. I wear skintight Lycra clothing and ride a $4,000, ultralight, carbon fiber, racing-type bike with drop handlebars, skinny tires, and a saddle that resembles a razor blade. I try to ride for at least an hour at a time four or five days a week, and on weekends I often take part in organized events in which I ride virtually nonstop for 100 miles or more. Yet I’m not a professional cyclist, and I would have trouble keeping up with the Tour de France racers even during their early morning warm-up while they are riding along with a newspaper in one hand and a latte in the other. […] Being a bike rider is part of who I am. When I am out on my bike, I feel like a cyclist. And you know, I’d be willing to bet that the feeling I have for the activity is not very different from [the professional bike racers].

It’s very different for me when it comes to, say, tennis. […] I don’t have the proper gear, and I have never played enough to become even competent. When I do pick up a (borrowed) racket and play, as I do from time to time, it always feels like I’m just dabbling. I never feel like a tennis player. I feel like an outsider who is just sticking his toe in the tennis waters. I do not know what it feels like to be a real tennis player. As a consequence of these two very different mental attitudes, I have become a pretty good cyclist, as average-Joe cyclists go, but I am terrible at tennis. The same is true for anyone and pretty much any human activity. Unless you get inside the activity and identify with it, you are not going to be good at it. If you want to be good at activity X, you have to start to see yourself as an X-er  – to act like an X-er.

A large part of becoming an X-er is joining a community of other X-ers. This often involves joining up with other X-ers, but it does not need to. It’s more an attitude of mind than anything else, though most of us find that it’s a lot easier when we team up with others. The centuries-old method of learning a craft or trade by a process of apprenticeship was based on this idea. [The video games scholar James Paul Gee, in his book What Video Games Have to Teach Us About Learning and Literacy, p. 18] uses the term semiotic domain to refer to the culture and way of thinking that goes with a particular practice – a term that reflects the important role that language or symbols plays in these “communities of practice,” to use another popular term from the social science literature. […]

In Gee’s terms, learning to X competently means becoming part of the semiotic domain associated with X. Moreover, if you don’t become part of that semiotic domain you won’t achieve competency in X. Notice that I’m not talking here about becoming an expert, and neither is Gee. In some domains, it may be that few people are born with the natural talent to become world class. Rather, the point we are both making is that a crucial part of becoming competent at some activity is to enter the semiotic domain of that activity. This is why we have schools and universities, and this is why distance education will never replace spending a period of months or years in a social community of experts and other learners. Schools and universities are environments in which people can learn to become X-ers for various X activities – and a large part of that is learning to think and act like an X-er and to see yourself as an X-er. They are only secondarily places where you can learn the facts of X-ing; the part you can also acquire online or learn from a book. […]

The social aspect of learning that goes with entering a semiotic domain is often overlooked when educational issues are discussed, particularly when dis cussed by policy makers rather than professional teachers. Yet it is a huge factor. […]

END QUOTE

In my blog MOOCtalk.org, I will explain what persuaded me to try to prove that the pessimism I expressed in the above passage about someone becoming an X-er through a remote experience like a MOOC might be misplaced, at least in part. But my focus here is describing mathematical thinking.

In many cases, the real value of being a mathematical thinker, both to the individual and to society, lies in the things the individual does automatically, without conscious thought or effort. The things they take for granted – because they have become part of who they are. This was driven home to me dramatically in the years immediately following 9/11, when I was one of many mathematicians, scientists, and engineers working on national security issues, in my case looking for ways to improve defense intelligence analysis.

My brief was to look at ways that reasoning and decision making are influenced by the context in which the data arises. Which information do you regard as more significant? How do you weight, and then combine, information coming from different sources. I’d looked at questions like this in pre-9/11 work – indeed that was the research that brought me from the UK to Stanford in 1987, and by the time the Twin Towers came down, I had written two research books and a number of papers on the topic. But that research focused on highly constrained domains, where the complexity was limited. The challenge faced in defense intelligence work is far greater – the complexity is huge.

I did not have any great expectations of success, but I started anyway, proceeding in the way any professional mathematician would. I could give you a list of some of the things I did, but that would be misleading, since I did not follow a checklist, I just started to think about the problem in a manner that has long become natural to me. I thought about it for many hours each day, often while superficially occupied with other life activities. I was not aware of making any progress.

Six months into the project, I flew to D.C. to give a progress report to the program directors. As I fired up my PowerPoint projection and copies of my printed interim report were passed around the crowded meeting room, I was sure the group would stop me half way through and ask me (hopefully politely) to get on the next plane back to San Francisco and not waste any more of their time (or taxpayers’ dollars).

In the event, I never got beyond the first content slide. But not because I was thrown out. Rather, the rest of the session was spent discussing what appeared on that one slide. I never got close to what I thought was my “best” work. As my immediate research report told me afterwards, beaming, “That one slide justified having you on the project.”

So what had I done? Nothing really – from my perspective. My task was to find a way of analyzing how context influences data analysis and reasoning in highly complex domains involving military, political, and social contexts. The task seemed impossibly daunting (and still does). Nevertheless, I took the oh-so-obvious (to me) first step. “I need to write down as precise a mathematical definition as possible of what a context is,” I said to myself. It took me a couple of days mulling it over in the back of my mind while doing other things, then maybe an hour or so of drafting some preliminary definitions on paper. The result was a simple statement that easily fitted onto a single PowerPoint slide in a 28pt font. I can’t say I was totally satisfied with it, and would have been unable to defend it as “the right definition.” But it was the best I could do, and it did at least give me a firm base on which to start to develop some rudimentary mathematical ideas. (Think Euclid writing down definitions and axioms for what had hitherto been intuition-based geometry.)

The fairly large group of really smart academics, defense contractors, and senior DoD personnel in that meeting room spent the entire hour of my allotted time discussing that one definition. Not because they were trying to decide if that was the “right” definition, or the best one to work with. In fact, what the discussion brought out was that all the different experts had a different conception of what a context is, and how it can best be taken account of – a recipe for disaster in collaborative research if ever there was.

What I had given them was, first, I asked the question “What is a context?” Since each person in the room besides me had a good working concept of context – different ones, as I just noted – they never thought to write down a formal definition. It was not part of what they did. And second, by presenting them with a formal definition, I gave them a common reference point from which they could compare and contrast their own notions. There we had the beginnings of disaster avoidance, and hence a step towards possible progress in the collaboration.

As a mathematician, I had done nothing special, nothing unusual. It was an obvious first step when someone versed in mathematical thinking approaches a new problem. Identify the key parameters and formulate formal definitions of them. But it was not at all an obvious thing for anyone else on the project. They each had their own “obvious things.” Some of them seemed really clever to me. Others seemed superficially very similar to mine, but on closer inspection they set about things in importantly different ways.

“Your work is not classified, so you are free to publish your results, if you wish,” the program director told me later, “but we’d prefer it if you did not make specific reference to this particular project.” “Don’t worry,” I replied, “I have not done anything that would be accepted for publication in a mathematics journal.” Which is absolutely the case. I had not done any mathematics in the familiar sense. I had not even taken some mathematical procedure and applied it. Rather, what I had done was think about a complex (and hugely important) problem in the way any experienced mathematician would.

I’ve had a number of similar experiences over the years, and though they appear on the surface to be widely different (from analyzing children’s fairy stories to looking at communication breakdown in the workplace to trying to predict the endings of movies like Memento to trying to make sense of the modern battlefield), at their (mathematical) heart they all have the same general pattern.

That then, is mathematical thinking. How do you teach it? Well, you can’t teach it; in fact there is very little anyone can teach anyone. People have to learn things for themselves; the best a “teacher” can do is help them to learn.

The most efficient domain to learn mathematical thinking is, perhaps not surprisingly (though it’s not such a slam-dunk as you might think) mathematics itself. Particularly well suited parts of mathematics for this purpose are algebra, formal logic, basic set theory, elementary number theory, and beginning real analysis. These are the topics I have chosen for my MOOC. Other topics could serve the same purpose, but would require more background knowledge on the part of the student. But it’s not about the topic. It’s the thinking required that is important.

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*One of the features of mathematical thinking that often causes beginners immense difficulty is the logical precision required in mathematical writing, frequently leading to sentence constructions that read awkwardly compared to everyday text and take considerable effort to parse. (The standard definition of continuity is an excellent example, but mathematical writing is rife with instances.) The opening paragraph is a parody of such writing. This comment was added a day after initial publication, when a letter from a reader indicated that he missed the fact that the opening was a parody, and complained that he found it difficult to read. That difficulty was, of course, the whole point of the opening, but that point is lost if readers don't recognize what is going on. So I added this remark.

The future of textbook publishing is us

In my May column, I announced my intention to give a free online course (a MOOC) this coming fall, and asked for assistance from the mathematical community.

That course, a school-to-university transition course, titled Introduction to Mathematical Thinking, is going ahead, with the first lecture on September 17. There is a brief description of the course, together with a short promotional video, on the Coursera website.

I have started a blog, MOOCtalk.org, to chronicle my experiences working with this new format and to provide a platform for feedback and discussion once the course gets going.

I also wrote a short textbook to accompany the course, Introduction to Mathematical Thinking. Though the textbook is not required for the course, some of my Stanford colleagues who gave the first generation of “Ivy League MOOCs” – just a few months ago, so fast has this new movement taken off – told me that many students want an old-fashioned physical book. On the other hand, all the transition textbooks I am familiar with are fairly pricey, which would put them well beyond many of the students who are likely to enroll. Moreover, none of them are designed to accompany a MOOC. So I decided to write one.

My two main criteria were: it had to be short (no more than 100 pages) and cheap (less than $10 in the US). The only option was to self-publish in print-on-demand format with Amazon’s CreateSpace service. I did not quite hit my page-limit; when I include the front material, the tally comes out at 102 pages, but that’s close enough. But I figured I’d cover my costs if I set the retail price at $9.99, just below my target.

The procedure is so ridiculously straightforward, I can see no reason why anyone should ever publish another textbook a different way, given the huge expense of textbooks. We authors have been typesetting our own manuscripts ever since Don Knuth first released TeX in 1978, and all that is required to produce a book with CreateSpace is to generate a PDF file that fits the page-size you select.

CreateSpace does not provide TeX support (by way of a style file), but they do provide sample pages for authors submitting manuscripts in Word, and I just played around with the page parameters in LaTeX until the output matched their samples for both odd and even numbered pages, which I checked by printing out copies of both, putting my output on top of theirs, and holding the two up to the light. (Low tech, but effective.)

In my case, I decided to produce my book in the standard 6 in x 9 in format, and the key LaTeX parameters I came up with are (for the record)
\oddsidemargin 1 in \evensidemargin .55 in
\marginparwidth .75in \marginparsep 7pt
\topmargin -.5in \headheight 12pt
\headsep .25in \footheight 12pt \footskip .35in
\textheight 7.5in \textwidth 4.95 in

When I submitted my final PDF file, CreateSpace’s automated checking system flagged the manuscript as possibly not being correctly formatted, but I pressed forward, since the next stage is that one of their employees examines the manuscript, and indeed that individual accepted it, confirming my suspicion that I was probably off by a millimeter or two, something that could upset an automated checking system but is close enough to pass a human eyeball test.

The point is, the whole process is so well designed, there is no reason why anyone who can use LaTeX should do anything other than self-publish from now on. With a very small number of exceptions, no one who writes a university-level textbook does so to make money. Our goal is to get material in front of students as quickly and cheaply as possible. If there were a way to do so that can save the students money, I am sure we would all want to do so, the more so given the way textbook costs have skyrocketed in recent years. With modern print-on-demand technology, we now can do just that.

You don’t need to know anything about publishing to do this. CreateSpace does for book publishing what TurboTax did for filing your tax return, and it does it in much the same way, by taking you through the entire process in a simple, step-by-step fashion, including cover design, securing an ISBN code, and selecting marketing channels.

For sure, the finished product is not quite as good as would be achieved with the professional expertise of a good publishing house. But to my mind, for a textbook, it’s close enough, especially when the resulting book can be sold for as little as a tenth the price a publisher would charge.

The one thing I paid someone else to do was copy-editing. I have written enough books to value highly the services of an experienced copy editor. (You might also want to pay an indexer. I did my own, but I have done so for several of my previous books.)

Of course, even with good copy editing, occasional errors creep through. Not long after my book was on the market, I was looking through one of my author’s copies (you have to buy them, but at an even lower price than the retail mark), and spotted a couple of small typos. A few minutes editing the LaTeX file, followed by a quick upload of the replacement PDF, and the correction was made, ready for the next person to buy a copy.

Returning to the MOOC now, let me re-iterate the request I made in my May column. I am giving my MOOC in the early fall to coincide with the many transition courses offered at colleges and universities across the US, in the hope that instructors of such courses will incorporate my MOOC in their courses in some way. My reason for this is that I think the only way to make a transition course MOOC work is to have enough participants who either are already familiar with the material (such as instructors) or else have direct access to such expertise (e.g., their students in a transition course). I see no other way for students struggling to understand the material to get the help, advice, and feedback they will need to progress. Social media provides various platforms for students to interact, to ask questions of one another and to comment on others’ work. But there has to be a mechanism for mathematical truth to find its way into the discussions!

So the key to making something like this work is, I think, to build up a Wikipedia- like community of instructors who, for five weeks each year, will make available their expertise to the thousands of students around the world who are taking advantage of a MOOC to obtain an education they would otherwise not have access to.

The benefit to the students in the transition classes given by MOOC-participating instructors is that their learning will assuredly be enhanced by acting as tutors for the students who are not so privileged. Both because teaching others is a powerful way to learn – as most of us discover when we become TAs at graduate school – and because those students will surely feel much more incentivized to understand by playing such a feel-good role.

Stay tuned to my MOOCtalk blog for updates on the project. And if you are an instructor giving a transition course this fall, please consider getting involved.

“Can’t we all get along?”

Unless you follow several mathematics education blogs or subscribe to certain Twitter feeds, you may well have missed the recent revival of the US Math Wars. The protagonists in the latest salvo are not the traditional foes, Mathematically Correct versus NCTM and MAA, but two groups who are making use of new technology in education, with Khan Academy squaring up against a number of web-savvy, younger mathematics and science educators who believe the US can and should do a lot better (and differently) in math ed than we do.

I summarized some of my own views on KA in an article in The Huffington Post on March 20 of this year, but this article focuses on another issue than the one I discussed there. Namely, the large numbers of fanatical supporters Khan has gathered, who respond to even the mildest and well meaning criticisms of anything Khan with venom, hatred, and personal attacks.

The first round of this new skirmish I was aware of (likely not the first to occur, since I check in only occasionally on developments in K-12 mathematics education, though now the excellent EdSurge provides me more useful news than I can possibly keep up with) was in February of this year when Mathalicious founder Karim Kai Ani posted a critique of KA pedagogy on his company blog. (Disclosure: I knew that posting was coming up, since Karim emailed me beforehand with questions about gamification and individually prescribed instruction, both of which he wanted to discuss in his article. I also voluntarily endorse his educational materials.)

The KA–Mathalicious deluge began within a few hours of the blog being published, with a lot of the venom unleashed on Y-Combinator’s Hacker News. From the nature of many of the comments, which were personal, and often visceral attacks on Ani rather than reasoned responses to what was a very well thought out and cogently presented argument, it’s a reasonable assumption that they were contributed by schoolkids. Which to my mind makes this phenomenon worth thinking about. For schoolchildren are what K-12 education is about. When a substantial number of our main customers speak, we should listen, even if the method by which the message is delivered leaves a lot to be desired.

Actually, I don’t know what percentage of the K-12 population is represented in those Hacker News (and other) blog posts, but what we do know is that Khan Academy has a large and devoted following. In some ways, I too am a fan of sorts, though less so than when I participated in a TV discussion with Sal back in 2010. For sure, I am dismayed by the huge number of mathematical errors and pedagogic shortcomings in his videos. (We’ve learned a lot about mathematics education in the past fifty years.) In the early days it was easy to overlook them – any new enterprise is buggy, particularly a one-person show like the early KA. But once the Silicon Valley millions started to roll in, and with those funds a staff, it would have been an easy Web exercise to crowdsource curriculum improvement/development to the many mathematics teachers and university mathematics education specialists who would surely have been eager to help.

My enthusiasm came not from the KA site’s contents, but from something else I saw: Sal Khan himself. Up close – and I have met him a few times – he comes across as a really nice, approachable guy with a great sense of humor. Well, so do a lot of people. But there is something else. Sal has the ability to project that identity and that personality over the Web, using just his voice and the trace of a digital pen on a tablet. In the Age of Social Media, that, it seemed to me, was powerful. And, at least to date, extremely rare.

Effective teaching is a human-to-human activity. Good teachers strive to achieve a connection with our students in the classroom. (I say “our students,” since I like to think I am a good teacher, albeit not at the K-12 level.) I’m told my voice conveys my enthusiasm for mathematics when I go on NPR as the Math Guy. If true, and some people seem to think so, then I can’t really take credit for it. I just speak into a microphone. No training, no rehearsing. Just me.

Sal Khan has that natural ability in spades. As he tells us (all Silicon Valley enterprises have a creation story, and in his case it’s true), Khan Academy began as his attempt to help his younger cousins with their math homework, by posting video tutorials on YouTube. He did not set out to build an education empire. People outside his family simply came across his videos on the Web and found them useful.

From a pedagogic perspective, his videos do not provide student learning, they deliver instruction – a distinction I discussed in my March Devlin’s Angle. It would be easy for me to critique them, and many have. (Read on.) But to me at least, providing learning is not where their real value lies.

Learning mathematics is hard. Very hard. It is easy to get discouraged and give up. Some of us, when we are learning math the first time, are lucky enough to have a parent, grandparent, uncle or aunt, older sibling, or family friend who can sit down alongside us and help us. I suspect that a great many of today’s professional mathematicians owe their eventual success in the subject to someone who mentored them in the early days.

But not everyone has such a person in their lives. At least, they did not until Sal Khan came along: friendly, non-threatening, patient, and a good explainer (actually not brilliant, but that might be all to the good, since a brilliant instructor could easily discourage a less-brilliant student). Above all, human. A regular guy. Just think about that for a moment. It’s a valuable weapon in the educational landscape.

Much of the attraction of KA came from its very amateurish nature. It really was just Sal Khan in his converted closet. Sure he did not really understand (as a mathematician does) a lot of the math he explains (though he knows an awful lot more than most people), nor was he trained in mathematics pedagogy. (Ditto for me, BTW – on both counts! – though I think I know more about both than Sal, which makes it possible for me to critique many of his lessons.) But that was a huge part of what made KA so popular. Millions of people around the world, young and old, whose experiences of math class was or had been awful, saw him as on their side. He was everyone’s friendly, helpful Uncle Sal. (I hope this does not come across as a hagiography. I have a lot of issues with KA. But I am trying to understand what leads to KA having such a passionately supportive fan base. I think there is something for us all to learn here.)

For millions of users of KA, what they find on the site is far better than anything they have had or are getting. For them, it’s not just a homework helper, it’s a lifeline. Their only lifeline.

Sure, they probably don’t really learn any mathematics. (I have yet to be convinced you can really learn mathematics over the Web, though that does not stop me wanting to try, as indicated by my other blog MOOCtalk.) The fact is, many people actually don’t want to learn mathematics, they want to pass a math exam. And if Sal Khan helps them do that – and millions say that he has done exactly that – then is it surprising they become KA fans? (And educational establishment enemies!)

Then Bill Gates comes along, and KA goes global. Expectations change. Now things have gotten more tricky. When a resource like KA becomes the primary vehicle by which millions of people acquire many or all of their mathematical skills, the stakes become dramatically higher than when it was a one-man homework-help service. Like it or not, ask for it or not, KA now has (in my view) an obligation to get things right. Doing so without destroying a major part of its appeal, is clearly going to be tricky.

Which brings me to the more recent skirmishes.

On June 18, two math professors, John Golden and David Coffey, posted to YouTube a parody of Mystery Science Theater 3000, which they called MTT2K, in which they watched and critiqued a KA video (an old one, as it happens) about multiplying and dividing negative numbers. To my mind, the target of their theatrical sarcasm was not KA, rather the reverence that many seem to have for KA, but many Khan fans (perhaps the reverent ones) seem to have reacted differently.

In any event, KA immediately took down the video, replacing it with two new videos, one on Multiplying Positive and Negative Numbers, the other on Dividing Positive and Negative Numbers. The two new videos appear to have been made in direct response to the MTT2K critique. A short while later, Khan released another new video, Why a Negative Times a Negative is a Positive, providing further elaboration.

Unfortunately, as John Jay High School (New York) physics teacher Frank Noschese noted, that third video was awfully similar to one produced some years earler by James Tanton. Things were escalating (or spiraling down, depending on your favored metaphor).

The pity was that critiques by knowledgeable teachers and pedagogy experts resulting in modifications to KA instructional materials is surely the way to take something that has value and make it even more valuable. But that tended to get lost in the MTT2K-parody sarcasm and the barrage of name calling that followed.

Incidentally, Noschese is a 2011

Presidential Award for Excellence in Math and Science Teaching awardee, and the author of an excellent science education blog Action-Reaction. He has commented and tweeted extensively (and to my mind constructively) on KA. I hope KA follows his blog and takes note of what he says.

Another experienced STEM teacher with excellent suggestions (for KA and for teaching in general)  is Dan Meyer, a former Google Education Fellow and now a PhD student in education at Stanford. He joined in the MTT2K exchange in his usually witty fashion in two blogposts, Bill Gates Just Put Out a Hit on John Golden and David Coffey, on June 20 and Sal Khan Comments On MTT2K In Chronicle of Higher Education, on June 28.

The latter was about an article in the Chronicle of Higher Education that day, summarizing the MTT2K affair. The Chronicle article was inspired in part by the article by Justin Reich in Education Week on June 22 about the “MTT2K Prize”, a cash prize to be awarded for the best video commentary on a Khan Academy video. (Actually it’s a great idea, if it can be done collaboratively with KA, absent any negativity.)

The final episode in this skirmish (at least so far, based on what I have seen) was an entry video to the MTT2K Prize competition on June 29, by Wired blogger Rhett Allain, an Associate Professor of Physics at Southeastern Louisiana University.

To my mind, Allain’s blog description is better than his videoed critique. Allain just does not come across on video as well as Khan does. Many teachers, and most academics, spend a lot of their time critiquing one another. We get used to it. It’s how we learn and improve. But when you put it on YouTube, it is viewed by millions of people totally unfamiliar with the process, and they can – and, in the cases I am citing, did – react negatively.

Which brings me back to my starting point. Education is hard, mathematics education particularly so. It takes a lot of different kinds of knowledge and expertise to get it right. The focal point is – and I think has to be – a single person, a teacher. Either in the flesh or over the Web. That teacher has to be able to connect to the students. Being a great classroom teacher does not make someone good at doing it on YouTube. In fact, most teachers come across pretty badly on video. But that does not matter if the people who are able to use the medium will listen to what they say.

Sal Khan’s strength is that he comes across extremely well on video (with or without his face on screen!). I just wish he would work (with real experts) on the content more.

But the real problem is not the stuff on the KA site. Flawed as it is, it is, as I noted earlier, a lot better than many people have, or ever had, access to. The fact that many of Khan’s fans describe him as “the best teacher ever” speaks volumes about the poor quality of the mathematics education that many receive. I’ve visited many math classrooms both in this country and around the world, and I’ve seen great math teaching. You won’t find it on KA. Instead, you will find something else, something unique and of value.

Sure, KA has lots of weaknesses and could be improved. That goes for any product. The real problem is that the US (and other nations) identify mathematics learning with instruction and passing procedural tests. In that world, KA meets a clear market need for instruction to help people pass procedural math tests.

In contrast, Ani, Noschese, Golden, Coffey, Meyer, Allain, and all the other KA critics in the educational world are interested in facilitating something quite different: real learning among their students.

Sal Khan says he is trying to move into the real, conceptual learning space as well, but so far I have not seen much that would qualify, and as I noted earlier, my own interest in trying out the MOOC format notwithstanding, I have yet to be convinced that it is possible over the Web.

Khan has something of real value to offer, most uniquely his ability to do well both locally and at a distance what many teachers find hard or impossible even in the classroom, namely connect with and inspire students. But he seems not to have deep conceptual understanding of mathematics or knowledge of the highly complex field of mathematics learning, and given his background it would be strange if he did. (For a good, reasoned series of articles pointing out the problems with KA, see the 2011 articles by Sylvia Martinez of Generation YES.)

On the other side, there are many teachers and education researchers who do have knowledge of mathematics and mathematical pedagogy, but are not able to connect well, at least on video, where they come across as cold or impersonal or condescending – video is a harsh medium.

If each party recognized that the others had something of value to offer, and if  we could get beyond the squabbles and the name-calling, we could produce something that benefits the people we all care about: the students.

Los Angeles police brutality victim Rodney King died recently. The words he famously uttered during the riots that followed his beating in 1992 seem equally pertinent to the current state of affairs regarding KA: “Can't we all get along?”

Telling stories with numbers

Ensuring that future citizens are quantitatively literate (as well as literate) should be the responsibility of every teacher and professor, but in the end it often falls on the mathematics instructors. In part this is because in the public, political, and bureaucratic minds, quantitative literacy is about numbers and numbers are the responsibility of mathematicians.  But in part too it is because all too often the mathematicians are the only ones not scared by numbers.

The trouble is, of course, as only those of us in the math biz know, for the most part we are not well versed in QL issues. Sure, along with millions of other people, we’ve read journalist Darrel Huff’s 1954 bestselling book How to Lie with Statistics, and academic Edward Tufte’s masterpiece The Visual Display of Quantitative Information, and some of us have likely managed to get hold of a copy of the excellent set of essays Statistics: A Guide to the Unknown, now into its N’th edition, though the publisher’s insanely high price can surely be motivated only by the desire that as few people as possible actually buy a copy. But that is about as far as it goes for most of us who find ourselves asked to teach a QL course.

Yet all of those excellent sources of QL educational material are mostly relevant to ensuring the future citizens can make intelligent use of the quantitative issues they face in their daily lives – making wise financial decisions, evaluating loan and credit-card agreements, buying houses, voting in elections, understanding graphs and charts in the media, and the like. What they miss is what is probably the single most important aspect of QL for people who use numerical data in the worlds of business and government: using numbers to communicate.

Every day, crucial business and political decisions are made on the basis of numerical data. Only rarely do the key decision makers produce that data; rather they rely on others, not only to produce it, but to present it to them.  Yet how many quants – the data producers – know how to present data effectively? To put it another way, how many of them know how to tell a story using numbers? With the educational QL focus on how to produce the numbers and how to present them using effective graphs and charts, what too often gets overlooked is how to communicate with the numbers themselves. I am referring primarily to what is surely the most ubiquitous numerical data preparation and presentation tool there is: the spreadsheet. For all the fancy graphics packages people use, I doubt that a single important business decision is ever made without the decision makers eyeballing a spreadsheet. For that, after all, is where the actual numbers can be found.

Or can they? Take a look at this example:

If you are a busy executive, with just a few minutes available to make a decision, how effective is that spreadsheet? Compare it with this one:

This is the same spreadsheet formatted differently. Some intelligent underlining and use of white spacing has provided structure that greatly aids comprehension. Actually, the formatting did not provide the structure, it simply represented in the spreadsheet printout the structure that the data already had, but which had been lost in the first presentation.

This example comes from a recent book, Painting with Numbers, by Randall Bolten, a lifelong quant based in Silicon Valley. As many high tech executives in the Valley will testify, Bolten is to the spreadsheet what Tufte is to the graph. Now, with the appearance of this book, his expertise is accessible to all. Aimed at the professional quant who has to present data to decision makers, much of the book goes well beyond what a high school or college QL instructor would require, but at under $20 for the Kindle edition, it is worth getting just to pull material from the early chapters (and perhaps the later one on PowerPoint presentations of spreadsheet data).

Much of what Bolten says boils down to adopting a common-sense attitude to the presentation of data. But as is so often the case, it can take someone else pointing something out before that common sense kicks in.

For example, simple Excel selections can make a huge difference to the intelligibility of the data. In the example below, right justification makes Version A much easier to grasp the overall picture at the first glance. 

As Bolten points out, Version A works so well because it takes advantage of the place-value representation of Hindu-Arabic numerals, which makes the leading digit the most significant. Simple, but very effective.

Likewise, how to represent the units can have a significant influence on readability. In the following example, Version F is much clearer than the other two.

None of this is rocket science – though more to the pity, for as Tufte pointed out, bad presentation of numerical data lay behind the disastrous decision to launch the space shuttle Columbia in 2003. It really is common sense. But having myself sat through many financial presentations as a university administrator, I know first hand that it is common sense that requires some prodding. Anyone faced with giving a QL course should devote some time to the crucial skill of being able to communicate effectively with numbers. As part of the language (sic) of mathematics, numbers too can tell stories. In today’s world, everyone should have the ability to ensure that they tell those stories as effectively as possible.

Math MOOC – Coming this fall. Let’s Teach the World.

Higher education as we know it just ended. Exactly what will take its place is not at all clear. All that can be said with certainty is that within a few short years the higher education landscape will look very different.

That is not to say that existing colleges and universities will suddenly go away, or indeed change what they do – though I think both will occur to varying degrees in due course. What is changing now is what classifies as higher education, who provides it, how they provide it, who will have access to it, how they will obtain it, and how it will be funded. Distance education, for many years the largely-ignored stepchild of the higher education system, is about to come of age.

This is not just my opinion. My own university, Stanford, recognizes what is going on, and is taking significant steps to lead and stay on top of the change, and a number of Silicon Valley’s famed venture capital firms, who make their fortunes by betting right on the future, have sunk significant funding into what they think may be key players in the new, higher ed world.

Last fall, Stanford computer science professor Sebastian Thrun used the Internet to open his on campus course in artificial intelligence to anyone in the world with Net access, and 160,000 students from 190 countries signed up. Some 22,000 of those students finished the course, receiving “certificates of completion” signed by Thrun (and co-teacher Peter Norvig of Google), but no Stanford credit. (For that, a student has to be on campus and officially registered; annual tuition is $40,050 and entry is fiercely competitive.)

Demonstrating the entrepreneurial spirit that Stanford faculty are famous for, Thrun promptly left Stanford to found a for-profit online university, Udacity. With Udacity receiving financial backing from a large Venture Capital firm, the MOOC – massive open online course – suddenly came of age. A short while later, two more Stanford computer science faculty, Andrew Ng and Daphne Koller, secured $16M of venture capital funding to launch a second Stanford spin-off company, Coursera, a Web platform to distribute a broad array of interactive courses in the humanities, social sciences, physical sciences, and engineering.

Initial courses offered on Coursera include, in addition to several from Stanford, offerings from faculty at the University of Michigan, the University of Pennsylvania, and Princeton. Stanford president John Hennessy appointed a blue-ribbon panel of Stanford faculty to develop a strategy for developing, and delivering, online courses. For free. To the world.

Yes, you read that correctly. The faculty, the universities, and the new platforms are making the courses available for free. All the funding is coming – for now – from for-profit investors and the private universities themselves. Why are they doing that? If you have to ask the question, you don’t really understand the Internet and how it changes everything. Think Napster and the music industry or Skype and the telephone industry. Like the settling of the American territories in the nineteenth century, the initial focus is on establishing a presence in the new land; monetization can come later – almost certainly in ways very different from today’s.

Computer-assisted, distance learning is not new, of course. Stanford was one of the universities that pioneered it the 1960s; many universities have for several decades offered adult professional education courses for a fee, largely to raise funds; and there are the for-profit online schools like the University of Phoenix. More recently, led by MIT, a number of universities started making recordings of their regular courses, together with course materials, available online for free. So what has changed now?

The answer is the platform and the target audience’s experience and expectations have changed. What has been missing so far is the active participation of the distant student in a learning community. Building on technology developed at Stanford to support flipped classroom experiences for its regular students, Udacity and Coursera have secured the major investments required to build scalable, robust platforms that can take the small learning seminar and create a similar experience across the Internet.

A generation that has grown up on the Web has taken to the new online medium like fish to water. During the term when Thrun made his AI course available online, most of the Stanford students enrolled in his class stopped attending his lectures and took their information delivery online, at times convenient to them.

Is this the beginning of the end of physical universities? I doubt it. Though online courses are excellent for in-career professional learning, the absence of being a member of a physical community makes them a poor substitute – arguably no substitute – for a traditional college or university when it comes to providing first-pass education. But what about the millions (make that billions) in the world who do not have access to a university education? “Let’s teach the world” is a buzz phrase you hear increasingly among the Stanford faculty these days. And Stanford is putting resources into making this attractive dream a reality.

What makes it fascinating to a faculty member, is figuring out how to take a learning experience that works in a small-group setting on a campus, and re-creating a similar – or equivalent – experience online. Having decided last December that I would offer a math MOOC this fall, I found myself at once faced with a number of challenges.

By far the greatest problem is how to provide the personal, expert feedback that is essential to good mathematics learning. Web delivery is fine for providing instruction, but that is just a part of learning, and a minor part at that, as I discussed in the March Devlin’s Angle. At first, it seemed an impossible task. But with Stanford and the now independent Coursera building innovative new platforms, I began to see the glimmer of opportunities. Over the coming months, I’ll use this forum to write about my progress. And hopefully get your assistance.

My focus for this first foray into this new educational landscape is the high school to university transition. As every university mathematics instructor knows, many students encounter difficulty going from high school math to college-level mathematics. Though the majority survive the transition, many do not. To help them make the shift, colleges and universities often have a transition course. I myself developed one of the first transition courses in the late 1970s, when I was teaching at the University of Lancaster in England.

Such courses typically comprise a mix of some elementary mathematical logic, proof techniques, some set theory through to an analysis of relations and functions, with a bit of elementary number theory and introductory real analysis thrown in to provide examples.

Given the problems students typically have when they meet this material for the first time, doing this at a distance is a challenge. Even if they did well at math in school, most beginning university students are knocked off course for a while by the shift in emphasis, from the K-12 focus on mastering procedures to the “mathematical thinking'' characteristic of much university mathematics. Clearly, offering such a course as a MOOC is a huge experiment.

This is where you come in. (I hope.) One of the things we’ve learned at Stanford from offering MOOCs, is that a key component is the creation of a strong online community. Learning is all about human interaction. The technology just provides the medium for that interaction. In offering my math transition MOOC at the start of the fall term, when many colleges and universities offer their own transition course, I am inviting any instructor who will be giving such a course, together with their students, to join me and my MOOC students online, making interaction with other students around the world a part of a much larger learning community.

The result could be a total failure. I won’t know until I try. On the other hand, anyone who joins me might just find themselves at the start of something major, new, and exciting. The online learning revolution is going to happen, and existing educational institutions are going to have to adjust to it, just as the music industry did to the iTunes revolution. Why not jump on the train as it is leaving the station?

I’m going to make my course just five weeks long, starting in early October. By incorporating participation in my Stanford course part of your students’ learning experience, everyone could benefit. For one thing, your students are likely to be inspired by being part of an educational revolution that for millions of less privileged people around the globe can quite literally be life changing.

Because they will be supported by being part of a physical learning community, with the personal support of you, their instructor, your students will be highly empowered, privileged members of that online community. They can take advantage of your support so that they can help others. And as we all know, there is no more powerful way to learn than to try to teach others.

For that student half way round the world, trying to improve his or her life through education – by learning to think mathematically – the potential benefit is, of course, far greater. Helping that unknown young (or not so young) person make that step might just help inspire your own students to put in that bit of extra effort to master that tricky new transition material. Everyone wins.

If my Stanford MOOC draws a student body in the tens of thousands, which it might, based on the experience of my colleagues here, there is no way I and a couple of graduate TAs can provide individual feedback to every student. But if instructors and their students across the US join me, then maybe we can collectively achieve something remarkable.

I am making my MOOC deliberately short, five weeks, so participation will leave most of the semester open for participating instructors to concentrate on giving their own course, perhaps using their students’ initial experience in the MOOC community as a springboard for the rest of the course.

By the time I post next month’s column, I hope to have more details available. In the meantime, I ask anyone giving a transition course this fall to consider joining me in this experiment. The only cost is our time. There is no need to make any advance commitment to me or to Stanford. At this stage, all I ask is that you consider joining me. I believe we will all benefit. Let’s teach the world.

When math and art meet

In 1991, two mathematicians proved that you cannot always hear the shape of a drum. There are different shaped drums that make the same sound.

What about hearing numbers? For instance, can you hear pi? The answer is yes, you can. In fact you can listen to two renderings, though it took a recent court ruling to make this possible.

The story begins on Pi Day (March 14, or 3.14) 2011, when New Scientist posted a video by a musician called Michael John Blake, in which he played a piano rendering of the first 31 decimal places of pi, played at a tempo of 157 beats per minute (314 divided by two).

The video immediately went viral, but a few hours later, YouTube was contacted by a lawyer representing jazz musician Lars Erickson, who claimed that Blake's work sounded very similar to his 1992 composition "Pi Symphony", which he had registered with the US copyright office. With a claim of copyright infringement, YouTube removed the video. But Blake decided to lodge an appeal.

Both musicians had produced their works by converting decimal digits to notes on the musical scale and then adjusted tempo, phrasing, and harmonies to give the resulting composition recognizable musicality. The basis for Blake’s appeal was whether it was legally possible to copyright such a representation of pi. This is clearly an interesting question, since books and articles talking about pi and visual art based on pi clearly are copyrightable. Indeed, this very blog post  automatically carries copyright.

One year later, on March 14 of this year, US district court judge Michael H. Simon, deliberately choosing to announce his decision on Pi Day, dismissed Erickson’s claim of copyright infringement. "Pi is a non-copyrightable fact, and the transcription of pi to music is a non-copyrightable idea," Simon wrote in his legal opinion. “The resulting pattern of notes is an expression that merges with the non-copyrightable idea of putting pi to music.” (Ideas are not copyrightable.) The only features of his work that Erickson’s registered copyright protected, the judge said, were the musical flourishes he added to give the result a pleasing sound, and on the face of it the flourishes the two composers added were different. Erickson disagrees with Judge Simon’s opinion on that point. Readers can make up their own mind.

In any event, the world now has access to two musical renderings of pi.

I’ve always been intrigued by artists who try to push the boundaries of their form, as indicated by an aside I made during my talk at Wonderfest 2010. As a mathematician, I’m particularly fascinated by attempts to interpret mathematics in different artistic media, novels, movies, TV shows, painting, sculpture, and of course music and dance.

Which brings me to the “Devlin’s Angle” column I posted back in January, when I mentioned the project I worked on with the choral group Zambra, where we set out to interpret some of my favorite mathematical equations in song. I ended my article by promising to say more about that project, but then other topics came up that seemed more pressing, and that promise was not fulfilled.

The equations/formulas we chose were Euler’s equation, Pythagoras’ equation, Area of a circle formula, Einstein’s energy equation, Leibniz’s series for pi, Newton’s second law of motion, and Euler’s polyhedron formula. You can find a description of the entire project on my website.

The stage performances of our show also involved dance, provided by math professor and dancer Karl Schaffer and members of his dance troupe MoveSpeakSpin, but unfortunately we did not have the funding for a video recording.

There is clearly considerable potential in the use of music and dance (and other artistic media) in school mathematics education. Someone else I am aware of, in addition to Schaffer, who is doing great things in this area is Malke Rosenberg with MathInYourFeet.

There is also a new TV series that includes some mathematics, Touch, on Fox TV. I commented on the portrayal of mathematics in that new series in a recent commentary in The Huffington Post. As I said there, I have positive and negative feelings about that particular portrayal, but surely anything that connects mathematics to an aspect of everyday life, particularly recreational activities and popular culture, provides an excellent opportunity for the mathematics educator faced with interesting students in a crucial subject whose many important applications are, to a large extent, hidden from public view.