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. […]
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.
*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.
After posting the article, I came across this great quotation regarding what good teachers do:
"[Good teachers] demonstrate by example what it is to think through a problem. Because the way experts think through a problem differs by field, so what professors are doing when they teach this way is communicating content knowledge but also initiating students into an intellectual discipline."
It's by Prof. Robert Art of Brandeis University, and is quoted in
Annie Murphy Paul's excellent "Brilliant Blog"
I am very excited for your course. I'm a biology/chemistry student currently and and am immersed in math but wanted to learn more about how I should/could be viewing math. That there is an actual way to frame the information more mathematically. There was no transition class at the start of my college career and have been sharing my excitement over your MOOC class with my husband after reading your wordpress blog. Thank you for the time and effort you are putting into this endeavor!
This is exactly the reason why I'm taking this course. I have no way to affirm or validate if I am already practicing "Mathematical Thinking". I've been trained heavily on qualitative analysis but I feel that, in order to "organize" my thoughts and contemplations, I would need to represent the "mess" in a way that goes beyond ordinary language. At least, this is one of my major concerns.
I am looking forward to a lot of insights and learnings from this course, Professor Devlin. As a fellow academic, I think that offering this through MOOC is a wonderful opportunity for university academics/researchers like me, who are unable to afford Stanford education, but very interested in learning further about mathematics and its applications in other disciplines.
Loved this particular blog. Read it twice already. I can still read it again :-). Can't wait for it to be 17th September.
Mathematics inspires me, through mathematical thinking, through this post, and through you... :)
Hi Prof. Keith,
I`ve enrolled to your MOOC course with big expectations, but they are already satisfied :) so, I feel like a Maths thinker already, and a doer in the sense that you say thinking "is" doing.
I find your ideas and examples -particularly gestures and enthusiasm, very different in flavor and sharp indeed.
Looking forward to having the course.
Thanks for all the brilliance you share.
I added a paragraph and an endnote after it became clear that not all readers recognized the parody of mathematical precision in my opening paragraph. What my DoD experience conformed for me was how anyone who has spent time thinking hard about something has something of value to offer. Just because something fails according to one metric, does not mean it is without value, and indeed could be of immense value. A person who derides someone else for "failing" should be cautious. One day they may find themselves working for that other person! :-)
In 1984, Robert B. Davis wrote in LEARNING MATHEMATICS of studies that were emerging from several researchers and universities: "They all show that many university students have actually learned far less than everyone had assumed. Students who have completed five or more years of studying mathematics in high school and college may still have very wrong ideas about VARIABLES (or 'unknowns', 'literals', etc., - the ubiquitous x's and y's of algebra). This despite the fact that these same students were thought to have learned about linear equations, systems of linear equations, quadratic equations, graphs of conic sections, and many other things THAT DEPEND UPON A CORRECT UNDERSTANDING OF VARIABLES. At first glance this may seem to be impossible: how can you learn something that depends fundamentally upon various things you don't know? Later in this book we shall see an answer: DIFFERENT TYPES OF 'LEARNING' ARE POSSIBLE. One type does depend upon certain skills, concepts and understandings. Another type does not. This is not entirely mysterious. A singer may sing German lieder without in fact possessing any real command of German as a language; the singing can be a matter of imitation. So in this example we see two different behaviors: the use of German as a language, where the words and phrases have meaning; and, second, the use of German as a set of sounds to be produced imitatively. This is probably a passable parallel to the two different mathematical behaviors, one of which builds on understanding, and the other of which does not. The implications of this unsuspected deficit in students knowledge are extremely far-reaching." (p. 4)
Hi Prof Devlin,
I appreciate the post you read. I signed up for classes in Coursera to fill up some of my free time and this one really attracted me. I cannot claim to be good in math, but I'm really looking forward to learning how to think like a mathematician. Despite my traumatic experience in college, I cannot suppress my love for learning, especially a subject that challenges me. I hope I'll be able to learn as much as I can from this class. Looking forward to September 17.
thank you for introducing your opinion in such a clever and appealing way.
looking forward to taking the course.
Beautiful and inspired anecdote. A story like this has a huge impact on students, and we need to use more of these as educators.
I am very curious... what appeared on that first slide? (if it is not classified).
Hector, Here is the definition I gave on my first slide. Just two sentences.
A feature F is contextual for an action A if F constrains A and may affect the outcome of A, but is not a constituent of A.
In particular, a situation, environment, or set of circumstances C is said to be a context for an action A if C constrains A or affects the outcome of A but is not a constituent of A.
Of course, those sentences contain several terms of art that require elaboration and definition. There is a whole formal apparatus within which the above definition operates: Situation Theory.
Situation Theory is a mathematically-based framework for describing and analyzing information flow that was developed at Stanford in the 1980s. (Not to be confused with John McCarthy's AI focused Situation Calculus, also developed at Stanford.) Situation Theory was initiated by Jon Barwise and John Perry, initially as a framework for natural language semantics. I was one of the team that developed it as a more general framework for analyzing information flow. I described the theory in my 2000 book Logic and Information.
After reading this article i dont have any doubt abt the course and value its going to offer me.
Thanks you professor, See u in the class
Yes, the concept seems to be on the line Gestalt theory in Psychology, that the sum of the constituents is less than the total. Here, context is part of the difference. Hope I am close to understanding the concept.
After reading this..I am excited about taking this course.it is what I was hoping for.
I will add, I think what a good teacher does is reignite the love of learning..all children start with it..discovery..that is learning.kd
Looking forward to your class. Thanks for the effort :)
Hello Mr Devlin... You wrote "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." Which I totally agree with. I just want to add a comment...
When my older sister started having children (she has 3) 10 years ago, my father told me "you should play with them and teach them everything you know, because children are like a sponge, they adsorb liquid information which is your example, if you teach them bad thing they will do bad things, and if you teach them good things they will do good things, do the family a favor and don't teach them bad thing..."... (I probably shouldn't add that last part, but even today I find it very funny)...
Today, I'm 24 years old, and I believe I can take that comment my father gave 10 years ago and modify it to fit the MOOC learning system:
"Mooc students are like a sponge, they will absorb liquid information from the teacher giving the course and use it to enhance their professional, academic, personal, and everyday life." Of course MOOC's are new, so the best thing a MOOC student can have is without doubt the willingness to learn, and that's why 50k-60k people enrolled to your Mathematical Thinking course, because we (yes I am enrolled) think we have the willingness to learn from someone that we don't know, we've never met, we don't have the same native language, we all have different aspects and goals in life. Yet, we all expect to learn as much as possible from you and your TA's in 7 weeks (which would be also amazing).
I have been trying for a very long time to get my students at a pre-service teacher institution not to teach 2X2=4, but let their students acquire mathematical concepts. I am so happy at my old age i can see what professor devlin is after in mathematical thinking.
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