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.
What is mathematical thinking?
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.
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.
[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 doing 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,” 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.
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. 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
discussed by policy makers rather than professional teachers. Yet it is a huge
factor.
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.
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.
When I start a project, my task is to
find a way of analyzing how context influences data analysis and reasoning in
highly complex domains. 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. 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.)
When the project got completed, 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.
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.
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, etc. 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.)
Mathematical thinking is a highly
complex activity, and a great deal has been written and studied about it. I
will give several examples of mathematical thinking, and to demonstrate two
pairs of processes through which mathematical thinking very often proceeds:
- Specializing and Generalizing
- Conjecturing and Convincing.
If students are to become good
mathematical thinkers, then mathematical thinking needs to be a prominent part
of their education. In addition, however, students who have an understanding of
the components of mathematical thinking will be able to use these abilities
independently to make sense of mathematics that they are learning. For example,
if they do not understand what a question is asking, they should decide themselves
to try an example (specialise) to see what happens, and if they are oriented to
constructing convincing arguments, then they can learn from reasons rather than
rules. Experiences like the exploration above, at an appropriate level build these
dispositions.
Enjoy Mathematical Thinking!
