What Motivates Math Learning? (Part 1)

For the sake of posting stuff on a regular basis,  I will divide this discussion into two parts.

This quote from former CCNY Professor (now Mercy College Dean) Alfred Posamentier was recently brought to my attention:

“The point is to make math intrinsically interesting to children. We should not have to sell mathematics by pointing to its usefulness in other subject areas, which, of course, is real. Love for math will not come about by trying to convince a child that it happens to be a handy tool for life; it grows when a good teacher can draw out a child’s curiosity about how numbers and mathematical principles work. The very high percentage of adults who are unashamed to say that they are bad with math is a good indication of how maligned the subject is and how very little we were taught in school about the enchantment of numbers.”

You can read the whole letter, too: It’s quite interesting. Professor Posamentier is under the impression that we can get more students to like math (or at least not fear it) if we pique the interest of children in mathematics while they’re in elementary school. This sounds like a great idea and a sensible one too, but what methods can teachers use to motivate math learning in elementary schools, especially if the teacher is not well versed in math beyond K-12 school curriculum? What about for math educators in secondary schools? I would like to address some things that I believe can be done at both levels to stimulate students’ interest in mathematics or, at the very least, remove some of the stigma that math is a difficult subject.

Consider the sequence of numbers 0, 2, 4, 6, 8, 10, x. What is likely the value of x? Chances are you guessed 12; if so, why? Well for starters, we are looking at a sequence of numbers that seem to fit the pattern of multiples of two. Something in our intuition tells us that the next logical step in the sequence is 2*6. Our minds seem to have a pattern recognition mechanism that tells us what is likely to follow given a series or prior statements or facts. It is, after all, the next logical step. My experience with intuitive problem solving in mathematics is a positive one that sort of opened the door to interest in problems that require more critical and ingenious problem solving.

Thinking about mathematics was not always entirely contingent on a large collection of definitions, theorems, proofs, and the likes; mathematics grew out of a series of intuitive arguments that called for more rigor as time progressed and the subject branches evolved. Without going into research paper-like detail, many topics in mathematics arose from intuitive questions stemming from preexisting mathematical or scientific frameworks. I am not asking for math education to revert back to Euler’s proofs (or lack thereof) in Institutiones calculi differentialis nor am I asking for the reader to prove each theorem as an exercise a la Rudin, but I am asking for math educators, regardless of formal education, to dig down into the heart of the symbols and numbers and explain intuitively what is going on in a subject, how it is believed to have come about, and some of the neat results and consequences. Granted, rote tasks like learning elementary multiplication tables are hard to motivate, but one can easily go into a discussion on elementary multiplicative patterns in numbers, e.g. any number whose sum of digits is divisible by three is itself divisible by three. I must acknowledge that within the realm of intuitive thought, one ought to use different teaching methods to reach a wider learning-style audience, i.e. visual learners, hands-on learners, etc. The point is to digress from repetitive computations and lifeless introductions to new material that fail to stress key ideas and concepts in mathematics and venture into a world of inquiry and critical thought using the language of nature’s book.

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5 thoughts on “What Motivates Math Learning? (Part 1)

  1. Well written as always, Jacob.

    I have a follow-up question. The conclusion of your post seems to be that educators should motivate math by placing it in a larger context, be it historical or conceptual. You quote Professor Posamentier as expressing this same sentiment, and certainly the idea has been around as long as there has been a “crisis” in math education.

    You also suggested in your post that a potential source of the problem is the fact that most math teachers aren’t familiar with math beyond the K-12 curriculum. This leads to my question:

    Certainly, it makes sense that a teacher without an advanced understanding of their subject math lacks the perspective to properly motivate students. At the same time, could there not be a societal problem where students are not motivated by “neat results” and learning for its own sake? To motivate, you need to presume that the students are, at some level, interested in learning and understanding. Children are taught to ask: “how is this important in real life? How can I use this on a daily basis?” But the time they reach math class, the idea of learning math because it’s interesting is very foreign to them, and most students can’t break out of this paradigm.

    Here’s a relevant quote from Stephen Fry: http://www.youtube.com/watch?v=wBZtoIEjiM4&t=5m

    That being said, (1) is this a legitimate problem? (2) how can one deal with it?, and, most importantly (3) assuming that it is a problem, is there any reason to try motivating students who are inoculated against motivation? Most of elementary school and high school math can be a little dull, and the only real motivation is the interest and curiosity of figuring things out. With that out of the equation, what is left to educators?

    A little anecdotal evidence to support this claim: some ethnic and religious communities that place great focus on “learning for its own sake” often produce motivated students, even though their teachers aren’t necessarily the most educated.

    Thank you for your time.

    • Thanks for the response Isaac. I think you definitely hit the nail on the head with your observation and it’s something I planned on addressing in Part 2 of my discussion on what motivates math learning.

      To add to your anecdotal evidence regarding “learning for learning’s sake”, I have come to notice that many students who are the most engaged in math learning are the curious, inquisitive kinds of students that seek explanations beyond problem answers. This is not to say that they are particularly interested in math over any other subject, but rather they exhibit an inherently inquisitive nature that permeates all of their studies.

      You are certainly touching on a really heavy and complex point regarding the purpose of learning and the desire for neat results. The thing is, I don’t know if blaming the student for being disengaged is the best way to characterize what could be a much larger issue at hand. Why can’t you learn something for the heck of it? I too ask myself this question, especially when I hear students (I’m talking university students too) say “Do I have to read this?” or “what’s the minimum number of “x” that I have to use?” or the dreaded “Is that gonna be on the test?.” I’m sure some argument can be made in support of the use of such questions, but what I find troubling is the overwhelming emphasis on satisfying the standards of merit and not the learning process and adventure. The meritocracy is what I find contributes the most to this problem, the problem that learning only valuable the context of external results. The meritocracy in America, not limited to education, is a societal framework that dictates that one must satisfy “X” to obtain “Y”; that’s cool and all, but what about when the institutions tell you that “you want “Y” ” or that “you need “Y” “; “without “Y”, you won’t amount to anything.?” Conscious of it or not, students are pressured by this system through various outlets and their reactions to are varied. The point I’m making is that there is a societal mainstay that influences this sort of behavior in students and it is likely that there are other such structures in place that work to disengage students from “learning for it’s own sake.”

      • Fair points. I guess another underlying question here is: why ARE we teaching math to people? And not just math: why do we teach science, English, social studies, etc?

        Is it because we need educated people to further the sciences and advance technology? If so, then we could probably just teach math and science to the most talented and motivated students. After all, the are certainly way more people trained in the sciences than there are science jobs.

        Is it because we need the layman to have a perfunctory command of arithmetic, science, language, and general knowledge so as to make educated political and life decisions? If so, then we should only teach the most applied, down-to-earth, useful information. People forget most of the math/science content they learn in school anyhow, so there’s no sense wasting their time with it to begin with. Why not make high school in a kind of trade school?

        Is it because we want to sharpen people’s critical thinking skills? For most people, math and science involve a lot of memorization. It is not often the case that they really apply critical thinking to these subjects. Maybe we should have logic/philosophy/contemporary-issues classes that helps students learn how to reason analytically in a pragmatic, user-friendly and easily applicable fashion? Learning the names of cell organelles or the algorithm to perform polynomial division involves very little critical thinking, or “thinking outside of the box”. Instead, like the SAT, it is about learning how to do certain types of tricky problems.

        Lastly, is it because we want people to be educated for its own sake? Because I would ask: “do we?” Many people believe in educating their kids to get them better career opportunities; how many care about learning for its own sake? In all seriousness, if this is not something we value as a society, why should we force it on children? There’s no inherent reason why learning for its own sake should be so important. There was a time when it was considered essential that kids learned proper etiquette at school. Nowadays, any curriculum with time set aside for etiquette classes would be seen as absurd. Why isn’t it absurd that the educational establishment forces this high-minded virtue of “learning as an end, and not just a means to an end” on the rest of society?

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