Why do people want to do things?
I’ve been asking myself this question a lot. If you believe that teaching is about more than simply imparting knowledge and skills, that it is also about inspiring students to achieve greatness through academic study, then you have to ask yourself: how do you give students experiences that will make them want to learn your subject and then apply it in a future career?
For many educators, the answer is often to show students that the subject is beautiful. For others, it is to show students that the subject is useful. Educators talk about motivation, and sometimes intrinsic vs. extrinsic motivation. These are all very useful conversations. I think they are also insufficient.
Let me give an example from my own life. I love playwriting. I wrote some plays that I consider decent while in college, and I would love to write some more plays and perhaps to see them performed. Yet I “don’t have time” to keep writing plays. Why not? Well, for one thing, I have a job doing something I love—education—and that job probably pays better. But I could still write plays as a hobby! I don’t, because there are other things I can do for fun, and because most of my friends are math/science-oriented and so I don’t have a community of people to talk to about it and encourage me. Instead, I go to see a lot of plays each year and content myself with occasionally imagining what plays I would write if I had the time.
My point is that even if someone likes doing something, that is not enough for them to dedicate their time to it. Just as we recognize this fact in our own lives, we should think about the many factors that play into our students’ decisions, both consciously and unconsciously:
- Do they see it as a productive/gainful use of their time?
- Do they have a supportive peer group? Will they be able to do it with others?
- Will it give them social status?
- Can they see themselves doing it in the future? Is that future self someone they like and admire? Do they have role models?
- Is there an established pathway for doing it that they understand?
- Do they expect success at it?
I believe that focuses on curriculum or study skills or content knowledge are all good steps, but they are insufficient to the task. Just because students learn that mathematics is useful for building bridges does not mean that they see themselves doing it. We need to consider the whole context surrounding a child, the whole environment that might encourage them to become a scientist (or not), and how we can make it more likely that they see math and science as a viable pathway.
Ultimately, I think we need to build their self-identity as scholars. To me, self-identity goes much farther than just motivation, be it intrinsic or extrinsic. Motivation plays a part, but self-identity is about how they see themselves. Indeed, I claim that without a resilient self-identity, all of our efforts to teach knowledge and skills are less effective.
Of course, saying that we should accomplish all these things is a far cry from specific proposals to do so. I hope to explore more about self-identity in future posts over the coming weeks: to try to give a better definition, and to give concrete thoughts for how to help students develop it.
Consider Learning 
Hi Dan. The identity question is something I have been starting to think about as well. You may or may not have heard that I have been tutoring students this semester in Danville Correctional Center (DCC), a medium-high security all male prison, this semester. My observations regarding this experience may shed light on the identity issue.
Some context:
A group of around 13 DCC students (most doing serious time for gang related violent crimes as teenagers or young men) are taking a 400 level Aerospace Engineering course on Robotics taught by an Illinois professor. They get one three-hour lecture per week and the opportunity to visit a “resource room” for a three hour tutoring session (office hours basically) once per week with volunteer tutors. These both occur within DCC. When I staff the resource room, they can get help with the mathematics involved, but when the other tutors, who have no technical expertise, are there, then they can receive little or no help with mathematics. I noticed early on that their math backgrounds, for the most part, were not strong. Many students did not know what sine or cosine were, for instance, and there were issues with using variables and basic algebra. The robotics class requires, however, fluency with trig, algebra, matrix algebra, and geometry. It is all about using sequences of coordinate systems to get robot arms with many rotators, joints, and arm segments to correctly pick up an object sitting within a 3D grid. Basically, lots of SO(3) algebra and using certain matrix forms to reduce the computational complexity.
My experience and observations:
As you would imagine the beginning was a bit rocky. The professor delayed the due date of the first homework assignment as many students were having to learn or relearn trig, algebra, matrices, right-hand-rule, etc. However, progress was made quite quickly and soon we were working with higher-level concepts like transformations between coordinate frames. One day we were going over a homework that was covering various properties of SO(3). Rather than avoid the overarching abstract concept (which, I fear, is what tends to happen with disadvantaged groups in education), I told the students what was really going on–that we were showing that SO(3) was a non-abelian group and I outlined in appropriate language the basic definition of a group and their relation to symmetry.
And here is the crucial observation: When I told them that what we had just covered was the same thing that a university student or grad student would have seen in their first day of their group theory course, then you could just see their faces light up. They had these big smiles that they seemed completely unable to control or suppress. It was clear that no one had ever talked to them in such a way before. Ever.
Fast forward a few weeks and the tutoring sessions had taken on a much different character. I was no longer the center of attention. Students were presenting their own solutions to complex robotics problems. Students, as often as I, were criticizing one another’s solutions and proposing new ones. To be honest, I had never worked in such gory detail with SO(3) from the engineering perspective, so I often genuinely did not know the answers or procedures sometimes. At one point we were all stuck on a part of a problem, and I noticed a student reading through the finer details of a proof in a chapter of the book in order to get a better clue of what was going wrong–a part of the chapter that even I had skipped when I had prepared for the tutoring session!!
Identity:
These experiences inspired the idea for me to apply to offer an ongoing math workshop at DCC for next semester. The idea is that we will be teaching DCC students about “beautiful” mathematics–NOT just the usual remedial mathematics that the system deems “appropriate” for inmates. Of course we will nurture those remedial skills as needed, but the real point is to create moments like the one I described above by not shying away from “fun” math like discrete math and proof writing, combinatorics, topology, etc. I recruited several math grad students to help because I want there to be a constant support system in place for math learning. Who knows? We may even have students working on open problems from within their cells in a year’s time.
The takeaway is that, if my intuition (based on my experiences at DCC so far) is correct, then I believe math can be used to positively shape and nurture identity in a powerful way, particularly among a severely disadvantaged population such as incarcerated men. But, yes, the intervention must be designed in a way that allows positive answers to at least a few of the bullets you mentioned in your post.
Also, if you only offer to teach a group of people certain “appropriate” (remedial) mathematics, then they will “get the message”. They are not stupid. And this, in and of itself, is dangerous. This is why we are refusing to offer a “basics only” math workshop.
This sounds really familiar to me. In particular, the validation that you are doing “advanced” or “college-level” work means a lot to students!
Dan,
This reminds me of thoughts I had after reading Papert’s _Mindstorms_. I don’t how the specific sequence happened, but when I was working with young kids & mathematics I became as interested in what it means for someone to identify as able to do mathematics as for them to do mathematics.
I think you’re talking about a later-stage notion: a durable, working identity as mathematician. But I believe Papert gave me the idea of mathphobia — of looking at the intense anxiety many children have around mathematics, and I found that (and his thinking more broadly) quite valuable.
Have you come across his writing before? I’m sure it’s not hard to notice that students have anxiety towards mathematics, but his particular strain of writing — which goes much broader than this particular note — I suspect you’d find interesting.
-Mike
I have certainly heard of Mindstorms but I’ve never read it. You’d recommend it, I take it?
Identifying yourself as able to do mathematics is critical, but I see it as part of your self-identity—it part of your ideas about what you are capable of doing. I actually want “self-identity” to be all-encompassing, because I having a broad term can get people thinking more generally than just thinking about motivation in isolation.
To be clearer about the relationship: being able to do mathematics seems part of developing a self-identity as a scholar of mathematics; and similarly, having such a self-identity gives you more resilience against being challenged. For example, I’ve run into hard problems, doubted my ability to solve them, and then thought to myself, “no, I’m a mathematician, this is something I can do.” That resilience helped keep me in mathematics.