How to help students be prepared to study mathematics
I was having a hard time coming up with an idea for a Monday post, so I asked Twitter for requests. One Twitter follower responded:
@RobertTalbert Anything but Pi Day!— Loop Space (@mathforge) March 13, 2016
I won't judge here, especially since I myself have never really understood the appeal of Pi Day among mathematics people. (Sorry!) But I will honor your wishes, @MathForge. And anyway, there was a clarification:
@RobertTalbert But seriously ... what are the main issues for university students studying maths that schools could better prepare them for?— Loop Space (@mathforge) March 13, 2016
@MathForge is from England, and there are cultural differences between UK and US mathematics students and how the subject is taught in schools before university, and in the university. Despite this, I'll take a stab at this from the US point of view.
I've been teaching in colleges and universities for almost 20 years now. When I first started, I used to think that the main issues for university mathematics students had to do with mathematical preparation -- their ability to perform basic arithmetic and algebra computations, their grasp of basic facts about logarithms, and so on. I still do think that this is an issue, and one that has gotten worse as the race to get more and more students into calculus has reached a fever pitch in the last few years.
However, I no longer see mathematical preparation as the issue among math students in the university. Rather, I see it as a symptom of something deeper. That "something deeper" has to do with students' conception of what mathematics is, in the first place and students' general abilities to manage complexity and information. I can break these down further into four interconnected issues.
First: Students' conception of what mathematics is, and what it is not. Ask most American mathematics majors why they are studying mathematics, and you are likely to get a combination of two answers: It was always easy for me, and I like that there's only one right answer. In other words the common conception, in my experience at least, that student hold about math coming into the university is that math is about getting a right answer to a simple problem exercise as quickly as possible and with as few mistakes as possible. This false conception leads to a host of other problems, for example memory loss (students learn computations and procedures just long enough to get right answers on a test) and the seemingly-universal hatred of "word problems". (There's also the issue that word problems as kids see them in school are generally pointless and stupid. and kids cannot take them seriously as a result.) Those of us who do math for a living, or use it in our work, obviously see mathematics very differently: it's about the apprehension and expression of patterns, the clear communication of essential universal truths (so much deeper than "getting right answers"), the thrill of discovering a language that describes things so beautifully and elegantly. I think if younger students could simply have experiences that point beyond the simple easy-right-answer approach -- or even just teachers who would say, "We are doing computations right now where you need to try to get the right answer but math is a lot bigger than this" -- those kids would be a lot better prepared for later study.
Second: Student's conceptions about mathematics is done, by professionals who study and use it. This is connected to but different from the first point. The naive conception is that mathematics is done by hand, with the purpose of getting a right answer to a simple exercise (that is just like the ones in the book or the lecture). Maybe there is a calculator involved. But real mathematicians -- and those who are not mathematicians but who use mathematics -- see it differently. Real mathematicians do a lot of guesswork and wishful thinking, a lot of intuitive reasoning and ad hoc rationalization that later will evolve into precise reasoning and argumentation. (I will never forget when one of my grad school professors said about one of his results, "The theorem is true but the proof isn't correct yet.") It is a collaborative process in which the right solutions, the correct proofs get hammered out as a dialogue between colleagues, and even then results are not correct due to some cosmic back of the book but rather simply by consensus. I believe this is starting to change in the lower grades, where kids are working together and reasoning about their work rather than simply checking to make sure it's right and moving on. But it's still an issue, as anybody who teaches introduction to proof can attest -- the wall that so many "good math students" hit when learning proof, when the real face of mathematics begins to show itself, is miles high and rock-hard for many former "A" students in high school math, simply because they aren't ready for the culture shock.
Third: Students' systems for managing time, tasks, and commitments. There's more to mathematical preparation than just mathematics itself. There is also the general skill of managing the information flow that a university education requires. I have said before that consistently, the biggest issue I face day in and day out with flipped learning or SBSG is time, task, and commitment management. Students coming into the university have had absolutely no training, and very little experience, with this kind of management. The vast majority of my students have no trusted system in place for organizing information that comes to them. So when something comes up that is hard -- and remember the conception is that math is not supposed to be hard for "good students" -- it gets treated the same way as something trivial. Do students break hard tasks up into multi-step projects and determine the next action? No. Do students put things on a trusted calendar and review it daily? No. (I still have students who write due dates down on their arms in pen instead of a calendar.) There is so much time spent in the lower grades teaching students about "study habits" -- how to take notes in a lecture (!), and so on. I wish that schools would throw all that out and teach the Getting Things Done philosophy to students, starting around the sixth grade (in the US; when kids are around 12 years old). If kids could be taught GTD explicitly and have them practice it day-in/day-out for 5--6 years before entering the university -- that would be revolutionary.
Fourth: Students' personal behaviors and beliefs about hard intellectual work. Related to all of the above are the dispositions that students have about doing intellectual work, especially when it's hard. Dispositions such as failure tolerance, patience, self-efficacy and so on. Two terms recently coined that give shape to this issue -- grit and growth mindset -- have their detractors, but I do believe that these concepts can and should be directly addressed when students are young and can learn from them. I know this for two reasons. First I have a 12-year old, and we have spent many tearful evenings working on math homework where the solution wasn't obvious after 10 seconds of review and the answer wasn't right the first time and she wanted to give up, or did give up. Second, with my students, every time I use standards-based/specifications grading there is a length period of adjustment where students have to learn that failing on an assessment means try again not game over -- they have no patience with failure because they never experienced it, or had to recover from it. I do think that SBSG and flipped learning address these dispositions, but it's a lifelong challenge, and students need to be exposed to these earlier than their freshman year.
I want to end this off by saying that I was an incredibly late bloomer when it comes to all of these issues. My conception of mathematics was essentially the same as that of a typical American high school student up until around my fourth year of graduate school when I was really hitting the deep waters of my doctoral thesis. And I don't feel like I really got my act together regarding issues 3 and 4 above until maybe 7--8 years ago. Looking back on my path to where I am now, I actually find it hard to believe that I made it here given how little I knew about my subject ad how to work with it.
So that's why I feel fairly strongly that I hope that schools, and the teachers who work in them, work hard to take these issues into account. One of the things I enjoy the most about working with K-12 teachers, either when I give professional development workshops or teach pre-service teachers, is that I feel like I can get this message across directly, and I get to know teachers who are really doing things right in this regard. These are foundational issues that not only should be addressed in the lower grades, they can be addressed without a huge overhaul in what is being taught. The Common Core State Standards for Mathematics are a big step in the right direction because they embody the kinds of thinking that I believe are conducive to these issues, but I'll save that politically-loaded topic for another time.