How to help students bridge the gap between highschool and university mathematics?

Question

Mathematics at highschools is quite different from that in universities. Instead of calculating numbers and finding solutions to specific problems, freshmen end up proving theorems and figuring out what happens in general. (By in general I mean, for example, that we are not very interested in the integral function of $xe^{\sin x}\log\cosh x$, but in the fact that every continuous function can be integrated and the integral satisfies the fundamental theorem of calculus.) I have heard statements like "I like math but I always hated when we had to prove stuff", which means that the student will hate mathematics the way it is presented and studied at universities, and studying mathematics would probably be a bad idea for the student. The way of thinking about mathematics as a whole and the collection of mathematical tools makes a big transition when one enters a university (at least in Finland), and it is easy for a teacher to make fallacious assumptions about students' proficiency in basic tools in doing and communicating mathematics.

How can I, as an instructor of some kind in a freshman course, help students adapt to the new situation? Is there something that I should be very careful about? In your experience, what is important when trying to get abstract ideas (and the need for them) across to freshmen? I understand that it is not uncommon that a freshman class is taught by someone who has long ago forgotten how a freshman thinks, and although (because?) the teacher has made his best to make the material mathematically elegant, the students get little grasp of what is going on. I would like to prevent this from happening in my class if I ever get assigned a freshman course. It would be great to hear experiences about supporting students in this transition.

Note: Educational systems are different in different countries, so "first year university mathematics" is not globally well defined. The transition I wanted to support the students in is the one from calculation-oriented mathematics to proof-oriented mathematics.

Answer 1

First it's a good sign that you're on this website and keen to make a difference. A passionate tutor is halfway there in my opinion. I've thought about what would have helped me as an undergraduate but everyone is different.

1) Break the theorem-proof-theorem-proof-theorem-proof-lemma-theorem cycle. I'm sure that many will argue that this is the bedrock of the subject but for me at least it quickly became overwhelming. I've thought about why it became overwhelming and I can think of two reasons. Firstly I needed time to aclimatise to the new maths and wanted to just calculate with each new theorem before moving on. Second it was just boring. There were lectures I probably could have understood if I hadn't switched off with the monotony of theorem proof.

2) Facilitate the students in forming abstractions and don't jump straight in with the abstract. I remember an abstract algebra class beginning with the axioms of arithmetic including commutativity and associativity. I remember not understanding what all the fuss was because I'd only ever dealt with real numbers and couldn't imagine anything being different. It was only until much later when I'd studied linear algebra did it dawn on me that things could be different!

3) Why say it with a letter when a number will do? Never start with a general case. If you can give examples with numbers do so. Do the matrix with number elements to calculate determinants or the differential equation with numbers where you might have put an $a$. Later on we then it's sunken in can you generalise.

4) Keep things in context. I remember calculating eigenvalues from characteristic equations and had no idea what I was doing. Much later I understood the topic thinking about transforms. The point being it wasn't hard after all but I had learnt it out of any meaningful context and so ended up lost.

These are just a few of my thoughts, everyone is entitled to their own opinion and disagree. If you want more then just let me know.

Answer 2

The more the students can reinvent theorems themselves, the better. This is the philosophy of math circles, and it's also how R.L. Moore ran his classes (http://legacyrlmoore.org/reference/mahavier1.html). He had a record number of PhD's come out of his classes I believe. (Though his legacy is problematic due to his racism.)

You can work somewhere in between lecture and discovery, guiding students. I tell my (lower level) students that we're going to build the quadratic formula together. And I get them to do most of the work. They feel pretty proud of that.

Answer 3

I appreciate the question and the problems it is trying to solve. I have the feeling that it is trying to "attack the symptoms rather than the cause", and that what is really needed is a more foundational approach. However, my perspective is firmly grounded in my experience as a student in America; the problems being mentioned may actually be different in Finland and other countries, and may not benefit from considering the following ideas.

I see one issue as "one course must fit all", when the intent is to provide a common foundation to be used in diverse environments. Yes, I want the students to be able to prove some useful results and be able to communicate those results to others. I also want them to have some approaches for tackling new problems. I also want them to recognize situations where they can apply methods that are available to get the answers they need. Finally, I want them to understand what directed questions to ask when they do not see how to work through a situation, be it solving a known problem, tackling an unfamiliar problem, or communicating how to think about concepts related to existing and new problems.

I think it should be made clear in the goals of the curriculum (and iterated on a small level for each course, and on a large level for each department) what are some of the expected outcomes ( e.g. you can learn where and how to ask for help, you can learn how to solve known problems, you can learn ideas on how to approach new problems, you can learn how to develop and communicate new technologies to deal with problems yet to be seen ). In delivering a particular lecture, the priority should be set for which one of the outcomes will be reinforced, and assign as a task outside the lecture something that helps progress to the other outcomes. If there are too many outcomes expected for a student, divide the class into those groups who wish to master that particular outcome, and have the groups perform and present to one another some results of their outcomes.

The major work lies, of course, in determining and properly expressing what outcomes are desired, as well as providing opportunities for achieving that outcome even in the event that only that outcome is achieved (that is, where possible arrange things so that achieving one outcome doesn't depend on achieving another outcome first, but so that achieving any of the outcomes will make it easier to achieve the others). To me, who am not a professional educator, this is the essence of programs in education: to find continually new ways of arranging and presenting material so that the connections within and without can be used and appreciated and properly communicated.

So, rather than viewing the situation as a gap between high school and university material, view it as introducing a new level of maturity where the student is taught that several outcomes result from mastery, and they should prioritize the set of outcomes in accordance with their (ever-changing) goals for their education. Ideally, learning styles and ability can also be factored in, but I don't know enough to talk sensibly about how to accommodate such in the classroom/lecture hall.

Gerhard "Ask Me About System Design" Paseman, 2015.03.13