Solving problems

Question

I have been telling my students to try to solve the problems on their as much as they can. As I tell them, it will help them be better at solving and understanding problems of Mathematics. This thing I learned from Khan Academy actually.

But students seem to have problems on this aspect. To me it seems like they don't know what to think. Before asking them to solve the exercises, I first do some problems on the board. As I do that, the students now can solve only the problems that was similar to the ones I did on the board. When questions come with a little tweak, they blank out. And then I ask them to think harder but they become blank.

So I would like to ask you how to tell them to think, let's say, mathematically in this context? Secondly, how should I encourage them to try to solve the problems on their own?

Answer 1

Perhaps they go blank when you ask them to think because it's been so long since they've tried to think in math class. Sadly, many (most?) students just try to memorize procedures.

You'll need to find problems that suck them in, so they really want to think. Also, you'll need to give them some strategies for problem-solving. How to Solve It, by George Polya has lots of great ideas. (Here's an online copy of the one-page outline he devised.) You can also find some excellent material at Avery Pickford's blog, by searching on "habits of mind."

Brainstorming helps, which means working in groups is much more effective than working solo. But it takes a lot of savvy to facilitate good groupwork. One principle for what types of problems are best used in this situation is that they have multiple solution paths, and a somewhat open answer. This sort of problem is often called a "rich task."

If you're teaching high school, there has been work done on "complex instruction," which refers to a suite of strategies used to get students thinking, while using lots of groupwork. I've found this work useful, although I am not comfortable using all their strategies with my college students.

One more note on your students' responses to your attempts so far: Students have taken math for many years by the time they reach you, and have expectations of math class. When you violate their expectations, there will be pushback. According to James Stigler, author of The Teaching Gap, much of it is deeply imprinted in us, as what school is. He writes, “The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom.”

Answer 2

I think that you may try this procedure. Go to the classroom. Teach them the material, if possible with a bit of historical and some amusing facts. But don't solve any problem. Tell the students to tell you to give a problem that they thinks is difficult. If someone gives you a problem, don't work it out fully on the board. You first ask him/her to tell you how far he/she has progressed. Then your job will simply be to involve all others of the class to the problem. If the student's approach is on the right path, just help him a bit and leave the rest upto him/her. If he/she is on the wrong track, produce an example which contradicts the student's procedure and say him/her to explain that using his/her theory.

For example if the student says that he proved that the interchange of limits in any case is always possible. You may then ask him to evaluate the limits $\displaystyle\lim_{x \to 0}\displaystyle\lim_{y \to 0}\dfrac{x^2}{x^2+y^2}$ and $\displaystyle\lim_{y \to 0}\displaystyle\lim_{x \to 0}\dfrac{x^2}{x^2+y^2}$.

This practice will help the students to develop their own thinking skills. If the students see that the teacher is taking special attention to those who can independently solve problems, they will (as is expected in general), try to put more effort in independently solving problems.

I think that in this respect the approach of this book will help you a lot.