# Focusing on definitions for understanding in secondary education

I am a teaching assistant at a school. My job is mostly to help them solve problems given by their teachers. Students are at high school level. I assume the curriculum In Brazil is similar to the one in US, except that we don't have the possibility of taking AP Calculus/Physics type of courses. You don't get to pick your courses, everyone is subject to everything at the same level.

I hope the answers are applicable despite of what specific things they are dealing with, but as a small breakdown we have

• Last year high school students are dealing with polynomials in algebra;
• Second year high school students are learning matrices;
• First year high school students are learning functions.

They all have geometry in a separate class, creating two math classes, but I don't know at what stage each of them are.

I see that they study hard and do many exercises, yet they lack basic understanding and their knowledge is almost completely procedural. As a remedy for this I've recommended them some of the strategies proposed by Polya, with a strong emphasis in understanding what are the terms involved and what is asked. Consequently, I'm constantly hammering the point that they know the definitions and how to apply them.

Is this sound? I know this is learned behavior and that it's uncommon to proceed from definitions, but I think this approach will result in better outcomes than they have now.

• What level and subjects are you teaching? What curriculum are the teachers using? – James S. Oct 28 '14 at 20:40

What you call proceed from definitions is quite similar to the teaching practices of the New Math or Modern Mathematics (UK) during the 1960s. Definitions, corollaries, theorems, etc. seem to us, as teachers, the perfect method to learn pure mathematics.

However, not every student in your class will want to become a mathematician or an engineer. Even those who show now interest in Mathematics, don't have the necessary background to learn in a significant way. Just because they are young or do not have the required cognitive level.

Polya strategies are fine. But just giving them to the students doesn't make a miracle. I bet you don't think in Polya when solving a problem. Nor your students. Yes, they can learn the rules and the steps, but most of them will fail to apply them in a new problem.

How can we handle it? Here are a couple of proposals. And I say just two, because there are a bunch of movements (not just Polya) which try to answer this question:

1. Obvious: one learn to solve problems by solving problems.
2. As we have a limited period for teaching (3-4 hours a week?), we should work efficiently. This means to choose the proper problems. And, better than problems, situations with a rich context. How do we know if problem, a collection of problems, or a situation is good as a learning experience?:

• Looking at how many different concepts and other math objetcs (properties, arguments, etc.) does it articulate, the more, the better.
• Looking at how many different ways do a concept or a math object appear. For example, algebraically, graphically, verbally, etc. Again, the more, the better.
• By using contexts that may be interesting for our students. Or real-life contexts.
• By using manipulatives (even if they are in high school). The thing is to make the situation appear as real as possible.

Hope this helps.

• Definitions, corollaries, theorems, etc. seem to us, as teachers, the perfect method to learn pure mathematics. However, not every student in your class will want to become a mathematician or an engineer. I know. I believe this is the best method for everything. It's just that in other areas it's not as clear what the definitions are as in mathematics. Thank you for your answer. – Mark Fantini Oct 29 '14 at 17:41
• Mathematics are crystal clear, no doubt. The process of taking some academic knowledge and transforming into something assimilable for a target group is called didactic transposition(Chevallard). You don't teach rational numbers to 12 years-old children. You teach them fractions. And in the process of transposition, usually, the logical reasoning chain of definitions, corollaries, etc. dillude. – Pablo B. Oct 29 '14 at 18:14
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