Can Conceptual Problem Solving be Taught in Math Class?

Consider the following math problem:

This is a practice test item from the Algebra NJSLA (link here). Many students struggle on this problem when I give it during our test prep period. At this point, they know all about rational and irrational numbers. They can simplify radicals, multiply them, add them, even change them into the fractional exponents to simplify. So why can’t they solve a conceptual problem involving them?

I have thought about this a lot the past few weeks, getting done with our state testing and focusing on getting my students ready for 10th grade Geometry. Many of them are experts at the rote procedures involved in Algebra problems like factoring, completing the square, finding slope, and so forth. Even on applied word problems, my students are able to identify the types of relationships present and choose the correct function type to model.

But when conceptual problems occur, I feel that I left a problem solving gap that is needing to be filled. And from looking at the NJ learning standards and Common Core Standards, these types of questions seem to be more abundant and regular than other procedural questions and applied questions. A new era is being ushered in of conceptual focus, which I love. However, we need the tools to get students ready for these types of problems. To find the answer, I thought back to my own math experience in college.

Generating Examples

An extremely common, but under utilized learning and problem solving technique is generating examples. In math proofs, if we are proving for something to be always true then we have to make sure to cover every possible example. Of course, we do this in general terms utilizing symbols. However, when we talk about proving something to not be true, it only takes one example. And with that example, we actually do not need abstract symbols, we can just use real numbers. Go ahead and find me a right triangle that does not follow Pythagorean Theorem, and we can disprove it and be famous!

Jokes aside, consider the question above again.

In this problem, we are not proving that the answer choices will always be rational. We only want one single example.

For this problem, there is no way that anyone could check all the possible number combinations for each answer choice, because there are an infinite amount of numbers. Furthermore, most 8th and 9th graders do not have the mathematical background to formally prove these conjectures. But they don’t need to, since only one result being rational will find the answer.

Therefore, an easy problem solving technique would be to assume a and b are specific numbers and plug them in. I encourage you to do the problem yourself, as it may help with the next section (the answer is D)

Examples and Counter-Examples

In my own teaching experience, we do not talk enough about generating our own examples and counter-examples. However, it is a huge part of discrete mathematics and a great problem solving technique. It is also a huge component of understanding. Just staring at math symbols sometimes does not paint the full picture. We need to spend time generating examples and having students create their own too.

Many of the times I passively write some examples on the board or have them pre-planned myself. However, rarely do I allow a student to share an example or non-example to a case presented. I recommend that you ask students to share examples and non-examples of introduced knowledge during the lesson. You might be quite surprised at their recommendations, and it will keep you on your toes since you will have to decipher if it fits or not.

I would also highly encourage you to build a lesson around generating examples. There are a few resourced online that you can find, but you can also make it basic by providing students with previous definitions and making them provide their own examples. Either way, it builds a skill for life in math.