When you learn a new concept, chances are that there’s some sort of procedure to follow in order to come up with an answer to a problem. This helps students when they are first learning, because it lets them follow clearly laid out steps that will culminate in the correct answer. For example, if we were trying to add two fractions together, we know that a common denominator is needed. As a result, students might be told that they should multiply each fraction by the other’s denominator, which will guarantee that the denominators are the same. It might even be written in a nice three-step method like this:
- Identify your fractions as a/b and c/d.
- Multiply a by d and c by b.
- Write the denominator as bd. Your new fraction should be of the form (ad+bc)/bd.
This is an easy-to-follow recipe that will produce the correct answer all of the time. It’s made so that you can go through the steps one line at a time and arrive at the answer you want. Students might even be encouraged to memorize this procedure.
I want to argue that this is a fundamentally flawed idea of how we should approach teaching students to solve problems.
First, let’s take a look at the example above. Imagine we wanted to add 1/2 and 3/4. The algorithm tells us that we should transform the two fractions in order to get 4/8 and 6/8, so that in total we get 10/8. This is the correct answer, but anyone who has taught this concepts knows that this isn’t the most efficient way to go about it. Instead, we notice that 2 is a multiple of 4, so we only have to change the first fraction in order to create common denominators. Doing so gives us 2/4+3/4=5/4, which is the same answer as above, but simplified. In fact, when doing these problems, students are usually required to simplify anyway, so the student would have to do more work after following the recipe.
This is a very simple example, but it illustrates an important point. Recipes blind students to shortcuts or other ways to solve a problem. Once a student is given a recipe, why should they look for a quicker way to do the problem? They know that the way they were given will work, so it usually isn’t worth the extra effort to look for a shortcut or another method. This is true even if the recipe takes longer to do! In this sense, I think we do students a great disservice if we only emphasize recipes.
It’s not that a recipe is bad. In fact, there are many wonderful recipes that solve many problems (more commonly known as algorithms in computer science and mathematics). However, the idea for those recipes is to feed them into a computer so that the computer can work on the result. They aren’t necessarily for students themselves to do. Plus, we don’t care if a computer takes a little longer to solve a problem (for the most part), because it can still do it fairly fast.
On the other hand, we want students to be able to solve all sorts of problems, and to use the techniques they’ve learned to tackle these problems. But recipes make students hone in on one way to solve a problem, without thinking about anything else. It offloads the thinking of a problem and reduces it to following predetermined steps. This could lead a student to completely miss the point of a problem, or to see an interesting connection if they weren’t strictly following an algorithm.
Some may scoff and say that seeing a better path to a problem rarely actually happens, but that’s because they aren’t trying hard enough. There are all sorts of tricks and techniques that one can use to solve problems without needing to go through recipes.
Another ripe example is the factoring of expressions, something that is the bane of many students. In secondary school, students focus on factoring quadratic polynomials, and there are a several cases that one has to consider in order to get the factorization just right. Since memory aids are allowed for students, these procedures are typically written down and some students may even analyze each expression to see which case it falls into.
This is a horrible way to go about learning factorization. It reduces the process to a classification problem, where students simply match their expression with the corresponding case, and follow the steps. This means that students will often miss shortcuts and other techniques that could be just as helpful to them as following the procedure!
The other problem with recipes is that they substitute knowledge for aptitude at following a procedure. Instead of knowing why the recipe works, the student becomes only responsible to use the recipe correctly. This encourages students to not even think about what they are doing, since they know the output is what they want. As a result, students aren’t thinking about the problem as much as going through the motions. This can lead teachers to thinking that a student knows why a concept works, when really the student can only tell you how it works.
My vivid example comes from learning how to complete the square in secondary school. The idea is that you want to factor a quadratic expression into something of the form a(x-b)2+c. When I first learned about it, the recipe seemed like magic (and not in a good way!). I had little idea about what was happening, but I did know that if followed the steps on my memory aid, I could solve the problem.
Did I look deeper in order to understand what was actually going on? Of course not. It was only years later that I looked at the technique again and saw that it did make sense and I should have understood it more when I was first introduced to it. However, since the recipe was there, I offloaded the responsibility of knowing in order to simply being able to use it.
I want to finish with the acknowledgment that recipes can be useful. They can speed problems up, particularly the one’s that are repeated over and over again (like finding the roots of a quadratic function). However, you should always ask yourself, “Could I work from first principles to get back to this point?” If the answer is “yes”, then you have understood the concept. If it’s “no”, then you should investigate that uncertainty! It’s an opportunity to learn. This is why I always try to ask the students I work with conceptual questions, because if the point of education is just to follow recipes, A.I. will replace us much sooner than we might want.