You can teach math in two ways, notes Rolin. Math can be presented as a series of rules that produce reliable results, or as a set of concepts that explain much about the world.
Much of our math teaching has been rule-based, and for good reason: the rules are reliable and consistent. But some of our students may be left behind unless we also teach the concepts behind the formulas. Rolin encourages math teachers to incorporate a strong mix of both types of teaching. He demonstrates how that might work with a common type of math problem.
A few years ago, I read What’s Math Got to Do with It? by Jo Boaler. And one chapter stood out to me because it was something that I hadn’t thought about before, but it did really fit well with what I’ve experienced as a math teacher. And because of that, it has influenced my teaching and it changed, a little bit, the way that I teach.
The chapter focused on the ways that girls are affected by the way that we teach math. I believe that God created males and females for distinct purposes and also to be different in many ways. But I also believe, at the same time, that both need to be treated equally and, especially in school, both need to be given equal opportunity. We do well on occasion to consider how the ways that God has made males and females to be different might actually keep us from treating them equal in the classroom.
And I think that that really comes through in math. That’s what I’ll be talking about a little bit here today. So the way that we teach and the differences between males and females really does need to be considered when we are teaching math because math is a prime example of a school subject that males and females have been found to respond to in different ways.
In my teaching and math, I found that two general approaches work pretty well or are pretty common in math teaching. One I call a rules-based approach and the other is more a concept-based or a “why” approach.
First, the rules-based approach. Usually it’s taught in a way that you do A and then you do B and you do those series of steps in order to achieve C, or the right answer. This method is used, I think, for a number of different reasons. It’s effective, it’s efficient, and it gets results in that students get the right answer. Students are to memorize the rules and when they do that and follow the rules correctly, they get their answers correct.
The second approach, which I call the concept method, is to focus more on why the math makes sense. The teacher in this method will take time to explain why this method works, why we do this. And in order to do that, give a more of a conceptual approach to the topic or to whatever is being learned at the time. This generally takes longer and is more difficult to achieve.
Math can be completely taught with a rules-based approach. So you can teach it all with “Do A and do B and you’ll get C.” And because of that, and for probably several other reasons, I think teachers tend to gravitate towards a rules-based approach to teaching math.
It’s efficient, like I said before, and it can actually be done without the teacher completely understanding the concept themselves. So they’re able to teach, “You do this, A and then you do B and then you get to C,” without understanding the concept behind it themselves. So I think, probably, even though a good mix should be used, we tend to lean more on the rules based side rather than the concept based.
How is this connected to how girls interact with mathematics? Well I’ve found, and what this book pointed out to me, is that girls more often than boys prefer to know the “why” of math rather than the how. Boys are often content to power through an assignment, get it done as quickly as possible, and then they’re satisfied by seeing that most of the questions have been answered correctly. Girls, even if they’re getting the questions correct and are able to follow the rules, will sometimes still be frustrated with a lesson that they don’t understand the “why” of. They can maybe follow the rules exactly, but they still will get frustrated at not understanding why this works and just being able to get the correct answers.
To demonstrate this, Boaler uses a simple but profound example that shows this well, and I’m going to borrow her example to demonstrate this. A typical high school math problem is multiply two binomials. So I’m going to use the example of X plus three times X plus seven to demonstrate this today. The rules-based method that’s usually used to teach this uses an acronym called FOIL. First, outside, inside, last. And it’s simply a use of the distribution property in mathematics. But it’s a way that students are hopefully able to remember it better. You multiply the first ones, then the outside, then the inside, and then the last. And you simply follow that rule; if you do it correctly, you’ll get the right answer every time you multiply two binomials together. So very simply… The rule has been followed. We’ve arrived at the correct answer.
But this is a question that the “why” was not explained at all. There was no explanation given of the—why it works to use FOIL to get to this point. So, Boaler suggests, why not use rectangles and areas to help give a little better idea of why the foil method works? So instead of writing it out as X plus three times X plus seven, let’s set it up instead as two rectangles. That’s… Two rectangles. One having a side of X plus seven, the other having a side of X plus three. This makes four individual rectangles within the diagram, and we can find the area of each one, which we’ll see corresponds to what we’re doing when we’re finding FOIL. So it’s really an area problem and it can be explained this way. The exact same answer is achieved. In fact, the method is really very similar when you compare the two, but it gives us a better picture of why FOIL works to multiply the two binomials together. This is just one example of a way that the why can be taught, the concept can be better taught, rather than just going by it with a rules based approach to the math.
And from my experience and reading, I think a method like this is going to benefit female students more than male students. These are generalities that I’m using. It’s not going to be the case every time, but I’ve found that girls really appreciate and do better with math they are understanding it and doing well with it. I believe that we’re doing the females in our classrooms a disfavor if we don’t explain “why” to the methods and we just stick to the rules.
This isn’t just a problem in our schools, our Mennonite or our Christian schools. Boaler cites a few different studies, and it seems to be a problem as a whole. I have noticed this in my classroom, as well. Since I’ve become aware of that, I’ve enjoyed looking for opportunities to explain things better to those that are looking for it; to all students, but I’ve especially noticed that there are some girls that really like the explanation better. That’s been fun to see, so it has fit with my experience, as well as being what I’ve read.
Let’s remember that while we are all equal in God’s eyes, he has created males and females to be different, and we see that they learn in different ways because of the differences that he’s given to us. By recognizing these trends, we can keep getting better at creating an environment in our classroom where everyone is likely to succeed.
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CONTRIBUTOR: Rolin Martin