How Important Is the “Why” in Math Class?
A MiddleWeb Blog
I recently listened to a podcast (Mr Barton Maths Podcast; Ed Southall: Part 1), and part of the conversation dealt with the importance of explaining the why as opposed to just instructing students on the steps or giving them an algorithm that works.
For example, when learning how to divide a fraction by a fraction, many students are taught “Keep, Change, Flip” – keep the numerator the same, change to multiplication and flip the denominator.
Is this sufficient or is it important for the students to know why “Keep, Change, Flip” works? (If you are interested in exploring the topic of dividing fractions by fractions, and why inverting and multiplying work, I have listed some resources below. There are several topics where I found the WHY to be particularly relevant, such as fractions, operations with signed numbers, multiplying binomials, and factoring).
The discussion on the podcast sparked two questions in my mind:
- What are the benefits of explaining WHY to our students?
- Is there a right way and a wrong way to explain the WHY to our students?
The Benefits of Explaining the WHY
1. Deepen Student Understanding
I recently taught students that when you multiply a negative by a negative your product is positive. I’m embarrassed to say that I did not teach them why it works, only that it does work. It never occurred to me to think about why it worked. I’ve taught it for so long I have forgotten to ask myself why.
As a result, I missed an opportunity to help students really understand the concept. There are so many great strategies I could have shared with my students – all related to why multiplying a negative with a negative has a positive product. The same goes for adding and subtracting signed numbers, which I taught in the same unit. (Here is a good resource from the NCTM Math Forum.)
2. Help Improve Student Retention
Often when students rely on memorizing problem solving steps without an understanding of why, they cannot retain the information. In fact, when my students were tested on adding/subtracting and multiplying signed numbers, some forgot the rules altogether or remembered them incorrectly. If I had given them a stronger foundation, they would have at least had the ability to reason through the problem.
In my Algebra 2 (with Trig) class I put this problem on the board:
Full disclosure: my students tend to struggle with fractions. I could hear them talking among themselves about how to solve the problem (without a calculator) of 1/2 ÷ 2/3. As a class we talked about “Keep, Change, Flip,” and some of them remembered being taught that phrase or something similar. However, even the students who remembered the phrase couldn’t quite remember how to execute those steps.
I showed them how “Keep, Change, Flip” worked and heard students say things like, “oh yeah” or “I remember now.” And they did. I am sure if I had given them a worksheet full of similar problems they could have worked them.
We then started talking about the WHY, and we discussed the convenience of making the denominator a 1 by multiplying by the reciprocal. (I shared what I learned from the podcast guy Ed Southall’s book Yes, But Why? Teaching for Understanding in Mathematics.) This seemed to make sense to the students.
I then asked them if they thought knowing why “Keep, Change, Flip” worked would help them remember how to work this type of problem in the future. About half of the students said it would, and about half thought it would make no difference.
During the course of the discussion some students said that they were not interested in knowing why; they just wanted to know the steps. One student said it was overwhelming when teachers tried to tell them too much. A few students said they did better if they knew why, and one student said that it was very important for them to be told why. This student felt like knowing why was necessary for them to be successful.
3. Improve Student’s Confidence
Many students recognize that memorizing steps is the not the same as understanding or mastery. Knowing why something works can help struggling students gain a sense of confidence.
My daughter shared with me recently that she never felt like she was good at math (despite always making A’s or B’s throughout her high school and college math classes) because she could follow the procedures, but she never understood why she was doing them. If she felt that way, that means that there could be some students in my class right now that are mastering the process/content but don’t feel like they are good at math because I haven’t sufficiently explained the principles behind what they are learning.
The Right and Wrong Way to Explain Why
I think in the past I have been guilty of too much too soon. The situation is like this: I get excited about explaining the why of fractions, Pythagorean Theorem, quadratic formula, or unit circle, and begin a lengthy explanation about the why. I look out at my students and everyone is either confused or uninterested.
After thinking about it, I realize this usually this happens if I try to explain why something works before I have given students a chance to practice how it works. For students that may be struggling, I think this cart-before-horse approach is a recipe for disaster.
In fact, a British article titled How can teachers help students overcome their fear of maths? states, “There is evidence that even strong students of maths can feel daunted and overwhelmed when there is too much information at once and not enough time to practise.” My recent conversation with my own students bear this out. They actually told me too much information too soon is overwhelming.
So timing is key. In the future, I plan to delay explaining the WHY until students have some experience practicing with the steps. At times students do ask why something works the way it does, and I think it can appropriate to address it with the whole class when that happens.
It’s My Job to Explain Why
As a teacher, it’s my responsibility to be ready with a clear and concise explanation for the WHY. With that goal in my mind, I plan to keep a notebook with all of my Why’s for each unit I teach. I already have student-friendly explanations about the Why of dividing fractions by fractions and multiplying a negative times a negative!
If you are curious about the subject, I have listed some resources here. And if you have ideas about teaching the Why from your own classroom, please share.
► Yes, But Why? Teaching for Understanding in Mathematics by Ed Southall
► NCTM’s The Math Forum: Ask Dr. Math