Words Pave the Way

I’m an abstract thinker. I sometimes think thoughts or feel feelings that I am unable to translate into elegant English, or any other language for that matter.  Sometimes I can’t even find a way to express them in inelegant English. When I was young and in middle school and high school, I believed-because it was drilled into my head-it was because I was a “not an language person”. I now realize that is just ridiculous, and furthermore, I understand how inaccurate and damaging such labels can be.

Math is a good subject of study for an abstract thinker, and that’s probably why I was drawn to it at a young age. The field is dedicated to taking complex thoughts and expressing them using letters, numbers, symbols and as few words as possible. Sometimes a single equation can represent a relationship that could take a paragraph or more to adequately explain using words.  These mathematical statements have a certain beauty and elegance.  It’s part of why I’m drawn to teaching developmental math, because I want to help others to learn to appreciate what math has to offer.

At some point during my teaching career, I started to realize that for some of my students, when they looked at equations, what they saw was the equivalent of “blah, blah, blah”.  They weren’t able to articulate the relationships that the equations represented.  They could read an equation like 3x+5=11, but it was like me reading text from a book written in German.  I’ve studied German in the past, I remember how to pronounce the words I see on the page, and I even recognize a few words, but not enough to really understand the meaning of a paragraph.  Recognizing that my students make it through their days in a world filled with numbers and most report that they do okay with numbers in real life, I wondered if learning math for some of my students was a language issue.  A matter of translation of math to English.

Let’s go back to that equation 3x+5=11. Some students can solve this by inspection or using guess and check, but are not able to solve the equation by using the additive and multiplicative properties of equality. When solving by inspection, they look at the equation and see that whatever is being added to 5 must be 6 in order to get 11. They ask themselves what number can be multiplied by 3 to give 6 and determine that x=2. This is a great mathematical thought process. However, if the students are unable to use the properties of equality, they will not be able to solve increasingly complicated equations.

In my mind, the words below are what I read when I look at the equation 3x+5=11.

I’m thinking of a number.

I multiply by three.

I add five.

The result is eleven.

What number was I thinking of?

I  found that when I changed an equation into the form of a math puzzle like this, almost all of my pre-algebra and elementary algebra students could solve it. Furthermore, when the students articulate the steps that they take in solving the puzzle, these are precisely the steps that would be used to solve the equation using the properties of equality.  Instead of asking what do I add to 5 to get 11, They start with the result 11 and undo adding 5 by subtracting 5 from 11 to get 6.  Then instead of asking what do you multiply by 3 to get 6, they undo multiplying by 3 and divide 6 by 3 to get 2.  There is something about the puzzle form that helps the students to focus on opposite operations. This is a subtle but really important distinction, that can really help students to understand the process of solving using the properties of equality.

Based on this observation I’ve changed the approach that I use in teaching, especially developmental math, to be much more language focused. I try to help my students to not just see the numbers, symbols and variables, but to understand and be able to articulate the relationships represented on the page.  I do this through emphasizing vocabulary and including vocabulary words on exams.  I also start many classes by asking students to write about the mathematics that they are learning.  For example: “What is a numerator? What is a denominator? What do they tell us?” My students are very resistant to these assignments at the beginning of the quarter, but there are always a few that “confess” at the end of the quarter, that they found these assignments to be very helpful in developing their understanding of the content.

I know that in the same way that I used to think that I was not “a language person,” that most of my students carry around the notion that they are not “math people.”  I do all that I can to convince them that learning math is about putting in effort using good strategies and do what I can to help them to develop those strategies.  However, I know it will take time for them to change their perceptions of themselves.  After all, it’s a hard shift to make and takes time. So, if my students choose to continue to see themselves as language people, that’s okay. I’ll just have to help them to learn the language of mathematics.