Thoughts on Math 3

Session 3 of How to Teach Math was on how making mistakes impacts math learning. It was eye-opening to learn that children who make mistakes and have a growth mindset (that is, they decide to learn from their mistakes and keep trying) learn more and experience more brain development than those who always "get it" immediately and rarely make mistakes. However, by the end of the session I was wondering if the proposed solution is really the answer.

How much should we celebrate mistakes?

First of all, I should be clear. The kinds of mistakes meant here are not errors in computation. We are talking about mistakes made in the process of figuring out the solution to a problem: trial and error, if you will. Think of a child putting together a puzzle that is challenging for him. He turns the pieces this way and that, he figures out what elements of the picture match at the edges, he might sort out colours or do the edges first, he tries a few different pieces in any given spot (those are the mistakes) before he finds the right one.

The problem in schools seems to be that the traditional way of teaching math involves a lot of exercises to develop fluency, but not a lot of problem solving (figuring out what needs to be done to find out the answer). When students are faced with open problems to solve, they freeze because they are so concerned with getting the right answer.

This really reminds me of the process I went through in learning to write. I came out of my high school years firmly believing that writing was "not my thing". I hadn't actually written very much by that point, but every time I did, it was agonizing. You see, I was a perfectionist that believed that not only did I have to say what I wanted to say, but I also had to say it right (no mistakes!). If I had to write something, it would come out of me word by agonizing word. The word would be deleted and replaced with a better word. Sometimes a sentence came out, and that sentence would be restructured until it was perfect. I would go on like this until the assignment was finished. After hours of torture like this, I would never want to see that piece of writing again. I don't blame my homeschooling mother. She did her best to give me the resources I needed to learn to write. And besides, she thought I was a good writer. (Any feedback I got from writing I did for others was always excellent...it was the difficult process that convinced me that I was bad at it.)

What possessed me to start taking courses, I'll never know. I wanted to learn, though, and in the process I had to write papers. The agonizing writing processes continued until one day I discovered freewriting in a random on-line search. I turned off the computer screen and started typing whatever was in my brain without regard for spelling, grammar, or even relevancy. When I turned the screen back on, I organized my thoughts, added a bit here, deleted a bit there, and I had a first draft to edit. It was a breakthrough moment. I still can't say writing was easy after that, but when paper after paper came back with good marks, I started to realize that I wasn't so bad at writing, after all. Now that I'm writing a blog, I'm finding myself actually enjoying it. (No kidding, say the readers of the reams of content I put out now...lol.)

What would have helped me in my process of learning to write, and how can I apply this to math?

1. The message that mistakes are part of the process should have been continually reinforced. I think that I would have scorned anyone telling me that "We like mistakes, mistakes are good." I was a born perfectionist. I didn't talk or walk until I was sure I could do it properly. The example in the course where a teacher asked a student to demonstrate his mistake on the board so the whole class could learn from it would have made me cringe. But I did need to learn not to be afraid of making mistakes, and that mistakes are part of the process of learning. I think the course's emphasis that mistakes are good might be helpful for some students (the ones that are discouraged because they make so many), but not for others.

2. I wish someone had helped me clarify my goal and focus on it. In writing, the end goal is to get the ideas that are in your mind onto the paper. (My problems came from trying to focus on perfect grammar and word choice at the same time.) In math, the goal is to solve the problem. Focus on that goal so much that mistakes along the way don't really matter to you. I was almost getting the message from the course that mistakes are our goal, since they are what we learn from. I don't agree. We want to have sufficiently challenging problems that we will have to make some mistakes along the way to solving them. We learn from our mistakes, but we don't focus on them. The goal is always to solve the problem.

3. I wish I had someone to give me some strategies that would have helped me to overcome my perfectionism during the process. In writing, the strategy that helped was turning off the screen so I could focus on saying what I wanted to say instead of saying it right the first time. Are there similar strategies that could work in math? How about starting with brainstorming, looking for clues in the wording of a problem that will help figure out what operations to use in solving it? Maybe coming up with two or three ways to try to solve it instead of just one would help a student not to get too hung up on getting it right the first time.

What do you think is the best way to help students get over a fear of making mistakes? Does it differ from child to child? Do you have any experiences that speak to this? I'd love to hear what your reactions were to this session.