My wife and I were talking mathematics the other day. She wanted to know why teachers were getting beat up over the teaching of mathematics. In Ontario, it boils down to one thing – EQAO scores. This is big media news as schools are compared to other schools; school districts compared to each other. And, teachers take the brunt of the scrutiny. Despite trite phrases like “Not the same way, not the same day”, students are expected to reach a certain level in grades 3, 6, 9 across the province on the same week, on the same test.
I reflected back to my own journey with mathematics. I have a degree (BMath) focussing on Combinatorics and Optimization and Computer Science. I love mathematics; I love problem solving; I love visualizing things. I’m the guy that goes to a restaurant and mentally determines the best value on the menu. I convert Celsius to Fahrenheit and kilometres to miles just for the heck of it.
But it wasn’t always that way. I distinctly remember in elementary school breaking down in tears over some of the mind numbing things that we had to do. One specific thing that sticks out was a multiplication table. In class, we worked the grid to go as far as 10×10. For homework, we had to extend it to 12×12. I spent a great deal of frustrating pain doing it and never got it. When we got to school the next day, we were expected to lay open our notebooks and we were checked to see how many got it right. Those of us who didn’t got additional pain by having to go to the blackboard and have our stupidity explained to us. For years, it seems, I just couldn’t do anything right when it came to mathematics. True story.
Then, it changed. My ship had come in, it seems. I think it was grade 6 or grade 7, I had a complete turnaround. I don’t recall whether it was explicit in the teaching or just an insight on my part. I became a mathematics genius. (Or, at least not the dummy that I was…) Calculations became just patterns; Geometry became puzzles; somehow I finally got mathematics. From that point on, every mathematics question became just another puzzle to be solved. I remember in Grade 10 how calculations became just measurements. We all had to learn how to use a slide rule for calculations. I remember one of the kids in the class had a $500 calculator (which today would be powerfully contained on a keychain). Our teacher impressed upon us the value of our tool and we were able to do calculations quicker than this calculator. Of course, this technology has been solidly dated but it was cool at the time.
I remember at university a professor talking about how important it was to appreciate those who loved mathematics. He used this example…
What do you do if you’re good at football? Practice, Practice, Practice
What do you do if you’re good at a musical instrument? Practice, Practice, Practice
What do you do if you’re good at baseball? Practice, Practice, Practice
What do you do if you’re good at mathematics? Do the odd numbered questions on page 37 and then go outside and practice baseball
Think about it for a bit.
It’s how so many people think and feel about mathematics. It’s solely about getting the right answer. It’s all or nothing.
As a student teacher, I remember my first mathematics placement. It was important to give a lot of homework every night and the logic was that if you did enough of the same problem, you’ll commit it to memory. I supposed it did commit an algorithm to memory.
On my second placement, I had a little more experience under my belt and I remember the first day going around the class asking if the students enjoyed Mathematics. The answer was surprisingly “yes”. When I asked why, the answer was “Because Mr. C. doesn’t give homework”. Interesting, I thought. I’ve got to see the end of this class. Well, it turned out that homework was assigned. But, it wasn’t doing the odd numbered questions… The instructions were simple – take one of the problems that we did in class today and redo it differently. The next day was interesting. Taking up homework was more of a discussion rather than doing more problems. These kids were actually talking mathematics. I had to learn more.
During our preparation period, I asked Mr. C. about it. He had an interesting philosophy. His logic was that kids hated mathematics because they were assigned to do problems that they were having challenges with in class. What do they do when they get home? They didn’t have him to ask for help. By reassigning a problem that they knew the answer to, the goal wasn’t to just solve another problem. It was to think deeply about something that they already knew.
That moment made me completely think about what I thought I knew about homework and it stuck with me forever.
Obviously, that particular technique doesn’t fit every day but it does make one reflect on the value of homework. Do you want to amplify a problem? Is the goal of mathematics just to get the right answer?
So, on to the scene recently we have this new application “PhotoMath“. Essentially, the app lets the camera take a picture of a problem and it solves it for you, including “showing your work”. How many times have you heard that in your mathematics life?!
Brian Aspinall and his students kicked the tires on the application and this generated a post from him here.
There have been various reports of people having success with the application and others delighting in finding a problem with its operation. I would suggest to anyone that it’s new. It can only get better.
Fast forward and imagine a mathematics classroom with this application in place. In the beginning, there will always be the question “Is this program doing it right?” What does it mean when it does? There are those already touting doom and gloom for teachers and the teaching of mathematics. Take a look at some of the CNN quotes that Brian pulled for his blog post.
There was a time when we forbid calculators in the classroom because it stole from the learning of mathematics. Then, we accepted them but banned graphing calculators because it was stealing the need to learn to graph. Now, they’re accepted and the discussion becomes not one of the value of them, but which of the ones on the market is best!
In the process, we haven’t killed the process of doing calculations; we’ve made it better. We haven’t killed graphing; we’ve made it better. I would suggest that PhotoMath or any of the applications bound to follow, understood and used properly, could make the problem solving process better.
Who wouldn’t want their own personal tutor at home? Who wouldn’t like relief from the dull activity of applying the same algorithm over and over until you get it? What teacher wouldn’t want students who think deeply and talk the story of mathematics instead of just doing the same problem over and over again.
This has real potential.