Arbitrary and Necessary Lessons in the Classroom (accidentally published in wrong blog)

The ideas in this article kind of resonate with some of my beliefs about math curriculum and teaching. For example, sometimes I ask myself what is necessary for students to learn about the different forms of linear equations. In one of our pre-practicum Math Assessment classes we took a test that tested on the different forms of an equation. I had to review the content with a peer because it had been a while since I had used these concepts, beside slope-intercept form and a bit of point-slope form. Now I imagine teaching these conventions in a class and I find it difficult to imagine having the time for students to discover the algorithms on their own. I suppose I could have an activity that guides them to this discovery; maybe do something interactive in my presentation to them so that it becomes more intuitive. In this particular example, I think something like Desmos would be quite useful. 

The other example I kept on thinking about is from a Math 9 class I observed over practicum where the students were learning how to solve linear equations and the teacher teaching the class "tricks". The one that I keep think about is cross multiplication. The teacher seemed to prefer cross multiplication to get rid of fractions over other methods, but I do not think the students understood at all what it was doing to the equation. I think when I learnt about solving equations we were not allowed to use cross multiplication unless we were able to tell the teacher what was happening. After they learnt about cross multiplication they started to incorrectly apply to equations that didn't even have fractions.

One thing that I observe in my practicum and I plan to incorporate into my lessons and unit plans is time for students to work through problems on their own and figure out the concepts. I found that this work time was where students learnt the most and I think it makes sense. Math takes inquiry and exploration to understand. This time gives students the opportunity to actually understand the necessary, whether it was given to them to memorize or if the work is a time for exploration for the students.