This year I have had quite a few students who are pushing themselves mathematically and doing a lot of great thinking. This also equates to making me think as well, and I really need that. Now a student has me thinking about PEMDAS, how it is taught, and how it is a coveted norm of mathematics.
One thing I have always struggled with when talking with students about PEMDAS is how to make it meaningful for students without pulling out the “This is the way it is” card. I don’t want it to become a memorization practice only, I really want students to understand why we do operations in a particular order.
One thing I have used and students understand (not surprising because I adopted this because of a student) is that multiplication implies “groups of.” I’m sure many pure mathematicians might frown upon this, or the fact that it also leads to repeated addition, but students really grasp the concept and start visualizing order of operations.
3 groups of 4 plus 2
When students think of it this way, they don’t want to add 4 and 2 and then multiply that by 3. This is also true when they see 2+3×4, the “groups of” thinking prevents them from adding 2 and 3. Is this just replacing a trick with a trick? It took me a few years to even accept this type of thinking for my students- I don’t want to provide a new crutch for students.
This all leads me to the other day, when a student was solving the equation 3X + 2 = 8. He was unsure of what to do, so I asked him what step he would try. He repled: “I would divide by 3 since I want to solve for X.” I immediately balked, that answer went against all of my mathematical fiber, but I tried to not show it visibly. I told him to go ahead and try his idea. This is what was produced:
Wait now! Hold on, that is the answer I expected, but not attained in the way I assumed it would be. I can’t tell you how many times I have used the last in- first out type of thinking with students to solve equations, and I expected seeing something like this:
But all of these provide the same solution! Is that a fluke? Some weird oddity that can’t be reproduced? It had me thinking of possible counter-examples and what else it could imply. Does order of operation last in-first out thinking absolute in solving equations? Or is it something like the standard addition algorithm that we accept as the best method and ignore others? My student gave me another example to consider later that hour.
This problem involved the area of a trapezoid. They wanted him to algebraically solve it for b2. I was very curious what he was going to do, and instead of starting with the parenthesis, so subtracting b1 (which is what I assumed he would do because of his previous reply), he told me he needed to distribute before solving. His work was this:
Which again produced the correct solution after he cleaned up the compound fraction. Once again his work produced the right answer without doing the expected procedure:
So currently I am rethinking approaches to solving equations. How will not using the standard approach effect him later on? I can see how trying some sort of variation on this with powers will be messy. I’m totally open to suggestions and comments on this, as it has my brain working on overload ATM.