This topic is something I meant to write about in January, but it got lost in the onslaught of third quarter. Anyway, summer is the perfect time to read through my reflection journal (maybe I’ll 180 blog next year?) and write something coherent.
Before winter break, my honors class was having a rough time writing expressions and equations. The two skills are highly similar, which created problems with over simplifying expressions to get a single number and attempting to avoid showing work when solving equations. None of these misconceptions are out of the ordinary, but the mistakes indicated a preference for procedural thinking over conceptual thinking. I wanted my students to see the power of expressions and equations to represent situations. Essentially, I wanted my students to build solid conceptual of expressions and equations to avoid thinking these topics are of no use to solving problems.
When we returned well rested and ready to go in January, I spent a couple days reviewing the basics of expressions and equations. I emphasized the importance of variables and constants in allowing people to represent situations perfectly without any words. To prove my point, I had everyone in class complete the following number trick I found on Dan Meyer’s blog:
Everybody pick a number.
Multiply it by four.
Divide by two.
Divide by two again.
Now subtract your original number.
On the count of three, everybody say the number you have.
After the inevitable surprise that everyone got the same number (0), we tried the trick two more times. With the same results, I asked students why the trick worked. A student shared an idea involving dividing by 2 twice will “reverse” multiplying by 4, then a few others tried to explain without too much success.
I sneakily said after a minute of dead silence in the room, “It looks like words are making this more complicated. What if we wrote an expression?… What would be the variable?”
With some coaxing from me, we eventually chose p as our variable to mean the number we choose at the start of the trick. I said, “Okay, so let’s do everything in the trick with the variable instead of a number.” Armed with a couple days of review, we were able to get the 4p+2 without any difficulty. When we encountered the divide by 2, I asked students how we could do this to our expression. I eventually gave the hint of the distributive property, then we proceeded to work through the remainder of the trick.
At the end, I paused after writing n – n and 0. Some students seemed a little silent. I could see some students working thinking through the steps they saw on the board and other students a little weirded out at what just happened. I emphasized that the power of what we just did was the fact that we just proved the trick will always work for integers! We represented everything in the problem with math and we proved with symbols what we struggled to explain with words. I asked if people were confused, but no one confirmed. I asked if anything we did was something they did not understand. No one confirmed. I asked who felt they understood what we did completely. Most students raised their hands.
I felt a bit elated as we ended the day on this positive note, but I was unprepared for what happened the next day of class. As I walked into class the following day, a couple kids came up to me and said they found other number tricks they wanted to use. I remember in particular that one girl said, “I was on Instagram last night looking for number tricks and I wrote this one down. Can we prove it?”
I loved this moment for multiple reasons. First, I thought a lot students might have been bored with the number trick from the previous day or maybe thought I tricked them into using expressions. Apparently not! Second, I liked that students were finding number tricks without prompting from me and wanted to prove them (not just use them to kill time). I know my use of the word prove might be a little far reaching here, but it’s really not so far removed from number theory when you think about it. Either way, my class proceeded to prove multiple number tricks in the days that followed and I could tell many students finally understood the power and meaning of expressions and variable.