Suppose I give a group of sophomores this image and asked them to find the value of the angle marked “x”.
Consider for a moment what method that you would use to solve this problem. (x = 121, in case that helps.)
Now, suppose I asked you to write out your solution and to “show your work.” What do you suppose it would look like?
I was a little surprised to see what I saw from my tenth graders, which was a whole lot of long hand arithmetic. Like this…
One-in-three had a mistake, which, in the midst of grading about 90 started to become an entire class worth of young people who were making mistakes doing a process that seemed fairly easy to circumvent (and by tenth grade, seems fairly cheap and easy to circumvent without much consequence.)
So, I asked why they were so intent on doing longhand arithmetic. The responses were fairly consistent.
1. Our math teachers have asked us to show your work and that’s how you do it.
2. It’s easier than using a calculator.
I will admit I was not prepared for either answer. (In retrospect, I’m not sure what answer I was expecting.) When I was asking students why they resisted the calculators knowing that they lacked confidence with the longhand, they said multiple times that they could show me that they “really did the math” without demonstrating the longhand. Also, one girl wondered why I would be advocating for a method that, as she put it, “makes us think less.”
They knew that I expected them to provide proof of their answers. Most of them were perfectly willing to provide the proof.
This student is starting to suspect that proof means words. So, he used words to describe the process.
The conversation was pretty engaging to the students. A variety of students chimed in, most of them willing to defend longhand arithmetic and the only “true” work to show. I had shown them a variety of different looks at the longhand (the ones picture here, among others… including some mistakes to illustrate the risk, as I see it.) Then I asked this question which quieted things down quite quickly:
“Okay… okay… you proved to me that you did the subtraction right. I’ll give you that. Which of them proved that subtracting that 146 from 180 is correct thing to do?”
At first, they weren’t sure what to do with that. Although, quickly enough they were willing to agree that none of the work got into explaining why 180-146 was chosen over, say 155+146 = x or something.
I tried to convey that by tenth grade, I’m really not looking for proof that students can do three-digit subtraction. I would very much prefer discussing why that is the correct operation. They didn’t seem prepared to hear this answer. Apparently we’re even…
To be fair, there was one example where a bit of the bigger picture made it into the work. Check it out:
I learned a lot today. I feel like I got a window into the students who are coming to see me. I ask them to explain, to prove, to show their work. Many of them willingly oblige, they just see an effective mathematics explanation differently than I do. It might be time to help the students get a vision of what explaining the solution to a math problem really looks like.
I would very much like your thoughts on this.