Stop whatever you’re doing and solve this equation:
How did you think about it?
Did you notice that the left side shows 2 times some number. If you noticed that, then noticed that it’s also easy to think about the right side as 2 times some number, you could think “Oh! x+3 has to be 7, so x has to be 4 to make this equation true.” Nice noticing!
What if I had asked it this way:
In the equation 2(x+3)=14, what is the value of x+3?
Duh, you might say. x+3 has to be 7 because 2 times 7 is 14.
I recently asked about 150 students enrolled in Algebra 1 at a community college the second question (In the equation 2(x+3)=14, what is the value of x+3?). I wanted in to see if they noticed that we could think of x+3 as one thing, as a quantity. 77% of the students applied the distributive property as their first step. About half of those students solved for x and were done. The other half plugged what they found for x into x+3 to report that value.
I’m more interested in strategy, but if you’re curious about how many students arrived at 7 for the value of x+3, it was only 43%. Of course, some ignored the x+3 part of the question and gave a solution of 4, but only 32%. This means that 25% of these students couldn’t solve the equation for either x or x+3.
Wow. This is serious, people.
Yes, these students are in a remedial class in community college, so I fully acknowledge their math struggles. These results might not be typical of all Algebra 1 classes. However, I recently taught workshop sessions to some students enrolled in the same Algebra 1 level course, and found the vast majority of them to be engaged, curious, and able to reason about numeric and algebraic expressions. Granted, we weren’t doing typical Algebra 1 stuff, and we only solved two equations in 4 hours. Even so, in the activities where it showed up, they still were triggered by the a(b+c) format to launch into the distributive property.
Here’s my takeaway: When do math students believe they’re allowed to think and reason and when do they think they’re required to follow a regimented prescription?
In an interview study with community college students enrolled in remedial classes, some colleagues and I found that students describe being good at math as remembering what to do*. Respondents told us that good math students have good memories, and their personal strategy to get better at math was to try to remember more. They rejected the idea that math was about thinking and reasoning. It’s funny that mathematicians would probably say the opposite, and are not all great remember-ers. They like that they are able to think and reason their way through problems and only need to remember a few things.
Last week I posed a question about the expressions 3(4+5) and 3(a+5). I encouraged my dear readers to think about the structure of these expressions, and how both show the product of 3 and a sum. This week I extend that idea to equations, and next week I’ll report on a problem about perimeter and the use of P=2l+2w versus P=2(l+w). Spoiler alert: Not a single student used the latter, even though the given perimeter was an even number.
My goal is to encourage us as math teachers to shift some class time from practicing equation solving techniques and talk to students about allowable algebra moves that help us make sense of expressions and equations.
This week’s Tweak: Use either of the above questions (straightforward solving for x or the sneaky x+3) as a quick formative assessment. Not for the students’ benefit, though. This is all about you. What do your students do? Do they think there’s anything else to do? After you see the results, have a little chat with students about ways to look at the equation in way that emphasizes the structure of the statement. Be brave and ask students this: “Did you know you algebra allows you to think about it a different way?”
Share your experiences here!
*Contact me if you’re interested in reading the study.