Yesterday I was putting together a few math projects when a Tweet caught my eye. The Tweet below started a short conversation that I thought was interesting.
From which do you think more is learned? Solving ten similar problems or solving one problem ten different ways. #mathchat
— David Wees (@davidwees) July 7, 2014
David’s Tweet had many responses. Most responses revealed that educators tend to side with solving one problem ten different ways rather than having students solve ten similar problems. I started to reflect on how teachers give assignments that ask students to complete repetitive problems that often reinforce procedural mathematical thinking. I also started to think how in an effort to provide practice, teachers may focus on procedural aspects first and then move towards practical application. I find this happens frequently with math concepts at the elementary level. What I don’t find often is the viewpoint that practicing procedural aspects can be embedded in solving specific problems multiple ways. This type of thinking reminds me of number collection boxes.
Regardless of the assignment I want to be able to give specific feedback. A larger problem that involves multiple steps can provide opportunities for teachers to pinpoint where misconceptions are and give direct feedback. This isn’t always possible with ten similar shorter problems. Below is an example of a few problems that you may find in a fifth grade classroom. I don’t condone using these types of problems as they are definitely utlized, but I think we need to ask what’s being assessed when students complete this type of problem? Students are simply asked to find the volume and show a number model. I appreciate how the problems ask students to show their number model, but these types of problems seem to measure procedural understanding. Do students know the formula? Yes, well then they can answer many of these problems, even 10 in a row.
I think the above problems have a place in the classroom, but shouldn’t necessarily be the norm. Usually these types of problems are found on homework sheets. The problem below which was adapted from a recent fifth grade test is more challenging, but gives students opportunities to showcase their own mathematical understanding and persevere. Some would say that these two problems are completely different. I would agree, but similar concepts are being assessed. They do look different and the second requires more skills to complete. Students need to be able to use their procedural understanding and apply it to the situation. Also, one key element that’s missing from the first problem is the student explanation. Students are required to show their mathematical thinking in the second problem. This is big shift and can reveal student misconceptions more clearly than the first problem. I struggled with the decision, but eventually had students work in groups to complete the problem below. Students were allowed to use any of the tools in the classroom to find a solution.
At first, all groups struggled with this problem. Near the end of class all the groups presented their findings. What’s interesting is that all the groups had different answers and ways in which they came to their conclusions. I was able to offer opportunities for students to see and ask questions about different math strategies. During the next class I was able to pull each group and give feedback. This activity took a good amount of time to complete, but I feel like it was worth the commitment.
Through this experience and others I’m continuing to find that it takes a “bridge” to connect the procedural and application pieces. At times I feel like there’s an assumption that if students are able to answer 10 similar procedural problems that they will be able to simply apply that knowledge in a multi-step problem. This isn’t always the case and sometimes the bridge doesn’t fully form immediately. Performance tasks, similar to the problem above can be one way in which teachers can help the transition from procedural understanding to practical application. Being able to apply that knowledge to a math performance task can be a challenge for some students. When teachers focus so much on the procedural, that’s the only context that students see and practice. A blend between procedural and application needs to be established within the classroom. I feel like activities like this help bridge this gap.
How do you bridge mechanical and conceptual understanding?