Reflecting on Math

Structuring homework and adding reflection result in greater enjoyment of math

For Middle Level

I have tried many approaches to homework over the years. I have analyzed its purpose, thought about how long an assignment should be (in time and in number of problems), how to grade it, the types of problems to give, and how to deal with late work. My current view of homework is that it is valid when it gives students meaningful, relevant practice and when it is scored in a way that encourages students to solve problems creatively. Practice and problem-solving skills are necessary for student success on tests, in class, and in life.

Traditional homework scoring methodologies that call for only one correct answer or solving method create reluctance in some students to creatively problem-solve. These students often don't complete their work, or they copy from a classmate. Also, when homework heavily affects a student's grade, some cheat to make sure their grades don't falter.

Multiple possibilities
I want my students to know that they're free to try different ways to solve problems, and there is no "right way" to attack a math problem. Before the start of this year, I resolved to help them dig into their homework, give it their best try, and see what happens. I implemented three changes to my approach to homework:

  • A new homework grading scale: I awarded two points for a completed assignment, one point for an assignment that is not finished, and zero for an assignment that is not attempted. I aimed to encourage creative thinking and problem-solving, and to give them a reason to stick with the work until it was done.
  • Reflection after each homework assignment: I asked questions that encouraged them to think about how they went about doing their work, what was hard, and how much effort they put into the work. The questions guided students to identify the skills they needed help with. I aimed for a more independent learning process. Clearer about what they needed to learn, students would be better able to pick up the missing pieces by asking questions after correcting homework or by asking me to explain more problems.
  • Greater attention to teaching routines: we modeled and practiced every one of our homework routines: heading papers; turning in homework; grading a peer's paper; reflecting with depth (when we were ready to do this). No procedure was too small to model, practice, and remodel, especially considering the large-scale changes I was instituting. Clear behavior expectations would support the increased rigor and results I wanted in their homework.

A good start
Things started out great. I used the first few class sessions to lay out the change in homework grading, all the while modeling our homework routines. We corrected homework in class, and I solved problems on the board that students may have struggled with. These practices quickly brought about a positive change. Missing homework was way down in comparison to years in which I used a traditional grading method. I also felt that, with less pressure, students began enjoying math more than in the past.

What about reflection?
After completing ten homework assignments, we again reviewed and listed the important elements to include in a homework assignment. The list included a heading with date, name, and hour; indication of the page numbers and problems assigned, and showing one's work. Then I asked, "What about reflection?" This started a class debate in each period about whether it's necessary to formalize a process for reflection or whether it's something they do automatically, inside their heads, without prompting or need for structure. It was clear we were treading on new ground. I was excited to see what would happen when the homework reflection questions were, after modeling, fully in effect!

The following day, I covered the new material quickly and kept the assignment short to allow time to model how to use a reflection question to think about a completed homework assignment. I handed out a question on a half sheet of paper. Then I called on volunteers to help me fill out a sample response on the overhead. At the same time, students completed their reflections on their own homework. I explained to my students that, from that point forward, a written response to a reflection question would be the final element of a completed math assignment. From then on, reflection became a regular part of each homework assignment. I had now instituted my three changes: incorporating reflection, a new grading framework, and carefully modeled routines.

Assessment results
In November, I randomly selected two students from each class to interview about the reflection process. The interviews consisted of two questions :

  • Was the reflection process worthwhile for you?
  • Would you continue to reflect in writing on your homework even if it were not required?

Valuing yet resisting reflection
The interviews yielded surprising results. Every student I interviewed explained that reflecting on their homework had been a worthwhile process, yet most students said that, if given the choice, they would not reflect on their homework in writing. The majority said they preferred to do their reflections verbally or internally. My take on this: they see the value of it and will do it, but it's not ingrained yet, nor is it something into which they want to put a lot of time. To them, reflection should involve thinking, sometimes speaking, but not writing.

In November, after conducting the interviews and surveying all my students about their feelings about homework reflection, it was clear they had begun to see its value. When I compared the homework grades before and after I introduced reflection into our regular homework routine, I noticed every class showed overall improvement after introducing reflection.

Academic improvement
Adding reflection helped the students to both understand and retain important information, and increased their internal feelings of competence and self-efficacy. For example, I received a lot more questions about the previous day's assignment after instituting reflection, and I don't think the questions were evidence of misunderstanding the material. On the contrary, these questions were a sign that students were identifying problems on their own and were more confident in their ability to work through something they didn't know well and that they were becoming more resilient when confronted with a challenging new concept. Students would ask me whether they selected the right formula under the circumstances, how I came to a certain conclusion, and whether they reached the goals they set for themselves. These lively discussions kept me on my toes and were proof to me that reflection questions were moving them toward greater problem-solving comfort.

Dean Wanless teaches math to 8th graders at Edgerton Middle School in Edgerton WI.

This article first appeared in the Origins' publication Developmental Designs: A Middle Level Newsletter, Winter 2010