# Reflecting on Math

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Structuring homework and adding reflection result in greater enjoyment of math

Structuring homework and adding reflection result in greater enjoyment of math

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

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