Written By: Lori Jane Dowell Hantelmann, Elementary Math Specialist
Regina Public Schools, SK Canada

My teaching philosophy has changed significantly since I began teaching. My students sit at tables and mathematical concepts are taught in a problem-solving context instead of being presented in isolation through discrete workbook pages. One change in my teaching is due to the classroom research into mathematics journals I completed recently for my Masters of Education. My students became active participants in their own learning as they confidently solved realistic problems and explained their ideas in mathematics journals, and I was excited to be part of this rewarding learning environment.

In past years, when attempting to have children write in a mathematics journal, I would read: "This was easy. I like math." My students were not able to successfully reflect or share what they understood about problem solving or mathematics through their writing. Frustrated, I began to read about using math journals in the classroom.

What is a math journal?

My students write in a notebook to answer open-ended questions using numbers, symbols, pictures, and words, and their writing can best be described as written conversations. A math journal is a place where every student has the opportunity to verbalize their math knowledge to their teacher, internally to themselves, and to their classmates. Students' writing becomes a source for social interaction as they read journal entries to partners and the whole class, talk about their learning and listen to others share different levels of mathematical reasoning.

What did I learn about using math journals?

My classroom research on mathematics journals led me to recognize four important steps needed to help students write reflectively about mathematics.

• Teachers need to model the writing process and the language of mathematics. First I modeled my own problem-solving process by thinking out loud as I solved problems and as I recorded my reflections on chart paper. Students soon began to contribute their own ideas about problem solving but continued to model the writing process by recording their comments on chart paper. Students copied these sentences into their math journals. Modeling the writing process took longer than I expected as students needed to become familiar with reflective writing and the language of mathematics, the words and symbols unique to mathematics. Once students became familiar with the vocabulary necessary to communicate in mathematics they began to independently express their own thoughts on paper.

• Teachers need to ask open-ended questions to guide students in their writing. I learned how to ask open-ended questions to help students think about their own understanding of problem solving and to guide their writing. I began my research by using a list of questions I found in Writing to Learn Mathematics, by J. Countryman (1992). Students answered my questions verbally at first and became comfortable sharing their thoughts and ideas with others. It was through their participation in our verbal discussions that students learned how to reflect upon their own knowledge of mathematics and to record their ideas on paper. I soon adapted the questions I found to better meet the needs of my students and to match the problems we were solving. Here are examples of questions I used in my research:
  1. Why was this problem easy?
  2. Would this problem be easier today than yesterday? Why?
  3. What did you do to solve this problem?
  4. Are numbers important in solving this problem? Why?
  5. Did graphs help you to solve the problem? Why?

• Students need to revisit similar tasks to increase their confidence as problem solvers and their knowledge of problem solving. As a teacher of young children, I quickly realized that involving students in rewriting similar problem-solving tasks to the problems they just solved was important in developing their confidence as problem solvers and in understanding the process. Children could not always solve the task independently the first time and were enthusiastic to help rewrite the task and solve it again. I noticed they were more successful in solving the second task.

• For example, we solved the Exemplars task "Space Creatures." On a new planet the astronauts discovered unusual creatures. The features they counted were 15 eyes, 3 horns, 12 legs and 7 arms. More of the creatures had scales than fur on their bodies. Draw your creatures and make a graph for each creature's features.

• The next week, we revisited a similar problem by writing our own version of "Space Creatures" called "Super Robots." Students were visiting a robot factory. They saw and counted 13 eyes, 10 legs, 8 knobs, 9 arms and 4 antennas. More bodies were triangles than rectangles. Draw a graph first for each of your robots and then draw a picture of your robots to match your graphs.

• Teachers need to support students in recognizing their individual problem-solving styles. Students discovered they were different in the strategies they used to solve problems. I guided students to solve "Space Creatures" by drawing pictures of the creatures before making their graphs. For "Super Robots," I asked students to make their graphs before drawing their pictures.

• Some of the students became frustrated when I asked them to begin their problem-solving task with a graph. I was curious to know why, so my first journal question asked students: "Is it easier to draw a picture first or draw a graph first?" Twelve children chose to draw their picture first and 11 children chose to create their graph first. Surprised with the split in their choices, I asked students to explain: "Why do you think it was easier to draw the graph or picture first?" The children wrote:

The picture was easier because you could count the objects better than the graph. (mathematics journal, Oct. 23, June)

It's easier to draw the graph first. Yes, because I knew what my Robot would look like. The graph helped me to count the features. (mathematics journal, Oct. 23, Cassy)

Their responses led me to realize the importance of drawing pictures in the problem-solving process for some children, but not all children. I needed to listen carefully to students so I would know how to best support them in recognizing their individual needs as problem solvers.

For four months I had supported students in becoming problem solvers and reflective writers in mathematics. In the end I questioned the students to see how they had changed since September. Students had become confident and independent writers, and they understood what it meant to be a problem solver. Two students wrote:

I can think of more answers. (mathematics journal, Dec. 3, Mitchel)

I've learned how to do harder problems. (mathematics journal, Dec. 3, Sam)

I now believe writing belongs in mathematics and is as important in developing students' mathematical knowledge as numbers and computation. It was my student-to-teacher interactions and my open-ended questions that guided students to write reflectively. I will always have a classroom of diverse learners and I now feel confident I can meet individual needs of students and lead them in their learning. Mathematics journals will guide my teaching.

References

Brewer, R. (1994-98). Exemplars. Underhill, Vermont: Exemplars.
Countryman, J. (1992). Writing to Learn Mathematics. Portsmouth, NH: Heinemann Education Books, Inc.