We're Off!

I devised my question:

"How can I improve my students' ability to effectively use problem solving strategies in a math journal format?"

For two weeks I modeled simple problems which we solved together, stressing the need to include pictures, numbers and words. Near the end of September I presented a very simple problem. The students' responses provided the baseline with which to compare subsequent work. More importantly, this baseline of problem-solving ability will help me to determine areas of strength and weakness for each child.

Megan was able to solve the problem using pictures, numbers and words independently. Shaina needed assistance to write the concluding sentence. Spencer, however, needed constant help to work through the steps. He sat staring at a blank page until help arrived. As a class, only five children achieved a Level 3; nine students failed to complete the problem.

During the patterning unit the students were encouraged to explain how they solved patterning problems in their journals. We had spent several days modeling how to continue patterns, create patterns and explain the patterns in writing. One third of the students were able to explain their patterns independently. It was a start, but we had a long way to go.

My next step was to present a problem that required the children to develop a strategy to find all the answers to a problem. I hoped they would use the commutative property of addition, a concept that I had previously taught. Only one student was able to apply the strategy independently. Afterwards Megan explained to the whole class why the strategy had worked. When I posed a similar problem a few days later, everyone immediately solved it using the commutative property. Although the students recognized the need to use pictures, numbers and words, the explanation section in their journals was the weakest link in the problem solving process.

We Round the Bend

They were ready, though, to try more complex problems. I introduced literature as the motivator. We read In the Henhouse together. The main character has hens that lay 2, 3, 4, 5 and 6 eggs each day. I posed the question, how many chicks will she have? I have included samples of their work.

I observed that everyone was immediately engaged. Each student seemed to understand the expectations. The students have developed some problem-solving strategies on which they can rely. Megan completed the task independently. Shaina and Spencer covered all areas, but needed assistance to communicate clearly. All but two students in the class wrote explanations.

One Gorilla is a beautiful picture book with a format similar to In the Henhouse. The story is more complex, involving the numbers 1 to 10. After reading the story and listing all the animals in the book, I asked them, how many animals does she love? The children returned to tables equipped with manipulatives and calculators and solved the problem.

The journals reveal that Megan thinks in a creative problem- solving way.

She considered multiple, or blended strategies to address the problem. She demonstrated the appropriateness of the strategy and explained her process clearly and completely. A few other students almost matched Megan in the thoroughness of their answers. The majority of the class did very well in the planning and implementing stages, but still needed help communicating their ideas.

In December and January I took a different approach. Each student completed a window frame, answering the question, "What do you know about money?" The answers were vague and stressed the functions of money. At the end of the unit I asked the question, "What did you learn about money?" My expectations were explicit. I wanted to read about three new things they had learned and I wanted to see the math terms we had learned in their answer.

Overall, I was pleased with the results. The following comments represent an overview of their responses.

Megan

Before: I know what money pays: bills. I know what I do with money- I buy things. I know who gives me lots of money, my mama and papa. I know where we get money, at the bank--200 pennies is 2 dollars. 4 quarters is a dollar.

After: 5 cents is a nickel, 10 cents is a dime, 1 cent is a penny, 25 cents is a quarter, 50 cents is a half dollar, 100 cents is a dollar.
I know what money is like, it's like adding, or you could subtract if it's too much or less.

Megan's second response demonstrates a thorough understanding of money values. She also explains the relationship of money to other math concepts.

Shaina

Before: You find money at the bank. You spend it and save it. You get it in your allowance. You spend your money at Snack Shack.

After: I know that money is real. I know that you can make a penny with 1 cent. I know that you can make a dollar with money. We worked with money. I can count money. I know that you can use money.

Spencer

Before: I know that money can buy groceries. My mom gets money from the bank. On my birthday I get an allowance. 5 cents makes 5 pennies.

After: I learned that a penny is 1 cent. I learned that a nickel is 5 cents. I learned that a dime is 10 cents. I learned that a quarter is 25 cents.

Both Spencer and Shaina offer explanations that involve some mathematical concepts. Spencer's response is more detailed than Shaina's, but both students are listing values without associating them to other mathematical concepts.

Most students in the class demonstrated confidence in their understanding of money values. I was surprised that no one mentioned the strategies we learned when counting money. I observed that most of the students needed to think and speak in mathematical language.

I wanted to further develop the students' use of math terminology. During our study of 3-D solids the students were required to explain whether they were able to build certain structures with geometric solids. They were directed to give reasons for their answers. They were also directed to use the math terminology found on the math word wall. The activity was successful; most students achieved a level 3 or higher. The lowest level of achievement was 2.

My next challenge was to provide opportunities to solve open-ended problems. "Bits and Bites" engaged them from the beginning.

Bites and Bites

You will need some of each ingredient to make a bag of bits and bites. There is a cost to each ingredient. The total cost of your bag must not be more than 90 cents. How much of each ingredient will you use? How will you keep track of how much you are spending?

Ingredients

• Raisins -- 1 cent each
• 1 spoonful of Cheerios -- 10 cents
• pretzels -- 5 cents each
• 1 Smartie -- 2 cents each

Students needed to sort each snack according to its value, determine the total value of each kind of snack, and then add these totals to arrive at the final sum. Over half the class independently completed the several steps necessary to solve the problem. Many children took their cues from stronger problem solvers. The students met the requirements, but once again, communication of ideas presented the greatest difficulty.

I like to revisit a problem to consolidate their understanding and give the students an opportunity to build on their knowledge. "Valentine Treats" follows the same format, but the values correspond to our multiplication unit. We had to count by 2, 3, 4, 5 and 10. Manipulatives, a counting chart and calculators were readily available. The students worked with greater confidence. They skillfully applied the strategies they had learned, but seemed to stumble when they had to explain their strategies in writing.

The Home Stretch

During the third term our problem-solving activities consolidated the lessons previously taught. We used versions of the Grade One Math Exemplars to compare their levels. I was quite pleased with the results; even the academically challenged students achieved Level 2 with relative ease.

Their final problem formed the culminating task of their major data management and probability unit.

You have a cup of Lucky Charms cereal. You need to find out how many of each kind of marshmallow treat you have. Graph the results and explain how you arrived at your answer.

Each student sorted the candy, displayed their data on a tally chart they created, and then transferred this data to a graph. Using squares of graph chart paper they planned their own graph and explained their procedure on the back.

After marking this activity I created my own tally chart to examine and compare the classroom results.

Level 1- 4 students
Level 2- 5 students
Level 3- 10 students
Level 4- 5 students

The results were a relief! Every student completed the assignment within the standards set for the Grade Two data management and problem-solving curriculum. I was elated to observe that every student followed procedures and used adequate strategies. Fifteen children achieved Levels 3 and 4. No one provided insufficient work. What a dramatic change from September! Then I had twenty-six timid, math-fearing students. Now I have a classroom full of confident, enthusiastic problem-solvers!

Reflections on Our Journey

I reflect on the progress we have made as a class this year. In September the students learned to use the strategy of pictures, numbers and words to solve one- step problems.

From this humble beginning, each student learned to search for multiple answers to a problem. They learned to solve open-ended problems with confidence. Finally, by the third term, they tackled problems from different math strands with ease. Several students provided rich and detailed explanations.

My students have traveled far this year, but I have traveled even farther. I learned to focus on each student's abilities. Every child can learn to organize, implement strategies and solve problems at their own level. When I set the standard high, I discovered that my students jumped to reach it. It was my task to provide the tools, and opportunities to improve their capabilities. We started by taking baby steps, but we were flying by the end of the year.

Mathematics is learned and applied more readily in integrated situations. Rich literature, math journals and window frames are excellent strategies to employ to integrate mathematics into other areas of learning. Students became engaged in an activity that had relevance to them.

I have also discovered that we, as a class, have moved beyond my original question. We began using journals as our only framework, but now, as capable problem-solvers, we can use many kinds of frameworks and apply these strategies to other subject areas, like science.

I was humbled and sometimes overwhelmed by the growth I have witnessed in my students. They accepted my learning challenge and raced ahead with such enthusiasm that I was often racing to keep up.

Solving problems with math journals has revitalized my passion for mathematics. I hope to build on my learning experiences this year to provide next year's students with an exciting problem-solving journey.

Bibliography

Barkins, Lori, et al. Problem Solving. Brantford: Grand Erie District School Board, 2002.

Burns, Marilyn. About Teaching Mathematics: A K-8 Resource. Sausalito: Math Solutions Publications, 2000.

Burns, Marilyn. Writing in Math Class. Sausalito: Math Solutions Publications, 1995.

Krpan, Cathy. The Write Math: Writing about Math in the Classroom. Toronto: University of Toronto Press, 2001.

Morozumi, Atsuko. One Gorilla. Hong Kong: Fairar, Straus and Giroux, 1990.

Nicol, Lynn. In the Henhouse. Toronto: Pearson Education Canada, 2002.