BUILDING CONFIDENCE THROUGH MATH PROBLEM-SOLVING
Linda Miller
My class of grade four students was not empowered. Their lack of confidence, when approaching assignments in any academic area, was very evident. They constantly asked for clarification as to the accuracy of their work or for assistance. They needed me by their side all the time. I wanted my students to become confident, capable, independent learners. I particularly wanted them to enjoy mathematics, to discuss mathematics using mathematical language, and to be able to explain what they were doing. This became the focus of my investigation.
I was taking my Mathematics Specialist Part III qualification at Brock University and became interested in a problem solving approach to Mathematics. The Ontario Curriculum Grades 1-8: Mathematics (1997) recognized that:
students’ attitudes have a significant effect on how they approach problem solving and how well they succeed in mathematics. Students need to understand that for some mathematics problems there may be several ways to get the correct answer… Teachers can help students develop the confidence they need by modeling positive attitudes in their approach to problems . . . By helping students to understand this, teachers can encourage them to develop the willingness to persist, to explore, and to take risks that they need to become successful problem solvers. (p.73)
I found two articles particularly helpful in clarifying my thinking about problem solving and the development of self confidence. The first was "Promoting a problem-posing classroom," (1997), by Lyn English. English wrote that one of the main strengths of a problem-posing learning environment is "…that it can empower children to explore problem situations. This atmosphere creates a context for learning that is more productive and enjoyable for mathematics learning." English went on the write that, "…problem-posing can encourage children to take greater responsibility for their learning and dissipate common fears and anxieties about mathematics learning."(p. 173)
The second article that helped me was "Enriching matchematical creativity in the elementary classroom," by McDougall and Kajander (1997). They stated "…that the exploratory process is important to learning to develop concrete levels of understanding." (p.5) Teachers need to give students appropriate time, tasks, and encouragement to think deeply and to talk about mathematical ideas. Children need manipulatives. They need time to explore and develop their problem solving abilities. Games are a motivating resource. Journal writing allows children to express the process they have used as they explored and to share their thinking processes.
Based on my reading, my question became, "If I teach a series of problem solving strategies to the students and allow time to practise each strategy, and if I provide lots of problem solving activities using manipulatives, real-life contexts and patterned activities, would I increase my students’ confidence, capabilities, and independence in mathematics and as overall learners ?
Conducting the Study
Over the next four months, I introduced the problem solving strategies one at a time. Every day I gave students a problem of the day. They had little booklets, which were novel to them, where they did some of their problems. They started using a math journal but writing and recording their thinking processes were difficult. After a colleague’s workshop, I gave them a sheet divided in half where they completed calculations on one side and explanations on the other and I believe that this helped them organize their thinking. However, this format has not been used enough yet to determine its success. Many of the activities they did for their numeration studies were game- oriented or presented in the form of patterning and algebraic format. I also gave the children lots of praise when they were successful. I encouraged them to share the work that they were doing with the principal. Samples of students’ work were shared and used as models to encourage and motivate the others.
As I conducted the study, I collected data about student achievement in mathematics in a number of ways. I kept the usual records of achievement on tests and assignments. I collected and compared samples of students’ work at the beginning, in the middle, and at the end of the project, and I observed how the level of support they required from me to complete work changed over time. In addition, I conducted a survey of student attitudes toward mathematics. I continued to meet weekly with my critical friends in the Mathematics Specialist course and talked through my experiences and data with them.
Findings
When I first started focussing on problem solving activities, I had to walk the children through every step of the process, even after I had taught them the introductory lesson on the particular problem solving teaching method. They definitely needed my constant input. Gradually, the class became split between those who could work on their own and those who still needed my constant, "Yes, you are doing it right," statements. I had to keep reminding them that they were capable and that they could do this. One day in December, though, I realized that I had just given them an assignment and no one had whined that they could not do it, nor did it appear that I was needed while they worked. It was as if we "had arrived". I remember that day vividly. It felt so good. When the period was over, I told them what had just happened. They were very proud of themselves. Then came the day when Philip came to me after math period and said, "You know, Mrs. Miller, I didn’t use to like math, but I like math now." I knew I was getting somewhere with them.
Early in January, another change was noted when I gave the students the following problem to solve:
An owner of a store wanted to decorate the front window with pictures of snowmen. Her design was finished when she decided to add colour to the snowmen. She wanted to use four colours: one colour each for the hat, head, middle section and bottom section of the snowmen. In how many different ways can the snowmen be coloured for the window display?
When I asked students to complete the activity, I was happy to see that the majority of them buckled down to work with great confidence that they could do it. Most, however, were not attacking the problem in a way that would bring about a solution. On the other hand, one very quiet student, named Jill, had made an organized chart and was working on the problem in such a way that she would actually find the answer. It was so exciting and rewarding to see her do this. I shared her strategy with the other students and they also ordered their thinking and come up with solutions. Jill’s confidence in her ability in Mathematics certainly rose as a result of this experience and that new-found confidence is now evident in all aspects of her work.
One week, I challenged students to design an Olympic Village. It was really an introductory lesson to pentominoes. Students were told that they had been hired to design a building for all of the Canadian athletes to live in while they were competing in the Winter Olympics. The building had to be made of 5 modules. A module was a cube. The modules had to be all the same size and had to be all on one level (no two-story buildings). Every edge of the modules was to line up with another edge. I was happy to see that everyone was able to do this task, after the initial introductory lesson using centicubes and centimetre graph paper.
Two days later, as students were completing their work on the Olympic Village, I told them that this question was from one of last year’s province wide tests. One of the students spoke out and said, "Yeah, but this was a lot easier than those questions."
I said, "Really? Well then, you’ll be surprised to hear that this was a question from the grade 6 province wide test."
Another student then said, "Wow, grade fours doing grade six work!" They all felt ten feet tall.
Then the first student blurted out, "But it was easier because you made us believe in ourselves and believe that math is easy."
Questionnaire Results
Near the end of the project, I gave the children a questionnaire on their feelings about mathematics. I asked them to tell me what math is to them. All but one of the students said that they had changed as math learners because now they liked math whereas they did not at the beginning of the year. Of the eighteen students who responded, ten said, "Math is fun." Seven said that it was numbers and all that came with number work. Other responses included, "Math is cool because it helps you to learn a lot of things," and, "Math is a subject that teaches you neat things to do with numbers." Perhaps the most thoughtful response came from one young lady who wrote, "Math is a learning experience that helps us learn or become learners. It helps us understand better and to have knowledge. I feel math is something you need in life, and it makes you smart and intelligent."
I also asked students to rank from one to five, with one being their favourite, the strands of mathematics identified in the Ontario Curriculum, Grades 1-8: Mathematics (1997). Here are the results:
| 1st. | 2nd. | 3rd. | 4th. | 5th. | 6th. |
|---|---|---|---|---|---|
numeration |
4 | 3 | 3 | 6 | 2 |
| geometry | 9 | 2 | 4 | 2 | 1 |
| measurement | 0 | 0 | 2 | 5 | 9 |
| pattern/algebra | 0 | 4 | 8 | 3 | 3 |
| data management | 4 | 10 | 1 | 2 | 1 |
I found the results interesting. Geometry was the favourite of all the strands and measurement the least favourite although it was almost tied for fourth. I wonder if this was because we were doing geometry the week of this survey and the children were having great success in the work. I also wonder if it was because we had not done enough measurement to make students feel confident in their abilities. We had done a great deal of data management work and they had really enjoyed the work they did on the computer.
Conclusions
If I had to evaluate the success of my research, I would say that it has been successful. The children are more confident and independent as learners. However, I wish to qualify that success. It has been at the expense of other subjects. I gave extra time to problem solving activities and mathematics and took time away from other subjects, especially from Environmental Studies. Sometimes, I carried on too long with an activity. The children do not understand the concept of a deadline and do not seem to care that they take too long at an activity. Next year I will survey the students at the beginning of the year, during the middle and at the end to evaluate their feelings about math. I should have done one right at the beginning. The children are still not proficient at recording their thoughts in their math journals but I no longer hear, "I did it in my head," and I no longer have to reply, "Tell me what you did in your head."
I have changed as a teacher of math this year. I finally feel capable and confident introducing and teaching problem solving strategies. I am still reworking my teaching strategies, but that is what teaching is all about, making connections and refining ideas. I believe even more strongly in the value of games and activities as learning tools. As The Ontario Curriculum Grades 1-8: Mathematics puts it, "The freedom to explore and the process of exploration itself are essential elements in the maturation of students’ capacity for mathematical reasoning." (p.76) I want my students to have the freedom to explore, but I need that freedom too. We will explore together.
Reference List
Clements, Douglas. (1997). "(Mis?)constructing constructivism." Teaching Children Mathematics, December. p. 198-200.
English, Lyn D. (1997). "Promoting a problem-posing classroom". Teaching Children Mathematics. November. P. 172-179.
McDougall, Douglas & Kajander, Ann. (1997). "Enriching mathematical creativity in the elementary classroom". OAME/AOEM Gazette, December. P. 5-9.
Ontario Ministry of Education and Training. (1997). The Ontario curriculum, grades 1-8, Mathematics. Toronto: Ontario Ministry of Education and Training. 1997.