Rationale

"I always enjoyed math as a child, crunching numbers in Mad Minute exercises, playing the train game to practice multiplication, or applying formulas to solve perimeter or area problems. I liked math because there was one correct answer. I could go home and study for a test by practicing my skills, memorizing concepts, number facts, and formulas, and know if I was doing it right by checking answers in the back of my text book. It was the ideal picture of rote learning and I did well in those types of learning situations." (Personal Journal, Oct. 2002)

Mathematics is no longer a task of mastering basic skills like multiplying or calculating perimeter and area. Math questions are open-ended and often require several steps to solve, with more that one correct way to answer them. According to the Ontario Curriculum(1997), students must not only develop a knowledge of basic skills, but knowledge of mathematical language, structure, and operations that will help them to reason, to justify their conclusions, and to express ideas about mathematics clearly. Students also need to be able to use math in connection with technology, in their daily lives, and eventually, in the workplace (p.5). Students are to be assessed on how well they solve problems, show understanding of concepts, apply math procedures, and communicate required knowledge (p.8). I could not teach math the way I was taught math, with a list of skills to memorize in time for the test, hoping that I would remember them when I came across a situation in my life where the skill may actually apply.

A focus in our school action plan this year is math literacy. Having discussed this at length during staff planning sessions, I immediately made the decision that not only did my students need help improving their math literacy, but I also needed to improve the way I taught math. I could see myself teaching only math concepts and simple procedures, and often leaving problem solving and communication until the end of a unit of study, where it would receive very little emphasis. Grade 3 and 6 Education Quality and Accountability Office(EQAO) provincial testing emphasizes the use of "Picture, Numbers, and Words" as a template for solving math applications. By communicating mathematical learning in these three ways, students can fully demonstrate their level of math literacy. I usually only gave my students the opportunity to answer in numbers. In teaching this way, students were learning that math was numbers, and hence, their problem solving and communication abilities in math were being hindered.

"Don't get me wrong. I liked rote learning because it was easy. I didn't have to put myself in a position where I would have to defend my learning. If I got the right answer, it said so in the back of the book. It was that simple. Of course, I applied what I learned in math regularly, but not consciously. My father designs and sells custom kitchen cabinets. He was always drawing custom cabinet designs in our home while I was growing up. I would always be drawing up designs for new bedroom suites, computer desks, or stereo cabinets in the hopes he would build one of my designs for me. Aside from his career, he had a hobby of building new homes for his family to live in. I lived through the designing and building of 5 new homes while I was growing up. I can remember sitting with my father, and drawing dream homes of my own, trying to put them to scale, and label the drawing correctly, the way he would do for the architect to draft up officially. For the most part, I did a pretty good job. Measurement and geometry skills, along with lessons involving basic operations came in handy during those drawing sessions. I could apply the learning if I was given the opportunity. Without recognizing, I did value math, which is why I probably put the effort into memorizing the skills I was taught. Being able to sit with my father, and communicate my ideas orally to him, as well as with pictures and numbers in my design was a valuable learning activity for me growing up in terms of my own math literacy." (Personal Journal, Oct. 2002)

Upon reflection, I concluded that teaching math as problem solving and communication could be best accomplished in real life problem situations, where students were allowed to discuss ideas, make calculations and conclusions together, and use language naturally. As they communicate ideas with each other, and work through problems, I felt that they would learn to clarify and consolidate their mathematical thinking the way I had when I sat and worked alongside my father designing houses.

Taking a Closer Look at Math in my Classroom

The first question I asked myself was, "Although I focused the majority of my math teaching time on concept and skill development, what parts of this math program already promote math literacy?" I couldn't accept the notion that my program was completely futile. I invited a colleague who assisted students with individual education plans for math to point out aspects of my program that she felt were productive in promoting math literacy. She observed that my use of a math lesson book, word walls, and modeling process during lessons were all worth the effort towards improving math literacy.

Each student in my room was given a math lesson book at the beginning of the year. At the beginning of each new "skills" lesson, such as multiplying fractions, or calculating the volume of a triangular prism, students would record any important vocabulary, the formula needed to perform the calculation, along with examples which often included corresponding diagrams to help them understand the new math concept. Students could use this book as reference material when they were attempting to solve more open-ended math problems, if we ever had the time to work on those types of questions.

My classroom also is decorated with a mathematics word wall. As new terms are taught or encountered during math lessons, they are put on our word wall as a reminder/tool for students to use when working through more difficult math problems.

Finally, my colleague felt that my modeling mathematical processes was also helpful in teaching students how to explain what they were doing when completing math questions. I not only model the process using numbers on the board, but talk about each step as I complete it on the board using appropriate vocabulary. Then, students are given an opportunity to do the same as we continue to practice on the board together. I will often prompt appropriate vocabulary while students are doing seatwork to help reinforce the importance of being able to explain what you are doing.

I recognized that although my program did offer some fostering of math communication, students were rarely given the opportunity to apply the skills and language studied to more complex math problems. I decided that it was important for me to implement more problem solving into my program.

"I need to give students an opportunity to problem solve. I teach them the basic skills, and give them opportunities to practice using appropriate vocabulary to explain simple math problems, but we never work on more complex, multi-step, open-ended problems. " (Personal Journal, Nov. 2002)

The Act, Reflect, Revise Cycle

I was familiar with a resource entitled, Puddle Questions- Assessing Mathematical Thinking. Puddle Questions are open-ended questions that pose authentic problems involving mathematics in the real world. Because of the open-ended nature, these questions have more than one right answer. I decided to have students work in teams to solve the problems using any skills they had acquired from across the five math strands. They would need to record diagrams, calculations, and written explanations to show their understanding and how they solved their problems. They could use any available resources within the school to do so.

My role would be to act as a facilitator to the problem solving process. I would introduce the problem by reading it aloud, re-iterate that I wanted to see their best work and most careful thinking, that there is more than one right answer so not to compare their work to other groups in the class, go over assessment criteria with them, and offer prompts throughout the process to keep them working. The resource also provides exemplars and examples to share with students for which I would also be responsible.

The first step was to divide students into groups. I decided to create groups that were not mixed ability. I felt that it would be more comfortable for students to work with others that had similar math skills. I hoped this would prevent one student being lost in the problem solving process because they did not understand what the rest of the group did. These groups were made according to first term report card grades.

Once groups were established, an example of the types of problems the students would be solving was introduced. The model question read, A Packaging Problem: Design and make a container that will hold 10 marbles snugly. This sample was great because it gave the students samples of work at different achievement levels. This gave them the opportunity to see the differences between a level 1 answer and a level 4 answer. Alongside this model, we also discussed the problem solving process. Students were reminded to show all their work. Many students were able to explain to me that showing all your work meant making calculations, explaining your thinking using good vocabulary, and drawing a diagram to help visualize their solution. They had encountered this process during E.Q.A.O. grade 6 testing two years earlier when they needed to answer questions using pictures, numbers, and words. We then discussed that answers were likely going to be multi-step, meaning more than one calculation, and possibly involving knowledge from more than one strand of math.

Upon completing this introduction to puddle questions, students were given their first question to solve. The first questions was entitled, "Garden Plotting." Students were asked to do the following:

Question:

Plan a vegetable garden with an area of 20 meters squared. Your garden needs 10 different vegetables. Make a scale drawing of your garden as it will look at harvest time. Write a report to explain you garden. Include calculations to show that 10 vegetables only take up the specified area.

Math Ideas:

• area
• units of measurement/conversion
• scale drawing
• spatial arrangements

Assessment:

• Is the plan workable? Practical?
• How accurate is the scale drawing?
• Are the calculations correct?
• How clearly do you explain your mathematical thinking?

Prompts:

• What different dimensions could your garden have?
• How will you choose plants for your garden?
• What scale will work well for your drawing?

Five out of seven assignments were completed with a low level response. Not much effort was put into making a workable plan for the garden. The approach to solving the problem was not workable in many cases due to inaccurate calculations. Drawings were not done to scale. Spacing was done based on personal preference/guessing rather than based on mathematical logic or research into the specific vegetables chosen. Students were reluctant to explain their thinking and process based on what was asked in the original question, which probably would have helped them to realize the mistakes in their solution.

I again shared exemplars for this problem with students, and allowed them to compare their answers with the ones in the book. Students were supposed to identify strengths and weaknesses in their own work, and use the information to help them improve on the next puddle question that I assigned. Students did not take this process seriously. While circulating, students were off topic which indicated to me the lack of value they placed in the activity. I knew that if they didn't recognize their mistakes in the first problem, that they would likely make the same ones if a second problem was assigned. I was ready to give up. How was I supposed to help students become better math problem solvers and communicators if they weren't interested? Furthermore, why weren't they interested?

Valuing the voices of others is essential in the action research process. The voices of those directly involved in your research is one of the best sources of validation and a critical part of the self-reflective process. This is known as a "second level validation" which occurs by obtaining feedback from persons who understand the context in which we work (McNiff et. al., 1996, 24). I was making the claim that my students weren't interested in math. I was consciously aware that this claim may be inaccurate, and so I openly discussed this concern with my students.

"I don't know how to multiply that well. I just put it in the calculator wrong. That's all." (Student Conversation, Dec. 2002).

"I didn't even get to help. My group didn't like any of my ideas. It's not my fault it was done wrong." (Student Conversation, Dec. 2002).

"There wasn't enough time to figure everything out." (Student Conversation, Dec. 2002).

"I didn't know we could do it that way, with a map scale." (Student Conversation, Dec. 2002.)

"I didn't know the right words, so I didn't write any." (Student Conversation, Dec. 2002).

"I still passed, right?" (Student Conversation, Dec. 2002).

After going through this conversation in my head several times, I came to the conclusion that students just didn't have enough basic skills to answer the questions much better than they did. Many students do not know their addition, and multiplication tables by heart, and so were unconsciously making miscalculations. Some even gave up on the process and hoped they had given me enough information to pass. Students were content that they had passed, and weren't too concerned that they didn't achieve a higher level. Aside from a lack of basic skills, students were not familiar with different ways to display their data, nor did they make the connection between math and scales they study on maps in Geography. So now what was I supposed to do? How can I teach them all of this stuff, and help them use it to solve more difficult open-ended math problems properly using pictures, numbers, and words?

A Living Contradiction

From December until February, I floundered in my thoughts. My research action had come to a sudden halt. I abandoned the process I had initially set out for myself, and worried daily about what I was going to do next. I was back to teaching just the skills; operations with fractions, translating algebraic equations, and others set out for me in the grade 8 curriculum. While working on operations involving fractions, I became aware of the extent to which many of my students did not know how to multiply. They were making error after error because of this. When I asked them who would be interested in some flash card practice, 14 of my 33 students raised their hands. This information, despite how disappointing it was to receive, was the spark I needed to dig myself out of the frustration with my research. I combined this new knowledge with my reflections back over the school year, and realized that the flaw in practice was not that I was teaching math wrong, but that I was teaching math the way I thought I had to based on the Ontario Curriculum. I had a "living contradiction" in my practice, just like Jack Whitehead explains. That was the cause of my discomfort.

Fortunately, Jack Whitehead was scheduled to visit our research group in early March. I had to prepare a journal entry for him to summarize my research to this point.

"Having students working on multi-step problems has not been successful in my classroom. Students lack the basic skills as well as the motivation to persist with the lengthy tasks. I took a vote today, and 14/33 students do not know their multiplication tables! It's no wonder they lack motivation. Problems are time consuming to begin with, never mind having to struggle with every simple operation they have to do to complete it. I realized this lack of basic skills when it took us 4 weeks to learn how to add, subtract, multiply, and divide fractions with minimal success. We practiced, and practiced, and practiced, and many students still did not understand how to do this type of operation. I felt this was mostly because of the frustration and errors they encounter due to the lack of knowing their multiplication tables. If they chose a wrong common denominator, their question would not work out. I needed to go back...go back and teach them the basic skills, and not feel guilty about not "covering the curriculum" exactly. This is my living contradiction. I know that my lessons and classroom should be organized around the needs of my students, but then there is the accountability factor in writing the report cards. If I taught students only what they needed to know, I wouldn't be able to evaluate them according to the grade 8 curriculum. I experienced this at the end of term 1 when I was scrambling to put math marks and comments together fairly, since I did not cover many expectations in each strand, and what I did cover was only indicative of students understanding of basic concepts. My research is not going the way I expected, but how can I meet the curriculum expectation of teaching math problem solving where students show their understanding of concepts, apply math procedures, and communicate required knowledge, when they can't even multiply! In this context, I was crazy to think I could teach my students how to see math in terms of "applications." Where do I go from here?" (Personal Journal, March 2002).

At the session with Jack Whitehead, we were asked to read over our personal journal entry, anddraw some conclusions about our work. The answer to my question, "Where do I go from here?" appeared easily upon reflection. I believed that my students were not successful at working on multi-step problem solving. By whose standards? Perhaps by the standards set out by the Ontario Curriculum, but not necessarily on a personal achievement level. Although I needed to evaluate students according to grade level expectations and ministry standards, I also needed to keep an objective attitude in terms of what I could reasonably expect from my students. I was judging the success of my research according to ministry standards as well, rather than realistically setting a success standard that I could reasonably meet within this school year with my students.

Next Steps

And so the cycle began again. My focus on math literacy had not changed. I didn't feel the need to re-articulate my question. I did feel that I needed to define, with my students' needs in mind, what improvement in math literacy would look like in my classroom. I now recognized that success for my students and in my practice is not what I had originally perceived. I thought that if the majority of my teaching was consumed by instructing students in basic math skills, that I wasn't doing a good job. But this was what my students needed most. Without the basic skills, they did not have the confidence and persistence to complete multi-task problems. They needed to learn the formulas, explore different organizers for data, and help to recognize and use appropriate math vocabulary to explain their knowledge. I wasn't planning on abandoning the expectations and philosophy of the Ontario Curriculum. I was, however, confronting my own conflicting values of trying to meet the needs of my students in the framework of a curriculum that they were not ready for in many instances.

Jack Whitehead discussed the notion of "creative compliance" with me. He encouraged me to continue teaching the basic skills and vocabulary to my students as I had been in the onset of my research. I do have to teach the Ontario curriculum but I know that I must recognize the need to adjust expectations to meet their needs. Teaching the basic skills would give students the tools they needed to foster confidence so they could begin to explain their understanding of math using pictures, numbers, and words.

My goals as I continue to research are:

1. Helping students to find value in mathematics.
2. Teaching students the basic skills and vocabulary to help them to become more confident in their ability to make calculations, and explain their knowledge using the calculations along with words, and pictures.
3. Helping students to become problem solvers.

I will no longer be assessing the success of my research according to the number of students who could answer multi-step problems with high level responses. Rather, my self-evaluation will be based on progress towards math literacy that my students are making.