Biographies
Patti Pogson teaches a grade three class, every other day, with her colleague, Kay Boss, who supported her in our math journal research. Marsha Lofthouse teaches a split grade five/six, every other day, and had the support of her teaching partners, Cheryl Coate and Sherry Cole.
We are both teachers of students who must complete the Grade 3 and Grade 6 EQAO and we were looking for a way to develop our professional knowledge as well as enrich our students' mathematics skills and abilities. Math Literacy is an area of focus on our school growth plan and we felt our students needed to find ways to express their thoughts to provide detailed math responses. We also knew that students needed to develop confidence to reason and think through the types of math problems they would be required to complete on the EQAO assessments.
Early in the school year, while looking at various possibilities to support our desire to improve our mathematics instruction, Christine Stewart hosted an Action Research information session. We both attended and listened to the information given. Chris described areas of focus for research and that there was funding for some special areas of interest. Math Literacy was one of these options. We decided, along with another colleague, Sandra Salerno, that we would investigate how math journaling could help our students strengthen math literacy skills. Our curiosity and ideas grew from that moment on.
How do we use math journals and problem solving strategies to improve math literacy skills?
Background
In order for students to be successful in math, they must learn how to solve problems and how to communicate, in writing, the steps taken to solve specific problems.
"Problem solving tasks prepare students to become inquisitive, competent, life-long learners. It also provides opportunities to link mathematics to other areas of the curriculum." Improving Student Learning Through Mathematical Literacy
"Problem solving is best learned and understood when modelled and when students are taught specific strategies to assist them in their investigations. When we teach children specific strategies to approach problem-solving situations, we are providing them with the tools that they need to practice and apply the mathematical skills they have learned to meaningful situations."(Barkans, Blake, Ricker & Suderman-Gladwell, 2002).
"When an entry is made in a math journal, it becomes a record of the experience received from the specific math exercise or problem solving activity." Math Journals For All Ages - Reflecting About Mathematics
"Some of the purposes of Math Journals are to increase confidence, to increase participation, to decentralize authority, to encourage independence, to replace quizzes and tests as a means of assessment, to monitor progress, to enhance communication between teacher and student and to record growth." (Countryman, 1992, as cited in Scales, 2000).
To assist students to develop mathematical knowledge and skills, we focussed our project on the research that showed that explicit teaching of problem solving and the use of math journals were appropriate strategies to strengthen math literacy skills.
Having seen, first hand, the way students struggled with the written part of EQAO -type math questions, we were interested in finding new methods to improve students' abilities to express their ideas clearly using pictures, numbers and words and to gain confidence in their abilities to solve these types of problems.
The Steps We Took:
We met together from time to time to brainstorm and investigate the skills we wanted our students to acquire. As a result of these brainstorming sessions, we felt it necessary to implement two separate teaching strategies.
- Explicit teaching of problem solving strategies, and
- Providing support for students to complete detailed math journal entries.
Explicit teaching of problem solving strategies included modelling and using The Problem Solver Strategies programme. This programme encourages students to approach each problem situation using the following four steps:
1. Find Out:
- What is the question you have to answer?
- What information does the problem give you?
2. Choose a Strategy:
- Act out or use objects
- Make a picture or diagram
- Use or Make a Table
- Make an Organized List
- Guess and Check
- Use or Look for a Pattern
- Work Backwards
- Use Logical Reasoning
- Make it Simpler
- Brainstorm
3. Solve It
4. Look Back:
- Read the question again.
- Look at the information given and the main question.
- Review your work.
- Is your answer reasonable?
Our next step was to model and teach the process described above. We clearly displayed the problem solving strategies in our classrooms so that they would always be available to our students. We felt confident that using the Problem Solver programme along with modelling a variety of responses would enable our students the confidence and encouragement they needed to give valuable responses to math problems.
Providing support for students to complete detailed math journal entries became our next focus. Writing appeared to be the important skill on which our students needed to focus. Support included the use of a math word wall, a math word pad, and a classroom display of the Problem Solver strategies (Appendix A, B & C). We also used homework assignments, brainstorming and oral practice. We introduced math exemplars and rubrics as a tool for students to compare and improve their responses. We provided suggestions to develop more detailed picture, number, and word answers. We allowed our students time to complete problems so that they would not feel rushed. As a follow-up to specific classroom problem solving sessions, we decided to send similar problems home with the students as homework assignments. After a few very discouraging results, where students became frustrated and gave up, both at home and at school, we decided to allow the students to work with partners or in small groups. A mathematics classroom, especially one that views students as young mathematicians, should include opportunities for social interaction (Kamii, 1985, as cited in Dawson, R. et al, 2003).
When students work, discuss, and write together, they have more opportunities to verbalize their thoughts, get reactions from others to their thinking, and hear other points of view. Having students work cooperatively reduces the isolation of working alone and gives individuals support for taking intellectual risks. Working in small groups can encourage otherwise shy or hesitant students to talk, which helps them clarify their ideas and prepare for writing. Writing in Math Class, Burns, 1995
We continued to feel confident that we could teach them the strategies and ensure enough practice in writing detailed math responses to enable them to be successful on the EQAO assessments.
In order to evaluate student responses, we needed tools for assessment. We decided on the math exemplars, the rubric for Problem Solving from Improving Students Learning Through Mathematical Literacy - Problem Solving (see appendix D), the rubric from our Math Curriculum document (see appendix E) and the Problem Solving rubric from Nelson's Mathematics, 2003 (see appendix F). Our subjective opinions on our student's progress and growth was also a major part of this evaluation.
Throughout this project, we met together from time to time to reflect and encourage each other to reach our goals and to compare written responses we were getting from our students. We continued to reflect and challenge ourselves as teachers to help, encourage and aid our students.
Our Research Findings
1. Some strategies are easier for students to use than others.
We found that different strategies caused different results in our students. After conducting a student survey, the results showed that "Make a Picture or Diagram" and "Act Out or Use Objects" became favourites with both age groups while "Working Backwards", "Guess and Check" and "Finding Patterns" were not. Ironically, the strategy "Make it Simpler", had both students and parents complaining that this math 'was too difficult'. We found that when the students 'shut down' on the mathematical part of the problem, the written part was often much shorter or not attempted at all.
2. You can use exemplars to level work.
We found that students could look at examples of student work from the Math Exemplar document and EQAO anchor booklets and could classify work according to the four levels. They could identify why a piece was levelled the way it was. Unfortunately, they did not consistently use this knowledge to improve upon their own independent work. Students knew what was needed to achieve a higher level, but, on their own, they seemed to lack the motivation and the effort needed to produce this type of work. When students were asked if these exemplars helped them to write more detailed answers, a grade six student commented, "No, because sometimes I'd already understood it and I didn't feel like writing, drawing and using numbers." One claimed that it was "just too much work" and a grade 3 student wrote that using the exemplars wasn't helpful because " I was already trying my best".
3. It is necessary to also educate the parents.
We discovered that parents are not yet convinced that this type of math is beneficial to their child. Homework assignments generated interesting comments from parents. One parent suggested, "This is too much to expect from Grade 6 students. I can't even do it." Another wrote, "I'm no good in math. We can not get all her math homework done because she can only get help from her grandpa and he can't help her all the time." A parent from the Grade 3 class wrote, "My daughter was very frustrated trying to figure this out and I believe this is a deterrent to an eight year old who should be concentrating more on the fundamentals of arithmetic instead of confusing problems."
4. Math Journals are an appropriate learning tool for students to develop math communication skills.
Looking through the students' journals for patterns or improvements provided us with more positive and encouraging results. We both found that responses gradually became more detailed as more problems were given and that there appeared to be an overall increase in the amount of effort put forth. Pictures, numbers and words seemed to become second nature.
"I like to solve math problems because it gets my brain working. It helps me understand more about math. I like the writing part because it helps me explain what I think about the problem." - Nicole - Grade 3
The most significant improvements were noticed when both of us changed the way students were asked to approach the problem. When we started out, the whole class would work on a problem together to focus on a particular strategy and then students would be left to solve a similar problem on their own. When we changed our approach and had students working in small groups, the results were dramatic. When students were given the chance to brainstorm with their peers and hear other suggestions and strategies being used, it appeared they felt more confident to share their own ideas and strategies, and became more motivated to try to write a detailed response about how they solved a problem. When asked how they preferred working, the majority of students from our classes choose working with a partner. "I like working with a partner because we can share what we think about the problem.", I like working with a partner. It makes it way easier!" "I liked working with a partner because you and your friend can put all your ideas together."
Brainstorming and sharing ideas with partners resulted in better results and happier students.
5. A positive and relaxed atmosphere makes a difference.
We discovered that students' responses were better organized, more detailed and longer in length when there was a positive, relaxed atmosphere present in the classroom. We found that if students felt that they had unlimited time, they made more of an effort to produce higher level results. As soon as time constraints were introduced, some students began to show signs of frustration, panic and then shut-down. These students seemed to adapt a generally negative attitude, their confidence levels appeared to plummet and their responses decreased in both detail and length. Students who were once showing positive signs of expressing their ability to be competent problem solvers and detailed writers, became discouraged and frustrated because they felt pressured to perform without given the time to think through the problem step by step. When asked how they performed on problems when a time limit was enforced, students replied, "I only got a few words in.", "Not good, but I tried my best.", I felt bad because I couldn't get my work done.", "I did not do well on it because I did not have enough time.
Our Conclusions and Next Steps
We felt that this research experience provided us with both positive and negative feedback and many opportunities for professional growth. We discovered that a safe, positive, and relaxed atmosphere is beneficial to encourage the best our students have to offer as well as setting up an environment where they can become competent risk takers. We found that students performed better when problems and responses were modelled. Brainstorming with peers was a critical motivator and working in small groups gave students the focus and direction they could not find on their own.
We strongly believe that our students can be competent problem solvers and can write in detail, using pictures, numbers and words, about how they solve problems using specific strategies. We have seen our students write detailed and lengthy responses and brainstorm with others to enhance their journaling. However, it also became very clear to us that forcing students to work independently on these types of questions and to work under rigid time constraints stifled students' creativity, created unnecessary frustration and resulted in an overall negative experience. It was not the type of questions asked that resulted in these feelings, but the pressured environment created when students were not allowed to brainstorm with others and when set time lines demanded compliance.
When we started our research project, we both were concerned about strengthening students' abilities to produce more detailed, complete answers on EQAO tests. Our intent was not to challenge the validity of the EQAO testing procedure. Although we believe that we have helped our students become more competent problem solvers and more confident in their abilities to write about how they solve math problems, we do not feel that we helped our students perform better on the EQAO test. The work our students produced in the classroom and then during the artificial setting of EQAO made us strongly question the type of math questions that students were being asked to answer and the manner in which they were being asked to answer them. It was unfortunate to see students who had performed exceptionally well in groups on exemplar questions and EQAO sample questions, struggle and give up during standardized testing sessions. As stated in the Elementary Teachers' Federation of Ontario (ETFO). Talking About Testing brochure, "The most important things that children learn -- to think critically and creatively -- cannot be assessed using a standardized test. During the five days of EQAO testing, students cannot interact with their teachers or with other students. This is not a normal classroom experience. It is abnormal and stressful."
Our action research allowed us to reflect on our current classroom practice when teaching math. Math literacy skills and math journaling are an area of focus in the Ontario Math Curriculum and on the provincial assessment and will continue to be a part of our math programming. The organization of math periods into longer blocks and the ability to work in small groups or with partners will become important components when working through math problems.
One of the expectations in our Ontario Math Curriculum is to relate mathematics to daily life situations. When faced with challenging problem solving situations, do we not encourage each other to seek advice and comfort from family, friends, or outside groups or agencies? If we don't expect others to solve problems alone, why should we force our students to face these problems alone? Some of our most beneficial times happened when we met with other teachers to discuss our findings, our challenges and our next steps. Working with others and sharing ideas was as beneficial to our well being as it was to our students when they worked with others.
Appendices
Appendices A, B, anc C: visual aids
Appendix D: Rubric from The Ontario Mathematics Curriculum 1997
Appendix E: Rubric from GEDSB Problem-Solving
Appendix F: Rubric from 2003 Nelson Mathematics
References
Barkans, Lori et al. Improving Student Learning Through Mathematical Literacy: Problem Solving. Written for Grand Erie District School Board, Brantford, Ontario. 2002.
Burns, Marilyn. Writing in Math Class. Sausalito: Math Solutions Publications, 1995.
Elementary Teachers' Federation of Ontario. Talking About Testing. Toronto: Allied Printing, 2003.
Hoogeboom, S. and Goodnow, J. The Problem Solver: Activities for Learning Problem-Solving Strategies. Chicago: Wright Group?McGraw Hill, 1987.
Kestell, Mary Lou, et al. Nelson Mathematics 4 - Evaluation Copy. Toronto: Nelson, 2004
Ontario Ministry of Education. The Ontario Curriculum, Grades 1-8: Mathematics. Toronto, 1997.
Ontario Ministry of Education. Helping Your Child Learn Math: A Parent's Guide. Queen's Printer for Ontario, 2003.
Prokaska, Lee. "School Offers Unisex Classes." The Hamilton Spectator 31 Mar. 2003: A3.
Russell, Deb. "Math Journals for All Ages: Reflecting About Mathematics." viewed 5 Apr. 2003 math.about.com/library/weekly/aa123001a.htm
Scales, Karen. "Networks: An On-line Journal for Teacher Research". Using Math Journals In A Grade 3/4 Classroom. Sept. 2000. Vol. 203, viewed April 8 2003 from www.oise.utoronto.ca/~ctd/networks/journal/Vol%203(2).2000sept/article2.html
