Actionresearch.ca -- Passion in Professional Practice II -- Nancy Carroll

What can I do to help my students communicate better in math?

Nancy Carroll

Biography

I began teaching in the fall of 1997. My first assignment was as a learning resource teacher at St. Williams Public School. In this role, I worked with individual students, helping them set and achieve personal goals as outlined in their individual education plans. I learned that my students needed appropriate supports to achieve, and that if their teachers don't set them up to achieve success, they will fail. My next teaching position was as a Reading Recovery teacher. This was an extremely valuable experience to me: I learned more about students' development and ability when given appropriate supports to learn. At the same time, I was working as a half-time primary-junior ESL teacher. Again, I had an opportunity to work with students in remediating their language skills. I had to develop strategies to help bring students from where they were to where they needed to be to achieve some success in their regular programs. These three positions gave me an opportunity to explore ways to assist learners overcome difficulties and gave me a hunger to improve students' learning in a bigger context.

Because my roles demanded that I travel from school to school, I had seen a wide variety of teaching styles among my colleagues, many of whom were happy to offer advice to me as a new teacher. This was sometimes difficult for me: I had a strong sense of the students' needs and my own values as a teacher, and often my colleagues' advice didn't match my own instincts about students' learning. I believed I needed to discover where my students were in their learning and then find ways to help them to use what they knew to improve their learning. It was this instinct that pushed me to explore new ways of helping the students understand and address their own needs as learners.

In the fall of 1999, I began teaching at Langton Public School. My class was a combined-grade class of twenty-six grade three and four students. At the beginning of the school year, I did some diagnostic assessment of the students and when I discovered a full continuum of skills and abilities in the students, I had an "a-ha" moment: I was really a teacher, this was really my classroom, and these were really my students. I set about making a plan to help them achieve success.

Abstract

After identifying a weakness in her grade 3 students' performance on the provincial mathematics assessment, the teacher-researcher set out to improve the ability of her new class of grade ones to communicate their thinking. Specific strategies were tried and modified through practice so students became more capable of sharing their thinking in math.

Background

Because my background is in Early Childhood Education, I was committed to creating a printrich learning environment, so when you entered my room, you could see an entire wall covered with math words, definitions, and samples of how to approach concepts and algorithms. Under my guidance, students used the "math wall" and occasionally some of my students would independently seek out information they needed from the "math wall" to complete clear, detailed math journals. I believed that this print-rich, math-focussed strategy would be sufficient to prepare my students for the Educational Quality and Accountability Office (EQAO) Grade Three assessment, and in May, I felt confident that my Grade Three students had performed well.

When the EQAO results arrived the next fall, I was anxious to see how my students had performed. I was quite shocked to learn that "communication" on the test was not a particular strength. During our class's day-to-day work, I knew that the students were able to communicate their thinking and learning about mathematics, but the test results indicated that there was a clear gap between what they were able to do with supported learning within our classroom and the way they responded on the test. Because of this, I knew I needed to focus my teaching on shifting responsibility from me to them so that they could independently use the supports without cues and reminders from me.

Over the course of the 2000-01 school year, I continued to work on developing student responsibility for their own supports with my now grade four class. For example, I decided that I would only add to our "math wall" as new concepts were mastered or as reinforcement for those students who hadn't fully mastered concepts studied. I also knew that students would need to engage in discussions with each other on a continuous basis and that as a class, brainstorming and modelling explanation for and with students would be focal point in our math program. At the end of the school year I viewed my students in the highest regard, especially as problem solvers who clearly articulated mathematical explanations and understanding of specific concepts.

This experience was really my introduction to action research. While this project was shared with others in a conversational setting, I didn't formalize the study with a written report. I knew that the action research process was a way for me to develop my practice and I resolved to continue to work as a reflective practitioner to build students' facility with using and applying classroom supports to their learning in all contexts.

When a chance arrived that I would be able to move back to the Primary division, I took it. In the fall of 2001, I started with reluctant group of grade one and two students. They knew lots of "stuff" but as a group were quite challenged by commitments to put anything on paper. There also seemed to be a weakness in making connections and forming knowledge deeper than fun facts. The students had studied countries the previous year, and were very good at telling me trivial information from that unit of study; for example, the class could explain what a kimono was or that chopin were shoes from Holland; they were not able, however, to explain that these countries were in fact different from Canada in terms of culture or location. Similarly, one morning, as the class was about to play a math game, Amy said, "I know how to multiply." She began to recite the two-times table. I asked her to explain to the class how to multiply, but she could not do any more than recite the facts. This showed me that she did not own an understanding of the process or language of this operation, that her "knowing" how to multiply was simply a rote recitation.

My move from the Junior division to the Primary division hadn't changed my goals as teacher of mathematics. I still believed that concepts needed to be presented to the students in real contexts that they could relate to. I still valued the importance of time for them as thinkers to work in small groups and pairs to work through the linguistics needed in order to clearly say what they meant. I still knew that modelling what mathematical explanations should look like was important. Finally, I knew that my students, even as young as they were, must take an important role in their own evaluation.

As I became more reflective as a teacher, keeping a journal and meeting with a local action research network, I looked for ways to improve my practice and help my students become more independent as learners. As always, my teaching was guided by the curriculum documents and developed through my instincts about what the students needed to be good learners. I searched for "a question", but in reality, my research had already begun as I started making changes to the way I taught students to develop their ability to understand and use the supports we had created for learning mathematics.

Communication is an essential part of mathematics and mathematics education"

- Principles and Standards for School Mathematics

As I started to plan my grade one and two program, I wanted to be sure that I was taking responsibility for helping them prepare for the provincial assessment they would write as grade three students. I remembered that my grade three students needed to be more independent in searching out and using supports available to them, and I made a commitment to begin building this type of independence in my combined grade one and two class.

During the summer of 2001, I engaged in much professional dialogue with a colleague and friend who would be teaching the same split in another school. From our dialogue, I decided that there was a place for some "sharpening of skills" but that the focus of the program would need to be based on having the children examine real life situations that they could interpret and articulate as solutions based on their own personal understanding

"When students learn mathematics, these do more than master basic skills, they acquire a concise and powerful means of communication"

- Ontario Curriculum-Mathematics

As a teacher in Ontario, I shared the same anxieties that many teachers have. Even with an extensive list of expectations across five strands of math, I knew where I wanted to go with my mathematics program; however, I had many uncertainties about how I would pull it all together.

What were my goals and how did I make them reality?

I want my students to know and use correct mathematical terms and symbols. I want them toapproach problems confidently and justify math solutions with clear explanations.

Developing My Plan For Instruction

I reflected on my own practice as mathematics teacher by asking the question: Where were the gaps in my instruction and what strategies could I use in order to help my students accomplish the goal of better communication in mathematics?

The strategies I would use would be to:

Seek out resources that will help me teach them to communicate better.
Model problem solving strategies and writing solutions.
Teach specific terms and symbols.
Provide a variety of math literacy resources for students to use.
Build time into the math program for students to work in small groups/partners.
Assess student communication of math concepts/vocabulary over time. I will address each of these strategies separately.

1. Seek out resources that will help me teach them to communicate better.

Examining a variety of resources became an important part of my planning process. Specifically,I wanted practical ways to engage my students in mathematical conversations and also samples of activities that would present opportunities for my students to use math terminology. The main resources I consulted were:

Stenmark, J.K., Thompson, V., & Cossey, R. (1986). Family Math. Berkeley: University of California, Lawrence Hall of Science.

Principles and Standards for School Mathematics, National Council for Teachers of Mathematics

Burns, M. (2000). About Teaching Mathematics (A K-8 Resource). Math Solutions Publications: Sausalito, California.

Goodnow, J., Hoogeboom, S., Moretti, G., Stephens, M., & Scanlin, (19-)A. The Problem Solver, Creative Publications: Pearson Education Canada

Burns, M. (1995). Writing In Math Class: A Resource for Grades 2-8. Math Solutions Publications: Sausalito, California.

Puddle Questions: Assessing Mathematical Thinking (1994). Creative Publications: Mountain View, California.

2. Model problem solving strategies and writing solutions.

With the reality of EQAO only one year away for my grade two students, I started modelling how to interpret and solve problems immediately in September. Most mornings, as part of our calendar routine, we would read a math problem and use a simple school template to organize a solution to the problem. Over time, a designated student took over the role of scribe. Initially, when students worked alone or with a partner, I provided them with a hard copy of the template.

In the beginning of the year, I used very simple problems. Our theme unit in October was Creepy Critters to coincide with Halloween. When sharing problems as a whole-group, I would generally create the organizer on the whiteboard. The problem would be printed in the designated area in our calendar display. As a class, we would discuss the problem, eliminating unimportant information, identifying key information, and working together toward a solution. I had modelled good skills, and here we shared them.

Sara bought some Halloween treats for her party. She bought 6 bags of treats. Sarah bought 2 bags of smarties and one bag of toffee. How many bags of gummy bears did she buy?

Sarah's Treats: Problem of The Day Creepy Critter's Unit : October, 2001

I observed that while students were able to articulate their work in conversation, I wasn't satisfied with the written explanations students were producing independently. As a whole group, with the opportunity to collaborate and cross-check information, clear detailed explanations usually resulted. Independently, students usually omitted important information such as naming problem solving strategies or vocabulary important to the clarity of explanations.

Conversations with a colleague, an early literacy teacher, convinced me that my students "needed to have their words" to improve in math. To reinforce the math words the students needed to have, I started to use a "think pad." This is a very simple strategy: I took a half-sheet of chart paper and printed the words "Think Pad" at the top. As the students worked through oral explanations, I would list the important words on the "think pad", modelling the writing for students. I would also ask the students key questions to make them use the words they had already made part of their oral lexicon; I would also write these words on the "think pad", again modelling the writing, and creating a reference which students could use during their independent work.

Over the course of the year, we worked through the process of solving problems but my role in modelling changed. Problem-solving sessions were led predominately by students. Often more than one student participated as leader, showcasing individual strengths with words, pictures, or numbers, as part of the group leadership for a particular problem. Prior to completing "the words" component, we would use the "think pad" to list important words that should be included in the explanations.

In January, during a unit on weather, I revisited number patterns, specifically counting by 3's.

It's Raining, It's Pouring

It has been raining day after day in the town of Langton. One centimetre fell on day one. Three centimetres fell on Day 2 and five centimetres fell on day three.

How many centimetres of rain fell on Day 4, Day 5, Day 6 and Day 8. How many centimetres of rain fell altogether in 8 days?

L Weins, 1999

As usual, students were seated in the carpeted areas in front of my whiteboard. The problem was printed on chartpaper so they could begin reading it immediately after sitting down. I read the problem with the children and then instructed them to discuss, with a partner, how they would solve the problem. Discussing the problem might have been considered a mistake by some. Several of the children started to argue about putting their solutions on the whiteboard, which led to a complete disruption at the carpet. Once I'd regained order, I suggested we create a group "think pad" together.

The following independent sample was generated from a grade two student.

S.J, : It's Raining, It's Pouring Sample Weather Unit: January, 2002

The "think pad" became a very functional visual aid for students. We used the "think pad" for written explanations of problems and math journal entries where students provided explanations of concepts and interpretations of data displays.

I started to use the "think pad" to help students interpret data displays better, often integrating mathematics with other subject areas. While studying Canadian authors, students recorded their preferences between two authors. After creating the graph, students discussed the words that should be in the written interpretation. The students began to understand that any bar graph has a generic set of words, but also that specific bar graphs rely on words that are specific to that topic. I was beginning to see two-tiered thinking through the class "think pads".

In the right side of the photo, you will see the "think pad". I just use long strips of paper so that I can move them around the room.

A student story: L.K. one of the students created a bar graph in the program "Graphers". She created an individual "think pad" and then used the notebook function to record a detailed interpretation of the data.(below)

3. & 4. Teach specific terms and symbols/ Provide a variety of math literacy resources for students to use.

Teaching specific terms and symbols didn't just happen during problem solving sessions "think pads" were used. Teaching math terminology became an integral part of many aspects of the entire program. I use several strategies to teach terms and symbols, and these become resources for students to use in a very print-rich environment.

Calendar

I introduced most new math concepts during my calendar routine which takes place during first period each morning. I designed my calendar program in such a way that I would focus on specific concepts each term.

Word Walls

I'm pleased with how much more my students used the "math word wall" than in previous years. I organized the words by strand and only added them as concepts were introduced (as I mentioned earlier).

My Word Wall

We use this word wall daily. As a math warm-up, we play simple games with the words not unlike those many teachers use in their language program. Last year, my students liked a tic-tactoe variation where students would print "math words" on the tic-tac-toe board and I would provide clues for 'X' and 'O' . Student Stories

Peter is a student who came to my class part way through the school year with fairly good mastery of English language when speaking. Not unlike many other students his age, Peter could explain mathematical concepts much better orally. Since we had used the word wall so frequently as a whole group, he was able to consider two thoughts to find the word he was searching for:

It starts with "c" and ends with "t".
It's in Numeration because it's about numbers .

Kayla is searching for misspelled words that interfered with the meaning of her explanation of a specific number pattern. Again, the math wall is an important tool for helping students bridge the gap between what they are able to produce orally and in writing.

Another one of my math "word walls" presents 2-d and 3-d shapes. In the past, I'd always used these types of visual aids, but now my display is presented in an organized way as we learn the concepts. I am using smaller icons, which allows me to put many more shapes on the wall, to use formal mathematics language, and to label each diagram.

Working With Words

Developing word sense through making word exercises is an important part of my language program. I found two strategies the most successful with students:

Mystery Words

To constantly review math words on our math wall, I started using math words as the 'mystery words' and words to build new words from. The 'mystery words' game functions like twenty questions, with the students asking me questions to try to narrow their understanding and express the specific word they're trying to guess. There are many variations of this game; for example, I would often reverse the game, giving students broad concepts and having the students create lists of words to explain the concept.

Word Building

The students use math words as a base and then add prefixes and / or suffixes to create bigger words. Here are two examples showing these games in my classroom:

?Reverse Mystery Words": Amy is glancing over her shoulder to look at the math wall. In this exercise students were asked to list ten measurement words on the mini-chalkboard.

"Word Building": In this example, a mixed group of (Grade One and Two students) each print the "add" on the chalkboards. I provided prompts so that they could create new words (e.g., add "ing", take off "ing" and add "ed".)

I continue to use and modify these strategies. For example, this year, I'm using miniwhiteboards with wipe-off markers. To begin math sessions, students routinely listen to clues in order to write "mystery numbers". Eventually, I want them to write clues for numbers, 2-d shapes and 3-d shapes, coins and time increments.

On-line resources and Technology Applications

I believe it is very important for my students to begin using computers quickly to help them convey information. Early in the school year, we established a link to Harcourt School Multi- Media Math Glossary at: http://www.harcourtschool.com/glossary/math2/index2.html/.

This glossary is an interactive mathematics dictionary organized by grade levels. Because it is very user-friendly, even grade one and two students can navigate the site following a quick lesson from the teacher. This incorporates information technology into the program and reinforces to my students that there resources for learning outside the classroom and in the world at large.

To further integrate math and computers, I have students visit the Harcourt glossary and maintain a math journal in word processing program like WordPerfect. I also have students use another program to demonstrate number sentences (for all operations), fractions, place value, descriptions and comparisons of 2-d and 3-d shapes, as well as various aspects of mapping and graphing.

5. Build time into the math program for students to work in small groups/partners.

Students in my class read a lot of picture books which relate to specific math concepts. I have students read this type of literature, discuss the math concepts, and develop paired journals about the concepts presented in the books. I'm cautious about student groupings for this kind activity. Although speaking about these concepts comes quite easily to most students, a capable scribe makes the process easier.

I always use these books:

Hutchins, P. (1986). The Doorbell Rang. Morrow.

Rylant, C. (1986). The Relatives Came. Illustrated by Stephen Gammell. Simon and Schuster.

Viorst, J. (1978). Alexander, Who Used to Be Rich Last Sunday. Ill. by Ray Cruz. Simon and Schuster.

Anno, M.. and Anno, M.. (1983). Anno's Mysterious Multiplying Jar. Putnam.

McGrath, B. (1994). M and M's Brand Chocolate Candies Counting Book. Charlesbridge.

Burns, M. (1994). The Greedy Triangle. Scholastic, Inc.

Schwartz, D. M. (1985). How Much Is a Million?. Lothrop, Lee, & Shepard.

You can also find some great resources on-line. The sites below provide a multitude of children's literature and professional resources that relate to using literature to teach mathematics.

http://www.carolhurst.com/subjects/math/math.html

http://www.geocities.com/Heartland/Estates/4967/math.html

http://users.anderson.edu/~roebuck/childlit.html

http://www.center.edu/pub/docs/chapter12.pdf

6. Assess student communication of math concepts/vocabulary over time.

Students often collaborate to assess math projects. Revising written explanations based on rubrics we create together is important to my students. We also share work in whole group sessions as well. I believe they need to hear the words they need to use but they also need experience opportunities where their peers can provide feedback for improving communication in mathematics. This year, I'm putting much more emphasis into the follow-up lessons and conferencing with individual and small groups of students after I teach concepts.

Final Thoughts and Reflections

When I began to work on this project, I asked "What can I do to help my students communicate better in math?" Identifying my goals and creating the instructional plan was very important. These two steps helped me become more aware of my development as a teacher, providing a series of things to try and reflect on in my practice. In addition, my involvement with action researchers and other reflective colleagues gave me a time and place to explore my questions and concerns, question my practice, and share my triumphs.

I know that the students' performance and independence has improved because the students' work provides evidence of that. I know that the strategies I have employed work in others' classrooms because colleagues with whom I've shared my strategies are reporting success in their own classrooms. I know that I continue to grow as a teacher and reflective practitioner: I am constantly assessing my students' needs for improvement and finding new ways to aid their improvement through my daily classroom work.

References