June 1996
By: Lillian Forsythe & Dr. Vi Maeers

Introduction /Background: A Journey
Questions/Quests: The Need For A Road Map
Teacher Rationale/Methodology: Planning The Trip
The Research Milieu: On Our Way
Research Data Findings from Fall, 1994 to Spring 1996: Highlights of the Excursion
Summary
Problems and/or Limitations
References
Parting Thoughts

Introduction /Background: A Journey

Although a new mathematics curriculum for kindergarten to grade five had just been developed for the province, there seemed to be little in the way of explanation as to exactly how the teacher enacted that curriculum in the classroom. Included were a scope and sequence, objectives, materials suggestions, timelines and references to help the teacher organize and implement the curriculum. Aspects of the teacher's role were not satisfactorily addressed: degree of teacher/student interaction, information on how students were to use materials, and sequencing of daily mathematical interactions. That was left up to the individual classroom teacher. As I discussed my role with other colleagues, I had a difficult time defining or articulating exactly what my role was and how I could maximize student learning using a manipulative mathematics approach. Thus, arose the desire to become involved in this research project. Application for funding was made to the Dr. Stirling McDowell Foundation For Research into Teaching during the winter of 1994 with the intention of beginning research into the teacher's role in manipulative mathematics instruction in the fall of that year.

I had completed classroom-based action research for my Master of Education degree in the fall of 1993 and felt strongly that information on classroom practice needed to come first of all from classroom teachers with a desire to examine what was occurring in the classroom. I had recently met Dr. Vi Maeers from the Faculty of Education, University of Regina, and felt that with her collaborative support we could together articulate the role of the teacher. I certainly needed the help of an outside observer to help document my role in classroom mathematics instruction. With ethics and approvals granted we were about to embark on a one-year (later it became two-year) journey into the world of action research focused on the role of the teacher.

A journey may be described as travel from one place to another, usually taking a rather long time, or passage or progress from one stage to another. Either definition applies to my personal journey in the primary mathematics field.

I began teaching primary mathematics over twenty years ago much like any other teacher, using the prescribed workbook and my newly acquired knowledge of teaching theory. Working within time constraints as a new teacher, seldom was the curriculum consulted since it was assumed the approved program for that school system covered the required material. That lasted for a number of years, but gradually I added to the workbook mathematics: I set up learning centers; I made up mathematical games; I tried using "found" materials to vary my teaching approach and make mathematics more fun. I was on a quest or journey that I did not totally understand but which kept niggling away at my thoughts about mathematics and student learning. During this time I had the uncomfortable feeling that I was doing a disservice to children and the mathematics subject by simply working rather blindly through a series of pages until (whether independently or with one-on-one help) each page was completed and clipped. I knew intuitively that there was more I should be doing to help children really understand mathematics and be able to apply that knowledge to new situations.

Some ten years ago, with administrative support, I stopped ordering workbooks and began the use of a truly manipulative approach to teaching primary mathematics. I had my curriculum, access to a number of mathematics programs, but most importantly the desire to build children's mathematical understanding by having children work first with concrete materials and then record what they had learned.

It is important to underline the role of administration in my journey through the change process and my growth as a teacher. My principal served as mentor, encouraging the change process, supporting my attempts to mesh my beliefs with practice, and discussing with me the successes and pitfalls of that practice as I implemented my "workbookless" mathematics. At the system level I worked with superintendents who were willing to listen, to support my efforts and foster change by providing assistance and approval. A new journey was underway! I enriched that journey through graduate classes to affirm my understandings of how I thought children learned, and to build a stronger theoretical basis for the judgments I was making. I continued to read professional literature to inform my practice and gradually modified and extended my instructional strategies through daily reflection on classroom happenings, documentation of occurrences and refinement of management and teaching techniques.

As my confidence grew and I found that mathematics really was fun for the children, I built in children's self-evaluation and more co-operative work. Over the last five years I have added a parental component so that parent and child working together could keep abreast of what we were learning in mathematics and apply that knowledge beyond the classroom walls. Not yet totally satisfied I now undertake further research...

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Questions/Quests: The Need For A Road Map

As in any journey this study had many issues to be considered. To my knowledge, seldom is a quest worth undertaking easily defined and arrived at. This research undertaking is complex and is an attempt to articulate how a teacher plans for, integrates, and implements theory into practice (or develop one's own theories) in the primary mathematics classroom. My question centers on both theory and practice and their interrelationships. I would like to articulate and define the teacher's role in the primary mathematics classroom. Two critical quests seemed important: an examination of my personal beliefs, and a search of research literature to help in defining teacher role.

Personal Beliefs

Do my knowledge and beliefs about children's learning, mathematics, and teaching have a sound basis? Are the daily interactions between teacher and students in my classroom enhancing their learning? Is that practice in keeping with what I believe about children's learning? Is my practice recursive as I implement new practices, modify these to meet my own or the students' needs, and once again reflect upon and assess their impact as I strive to make my classroom a context for maximizing student learning?

Is it the teaching that makes the difference in children's understanding of mathematics, factors related to the children who are doing the learning, or is it some other combination of components which influences learning in mathematics? But what then is the teacher's role? Where does curriculum enter into this and whose curriculum is implemented? Does that development also apply to the teacher and her life experiences? Analyzing the relationship between theory and practice becomes similar to examining a road map, choosing options, exploring new territories, perhaps backtracking, but always attempting to move forward to new and better teaching and learning. One is on the road to change and improved practice but seeking to gain a better understanding of the whys and wherefores of that endeavor.

Research Literature

To approach this topic and generate ideas worth consideration, reading of the research literature, reflection, and certainly some further examination of classroom practice (as this is a very personal journey), are necessary to aid in arriving at a coherent destination. There is certainly no one appropriate "theory" which I feel links knowledge and beliefs directly to practice. I have a number of beliefs about how I feel children learn, but what combination of factors may be involved in maximizing that learning requires further study. Researchers and classroom teachers have not always been in agreement in relating teaching and learning, as each has approached the topic from a very different perspective. There is certainly much to be gained by both researcher and teacher as we share viewpoints and experiences, and make comparisons between theory and practice. Perhaps, with more collaborative research, we may find ways to bring more consensus to the theory-practice linkage.

I have some strong personal beliefs which influence what I do in my mathematics classroom. How do they impact on my role as a mathematics teacher? I believe:

• for intelligent learning of mathematics, understanding precedes rote memorization of facts and algorithms;

• that "scaffolding" by the teacher will help to build on a child's previous learning;

• that children require multiple experiences in a variety of contexts to construct knowledge

• that in children's developmental levels, concrete precedes abstraction;

• that teachers play a facilitating role in a child's explorations and learnings;

• that there is real value in integrating mathematics into a child's everyday world and across the curriculum;

• that involving children in cooperative work with peers and parents enhances their learning:

• that children should be involved in decision-making in the classroom.

The above beliefs lead to empowerment of both teacher and student as defined by Saifer (1990) "to give someone the ability to have control over a situation, themselves, or their lives. Children are empowered when they are given choices and encouraged to make meaningful decisions" (p.188). Through involvement by children and parents working in cooperation with the teacher we can make learning fun and improve attitudes towards mathematics.

Several research articles were of interest as I began analysis of my research data. One was the Exemplary Grade 1 Mathematics Teaching work of Ciupryk et al (1989) who examined the role and classroom practice of an Australian teacher identified as "exemplary." She had fourteen years of classroom experience. Through observation and interview they attempted "to isolate those attributes and characteristics of an exemplary first-grade teacher whose teaching of mathematics appeared to set her apart from her peers" (p.43). Among the strengths they observed were:

• knowledge of both curricula and child development,

• integration across the curricula,

• appropriate programming for students,

• personal beliefs and values of the teacher which included her warm personality, awareness, caring, setting of classroom climate and student expectations, and communication skills.

In their article Sternberg and Horvath (1995) attempted to construct a categorization model to identify key characteristics of an "expert teacher." They looked at the work of other researchers to identify areas where expert and novice teachers differed in order to develop three major categories (knowledge, efficiency and insight) into which characteristics might fit. Within knowledge were pedagogical, practical and tacit areas which covered curricular, instructional, management and contextual awareness that would impact on teaching and learning. Expert teachers' efficiency included skills in planning, monitoring and evaluating. Teachers appeared to carry out routines almost automatically. Insight allowed teachers to select and combine information as needed and use more creative problem solving. Would the results of my study identify any of these characteristics of teacher or classroom milieu?

Vacc (1993) examined discussion as a technique for mathematics instruction. She identified five attributes of teachers and/or their practice in "creat(ing) environments in which students feel free to:

a) share their beliefs and opinions,
b) ask what, how, and why questions,
c) take risks,
d) hypothesize, and
e) make mistakes." (p.225).
Teachers need to create a classroom climate or atmosphere where students feel free to participate and discuss, and also include activities where students are encouraged to question, provide input, and interact with peers and the teacher.

Lubinski (1993) examined problem solving in a case study of a grade one teacher's decision-making. The teacher's beliefs and pedagogical knowledge figured strongly in those decisions. Lots of verbal responses were encouraged; there were a vast number of materials available; many student explanations were involved; writing in mathematics was encouraged; and there was emphasis on student thinking. All of these are areas which are emphasized in the Saskatchewan Elementary Mathematics curriculum.

In Ball's (1993) third-grade teaching experience, she examines some of the problems in representing mathematical concepts to children, building a sense of classroom community, and respecting students as mathematical thinkers. She stresses the need to have children read, write and create things that are of interest and/or matter to them. Here we are valuing context and authenticity in the classroom community. She also emphasizes the importance of students using conjecture, experimentation, and argument to build mathematical understanding.

One other article of interest was that of Rosaen (1992) in which she examined research on the potential of instructional materials to support teacher effectiveness. Although she looked at many subject areas, mathematics among them, she noted that to make full use of the potential of materials and resources, teacher knowledge, skills, disposition, and context all have impact and affect the type of resources used as teachers strive to improve practice. Teacher work load affects both the kind and use of resources.

Schwartz's (1994) large-scale study of teacher knowledge and beliefs and their relationship to teaching practice revealed that beliefs, and beliefs and mathematics understanding together had a significant impact on practice, while mathematics understanding alone did not have a significant impact. According to Schwartz the beliefs-mathematics understanding relationship profile includes: teachers with constructivist stances, a belief that problem solving is a context for development of computation skills, and children's natural development as a determining factor for sequence of instruction.

Resnick et al (1991) were involved in a study of grade one and two children where a reasoning-based program was instituted developed on the following principles:

• develop children's trust in their own knowledge,

• make use of children's informal knowledge in the classroom,

• keep records of class discussions and conclusions,

• introduction of key mathematical structures as early as possible,

• talking about mathematics as well as doing it, and

• encouraging everyday problem finding and problem solving. In this long term project they are finding that student scores can be effectively raised using this approach.

Peterson et al (1991) also examined making use of children's prior knowledge in their study of some forty first-grade teachers. Half the group of teachers were given inservice to enhance their ability to use children's problem-solving strategies in mathematics instruction. Three key elements were recognized: multiple solutions and/or strategies were encouraged; teachers had an expansive view of children's knowledge and thinking; and teachers' programs had a problem-solving focus. Use of problem solving facilitated the development of mathematical abilities of disadvantaged children.

Young's (1991) handbook on helping children become risk-takers suggests that the teacher's role is important to promoting risk-taking, that problem solving is important as is the classroom setting, scheduling and emphasis. Again the teacher is pivotal to maximizing the ability of children to feel free to take risks in their learning and problem solving.

Some recent research studies in mathematics education have explored constructs of enactivism, or the enactive approach to cognition, as a way to explain classroom learning (Maeers, 1995; Davis, 1994; Kieren, 1994). In enactivism both the subject and the world are thought to mutually specify each other, to co-emerge together, and both are considered integral to learning. A teacher following an enactivist stance would play an integral role in enabling learning to occur. Without question, this teacher would plan an intended learning environment based on children's experiences, needs and abilities, but this teacher would realize that it is the learner who chooses a personal viable mathematical pathway to suffice the conditions for maintenance of mathematical integrity. Thus one learner interacts with the learning activity, with the teacher, and with other learners, to bring forth personal meaning, which is now shared with others in the learning community.

We learn from cognitive anthropological theory (Lave, 1988; Brown, Collins & Duguid, 1989) the importance of a learning community and the interaction of others within the community. Mathematical concepts learned in social interactive situations are always shaped by the culture of authentic mathematical behavior which the community generates. Within such a community meaning exists for the learner to the extent that the learner is able to interact within the culture and negotiate meaning. 'Cognitive apprenticeship' is a term used by Brown et al. to describe a process of "enculturation into authentic practice through activity and social interaction in a way similar to that evident--and evidently successful--in craft apprenticeship" (p.37). Cognitive apprentices in a mathematics classroom become involved in an authentic mathematical activity where they assume the behavior of mathematicians (Lampert, 1986; Fellows, 1991).

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Teacher Rationale/Methodology: Planning The Trip

With the implementation of the Saskatchewan Elementary Mathematics curriculum province-wide over a short period of time, it seems that a logical step to try gaining a better understanding of the ways in which a primary teacher puts theory into practice, defines her role as teacher and its impact on student learning, should be investigated. As we move to more holistic, child-centered patterns of literacy in our school programs, there is a need to help teachers identify the key elements in their role, to make changes in assessment so consistency is maintained and it becomes a part of instruction, and whenever possible have all the players in the learning process involved. Including assessment and evaluation as part of the instructional process rather than as a separate entity provides for unity and on-going development across all subject areas. An apt analogy is used by Anthony et al (1991) -- if both process and product are assessed, they become more than "snapshots" of achievement; rather they become "videotapes" of performance. As we make changes in curricula and instruction we must also adapt our assessment practices to reflect what we know about children's learning and to enlarge the scope of what and how we evaluate.

Using a classroom-based ethnographic study I hope to identify what I do as teacher as I plan for and implement my mathematics program in a grade one classroom. Through use of videotape, collaboration with Vi Maeers from the University of Regina, parent interviews and feedback, journal writing and reflection I wish to gain an insight into mathematics teaching. Students will be actively involved in doing mathematics, but also in self-evaluation, math journaling, and math IMPACT homework. Excerpts from student work will be used to validate observations. Anecdotal notes regularly made by myself as their teacher will provide on-going documentation of classroom activity and student learning. Since this is collaborative research the field notes made and response journaling with Vi Maeers will be invaluable tools in helping me reflect and articulate the findings of the research as I examine how theory and practice mesh.

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The Research Milieu: On Our Way

Having made the decision to be involved in another research project aimed at improving practice, and having not only received funding but also the required clearances and approval for that research, we were on our way travelling a road which would become long and involved as our research extended over one and then two years before a destination was in sight. Over the two years of this study, there have been a total of forty-seven grade one students involved in the study. These students come from a variety of backgrounds and socio-economic situations and are a multi-cultural group. Located in a middle-class area of an urban centre, the school provides educational opportunities for students from single-parent to two-parent families, upper middle class to low economic households, and a slightly higher than average mix of male to female in the grade grouping. The range of abilities in each year covered about a three-year span although all were grade one students. Adapting material was a necessity.

The classroom was organized with tables and chairs, ice cream buckets for personal supplies, and a range of centres for free choice by students. A mathematics lab was in operation next door to the classroom allowing for flexibility in mathematics programming. Students were initially introduced to new areas of mathematical content as a group; then adaptations were made to meet individual needs; often centre work became the focus as students used a variety of materials to build understanding of math concepts. As children progressed through the year, four or five centres offered reinforcement of previous learning while new work was incorporated at a maximum of two centres to allow for teacher-student interactions and enhancement of learning.

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Research Data Findings from Fall, 1994 to Spring 1996: Highlights of the Excursion

After examining the field notes provided from Vi's observations, my own reflective journaling, parent feedback and information provided from three parent interviews on the role of the teacher, and student work samples and feedback, I feel there exist some commonalities from the various sources. In this collaborative manipulative mathematics research examining teacher's role, we looked at each of the participants' perspectives individually and then drew some common themes from this analysis. The following are the verbatim phrases and ideas expressed by each source:

From Many Perspectives

1. Through (my collaborative partner) Vi's eyes, the following were highlighted:
   a) Community of Learners
   b) making productive use of time
   c) child-centred, collaborative interactions
   d) teacher knowledge of curriculum
   e) connecting theory and practice
   f) student input
   g) class management
   - listing skills for math
   - use of teachable moments
   - peer tutoring and cooperative learning
   h) caring, trusting, thoughtful atmosphere in the classroom
   I) student expectations
   j) teacher attitude and love of teaching
   k) Problem of the Day

2. Through parents' eyes, the following were highlighted:
   a) student understanding
   b) integration
   c) mathematics more than rote memorization-connections
   d) extension of math beyond the classroom-parent involvement
   e) problem solving and sharing of strategies
   f) application aspect of mathematics
   g) math seen as a positive - "fun"
   h) homework tasks allow for parent-child interaction
   I) teacher facilitates children's learning
   j) mathematics is fun as students are interested in it
   k) taking children beyond curriculum and parent expectations
   l) creativity and imagination
   meeting) Parent Nights

3. Through students' eyes, we see simpler more immediate and personal ideas:
   a) Math is fun!
   b) We get to say how we did! (student reflection)
   c) problem of the day is a highlight
   d) self-assessment
   e) planning and decision-making (student planning input)
   f) manipulatives
   g) computers and calculators (technology)
   h) Parent-Child Nights to show all the stuff we know
   I) portfolio proud
   j) favorites-Problem of the Day
   k) centre work

4. Through the teacher's eyes we view many aspects of the on-going day-to-day operation of classroom practice:
   a) on-task, busy right away
   b) regular school routine
   c) cooperative work
   d) Problem of the Day, challenge
   e) discussion, strategies
   f) integration
   g) love of math, fun
   h) activity-based, many and varied experiences
   I) student interest
   j) observation, diagnostic tasks, progress, thinking
   k) student understanding-manipulatives
   l) on-going planning
   m) student input into planning
   n) parent involvement
   o) system involvement
   p) pride, self-esteem
   q) student assistance (peer work)
   r) extension
   s) Parent Night, Buddies, parents and volunteers, feedback
   t) teacher-as-learner, too
   u) teachable moments
   v) adapting curriculum to meet needs
   w) using the environment
   x) Community of Learners

Common Themes
Common themes emerging from each of the perspectives may be organized around four main areas or commonplaces as outlined by Schwab (1983): the student, the teacher, the curriculum and the milieu. These follow the same format as the video THINKING MATHEMATICALLY: The Teacher's Role As Facilitator of Children's Mathematical Learning, produced in conjunction with our research.

I. Students

a) The classroom as a Community of Learners
Building a Community of Learners pays dividends throughout the year as students and teacher are together a group of "teachers and learners" where each has a chance to help and learn from one another. In striving to nurture a sense of community everyone becomes "included"; there is commitment to each other and the class as a whole; there is a growing sense of trust and caring among the participants as each respects, listens, and consults with peers and teacher. Collaboration and consensus guide the classroom interactions as we plan, work and grow together. Communication is necessary as we share interests and experiences within the classroom milieu. As we worked on Hundred Day activities this year the learning was not limited to the students alone; from my journal of February 9, 1996, I write, "Most were surprised at how few it took to make 100 once it was in the baggie (popcorn seeds). That included me!" Working in pairs, we had counted and bagged 100 of each of the available materials to gain an appreciation of what 100 was. We bagged 100 sunflower seeds, straws, cubes, styrofoam peanuts, beans, pasta, etc. Teacher and students certainly learned together.

My usual communication with children early in September as we begin our year together is, "There is not one teacher and twenty-four learners here but rather, twenty-five teachers and learners who can help one another be the best we can be. Each of you is both a teacher and learner at various times, who can help one another and ask for help if needed. Together we can contribute to each other's success during the school year."

Vi sums it up in her field notes as..."Lillian insists that `we're all teachers' in this room. She encourages children to cooperate, help each other, and share what they know." In my notes of Oct. 13, 1994, I write "...we created our string webs. What an observational experience! Many cooperated and helped one another in this individual task We are becoming a caring Community of Teachers and Learners!"

b) Value in hands-on exploration; constructing knowledge
We know from experience that young children require many hands-on experiences with materials as they construct understanding. Explorations with different materials maintain student interest and satisfy their natural curiosity as they experiment. Since gaining understanding takes time, it is important to provide experiences with a variety of contexts. Previous experiences become the foundation on which to build further learning. Affirmation of this comes from Vi's field notes of October 4, 1994, "Lillian tells me...children learn best from a hands-on, manipulative (and) collaborative approach...." On January 12, 1996, Vi's notes include "Lillian gives the children many opportunities to physically interact with materials. Today the children worked in groups with green, red, blue and yellow plasticine, which they rolled into circle shapes and partitioned into halves, fourths, and thirds. I was amazed at how many of these grade one children could correctly partition into thirds." This is further affirmed by parent input as they see a hands-on approach being "... a positive influence, setting no limits to student learning and extending capabilities."

c) Math is fun! It's challenging, yet builds self-esteem.
Students continually report, "I love math." as they write in their math logs or do self-assessments. They report that "Math is fun!" as they are involved in hands-on purposeful learning. It is not the particular materials, but rather the children's interactions with those materials that provides the motivation, challenge, enhanced self-esteem and learning growth. Children manipulate the materials, discuss strategies, share insights, record their findings and assess their own learning in an all-encompassing way to maximize mathematics learning. Parents see "math talk" as a way of allowing student "interaction (and growth of) communication skills." They are often surprised at the depth of student understanding of concepts which they do not recall learning at such a young age. Parents also report that mathematics learning using a hands-on approach allows learning to "stick with them (students) as there is a concrete aspect that children readily hang on to."

d) A collaborative undertaking to build understanding
The teacher may do the planning for learning, but children, too, have a chance for input at our end-of-month assessment sessions where there is a chance for all to have input into the highs and lows of the past monthly theme and to make suggestions for activities which may be incorporated into the upcoming work. This provides students with a vested interest in what and how they will be learning. From my journal notes of October 3, 1995, I have written, "It's amazing how many said that Problem of the Day was a highlight of our month of September when we did a recap of the month. I was pleased!" Not only had the children had an opportunity to communicate likes and dislikes, but I got the sense that the challenge of problem solving whether individually or as a group offered these children a chance to test their limits, collaborate and have success. In Vi's Fall, 1995 (Day 34) notes, she indicates that the children have expressed "...Problem of the Day is the most favorite event in the classroom." She continues with "Why? My observations and notes reveal that during Problem of the Day all children are focused on a single topic; they all can and do relate their talk to the topic and to each other. All contributions are valued. Children feel good about contributing."

The chance to have input into planning also ensures that student interests are highlighted, that their personal learning styles are acknowledged, and that together we can build personal values and skills as teachers and learners working together. Student input also helps me, as the teacher, assess the planned mathematics curriculum and whether it is meeting the needs of my students. Parent feedback indicates that they also see the strength of a manipulative focus "...to help them (students) understand the concepts behind the manipulations and apply that knowledge." Interactions between students and teacher as we work together serve to build bridges to learning.

e) Self-assessment, Peer-assessment, Logs and Portfolios
In keeping with my belief that we are a Community of Learners, assessment must be linked to instruction and cannot be only a teacher-directed activity. Students in my class take pride in being able to self-assess at the end of our mathematics lessons by reflectively examining personal performance and indicating the degree of success achieved through a thumb-up, sideways or down assessment. Problem solving and goal setting for future activities then follow with the input coming not from the teacher, but rather from the students who are doing the individual assessment. Other assessments (happy faces, conferencing about student work, and peer assessment) allow for student input and problem solving as well as reflective insights and peer interactions. Students even at grade one are able to look at personal and peer performance and objectively provide helpful and insightful feedback. From my journal of September 28, 1994, I note "self-evaluation (student) was really reflective. Those off-task or experiencing problems really could see what didn't go well and how they might change."

Logs and student portfolios are a source of pride for these students. Entirely student-chosen, portfolios are taken home by the students to share with parents prior to our three-way conferences. Student-choice ensures that students are aware of what and why materials are included in their portfolios and allows them to take the initiative in sharing their work with parents. This is a great way to build student self-esteem! From my January 11, 1996, journal notes I have written "...started our math logs. The kids really like the idea of a math "journal" as they call it. We'll hopefully use it regularly as we record our math experiences..."

This provides an on-going record of our math work and allows students to respond to the mathematics they are involved in regularly. Children often express their love of mathematics and the activities they are involved in, when they reflect, self-assess, goal set, or record their feelings in Friday reports or in their journals and logs.

II. Teacher

a) Love of teaching
The journaling I do points directly to the enjoyment I experience in my interactions with students on a daily basis. At times, instruction may not work out as planned, but all interactions become learning experiences for future work with my students. Every opportunity needs to be taken to "ask" students for input rather than to "tell" them what you perceive as being important. Remembering to "guide" students through mathematical experiences rather than always playing the role of "director" is important for then you have the enjoyment of watching students make discoveries and add new perspectives to their mathematical knowledge. From my November 2, 1994 notes, I write "...Children did a great job of carving, cleaning out, and sorting pumpkin seeds by tens. I was amazed that they could help add up the total seeds (2024). Wow!" Not only were the children making discoveries about how many seeds really were in the pumpkins but we were enjoying the experience and learning together.

How much of the attitude I have for my subject matter (mathematics) rubs off on my students? There appears to be a direct relationship between my love of mathematics and children's sense of "Mathematics is fun!" They continue to report in their math logs on the fun they had and how much they like mathematics. The students also point out mathematics in the environment on a regular basis indicating the connections they are making to the real-world context for mathematics. On October 7, 1994, I note that "...we also looked around our room and identified all the places we have numbers."

Challenging students to try different strategies, share experiences, and pool resources builds a love of mathematics in the classroom. Students often report that Problem of the Day is a highlight of their mathematical experiences.

b) Knowledge of curriculum and student development
As the teacher of a class of students of varying abilities and at different stages of development, I must plan for a mathematics program that will meet the learning styles and needs of all my students. That's a tall order! For each class it will be different and present new challenges. I can most successfully meet student needs with a hands-on manipulative approach. Using this method of instruction I can circulate amongst the students and use questioning and extension to challenge students. I must be aware of not only curriculum content and mathematical knowledge but also of children's developmental levels. Vi noted on January 19, 1996 "Lillian plans an activity to enable development of spatial awareness and also problem solving-a tessellation activity where the children fit tiles together to cover an area, a task that was self-correcting. Children knew if they were right if the pieces fit together." Vi (October 4, 1994) alludes to the many levels of teacher knowledge required when she writes about the work of Shulman as she responds to what she has learned in her discussions with me and in observing in the classroom "watching Lillian work with her children."

c) Valuing student input
Not only is there student input into planning and assessment, but there is a genuine effort made to value student input when problem solving, sharing strategies, and discussing mathematics learning. Students are free to risk putting forth their ideas, trying alternate solutions, and recording their solutions. The teacher's role here is to question, probe, and facilitate student thinking to extend learning. There is no "right" answer which is praised above others, but rather many approximations and ways of arriving at possible solutions. Productive use of time, staying on task and maximizing human and other resources all play a role in productive student learning. Again from Vi's field notes we have her observation that "...she encouraged their input regarding decisions about their learning."

d) Planning for learning
Having a thorough knowledge of curriculum is just the first step in planning for children's mathematical experiences. My personal mathematics background, knowledge honed from years of experience, and practical knowledge will affect whether I can project a "love of mathematics" into the classroom experience. I must also be aware of the background experiences and mathematics knowledge of the class in order to successfully plan to meet their needs. Children's interests and input will gradually be added to the mix. Other types of knowledge and experience which help in classroom management, monitoring and assessing student needs, and flexibility of planning and implementing will also impact on the classroom experience.

Long range and day-to-day planning ensure that curriculum objectives are met and that scope and sequence are spiraled throughout the year to reinforce and enhance learning. Vi noted "When Lillian plans for the children's math learning she often organizes centres. Prior to working in centres, Lillian spent interactive whole-class time developing concepts." Personal beliefs about use of technology (calculators, computers and audio-visual materials), the need for manipulatives to build understanding, and thematic planning to make connections for students, are ways of helping students apply their mathematical learning beyond paper-pencil tasks. One parent insisted that this approach made mathematics an "adventure" and that this was "innovative" and allowed for mathematics "appreciation and application." Independence is a skill that can be promoted through the use of centres in mathematics. In November I note that "I'm busy setting up centres for next week to allow for student independence-I will work at one centre with students."

e) Setting standards
Teacher expectations can play a role in student achievement. The attitude that students can do the work; that students should find mathematical learning fun; that students should take on ownership for their learning, all play a part in setting classroom climate. There is a sense of pride in accomplishing tasks and learning new concepts. Spiraling the curriculum so that we revisit concepts and consolidate learning builds student self-esteem. Student involvement in assessment acknowledges their role in their own learning. Becoming a Community of Learners where all are both teachers and learners encourages student recognition of their own strengths and areas where they may need further help. Regular goal setting and conferencing increases student awareness and helps them focus on the task at hand. Providing children with a focus for the day [Today we will be looking for good examples of cooperation.] provides for strong community building as a different aspect of a Community of Learners is highlighted throughout the weeks and year. In Vi's notes in what she termed our "Clock Day" in November, 1995 she writes, " Occasionally, when appropriate, Lillian has children complete worksheets (eg. clock faces with hour times filled in). Children have to write the time on the line. During this at-desk work time, Lillian walked around encouraging, supporting, checking for:

a) on-task behaviour,
b) :00 in the hour time,
c) right answers.
Lillian encouraged all kids, not just those who got it right."

Another aspect of setting standards focuses on starting where children are and building on what they already know. It may be part of "just good teaching" but asking children what they already know about a particular concept provides a basis for where you need to begin. The printed curriculum gives guidelines for what students may learn at any particular grade, but we all know that the grade one student comes to the classroom with a unique set of experiences and at a variety of different developmental levels. The printed curriculum need not limit how far students progress in any particular area. Grids are not part of the grade one mathematics curriculum, but through integration of map symbols, grids become one of the very positive things my grade ones learn. Starting from physically placing ourselves on large floor grids using specific coordinates (A6, D2, H7...) my students quickly progress to building their own grids from scratch using pieces of one-inch chart paper. In our assessments, they continually request grids as one of the new mathematics centres integrated into our themes. Other concepts can be successfully extended by incrementally building on what they know, challenging their thinking, and expecting that they will have success. Parents report that a hands-on interactive mathematical experience "...sticks with them, is a positive experience, sets no limits and extends capabilities." Vi's notes of October 4, 1994 as she reflects on her classroom observations, also indicate: "The experience provides the occasion for children's learning - it does not dictate what is learned. The experience is structured to satisfy the children's learning needs."

f) Integration
Integrating mathematics into other thematic content helps students see mathematics as a useful tool beyond the scope of the mathematics class and the schoolroom. The importance of this is stressed in my September 11, 1995 entry, "I took my prep time to cut up apples for tasting. The kids loved it! The line-up to paste their favorite (apple pictures) on the graph was a bit chaotic but the physical line-up was great so we could visually see which was the favorite." Mathematics graphing became one small but important part of our fall Apple theme.

In Vi's September, 1995 notes on our sixteenth day of school, she observes "Lillian spent most of the morning on L words-leaf, lamp, laugh, lamb, last, etc. She had drawn pictures of the words and children identified the words. They listened to L sounds, practised writing the letter L, and they discriminated from an assortment of words which began with L and which did not. Throughout this entire experience, the children were engrossed, interested, sharing info with each other, and learning." It may not have been mathematics at that moment but as the morning progressed we would count pictures, numbers of the letter L in our books, and make connections.

g) Extension
Mathematics goes far beyond classroom activities. We regularly take trips, have speakers come into the class, and make use of volunteers to extend mathematics learning. A retired volunteer enhances student mathematics learning through his willingness to visit the class two mornings per week to work with the students on computers. What a valuable and popular asset he has become to these students! Parents and other volunteers who have the time to come in for special projects or to work with students on a regular basis certainly support any mathematics program. They can often give that needed one-on-one reinforcement to consolidate individual student learning or provide enrichment and challenge for students who are moving through content at an accelerated pace. Just playing mathematics games with students makes math learning fun! As well as in-school mathematics extensions, many possibilities exist for incorporating math activities beyond school. Looking for math in the newspaper, looking for number in the environment, connecting literature and mathematics, and providing games and math homework all reinforce mathematics applications. My November 9, 1995 entry indicates my use of such strategies "...I'm busy designing games as homework tasks." Another way to maximize student learning is to make use of "teachable moments" whenever possible. Vi's notes confirm this when she comments on the use of the Thanksgiving story as a springboard for examining how long ago it was that the Pilgrims came to America.

h) Involving parents
For a number of years now I have involved parents and students in a weekly homework task (Math IMPACT) to allow parents a greater understanding of concepts being taught, a knowledge of how their child thinks through mathematics, and to allow for student self-esteem and demonstration of growing mathematics understandings. Parents report that the tasks allow for parent-child interactions and communication about what is happening at school and that often the homework time becomes a "special time" or a "quiet time" and allows for "real world-home application" of mathematics. Vi, too, reports "Math IMPACT connects the home and classroom in a very real way. I am amazed at what the children and parents accomplish together. This aspect of Lillian's math program is extremely beneficial in:

a) supporting what Lillian is doing in class, and
b) extending and challenging the children to explore new math horizons."

Parent Mathematics Evenings and Parent-Child Math Nights are both popular aspects of extending mathematics. These are reported to be "fun and enjoyable." Students love to demonstrate to Mum and Dad exactly how they work through centres, what they know, and how they record their learning. Parents have, not surprisingly, expressed the wish that these occur more than twice a year. These evenings are seen as a way of involving parents and helping them to better understand how mathematics instruction is occurring and what children are learning. Vi attended our Parent-Child Night and talked to parents about Parent Math Evenings, Parent-Child Math Nights and the Math IMPACT program. She wrote in her notes of January 22,1996 that "Math IMPACT enables parents to follow the major math concepts (what children are learning) being developed in class and to work with and extend their children's learning at home." She continues, "Parent Night provides parents with a window into how children are learning. It was obvious to me that some parents were reluctant to 'take risks' and explore with the materials; some wanted to know if they were right; and some did not know how to build a graph. These Parent Nights are invaluable for parents to understand new instructional approaches and thus be stronger advocates of Lillian's program."

III. Curriculum

a) Making math connections
Curriculum is more than the printed document outlining expectations for student learning at a particular grade level. My conception of curriculum is much broader, encompassing the pre-active planning phase for student learning, the interactive phase where teacher and students are jointly involved in mathematical learning, and the post-active reflective stage where students and teacher reflect on the learning situation and make decisions for further learning. Throughout these stages the teacher serves the role of facilitator of student learning, but the students' engagement with appropriate learning experiences will help mediate what and how the learning will occur. Connecting mathematical learning to students' previous experiences, "scaffolding" and challenging them to extend their learning, and helping students see the applications to other learning and their world will provide ladders to making math meaningful. As a parent stated, "...A hands-on approach lets them see the applications and connections...In my experience math had no bearing in real life; there was school and then there was life...with no carryover." Vi noted the connections made when we shared a "Tooth Fairy" story and then built a variety of graphs about teeth. "These graphs included: cavities, lost teeth, flossing or not, dentist visits, toothpaste type, and brushing. Children who have lost teeth have drawn pictures of these teeth and entered them on tooth charts. The charts are mounted on the wall between the math lab and the classroom. Regularly, on the way to the math lab, children are asked to stop and examine the charts." As children collected and filled in the data the results and interpretations changed. We were making a connection to the children's daily lives.

b) Working together
Teaching mathematics involves active participation and interaction with the students who are interacting with the materials. The teacher's role is one of facilitating learning by asking questions, knowingly involving students in explaining their thinking, posing problems and challenges, and promoting the use of a problem-solving, discovery approach to learning. As parents comment "...There are no limits; manipulative materials provide a positive experience; and active learning extends student capabilities." Together students and teacher can make working together enjoyable and worthwhile and as one parent termed it "make it an adventure." In her November, 1995 notes Vi writes: "I realize that after many observations in Lillian's room that I also place Problem of the Day as my favorite event. [My class often report it as #1.] Why?
1. It brings me up to date and summarizes for me what Lillian has been teaching and what new concepts she is intending the children to learn;
2. It enables me to see the children interacting with Lillian, the topic, each other; and to hear each contribution as it is validated, challenged and/or extended. The Problem of the Day is a time for consolidation and connection to previous knowledge and to new concepts." It involves students and teacher doing mathematics together.

c) Active learning
Curriculum and program documents are merely guidelines to the activity that will occur in the classroom. The teacher and the class will set the pace and the scope of the curriculum, adjusting to meet the needs of the individual members of the class. Moving too quickly, using only paper-pencil tasks, and failing to use manipulatives to build understanding will discourage some students, give mathematics a sense of isolation, and decrease student interest. Parents report that in the past when a hands-on approach was not used, "math was merely teaching skills such as adding and subtracting without focusing on why they were useful." Promotion of active learning means "...enjoyment, real-world applications, interactions, good progress and math that is way more advanced than when I was in school." Vi, too, in her notes from Day 34 of our school year comments on the children's focused and active learning, "...In the grid activity the children demonstrated adequate conduct by providing viable solutions to the problem. All children were focused on task and on Lillian as the facilitator of the conversation. The answers did not appear directed at pleasing Lillian, but rather at providing a 'does it work?' solution-attention riveted on the task/problem/activity and on listening to Lillian and their peers." What she had observed that morning was a group task where we integrated our map symbols into mathematics by learning about co-ordinates and how to use them on our grid. All children participated in either identifying coordinates or coming up to the whiteboard and drawing on a map symbol in the appropriate place-active learning at its best as all checked to see if we were correct as we progressed through a series of problems!

d) Meaning-making and connections
The use of interactive experiences between student-teacher and peers encourages the exchange of ideas and ways of thinking through problems. Cooperative work enhances the opportunities students have to share and scaffold or piggyback on others' ideas. This is a definite asset as students willingly share strategies as they work together. Students report that they like to be teachers and often ask for the chance to help their peers when their own tasks are complete. Parents indicate that "math talk" is valuable as there is "interacting and use of communication skills."

The teacher's role in promoting mathematical meaning-making is evident in the questioning that occurs, the use of the teachable moment, and the oft recognized chance to pull mathematics out of experiences in class, on the playground, on trips, or from student experiences. Having students recall and continually revisit concepts as in our centre work also promotes making connections. Student self-assessment and month-end input into assessment and planning for instruction also helps students assume ownership for their mathematical learning. Parents indicate that students develop an "appreciation for mathematics and see it all around."

Vi observed our grid work in November of 1995. She jotted down notes about valuing student input and the teacher's role in helping children make meaning as we solved our Problem of the Day. She notes "...obviously there were some right answers (eg. F2=railway). Children could choose an object (eg. mountain, river, etc.), say what it is, give position on the grid and then point to it on the whiteboard. They could also say F2 and ask others what it was. Children volunteered answers, were eager to answer, all had their hands up and all contributed adequately to this conversation on grids." Children made connections between our map skills work and mathematics grids as we progressively built on previous experiences.

IV. Milieu

a) Classroom climate as a Community of Learners
Teacher and students together are responsible for developing the classroom environment or milieu. The physical arrangement can be used to promote student interactions. Tables and chairs in my classroom make varied groupings possible. Personal belongings in an ice cream pail allow for flexible movement of students as groupings change frequently. Easily accessible manipulatives allow students to use materials as needed to assist in their work.
The "values" which are built into a Community of Learners will help determine how children interact as they develop mathematical concepts. Including everyone, building a sense of collaboration among peers, and ensuring that the classroom is a safe place will help build community. Stressing the view that we learn from our mistakes enhances risk-taking and frees children to see that in spite of differing strengths we can all learn together. As has been frequently noted this allows all to be both teachers and learners. Children need to recognize that there are many approaches to arriving at a solution. The chance to try and then share their strategies benefits all students. Respect for one another, cooperation, sensitivity to each other's feelings and the willingness to share become the basis for classroom climate. Praise, wise use of questioning, and the chance for student input and assessment strengthen self-esteem and a sense of oneness within the classroom. In our goal setting conferences, mathematics is often named as a positive or something children see themselves as being good at.

Vi's notes on our class grid work is an example of the way all children can be included in our Community of Learners free to risk and grow. In other November 1995 notes, Vi comments on a tile activity the class are working on. Children were asked to "...find a number of ways in which to arrange four tiles to be as different as possible. As the pieces were given out instructions were clarified (eg. whole sides touching). Children accomplished the task and transferred constructions to one-inch grid paper (colored). A very successful activity! Each child made at least two different constructions. Some made many." Further sharing of the constructions enhanced the learning of all students and showed many possibilities. Children became both teachers and learners within the class.

b) Parent involvement
Opening the classroom to visits and making the effort to help "educate" parents about mathematics learning is one of my responsibilities. Through Parent Nights, Orientation meetings, newsletters, communication books, and homework tasks, I strive to make mathematics informative and fun for parents, too. Many parents report that they did not have positive mathematical experiences during their school years. Through our efforts, we can help to change their perceptions about mathematics and together make their children's experiences "special." Parents report that their experiences in mathematics did not include application: "They were teaching skills but not why those skills were useful." Parents also report that a problem-solving approach "allows for home extension, building of student self-esteem, and interactions between parents and their child."

c) Volunteers and buddies
A chance to work with volunteers extends mathematics learning for all of us. For the teacher, it means having extra hands and eyes; for the volunteer, it allows for a feeling of satisfaction in helping another generation of students learn to be the best they can be, besides keeping abreast of what's happening in our schools (that was expressed by a retired volunteer who comes in twice a week to work with my students on computers); for students, it is one more opportunity to learn mathematics. My students love their computer experiences and the chance to work with parent volunteers completing tasks or playing math games. Together we all gain! The same argument can be made for the use of an older group of students as "Buddies" to young learners. Young children enjoy the interactions as Buddies read, work on mathematics tasks, and act as transcribers for early stories, assessments and recordings of mathematics learning. My journal notes of February 6, 1996 state: "Work with our Buddies was great! The kids report that they love working with their Buddy doing math. They also tell me they enjoy reading math stories and doing task cards."

d) Trips
Helping students to see that mathematics occurs everywhere is one of my goals. By doing mathematics beyond the classroom walls, students are increasingly informed about the uses of mathematics. On our walk to the store to purchase pumpkins we looked for number and made decisions about whether those numbers were used to identify or for counting purposes. What awareness! Students now point out numbers wherever we go!

e) Enhanced facilities
The literature certainly acknowledges the benefits of enhanced facilities to making a process-oriented mathematics program effective. This may be in the area of materials and resources, extra funding or in physical arrangements. Having been involved in a hands-on approach to mathematics for ten years or more [workbookless!], I see the value in accumulating a variety of resources for use in my teaching. To promote the use of manipulatives to building student understanding of mathematics concepts, I arranged for the organization of a Math Lab in my school. Into this room went tables, chairs, accessible storage and shelving, print and manipulative resources, charts, etc. This allows teachers and students to access materials from a central place and to make use of the room for regular class instruction. Having the resources readily available saves planning time for teachers and encourages use of manipulative experiences.

f) Reaching out to the community
Providing inservice sessions, promoting the use of a mathematics lab, and welcoming visitors to the classroom are ways of promoting a rich mathematical experience for students. Involvement with the University of Regina's pre-service programs on a regular basis, welcoming undergraduate students to the classroom, inviting other teachers to visit our classroom, and opening the classroom to parents and other visitors are ways of promoting mathematics learning. Vi notes in October, 1995, "I am impressed by Lillian's energy and commitment to helping others understand the changes in the mathematics curriculum. She was off to Newfoundland to NCTM (National Council of Teachers of Mathematics conference), was presenting at SMTS (Saskatchewan Mathematics Teachers Society-Sciematics), did a radio talk show with Rita (She was the chairperson of the NCTM Conference in St. Johns) all in October, 1995. She speaks on topics such as changes occurring in mathematics, the importance of parental involvement in education, and communication between home and school. She also spends each afternoon writing math and science assessment tasks and end-of-year grade one tests." What Vi says is somewhat true as I have a real love for the science-mathematics area and hope I can communicate that to my students and to others.

g) System support
An often unrecognized factor in how "teacher role" is played out in the classroom, is the support of administration and the school division. Over the years, I have often documented in my journaling, the role of principal as mentor, colleagues as supporters and fellow travellers on the road to change, and the role of administrators in allowing innovative changes to occur. I can report that within our school division, there is an emphasis on grassroots involvement in the change process, and support for individual growth as professionals. As well, work with the Faculty of Education personnel and students adds fresh perspectives to encourage and enhance willingness to make changes and grow professionally. Support in this research came from the University of Regina as Vi made the commitment to visit and collaborate with me as I examined teacher role. This research was made possible through the financial support of the Dr. Stirling McDowell Foundation.

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Summary

This research study certainly highlighted for Vi, parents, students and myself some of the factors which are of importance to identifying teacher role in the mathematics classroom. These factors, however, will vary from classroom to classroom. As documented by both my collaborative partner and through parent feedback the following are critical to teacher role in my classroom:

• engagement of students in building mathematical understanding through hands-on, experiential learning;

• involvement of all the players-students, parents, and teacher in collaborative interactive learning to promote a positive view of mathematics and help students make the connections and apply their knowledge beyond the classroom;

• involving parents in the process through Parent evenings, communications, and Math IMPACT so they communicate with their children about mathematics learning and gain an appreciation of what and how mathematics is approached in the classroom context;

• the use of teacher facilitation and "math talk" such as sharing problem-solving strategies, peer and student-teacher interactions, questioning, and challenging students to extend learning;

• making mathematics learning "fun" and interesting so that students enjoy it and develop a positive attitude towards mathematics;

• providing opportunities for students to extend their learning beyond curricular expectations through mental math, estimation, problem solving and the setting of high expectations for student learning and classroom work.

Participating collaboratively in this study has not altered my personal beliefs, but it has certainly 'illuminated' or highlighted aspects of my practice which only stood out as significant through documentation and use of feedback from Vi and parents. This research affirmed many of my beliefs about children's learning while enhancing my perception of myself as a constructivist teacher-one believing that children construct their own knowledge through multiple experiences, that children need both autonomy and direction, that children's interests, curiosity and abilities should be considered in planning, and that children should become equal partners in the teaching-learning process. My appreciation of the role of parents in education was again affirmed, as was the importance of having students play a role in planning for instruction and assessment. Integration and extension of mathematics activities in and beyond the school shone brightly as I analyzed data from all sources.

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Problems and/or Limitations

This study is just one way of examining teacher role in the primary mathematics classroom. It is unlikely that this study is generalizable to other classrooms. Certain characteristics and factors may be evident in many classrooms but the effect of each will vary. Although we have discussed the teacher's role from various perspectives, each classroom and each new year will impact on the mathematics teaching and thus the role of the teacher. Striving to identify critical factors was a useful exercise, but only highlighted how many factors impinge on each classroom and teacher.

Not only did this study provide an opportunity to examine my practice from both a personal and outside perspective, but the study in and of itself increased my awareness of my practice. In this study, teacher role data was analyzed through the commonplaces: students, teacher (experience, background, knowledge, insights and efficiencies), curriculum and milieu. A similar form of analysis could be used in other studies, but results may emerge in a variety of ways dependent on individual classroom dynamics.

Not only has the interactive, hands-on approach to mathematics learning been emphasized in this study, but the degree of parent involvement stood out as an important factor in making student mathematics learning and application fun. Will this occur elsewhere?

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Recommendations, Implications and Areas for Further Research

Identification of some of the "critical" factors which combine to highlight the role of the teacher was possible in a study such as this. Perhaps even more useful would be other similar studies to examine consistency of certain factors in teachers of mathematics whose programs appear to be successful. Many of the characteristics identified in research literature may have been evident in the classroom, but the impact of each factor is difficult to judge. Often it is the "whole package" which together makes the classroom milieu a healthy learning environment.

"Of particular note in this study was the importance of Problem of the Day as documented by students, Lillian and Vi. What factors impinge on Problem of the Day to make this such an important part of Lillian's math program?" Vi wrote at the end of the study. This would be worth pursuing to find out why and how it develops and why it is so popular.

In this study parental involvement was critical. Parents can be strong advocates of a program if they feel involved in it, if they understand what is happening in the classroom and they feel they play an important part in their children's learning. An important follow-up study could investigate parent participation (or lack of it) in classroom learning, their associated feeling of involvement, and with their children's sense of accomplishment and success in math.

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References

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Parting Thoughts

This is a personal expression of my thoughts on this research undertaking. Often the words I use to describe my research are a synthesis of the journey through the project.

What is Mathematics Teaching and Learning?

Student-empowerment
initiating activities, seeking knowledge
cooperating, discovering, actively learning
sharing, discussing, problem-solving
freely thinking, creatively working
building understanding.

Teacher-facilitation
modeling, instructing, moving onward
encouraging, questioning, challenging
scaffolding, observing, caring, providing learning contexts
allowing student sharing and decision-making
following student interests.

Interactive mathematics
suited to student needs, collaborative
manipulative, discovery-based, creative
making connections, moving from concrete to abstract
building on student knowledge, providing variety
integrated across the curricula.

Mathematics in my classroom
meaningful, stimulating, liberating,
open-ended, allowing for student extension
absorbing, inspiring, involving student-input
thinking, predicting, problem-solving
responsive and reflective.