Professional Learning Goals

• Learned more about what it means to develop algebraic reasoning, and to purposefully plan for this development
• Learned more about how to assess and track our students’ algebraic reasoning
• Saw the effects guided play can have on our students’ algebraic reasoning
• Learned more about designing and implementing materials and/or assessment tools related to algebraic reasoning
• Effectively used these materials and/or assessment tools and were able to implement easily
• Better understood how spatial reasoning and algebraic reasoning connect

Activities and Resources

Based on the research we have studied, we co-planned, co-taught, co-debriefed and engaged in professional dialogue about the action research we conducted. We documented student learning and growth, and reflected on how that documentation represented learning. Student learning occurred within large group settings and small group settings, as well as through guided play and provocations.

We deepened our own mathematical content knowledge together through professional readings and through professional learning opportunities with experts in the area (Tara Flynn, Petra LeDuc, Cathy Bruce).

During our time together, we interviewed and documented students’ thinking to inform next steps.

Unexpected Challenges

• Since there is a limited body of research related to algebraic reasoning in relation to young students, we had to co-create both the experiences we wanted to present for our students, and a basic pathway of how algebraic reasoning develops over time.
• We found it difficult to know when students were using algebraic reasoning, spatial reasoning or skills related to numbers and counting. We thought we would be only helping develop algebraic skills, but many were also developing/using counting skills and spatial reasoning.
• Developing algebraic reasoning didn’t happen in the few days that we have release time for (need many formal/informal experiences every day)
• Viewing the learning for each student is complicated. We looked at each student’s learning across all four frames, and across several overall expectations from within the Literacy and Mathematics frame. Our goal was to help make this easier for us, but the more we noticed, the more we realized how interconnected the learning is, yet how individualized the learning is for each specific student.
• Seeing students’ strengths from what they can do (asset model of thinking) as opposed to the traditional deficit model of thinking (finding gaps)

Enhancing Student Learning and Development

Our research lead us to four key messages that guided our project and shaped the way we think about spatial thinking and learning:

1. Children from an early age are capable of complex thought, including algebraic reasoning.
2. Students need a variety of experiences to make connections between spatial reasoning and algebraic reasoning.
3. Algebraic reasoning can be fostered through guided play experiences, hands-on learning and other kinesthetic opportunities.
4. Balance and equality are essential to thinking algebraically.

We created a list of look-fors based on some of our tasks. These look-fors illustrate a potential development of students’ algebraic reasoning (see Algebraic Reasoning Trajectory). We found that knowing a possible trajectory was very helpful for us to know what experiences we should be providing for our students, and what to look for/listen for as they were engaged in the tasks.

Viewing our students as capable and competent helped us think about the kinds of experiences we wanted to provide for our students to develop algebraic reasoning. We noticed increased mathematical dialogue, increased engagement, and students’ views of themselves and each other as capable and competent budding mathematicians.

Sharing

On June 12, we will be sharing our work with a variety of stakeholders (principals, superintendents, DECEs, consultants, instructional coaches) in the format of a public research lesson. We enjoyed sharing our findings at the Ontario Association for Mathematics Excellence (OAME) conference in Toronto on May 4 (PLEASE CLICK HERE TO VIEW SLIDESHOW). We plan on sharing our learning with our teaching partners and other kindergarten/Grade 1 teachers at our school and are hoping to share our journey with other kindergarten/Grade 1 teachers across our board as Professional Development.

Project Evaluation

Each of our action-research members provided insight on our journeys:

Kristin Willms (Instructional Coach): I have already learned so much about algebraic reasoning with young learners even though I feel like we’ve just started to “scratch the surface” with this project. At the beginning of the project when we first started talking about what we were interested in learning about this year in September, I couldn’t really imagine what it would look like to inquire about four- and five-year-old’s algebraic thinking. Now, as we are deeply immersed in the inquiry, I am definitely starting to see how this important underpinning of mathematical thought is such an important and interesting area of research. Our Kindergarten Program Document states that kindergarten children are “capable and competent.” This belief has permeated my thinking throughout the course of this inquiry. I do see how our youngest learners are capable of engaging in tasks that allow them to develop their algebraic reasoning and they can do so with success. When Kristy and Sue presented their 3-Part Lesson at our math AQ one night and they had the title up on the screen, “Algebraic Thinking in Kindergarten,” I overheard a couple of comments in the room that sounded slightly flabbergasted (i.e., “Algebra in Kindergarten? What?”). I am proud of the work we have done and continue to do to break down some of those myths that are young students aren’t capable of such complex thought. I am also looking forward to learning more even after this as we continue our journey together with this.

Melanie Otta (Kindergarten Teacher): I have learned about the importance of giving the students experiences either through provocations or play or through teacher-guided lessons in order to explore the idea of equality. I found that I have begun to make intentional observations to discover where they are in their understanding of equality. I am focusing on using the mathematical vocabulary and getting them familiar with the terms, “more,” “less” or “equal.” I was surprised with how well my kindergarten students did with finding the missing part of an equation, which came out during the mystery bag challenge and how well they composed different ways of making a number or a quantity, which helped to drive my next steps.

Pam Schonewille (Grade 1 Teacher): This project really opened my eyes to see the development of spatial and algebraic reasoning through kindergarten and Grade 1. As a Grade 1 teacher, it is so beneficial to see what experiences our youngest learners have. Watching how some of my students engaged in the tasks, explained their thinking and applied concepts across other strands and problems was fascinating. One student in particular had some fragile understandings throughout this journey. It was rewarding to see him using the relational (Cuisenaire) rods to demonstrate his thinking about a problem and to know that he is moving forward in his understanding of equality. My students love playing the train game and creating new popsicles with different amounts of cubes during their free choice time.

Susan Di Teodoro (Kindergarten Teacher): One thing that I really noticed during the course of this project was the growth of relationships.  The first and most important one was with my students. They love challenges and truly think of themselves as competent mathematicians. The second most important relationship that was strengthened was with my DECE partner as she was just as excited as I was and we spent much time after school discussing what we had seen and next steps for our students and especially what surprised us and why. My principal and LRT, as well as many fellow teachers in my school, were interested in our discoveries and started popping in more often. Our parents were also engaged and often asked for explanations after the convoluted stories of our kindergarten students were told. Very last was the support and questions we got from our professional learning community through Twitter.

Kristia Penlington (Kindergarten Teacher): As I reflect upon my journey over the past two years, I have spent time reflecting on myself as a learner. I have become a much better listener. By listener, I mean that I am looking at what the children “can” do.  I have taken time to quietly observe the children during play and noticed what they were doing and saying. I noticed that when the children play, everything they do has a purpose. They communicate what they have learned and what they know through their play. This reminds me of Loris Malaguuzzi’s famous poem, “The child has a hundred languages…”  Children do not only communicate their math knowledge through worksheets, they show you every day through their symmetrical towers made from blocks, the intricate way they set the table in the kitchen, how they use space on the paper when they paint. By “listening,” I have become a better observer as well. I step back and observe. I guide and ask questions when the time is right, sometimes silence and documentation is all that is needed. I believe that every mess tells a story! I noticed a mess and my students taught me, “This was not balanced (pointing to a rock and the gems). The sides were unequal and the scale tipped over.” I have taken a spatial and algebraic lens into our play. Children speak 100 languages. What are they telling us mathematically? They are communicating their math knowledge through what they say and what they do. Are you listening?

Mark Chubb (Instructional Coach):  Throughout our time together over the past two years, I have come to see just how capable young mathematicians can be. Together, we have created and implemented many lessons where our students are developing both in their specific content and in their mathematical reasoning. This year, we started with a very challenging question, “What does algebraic reasoning look like for young students?”. With this question, we have come to see what it looks like to create a progression of experiences that will help students develop the underpinnings of important mathematics. Specifically, we learned that looking at complex mathematics with students from the early years doesn’t mean just doing the same things with smaller numbers. Instead, we’ve learned that there are much better experiences where students can access their spatial reasoning to help them make connections between concepts and visuals. As a coach, I have been impressed with the level of pedagogical dialogue within our group. Coming together to create and plan specific lessons (lesson study) is a powerful way for teachers to learn!