President's Message - September 2011

Teachers Preparing for Knowledge Building Classrooms - Aligning, Curriculum, Instruction, and Assess

In my role as mathematics consultant, I have the privilege of working and learning alongside teachers of mathematics. Through co-planning, co-teaching, and codebriefing, we work collaboratively to implement teaching approaches and strategies that engage students in rich mathematical experiences, enabling them to develop understanding of concepts as they problem-solve, select tools and strategies, represent their thinking, reason, make connections, and communicate and reflect upon their understanding. During co-planning, we develop a learning goal that reflects the important mathematics or big ideas, and is based on curriculum and mathematical process expectations. Once we articulate the learning goals and anticipated success criteria, we select a problem that will provide an entry point for all students, enabling them to engage in and make sense of the mathematics identified in the learning goal. This problem is crafted to be in the "zone of proximal development" for students, challenging and engaging, yet accessible. In preparation for our coteaching, we, as learners, do the problem so we can anticipate student solutions. This work naturally prompts us to consider the role of assessment for learning, which is, "the process of seeking and interpreting evidence for use by learners and their teachers to decide where the learners are in their learning, where they need to go, and how best to get there" (Assessment Reform Group, 2002, p. 2). As we work to align the learning goal and anticipated success criteria, we prepare "our eyes to see and ears to hear student thinking" (Ontario Ministry of Education, 2005). In our effort to elicit information about student learning, we consider assessment strategies that will help us gather data during the lesson. As we coteach, we observe students as they engage in the mathematics that emerges while solving the problem. We listen as students pose questions to one another as they clarify and expose thinking. As co-teachers, we then facilitate the analysis of solutions and consolidate the learning through an interactive summary, which we expect will reflect our anticipated success criteria. During the co-debrief session, we triangulate our observations and conversations, and student products (Ontario Ministry of Education, 2010, p. 34) to clearly understand what a student knows and can do. At times, we discover that the student thinking is unclear or lacks sufficient detail for our triangulation. This valuable information serves as a guide for feedback in an effort to move student thinking forward, and prompts us to consider next steps for instruction. As we engage in this process collaboratively, we develop the content knowledge for teaching (CKT-M) needed to assess and respond to students' mathematical thinking.

A Classroom Example

During a co-teaching cycle with a Grade 9 Applied Mathematics teacher, we co-planned a lesson focused on proportional reasoning. In an effort to assess students' representations of proportional thinking and repertoire of problem-solving strategies, we selected and adapted a question from the Grade 9 Applied EQAO Winter Assessment (2006), #2b.

Paul's Quilt
Paul's grandmother is sewing a quilt for him. She asks him to cut red and white pieces. Every 5 pieces in the quilt consist of 2 red pieces and 3 white pieces. If the quilt has a total of 60 pieces, how many pieces are there of each colour? Show your work.

The original problem presented a quilt with a total of 300 pieces. We changed this number to 60, recognizing that this adjustment would enable students to access the problem and solve it, using a variety of tools and strategies.

In preparation for our observations and conversations with students, we considered the way in which we would assess students' thinking.

• What mathematical thinking is evident in the solutions?
• What is not clear?
• What questions, if any, should we pose to further elicit student thinking?
• What feedback might we provide?

During the lesson, I circulated as students solved the problem.

I observed Maria as she created her model (Sample 1), using red and white tiles. It appeared evident from her model that she u n d e r s t o o d t h e proportional relationship (2:3) and the whole (60). In observing Maria, I noticed that she applied her understanding of a proportional relationship by maintaining the proportion to generate the number of red and white pieces. I watched her skip-count to determine the total number of red and white pieces. Maria represented her solution as a ratio, noting "by 2's" and "12 groups." I asked her to share the significance of these notes. Maria indicated that she skip counted by 2's, 12 times in order to determine the number of red pieces. Through observations, conversations with Maria, and by examining her product, it was evident that she had an understanding of the proportional relationship and could solve the problem, using concrete materials. The coteacher and I would need to support her in further annotating her mathematical thinking on her product.

As the co-teacher and I examined Sample 2, created by Josh, we noticed how he used the given parts to create a whole that would be multiplied by 12 to find the total number of pieces in the quilt. This scale factor was used to determine the number of red and white pieces in the quilt. Clearly, Josh's solution exposed his proportional reasoning.

As Hakim worked, the co-teacher and I watched as he used the parts given in the problem to create a fraction 2/3 (Sample 3). I wondered whether he would make the common error of comparing this fraction with a fraction involving the whole. To our surprise, we watched as Hakim created equivalent fractions. In my zeal to understand Hakim's thinking, I ask him, "Tell me about your strategy." He proceeded to describe his "test and check" method that had him use easy numbers to multiply by to find the right total. This systematic trialand- error strategy, unbeknownst to Hakim, enabled him to find the scale factor, and then solve the problem. As teachers, we knew that we would have to address the precision in mathematical form in his solution and agreed that the discussion would focus on the concept of equality.

During our Consolidation, we took the opportunity to have students compare and contrast their solutions. Students quickly identified commonalities across the solutions. They saw the parts, 2 and 3, the wholes, 5 and 60, and the 12, which we named the scale factor. In prompting students to find the 12 in the model from Sample 1, students suggested that we rearrange the 6 x 10 array to form a 12 x 5 array so that the scale factor would be easily seen. We introduced the concept of variable as a placeholder for the unknown number of pieces of, for example, red pieces. We also introduced a proportion equation, , and asked students to identify the similarities and differences between this representation and the representations presented in the student solutions. Students made connections with ease, once again, identifying parts and wholes, and determined the scale factor, which would then be used to find x = 24.

Using Assessment for Learning to Inform Instruction

During the co-debrief, the co-teacher and I discussed the mathematical thinking that was evident within the solutions and how our observations and conversations with students opened the window to students' understanding of proportional relationships. Through further deconstruction and analysis of the solutions, we discussed next steps for instruction which, for some, included formalizing the mathematical thinking that was shared during conversations, for others, precision in communicating mathematical thinking, and for the entire class, further supporting students in generalizing a proportional relationship using an equation. The next day's lesson would then provide students with an opportunity to use an elastometer to further develop conceptual understanding of proportions and enable a transition to representing proportional relationships using an equation. For more information on creating and using an elastometer, visit www.edugains.ca/resources/ ProfessionalLearning/SelfDirectedPLM/ClassroomIn structionalStrategies/index.html?movieID=3..

Creating the Conditions for Success

Through observations, conversations, and products, we create the ideal conditions for uncovering what students truly understand and can do. As Fullan, Hill, and Crevola (2006) suggest, "Assessment for learning is information gathered today about what needs to be done tomorrow," so let us "prepare our eyes to see, ears to hear" and voice to respond.

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References
Assessment Reform Group. (2002). Assessment for learning: 10 principles. Retrieved from www.assessment-reformgroup. org
Fullan, M., Hill, P., & Crevola, C. (2006). Breakthrough. Thousand Oaks, CA: Corwin.
Ontario Ministry of Education. (2010). Growing success: Assessment, evaluation, and reporting in Ontario schools – First edition, covering grades 1 to 12. Toronto: Queen's Printer for Ontario.
Ontario Ministry of Education, Literacy and Numeracy Secretariat. (2005). Coaching institute for literacy and numeracy leaders. Toronto: Queen's Printer for Ontario.

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