# Capacity Building Series

## The Capacity Building Series is produced by the Literacy and Numeracy Secretariat to support leadership and instructional effectiveness in Ontario schools. Excerpts from each monograph are listed below, and the images are hyperlinked to the full document.

The challenges [of learning about fractions] begin at an early age (Bruce & Flynn, 2011), and, over time, a lack of fractions understanding often forces students to memorize rules “to get by” because the foundations are not in place. And the rules, when not supported with conceptual understanding, become muddled.

*Fractions Across the Curriculum, *pg. 1

In order to know what questions to ask to move the mathematical ideas forward, it is critical that teachers continually work to develop their knowledge of mathematicsfor-teaching as they connect this understanding to the curriculum. By listening attentively to students’ ideas and keeping the learning goal and the big mathematical ideas in mind, we are able to identify and develop the important ideas in the students’ discourse.

*Asking Effective Questions, pg. 1.*

The development of students’ mathematical communication shifts in precision and sophistication throughout the primary, junior and intermediate grades, yet the underlying characteristics remain applicable across all grades. During whole-class discussion, teachers can use these characteristics as a guide both for interpreting and assessing students’ presentations of their mathematical thinking and for determining discussion points:

about problem details, relevant choice of method or strategy to solve the problem, accurate calculations**precision**that show how the details of the mathematical task/problem are addressed in the solution**assumptions and generalizations**in terms of logical organization for the reader’s ease of comprehension, requiring little or no reader inference**clarity**- a
that consists of an interplay of explanations, diagrams, graphs, tables and mathematical examples**cohesive argument** that explain and justify mathematical ideas and strategies with sufficient and significant mathematical detail**elaborations**- appropriate and accurate use of
for labelling graphs and diagrams.**mathematical terminology, symbolic notation and standard forms**

*Communication in the Mathematics Classroom, pgs. 1-2.*

Teachers can help students achieve their potential as learners by providing learning and consolidation tasks that are within the student’s “zone of proximal development.” The zone of proximal development, a phrase coined by the psychologist, Lev Vygotsky, refers to the student’s capacity for learning. Technically, it is “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers” (Ontario Ministry of Education, 2005a, p. 61). Identifying the student’s zone of proximal development is of paramount importance if differentiated instruction is to achieve its maximum impact.

This monograph focuses on differentiating instruction in the mathematics classroom. It describes several classroom strategies for differentiating mathematics instruction – namely, focusing instruction on key concepts, using an instructional trajectory or learning landscape for planning and designing open and parallel tasks.

*Differentiating Mathematics Instruciton, pg. 1.*

### “Numeracy is not the same as mathematics, nor is it an alternative to mathematics. Rather, it is an equal and supporting partner in helping students learn to cope with the quantitative demands of modern society.” (4)

### Numeracy is about doing the math – about recognizing and using mathematics – in a variety of contexts that range from the everyday to the unusual; it’s about being able to use mathematics as a tool to explore problems and situations. There is a vast array of opportunities to explore numeracy in the curriculum in every subject area.

*Supporting Numeracy, pg. 2.*

Growing evidence indicates that early mathematics plays a significant role in later education. From an analysis of six longitudinal studies, Duncan and colleagues found that early mathematics skills were more powerful predictors of later academic achievement in both mathematics and reading than attentional, socioemotional or reading skills (2007, p. 1428).

In addition, the differences in mathematical experiences that children receive in their early years “have long-lasting implications for later school achievement, becoming more pronounced during elementary school … and continuing on into middle school and high school” (Klibanoff, 2006, p. 59). Such findings raise a critical question:

### How can educators take advantage of the mathematical knowledge and experience that children bring to early primary classrooms?

*Maximizing Student Mathematical Learning in the Early Years, pg. 1-2.*

## Preparing for Consolidation

- WHAT MATHEMATICS IS EVIDENT IN STUDENTS’ COMMUNICATION (ORAL, WRITTEN, MODELLED)?
- WHAT MATHEMATICAL LANGUAGE SHOULD WE USE TO ARTICULATE THE MATHEMATICS WE SEE AND HEAR FROM STUDENTS (i.e., MATHEMATICAL ACTIONS, CONCEPTS, STRATEGIES, MODELS OF REPRESENTATION)?
- WHAT MATHEMATICAL CONNECTIONS CAN BE DISCERNED BETWEEN STUDENTS’ DIFFERENT SOLUTIONS? HOW ARE THE SOLUTIONS MATHEMATICALLY RELATED?
- WHAT MATHEMATICS (i.e., CONCEPT, ALGORITHM, STRATEGY, MODEL OF REPRESENTATION) ARE THE STUDENTS USING IN THEIR SOLUTION? HOW DOES THE MATHEMATICS IN THE SOLUTION RELATE TO THE MATHEMATICS LESSON LEARNING GOAL?
- WHICH SOLUTIONS ARE CONCEPTUALLY-BASED? WHICH SOLUTIONS HAVE AN EFFICIENT METHOD OR ALGORITHM? WHICH SOLUTIONS INCLUDE A MATHEMATICAL GENERALIZATION?
- HOW ARE THE SOLUTIONS RELATED TO ONE ANOTHER, MATHEMATICALLY? HOW ARE THE SOLUTIONS RELATED TO THE MATHEMATICS LEARNING GOAL OF THE LESSON?

Mathematically literate students demonstrate the capacity to “formulate, employ and interpret mathematics” (OECD, 2012, p. 4); they view themselves as mathematicians, knowing that mathematics can be used to understand important issues and to solve meaningful problems, not just in school but in life. By extension, the physical environment for mathematics learning should include:

- Spaces where students can use
**manipulatives**to solve problems and record their solutions. - Board and/or wall space to
**display student solutions**for Math Congress and Bansho – student solutions should be easily visible from the group gathering space. - Space to post
**co-created reference charts**such as glossary terms and past and current summaries of learning that specifically support the development of the big ideas currently under study. - Instructional
**materials organized in such a way as to provide easy selection and access for all student**s; may include mathematics manipulatives, calculators and other mathematical tools, mathematical texts, hand-held technology.

*The Third Teacher, pg. 2.*

## What makes documentation pedagogical?

Educators make important decisions regarding why, where and with whom they share their documentation, depending on its purpose. But what makes documentation pedagogical? In the words of educators participating in a ministry professional learning session for teaching in challenging circumstances, “[We] are using what has been recorded to reveal the learning within the documentation.” They suggest that what makes their team’s documentation pedagogical is “discussing the wonderings they have about the documentation, the inferences they are making from it and where they need to go next in the learning.” In essence, then, part of what makes documentation pedagogical is the careful, iterative process of examining and responding to the interplay between learning, the educator’s pedagogical decisions, and the student’s role and voice in the learning.

*Pedagogical Documentation Revisited, pg. 2.*