Introduction

• asking questions that help students understand new mathematical concepts andskills;• identifying and addressing misconceptions;• making “at the moment” decisions about where to go next to best build the mathematicalknowledge of the small group;• monitoring group processes, routines, and behaviours;• encouraging students to make collaborative decisions (e.g., about which manipulativesto use, what strategies to use to solve a problem);• encouraging individual responsibility in each of the students working in a group;• promoting pair or small-group discussion;• pairing an English language learner with a peer who speaks the same first languageand also speaks English, and allowing the students to converse about mathematicalideas in their first language;• making modifications or providing extensions for individuals or groups of students;• facilitating discussions in which students share their ideas, strategies, and solutionswith others;• encouraging students to consider the appropriateness, effectiveness, and accuracyof different strategies.For an example of shared mathematics in the primary grades, see “Problem-SolvingVignette – Grade 1” in Chapter 5: Problem Solving, in Volume Two. The characteristicsof this lesson that make it an example of shared mathematics are as follows:• Students work in pairs to determine the different numbers of gerbils that can be putin two cages.• The teacher circulates around the classroom to facilitate and monitor discussions.• At different points in the lesson, students discuss the problem together as a class,sharing their strategies for sorting the gerbils.• Students make the decision about what manipulatives to use to help solve theproblem.Chapter 5 also contains an example of shared mathematics at the junior level:“Problem-Solving Vignette – Grade 4”.68 A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Volume One