Introduction

• working with concrete materials to represent abstract conceptsFor example, students working with geoboards to determine the smallest perimeterfor the greatest area may develop a richer understanding of the interrelationsbetween area and perimeter.• using investigation and inquiry to explore problemsStudents who engage in problem solving build a repertoire of reasoning skills andstrategies. For example, asking students to find all the ways to combine two setsof blocks to make the sum of 10 encourages them to reason about the patterns andrelationships in the numbers from 1 to 10 and eventually to extend their reasoningto all the numbers between 10 and 20, then 20 and 30, and so on, to 100. The patternsand relationships in the numbers from 1 to 10 permeate our entire number system.Similarly, older students who investigate the relationships between the fraction 1 ⁄4,the decimal 0.25, and the percentage 25% gain a more complex understanding of theproportional relationships within our number system.• interacting with other studentsStudents who work together to solve problems learn from one another as theydemonstrate and communicate their mathematical understanding.• exploring mathematical concepts in a variety of ways, including thekinaesthetic, the artistic, or the musicalStudents who have the opportunity to learn in a variety of ways, including their ownpreferred learning style, are more likely to be favourably disposed to mathematicsand more likely to retain their knowledge.• making connections with the outside environment and their home lifeFor example, students who go for “geometric shape” walks or search for naturallyoccurring number patterns in the school or in nature (e.g., the Fibonacci sequencefound in the arrangement of seeds in a sunflower: 1, 1, 2, 3, 5,...) are more likely tobe engaged in the learning process.• engaging in student talkStudents who talk to others about their mathematical understandings are compelledto explain their reasoning and revisit their strategies.• working in blocks of timeMathematical investigation is enhanced when students have ample time to exploreand consolidate mathematical ideas.Principles Underlying Effective Mathematics Instruction 27