Introduction

Research indicates that procedural skills are bestacquired through understanding rather than rotememorization. See also automaticity and conceptualunderstanding.procedures. See mathematical procedures.prompt. An open-ended phrase given tostudents for completion (e.g., “The steps Ifollowed were . . .”). Prompts play a crucial rolein providing guidance for students’ reflectionsand eliciting deeper thinking and reasoning. It isimportant that the person prompting not providetoo much information or inadvertently solve theproblem. See also scaffolding.proportional reasoning. Reasoning that involvesan understanding of the multiplicative relationshipin size of one object or quantity comparedwith another. Students express proportional reasoninginformally using phrases such as “twice asbig as” and “a third of the size of”.provincial standard. Level 3 of the four levelsof achievement, as defined in the Ontario curriculumdocuments for the various subjects; the levelat which students are expected to achieve in eachgrade. See also achievement level.quantity. The “howmuchness” of a number. Anunderstanding of quantity helps students estimateand reason with numbers, and is an importantprerequisite to understanding place value, theoperations, and fractions.questionnaire. See attitudinal survey.recomposition of numbers. The putting backtogether of numbers that have been decomposed.For example, to solve 24+27, a student mightdecompose the numbers as 24+24+3, thenrecompose the numbers as 25+25+1 to give theanswer 51. See also composition of numbers anddecomposition of numbers.regrouping. (Also called “trading”.) The processof exchanging 10 in one place-value position for1 in the position to the left (e.g., when 4 ones areadded to 8 ones, the result is 12 ones or 1 ten and2 ones). Regrouping can also involve exchanging1 for 10 in the place-value position to the right(e.g., 56 can be regrouped to 4 tens and 16 ones).The terms “borrowing” and “carrying” are misleadingand can hinder understanding.relationship. In mathematics, a connectionbetween mathematical concepts, or between amathematical concept and an idea in anothersubject or in real life. As students connect ideasthey already understand with new experiencesand ideas, their understanding of mathematicalrelationships develops.remainder. The quantity left when an amounthas been divided equally and only whole numbersare accepted in the answer (e.g., 11 divided by 4is 2 R3). The concept of a remainder can be quiteabstract for students unless they use concretematerials for sharing as they are developing theirconceptual understanding of division. When studentsuse concrete materials, they have little difficultyunderstanding that some items might be leftafter sharing.repeated addition. The process of addingthe same number two or more times. Repeatedaddition can be expressed as multiplication(e.g., 3+3+3+3 represents 4 groups of 3, or 4 x 3).repeated subtraction. The process of subtractingthe same subtrahend from another number two ormore times until 0 is reached. Repeated subtractionis related to division (e.g., 8–2–2–2–2=0 and8÷2=4 express the notion that 8 can be partitionedinto 4 groups of 2).representation. See mathematical model.rich mathematical experiences. Opportunitiesfor students to learn and apply mathematics ininteresting, relevant, and purposeful situations.Rich mathematical experiences often involveproblem solving and/or investigations.Glossary 99