elationships is fundamental to understandingmathematics.pattern structure. The order in which elementsin a pattern occur, often represented by arrangementsof letters (e.g., AAB AAB AAB).performance task. A meaningful and purposefulassessment task in which students are requiredto perform, create, or produce something. Thesetasks are generally authentic insofar as theysimulate authentic challenges and problems.A performance task usually focuses on processas well as on product.place value. The value given to a digit in anumber on the basis of its place within the number.For example, in the number 444, the digit 4 canrepresent 400, 40, or 4.planning. In the mathematics classroom, theworking out beforehand of instructional activitiesthat support students in achieving the expectationsoutlined in the curriculum policy document fortheir grade. There are three major planning formats:– long-term plan. A year-long plan in whichconcepts are organized in a meaningful andlogical order that fosters mathematical growth.– unit plan. A series of lessons that build towardsthe conceptual understanding of a big ideaand key concepts in a single strand or severalstrands of mathematics.– daily lesson plan. A specific lesson that is partof the larger unit plan and that builds towardsunderstanding of a key concept.portfolio. A folder or other container thatholds a selection of a student’s work related tomathematics, produced over the course of a termor year. The selection can include a range of items,such as paper-and-pencil tasks, drawings, solutionsto problems, and journal entries, that reflectthe student’s typical work and best efforts andtogether show the student’s learning progressover time.prerequisite understanding. The knowledge thatstudents need to possess if they are to be successfulin completing a task. See also prior knowledge.prior knowledge. The acquired or intuitiveknowledge that a student possesses prior toinstruction.problem posing. An instructional strategy inwhich students develop mathematical problemsfor others to solve. Problem posing, done orallyor in writing, provides an opportunity for studentsto relate mathematical ideas to situations thatinterest them.problem solving. Engaging in a task for whichthe solution is not obvious or known in advance.To solve the problem, students must draw on theirprevious knowledge, try out different strategies,make connections, and reach conclusions. Learningby inquiry or investigation is very natural foryoung children.problem-solving model. A process for solvingproblems based on the work of George Polya. Thesteps in the problem-solving model – understandingthe problem, making a plan, carrying out the plan,and looking back – should be used as a guide,rather than as prescribed directions, to helpstudents solve problems.problem-solving norms. Guidelines that havebeen developed in a classroom for problem solving –for example, regarding the use of manipulativesor the choice of whether to work with a partneror independently. See also problem-solving model.problem-solving strategies. Methods used fortackling problems. The strategies most commonlyused by students include the following: act it out,make a model with concrete materials, find/use apattern, draw a diagram, guess and check, use logicalthinking, make a table, use an organized list.procedural knowledge. Knowledge that relatesto carrying out a method (procedure) for solving aproblem and applying that procedure correctly.98 A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Volume One
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
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- Page 119 and 120: Payne, J.N. (Ed.). (1990). Mathemat