Introduction

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