When planning for instruction, teachers need to consider how to keep students workingin this zone. As students are introduced to new concepts, their thinking is stretched,and they experience a state of discomfort known as cognitive dissonance. This is thepoint at which they need the most support. Teachers give this support through askingquestions, guiding discussion and dialogue, and providing appropriate activities.The following chart provides a summary of the impact on the student of learningbelow, within, or above the zone of proximal development.Vygotsky’s Zone of Proximal Development (as related to mathematics)Below the Zone ofProximal Development• The student can complete thetask independently.• The student does not gain newknowledge, but the task maybuild confidence and fluencyand may help to consolidatepreviously learned concepts.• The student practises thelearning/skill/concept to deeplyentrench understanding.• The student’s learning isproficient and automatic.• Tasks may be too easy if thestudent remains in this zonelonger than is appropriate.In the Zone ofProximal Development• The student’s learning is supportedso that the studentcan move to a higher level ofunderstanding (i.e., teacherassistance is required to ensurethat problems at an appropriatelevel are presented, and thatmodelling, guidance, andquestioning occur duringthe task, as necessary).• The student’s learning is linkedto prior knowledge.• New learning occurs. Theexperience is challengingenough to trigger newunderstandings.• The student contributesmeaningful talk and actionduring the construction ofnew knowledge.• The learning tasks are”just right”.Beyond the Zone ofProximal Development• The student can completethe task only through relianceon procedures at the expenseof conceptual understanding(e.g., performs step-by-steplong division withoutunderstanding).• There is no new learning;understanding is limited;the student cannot generalizeand apply the knowledge tonew situations.• The student may disengagefrom the learning process.• Tasks are too difficult and leadto frustration.30 A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Volume One
Respect How Each Student LearnsTeachers need to respect how students learn by taking into considerationthe balance of learning styles, attitudes, preferences,cultural backgrounds, and special needs of students in the classroom.Keeping students engaged in the learning process involvesboth maintaining their interest and sustaining their understandingof the task they are undertaking.Classrooms and instructional approaches that are conducive tomathematical learning include:• a variety of visual displays (e.g., math word walls, strategywalls, students’ work);• oral reminders that emphasize the connections with priorknowledge and similar learning experiences;“Patterns of learning may varygreatly within a classroom.Teachers need to plan fordiversity, give students tasks thatrespect their abilities, usedynamic and flexible grouping forinstruction, and provide ongoingassessment.”(Expert Panel on Literacy and NumeracyInstruction for Students With SpecialEducation Needs, 2005, p. 4)• organizers such as mind maps and webs for a particular concept;• prompts such as questions that the teacher poses as well as access to promptsthat help students become “unstuck” (e.g., multiplication grids, T-charts);• a range of learning experiences that include opportunities for shared, guided,and independent mathematics;• the use of a three-part lesson format that includes “Getting Started”, ”Working onIt”, and “Reflecting and Connecting” segments;• a focus in all lessons on connecting conceptual understanding with proceduralknowledge;• care in ensuring age-appropriate pacing and duration of lessons;• attention to the special needs of students (e.g., through adaptations of lessons andof teaching and assessment strategies and other accommodations – see the appendixto this chapter, entitled “Accommodations and Modifications”. See also the SpecialEducation Companion in the Ontario Curriculum Unit Planner, available atwww.ocup.org);• strategies that support English language learners (e.g., using simple and familiarlanguage; using manipulatives, visuals, and gestures to explain concepts; explainingcultural content that may be unfamiliar to newcomers to Ontario);• flexibility in the teaching of a lesson (to follow the needs of the class, not simplyto follow the lesson plan);• a variety of materials that students can use depending on their particular needsor interests (e.g., one student may feel more comfortable using base ten blockswhile another favours counters);• appropriate “wait time” that allows students to think through a problem beforeresponding.Principles Underlying Effective Mathematics Instruction 31
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Gavin, M.K., Belkin, L.P., Spinelli
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Tank, B., & Zolli, L. (2001). Teach
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Leadership ResourcesBurns, M. (Ed.)
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ReferencesAdams, L., Waters, J., Ch
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Ginsberg, H.P., Inoue, N., & Seo, K
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Payne, J.N. (Ed.). (1990). Mathemat