A Guide to Effective Instruction in Mathematics

1 A Guide to EffectiveInstruction in MathematicsA Guide to EffectiveInstruction in MathematicsKindergarten to Grade 6A Resource in Five Volumes from the Ministry of EducationVVoolluummee OOnneeFFoouunnddaattiioonnss ooff MMaatthheemmaattiiccss IInnssttrruuccttiioonnEvery effort has been made in this publication to identify Mathematics resources and tools( , manipulatives) in generic terms. In cases where a particular product is used by teachersin schools across Ontario, that product is identified by its trade name, in the interests of clarity. Reference to particular products in no way implies an endorsement of those productsby the Ministry of publication quivalente est disponible en fran ais sous le titre suivant : Guide d enseignement efficace des math matiques, de la maternelle la 6eann ..vIntroduction .. vii1. Achieving and Sustaining Improvement ..1A Framework for School Improvement.

2 3 Improvement Planning ..8 Working Together to Improve Mathematics Instruction .. 10 Professional Learning .. 16 Home and Community Connections .. 182. Principles Underlying Effective Mathematics Instruction .. 21 Foster Positive Mathematical Attitudes .. 24 Focus on Conceptual Understanding .. 24 Involve Students Actively in Their Learning .. 26 Acknowledge and Utilize Students Prior Knowledge .. 28 Provide Developmentally Appropriate Learning Tasks .. 28 Respect How Each Student Learns .. 31 Provide a Culture and Climate for Learning .. 32 Recognize the Importance of Metacognition .. 33 Focus on the Significant Mathematical Concepts ( Big Ideas ) .. 33 Diversity and Effective Instruction .. 34 Appendix .. 413. Planning the Mathematics Program .. 45 Long-Term Planning .. 48 Unit or Short-Term Planning .. 49 Daily Lesson Planning .. 51 Appendices.

3 594. Instructional Approaches .. 63 Shared Mathematics .. 66 Guided Mathematics .. 69 ContentsivA Guide to Effective Instruction in Mathematics , Kindergarten to Grade 6 Volume OneIndependent Mathematics .. 71 Appendices .. 73 Professional Resources .. 77 Glossary .. 87 References .. 103vPrefaceThe past ten years have seen tremendous change worldwide in literacyand numeracyinstruction. Countries such as England, the United States, and Australia have devel-oped and put into practice large-scale, research-based initiatives to improve instruc-tional and assessment practices, instructional leadership skills, and accountability inthe areas of reading and writing and Mathematics . The Ontario Ministry of Educationhas implemented similar initiatives focused on improving student achievement inreading, writing, and Mathematics . The supports provided under these initiatives arebased on research that points to two key factors in improving the achievement levelsof students: developing teacher expertise in Effective Instruction , and developing andimplementing clear improvement plans.

4 This five-volume reference Guide focuses on Effective Instruction . It contains infor-mation derived from research on instructional and assessment practices and sup-ports that have proved successful in improving student achievement in Mathematics .*This Guide builds on the strengths that exist in Ontario s education system andseeks to develop increased capacity in the teaching of Mathematics at the school level. A broad consensus now exists among researchers and educators on the knowledge andskills children need in Mathematics , the experiences that advance the development ofmathematical skills and understanding, and the basic components of an effectivemathematics program. But for many teachers, both new and experienced, there continues to be a gap between theory and practice. How can current research aboutteaching and learning Mathematics be brought to life in the classroom? What skillsand knowledge can best help teachers meet their commitment to help every childbecome mathematically proficient?

5 This Guide is designed to answer those questions by providing teachers with practi-cal, tested approaches and strategies informed by current research and practice. It* This Guide is based in part on research and advice set out in two expert panel reports published by the Ministry of Education: Early Math Strategy: The Report of the Expert Panel on Early Math in Ontario, 2003and Teaching and LearningMathematics: The Report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario, been developed by a team of experts in Mathematics education highly experi-enced teachers, Mathematics education researchers, consultants, and administratorsfrom across the province who came together for the purpose of gathering evidence-based information, resources, materials, and supports related to Effective practices in Mathematics Instruction . The Guide is intended to help classroom teachers andother educators as they work to enhance students understanding of Mathematics in Kindergarten to Grade similar reference Guide , reflecting research and Effective practices in mathematicsinstruction in the French language, has been developed for educators in Ontario sFrench-language schools.

6 ViA Guide to Effective Instruction in Mathematics , Kindergarten to Grade 6 Volume OneviiIntroductionThe Importance of Early Math SuccessMathematics competency is a powerful predictor of future economic success for individuals and for society. The demands of technology, a global market, and informeddemocratic decision making require a level of mathematical literacy unparalleled in thepast. But success in Mathematics requires more than just computational also requires the ability to apply Mathematics in solving problems, to process information from a variety of sources and technologies, and to access and use quantitative information to make informed decisions (National Council of Teachers of Mathematics , 2000). Students with a poor understanding of Mathematics will havefewer opportunities to pursue higher levels of education, to compete for good jobs,and to function as informed and intelligent citizens (Kilpatrick & Swafford, 2003).

7 Some knowledge of Mathematics is essential for most occupations, and many requirea more sophisticated level of knowledge. In addition, of course, an understanding of Mathematics can be personally satisfying and empowering. For students to be successful in later Mathematics endeavours and to use mathematicseffectively in life, they must have a sound understanding of elementary mathematicsconcepts, a positive attitude towards learning Mathematics , and the belief that anunderstanding of Mathematics is attainable (Kilpatrick & Swafford, 2003). The Ontario ContextNewcomers to Ontario from countries other than Canada now represent almost 25 percent of Ontario s population, with 18 percent of the population speaking neitherEnglish nor French as a first language. In some school communities, more than75 languages and varieties of English are spoken. This diversity in student back-ground has implications for math Instruction .

8 Teachers need to ensure that theirinstructional approaches, selection of resources, and classroom practices reflect theneeds of the changing Beliefs Underlying the Development of This GuideThe following principles, or beliefs , which guided the work of the Expert Panel onEarly Math and the Expert Panel on Mathematics in Grades 4 to 6 in Ontario, arereflected in this five-volume reference 1: All students can be successful in students can learn Mathematics . Mathematical proficiency should not be restrictedto a select few. A variety of teaching strategies, student groupings, resources, andadministrative and parental supports can be used to ensure that all students learnmathematics. All students should have a common foundation in Mathematics andaccess to Instruction of the highest quality. The material in this reference Guide includes suggestions that will provide learning opportunitiesin Mathematics for all 2: Mathematics Instruction should be based on the evidence of soundresearch that has been verified by classroom findings of international studies, information from continuing research in Mathematics education, and a new understanding of how students learn have allprompted the Mathematics education community to examine what methods ofinstruction are most Effective .

9 There is now, for example, a general consensus thatinstructional approaches that focus on developing students understanding of thefoundational concepts of Mathematics through problem solving benefit students material in this reference Guide is informed by the results of research and by the experience of expert classroom 3: Helping students develop a fundamental understanding of mathematicalconcepts and a positive attitude towards Mathematics in the primary and junior grades will give them a foundation for life-long learning in understanding of fundamental concepts in Mathematics in the primary and juniorgrades is developed through an Effective Mathematics program that includes qualityinstructionand an environment that fosters a community of Mathematics learners. This reference Guide offers advice, tools, and instructional strategies that will help educatorsprovide expert Instruction , building on a child s intuitive understanding of Mathematics and establishing a solid foundation in Guide to Effective Instruction in Mathematics , Kindergarten to Grade 6 Volume OneBelief 4: The teacher is the key to a child s success in developing teacher s ability to deliver Effective Mathematics Instruction is the most powerfulfactor in determining how well students learn Mathematics .

10 Effective mathematicsinstruction is enhanced when teachers develop and deepen their own understandingof Mathematics , of student learning, and of strategies that promote mathematical proficiency. This understanding can help to ensure that teachers are informed andcritical thinkers who are able to make wise choices about activities, strategies, andresources and who are able to provide a comprehensive program that supports children s development of mathematical goal of this reference Guide is the enhancement of teachers knowledge and skills in thearea of Effective Mathematics Instruction . Belief 5: Effective Mathematics Instruction occurs when Instruction is supported by the community, through the cooperation of instructional leaders at both the school and board levels, of parents, and of other members of the community at math Instruction does not occur in isolation. It involves not only classroomteachers, but all partners in education, including parents.