Early Math Strategy, The Report Of The Expert Panel On .
Contents1Introduction . 12The Ontario Context . 33The Teaching and Learning of Mathematics . 7Characteristics of the Early Mathematics Learner . 7Developmental Aspects of Learning . 7Building on Children’s Prior and Intuitive Knowledge of Mathematics . 8Learning by Doing and Talking . 10Summary of Characteristics of the Early Learner . 11Characteristics of an Effective Early Grades Mathematics Program . 11Overview of Learning Environment and Framework for Teaching . 11The Learning Environment . 11An Effective Teaching Framework . 31Assessment and Evaluation . 38Mathematics for All . 45Summary of an Effective Mathematics Program . 484Developing and Sustaining Teacher Expertise . 49Characteristics of Effective Professional Development Models . 49Implementation and Research . 55Role of the Principal and Other Administrators . 55The Importance of the Lead Teacher . 58Early Math Strategy Implementation: Feedback and Research . 605Summary and Conclusions . 65Glossary . 69References . 73Cette publication est également offerte en français sous le titre suivant : Stratégie de mathématiques au primaire –Rapport de la table ronde des experts en mathématiques, 2003.This publication is available on the Ministry of Education’s website at http://www.edu.gov.on.ca.
Early Math Strategy Expert PanelRuth Dawson (Co-chair)Co-ordinator, Halton District School BoardChris Suurtamm (Co-chair)Assistant Professor, University of OttawaFaculty of EducationPat BarltropTeacher, Primary Grades, Toronto District School BoardJane BennettConsultant, Halton District School BoardRalph ConnellyProfessor, Brock UniversityFaculty of EducationMichelle FerreiraConsultant, Durham District School BoardJohanne GaudreaultConsultant, Conseil scolaire de district catholique duCentre-EstRichard GauthierDirector (retired), French-language Education Policy andPrograms Branch, Ministry of EducationHeather HutzulTeacher, Kindergarten, York Catholic District School BoardÉmilie JohnsonConsultant (retired), Conseil scolaire de district catholiqueFranco-NordAnna JuppConsultant, Toronto District School BoardLaurie MoherConsultant, Kawartha Pine Ridge District School BoardEric MullerProfessor, Brock UniversityDepartment of MathematicsBarry OnslowAssociate Professor, University of Western OntarioFaculty of EducationFrancine PaquetteTeacher (retired), Kindergarten, Conseil scolaire de districtcatholique du Centre-EstEstelle RondeauTeacher, Primary Grades, Conseil scolaire de districtcatholique Franco-NordDemetra SaldarisPrincipal, Halton District School BoardJoanne SimmonsConsultant, Toronto District School BoardLyn VauseConsultant, Simcoe Muskoka Catholic District School BoardNancy VézinaProfesseure adjointe, Faculté d’éducationUniversité d’OttawaTH ERE P O R TO FT H EEX P E R TPAN E LO NEA R L YMAT HI NON TA R I Oiii
Early Math Experts Addressing the PanelivEA R L YDr. Alex LawsonAssistant Professor, Lakehead UniversityFaculty of EducationCraig FeatherstoneConsultant, Halton District School BoardDr. Lynne OuthredSenior Lecturer, Macquarie University, Sydney, Australia.Writer and researcher on the Count Me In Too project,New South Wales, AustraliaMAT HST R AT E G Y
1IntroductionHaving identified the early grades of schooling as critical for a solid foundation in thebasics of reading and mathematics, the Ontario government has been developing strategiesaimed at improving achievement in reading and mathematics among children from JuniorKindergarten to Grade 3. In May 2002, the Ontario government announced that it wouldspend 25 million both to expand the year-old Early Reading Strategy and to create anEarly Math Strategy that would help primary school students to improve their basicmathematics understanding and to begin to develop the mathematical skills neededin the twenty-first century.Success in mathematics in the early grades is critical. Early mathematicsunderstanding has a profound effect on mathematical proficiency in thelater years. The Early Years Report by McCain and Mustard states: “We now know thata substantial base of mathematical understanding is set in the first few years” (1999, p. 9).A positive attitude towards mathematics, an understanding of key concepts, andmathematical skills must be developed in the early grades. Hence, the primary teacherplays an extremely influential and important role.Research and experience have shown that the best way to raise student achievement isthrough a combination of doing intensive subject-focused work with teachers and settingimprovement targets for student learning.As part of the Ontario government’s new strategy, a panel of experts in the learning andteaching of mathematics in the early grades was assembled to provide information andguidance on how best to support teachers and students. This panel was composed ofpractitioners and researchers from both the English-language and French-languageeducational systems as well as from various regions across Ontario. The practitioners andresearchers on the panel assessed the current research-based knowledge of effectivemathematics instruction and identified key components of an effective mathematicsprogram and strategies for successful program implementation. There was muchconsensus on these key components in mathematics education.1
Panel members compiled research and discussed best practice in the area of earlymathematics education. Their meeting was an important opportunity to focus on earlymathematics, and it provided an essential first step in focusing on mathematics in theearly grades. This report summarizes the results of the panel meeting and therecommendations made by the panel.2EA R L YMAT HST R AT E G Y
2The Ontario ContextThe Ontario Curriculum, Grades 1– 8 provides a framework for learning for students in allpublicly funded English- and French-language elementary schools in Ontario. It outlinesthe knowledge and skills that students are expected to acquire by the end of each grade.Annual province-wide assessments of students’ performance in reading, writing, andmathematics are conducted in Grades 3 and 6 as one measure of how well studentsachieve the expectations.The current mathematics curriculum (The Kindergarten Program, 1998; The OntarioCurriculum, Grades 1– 8: Mathematics, 1997 ) includes a broad range of knowledge andskills. The Grades 1– 8 curriculum gives descriptions of the knowledge and skills requiredfor each grade and links them with descriptions of achievement levels – different degreesof achievement of the curriculum expectations. This ensures consistency of expectationsacross the province and facilitates province-wide testing. Four levels of achievement arerepresented in the achievement chart, with level 3 considered as the provincial standardof achievement. The mathematics curriculum is divided into five strands: Number Senseand Numeration, Measurement, Geometry and Spatial Sense, Patterning and Algebra,and Data Management and Probability.Yearly province-wide assessments are conducted by the Education Quality and AccountabilityOffice (EQAO). Results are reported by school, by school board, and by the overall system.Over the five years from 1997–1998 to 2001–2002, system-wide results show that, whileoverall trends are generally moving in the right direction, large percentages of studentsare not achieving the provincial standard. Improving the mathematics education of allstudents is worthwhile.Provincial Grade 3 EQAO Results – MathematicsPercentage of Students Achieving at Levels 3 and 4on Grade 3 EQAO AssessmentsEnglish-languageboards1997–19981998 –19991999 –20002000 –20012001–200243565761583
The Early Math Strategy – like the Early Reading Strategy – is being undertaken by theMinistry of Education in response to this information. It has two complementary goals: to reinforce the accountability of school boards and schools for improving studentachievement through the setting of improvement targets, and to provide targeted supports to help elementary teachers, principals, and schoolboards work together to achieve consistently effective classroom teaching andassessment practices to support students’ learningThe supports being provided for the Early Math Strategy are founded on the belief, wellattested in research, that teachers’ knowledge of a subject and the skills that teachers canapply in the classroom have the most impact on students’ learning. The Ministry ofEducation is supporting this strategy with many initiatives: An Expert Panel was established to share research and best practice of teaching andlearning mathematics in the early grades and to summarize its findings in this report. Learning resources in early math will be provided for students in the classroom. In-depth training of instructional leaders (“lead teachers”) will be provided to ensureconsistent implementation of the strategy across the province. Training materials will be provided for all elementary teachers to help them identifyand use effective instructional practices and assessment strategies. Guides will be provided to parents on how they can help their children learn to domathematics. Beginning in 2003– 04, school boards will be required to establish targets forimprovement for students taking Grade 3 mathematics.Target setting has been required since 2001 under the Early Reading Strategy and will berequired as part of the Early Math Strategy, beginning in 2003– 04, as noted. Boards andschools will set three-year targets for improvement, will develop plans to reach thesetargets, and will report on the plans to their communities and the ministry. Boards andschools will measure their success in reaching their targets and then adjust and refinetheir plans on the basis of information from school-based data and the EQAO Grade 3mathematics assessment.4EA R L YMAT HST R AT E G Y
The early grades of schooling in mathematics are critical for the later success of all students.There are special issues in French-language schools that need to be taken into consideration,because in many regions such schools operate in a minority-language environment. Thelack of an adequate foundation in French on the part of a number of children attendingthese schools has an impact on how mathematics is learned in the early grades and onwhat teaching strategies are used. The net improvement in Grade 6 EQAO results forFrench-language students seems to indicate that the acquiring of language skills by thesestudents has a positive effect on student performance in mathematics. In later sections ofthis report, specifically those on the characteristics of the early math learner and on theframework for teaching, this issue will be more fully addressed.TH ERE P O R TO FT H EEX P E R TPAN E LO NEA R L YMAT HI NON TA R I O5
3The Teaching and Learningof MathematicsAny discussion of the teaching and learning of mathematics in the early grades needs toconsider several significant elements. In this report, these elements have been organizedby considering two central components. These components are: the characteristics ofthe early mathematics learner and the characteristics of an effective early gradesmathematics program. Within the discussion of an effective mathematics program,the following elements will be elaborated: a sound learning environment for earlymathematics, a framework for teaching, the role of assessment, and approaches forensuring mathematics for all students.Characteristics of the Early Mathematics LearnerMany elements must be acknowledged about the early learner of mathematics. Theseinclude recognition of the developmental aspects of learning, the importance of buildingon prior mathematical understanding, and the essential fact that children learn mathematicsprimarily through “ doing, talking, reflecting, discussing, observing, investigating,listening, and reasoning” (Copley, 2000, p. 29).Developmental Aspects of LearningResearch across cultures and across socio-economic groups indicates that children gothrough stages of conceptual development that are identifiable (Clements, 1999). Severalresearchers have established continuums that delineate some of the growth points thatchildren go through as they acquire various concepts in some areas of mathematics (Clarke& Clarke, 2002; Clements & Sarama, 2000; Griffin, Case, & Siegler, 1994 ). There is currentlyno agreed-upon continuum, because of the complexity and number of mathematicalconcepts. However, researchers do agree that children go through different stages ofmathematical development. Also, there is considerable individual variation from child tochild, and recognition of this variation is key to establishing the most effective learningenvironment. For the teaching and learning processes to be successful, it is important thatthe child’s existing conceptual understanding of mathematics be recognized. Childrenneed to encounter concepts in an appropriate manner, at an appropriate time, and witha developmentally appropriate approach.7
Developmentally appropriate means challenging but attainable for most childrenof a given age range, flexible enough to respond to inevitable individualvariation, and most important, consistent with children’s ways of thinkingand learning. (Clements, Sarama, & DiBiase, in press)Consequently, teachers must recognize the child’s level of cognitive, linguistic, physical,and social-emotional development. The most effective learning takes place when theseaspects of development are taken into consideration. This means that the child needs tobe cognitively capable of taking on the mathematical task at hand; able to comprehendthe language of instruction; have sufficient fine motor control to complete the task; andbe emotionally mature enough for the demands of the task so that frustration does nothamper the learning situation (Sophian, in press).French-language schools, in particular, have implemented two programs, called Actualisationlinguistique en français and Perfectionnement du français, that address the need for somechildren to acquire the necessary language skills in French to pursue the curriculumexpectations. Teachers play an important role at this critical language-development stage,in creating the classroom environment and using teaching strategies whereby childrenlearn the language while being positive about learning mathematics.Teachers should take into consideration the conceptual and developmental levels of theirstudents in providing general guidelines for furthering children’s development of conceptsin mathematics. Teachers also need guidance in organizing the curriculum to allow formore time to be spent on important concepts in mathematics and to link such conceptswith the developmental levels of their students.Building on Children’s Prior and Intuitive Knowledge of MathematicsChildren arrive at school with a variety of backgrounds and experiences and with moremathematical knowledge than was previously thought, regardless of different socioeconomic situations (Ginsburg & Seo, in press). Children may not immediately communicatethis understanding, but research on early stages of learning indicates that children beginthe process of making sense of their world at a very young age, and this includes makingmathematical sense (National Association for the Education of Young Children [NAEYC], 2002;Wright, Martland, & Stafford, 2000). Young children have a natural inquisitivenessabout mathematics, and teachers can build on this inquisitiveness to helpstudents develop the positive attitudes that often occur when one understandsand makes sense of a topic. Research indicates that, when young children use8EA R L YMAT HST R AT E G Y
mathematics to explore their world, their understanding can be quite complex andsophisticated (Ginsburg & Seo, in press). These initial understandings have powerfuleffects on how the child will assimilate and accommodate new knowledge (Bredekamp& Rosegrant, 1995).Children also bring diverse cultural or linguistic contexts to their mathematics understanding.These linguistic, cultural, and community backgrounds, which result in distinctive approachesto learning and lead to a variety of prior knowledges in children, need to be recognizedand valued. Research across cultures and across socio-economic groups indicates that allchildren go through various stages of conceptual development. Although the ages at whichsome children go through these stages is extremely variable, depending on their priorexperiences, the actual sequence has some consistency (Bredekamp, Bailey, & Sadler, 2000;Clements, 1999; Ginsburg & Seo, 2000).The mathematics that children bring to school should be valued and utilized inthe classroom. Research points out that one of the difficulties in trying to improve theteaching of early mathematics is that teachers tend to underestimate the capabilities ofyoung children when it comes to mathematics and may not have the knowledge to focuson important mathematical experiences (Sarama & DiBiase, in press; Sophian, in press).For instance, teachers may not realize that developing a solid understanding of thequantitative value of a number is as important as number identification and countingfeatures.Children need to see mathematics as sensible, and they do so when the mathematics theyare learning in school connects with their intuitive sense of mathematics and with theunderstanding of mathematics that they bring with them to the classroom. Quite oftenstudents see school mathematics as different from the mathematics that they experienceoutside of school.Paul Cobb (as cited in Yackel, 2001) gives an example of such a situation.He conducted mathematical interviews with Grade 1 and 2 children andasked, “Do you have a way to figure out how much is 16 9?”Children used a range of methods including counting to solve the problem.Almost all of them were able to find the answer of 25. Later, students weregiven the same problem embedded in a typical page from a school text witha vertical format and instructions at the top for the standard North Americanaddition algorithm with carrying.TH ERE P O R TO FT H EEX P E R TPAN E LO NEA R L YMAT HI NON TA R I O9
This time many children attempted to use the standard school algorithm withcarrying. While they had originally given correct answers some now had errors.The errors they made were ones that primary teachers would recognize, withsome answers such as 15 or 115. When Cobb discussed the answer of 15 withone child and asked her whether both her original answer of 25 could beright as well as he
Early Math Strategy that would help primary school students to improve their basic mathematics understanding and to begin to develop the mathematical skills needed in the twenty-first century. Success in mathematics in the early grades is critical. Early mathematics understanding has a profound effect on mathematical proficiency in the later years.The Early Years Reportby McCain and Mustard .
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