province of new brunswickqueen elizabeth high school ken macinnis, elementary teacher, sir charles...

iOutcomes

PROVINCE OF NEW BRUNSWICK

John Hildebrand, Mathematics Consultant,Department of Education

Carolyn Jones, Mathematics Teacher,Fredericton High School

Joan Manuel, Mathematics/Science Supervisor,School District 10

Jack Powers, Vice-Principal,MacNaughton Science and Technology Centre

PROVINCE OF NEWFOUNDLAND ANDLABRADOR

Patricia Maxwell, Program Development Specialist,Department of Education

Sadie May, Mathematics Teacher,Deer Lake-St. Barbe South Integrated School Board

Ed Somerton, Mathematics Consultant,Department of Education

Donald Squibb, Mathematics Teacher,St. James Regional High School

PROVINCE OF NOVA SCOTIA

Beth Calabrese, Mathematics Teacher,Queen Elizabeth High School

Ken MacInnis, Elementary Teacher,Sir Charles Tupper Elementary School

Richard MacKinnon, Mathematics Consultant,Department of Education and Culture

Linda Wheadon, Mathematics Teacher,Horton District High School

PROVINCE OF PRINCE EDWARDISLAND

Jerry Coady, Mathematics Teacher,Bluefield High School

Byron Cutcliffe, Senior High Mathematics/ScienceConsultant, Department of Education

Wayne Denman, Mathematics Consultant,Department of Education

Bill MacIntyre, Elementary Mathematics/ScienceConsultant, Department of Education

AcknowledgmentsThe Departments of Education of New Brunswick, Newfoundland and Labrador, Nova Scotia and PrinceEdward Island gratefully acknowledge the contributions of the following groups and individuals toward thedevelopment of this document:

• The regional mathematics common curriculum committee, which has overseen the common curriculumdevelopment and provided direction with respect to the completion of this foundation document. Currentand past members include the following:

• The mathematics foundation document working group, comprising teachers and other educators in theprovince of New Brunswick, which served as lead province in drafting and revising the document.

• The provincial working groups, comprising teachers and other educators in New Brunswick, Newfoundlandand Labrador, Nova Scotia and Prince Edward Island, who provided feedback and input to the documentduring the development and revision process.

• The educators, parents and other stakeholders, who contributed many hours to the validation processwhich led to the finalization of the Foundation for the Atlantic Canada Mathematics Curriculum.

Acknowledgments

iiiOutcomes

Table of ContentsVision ................................................................................................................................................. v

IntroductionPurpose of Document ..........................................................................................................................1Curriculum Focus ................................................................................................................................1A Common Approach .........................................................................................................................2

OutcomesEssential Graduation Learnings ............................................................................................................3Introduction to Curriculum Outcomes .................................................................................................7

• Vision ........................................................................................................................................7• Unifying Ideas ...........................................................................................................................7

Curriculum Outcomes .......................................................................................................................10• Strands and General Curriculum Outcomes ............................................................................. 10• Key-Stage Curriculum Outcomes .............................................................................................12

Contexts for Learning and TeachingLearning and Teaching Mathematics .................................................................................................27

• Principles of Learning and Teaching Mathematics ....................................................................27• Resource-Based Learning ......................................................................................................... 28• Balancing Procedural and Conceptual Knowledge ................................................................... 28• Estimation and Mental Computation ....................................................................................... 29• Homework ..............................................................................................................................30

The Learning Environment ................................................................................................................ 31Equity and Diversity ........................................................................................................................... 32

• Diversity in Student Needs ......................................................................................................32• Cognitive Development ...........................................................................................................32• Gender and Cultural Equity .....................................................................................................34

Assessing and Evaluating Student Learning ........................................................................................ 35• Assessment ..............................................................................................................................35• Evaluation ...............................................................................................................................35• Reporting ................................................................................................................................36• Guiding Principles ...................................................................................................................36• Assessing Student Learning in the Mathematics Classroom ......................................................36• External Assessment ................................................................................................................ 37• Program and System Evaluation ..............................................................................................37

Resources ..........................................................................................................................................38• Professional Resources ............................................................................................................. 38• Manipulative Materials ............................................................................................................38• Technology .............................................................................................................................39

References ....................................................................................................................................40

Table of Contents

vOutcomes

VisionThe Atlantic Canada mathematics curriculum

is shaped by a vision which fosters the development

of mathematically literate students who can extend

and apply their learning and who are effective participants

in an increasingly technological society.

Vision

1Outcomes

Further, a mathematics curricu-lum should reflect the followingrealities about the nature ofmathematics itself:

• To learn and understandmathematics is to “do”mathematics, i.e., studentsmust take an active role intheir study of mathematics.

• Mathematics is a means bywhich we interpret the worldand must be regularly con-nected to meaningful applica-tions.

• Mathematics instruction, andmathematics itself, have beengreatly affected by changes intechnology.

An appreciation of the impor-tance of mathematical literacyand of the nature of mathematicsitself needs to permeate thebreadth and depth of the math-ematics curriculum at all instruc-tional levels.

IntroductionPURPOSE OFDOCUMENT

he development of thisfoundation documenthas been motivated by

the Atlantic Canada commoncurriculum project in mathemat-ics. Within this context thedocument is intended to

• provide both educators andmembers of the general publicwith an outline of the philoso-phy, scope and outcomes ofmathematics education inAtlantic Canada

• provide a foundation uponwhich educators and otherswill make decisions concerninglearning experiences

• serve as a guide to inform thesubsequent development ofdetailed curriculum guides atall grade levels from schoolentry through grade 12

• articulate a progression andcontinuity in the developmentof mathematical skills andconcepts from school entrythrough grade 12, and en-courage discussion amongmathematics educators atvarious grade levels regardingthis development

he philosophy and out-comes of the AtlanticCanada mathematics

curriculum are based firmly onthose articulated by the NationalCouncil of Teachers of Mathemat-ics (NCTM) in its Curriculum andEvaluation Standards for SchoolMathematics (1989).

The Atlantic Canada mathematicscurriculum is shaped by a visionwhich fosters the development ofmathematically literate studentswho can extend and apply theirlearning and who are effectiveparticipants in an increasinglytechnological society.

The mathematically literatestudent will

• appreciate the utility and valueof mathematics

• demonstrate mathematicalpower, i.e., display confidenceand competence in his/herability to do mathematics

• be a mathematical problemsolver

• communicate mathematically

• reason mathematically

CURRICULUM FOCUS

Introduction

T T

2 Foundation for the Atlantic Canada Mathematics Curriculum

inces have equal weight indecision making. Each provincehas procedures and mechanismsfor communicating and consult-ing with education partners, andit is the responsibility of theprovinces to ensure thatstakeholders have input intoregional curriculum development.

Each foundation documentincludes statements of essentialgraduation learnings, generalcurriculum outcomes for that coreprogram, and correspondingoutcomes at the end of key stages(entry-grade 3, grades 4-6,grades 7-9, grades 10-12).Essential graduation learnings andgeneral curriculum outcomesprovide a consistent vision for thedevelopment of a rigorous andrelevant core curriculum. Inaddition to this foundationdocument, teachers will receivesupporting curriculum guides forthe grade levels at which theyteach.

A COMMON APPROACH

n 1993 work began on thedevelopment of commoncurricula in specific core

programs. The Atlantic ministers’primary purposes for collaborat-ing in curriculum developmentare to

• improve the quality of educa-tion for all students throughshared expertise and re-sources

• ensure that the educationstudents receive across theregion is equitable

• meet the needs of bothstudents and society

Under the auspices of the Atlan-tic Provinces Education Founda-tion, development of Atlanticcommon core curricula formathematics, science, Englishlanguage arts and social studiesfollows a consistent process. Eachproject requires consensus by aregional committee at desig-nated decision points; all prov-

General curriculumoutcomesare statements which identifywhat students are expectedto know and be able to doupon completion of study ina curriculum area.

Key-stage curriculumoutcomesare statements which identifywhat students are expected toknow and be able to do by theend of grades 3, 6, 9 and 12,as a result of their cumulativelearning experience in acurriculum area.

I

3Outcomes

Outcomes

Essential graduationlearnings (EGLs)are statements describing theknowledge, skills and attitudesexpected of all students whograduate from high school.Achievement of the essentialgraduation learnings willprepare students to continue tolearn throughout their lives.These learnings describeexpectations not in terms ofindividual school subjects butin terms of knowledge, skillsand attitudes developedthroughout the curriculum.They confirm that studentsneed to make connections anddevelop abilities across subjectboundaries if they are to beready to meet the shifting andongoing demands of life, workand study today and in thefuture. Essential graduationlearnings are cross-curricular,and curriculum in all subjectareas is focussed to enablestudents to achieve theselearnings. Essential graduationlearnings serve as a frameworkfor the curriculumdevelopment process.

ESSENTIAL GRADUATION LEARNINGS

FIGURE 1 – Relationship among Essential GraduationLearnings, Curriculum Outcomes & Levels of Schooling

Aesthetic Expression Citizenship Communication Personal Development Problem SolvingTechnological Competence

Socia

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Curriculum Outcomes

Leve

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EssentialGraduationLearnings

Curriculum outcome statementsarticulate what students are expected to know and be able to do inparticular subject areas. These outcome statements also describe theknowledge, skills and attitudes which students are expected todemonstrate at the end of certain key stages in their education as aresult of their cumulative learning experiences at each grade level inthe entry-graduation continuum. Through the achievement ofcurriculum outcomes, students demonstrate the essential graduationlearnings.


G G

raduates from the public schools of Atlantic Canada will be able todemonstrate knowledge, skills and attitudes in the followingessential graduation learnings. Provinces may add

additional essential graduation learnings as appropriate.

Aesthetic Expression

raduates will be able to respondwith critical awareness to variousforms of the arts and be able to

express themselves through the arts.

Graduates will be able, for example, to

• use various art forms as a means of formulatingand expressing ideas, perceptions and feelings

• demonstrate understanding of the contributionof the arts to daily life, cultural identity anddiversity, and the economy

• demonstrate understanding of the ideas,perceptions and feelings of others as expressedin various art forms

• demonstrate understanding of the significanceof cultural resources such as theatres, museumsand galleries

The mathematics curriculum contributes to thedevelopment of aesthetic expression in at least twoways. First, measurement and geometry play asignificant role in terms of shape and perspective inthe visual arts, architecture and other media.Second, the elegance, power and efficiency ofmathematical symbolism and representations areaesthetically satisfying to many.

Citizenship

raduates will be able to assesssocial, cultural, economic andenvironmental interdependence in

a local and global context.


• demonstrate understanding of sustainabledevelopment and its implications for theenvironment

• demonstrate understanding of Canada’s politi-cal, social and economic systems in a globalcontext

• explain the significance of the global economyon economic renewal and the development ofsociety

• demonstrate understanding of the social,political and economic forces that have shapedthe past and present and apply thoseunderstandings in planning for the future

• examine human rights issues and recognizeforms of discrimination

• determine the principles and actions of just,pluralistic and democratic societies

• demonstrate understanding of their own andothers’ cultural heritage, cultural identity andthe contribution of multiculturalism to society

The citizenship EGL is a major focus of the math-ematics curriculum in terms of applications ofmathematics. Mathematical applications are ofconsiderable significance in relation to keyunderstandings with respect to government,society and environment. Examples of theseapplications would range from measurement andgeometry in geography to exponential relations inpopulation dynamics and economics and statisticalissues in election polling.

G

5Outcomes

Communication

raduates will be able to use thelistening, viewing, speaking,reading and writing modes of

language(s) and mathematical and scientificconcepts and symbols to think, learn and com-municate effectively.


• explore, reflect on and express their own ideas,learnings, perceptions and feelings

• demonstrate understanding of facts and rela-tionships presented through words, numbers,symbols, graphs and charts

• present information and instructions clearly,logically, concisely and accurately for a varietyof audiences

• demonstrate a knowledge of the second officiallanguage

• access, process, evaluate and share information

• interpret, evaluate and express data in everydaylanguage

• critically reflect on and interpret ideas presentedthrough a variety of media

Communication is one of the unifying ideas withinthe mathematics curriculum. Not only doesmathematics represent a major means of commu-nication in the sciences and related disciplines, butthe mathematics curriculum provides students witha wide variety of communication tools, includingdiagrams, graphs and tables, as well as a broadrange of mathematical symbolism. Through thestudy of mathematics students will reflect uponand clarify ideas and relationships; express math-ematical ideas orally and in writing; and use theskills of reading, listening and viewing to interpretand evaluate mathematical ideas.

Personal Development

raduates will be able to continue tolearn and to pursue an active,healthy lifestyle.


• demonstrate preparedness for the transition towork and further learning

• make appropriate decisions and take responsi-bility for those decisions

• work and study purposefully both independ-ently and in groups

• demonstrate understanding of the relationshipbetween health and lifestyle

• discriminate among a wide variety of careeropportunities

• demonstrate coping, management and inter-personal skills

• demonstrate intellectual curiosity, an entrepre-neurial spirit and initiative

• reflect critically on ethical issues

The mathematics curriculum contributes to per-sonal development in a number of ways. First, itdemonstrates the central role of mathematics in awide variety of career options. Second, it providesa vehicle by which students can learn to work onproblems both cooperatively and independently.Finally, the study of mathematics will help developthe intellectual curiosity, persistence and habits ofmind which will serve individuals as lifelonglearners.

G G


Problem Solving

raduates will be able to use thestrategies and processes needed tosolve a wide variety of problems,

including those requiring language, and math-ematical and scientific concepts.


• acquire, process and interpret informationcritically to make informed decisions

• use a variety of strategies and perspectives withflexibility and creativity for solving problems

• formulate tentative ideas and question their ownassumptions and those of others

• solve problems individually and collaboratively

• identify, describe, formulate and reformulateproblems

• frame and test hypotheses

• ask questions, observe relationships, makeinferences and draw conclusions

• identify, describe and interpret different pointsof view and distinguish fact from opinion

Problem solving is a unifying idea within themathematics curriculum. Through the study ofmathematics, students will learn and apply a largevariety of information processing and interpretationskills, problem-solving strategies and critical andcreative thinking skills. Ultimately, problem solvingis the primary reason for doing mathematics.

Technological Competence

raduates will be able to use avariety of technologies, demon-strate an understanding of techno-

logical applications and apply appropriatetechnologies for solving problems.


• locate, evaluate, adapt, create and shareinformation, using a variety of sources andtechnologies

• demonstrate understanding of and use existingand developing technologies

• demonstrate understanding of the impact oftechnology on society

• demonstrate understanding of ethical issuesrelated to the use of technology in a local andglobal context

The mathematics curriculum will contribute to thedevelopment of technological competence inseveral ways. Students should not only regularlyuse calculators, graphics calculators and computersoftware, but they should also learn when it isappropriate to use such technologies. As well,students will be open to the use of new technologyand develop an appreciation of the impact oftechnology on the study of and nature of doingmathematics.

G G

7Outcomes

VISION

The Atlantic Canada mathematicscurriculum is shaped by a visionwhich fosters the development ofmathematically literate studentswho can extend and apply theirlearning and who are effectiveparticipants in an increasinglytechnological society.

UNIFYING IDEAS

To help accomplish this vision andto contribute significantly toaiding students in meeting theessential graduation learnings forAtlantic Canada, the mathematicscurriculum is focussed aroundfour unifying ideas. These unify-ing ideas permeate the math-ematics curriculum across topicsand grades and reiterate the keycurriculum standards identified bythe National Council of Teachersof Mathematics.

Upon graduation from highschool, all students will

• demonstrate confidence andcompetence in their ability touse problem-solving ap-proaches to investigate andunderstand mathematicalcontent, to use mathematicsto solve both routine and non-routine problems from withinand outside mathematics, torecognize multiple solutions toproblems and to apply math-ematical modelling to real-world problem situations

• be able to communicatemathematically: students mustreflect on and clarify theirmathematical thinking,formulate definitions andexpress generalizations,recognize multiple representa-tions of concepts, expressideas orally and in writing,read mathematics with under-standing, ask clarifying andextending questions, andappreciate the economy,power and elegance of math-ematical notation

• be able to reason mathemati-cally: students must be able todemonstrate logical reasoningskills that include making andtesting conjectures, formulat-ing counterexamples, judgingthe validity of arguments andconstructing simple, validarguments

• value mathematics: studentswill understand how math-ematics and historical situa-tions have interacted and howthis has influenced their lives;they will be able to identifyand use connections bothwithin mathematics andbetween mathematics andother disciplines and see howthese connections impact ontheir world

These unifying ideas influence allthe key-stage curriculum out-comes which follow (see pp. 12-25) and which are further elabo-rated in detailed curriculumguides. The vision of mathematics

INTRODUCTION TO CURRICULUM OUTCOMES

education is that all students willwork toward the same curricularoutcomes. The key-stage curricu-lum outcomes are arranged bycontent strands (see pp. 10-11)which organize the skills andconcepts of an appropriate bodyof mathematical knowledge forschool students.

Problem Solving

“To solve a problem is to find away where no way is known off-hand, to find a way out of adifficulty, to find a way aroundan obstacle, to attain a desiredend, that is not immediatelyattainable, by appropriatemeans.”(G. Polya quoted by Krulik and Reys in the NCTMCurriculum and Evaluation Standards, p. 75)

Problem solving should be amajor focus of all mathematics,given that the purpose of math-ematics is to make sense of theworld. As people encounter theworld throughout their lives, theyare faced with “problems” whichrequire solutions if they are tomeet their goals. Mathematicalproblem solving serves not only toanswer questions raised in every-day life, but also to deal withissues from professions such asbusiness and engineering, in otherdisciplines such as the physicaland social sciences, and whichextend and connect mathematicstheory itself.


Communication

“Mathematics can be thought ofas a language that must bemeaningful if students are tocommunicate mathematicallyand apply mathematicsproductively.”(NCTM Curriculum and Evaluation Standards, p.6)

Communication plays a major rolein the development of mathemati-cal understanding. While talkingoften precedes writing, bothverbal and written communicationare of great value with respect tochildren constructing knowledge,learning alternative ways to thinkabout ideas and clarifying theirown thinking. Communicationallows students to construct linksbetween their informal notionsand the abstract language andsymbolism of mathematics;recognize the importance ofcommon definitions; achievedeeper understandings of con-cepts and principles; and clarify,refine and consolidate theirthinking.

Students need to be daily encour-aged to discuss and write aboutmathematical ideas. They mustengage in such activities asdescribing mathematical proc-esses, explaining and differentiat-ing among mathematical con-cepts, explaining their reasoningand evaluating the responses ofothers. Through the process ofregular communication the use ofthe language and symbols ofmathematics becomes natural.

As well as communicating toothers mathematically, it is in-creasingly necessary that studentsread mathematics effectively.

Problem-solving activities need toinclude an appropriate combina-tion of situations from the rela-tively clearcut to the more com-plex and/or abstract. Studentsneed to experience sufficientsuccess in solving problems togain confidence in doing math-ematics, persevere in tasks thatrequire extended effort andbecome encouraged to takeintellectual risks when faced withnovel situations. Problem solvingmust provide opportunities forstudents to explore alternate linesof approach and to appreciatethat there may well be more thanone possible outcome.

Ultimately, problem solving needsto be seen as not a distinct topicbut rather as a process that mustpermeate the entire mathematicsprogram. It will serve as a meansto introduce new mathematicalcontent, develop the understand-ing of concepts and facility withprocedures and apply that whichis already known. Problem-solvingstrategies, including the applica-tion of technological resources,should be regularly modelled byteachers and peers so that theybecome increasingly integrated bystudents as a basis for “doingmathematics.”

“The curriculum must givestudents opportunities to solveproblems that require them towork cooperatively, to usetechnology, to address relevantand interesting mathematicalideas, and to experience thepower and usefulness ofmathematics.”(NCTM Curriculum and Evaluation Standards, p. 75-76)

There is a vast difference be-tween reading a work of fictionand reading a piece of math-ematical writing, yet doing thelatter well is necessary if printmathematical materials are to bevaluable resources. Reading formathematical understandingdoes not readily lend itself toskimming. Rather each part ofmathematical statements mustbe carefully considered formeaning and readers must takefull advantage of accompanyingillustrative material.

“The ability to read, write,listen, think creatively, andcommunicate about problemswill develop and deepenstudents’ understanding ofmathematics.”(NCTM Curriculum and Evaluation Standards, p.78)

Reasoning

“Mathematics is reasoning.One cannot do mathematicswithout reasoning.”(NCTM Curriculum and Evaluation Standards, p.29)

Because reasoning is fundamen-tal to the knowing and doing ofmathematics, it is important thatreasoning be a regular focus ofmathematical activity. Teachersshould consistently model theuse of mathematical reasoningand engage students in discus-sions and tasks designed todevelop their capacities to reasonmathematically.

The development of logicalreasoning is tied to the intellec-tual and verbal development of astudent. The development of thestudent’s ability to reason is a

9Outcomes

process that grows out of manyexperiences with such activities asanalyzing problems, makingconjectures, gathering evidenceand building arguments tosupport ideas. Students need tofocus in numeric, spatial andalgebraic contexts on suchquestions as: Are we sure? Howcan we be sure? What conditionscould make you sure? Whatobstacles are there to being sure?

In the early grades, the develop-ment of mathematical reasoningability can be fostered by ofteninvolving students in the kind ofinformal thinking, conjecturingand validating that helps studentsto see that mathematics makessense. In later grades, as thedepth and complexity of math-ematical content is increased,these informal reasoning activitiesshould be expanded to includemore formal recognition of theinterplay between conjecturingand inductive reasoning and theimportance of deductive verifica-tion.

Reasoning is an integral part ofthe study of mathematics anddemonstrations of reasoningshould be openly valued. Stu-dents at all levels should beencouraged to justify theirsolutions, thinking processes andconjectures in a variety of ways.

“Conjecturing anddemonstrating the logicalvalidity of conjectures are theessence of the creative act ofdoing mathematics. To givemore students access tomathematics as a powerful wayof making sense of the world, it

is essential that an emphasis onreasoning pervade allmathematical activity.”(NCTM Curriculum and Evaluation Standards, p.81)

Connections

“We can no longer afford to letmathematics remain an isolateddiscipline nor to permitcontinued fragmentation withinthe mathematics curriculumitself into isolated courses,separate topics, anddisconnected bits of knowledge.”(Steen, Lynn Arthur, “Teaching Mathematics forTomorrow’s World,” Educational Leadership, Sept.1989, p. 18)

Teachers of mathematics shouldstrive to make natural connectionsamong mathematical conceptsand representations, and toemphasize the natural connec-tions that exist across curricula.

Inherent in this integrative focus isthe potential for importanteducational benefits for bothteachers and students which havestrong foundations in recognizedprinciples of learning and cogni-tive development. These benefitsinclude, but are not limited to,

• the use of active, experientiallearning

• an increase in student motiva-tion and interest

• the teaching of mathematics incontext and with relevance

• the development of higherorder thinking skills

• the development of coopera-tive and collaborative interper-sonal skills

• a strengthening of students’understanding of mathematicsconcepts and principles

• an understanding of theinterdependent nature of theworld

• the development of criticalproblem-solving skills

As well as highlighting connec-tions among mathematical ideasand with other areas of the schoolcurriculum, the mathematicsprogram should also addressconnections with career areas andother aspects of the everydayworld.

“It is important that childrenconnect ideas both among andwithin areas of mathematics.…When mathematical ideas arealso connected to everydayexperiences, both in and out ofschool, children become awareof the usefulness ofmathematics.”(NCTM Curriculum and Evaluation Standards, p. 32)


The outcomes for the mathemat-ics curriculum are organized interms of four content strands:

• number concepts/number andrelationship operations

• patterns and relations

• shape and space

• data management and prob-ability

One or two general curriculumoutcomes (i.e., statements whichidentify what students are ex-pected to know and be able to doupon completion of study in acurriculum area) are identified foreach of these strands. The generalcurriculum outcomes (GCOs) arethen further elaborated (pp. 12-25) in terms of key-stage curricu-lum outcomes (i.e., outcomes atthe end of each of grades 3, 6, 9and 12).

The content strands and generalcurriculum outcomes are detailedas follows:

Number Concepts/Numberand RelationshipOperations

• Students will demonstratenumber sense and applynumber theory concepts.

• Students will demonstrateoperation sense and applyoperation principles andprocedures in both numericand algebraic situations.

Patterns and Relations

• Students will explore, recog-nize, represent and applypatterns and relationships,both informally and formally.

Shape and Space

• Students will demonstrate anunderstanding of and applyconcepts and skills associatedwith measurement.

• Students will demonstratespatial sense and apply geo-metric concepts, propertiesand relationships.

CURRICULUM OUTCOMES

Data Management andProbability

• Students will solve problemsinvolving the collection,display and analysis of data.

• Students will represent andsolve problems involvinguncertainty.

It is critical that the unifying ideasoutlined in the previous section(i.e., problem solving, communi-cation, reasoning and connec-tions) strongly influence, in factpermeate, the outcomes articu-lated for the content strands. Asindicated in the diagram follow-ing, this integration of the strandsand the unifying ideas takes placein the development of the key-stage curriculum outcomes.

It must be noted that, while thekey-stage curriculum outcomesare intended as targets for allstudents, all students will not beexpected to achieve them at asingle level of performance. Aswell, there will be an additionalsmall percentage of students whowill see their outcomes signifi-cantly altered in individualeducational programs.

STRANDS AND GENERALCURRICULUMOUTCOMES

11Outcomes

FIGURE 2 – Integrating Strands and Unifying Ideas

The key-stage curriculumoutcomes reflect theinfusion of the unifyingideas into the contentstrands.

Unifying IdeasProblem Solving,Communication,

Reasoning,Connections

StrandsNumber Concepts/

Number &Relationship Operations,

Patterns & Relations,Shape & Space,

Data Management& Probability

Key-StageCurriculumOutcomes


GCO: Students will demonstrate number sense andapply number theory concepts.

Elaboration: Number sense includes understanding of number meanings, developingmultiple relationships among numbers, recognizing the relative magnitudes of numbers,knowing the relative effect of operating on numbers and developing referents formeasurement. Number theory concepts include such number principles as laws (e.g.,commutative and distributive), factors and primes, number system characteristics (e.g.,density), etc.

By the end of grade 3, students will be expected to

construct and communicate number meanings,and explore and apply estimation strategies, withrespect to whole numbers

concretely explore common fractions and deci-mals in meaningful situations

read and write whole numbers and demonstratean understanding of place value (to four places)

order whole numbers and represent them inmultiple ways

apply number theory concepts (e.g., place valuepattern) in meaningful contexts with respect towhole numbers and commonly used fractions anddecimals

By the end of grade 6, students will have achieved theoutcomes for entry-grade 3 and will also be expected to

demonstrate an understanding of number mean-ings with respect to whole numbers, fractions anddecimals

explore integers, ratios and percents in common,meaningful situations

read and write whole numbers and decimals anddemonstrate an understanding of place value (tomillions and to thousandths)

order whole numbers, fractions and decimals andrepresent them in multiple ways

apply number theory concepts (e.g., prime num-bers, factors) in relevant situations with respect towhole numbers, fractions and decimals

KEY-STAGE CURRICULUM OUTCOMES

13Outcomes


demonstrate an understanding of number mean-ings with respect to the real numbers

order real numbers, represent them in multipleways (including scientific notation) and applyappropriate representations to solve problems

demonstrate an understanding of the real numbersystem and its subsystems by applying a variety ofnumber theory concepts in relevant situations

Some post-secondary intending students will be expected to

explain and apply relationships among real andcomplex numbers

GCO: Students will demonstrate number sense andapply number theory concepts.

Elaboration: Number sense includes understanding of number meanings, developingmultiple relationships among numbers, recognizing the relative magnitudes of numbers,knowing the relative effect of operating on numbers and developing referents formeasurement. Number theory concepts include such number principles as laws (e.g.,commutative and distributive), factors and primes, number system characteristics (e.g.,density), etc.


demonstrate an understanding of number mean-ings with respect to integers and rational andirrational numbers, and explore their use in mean-ingful situations

read, write and order integers, rational numbersand common irrational numbers

represent numbers in multiple ways (including viaexponents, ratios, percents and proportions) andapply appropriate representations to solve prob-lems

apply number theory concepts in relevant situa-tions and explain the interrelated structure ofwhole numbers, integers and rational numbers



demonstrate an understanding of the connectionbetween relevant, concrete experiences and themathematical language and symbolism of the fourbasic operations

recognize and explain the relationships among thefour basic operations

create and model problem situations involvingwhole numbers, using one or more of the fourbasic operations

demonstrate proficiency with addition and sub-traction facts

apply computational facts and strategies withrespect to the four basic operations and modeladdition and subtraction in situations involvingwhole numbers

apply estimation techniques to predict, and justifythe reasonableness of, results in relevant problemsituations involving whole numbers

select and use appropriate computational tech-niques (including mental, paper-and-pencil andtechnological) in given situations


model problem situations involving whole num-bers and decimals by selecting appropriate opera-tions and procedures

model problem situations involving the additionand subtraction of simple fractions

explore algebraic situations informally

apply computational facts and procedures (algo-rithms) in a wide variety of problem situationsinvolving whole numbers and decimals

apply estimation techniques to predict, and justifythe reasonableness of, results in relevant problemsituations involving whole numbers and decimals

select and use appropriate computational tech-niques (including mental, paper-and-pencil andtechnological) in given situations

GCO: Students will demonstrate operation sense andapply operation principles and procedures in bothnumeric and algebraic situations.

Elaboration: Operation sense consists of recognizing situations in which a givenoperation would be useful, building awareness of models and the properties of anoperation, seeing relationships among operations and acquiring insights into the effectsof an operation on a pair of numbers. Operation principles and procedures wouldinclude such items as the effect of identity elements, computational strategies andmental mathematics.

15Outcomes


explore and explain, using physical models, theconnections between arithmetic and algebraicoperations

model problem situations involving rationalnumbers and integers

apply computational procedures (algorithms) in awide variety of problem situations involvingfractions, ratios, percents, proportions, integersand exponents

apply operations to algebraic expressions torepresent and solve relevant problems

apply estimation techniques to predict, and justifythe reasonableness of, results in relevant problemsituations involving rational numbers and integers

select and use appropriate computational tech-niques in given situations and justify the choice


explain how algebraic and arithmetic operationsare related, use them in problem-solving situations,and explain and demonstrate the power of math-ematical symbolism

derive, analyze and apply computational proce-dures (algorithms) in situations involving all repre-sentations of real numbers

derive, analyze and apply algebraic procedures(including those involving algebraic expressionsand matrices) in problem situations

apply estimation techniques to predict, and justifythe reasonableness of, results in relevant problemsituations involving real numbers


apply operations on complex numbers to solveproblems

GCO: Students will demonstrate operation sense andapply operation principles and procedures in bothnumeric and algebraic situations.

Elaboration: Operation sense consists of recognizing situations in which a givenoperation would be useful, building awareness of models and the properties of anoperation, seeing relationships among operations and acquiring insights into the effectsof an operation on a pair of numbers. Operation principles and procedures wouldinclude such items as the effect of identity elements, computational strategies andmental mathematics.



describe, extend and create a wide variety ofpatterns and relationships to model and solveproblems involving real-world situations andmathematical concepts

explore how a change in one quantity in a rela-tionship affects another

represent mathematical patterns and relationshipsin a variety of ways (including rules, tables andone- and two-dimensional graphs)

solve linear equations using informal, non-alge-braic methods


recognize, describe, extend and create patternsand sequences in a variety of mathematical andreal-world contexts (e.g., geometric, numeric andmeasurement)

use patterns to solve problems

represent mathematical patterns and relationshipsin informal ways, including via open sentences(i.e., statements with missing addends)

GCO: Students will explore, recognize, represent andapply patterns and relationships, both informally andformally.

Elaboration: Patterns and relationships run the gamut from number patterns and thosemade from concrete materials to polynomial and exponential functions. Therepresentation of patterns and relationships will take on multiple forms, includingsequences, tables, graphs and equations, and these representations will be applied asappropriate in a wide variety of relevant situations.

17Outcomes


analyze, generalize and create patterns andrelationships to model and solve real-world andmathematical problem situations

analyze functional relationships to explain how thechange in one quantity results in a change inanother

represent patterns and relationships in multipleways (including the use of algebraic expressions,equations, inequalities and exponents)

explain the connections among algebraic andnon-algebraic representations of patterns andrelationships

apply algebraic methods to solve linear equationsand inequalities and investigate non-linear equa-tions


model real-world problems using functions, equa-tions, inequalities and discrete structures

represent functional relationships in multiple ways(e.g., written descriptions, tables, equations andgraphs) and describe connections among theserepresentations

interpret algebraic equations and inequalitiesgeometrically and geometric relationships algebrai-cally

solve problems involving relationships, usinggraphing technology as well as paper-and-penciltechniques

analyze and explain the behaviours, transforma-tions and general properties of types of equationsand relations

perform operations on and between functions


describe and explore the concept of continuity of afunction

investigate limiting processes by examining infinitesequences and series

make connections among trigonometric functions,polar coordinates, complex numbers and series

GCO: Students will explore, recognize, represent andapply patterns and relationships, both informally andformally.

Elaboration: Patterns and relationships run the gamut from number patterns and thosemade from concrete materials to polynomial and exponential functions. Therepresentation of patterns and relationships will take on multiple forms, includingsequences, tables, graphs and equations, and these representations will be applied asappropriate in a wide variety of relevant situations.



measure and understand basic concepts andattributes of length, capacity, mass, area and time

identify and use non-standard and standard unitsof measurement and appreciate their role incommunication

estimate and determine measurements in every-day problem situations and develop a sense of therelative size of units


extend understanding of measurement conceptsand attributes to include volume, temperature,perimeter and angle

communicate using standard units, understand therelationship among commonly used SI units (e.g.,mm, cm, m, km) and select appropriate units ingiven situations

estimate and apply measurement concepts andskills in relevant problem situations and select anduse appropriate tools and units

develop and apply rules and procedures for deter-mining measures (using concrete and graphingmodels)

GCO: Students will demonstrate an understanding ofand apply concepts and skills associated withmeasurement.

Elaboration: Concepts and skills associated with measurement include making directmeasurements, using appropriate measurement units and using formulas (e.g., surfacearea, Pythagorean Theorem) a/o procedures (e.g., proportions) to determinemeasurements indirectly.

19Outcomes


measure quantities indirectly, using techniques ofalgebra, geometry and trigonometry

determine measurements in a wide variety ofproblem situations and determine specified degreesof precision, accuracy and error of measurements

apply measurement formulas and procedures in awide variety of contexts


demonstrate an understanding of the meaning ofarea under a curve


demonstrate an understanding of the concept ofrates; use direct and indirect measurements todescribe and make comparisons and read andinterpret scales; and describe how a change in onemeasurement affects other, indirect measurements

communicate using a full range of SI units (e.g.,mm, cm, dm, m, hm, Dm, km) and select appro-priate units in given situations

estimate and apply measurement concepts inrelevant problem situations and use tools and unitswhich reflect an appropriate degree of accuracy

develop and apply a wide range of measurementformulas and procedures (including indirectmeasures)

GCO: Students will demonstrate an understanding ofand apply concepts and skills associated withmeasurement.

Elaboration: Concepts and skills associated with measurement include making directmeasurements, using appropriate measurement units and using formulas (e.g., surfacearea, Pythagorean Theorem) a/o procedures (e.g., proportions) to determinemeasurements indirectly.


GCO: Students will demonstrate spatial sense andapply geometric concepts, properties andrelationships.

Elaboration: Spatial sense is an intuitive feel for one’s surroundings and the objects inthem and is characterized by such geometric relationships as i) the direction,orientation and perspectives of objects in space, ii) the relative shapes and sizes offigures and objects and iii) how a change in shape relates to a change in size.Geometric concepts, properties and relationships are illustrated by such examples as theconcept of area, the property that a square maximizes area for rectangles of a givenperimeter, and the relationships among angles formed by a transversal intersectingparallel lines.


explore and experiment with geometric shapesand relationships (including the orientation andperspectives of objects)

describe, model, draw and classify 2- and 3-Dfigures and shapes

investigate and predict the results of combining,subdividing and transforming shapes

relate geometric ideas to number and measure-ment ideas and recognize and apply geometricprinciples in real-world situations


identify, draw and build physical models of geo-metric figures

describe, model and compare 2- and 3-D figuresand shapes, explore their properties and classifythem in alternative ways

investigate and predict the results of transforma-tions and begin to use them to compare shapesand explain geometric concepts (e.g., symmetryand similarity)

solve problems using geometric relationships andspatial reasoning

21Outcomes


extend spatial sense in a variety of mathematicalcontexts

interpret and classify geometric figures, translatebetween synthetic (Euclidean) and coordinaterepresentations, and apply geometric properties andrelationships

analyze and apply Euclidean transformations,including representing and applying translations asvectors

represent problem situations with geometric models(including the use of trigonometric ratios andcoordinate geometry) and apply properties offigures

make and test conjectures about, and deduceproperties of and relationships between, 2- and 3-Dfigures in multiple contexts

demonstrate an understanding of the operation ofaxiomatic systems and the connections amongreasoning, justification and proof


represent and apply vectors in three dimensionsalgebraically and geometrically

explore and apply, using multiple representations,circles, ellipses and parabolas and, in 3-D, spheresand ellipsoids


construct and analyze 2- and 3-D models, using avariety of materials and tools

compare and classify geometric figures, under-stand and apply geometric properties and rela-tionships, and represent geometric figures viacoordinates

develop and analyze the properties of transforma-tions and use them to identify relationshipsinvolving geometric figures

represent and solve abstract and real-worldproblems in terms of 2- and 3-D geometricmodels

draw inferences, deduce properties and makelogical deductions in synthetic (Euclidean) andtransformational geometric situations

GCO: Students will demonstrate spatial sense andapply geometric concepts, properties andrelationships.

Elaboration: Spatial sense is an intuitive feel for one’s surroundings and the objects inthem and is characterized by such geometric relationships as i) the direction,orientation and perspectives of objects in space, ii) the relative shapes and sizes offigures and objects and iii) how a change in shape relates to a change in size.Geometric concepts, properties and relationships are illustrated by such examples as theconcept of area, the property that a square maximizes area for rectangles of a givenperimeter, and the relationships among angles formed by a transversal intersectingparallel lines.


GCO: Students will solve problems involving thecollection, display and analysis of data.

Elaboration: The collection, display and analysis of data involves i) attention to samplingprocedures and issues, ii) recording and organizing collected data, iii) choosing andcreating appropriate data displays, iv) analyzing data displays in terms of broadprinciples (e.g., display bias) and via statistical measures (e.g., mean) and v) formulatingand evaluating statistical arguments.


collect, organize and describe relevant data inmultiple ways

construct a variety of data displays (includingtables, charts and graphs) and consider theirrelative appropriateness

read, interpret, and make and modify predictionsfrom displays of relevant data

develop and apply measures of central tendency(mean, median and mode)

formulate and solve simple problems (both real-world and from other academic disciplines) thatinvolve the collection, display and analysis of dataand explain conclusions which may be drawn


collect, record, organize and describe relevantdata

construct concrete and pictorial displays of rel-evant data

read and interpret displays of relevant data

generate questions, develop and modify predic-tions and implement plans with respect to dataanalysis

23Outcomes


understand sampling issues and their role withrespect to statistical claims

extend construction (both manually and via appro-priate technology) of a wide variety of data displays

use curve fitting to determine the relationshipbetween, and make predictions from, sets of dataand be aware of bias in the interpretation of results

determine, interpret and apply as appropriate awide variety of statistical measures and distributions

design and conduct relevant statistical experiments(e.g., projects with respect to current issues, careerapplications, a/o other disciplines) and analyze andcommunicate the results using a range of statisticalarguments


test hypotheses using appropriate statistics


be aware of sampling issues and understandprocedures with respect to collecting data

construct various data displays (both manually andvia technology) and decide which is/are mostappropriate

draw inferences and make predictions from avariety of displays of real-world data (including viacurve-fitting with respect to scatterplots)

determine, and apply as appropriate, measures ofcentral tendency and dispersion (e.g., range)

demonstrate an appreciation of statistics as adecision-making tool by formulating and solvingrelevant problems (e.g., projects with respect tocurrent issues a/o other academic disciplines)

make convincing statistical arguments and evalu-ate those of others

GCO: Students will solve problems involving thecollection, display and analysis of data.

Elaboration: The collection, display and analysis of data involves i) attention to samplingprocedures and issues, ii) recording and organizing collected data, iii) choosing andcreating appropriate data displays, iv) analyzing data displays in terms of broadprinciples (e.g., display bias) and via statistical measures (e.g., mean) and v) formulatingand evaluating statistical arguments.



conduct informal investigations of chance andestimate probabilities with respect to games andother simple, everyday situations

GCO: Students will represent and solve problemsinvolving uncertainty.

Elaboration: Representing and solving problems involving uncertainty entailsi) determining probabilities by conducting experiments a/o making theoreticalcalculations, ii) designing simulations to determine probabilities in situations which donot lend themselves to direct experiment and iii) analyzing problem situations so as todecide how best to determine probabilities.


explore, interpret and make conjectures abouteveryday probability situations by estimatingprobabilities, conducting experiments, beginningto construct and conduct simulations, andanalyzing claims which they see and hear

determine theoretical probabilities using simplecounting techniques

demonstrate an understanding of the relationshipbetween the numerical expression describing aprobability and the events which give rise to thenumbers

25Outcomes


design and conduct experiments a/o simulations tomodel and solve a wide variety of relevant probabil-ity problems, and interpret and judge theprobabilistic arguments of others

build and apply formal concepts and techniques oftheoretical probability (including the use of permu-tations and combinations as counting techniques)

understand the differences among, and relativemerits of, theoretical, experimental and simulationtechniques

relate probability and statistical situations


create and interpret discrete and continuousprobability distributions and apply them in real-world situations


make predictions regarding and design and carryout probability experiments and simulations, inrelation to a variety of real-world situations

derive theoretical probabilities, using a range offormal and informal techniques

determine and compare experimental and theo-retical results

relate a variety of numerical expressions (ratios,fractions, decimals, percents) to the correspondingexperimental or simulation situation

GCO: Students will represent and solve problemsinvolving uncertainty.

Elaboration: Representing and solving problems involving uncertainty entails i)determining probabilities by conducting experiments a/o making theoreticalcalculations, ii) designing simulations to determine probabilities in situations which donot lend themselves to direct experiment and iii) analyzing problem situations so as todecide how best to determine probabilities.

27Outcomes

Instructional settings shouldinclude varied learning con-figurations—whole class, smallgroup, pairs and individual—as well as varied activities,including investigations,discussions, oral presentationsand projects.

• Students learn best in anenvironment which supportsexploration, investigation,critical and creative thinking,risk taking, reflection andother higher order thinkingskills.

Learning mathematics is aconstructive rather than apassive activity. Students needto learn mathematics in aclimate in which they feel freeto explore, knowing that theirefforts, ideas and approacheswill be valued by both theirteacher and their classmates.Students should be encour-aged to raise questions andinvestigate answers, therebyinfluencing the direction oftheir learning.

Contexts for Learningand Teaching

The following principles oflearning and teaching mathemat-ics are based on current researchand best practice. They areintended to guide and supportdecisions relating to the teachingand learning process, includingcurriculum and classroom organi-zation, assessment and reporting.

• Students should be viewed asindividuals who develop andlearn via different styles and atdifferent rates.

Students’ strengths, needs andbackgrounds must be recog-nized and addressed through avariety of teaching methods.All students should be consist-ently and appropriatelychallenged. They must beprovided the time, resourcesand feedback necessary toreach their potential.

• Students benefit, both sociallyand intellectually, from avariety of learning experiences,both independent and incollaboration with others.

Contexts for Learning and Teaching

LEARNING AND TEACHING MATHEMATICS

• Students learn best when ideasare approached in a variety ofways.

Multiple representations ofmathematical ideas enhancestudent understanding. (Forexample, relations may bereadily represented bothgraphically and via equationsor inequations.) As well,students should be encour-aged to represent their ownunderstandings in variousways.

• Students learn best in anenvironment that nurturespositive attitudes, sustainedeffort, self-discipline and thedevelopment of an appropri-ate degree of autonomy.

Students should be praised foreffort, original thought andperseverance. If these arequalities which the studentsare to perceive as important,they must be reflected as wellin assessment practices.

PRINCIPLES OFLEARNING ANDTEACHINGMATHEMATICS


• Students’ learning is mosteffective when standards ofexpectation are made clearand explicit.

Clearly stated expectationsenhance academic achieve-ment and feelings of self-esteem. Well-articulatedoutcomes allow students tomonitor their daily work andimprove their learning.

RESOURCE-BASEDLEARNING

As students and schools enter the21st century, they find themselvesin an era of rapid change andrapidly expanding knowledge. It isno longer adequate or realistic forstudents to acquire a select bodyof knowledge and expect it tomeet their needs as citizens of thenext century. The need for lifelonglearning is shifting the emphasisfrom a dependence on the ‘what’of learning to the ‘how’ of learn-ing—today’s students must ‘learnhow to learn.’

Resource-based learning is aphilosophy which stresses a shiftfrom the use of a single resourcein the classroom to the use of awide variety of print and non-print resources. Resources includemulti-media, telecommunicationand human, as well as a widevariety of print resources, both inthe classroom and in libraries/resource centres. This philosophyof learning complements theapproach to mathematicalinstruction identified in thisdocument and is characterized by

• students actively participatingin their learning

• teachers acting as facilitators oflearning, continuously guiding,monitoring and evaluatingstudent progress

• varying locations for learning

• learning experiences based oncurriculum outcomes

• learning strategies and skillsidentified and taught withinthe context of relevant andmeaningful units of study

• teachers working together tofacilitate resource-basedlearning across grade levelsand subject areas

Resource-based learning has manyadvantages. With students at thecentre of the instructional process,they will

• acquire skills and attitudesnecessary for independent,lifelong learning; they learnhow to learn, one of thefundamental aims of education

• interact in group work, sharingand participating in a variety ofsituations

• think critically and creatively,experimenting and taking risksas they become independentand collaborative problem-solvers and decision-makers

• make choices and acceptresponsibility for these choices,thereby making learning morerelevant and personal

BALANCINGPROCEDURAL ANDCONCEPTUALKNOWLEDGE

Many view mathematics as a setof rules and procedures. Funda-mentally, however, mathematicsis a set of ideas. The intent of thiscurriculum is to ensure thatstudents understand these ideas,not just master the rules andprocedures. The re-creation,transfer and application of ideas isenhanced when students assimi-late them into a conceptualframework. At the same time, it isessential that students accomplisha certain level of skill proficiencyso that they have the tools tosolve interesting and relevantproblems.

Students should be encouragedto invent and examine variousstrategies that might help themdevelop skills. This suggests thatthe teacher encourage diverseapproaches rather than requiringa single “best” approach for allstudents to use. For example,some students may solve theequation 6x+36=90 by firstsubtracting 36 from each sideand then dividing by 6. Othersmight initially divide both sides ofthe equation by 6 and thensubtract 6 from each side. Gener-ally, those who are most success-ful in mathematics are those whouse different strategies in differentsituations.

Practice of skills is usually moreeffective if the practice arises inmeaningful contexts. For exam-ple, rather than requiring stu-dents to complete a series ofunrelated exercises, teachers

29Outcomes

could structure game formats inwhich the fast pace encouragesproficiency with skills, as well asproviding situations involvingrelated questions so that patternsand underlying ideas becomeclearer.

More importantly, skills must beembedded in, and arise from,conceptual situations as much aspossible. It is of little value to astudent to be able to use a treediagram if he or she does notunderstand that the tree diagramis valuable in certain types ofcounting situations. Thus, asmuch attention must be paid towhen such diagrams are useful asto how to create them.

Mathematics is much more thanprocedures, with a bit of concep-tual learning thrown in. Curriculaand instructional practice for thefuture must adjust this balance toplace greater emphasis onconceptual understanding.

ESTIMATION ANDMENTAL COMPUTATION

The NCTM Curriculum andEvaluation Standards’ view ofcomputation extends far beyondthe traditional paper-and-pencilalgorithms. When faced with anumerical problem, a decisionabout calculation procedures hasto be made. As illustrated infigure 3 (NCTM Curriculum andEvaluation Standards, p. 9),students should have a repertoireof computational techniquesavailable and must learn to selectan appropriate technique in eachgiven situation.

Both estimation and mentalcomputation (more often re-ferred to as “mental math”) aredone “in one’s head” and havebeen deemed important skills bythe NCTM. Computing mentallyto find approximate or exactanswers is frequently the mostappropriate and useful of allmethods. There are specificstrategies and algorithms formental computation that mustbe taught and practised regularly.

Estimation and mental math arenot topics that can be isolated asa unit of instruction; they mustbe integrated throughout thestudy of mathematics. It isimportant to remember that

• skill in mental computationtakes time to develop

• mental computation must bepractised regularly

• mental computation is bestdeveloped in context

• a variety of strategies for onecomputation should beencouraged and shared

• students need to understandwhy particular procedureswork; they should not betaught computational trickswithout understanding whatthey are doing

• a thorough understanding of,and facility with, computationis a valuable prerequisite foreventual use in algebra


ProblemSituations

Estimate

Use MentalCalculation

Use aPaper-and-Pencil

Calculation

Use aCalculator

Use aComputer

ApproximateAnswer

Appropriate

Exact AnswerNeeded

CalculationNeeded

FIGURE 3


HOMEWORK

Homework is an essential compo-nent of the mathematics pro-gram, as it extends the opportu-nity for students to think math-ematically and to reflect on ideasexplored during class time. Theprovision of this additional timefor reflection and practice plays avaluable role in helping studentsto consolidate their learning.

Traditionally, homework hasmeant completing ten to twentydrill and practice questionsrelating to the procedure taughtin a given day. With the increasedemphasis on problem solving,conceptual understanding andmathematical reasoning, how-ever, it is important that home-work assignments change accord-ingly. More assignments involvingproblem solving, mathematicalinvestigations, written explana-tions and reflections, and datacollection should reduce some ofthe basic practice exercises givenin isolation. In fact, a goodproblem can sometimes accom-plish more than many drill-oriented exercises on a topic. For

example, instead of a number ofseparate questions, problemssuch as the following might beconsidered:

• Write as many addition andsubtraction sentences as youcan that produce the answerof 10.

• Predict how often you will get“tails” if you flip a penny fiftytimes. Try it. Would it differ ifyou used a loonie? Check tosee and write about yourfindings.

• A bike lock has four differentnumbers in the combination.The sum of the last twonumbers is the first number.The sum of all the numbers is15. What might the combina-tion be?

• Why does trying to obtain thetangent of 270° from acalculator produce an errormessage?

As is the case in designing alltypes of homework, the needs ofthe students and the purpose ofthe assignment will dictate thenature of the questions included.Homework need not be limitedto reinforcing learning; it pro-vides an excellent opportunity torevisit topics explored previouslyand to introduce new topics.

Homework provides an effectiveway to communicate withparents and provides parents anopportunity to be activelyinvolved in their child’s learning.By ensuring that assignmentsmodel classroom instruction andsometimes require parentalinput, a parent will have a clearerunderstanding of the mathemat-ics curriculum and the progressof his or her child in relationshipto it. As Van de Walle (1994, p.454) suggests, homework canserve as a parent’s window to theclassroom.

31Outcomes

The NCTM Professional Standards(p.3) lists five major shifts in theenvironment of mathematicsclassrooms (as shown in the tablebelow).

Structurally the classroom envi-ronment will need to provide theflexibility to accommodatevarying student needs andlearning styles and to facilitatethe active involvement of stu-dents in the development ofmathematical concepts and skills.Movement among whole class,small group and independentexperiences will be essential toprovide all important contexts forlearning.

As well, for students to develop abelief in themselves as math-ematical thinkers, they must learnin a climate in which there is

• a genuine respect for others’ideas

THE LEARNING ENVIRONMENT

he call for change inmathematics instructionis characterized by not

only what mathematics is taughtbut also the manner in which it istaught. What students learn isfundamentally connected to howthey learn it. A classroom environ-ment in which optimal learningcan take place must be estab-lished and maintained; it formsthe learning foundation uponwhich skills and attitudes areconstructed.

There are four unifying ideas thatare central to the overall view ofmathematics, as outlined in theNCTM Curriculum and EvaluationStandards (1989). They addressproblem solving, reasoning,communication and connections,and are common across all gradelevels (entry-grade 12). In orderto meet the requirements of theseideas, classrooms must look verydifferent from those of the past.

• a valuing of reason and sense-making

• pacing and timing that allowstudents to puzzle and to think

• the forging of a social andintellectual community

Students, teachers and parentseach play a significant role increating this learning environ-ment. Students have a responsi-bility with regard to participation,behaviour and work ethic. Teach-ers convey their attitude toward,and their value of, mathematicsthrough their classroom presenta-tion, responses to students’ ideasand solutions and in their assess-ment practices. Positive attitudesof parents encourage students topursue and persist at studies ofmathematics. The entire climateor environment must be condu-cive to the fostering of math-ematically-thinking students.

Moving away from:classrooms as collections of individu-als

the teacher as the sole authority forright answers

primarily memorizing procedures

an emphasis on mechanistic answerfinding

treating mathematics as a body ofisolated concepts and procedures

Toward:classrooms as mathematical commu-nities

logic and mathematical evidence asverification

mathematical reasoning

conjecturing, inventing and problemsolving

connecting mathematics, its ideasand its applications

Summary of Shifts in Instructional Practices


T


DIVERSITY INSTUDENT NEEDS

Every classroom comprisesstudents at many differentcognitive levels. Rather thanchoosing a certain level at whichto teach, a teacher is responsiblefor tailoring instruction to reachas many of these students aspossible. In general, this may beaccomplished by assigningdifferent tasks to different stu-dents or assigning the sameopen-ended task to most stu-dents.

Sometimes it is appropriate for ateacher to group students byinterest or ability, assigning themdifferent tasks in order to bestmeet their needs. These group-ings may last anywhere fromminutes to semesters but shouldbe designed to help all students,whether strong, weak or average,to reach their highest potential.

There are other times when anappropriately open-ended taskcan be valuable to a broadspectrum of students. For exam-ple, asking students to make upan equation for which the answeris 5 allows some students to makeup very simple equations whileothers can design more complexones. The different equationsconstructed can become the basisfor a very rich lesson, from whichall students come away with abetter understanding of what thesolution to an equation reallymeans.

Entry - grade 12 mathematicsteachers must be provided withresource materials that includemany open-ended tasks fromwhich to choose. In addition,they need access to a broadspectrum of resources whichfocus on the needs of particularsegments of the student popula-tion.

COGNITIVEDEVELOPMENT

To match instruction to thestudent’s level of intellectualmaturity, teachers need to befamiliar with the stages of cogni-tive development, the student’sability to think mathematicallyand the student’s preferredlearning style. While there aregeneral patterns of developmentas students move through anumber of age spans, it must beemphasized that these stages aresimply reference points forteachers. Not all students movethrough these stages at the samerate, nor is any given studentnecessarily at the same stage in allareas within the mathematicsprogram (e.g., number conceptsvs. spatial sense). It is thereforeimportant for teachers at all levelsto build flexibility into theirprograms to help address indi-vidual developmental variations.

EQUITY AND DIVERSITY

Primary/Elementary (AgesFive to Ten)

Students initially learn mosteffectively through first-handexperience. In school they needtime and space for active explora-tion. As their language fluencygrows, so does their ability torepresent their thoughts symboli-cally through writing, drawing,graphing and making models.Teachers should ensure thatstudents experiment with avariety of ways to represent theirknowledge and understanding.

Students use direct experience,objects and visual aids to helpunderstanding. First-hand experi-ences are necessary, since stu-dents better understand symbol-ism if they have had adequate,meaningful experiences in theconcrete and pictorial modes.

Students will begin to think abouttheir own thinking and to developmore sophisticated strategies forlearning and for rememberingwhat they have learned.

“[At this stage] adevelopmentally-appropriatecurriculum encourages theexploration of a wide variety ofmathematical ideas in such away that children retain theirenjoyment of, and curiosityabout, mathematics. Itincorporates real-worldcontexts, children’s experiences,and children’s language indeveloping ideas. It recognizes

33Outcomes

that children need considerabletime to construct soundunderstandings and develop theability to reason andcommunicate mathematically. Itlooks beyond what childrenappear to know to determinehow they think about ideas. Itprovides repeated contact withimportant ideas in varyingcontexts throughout the yearand from year to year.”(NCTM Curriculum and EvaluationStandards, p.16)

Middle Level/Junior High(Ages Ten to Fourteen)

During these years many studentsbegin to think in abstractions.They are better able to under-stand the symbolic nature ofmathematics and to use symbolsto represent number relations andother mathematical abstractions.It should be noted that, althoughthey are beginning to develop theability to “manipulate” thoughtsand ideas, they still need hands-on experiences. The way in whichstudents process informationleads to a greater success insolving abstract problems.Accumulated knowledge, com-bined with logical conceptualconnections, allows solutions tomulti-stage problems.

Another trend at this time is thedevelopment of hypotheticalthinking and appreciation of themultiple possibilities that arepresented by given circum-stances. It is important to respectdifferent ways of presenting ideas.Positive encouragement in a low-risk environment is required in

order for students at this age todevelop their talents.

“As vast changes occur in theirintellectual, psychological,social, and physicaldevelopment, students [in thisstage] begin to develop theirabilities to think and reasonmore abstractly. Throughout thisperiod, however, concreteexperiences should continue toprovide the means by which theyconstruct knowledge. From theseexperiences they abstract morecomplex meanings and ideas.The use of language, bothwritten and oral, helps studentsclarify their thinking and reporttheir observations as they formand verify their mathematicalideas.”(NCTM Curriculum and EvaluationStandards, p.68)

High School (Ages Fourteento Nineteen)

During these years, thoughtprocesses become more preciseand analytical, while interestsbecome more specific andspecialized.

Students may use abstract rules tosolve problems but will needassistance and experience inrecognizing contexts for theapplication of such rules. It isimportant to note that the abilityto apply formal operational skillsvaries according to the degree ofexperience in an area; therefore,students need active involvementin constructing and applyingmathematical ideas in meaningfulcontexts. Also, during these years

students often prefer to do an in-depth investigation of an area ofinterest and choice.

As students continue to refinetheir thinking abilities, theydemonstrate greater awarenessof the complexity of issues andmay reject simplistic explana-tions. Increased life experienceprovides more and new opportu-nities for refinement of previ-ously-learned reasoning andthinking skills. Students developthe ability to move freely be-tween the concrete and theabstract, yet still need instruc-tional strategies providing both.

“The role of students in thelearning process…should shiftin preparation for theirentrance into the work force orhigher education. Experiencesdesigned to foster continuedintellectual curiosity andincreasing independence shouldencourage students to becomeself-directed learners whoroutinely engage inconstructing, symbolizing,applying, and generalizingmathematical ideas…[Further,] teachers and students[should] become naturalpartners in developingmathematical ideas and solvingmathematical problems.”(NCTM Curriculum and EvaluationStandards, p.128)



GENDER ANDCULTURAL EQUITY

The mathematics curriculum andmathematics instruction must bedesigned to equally empowerboth male and female students,as well as members of all culturalbackgrounds. Ultimately, thisshould mean that not only shouldenrolments of students of bothgenders and various culturalbackgrounds in public schoolmathematics courses reflectnumbers in society, but also thatrepresentative numbers of bothgenders and the various culturalbackgrounds should move on tosuccessful post-secondary studiesand careers in mathematics andmathematics-related areas.

Two obvious goals for mathemat-ics curriculum and mathematicsinstruction which will helpfacilitate gender and culturalequity are i) to portray thehistorical development of math-ematics as involving both menand women of a wide variety ofcultures and ii) to ensure that theinstructional environment isuniformly supportive of membersof both genders and all culturalgroups.

The first of these goals might beachieved by employing suchinstructional strategies as

• examining the lives of associ-ated mathematicians (includ-ing men and women ofvarious backgrounds) whenintroducing new mathematicaltopics in the classroom

• including the study of the livesof mathematicians such asSophie Germain andSubrahmanyan Chandrasekharas potential topics for indi-vidual research

Strategies designed to assist inthe accomplishment of thesecond goal include

• using curriculum materialswhich present a positive imageof mathematics and mathema-ticians

• employing an appropriateinstructional balance betweenindividual and group work

• ensuring equitable sharing ofclassroom resources (includingteacher attention and support)

• monitoring classroom instruc-tion for teacher bias

• facilitating contact betweenstudents and those working inmathematical professions

• reinforcing the belief that “allstudents can do mathematics”

35Outcomes

about what is really valued—whatis worth learning, how it shouldbe learned, what elements ofquality are considered mostimportant and how well studentsare expected to perform.

Teacher-developed assessmentsand evaluations have a widevariety of uses, such as

• providing feedback to improvestudent learning

• determining if curriculumoutcomes have been achieved

• certifying that students haveachieved certain levels ofperformance

• setting goals for future studentlearning

• communicating with parentsabout their children’s learning

• providing information toteachers on the effectivenessof their teaching, the programand the learning environment

• meeting the needs of guid-ance and administration

ASSESSMENT

To determine how well studentsare learning, assessment strategieshave to be designed to systemati-cally gather information on theachievement of the curriculumoutcomes. In planning assess-ments, teachers should use abroad range of strategies in anappropriate balance to givestudents multiple opportunities to

ASSESSING AND EVALUATING STUDENT LEARNING

demonstrate their knowledge,skills and attitudes. Many types ofassessment strategies can beused to gather such information,including, but not limited to,

• formal and informal observa-tions

• work samples

• anecdotal records

• conferences

• teacher-made and other tests

• portfolios

• learning journals

• questioning

• performance assessment

• peer- and self-assessment

EVALUATION

Evaluation involves teachers andothers in analyzing and reflectingupon information about studentlearning gathered in a variety ofways. This process requires

• developing clear criteria andguidelines for assigning marksor grades to student work

• synthesizing information frommultiple sources

• weighing and balancing allavailable information

• using a high level of profes-sional judgment in makingdecisions based upon thatinformation

Evaluationis the process of analyzing,reflecting upon andsummarizing assessmentinformation and makingjudgments or decisions basedupon the information gathered.

Assessmentis the systematic process ofgathering information onstudent learning


ssessment andevaluation areessential compo-

nents of teaching and learning inmathematics. Without effectiveassessment and evaluation it isimpossible to know whetherstudents have learned, whetherteaching has been effective orhow best to address studentlearning needs. The quality of theassessment and evaluation in theeducational process has a pro-found and well-established link tostudent performance. Researchconsistently shows that regularmonitoring and feedback areessential to improving studentlearning. What is assessed andevaluated, how it is assessed andevaluated and how results arecommunicated send clear mes-sages to students and others

A


REPORTING

Reporting on student learningshould focus on the extent towhich students have achieved thecurriculum outcomes. Reportinginvolves communicating thesummary and interpretation ofinformation about studentlearning to various audiences whorequire it. Teachers have a specialresponsibility to explain accu-rately what progress studentshave made in their learning andto respond to parent and studentinquiries about learning.

Narrative reports on progress andachievement can provide infor-mation on student learning whichletter or number grades alonecannot. Such reports might, forexample, suggest ways in whichstudents can improve theirlearning and identify ways inwhich teachers and parents canbest provide support.

Effective communication withparents regarding their children’sprogress is essential in fosteringsuccessful home-school partner-ships. The report card is onemeans of reporting individualstudent progress. Other meansinclude the use of conferences,notes and phone calls.

GUIDING PRINCIPLES

In order to provide accurate,useful information about theachievement and instructionalneeds of students, certain guidingprinciples for the development,administration and use of assess-ments must be followed. Principlesfor Fair Student Assessment Prac-tices for Education in Canada(1993) articulates five fundamen-tal assessment principles asfollows:

• Assessment methods shouldbe appropriate for and com-patible with the purpose andcontext of the assessment.

• Students should be providedwith sufficient opportunity todemonstrate the knowledge,skills, attitudes or behavioursbeing assessed.

• Procedures for judging orscoring student performanceshould be appropriate for theassessment method used andbe consistently applied andmonitored.

• Procedures for summarizingand interpreting assessmentresults should yield accurateand informative representa-tions of a student’s perform-ance in relation to the curricu-lum outcomes for the report-ing period.

• Assessment reports should beclear, accurate and of practicalvalue to the audience forwhom they are intended.

These principles highlight theneed for assessment whichensures that

• the best interests of thestudent are paramount

• assessment informs teachingand promotes learning

• assessment is an integral andongoing part of the learningprocess and is clearly relatedto the curriculum outcomes

• assessment is fair and equita-ble to all students and involvesmultiple sources of informa-tion

While assessments may be usedfor different purposes and audi-ences, all assessments must giveeach student optimal opportunityto demonstrate what he/sheknows and can do.

ASSESSING STUDENTLEARNING IN THEMATHEMATICSCLASSROOM

Student Assessment

“Testing to assign grades is oneof the most common forms ofevaluation. But assessment is amuch broader and basic task,one designed to determine whatstudents know and how theythink about mathematics.”(NCTM Curriculum and EvaluationStandards, p. 203)

Effective assessment of studentlearning in mathematics provideseducators with important infor-mation about the instructionalneeds of students and aboutstudent achievement of the

37Outcomes

curriculum outcomes. NCTM haspublished Assessment Standardsfor School Mathematics which maybe used as a guide to enableteachers to match assessmentpractice with the Atlantic Canadavision for mathematics. Theseprinciples provide a basis for bestpractice and are consistent withthe Principles for Fair StudentAssessment Practices for Educationin Canada.

Alignment with Curriculum

Tasks and methods for assessingstudents’ learning must bealigned with the curriculumoutcomes and instructionalpractice. An assessment programthat mirrors instruction and isinterwoven with it is vital for aclear, reliable and valid picture ofhow students are progressing andof how well instruction meets astudent’s needs.

Assessment instruments chosenneed to reflect the relativeemphases given in instruction totopics and to processes. Forexample, an effective programassesses the student’s progresswith the skills and concepts thatare stressed in a particular instruc-tional unit, and it includes ongo-ing assessment of the processcomponents—problem solving,communication, mathematicalreasoning and making math-ematical connections—to theextent that these processes arepart of the instructional unit.

For assessment to be aligned withinstruction, it is necessary thatassessment tasks involve the useof the same materials and tech-

nologies used regularly in theclassroom. The routine use ofcalculators, computers andmanipulatives is integral to thesuccess of this curriculum; there-fore, assessment tasks incorporat-ing the use of these materials andtechnologies are necessary.

EXTERNAL ASSESSMENT

Administration of externallyprepared assessments is on alarge scale in comparison toclassroom assessments and ofteninvolves hundreds, sometimesthousands, of students, allowingfor use of results at the provincial,district and/or school levels.Depending on the comprehen-siveness of the assessment,information can be used for all ofthe same purposes as classroom-based assessment, but it can alsoserve additional administrativeand accountability purposes suchas for admissions, placement,student certification, educationaldiagnosis and program evalua-tion. External assessments offercommon standards for assess-ment and for administration,scoring and reporting whichallow for comparison of resultsover time.

As part of the regional agenda,development of external assess-ments in the core curriculumareas is being undertaken. Gener-ally, external assessment includesassessments prepared by depart-ments of education, national andinternational assessment groups,publishers and research groups.Each provincial department ofeducation makes decisions on

whether or not to administerexternal assessments.

PROGRAM AND SYSTEMEVALUATION

The results from both externaland internal assessments ofstudent achievement can be usedto varying degrees for programand system evaluation. Externalassessment results, however, aremore comparable across variousgroups and are therefore morecommonly the basis for thesetypes of evaluations.

In essence, the main differencebetween student evaluation andprogram and system evaluation isin how the results are used. Inprogram evaluation marks orscores for individual students arenot the primary focus of theassessment—it is the effective-ness of the program that isevaluated, and the results areused to show the extent to whichthe many outcomes of theprogram are achieved.

When results are used for systemevaluation, the focus is on howthe various levels and groupswithin the system, such asclassrooms, schools, districts andso on, are achieving the intendedoutcomes. In many ways studentand program evaluation are verymuch the same in that bothemphasize obtaining studentinformation concerning theirconceptual understanding, theirability to use knowledge andreason to solve problems andtheir ability to communicateeffectively.



MANIPULATIVEMATERIALS

The use of manipulative materialsin the mathematics classroomsupports the development ofunderstanding in students ofvarious ages and developmentallevels. Manipulatives may be usedto introduce new concepts, verifyresults and/or provide remedialhelp.

Every student should have theopportunity to explore withmanipulative materials.Manipulatives should clearlyrepresent the concept beingtaught; nevertheless, a particularmanipulative may be sufficientlyflexible for use in teaching avariety of concepts.

When possible, a variety ofmanipulatives should be used todevelop each concept, thedifferent representations helpingstudents to generalize mathemati-cal ideas. Students should as wellexpress their understanding ofthe connection among themanipulatives, the mathematicalconcept and the mathematicalsymbols used to denote it.

PROFESSIONALRESOURCES

Resources with which all teachersof mathematics should be familiarare publications of the NationalCouncil of Teachers of Mathemat-ics (NCTM):

• Curriculum And EvaluationStandards for School Mathemat-ics (1989)—represents NCTM’svision of what the mathematicscurriculum should include interms of content priority andemphasis

• Professional Standards ForTeaching Mathematics (1991)—provides standards regardingwhat teachers must know inorder to teach toward the newcurriculum standards, recom-mendations for professionaldevelopment and suggestionsfor the evaluation of theteaching of mathematics

• Assessment Standards For SchoolMathematics (1995)

• Addenda to the Standards

• Yearbooks

Professional publications helpteachers by providing substantiveideas about mathematics teaching.Four recommended journals fromNCTM are

• Teaching Children Mathematics(K-5)

• Mathematics Teaching In TheMiddle School (grades 6-8)

• Mathematics Teacher (grades 9-12)

• Journal for Research in Math-ematics Education

RESOURCES

he challenge facingteachers today is im-plementing a math-

ematics program that will preparestudents for a rapidly changingworld. There is an increasing needfor a variety of resources tosupport the teaching of thecurriculum at all levels (entry-grade 12).

Students’ learning resources andtechnologies used must supportboth the philosophy and recom-mended approaches of themathematics curriculum andshould be used in conjunctionwith a variety of other availableteaching tools. These tools mayinclude professional resources,kits, games, activity cards, ma-nipulative materials, calculatorsand computer software. Whenpurchasing commercial materials,it is important to be selective,choosing those that complementthe curriculum focus and out-comes.

It should also be clear, given theneed to implement new math-ematical content, new instruc-tional techniques and instructionalmaterials such as those identifiedhere, that on-going professionaldevelopment will be needed.

T

39Outcomes

TECHNOLOGY

“... New technology not only hasmade calculations and graphingeasier, it has changed the verynature of the problemsimportant to mathematics andthe methods mathematicians useto investigate them.”(NCTM Curriculum and EvaluationStandards, p. 8)

Modern computing technologieshave changed the way everyonelearns and works. For many years,they have served as a catalyst forreform efforts in mathematicseducation. In today’s elementaryand secondary schools, they canand will continue to influencehow mathematics is learned andtaught, not only by making, forexample, calculations andgraphing easier and more man-ageable, but also by altering thevery nature of what mathematicsis important to learn. Newproblems as well as innovativeways of investigating them nowbecome possible.

NCTM’s Curriculum and EvaluationStandards for School Mathematicsacknowledges the impact thattechnology has on mathematicsand its many uses. The CurriculumStandards recommend the use oftechnology i) to enhance theteaching and the learning ofmathematics and ii) to relateschool mathematics to the worldin which the students live throughdeveloping and interpretingmathematical models.


The introduction of computers,graphics calculators, videotechnology and other technolo-gies into the mathematics class-room allows students to

• explore situations with compli-cated numbers which previ-ously would have been be-yond their capabilities

• explore individual or groups ofrelated computations orfunctions quickly and easily

• create and explore numericand geometric situations forthe purpose of developingconjectures

• perform simulations of situa-tions which would otherwisebe impossible to examine

• easily link different representa-tions of the same information

• model situations mathemati-cally

• observe the effects of simplechanges in parameters orcoefficients

• analyze, organize and displaydata

All of these situations enhancediscovery learning and problem-solving potential. At the sametime, teachers have the opportu-nity to use technology to commu-nicate with fellow mathematicsteachers, to share lessons withexperts and to expose theirstudents to information that wouldotherwise be inaccessible.

As more sophisticated technologybecomes accessible in classrooms,its productive use in support of themathematics curriculum will needto be considered. In making suchdecisions, the opportunity forimproved instruction and learningshould be the guiding principle.

The effective use of technology inthe mathematics classroom is noteasy to achieve. Students will needto learn to make judgments as towhen the use of technology isappropriate and when it is not. Inall situations, it is imperative that itshould be used both as a tool toinclude, rather than exclude,students and as a means ofcreating new teaching strategies.


ReferencesCentre for Research in Applied Measurement andEvaluation. Principles for Fair Student AssessmentPractices for Education in Canada. Edmonton, AB:University of Alberta, 1993.

National Council of Teachers of Mathematics.Assessment Standards for School Mathematics.Reston, VA: NCTM, 1995.

National Council of Teachers of Mathematics.Curriculum and Evaluation Standards for SchoolMathematics. Reston, VA: NCTM, 1989.

National Council of Teachers of Mathematics.Estimation and Mental Computation: 1986 Yearbook.Reston, VA: NCTM, 1986.

National Council of Teachers of Mathematics.Professional Standards for Teaching Mathematics.Reston, VA: NCTM, 1991.

Steen, et al. “Teaching Mathematics for Tomor-row’s World.” Educational Leadership, (September,1989).

Van De Walle, John. Elementary School Mathematics:Teaching Developmentally. 2nd ed. New York:Longman, 1994.

province of new brunswickqueen elizabeth high school ken macinnis, elementary teacher, sir charles...

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