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First Nations, Métis, and Inuit PerspectivesFirst Nations, Métis, and Inuit students in Manitoba come from diversegeographic areas with varied cultural and linguistic backgrounds. Studentsattend schools in a variety of settings, including urban, rural, and isolatedcommunities. Teachers need to recognize and understand the diversity ofcultures within schools and the diverse experiences of students.First Nations, Métis, and Inuit students often have a whole-world view ofthe environment; as a result, many of these students live and learn best in aholistic way. This means that students look for connections in learning, andlearn mathematics best when it is contextualized and not taught as discretecontent.Many First Nations, Métis, and Inuit students come from culturalenvironments where learning takes place through active participation.Traditionally, little emphasis was placed upon the written word. Oralcommunication along with practical applications and experiences areimportant to student learning and understanding.A variety of teaching and assessment strategies are required to build uponthe diverse knowledge, cultures, communication styles, skills, attitudes,experiences, and learning styles of students. The strategies used must gobeyond the incidental inclusion of topics and objects unique to a culture orregion, and strive to achieve higher levels of multicultural education (Banksand Banks).Affective DomainA positive attitude is an important aspect of the affective domain that has aprofound effect on learning. Environments that create a sense of belonging,encourage risk taking, and provide opportunities for success help studentsdevelop and maintain positive attitudes and self-confidence. Students withpositive attitudes toward learning mathematics are likely to be motivated andprepared to learn, participate willingly in classroom learning activities, persistin challenging situations, and engage in reflective practices.Teachers, students, and parents* need to recognize the relationship betweenthe affective and cognitive domains, and attempt to nurture those aspectsof the affective domain that contribute to positive attitudes. To experiencesuccess, students must be taught to set achievable goals and assess themselvesas they work toward reaching these goals.Striving toward success and becoming autonomous and responsible learnersare ongoing, reflective processes that involve revisiting the setting andassessment of personal goals.*In this document, the term parents refers to both parents and guardians and is used with the recognitionthat in some cases only one parent may be involved in a child’s education.Introduction3

Early ChildhoodYoung children are naturally curious and develop a variety of mathematicalideas before they enter Kindergarten. Children make sense of theirenvironment through observations and interactions at home, in daycares,in preschools, and in the community. Mathematics learning is embedded ineveryday activities, such as playing, reading, storytelling, and helping aroundthe home.Activities can contribute to the development of number and spatial sense inchildren. Curiosity about mathematics is fostered when children are engagedin activities such as comparing quantities, searching for patterns, sortingobjects, ordering objects, creating designs, building with blocks, and talkingabout these activities.Positive early experiences in mathematics are as critical to child developmentas early literacy experiences are.Mathematics Education Goals for StudentsThe main goals of mathematics education are to prepare students toQQQQQQQQQQQQcommunicate and reason mathematicallyuse mathematics confidently, accurately, andefficiently to solve problemsappreciate and value mathematicsmake connections between mathematicalknowledge and skills and their applicationsMathematics educationmust prepare studentsto use mathematics tothink critically aboutthe world.commit themselves to lifelong learningbecome mathematically literate citizens, usingmathematics to contribute to society and to think critically about the worldStudents who have met these goals willQQ4gain understanding and appreciation of the contributions of mathematics asa science, a philosophy, and an artQQexhibit a positive attitude toward mathematicsQQengage and persevere in mathematical tasks and projectsQQcontribute to mathematical discussionsQQtake risks in performing mathematical tasksQQexhibit curiosityG r a d e 4 M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s

Conceptual Framework for Kindergarten to Grade 9 MathematicsC OTheN C EchartP T U AbelowL F R providesA M E W O R anK FoverviewO R K - 9 ofMhowA T H mathematicalEMAT ICSprocesses and thenatureof mathematics influence learning outcomes.The chart below provides an overview of how mathematical processesand the nature of mathematics influence learning ATICSPatterns and Relations Patterns Variables and EquationsCHANGE,Shape and SpaceCONSTANCY, NUMBER SENSE, PATTERNS,RELATIONSHIPS,SPATIAL SENSE,UNCERTAINTY Measurement3-D Objects and 2-DShapesGENERAL LEARNING OUTCOMES,SPECIFIC LEARNING OUTCOMES,AND ACHIEVEMENT INDICATORSTransformationsStatistics and Probability Data Analysis Chance and UncertaintyMATHEMATICAL PROCESSES:COMMUNICATION, CONNECTIONS, MENTALMATHEMATICS AND ESTIMATION, PROBLEMSOLVING, REASONING, TECHNOLOGY,VISUALIZATIONMathematical ProcessesThere are critical components that students must encounter in mathematics toachieve the goals of mathematics education and encourage lifelong learning inmathematics.Conceptual Framework for K-9 Mathematics7Students are expected toQQQQcommunicate in order to learn and express their understandingconnect mathematical ideas to other concepts in mathematics, to everydayexperiences, and to other disciplinesQQdemonstrate fluency with mental mathematics and estimationQQdevelop and apply new mathematical knowledge through problem solvingQQdevelop mathematical reasoningQQselect and use technologies as tools for learning and solving problemsQQdevelop visualization skills to assist in processing information, makingconnections, and solving problemsIntroduction5

The common curriculum framework incorporates these seven interrelatedmathematical processes, which are intended to permeate teaching andlearning:QQQQQQQQQQQQQQCommunication [C]: Students communicate daily (orally, through diagramsand pictures, and by writing) about their mathematics learning. They needopportunities to read about, represent, view, write about, listen to, anddiscuss mathematical ideas. This enables them to reflect, to validate, and toclarify their thinking. Journals and learning logs can be used as a record ofstudent interpretations of mathematical meanings and ideas.Connections [CN]: Mathematics should be viewed as an integrated whole,rather than as the study of separate strands or units. Connections mustalso be made between and among the different representational modes—concrete, pictorial, and symbolic (the symbolic mode consists of oral andwritten word symbols as well as mathematical symbols). The process ofmaking connections, in turn, facilitates learning. Concepts and skills shouldalso be connected to everyday situations and other curricular areas.Mental Mathematics and Estimation [ME]: The skill of estimation requiresa sound knowledge of mental mathematics. Both are necessary to manyeveryday experiences, and students should be provided with frequentopportunities to practise these skills. Mental mathematics and estimationis a combination of cognitive strategies that enhances flexible thinking andnumber sense.Problem Solving [PS]: Students are exposed to a wide variety of problemsin all areas of mathematics. They explore a variety of methods for solvingand verifying problems. In addition, they are challenged to find multiplesolutions for problems and to create their own problems.Reasoning [R]: Mathematics reasoning involves informal thinking,conjecturing, and validating—these help students understand thatmathematics makes sense. Students are encouraged to justify, in a varietyof ways, their solutions, thinking processes, and hypotheses. In fact, goodreasoning is as important as finding correct answers.Technology [T]: The use of calculators is recommended to enhance problemsolving, to encourage discovery of number patterns, and to reinforceconceptual development and numerical relationships. They do not, however,replace the development of number concepts and skills. Carefully chosencomputer software can provide interesting problem-solving situations andapplications.Visualization [V]: Mental images help students to develop conceptsand to understand procedures. Students clarify their understanding ofmathematical ideas through images and explanations.These processes are outlined in detail in Kindergarten to Grade 8 Mathematics:Manitoba Curriculum Framework of Outcomes (2013).6G r a d e 4 M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s

StrandsThe learning outcomes in the Manitoba curriculum framework are organizedinto four strands across Kindergarten to Grade 9. Some strands are furthersubdivided into substrands. There is one general learning outcome persubstrand across Kindergarten to Grade 9.The strands and substrands, including the general learning outcome for each,follow.NumberQQDevelop number sense.Patterns and RelationsQQPatternsQQQQUse patterns to describe the world and solve problems.Variables and EquationsQQRepresent algebraic expressions in multiple ways.Shape and SpaceQQMeasurementQQQQ3-D Objects and 2-D ShapesQQQQUse direct and indirect measure to solve problems.Describe the characteristics of 3-D objects and 2-D shapes, and analyzethe relationships among them.TransformationsQQDescribe and analyze position and motion of objects and shapes.Statistics and ProbabilityQQData AnalysisQQQQCollect, display, and analyze data to solve problems.Chance and UncertaintyQQUse experimental or theoretical probabilities to represent and solveproblems involving uncertainty.Introduction7

Learning Outcomes and Achievement IndicatorsThe Manitoba curriculum framework is stated in terms of general learningoutcomes, specific learning outcomes, and achievement indicators:QQQQQQGeneral learning outcomes are overarching statements about what studentsare expected to learn in each strand/substrand. The general learningoutcome for each strand/substrand is the same throughout the grades fromKindergarten to Grade 9.Specific learning outcomes are statements that identify the specific skills,understanding, and knowledge students are required to attain by the end ofa given grade.Achievement indicators are samples of how students may demonstratetheir achievement of the goals of a specific learning outcome. The range ofsamples provided is meant to reflect the depth, breadth, and expectations ofthe specific learning outcome. While they provide some examples of studentachievement, they are not meant to reflect the sole indicators of success.In this document, the word including indicates that any ensuing items must beaddressed to meet the learning outcome fully. The phrase such as indicates thatthe ensuing items are provided for illustrative purposes or clarification, andare not requirements that must be addressed to meet the learning outcomefully.SummaryThe conceptual framework for Kindergarten to Grade 9 mathematics describesthe nature of mathematics, the mathematical processes, and the mathematicalconcepts to be addressed in Kindergarten to Grade 9 mathematics. Thecomponents are not meant to stand alone. Learning activities that take placein the mathematics classroom should stem from a problem-solving approach,be based on mathematical processes, and lead students to an understandingof the nature of mathematics through specific knowledge, skills, and attitudesamong and between strands. Grade 4 Mathematics: Support Document for Teachersis meant to support teachers to create meaningful learning activities that focuson formative assessment and student engagement.8G r a d e 4 M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s

AssessmentAuthentic assessment and feedback are a driving force for the suggestionsfor assessment in this document. The purposes of the suggested assessmentactivities and strategies are to parallel those found in Rethinking ClassroomAssessment with Purpose in Mind: Assessment for Learning, Assessment as Learning,Assessment of Learning (Manitoba Education, Citizenship and Youth). Theseinclude the following:QQassessing for, as, and of learningQQenhancing student learningQQassessing students effectively, efficiently, and fairlyQQproviding educators with a starting point for reflection, deliberation,discussion, and learningAssessment for learning is designed to give teachers information to modifyand differentiate teaching and learning activities. It acknowledges thatindividual students learn in idiosyncratic ways, but it also recognizes thatthere are predictable patterns and pathways that many students follow. Itrequires careful design on the part of teachers so that they use the resultinginformation to determine not only what students know, but also to gaininsights into how, when, and whether students apply what they know.Teachers can also use this information to streamline and target instructionand resources, and to provide feedback to students to help them advance theirlearning.Assessment as learning is a process of developing and supportingmetacognition for students. It focuses on the role of the student as the criticalconnector between assessment and learning. When students are active,engaged, and critical assessors, they make sense of information, relate it toprior knowledge, and use it for new learning. This is the regulatory process inmetacognition. It occurs when students monitor their own learning and usethe feedback from this monitoring to make adjustments, adaptations, and evenmajor changes in what they understand. It requires that teachers help studentsdevelop, practise, and become comfortable with reflection, and with a criticalanalysis of their own learning.Assessment of learning is summative in nature and is used to confirm whatstudents know and can do, to demonstrate whether they have achieved thecurriculum learning outcomes, and, occasionally, to show how they are placedin relation to others. Teachers concentrate on ensuring that they have usedassessment to provide accurate and sound statements of students’ proficiencyso that the recipients of the information can use the information to makereasonable and defensible decisions.Introduction9

Overview of Planning AssessmentAssessment for LearningWhy QQQQQQQUsing theInformationQQQQQQto enable teachers todetermine next steps inadvancing student learningeach student’s progressand learning needs inrelation to the curriculumoutcomesa range of methods indifferent modes that makea student’s skills andunderstanding visibleaccuracy and consistencyof observations andinterpretations of studentlearningclear, detailed learningexpectationsaccurate, detailed notesfor descriptive feedback toeach studentprovide each studentwith accurate descriptivefeedback to further his orher learningdifferentiate instruction bycontinually checking whereeach student is in relationto the curriculum outcomesprovide parents orguardians with descriptivefeedback about studentlearning and ideas forsupportAssessment as LearningQQQQQQQQQQQQQQQQQQQQQQto guide and provideopportunities for eachstudent to monitor andcritically reflect on his orher learning and identifynext stepseach student’s thinkingabout his or her learning,what strategies he orshe uses to support orchallenge that learning,and the mechanisms heor she uses to adjust andadvance his or her learninga range of methods indifferent modes that elicitthe student’s learning andmetacognitive processesaccuracy and consistencyof a student’s selfreflection, self-monitoring,and self-adjustmentengagement of thestudent in consideringand challenging his or herthinkingAssessment of LearningQQQQQQQQQQQQthe student records his orher own learningprovide each studentwith accurate, descriptivefeedback that will help himor her develop independentlearning habitshave each student focuson the task and his or herlearning (not on getting theright answer)provide each studentwith ideas for adjusting,rethinking, and articulatinghis or her learningQQQQQQto certify or inform parentsor others of student’sproficiency in relationto curriculum learningoutcomesthe extent to which eachstudent can apply thekey concepts, knowledge,skills, and attitudes relatedto the curriculum outcomesa range of methods indifferent modes thatassess both product andprocessaccuracy, consistency,and fairness of judgmentsbased on high-qualityinformationclear, detailed learningexpectationsfair and accuratesummative reportingindicate each student’slevel of learningprovide the foundation fordiscussions on placementor promotionreport fair, accurate, anddetailed information thatcan be used to decide thenext steps in a student’slearningprovide the conditions forthe teacher and student todiscuss alternativesthe student reports his orher learningSource: Manitoba Education, Citizenship and Youth. Rethinking Classroom Assessment with Purpose in Mind: Assessment forLearning, Assessment as Learning, Assessment of Learning. Winnipeg, MB: Manitoba Education, Citizenship and Youth,2006, 85.10G r a d e 4 M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s

Instructional FocusThe Manitoba curriculum framework is arranged into four strands. Thesestrands are not intended to be discrete units of instruction. The integrationof learning outcomes across strands makes mathematical experiencesmeaningful. Students should make the connection between concepts bothwithin and across strands.Consider the following when planning for instruction:QQQQQQQQQQQQQQQQQQRoutinely incorporating conceptual understanding, procedural thinking,and problem solving within instructional design will enable students tomaster the mathematical skills and concepts of the curriculum.Integration of the mathematical processes within each strand is expected.Problem solving, conceptual understanding, reasoning, making connections,and procedural thinking are vital to increasing mathematical fluency, andmust be integrated throughout the program.Concepts should be introduced using manipulatives and graduallydeveloped from the concrete to the pictorial to the symbolic.Students in Manitoba bring a diversity of learning styles and culturalbackgrounds to the classroom and they may be at varying developmentalstages. Methods of instruction should be based on the learning styles andabilities of the students.Use educational resources by adapting to the context, experiences, andinterests of students.Collaborate with teachers at other grade levels to ensure the continuity oflearning of all students.Familiarize yourself with exemplary practices supported by pedagogicalresearch in continuous professional learning.Provide students with several opportunities to communicate mathematicalconcepts and to discuss them in their own words.“Students in a mathematics class typically demonstrate diversity in the waysthey learn best. It is important, therefore, that students have opportunities tolearn in a variety of ways—individually, cooperatively, independently, withteacher direction, through hands-on experience, through examples followedby practice. In addition, mathematics requires students to learn conceptsand procedures, acquire skills, and learn and apply mathematical processes.These different areas of learning may involve diffe

Grade 4 Mathematics: Support Document for Teachers provides various suggestions for instruction, assessment strategies, and learning resources that promote the meaningful engagement of mathematics learners in Grade 4. The document is intended to be