western canadian teacher guide - sd67 (okanagan...

Kindergarten

Western

Teacher GuideWestern Canadian

Assessment Support

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Assessment Support

Complete Teacher Guide Package

Anthony Azzopardi

Sandra Ball

Bob Belcher

Judy Blake

Steve Cairns

Daryl Chichak

Lynda Colgan

Marg Craig

Jennifer Gardner

Florence Glanfield

Linden Gray

Pamela Hagen

Dennis Hamaguchi

Angie Harding

Peggy Hill

Auriana Kowalchuk

Gordon Li

Werner Liedtke

Jodi Mackie

Kristi Manuel

Lois Marchand

Cathy Molinski

Bill Nimigon

Eileen Phillips

Evelyn Sawicki

Shannon Sharp

Martha Stewart

Lynn Strangway

Mignonne Wood

Publishing TeamSusan GreenRosalyn SteinerShari SiamonStephanie CoxJudy WilsonNicole Argyropoulos

PublisherClaire Burnett

Team LeaderAnne-Marie Scullion

DesignWord & Image Design Studio Inc.

Program Advisers

Pearson Education thanks its Program Advisers, who helped shape the visionfor Addison Wesley Mathematics Makes Sense through discussions and reviewsof prototype materials and manuscript.

Field Testers

Pearson Education thanks the teachers and students who field-tested Addison Wesley Math Makes Sense prior to publication. Their feedback and constructive recommendations have been most valuable in helping us to develop a quality mathematics program.

Kindergarten Reviewers

Program Reviewers

Debbie BaileyCalgary Board of Education, AB

Bonnie BonfontiPrimary Teacher, BC

Cathy BromleyDurham District School Board,ON

Elvira CastanedaCalgary Roman Catholic SeparateDistrict, AB

Kanja ChenToronto District School Board, ON

Ralph ConnellyBrock University, ON

Barbara DuewelHastings and Prince EdwardDistrict School Board, ON

Lorelei GibeauEdmonton Catholic SeparateSchool District, AB

Margaret HalfertyCalgary Roman Catholic SeparateDistrict, AB

Maxine HardyCape Breton Victoria-RegionalSchool Board, NS

Cathie KlaassenLimestone District School Board,ON

Mary A. LeBlancHalifax Regional School Board, NS

Christine PazzanoToronto District School Board, ON

Anne PowersLimestone District School Board,ON

Shannon SharpNorth Vancouver School District,BC

Lynn StrangwayToronto District School Board, ON

Vera TeschowPeel District School Board, ON

Marie ThomRichmond School District 38, BC

Jennifer TravisPeel District School Board, ON

Cathy VachonOttawa-Carleton Catholic SchoolBoard, ON

Jodi WistSt. Paul’s Roman CatholicSeparate School Division, SK

Assessing Student Progress

A. Observe and L i s ten 1

B . Ora l Ques t ion ing and Conferenc ing 5

C . Work Samples and Co l l e c t ions 8

D. Re f le c t ion and Se l f -Assessment 10

E . Learn ing Sk i l l s and D i spos i t ions 12

F. Assess ing A l l Learners 13

G. References 14

Problem-Solving Investigations

Teach ing a Prob lem-Based Program 15

How Can We Make a Pa t te rn? 16

How Many Ways Can You Make 5? 18

How Can We A l l Take a Turn? 20

How Can We Share the Puppe ts? 22

How Many o f Each Cou ld There Be? 24

C O N T E N T S

Assessment Masters

Assessment Mas ter 1 .0 : D iagno s t i c Chec k l i s t 26

Assessment Mas ter 1 .1 : D iagno s t i c Con f e r e n ce f o r S e l e c t ed Ch i l d r e n 27

Assessment Mas ter 1 .2 .1 : Ongo i ng Obse r va t i o n s Chec k l i s t 28



Assessment Mas ter 1 .3 : Deve l opmen t a l Cha r t 31






















Assessment Mas ter 7 : Obse r va t i o n s R e co rd 1 53

Assessment Mas ter 8 : Obse r va t i o n s R e co rd 2 : Cen t r e s a nd P r a c t i c e Wo r k 54

Assessment Mas ter 9 : Con f e r e n ce P r omp t s 55

Assessment Mas ter 10 : L anguage t o L i s t e n F o r 56

Assessment Mas ter 11 : L ea r n i ng S k i l l s 57

Assessment Mas ter 12 : Se l f - A s s e s smen t I n t e r v i ew 58

Assessment Mas ter 13 : Obse r va t i o n s Chec k l i s t : S o l v i ng a P r ob l em 59

Assessment Mas ter 14 : Un i t S umma r y 60

Assessment Mas ter 15 : Obse r va t i o n s Chec k l i s t : Cu r r i c u l um E xpec t a t i o n s 61

Effective assessment involves ongoing, consistent, andfocused observations. For a complete picture, weobserve children at different times of the day, indifferent settings, doing different kinds of work.

Jean Kerr Stenmark & William S. Bush (eds.) (2001), p. 38

E f fec t i ve Observa t ion

Skilful observation is one of the keys toeffective teaching. Teachers routinely adjusttheir lessons, offer support and feedback tochildren, modify tasks, ask probing questions,and make plans for future activities based onwhat they notice. In the course of a mathematicscircle or activity, kindergarten teachers maymake literally hundreds of observations. Someof these observations become written records,others are more fleeting; teachers observe,assess a learning situation, and act on what theynotice, almost simultaneously. For manyteachers, the most challenging part of observingis documenting what they see and hear.

Experienced kindergarten teachers havedeveloped a variety of observing and recordingstrategies.

Emphasize active, hands-on explorations and problems

• Ensure that children are actively engaged inexploring and problem solving. When childrenparticipate actively, there is somethingimportant to observe—they reveal theirthinking and the strategies they are using.

Focus

• Focus observations on a small number ofbehaviours or expectations that are central tothe content of the lesson. Teachers whoobserve effectively plan their observations at

the same time as they plan the lesson, andclearly identify some of the behaviours thatwill indicate children are developing—or notdeveloping—the necessary concepts andstrategies.

• Give children’s understanding of conceptsand their use of strategies priority inobserving. Most teachers find that they areable to gather a great deal of informationabout the procedures and skills that childrenare developing from work samples and otherrecords; on the other hand, observation maybe the only way to gain insight into a child’suse of strategies.

• Focus attention on 2 or 3 children in eachlesson. While most teachers observe severalchildren in any one lesson, they often select2 or 3 as the focus for their observations andrecording. In this way, they ensure that theygather evidence about all children over each2- to 3-week period.

Question and prompt

• Use thoughtful questions and prompts tolearn more about children’s thinking as theywork. Simple prompts, such as, “Tell meabout your thinking,” “Show me whatyou’ve tried,” or “How did you decide…?”can help to elaborate and clarifyobservations.

Set up a recording system that suits their needs

• Make anecdotal notes on self-stick notes,labels, or index cards that are prelabelledwith children’s names, or use an electronicgrid.

• Use a checklist that identifies specificbehaviours associated with the lesson,simply entering a check mark, date, or codeto indicate what has been observed.

A S S E S S I N G S T U D E N T P R O G R E S S

Assessing Student Progress 1

Observe and ListenA

2 Assessing Student Progress

• Use assessments or developmental charts toguide observations and document behaviours.Teachers often use a highlighter pen to recordobservations about individual children.

Seize the moment

• Be flexible. Not all observations can beplanned and focused. Often, children revealtheir mathematical thinking anddevelopment in unexpected ways and atunexpected times—sometimes even duringother subjects or outside of class. Someteachers take time routinely to write onesentence at the end of the morning and onesentence immediately after school to capturethese important events and insights. Overtime, this can become a rich source ofinformation.

Look back

• Reread past observations frequently,especially just before focusing observationson a particular child or group of children.This helps to provide context and continuity,and can reduce the amount of writingrequired in some cases.

Us ing the Observa t ion Fea tures and Too l s

Observation is the key to assessment inkindergarten. In Math Makes Sense, the greatestnumber of assessment masters is designed tosupport this important area of assessment.

Launch to the Year andDiagnos t i c Check l i s t

The Teacher Guide for kindergarten begins withan opening activity, Launch to the Year (Unit 1,page 8), designed to take place over severalsessions. Assessment Master 1.0: Diagnostic Checklist(page 26) identifies a small number of specificbehaviours teachers can watch for anddocument as children talk about what they seein a series of pictures, compare the pictures to

their own experiences, and offer ideas aboutwhat might come next in a sequence. It isexpected that any kindergarten class willcontain children with a wide range ofexperiences. Documenting some of thoseexperiences closely related to mathematicaldevelopment can help teachers plan theirprograms, particularly the initial activities.

Assessment fo r Learn ing andOngo ing Observa t ions Check l i s t

Each Math Circle includes the feature“Assessment for Learning,” which cues teachersto 2 or 3 key behaviours they may note aschildren participate in lesson activities. Over thecourse of a unit, these behaviours focus on mostof the expectations associated with the unit andreflect various aspects of achievement, includingunderstanding concepts, applying procedures,


solving problems, and communicating. Mostimportant, this feature includes “What to Do,”specific suggestions teachers can use withchildren who may be having difficulty or needanother approach to concepts.

The assessment master Ongoing ObservationsChecklist (beginning pp. 28, 33, 37, 41, 45, 50) isassociated with each unit. The behaviours inthe “Assessment for Learning” feature arereproduced in this checklist, providing anefficient way for teachers to note children’sprogress. Teachers can record their observationsusing symbols or notes that are mostmeaningful in terms of their own recordkeeping. Some teachers record levels ofachievement; others prefer simple check marksand/or pluses; still others choose to note thedate on which the behaviours were observed.The Ongoing Observations Checklists can besummarized on Assessment Master 14: UnitSummary (page 60), using a different copy foreach unit.

Other Observa t ion Too l s

Assessment Master 7: Observations Record 1 (page 53)provides a format that teachers can use torecord anecdotal notes about 2 or 3 childrenwho are working together or in closeproximity. The form is appropriate for anyactivity. It includes space for teachers to notetheir assessments of how well the activityworked and ideas about adjustments theymight make in future.

Assessment Master 8: Observations Record 2: Centresand Practice Work (page 54) is designed toaccumulate observations about a child’sdevelopment over several activities. The formalso includes space for teachers to explicitlyidentify any patterns they have noticed in thechild’s development and to make notes aboutpotential next steps.

In each unit, the assessment masterDevelopmental Chart (pp. 31, 35, 39, 43, 48, 52) isprovided where teachers can record theirobservations and assessments about the extentto which a child is showing progress towardsthe expectations associated with that unit. Inthis case, observations are likely to becombined with other evidence (e.g.,conferences, work samples).


Assessment Master 10: Language to Listen For (page56) lists some of the vocabulary related tomathematics that teachers are likely to hear aschildren engage in and talk about a variety oflearning activities. Teachers can use thisassessment as a reference, or as a way ofdeveloping a class or individual profile, byhighlighting language as they hear it.

Assessment Master 13: Observations Checklist:Solving a Problem (page 59) is designed toallow teachers to record a few keyobservations about children as they engage inproblem-solving activities. The form can beused for the entire class or can focus on a fewchildren in each session.

Assessment Master 15: Observations Checklist:Curriculum Expectations (comprehensive) (page61) focuses on the overall expectations fromthe kindergarten curriculum and providesspace for teachers to accumulate observations,assessments, and evaluations of children’sprogress towards achieving theseexpectations. While the form focuses on theoverall expectations, the specific expectationsare also provided.

Oral questioning plays a critical role inassessment:• in prompting class and group discussion

where teachers are able to stimulate studentthinking and gain insights into their levels ofunderstanding

• in individual interviews or conferences, bothformal and informal, where teachers promptand support children as they explain andjustify their thinking, the procedures they areusing, and their results and conclusions

E f fec t i ve Ques t ion ing andConferenc ing

Preparation

• Be prepared. Keep a short list of keyquestions readily available to use whenopportunities for a conversation or shortconference arise. Identify key questions formore formal interviews or conferences aheadof time.

Focus

• Focus carefully on the child you are speakingwith. It is often difficult to give one child fullattention in a busy classroom, and it is easyto become distracted. Children know quicklywhen an adult has stopped paying closeattention, and they quite logically concludethat what they are saying is not interesting orimportant to that adult.

Accept, rephrase, and model

• Accept the language that children usewithout criticism or correction. At the sametime, rephrase some of the key ideas andmodel appropriate terminology, so that theybecome accustomed to hearing the languageof mathematics.

Emphasize thinking

• Ensure that children understand that it istheir thinking that is most interesting to you.Children need to see that their thinking isvalued more than a particular solution orapproach. Offer encouragement through closeattention, positive body language, andnonjudgmental comments.

Allow wait time

• Allow wait time after a question. Ifchildren’s thinking is valued, they need timeto think. After an initial response, childrenneed time to think again—to reflect, self-correct, or confirm what they have said.Some teachers count slowly and silently to10 to ensure that they do not interrupt achild’s thinking.

Choose when to scaffold

• Whether teachers provide scaffoldingdepends on the purpose and context of thequestioning. While it is important to benonjudgmental, it is also important to helpchildren clarify misconceptions and developunderstanding. Putting forward ideas andsuggestions tentatively (e.g., “I wonder howwe could . . .” “What if . . . ?) and rephrasingto clarify what a child has said respects thechild and may help that child reach a newlevel of understanding.

Invite children’s questions

• Invite children to pose their own questions.Teachers often learn more about children’sunderstanding from the questions they askthan from the responses they offer.


Oral Questioning and ConferencingB


Explore different ways of approaching the same question

• Try to find a way of phrasing key questionsthat is comfortable for the child. Forexample, instead of asking: “Tell me about . . . .” consider “How would you explain . . .to someone who was new to our class?” or“What is the most important thing you’velearned about . . . ?”

Incorporate demonstrations

• Do not confuse children’s linguistic abilitieswith their mathematical understanding. ESLchildren or children who have delayedlanguage competencies are often better ableto show or demonstrate their thinking thanto talk about it.

• Language is abstract; young children need towork from the concrete. All children,regardless of their language competence,should have access to manipulatives duringinformal questioning and conferences orinterviews.

Us ing the Ques t ion ing andConferenc ing Too l s

Diagnostic conference

Each unit includes a diagnostic conferenceoutline and recording assessment, Diagnostic

Conference forSelected Children(pp. 27, 32, 36,40, 44, 49). This outline isintended for usewith selectedchildren—those whosedevelopment isa source ofconcern at amidway pointin the unit.

However, teachers may also choose to use theseconferences with other children, as well, togain insight into their thinking. Throughcarefully scaffolded questions and prompts,the conference outline leads children throughsome basic activities intended to demonstratethe extent to which they are able to deal withunit content. Teachers are provided withconference prompts and observation cues.

Conference prompts

• Assessment Master 9: Conference Prompts (page55) offers a variety of prompts and questionsthat teachers can use in conferences,interviews, discussions, or informalconversations with children. The questionsand prompts are organized under fourcategories: Problem Solving, UnderstandingConcepts, Accuracy of Procedures, andCommunication. Several options areprovided under each of these headings; theyare intended to be samples only and do notconstitute a conference outline. Teachers canadd to these from their own repertoires.


Ques t ion ing in the Lessons

Each Math Circle includes prompts andquestions designed to stimulate children’sthinking, provide scaffolding as needed, andoffer insights into their development. Thesequestions are particularly prominent in thesections “Get Started,” where questions help toprompt and reveal prior knowledge; in “Show and Share,” where they are intended to prompt discussion and offer scaffolding;

in “Connect,” where questioning is used tohelp children formalize their understanding ofthe concept they have been working with,

as well as in “Reflect,” where children considertheir own learning.

Note that Assessment Master 10: Language to ListenFor (page 56) and Assessment Master 12: Self-Assessment Interview (page 58) also providesupport for questioning and conferencing.


Collections of children’s work over time allowteachers, children, and parents to see concreteevidence of progress. Analyzing several worksamples together helps to create an overallpicture of strengths and needs.

At kindergarten, work that is kept in a portfolioor collection often needs to be accompanied bybrief notes written by the teacher (sometimesdictated by the child) so that the sample is stillmeaningful after a period of days or weeks.Young children are unlikely to recall each pieceof work; the teacher needs to capture theirthinking immediately as the work is done. It isoften difficult to make sense of a written samplewithout hearing the child’s rationale. For youngchildren, particularly, as well as ESL childrenand others who have not developed facilitywith written English, work samples may notoffer an accurate representation of theirlearning.

E f fec t i ve Co l l e c t ions

Be purposeful and parsimonious. Each item in acollection of work should illustrate somethingimportant about a child’s development orprogress. When the number of work samplesbecomes too large, key evidence of learning canbe lost in an overwhelming amount of detail.Further, children often need and want to taketheir samples home.

Engage children in selecting work to collect. Evenvery young children can help to make decisionsabout what evidence of their learning theywant to display.

Incorporate notes about children’s thinking andreflections. Notes can be based on questions suchas: “What does this work show about the mathyou learned?”

Consider a range of material that illustrates variousaspects of a child’s achievement, as well as variousexpectations. Try to incorporate material that willoffer insights into children’s attitudes anddispositions, their understanding of concepts,and their use of strategies, as well as theirapplication of procedures.

Use the portfolios or collections as the basis forconferences with children and their parents. Having arange of concrete examples at hand helps tomake conversations about children’s progressmore concrete and purposeful.

In assessing portfolios, look for patterns and trends.Try to make an overall or holistic assessment ofthe child’s progress.

Us ing the Too l s fo r Rev iewingWork Samples and F i l e s

Children’s work samples are often meaningfulonly when accompanied by a teacher’sobservations, and/or children’s comments.Observations about children’s work samplesand portfolios can be recorded using several ofthe tools. For example:

Assessment Master 7: Observations Record 1 (page 53)may have samples of children’s work attached.

Assessment Master 8: Observations Record 2: Centresand Practice Work (page 54) may include samplesof the same child’s work over several activitiesto illustrate and elaborate the teacher’s notes.

Assessment Master 9: Conference Prompts (page 55)can provide a teacher with questions andprompts for discussing a particular piece ofwork with a child.

Work Samples and CollectionsC

Assessment Master 12: Self-Assessment Interview(page 58) can focus on a particular sample ofwork or a small collection.

Assessment Master 13: Observations Checklist:Solving a Problem (page 59) can also beaccompanied by any written evidence ofvarious children’s work.

Us ing Work Samples toSummar ize

Children’s work samples and portfoliosprovide important evidence when teachers aresummarizing progress using the DevelopmentalCharts associated with each unit, AssessmentMaster 14: Unit Summary (page 60), or AssessmentMaster 15: Observations Checklist: CurriculumExpectations (page 61). When using any of theseforms, teachers may consider samples ofchildren’s work, along with their observationand conference notes, children’s self-assessment, and other evidence such asphotographs when summarizing a child’sprogress.



Opportunities to reflect and talk about theirthinking and the activities they do are helpfulfor even very young children. They improvetheir learning and establish the important habitof reflection. Children begin to be aware ofwhat is important in their learning and come tounderstand their own progress. Children whohave opportunities to reflect and discuss theirlearning increase their sense of control andresponsibility.

Writing can be an important learning tool inmathematics. Even very young children canstart to record their thinking, using acombination of pictures, numbers, or words.

E f fec t i ve Re f le c t ion and Se l f -Assessment

To become effective at reflecting and self-assessment, children need frequent oralmodelling and guided practice. Modellinghelps children to develop the language ofreflection; they benefit from hearing and seeingexamples from their peers, as well as fromadults. A collaborative class journal is anexcellent way of initiating reflection. “Showand Share” activities throughout Addison WesleyMathematics Makes Sense, along with “Connect”and “Reflect,” highlight the importance ofbringing children together to share theirthinking and summarize what they havelearned.

Provide a variety of formats and media for self-assessment—oral, written, visual, kinesthetic—so that all children are comfortableparticipating. All children should haveopportunities to reflect on their learning,including those with limited English or writtenskills. Self-assessment can be as simple as,“One thumb up if you think you can do this onyour own. Two thumbs up if you are ready toteach someone else.” Teachers often askchildren to cover their eyes as they do this, sothat only the teacher sees a child’s response.Some teachers provide children with cards thathave different colours on each side. When anew concept is being introduced, they pauseand ask children to show one colour if they areready to work on their own, the other colour ifthey want to see another example. Children canalso use simple symbols, such as happy faces,to show places in their work where they werefeeling confident.

Model simple frames to help children getstarted, for example:• Today, I was happy that I . . .• You might be surprised to see that I . . .• I am proud of . . . • I want to get better at …• I found out that . . .• Look at these two pieces of work. What do

you notice? What can you do now that youcouldn’t do before?

Reflection and Self-AssessmentD


Us ing the Re f le c t ion and Se l f -Assessment Too l s

Self-assessment interview

• Assessment Master 12: Self-Assessment Interview(page 58) provides a series of prompts thatteachers can use to invite children to reflecton their own learning. The form includesquestions about learning skills. Forexample: “Tell me how you got alongworking today. Did you share? Help? Taketurns? Listen?” Prompts about the child’swork are also included. Generally, this formis best used in connection with an activity achild has just completed.

Self-assessment in the lessons

• Within the lessons, children are frequentlyasked to think about and verbalize what theyare learning. Each Math Circle includes aspecific section, “Reflect,” where questioningis designed to prompt and support thisimportant aspect of learning. As well ashaving opportunities to think about andrespond to questions about their learning,children have opportunities to listen to theideas of their classmates. Listening to thewords of their peers (mediated by theteacher) often provides more powerfulmodelling than other adults can give.


Learning Skills and DispositionsELearning skills and dispositions enablechildren’s achievement and success. Childrenneed to develop the skills that supportlearning, for example, working independently,showing initiative, completing work, andparticipating in classroom activities.

It is also important for children to developpositive attitudes towards mathematics, tovalue and appreciate its power and itsapplication to their daily lives, and to enjoymathematical activities.

E f fec t i ve Assessment o fLearn ing Sk i l l s and D i spos i t ions

Document children’s learning skills anddispositions separately from their mathematicaldevelopment and report on them separately, asoutlined in provincial or local guidelines.

Make assessment of these skills anddispositions clear to children. Take time todiscuss their importance and collaborate withchildren to develop sample lists anddescriptions of behaviours. Ask: “What wouldit look like/sound like if …?”

Emphasize the importance of learning skillsand dispositions to children and their families.Children who are unable to develop theseskills and attitudes are unlikely to besuccessful, particularly when they encounterchallenging material or extended tasks.

Encourage children to assess their learningskills and to set personal goals. Focus on oneskill at a time, and on a short list of behavioursthat children can reflect upon. For example, aclass may focus on taking initiative (or, inchildren’s words, trying things for myself), and

identify the following specific behaviours theywill try: asking questions (showing curiosity);looking for new problems; trying out manydifferent centres/practice activities. Over time,learn to look for evidence of related behaviours.

Connect the learning skills that children arefocusing on in mathematics with those they aredeveloping in other parts of their day. Wherepossible, use the same recording or self-assessment formats and strategies, whileensuring that you include specific examplesfrom mathematics. Children need to see theconnections among various parts of their dayand their learning.

Us ing the Too l s fo r Assess ingLearn ing Sk i l l s

Assessment Master 11: Learning Skills (page 57)provides a format that teachers can use togather evidence of children’s developinglearning skills—the behaviours anddispositions that enable them to becomeeffective learners. The form is designed for

collectingobservationsabout one childover time, andcan become partof a teacher’songoing recordsfor that child.


Assessing All LearnersFImproved formative assessment helps low achieversmore than other children and so reduces the range of achievement while raising achievement overall.

Paul Black & Dylan Wiliam (1998), p. 141

Assessment needs to be fair and valid for alllearners. Teachers also need to be sensitive to theimpact assessment can have on children who arestruggling. Teachers should develop a numberof practices and strategies for assessing andsupporting children whose achievement may notbe validly reflected in specific types of tasks.

Ensure that assessment opportunities include arange of ways that children can represent theirlearning. In most cases, it is the children’slearning that is important, not their ability torepresent it in one particular way.

Provide opportunities for ESL children, andothers with limited language skills, todemonstrate their learning with pictorial andconcrete materials. Provide support such asnumber cards or word cards that children can use.

Conduct frequent one-to-one conferences sothat you can clarify and extend children’slearning. Many children—particularly youngchildren—are not comfortable or confidentabout sharing their mathematical ideas in agroup setting.

Look for patterns in children’s work that mayhelp to identify those who are having troublewith an aspect that is unrelated to mathematics,such as fine motor control, attention, orlanguage. Where there is a discrepancybetween what you observe during classroomactivities and what children record when theyare working independently, the problem maybe associated with literacy.

Ensure that children are not asked to performassessment tasks that they are incapable ofcompleting. This can have extremely damagingeffects on their motivation and attitude towardsmathematics. Try to select tasks that all childrencan attempt, and then provide adaptations andscaffolding to support them; for example, breakthe task into smaller steps, have the child workwith a peer or an adult, provide help readingand scribing. In some cases, you may need tomodify the task. If children require frequentmodifications, you may need to develop anindividual plan, according to provincialguidelines. Note: This will affect how youreport on the child’s mathematical progress.

A problem must begin where children are. Everystudent in the class should be able to demonstrate some knowledge, skill, or understanding.

John A. Van de Walle (2004), pp. 66–67

Make materials readily available and emphasizethat all good mathematicians use objects anddrawings to help their thinking. Childrenshould never feel that using concrete materialsis an indication of weakness.

Emphasize that you are most interested inchildren’s thinking rather than in their particularanswers or solutions. If you are able to havestruggling learners share their thinking, youwill gain insights that you can use to help them.

Provide concrete opportunities for children to seeand hear how you use assessment results to helpthem learn. Try to instil confidence that theseresults will help them improve. For example,include descriptive feedback, and share strategiesfor improvement: “Here’s what I noticed aboutyour thinking today.…” “Here’s one thing I thinkwe can do that will help you learn to….”


Assessment Reform Group, U.K. (1999).Assessment for Learning: Beyond the Black Box.Cambridge, UK: University of CambridgeSchool of Education.

Assessment Reform Group, U.K. (2002)News/events: Current activities.http://www.assessment-reform-group.org.uk

Black, Paul & Dylan Wiliam (1998). Inside theBlack Box: Raising Standards through ClassroomAssessment. London, U.K.: King’s CollegeLondon School of Education.

Black, Paul & Dylan Wiliam (1998).Assessment and Classroom Learning.Assessment in Education, 5 (1), pp. 7–74.

BC Ministry of Education. BC PerformanceStandards: Numeracy (2000; 2002). Victoria,BC: Ministry of Education. Available atwww.gov.bc.ca/bced

Harlen, Wynne & Ruth Deakin Crick (2002).Review: What is the evidence of the impact ofsummative assessment and tests on students’motivation for learning? Presentation,Invitational Conference, Assessment ReformGroup, March 5, 2002.

Marzano, Robert J., Debra J. Pickering, & JaneE. Pollock (2001). Classroom instruction thatworks: Research-based strategies for increasingstudent achievement. Alexandria, VA: ASCD.

National Education Association in collaborationwith Rick Stiggins and the AssessmentTraining Institute (2003). Balanced Assessment.Washington, DC: National EducationAssociation.

Stenmark, Jean Kerr & William S. Bush (eds.)(2001). Mathematics Assessment: A PracticalHandbook. Reston, VA: NCTM.

Stiggins, Rick (2001). Making ClassroomAssessment Instructionally Relevant. Paperpresented at the annual meeting of theAmerican Educational Research Association,Seattle, 2001.

Van de Walle, John A. (2004). Elementary andMiddle School Mathematics: TeachingDevelopmentally (fifth edition). Toronto:Addison Wesley Longman.

ReferencesG

Teaching a Problem-Based Program

P R O B L E M - S O L V I N G I N V E S T I G A T I O N S

Children learn mathematics throughout MathMakes Sense for kindergarten by exploringproblems. Problem solving is not a specialevent; it is integrated into all Math Circles andOngoing Centres activities. The problemspresented on Big Math Book, pages 60 to 64, arenot intended to be a discrete unit; they can beused at any time during the year to support andextend children’s learning.

The problems presented on these pages reflectchildren’s daily experiences and engage them inchallenging and open-ended investigations.Each problem draws on problem-solvingdispositions and strategies, as well as keyconcepts children are developing. The problemsinvolve important mathematical ideas and havemultiple entry points so that all children can beinvolved in exploring solutions. They requiresome persistence; there is not one easy-to-findcorrect answer. They help children to see thatmathematics is a way of making sense out oftheir daily experiences.

The problems presented on the Big Math Bookpages can be used in a variety of ways, butshould not be viewed in isolation of children’sdaily experiences. For example:• When an appropriate situation arises in the

classroom, such as finding a way to share anodd number of things, teachers can workwith the children to solve the classroomproblem, then present the Big Math Bookproblem to practise and connect (formalize)the experience. These opportunities arehighlighted under the heading “Connect tothe Classroom.”

• Teachers can introduce the problems in the context of relevant units in the program,as a way of extending, integrating, andconnecting children’s understanding. Someopportunities are identified under theheading “In the Units.”

• Teachers can use the Big Math Bookproblems at intervals throughout the yearto model problem solving before engagingthe children in a real classroom problem.The Big Math Book problem becomes aGetting Started activity where the teacherand children work together; children thenwork independently or in pairs to explorea contextualized problem.

• Stories offer the introduction to manyproblems. For each Big Math Book problem,one or two books are suggested as a way ofstarting or extending the experience, under“Start with a Story!”

• Problem solving develops over time.Revisiting problems is an important part ofthis development. For each Big Math Bookproblem, the heading “Revisit the Problem”suggests a number of related problems andproblem situations teachers can usethroughout the year.

Problem-Solving Investigations 15

Start with a Story

Use the context of a story that the children have recently enjoyedto pose a similar problem. For example, after reading The 3 LittlePigs, display two colours of bricks or blocks, and say: “The thirdlittle pig wanted to make a fence with a pattern using these 2colours. What pattern could he make?”

Connect to the Classroom

Make paper hats for the children that resemble the hats in thepicture on Big Math Book, page 60—some children will have pointedhats, others will have crowns. Provide stickers, stamps, andcrayons so they can decorate their hats. After looking at the pictureand talking about the patterns you can make with the 6 hatsshown, have children put on their own hats and explore otherpatterns they can create.

Note: Do not allow children to exchange hats for sanitary reasons.

Present a similar problem using blocks or other classroom objectsthat children enjoy playing with.

G e t S t a r t e d : P r e s e n t t h e P r o b l e m

Display Big Math Book, page 60, and ask:■ What do you see in the picture? (children and teacher; children

wearing hats)■ How many kinds of hats are there (2)■ How many children are wearing pointed hats? (3)■ How many are wearing crowns? (3)

Read the text to the children: “How can we make a pattern?”Reread it, inviting children to chime in. Ask:■ What are we supposed to do? (make a pattern; arrange the

children in a pattern)■ What do we know? (2 kinds of hats; 3 pointed hats and 3 crowns)

E x p l o r e : Wo r k o n a S o l u t i o n

Ask:■ How could we help the teacher solve this problem? What could

she use to arrange the children in a pattern? (Children maysuggest pointed hats/crowns or girls and boys.)

How Can We Make a Pattern?

IN THE UNITS

This problem focuses onpatterning. Opportunitiesto pattern are important.They allow children thechance to predict, extend,and represent their thinkingin a variety of ways. Thisproblem can be used withUnit 4.

Problem

Solving

16 Problem-Solving Investigations

■ How could we get started? (We could act the problem out; wecould draw pictures.)

Work with the children to create a pattern using one of theirsuggestions. For example, if they suggest acting it out, ask:■ Who will be the children? How many do we need? (6)■ How many will have pointed hats? (3) How will we know which

ones are pretending to have those hats? (Give them an orangepiece of paper to hold.)

■ How many will have crowns? (3) How will we know which onesthey are? (Give them a blue piece of paper to hold.)

Prompt children as they help you create an a-b pattern based on thehats (e.g., AB, AB, AB). After you have created a pattern together, ask:“If we had more children wearing these hats, what would come next?”

Model recording the pattern, using cutouts that resemble the twotypes of hats. Manipulating the cutouts will help children see thepattern. Ask: ■ What if there were more children wearing pointed hats and

crowns? Could you make a different pattern? (yes)

Provide children with recording materials and manipulatives, andhave them work in pairs or small groups to create another patternusing pointed hats and crowns.

Bring children together to share their patterns. Ask:■ How did you make your pattern? Tell me about your thinking.■ Does anyone have the same pattern? How did you think of your

pattern? How did you check it to make sure it was a pattern?

Invite all the children to show and talk about their patterns.

C o n n e c t

As children offer their patterns, make a record by drawing them onchart paper or manipulating cutouts of the types of hats. Ask:■ How many different patterns did we have altogether?■ What is your favourite pattern? Why?

Emphasize that there are many different ways to make a patternusing just 2 elements.

R e f l e c t

Invite children to retell the problem and what they did. You maywish to record this conversation as a collaborative journal.

REVISIT THE PROBLEM

Variations of this problemcan be revisited throughoutthe year with sets of any 2types of objects, stickers,or stamps. Ensure thatchildren have enough forat least 3 repetitions oftheir pattern core.

WHAT TO LOOK FOR

■ restates the problem■ offers ideas about how to

solve it■ tries out one strategy■ shows willingness to

persevere■ finds one or more

solutions■ explains their solution

Teachers can useAssessment Master 13:Observations Checklist:Solving a Problem to recordobservations.


Start with a Story

Use the context of a story that the children have recently enjoyedto pose a similar problem. For example, Benny’s Pennies by PatBrisson (Dragonfly Books, 1995) prompts children to think aboutall of the combinations of 5 pennies.


Present a similar problem using classroom objects that childrenenjoy playing with. It is best to use objects that are uniform; forexample, 4 miniature cars; 6 blocks; 5 teddy bear counters; 6 toothpicks.


Display Big Math Book, page 61, and ask:■ What is the girl doing? (playing with teddy bears; arranging her

teddy bears; putting teddy bears in groups)■ How many teddy bears are there in this part of the picture? (point

to one group) (5)■ What about this group? (point to another group) (5)■ How are these 2 groups the same? (both have 5 teddy bears)■ How are they different? (teddy bears are lined up in different

ways; organized differently; this one has … and that one has …)

Continue until children have counted the bears in each group andconcluded that there are 5 in each.

Pose the problem to the children: “How many ways can you make 5?” Then invite children to chime in. Ask:■ How many ways has the girl arranged the teddies to make 5?

(4 ways)■ Tell me how they are arranged in each group (2 in one line and

3 in another; 2 lines with 2 and 1 in the middle; 1 line of 5; she ismaking one line of 3 across and one line of 2).


Ask:■ Can you think of another way to arrange 5 teddies? How could

you figure that out? (take 5 counters or Snap Cubes and try outdifferent ways; draw pictures)

How Many Ways Can You Make 5?

IN THE UNITS

This problem focuses onnumber and geometry,particularly on the conceptsthat sets can be broken upand reassembled (part-part-whole), and that the numberof objects does not changewhen they are rearranged(conservation of number).The problem can beintroduced during Unit 2 orUnit 6. It can also be usedduring the geometry unit,Unit 3, to help childrenpractise positionallanguage while they aredescribing their solutions.

Problem

Solving


Work with the children to create other representations of 5 usinguniform objects. Follow the children’s directions, and have themcompare each new arrangement with the picture and with otherarrangements to make sure it is different. Remind children to countthe number of objects each time to make sure there are exactly 5.

Model recording the arrangements.

Tell children they are going to explore ways to make a differentnumber: 4. Ask: ■ How many ways do you think you will be able to make 4? Tell me

about your thinking.

Provide children with manipulatives (all identical) and recordingmaterials. Have them work in pairs or small groups to find as manyways to make 4 as they can.

S h o w a n d S h a r e

Bring children together to share what they found out. Ask:■ Who can show us one way to make 4?■ Does anyone else have the same way to make 4?

Continue until all the children have had a chance to show one oftheir ways to make 4. As each way is shown, have the children countthe objects together: 1–2–3–4.

C o n n e c t

As children offer their solutions, make a record by drawing them onchart paper. Ask:■ How many different ways did we find altogether?■ How did you find so many ways? How did you know when there

was a new way? ■ Do you think we found all the ways to make 4? Tell me about your

thinking.

Emphasize that there are many different ways to make 4 and manydifferent ways to describe them.

R e f l e c t

Invite children to retell the problem and what they did. You maywish to record this conversation as a collaborative journal.

REVISIT THE PROBLEM

Variations of this problemcan be revisited throughoutthe year with any set ofidentical objects. For mostchildren, 4 to 6 objects willbe enough to challenge theirthinking and persistence.Children who are havingdifficulty can work with 3 objects.

This problem can also bepresented in the context ofdot cards, dominoes, andother visual–spatial materials.

WHAT TO LOOK FOR







Start with a Story

Use the context of a story that the children have recently enjoyedto pose a similar problem. For example: “After Goldilocks runsaway, the 3 bears decide to go for a bike ride, but there are only 2bikes. What should they do?” Other possibilities include: MonsterMusical Chairs by Stuart Murphy (HarperTrophy, 2000), and OddNumber 13 by Adria Klein, Addison Wesley Little Books, Earlylevel (Addison Wesley, 2002).


Present a similar problem within the classroom context. Forexample, 2 children can play a particular game, but 3 children wantto play. What should we do? Two children can be at the watertable, but 5 children want to be there. What should we do? Sixchildren want to play on the swings, but there are only 3 swings.

This problem can also introduce discussions about socialresponsibility, particularly fairness, taking turns, and sharing.


Place 2 chairs at the front of the room. Invite 2 children to come and sitin the chairs. Then invite 2 more, and ask: “How could all 4 childrenshare these chairs?” (take turns; go 2 at a time) Invite 2 more childrenand ask: “Could 6 children share these chairs?” (Yes, go 2 at a time.)

Display Big Math Book, page 62, and ask:■ What are the children doing? (playing in the snow; getting ready

to go on a toboggan; waiting their turn on a toboggan)■ How many children are there? (7)■ How many can go on the toboggan? (3)

Pose the problem to the children: “What should we do?” Ask:■ How is this like when we had 6 people and 2 chairs? (They need to

take turns.)■ Is there a problem? Tell me about it. (Seven children want to be on

the toboggan but it will only take 3.)■ Why is this a problem? (Everyone should have a turn; some

children will have to wait; things won’t come out even.)■ What do we know? (There are 7 children; there are 3 places on the

toboggan.)

How Can We All Take a Turn?

IN THE UNITS

This problem focuses onnumber, 1:1 correspondence,as well as part-part-wholerelationships, depending onthe complexity of the problemyou pose, and can be usedwith Unit 6.

Problem

Solving



Ask:■ How could we help the children solve this problem? How could we

get started? (We could use people and chairs and act the problem out;we could draw pictures; we could use counters for the children.)

Work with the children to act out one solution to the problem. Createa “toboggan” by making masking tape squares on the floor or using 3 chairs. Invite 7 children to be the children in the picture.

Ask the children in the toboggan to pretend they are going downhill.■ How many children have a ride? (3)■ How many children have not had a ride yet? (4)■ What should we do next? (Put some different people in the toboggan.)■ What about the ones that have had one ride? (Make them stand

somewhere else so we don’t give them a second ride untileveryone has had a turn.)

Continue following children’s directions to act out the scenario untilall 7 have had a “ride.” Model recording how you solved the problemtogether.

Provide children with recording materials and manipulatives, andhave them work in pairs or small groups to show another way tosolve the problem.


Bring children together to share their solutions. Ask:■ Who can tell us about their answer? How did you share? Tell us

about your thinking. Was that a fair solution?■ Does anyone have the same answer? ■ Does anyone have a different answer? Tell us about it.

C o n n e c t

Make a record of children’s solutions on chart paper. Ask:■ How many different ways did we find altogether?■ What is your favourite solution? Why?

Emphasize that there are many different answers to this problem andmany different ways to work on it.

R e f l e c t

Invite children to retell the problem and some of the solutions. Youmay wish to record this conversation as a collaborative journal.

REVISIT THE PROBLEM

Variations of this problemcan be revisited throughoutthe year with any similarsituation. For example: ■ There are two swings.

Five children want to playon the swings. Whatshould we do?

■ Three people can playthis game. Seven childrenwant to play. Whatshould we do?

■ Three children can be atthe sand table. Fivechildren want to build inthe sand. What shouldwe do?

WHAT TO LOOK FOR




solutions■ explains a solution



Start with a Story

Almost any story where two children or characters have somethingthey could share can provide a context for this problem. For example,after reading Dragon Sandwiches by Gwendolyn MacEwen (FireflyBooks, 1988), ask: “How could two people share three sandwiches?”


Sharing is a natural way to introduce this concept.

Create a real classroom problem. Invite 2 children to help you. Givethem 2 toys, and ask: “Show us how you could share these 2 toys.”Invite 2 different children then, and give them 4 toys, asking again,“Show us how you could share 4 toys.” (2 each) Invite them to sharetheir thinking—how did they decide to take 2 each? Next, invite twodifferent children and give them 3 toys. Ask again: “How could youshare these 3 toys?” Invite other children to offer suggestions as well.

Repeat the activity with objects like crackers that can be broken in half.


Display Big Math Book, page 63, and ask:■ What are the children doing? (getting ready to play with finger

puppets)■ How many children are there? (2)■ How many puppets are there? (5)

Read the problem to the children: “How can we share?” Reread it, inviting children to chime in. Ask:■ What do we have to find out? (how to share the puppets)■ What do we know? (5 puppets; 2 children)


Ask:■ How could we get started? What would help us solve this

problem? (We could use our own puppets; we could act it out; wecould just figure it out in our brains; we could draw pictures; wecould use counters for the puppets …)

If this is the first time that children have encountered this type ofproblem, choose one of their suggestions and begin working on the

How Can We Share the Puppets?

IN THE UNITS

This problem focuses onnumber and part-part-wholerelationships, as well asdivisibility. It is connected toodd and even numbers andthe idea of doubles. It canbe introduced any time afterchildren have practisedcounting to 5 in Unit 2. It isalso related to visual–spatialthinking and can be usedwhen children are workingwith beginning concepts ofgeometry in Unit 3.

Problem

Solving


problem. For example, if they decide to act it out, choose 2 childrenand 5 counters or other objects, and ask:■ What should we do first? (Have each person take one puppet.)■ How many puppets do we have left? (3)■ What should we do next? (have each person take another puppet)■ How many do we have left? (1)■ What should we do now? (take turns with it, give it to the

biggest/smallest, give it to someone else …)■ Would that be fair? Tell me about your thinking.■ Can you find another way for 2 children to share 5 puppets?

Provide children with recording materials and manipulatives andhave them work in pairs or small groups to find another solution.


Bring children together to share their solutions. Ask:■ Who can tell us about their answer? How did you share? Tell us

about your thinking. Was that a fair solution?■ Does anyone have the same answer? ■ Does anyone have a different answer? Tell us about it. Show us

what you did.■ Did anyone have a different solution?

Ask:■ What if the problem was about crackers instead of puppets?

Have children work in pairs or groups to find a way for 2 childrento share 5 crackers.

C o n n e c t

Bring children together to share their solutions. As children offertheir solutions, make a record by drawing them on chart paper. Ask:■ How many different ways did we find?■ Which ones were the same as our ways to share puppets?■ Were any of them different? Would that work for the puppets too?■ Which way do you think is best for puppets? Which way do you

think is best for crackers? Why?


R e f l e c t


REVISIT THE PROBLEM

Variations of this problemcan be revisited throughoutthe year with any oddnumber of objects. Vary theactivity so that sometimesthe objects can be brokenor divided in half, whileother times they cannot.

WHAT TO LOOK FOR







Start with a Story

Almost any story can be used to introduce or extend this problem.For example, Just Plain Fancy by Patricia Polacco (Dragonfly Books,1994) features one fancy egg. After reading the story, pose thefollowing problem: “Ruth and Naomi have 4 eggs. Some are plainand some are fancy. How many of each could there be?” Afterreading Miss Spider’s Tea Party by David Kirk (Scholastic Canada,1994), pose the problem: “There are 4 guests. Some are moths andsome are ladybugs. How many of each could there be?”


Introduce or extend the story using classroom objects. For example,show children a bag or box in which you have placed some blue blocksand some red ones. Pose the problem: “I have 3 blocks in this bag.Some are blue and some are red. How many of each could there be?”


Display Big Math Book, page 64, and ask:■ What do you notice in this picture?■ What does the teacher have in the box? (skipping ropes and balls)

Pose this problem to the children:■ There are 5 things in the box. ■ Some are skipping ropes and some are balls.■ How many of each could there be?

Now invite children to chime in. Ask:■ What do we have to find out? (how many ropes and balls there

could be)■ What do we know? (There are 5; there are some ropes and some balls.)


Ask:■ How could we get started? What would help us solve this

problem? (We could get our own ropes and balls from the gym; use blocks or counters for the ropes and balls; draw pictures …)

If this is the first time that children have encountered this type ofproblem, choose one of their suggestions and begin working on theproblem. For example, if they decide to work with blocks or counters, ask:

How Many of Each Could There Be?

IN THE UNITS

This problem focuses onnumber and the concept ofpart-part-whole—that aquantity or set can be brokeninto parts and then put backtogether. It is connectedconceptually to conservationof number. Children developthis concept over time. Thisproblem can be introducedduring Unit 2, once childrenhave practised counting to 5.

Problem

Solving


■ Which ones will be the skipping ropes? Which ones will be the balls?■ How many will we need for skipping ropes? (no more than 5; there

are only 5 things altogether)

Note: This will not be obvious to many children at this stage in theirdevelopment. Some children may decide there are more than 5.Accept their current understanding, but provide opportunities forthem to hear how others are thinking about this. Ask:■ How many for balls? (no more than 5)■ What is one way we could put together the skipping ropes and

balls to make 5 things altogether?■ Can you find another way?

Provide children with recording materials and manipulatives andhave them work in pairs or small groups to find another solution.


Bring children together to share their solutions. Ask:■ Who can tell us about their answer? How many skipping ropes?

How many balls?■ How did you figure that out? Tell us about your thinking. Show us

what you did.■ Does anyone have the same answer? How did you figure that out?■ Does anyone have a different answer? Tell us about it. Show us

what you did.■ Did anyone have a different solution?

As children offer their solutions, make a record by drawing them onchart paper. Then ask each pair or group to see if they can findanother answer—an answer that is different from their first solution.

C o n n e c t

Bring children together again to share solutions. Add their ideas toyour chart. Ask:■ How many different ways did we find?■ What are some of the strategies you used to find answers? ■ What other way could you have worked on this problem?


R e f l e c t


REVISIT THE PROBLEM

Variations of this problemcan be revisited throughoutthe year. It is important thatchildren be able to drawthe objects you choose.Possibilities might include:■ carrot sticks and apples■ boys and girls■ puppies and kittens

WHAT TO LOOK FOR







Unit 1: Data Management

Copyright © 2004 Pearson Education Canada Inc. The right to reproduce this page is restricted to the purchasing school. 26

Date:

Diagnostic Checklist

Name Follows simple pattern (e.g., I see a _____ )

Notices detail in picture

Shows awareness of number

Uses simple positional words (e.g., on, beside)

Identifies attributes (e.g., colour, size)

Identifies things that are same / different

Uses sequence words (e.g., next, after)

Assessment Master 1.0



Name: Date:

Diagnostic Conference for Selected Children

This outline is intended for use with children whose progress is a concern at the midway point of the unit (e.g., Math Circle D). It can be used with an individual child or a small group of children who appear to be having difficulty with basic concepts and procedures.

CATEGORIES OBSERVATIONS AND COMMENTS

Reasoning; applying concepts Show the child three objects of different colour and size. Pick up one object and describe it in very simple terms, for example: • I can tell you a lot of things about this. This is a…. It is [colour]. It feels

[smooth/rough/bumpy…] in my hand. It is [size]. Look at this part—it looks like a…. I could…with this.

Say: • Now it is your turn. Pick one of these and tell me about it. Tell me as

much you can. (Prompt for colour, size, how it feels, what it looks like, how the child could use it.)

Accept the child’s description without correction. Rephrase to model appropriate language. Model identifying a difference between the two objects, for example: • Yours has wheels and mine doesn’t. Ask: • Can you see another way they are different? (Prompt for colour, shape,

size…) If the child is able to talk about one or more differences, model and prompt for similarities, for example: • Both your… and my… are very small. Can you find another way they are

alike? Notice the child’s confidence and ability to • identify ways two objects are different • identify ways two objects are similar

Accuracy of procedures Ask: • How many objects/things do we have altogether? (3) • What colour is this one? • What about this one? • Which one is bigger/smaller? Notice the child’s confidence and ability to • count to 3 • identify colours • determine which of two objects is bigger/smaller

Communication Say: • Thank you for your hard work and thinking! Let’s make a list of what we

did and what we noticed. [Allow child to retell the activities freely; prompt if stuck.]

Notice the child’s confidence and ability to • use appropriate language (e.g., colour words; big/small; same/different)




Date:

Ongoing Observations Checklist

Math Circle A Math Circle B Math Circle C

Name describes colour, shape, size

tells how two things are the same

sorts by colour, shape, size

identifies objects that are same/ different given an attribute

uses prior experience to solve problems

uses a variety of words to describe attributes

Assessment Master 1.2.1



Date:


Math Circle D Math Circle E Math Circle F

Name matches objects to describing words

knows difference between common opposites

knows same/ different in many contexts

explains sorting rule

places self on people graph to answer simple question

suggests questions for creating graph




Date:


Math Circle G Math Circle H Math Circle I

Name begins at base; places 1 object per square

talks about what graph shows

creates a simple picture graph

tells which column has more, fewer (same)

identifies events that might happen




Name: Date:

Developmental Chart This assessment can be used to summarize children’s development in terms of the outcomes associated with Unit 1.

Progress in relation to learning expectations not yet evident

developing; with

assistance

achieved; some

support

fluent; independent

COMMENTS

Reasoning; applying concepts sorts objects using a single attribute

compares objects (explains how they are alike and different)

compares data in two categories using words such as more, less, the same

Accuracy of procedures describes characteristics of objects (colour, shape, size)

with assistance, places objects on concrete graphs or pictographs (begins at base; leaves no spaces, one-to-one correspondence)

collects (with assistance) first-hand information

Problem solving shows curiosity: explores and experiments with concrete objects

Communication uses a variety of simple words to describe colour, shape, size


Unit 2: Number Sense


Name: Date:




Reasoning; applying concepts Show the child 3 or 4 objects (e.g., blocks) that are all the same. Say: • Look at these blocks. I wonder how many there are. Let’s count them

together. Count with the child, moving each block as it is counted. When you have finished counting, ask: • How many blocks do we have? If the child does not associate the number of blocks with the last number counted (cardinality), say: Let’s count them together again: 1, 2, 3, 4. We have 4. Ask: • What if I took one of the blocks and put it away? How many would there

be then? Support the child in counting as needed again, moving each block as it is counted and emphasizing the last number: 3—there are 3 blocks—we counted 3. Next, take the three blocks and rearrange them, with more space between them (i.e., spread out more). Ask: • How many are there now? If the child does not recognize that the

number has not changed, count together, again emphasizing the last number.

Place the 3 blocks very close together and ask again: How many now? Notice the child’s confidence and ability to • use 1:1 correspondence to count • use the principle of cardinality (last number counted is the quantity) • understand the principle of conservation of number (the number doesn’t

change when you rearrange the same set of objects)

Accuracy of procedures Show the child a collection of blocks similar to the ones you are using. Point to your set of 3, and say: • We have 3 blocks here. Can you make another set of 3 blocks? Notice the child’s confidence and ability to • count to 3 • create a matching set of 3



Notice the child’s confidence and ability to • use number words




Date:


Math Circle A Math Circle B Math Circle C Math Circle D

Name is aware of number

matches 1:1

counts to 5

knows that last number is quantity: cardinality

counts to 5

shows 5 in various ways

counts back from 5

under-stands conser-vation




Date:


Math Circle E Math Circle F Math Circle G Math Circle H

Name counts to 10

constructs sets of 10

follows rules of counting games

counts back from 10

recognizes numerals 1 to 5

sorts numerals by shape

recognizes numerals 6 to 10

writes some numerals




Name: Date:

Developmental Chart This chart can be used to summarize children’s development in terms of the outcomes associated with Unit 2.


developing; with

assistance

achieved; some

support

fluent; independent

COMMENTS

Reasoning; applying concepts sorts, classifies, and counts objects in a set

builds and compares sets of objects (more, greater, fewer, less, same, equal)

uses number in daily experiences

Accuracy of procedures matches objects with 1:1 correspondence

counts to 10

recognizes numerals to 10

Problem solving explores a variety of physical representations of numbers

Communication uses some mathematical language (e.g., same, more/less than, greater/fewer than)


Unit 3: Spatial Sense and Geometry


Name: Date:




Reasoning; applying concepts Problem solving Have a small collection of objects in a box or bag. These can be regular geometric solids, or boxes, cans, or balls. You will need more than one of each, but they can be different sizes. Hand the child a block or box (a rectangular prism) and say: • Here is a _____. What can you tell me about it? Children may start by identifying colour and size. Compliment them for any attributes they are able to identify, and prompt for spatial features if the child is not able to describe them: • Does it have corners? (yes) • Are the sides flat or curved? • Can you count the sides? (6) • What other objects have the same shape as this one? (blocks, boxes) • Can you find other objects with the same features in this box/bag?

(provide support as needed) When the child has 3 or more blocks/boxes, ask: • What could you build with these? (a tower, a building …) • Show me how that would work. Introduce a ball or sphere and ask: • Could we put this in your building? (no, it will fall off) • Why won’t it work? (it is too curvy) Confirm that the blocks will stack but the ball will not. Ask: • What can the ball do that the blocks can’t do? (roll) Next, introduce a cylinder/can and ask: • How could we add this to your building? (by stacking it on the flat side) • It’s a bit tricky! What if we put it this way (curved side down)? (it won’t

stay) Why? (it isn’t flat on that part; it is curvy) If the child is working confidently with you, point to a block and a cylinder and ask: • How are these alike? (you can build with them; they both have a flat part;

they can stack) (Prompt by stacking them if needed.) • How are they different? What can this one (cylinder) do that this one

(block) can’t do? (roll) Repeat with the sphere and cylinder. Notice the child’s confidence and ability to • identify characteristics of 3-D objects • sort 3-D objects (e.g., find others the same) • build with objects


did and what we noticed. [Allow child to retell the activities freely; prompt if stuck.] Tell me about the building/tower you made.

Notice the child’s confidence and ability to • use language to describe attributes and position




Date:



Name follows 1-step directions about position

describes position of an object

describes some attributes

sorts objects on one attribute

builds with objects

describes attributes of objects

describes attributes of 2-D shapes

sorts 2-D shapes on one attribute




Date:



Name constructs 2-D shapes

describes attributes of 2-D shapes

identifies triangles

creates triangles on a geoboard

combines figures to make a picture

names figures

identifies figures in their world

uses simple language of position




Name: Date:



developing; with

assistance

achieved; some

support

fluent; independent

COMMENTS

Reasoning; applying concepts sorts and describes 3-D objects

Accuracy of procedures follows 1-step oral instructions involving position

identifies 3-D objects

describes relative position of 3-D objects

Problem solving builds 3-D objects

shows curiosity; explores and experiments with 3-D objects

Communication uses some spatial and positional language (e.g., round, over, under, beside, between, inside, outside)


Unit 4: Patterning


Name: Date:


This outline is intended for use with children whose progress is a concern at the midway point of the unit (e.g., Math Circle C). It can be used with an individual child or a small group of children who appear to be having difficulty with basic concepts and procedures.


Reasoning; applying concepts Show the child a collection of concrete objects that vary in one obvious way. For example, red and white blocks; circles and squares (similar size, same colour). Say: • Look at these. They are all mixed together. How could we sort them into

groups? Could you sort them out for me? Watch as the child sorts them. Prompt the child to talk about what he or she is doing: • Tell me what you did. How did you sort them out? If the child is unable to sort them, say: • Here’s one way. We could sort [the colours]. We could put all the [red

ones] together and all the [white ones] together. Say: • How could we use these to make a pattern? Let’s make a

pattern together. Allow the child to suggest a pattern. If unable to, begin creating a simple AB pattern, asking for help as you work, for example: What should we put next? Notice the child’s confidence and ability to • sort objects by one simple attribute • use objects that differ on one attribute to make a simple pattern

Accuracy of procedures Once you and the child have created a pattern with at least 3 repetitions, ask: • What comes next? How do you know? • Can you add some more [blocks] to this pattern? • What could we name our pattern? (How we could describe it?) [e.g., red,

white; red, white] Notice the child’s confidence and ability to • predict the next element • extend a simple pattern • label a simple pattern

Problem solving If the child is able to contribute to, extend, and talk about a simple pattern, ask: • Could we make a different pattern with the same [blocks]? How would that work? Show me…. Support and prompt as needed. If the child has trouble, offer prompts such as: • What if we started with [2 red ones]? Notice the child’s confidence and ability to • ask questions as needed to get information that will help him or her • persevere to create a new pattern • rearrange the same materials to create a new pattern



Notice the child’s confidence and ability to • use appropriate language (e.g., colour words, pattern, first, next)


Unit 4: Patterning


Date:



Name identifies a pattern

tells how elements are different

describes a simple pattern

tells how 2 patterns are the same

follows an action pattern

creates simple action patterns


Unit 4: Patterning


Date:



Name recognizes differences in sound; rhythm

copies simple sound patterns

predicts next element

identifies pattern core

recognizes patterns around them

creates own patterns


Unit 4: Patterning


Name: Date:



developing; with

assistance

achieved; some

support

fluent; independent

COMMENTS

Reasoning; applying concepts describes patterns orally

extends patterns using actions and manipulatives

Accuracy of procedures recognizes patterns (colour, shape, size, position, actions)

reproduces patterns using actions and manipulatives

Problem solving creates patterns using actions and manipulatives

takes risks; explores and experiments with patterns

Communication uses a variety of simple, appropriate language to label and discuss patterns involving colour, shape, size


Unit 5: Measurement


Name: Date:




Reasoning; applying concepts Accuracy of procedures Collect 3 or 4 objects that are different lengths (e.g., pencils, crayons) and a shoebox lid or tray you can use for a common baseline. Show the child two objects, resting side by side on a common baseline, and ask: • What can you tell me about these two objects? (Prompt: How are they

the same? e.g., same colour, both pencils … How are they different? e.g., different lengths, one is longer …)

• Which one is longer? How can you tell? (Yes, when you compare them side by side with the bottom edges, they are exactly the same; the top of this one is farther than the top of that one.)

• What if I move them like this [move the longer one away from the baseline]—now which one is longer? (Yes, this one is still longer—moving it does not change how long it is.)

• What about like this? [Now put the longer one back on the baseline but move the shorter one so the top is positioned past the top of the longer one.] (Yes, it is still shorter. We just moved it; we didn’t make it longer.) If the child has difficulty determining that the longer one is still longer, say: Let’s put it back here [on the baseline]—now is it shorter or longer?

Move the shorter object a short way at a time, checking after each move. If the child continues to have trouble, mediate by saying: • I can tell that it is always shorter because I put them side by side. When I

move it, it is still shorter, even though it looks different. Any time I want to check, I can just put it back beside the other one again to be sure.

• Does moving the objects change which one is longer? Tell me about your thinking. [Option for children with limited language] Show me.

Note: If the child is having difficulty, repeat the activity with two objects that are very different in length. If the child is proceeding comfortably, continue. Hand the child a different pair of objects and ask: • How could we find out which of these is shorter? (Put them side by side;

look at them together; put them in shoebox or on the tray.) • Show me how you would figure that out. • What if we moved the shorter object, could we make it longer? (No, it is

always shorter than the other one, no matter where we put it.) Notice the child’s confidence and ability to • compare length using a baseline • understand conservation of length • identify objects that are shorter and longer

Communication Say: Thank you for your hard work measuring and thinking! Let’s make a list of what we did and what we noticed. [Allow child to retell the activities freely; prompt if stuck.] Notice the child’s confidence and ability to • use simple measurement terms (shorter, longer)


Unit 5: Measurement


Date:



Name identifies differences in size

uses simple language to describe size (e.g., big, small)

uses a common surface to compare height

uses taller and shorter correctly

compares length of 2 objects directly

sees need for common baseline


Unit 5: Measurement


Date:



Name is developing concept of conservation of length

suggests a way to solve a simple problem

recognizes that containers have capacity

compares capacity by emptying one container into another

recognizes that an object has mass

uses heavier and lighter correctly


Unit 5: Measurement


Date:


Math Circle G Math Circle H Math Circle I Math Circle J

Name recalls and sequences events

uses sequence words (e.g., before/after, first, next)

compares duration of events (takes longer/ shorter time)

uses simple language of duration (e.g., long/ short time)

realizes that money has value

names pennies, nickels, dimes

realizes that different coins have different value

uses language such as cents, pay, buy, more, less, and names of coins


Unit 5: Measurement


Name: Date:



developing; with

assistance

achieved; some

support

fluent; independent

COMMENTS

Reasoning; applying concepts shows understanding of conservation of length/ height

shows understanding that objects can be measured and compared by attributes such as length, height, mass, capacity

shows awareness of the passage of time (e.g., long/ short time)

recognizes that money has value

Accuracy of procedures arranges objects in order of size, by length or height (linear)

classifies objects by linear attributes (e.g., short/long)

compares two objects by mass (heavier/lighter)

Problem solving explores the exchange of money in play activities

measures (non-standard units, with assistance) to collect first-hand information to solve problems (e.g., how to check which of two objects is longer)

covers a surface with a variety of objects

Communication uses simple measurement terms (e.g., tall/short; big/small; empty/full; heavy/light; hot/cold; less/more)


Unit 6: Numbers and Applications


Name: Date:




Reasoning; applying concepts Accuracy of procedures Begin by taking the child on a number hunt around the classroom. Say: • We are going to walk around and see what numbers we can find. We will

take turns. First, I will find a number; then it will be your turn. • Here’s my first number. See, I found it on the…. Now it is your turn.

What number can you find? Note the child’s awareness and ability to read numerals. Begin with small numbers. Say: • Now we are going to look for numbers in a different way! We are going

to look for groups of things we can count. I’ll start. I see 5 books right here: 1–2– 3–4–5. [Touch each book as you count it.]

• Now it’s your turn. What can you find to count? [Support the child in counting orally as needed; in some cases, you may need to count together. Prompt the child to touch each object as it is counted.]

If the child is counting comfortably, begin looking for larger groups (e.g., 11 to 20). Next, explain that you are going to play a game. You will take turns flashing a dot card with 1, 2, 3, 4, or 5 dots, and see if the other person can remember the number without counting. Allow the child to begin, by quickly flashing a dot card, then turning it face down. Say: • I think it was ___ dots. I still have the picture in my mind. There were ___

rows [describe the pattern you recall]. • Now it is my turn. Here’s my card. [Quickly show and hide the card so

that there is not time to count.] How many dots did you see? If the child answers correctly, ask: • Can you remember the picture of the dots in your mind? Tell me about it. Continue with other cards to 5. If the child is having difficulty, try alternating just two cards. Notice the child’s confidence and ability to • show awareness of numbers and sets in the environment • use 1:1 correspondence to count • count orally to 10 • use the principle of cardinality (last number counted is the quantity) • see a visual array as a quantity (subitize)

Communication Say: Thank you for your hard work and thinking! Let’s make a list of what we did and what we noticed. [Allow child to retell the activities freely; prompt if stuck.] Notice the child’s confidence and ability to • use number words




Date:



Name is aware of number

uses numbers in play and routines

uses ordinals to 5

talks about use of number

counts orally to 10

reads numerals to 10

recognizes numbers to 10 (different ways)

uses spatial cues for quantity (subitize)




Date:



Name sees some combinations that make 5/10

“sees” a visual array as a quantity (subitize)

begins to use 5 and 10 as anchors

compares to 5 and 10

counts orally to 20

matches 1:1 as he or she counts

counts orally to 30

uses numbers in play and routines




Name: Date:



developing; with

assistance

achieved; some

support

fluent; independent

COMMENTS

Reasoning; applying concepts shows awareness of number; uses numbers in play and routines

estimates and counts to identify sets with more, fewer, or the same number of objects

explores a variety of representations of numbers (0-9)

begins to use 5 and 10 as anchors; aware of some combinations that “make 5” or “make 10”

Accuracy of procedures matches objects with 1:1 correspondence

counts orally (to 10)

recognizes numerals to 10

orders sets of like objects by the number in the set

Problem solving uses the processes of joining (addition) and separating (subtraction) in solving problems (role-playing; using manipulatives)

Communication uses some mathematical language (e.g., same, more, fewer)



Date:

Observations Record 1 Teachers can use this form to record observations about 2 or 3 children who are working on the same activity. Observations can include notes about any work samples children may produce. Unit/Activity: _____________________________________________________________ Name:_____________ Observations

Name:_____________ Observations

Name:_____________ Observations

Ways to adjust or enrich this activity in future

Assessment Master 7


Name:

Observations Record 2: Centres and Practice Work

Teachers can use this form to record observations about a child over several activities. Notes can include comments about any work samples produced. Observation period from ________to ___________ Activity/focus Observations

Activity/focus Observations



Patterns observed; ideas for next steps Observations

Assessment Master 8


Name: Date:

Conference Prompts Teachers can select and develop questions and prompts such as the following to use during conferences, as well as when children are working on various activities. Answers will often provide evidence of more than one category. The questions are examples; they are not intended for use in sequence or as an overall outline. Problem solving • Tell me about the problem. What are you trying

to do? • What have you tried? How did that work? • How did you get started? • What part was easy? • Were any parts hard for you? Tell me about them. • Show/tell me about your thinking. • How do you feel about the way your work

turned out? • Did you solve the problem? How did you know

when it was solved (or when you were finished)? • Can you think of another way to solve/work on

this problem? • Is this like any other problems we have solved?

How are they the same? • Could you help someone else solve a problem like

this? What would you tell/show them?

Reasoning; applying concepts • What can you tell me about …? • Tell me something that you have learned about …. • Tell me about/show me your thinking. • How do you know that …? • What do you think/predict will happen if …? • How could you explain … to someone who is just

learning about it? • What is the same/different about …? • Does that make sense to you? Tell me

why/why not. • How did you know/decide …? • What did you find out about …? • Could you show what you found out in

another way? • What would you like to learn about?

Accuracy of procedures • How many …? • Show me how you …. • What did you do? • What answer/solution did you find? How did you

decide it was a good answer? • How could we check? • What would you like to learn how to do?

Communication Can be observed as children respond to questions such as those listed above in speech, drawings, with manipulatives, or by writing. • What do you call that? Does it have another name? • How could you tell/show someone what you

found out? • Tell/show me what you did. • How do you know that …?

Assessment Master 9


Name: Date:

Language to Listen For Teachers can use and add to the following word lists to help them listen for and record some of the mathematical language children are developing. Children are not expected to know and use all of these in kindergarten—these are examples of language that help to indicate that children are developing mathematical understanding. Position above near below far in beside out next to inside over outside under up on top of down underneath front in front of back behind top between bottom forward toward backward

Attributes (colour, shape, size) colour words round tall flat short bumpy big curved/curvy small straight empty pointed full edge heavy circle light square hot rectangle cold triangle sharp like a box smooth like a ball same different comparatives (-er) and superlatives (-est)

Sequence/Time tomorrow first today second yesterday next before then after short time long time

Number number words (cardinal) match how many add more take away less/fewer estimate same fair share equal check number count first, second, third (ordinal)

Assessment Master 10


Name:

Learning Skills Teachers can use this form to accumulate observations about a child’s learning skills over a period of time.

Observation notes Observation notes Observation notes

Learning skill Date: Date: Date:

follows routines and instructions

interested and curious; questions; explores

shows self-confidence; becoming independent

completes tasks

takes turns; shares materials

participates in class activities and discussions; shares ideas

listens to others



Name: Date:

Self-Assessment Interview Teachers can record the child’s self-assessment. It is not essential to ask all questions in a single interview. If possible, attach the child’s drawing, a photograph, or other representation.

Unit/Activity ____________________________________________________________ Tell me how you got along working with others today. Did you • Share? yes sometimes no • Help? yes sometimes no • Listen? yes sometimes no • Take turns? yes sometimes no

Observation Notes

Show me the work you did today. (Note: May ask child to display, draw, or recount orally) Tell me about your thinking.

How did it turn out? (How did you like your work?) Tell me something you are proud of.

What can you tell me about what you learned? (found out)

Tell me about something you would like to learn/get better at in math. How could I help you with that?



Date:

Observations Checklist: Solving a Problem

Teachers can use the following checklist to record observations of children’s problem solving using notes or symbols such as check marks.

Name Restates

problem Offers ideas about how to solve it

Tries out strategies

Willing to persevere

Arrives at reasonable solution

Explains solution



Name: Date:

Unit Summary Unit/Strand _______________________________ Review assessment records to create a summary statement about the child’s achievement in relation to each aspect of mathematics for this unit/strand. Depending on the reporting requirements in your region, you may also choose to provide achievement levels for each aspect and/or overall.

Observation Notes

Reasoning; applying concepts

Accuracy of procedures

Problem solving

Communication

Learning skills:

Self-assessment:

Strengths:

Needs:

Next steps: At school: At home:



Name: Date:

Observations Checklist: Curriculum Expectations

This form can be used to accumulate observations about a child’s mathematical development.

By the end of kindergarten, children should be able to meet the following general outcomes for each of the strands.

Check Observations and Comments

Number Describe orally, and compare quantities for 0 to 10, using number words in daily experiences. For example: • count objects in a set (0-10) • build and compare sets (more/less than; greater/fewer than;

same; equal) • order two sets of like objects based on number • explore representation of numbers to 9 (computer, calculator) • represent processes of addition/subtraction through role-

playing and manipulatives

Patterns and Relations Identify and create patterns arising from daily experiences. For example: • sort objects, using a single attribute • recognize and reproduce a pattern (actions, manipulatives) • recognize and reproduce a pattern (actions, manipulatives) • create and extend a pattern • describe orally, a pattern

Shape and Space: Measurement Demonstrate awareness of measurement. For example: • classify and describe linear attributes (e.g., long/short) • arrange objects in order of size (length/height) • cover a surface with a variety of objects • use words full, empty, less/more to talk about volume and

capacity; heavier/lighter to compare mass of 2 objects • use terms long/short time to talk about duration of events • use words like hot/hotter; warm/warmer; cool/cooler;

cold/colder to talk about temperature • exchange play money for objects in a play store

Shape and Space: 3-D Objects and 2-D Shapes Sort, classify, and build real-world objects. For example: • identify, sort, and classify 3-D objects in environment • describe and discuss objects (e.g., big, little, round, like a box,

like a can) • build 3-D objects

Shape and Space: Transformations Describe, orally, the position of 3-D objects. For example: • describe the relative position of 3-D objects (e.g., over, under,

beside, between, inside, outside)

Statistics and Probability Collect and organize, with assistance, data based on first-

hand information. For example: • collect, with assistance, first-hand data • construct, with assistance, a concrete/object graph (1:1) • compare data in two categories using words such as more,

less, the same


Pat Dickinson

Michelle Jackson

Sharon Jeroski

Carole Saundry

Maureen Dockendorf

Charlotte MacKay

Craig FeatherstoneMaggie Martin ConnellTrevor Brown

Assessment ConsultantSharon Jeroski

Primary Mathematics and Literacy ConsultantPat Dickinson

Elementary Mathematics Adviser John A. Van de Walle

British Columbia Early Numeracy Adviser Carole Saundry

Ontario Early Math Strategy Adviser Ruth Dawson

Copyright © 2004 Pearson Education Canada Inc.

All Rights Reserved. This publication is protected bycopyright, and permission should be obtained fromthe publisher prior to any prohibited reproduction,storage in a retrieval system, or transmission in anyform or by any means, electronic, mechanical,photocopying, recording, or likewise. Forinformation regarding permission, write to thePermissions Department.

The information and activities presented in thisbook have been carefully edited and reviewed.However, the publisher shall not be liable for anydamages resulting, in whole or in part, from thereader’s use of this material.

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