Contentsmathematics for teachingto improve their instruction. These materials were not designed as...

CCoonntteennttssOOvveerrvviieeww ooff TThhee LLiitteerraaccyy aanndd NNuummeerraaccyy SSeeccrreettaarriiaatt PPrrooffeessssiioonnaall LLeeaarrnniinngg SSeerriieess .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11

GGeettttiinngg OOrrggaanniizzeedd .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 44

SSeessssiioonn AA –– MMooddeelllliinngg aanndd RReepprreesseennttiinngg .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 77Aims of Numeracy Professional Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Learning Goals of the Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Book Walk: Tabbing the Volumes (1, 3, 4, and 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Warm Up – What Ways Do We Use Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Modelling and Representing Multiplication – Problem #1 . . . . . . . . . . . . . . . . . . . . . . . . 11

SSeessssiioonn BB –– CCoonncceeppttuuaall DDeevveellooppmmeenntt .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1133Problem Solving to Develop Conceptual Understanding . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Warm Up – A Math Congress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

The Concepts of Multiplication – Problem #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

A Gallery Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

SSeessssiioonn CC –– EExxpplloorriinngg AAlltteerrnnaattiivvee AAllggoorriitthhmmss .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1166Applying Student-Generated Algorithms and Analysing Standard Algorithms . . . . . . . . . 16

Warm Up – Partitive and Quotative Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Student-Generated and Standard Algorithms – Problem #3 . . . . . . . . . . . . . . . . . . . . . . . 17

Bansho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

SSeessssiioonn DD –– CCoommmmuunniiccaattiinngg MMaatthheemmaattiiccaall TThhiinnkkiinngg .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2222Estimating Decimal Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Warm Up – “All About Place Value” Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Making the Strategies and Math Talk Explicit – Problem #4 . . . . . . . . . . . . . . . . . . . . . . 23

Professional Learning Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

RReeffeerreenncceess .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2277

RReessoouurrcceess ttoo IInnvveessttiiggaattee .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2288BLM “All About Place Value” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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OOvveerrvviieeww ooff TThhee LLiitteerraaccyy aanndd NNuummeerraaccyy SSeeccrreettaarriiaatt PPrrooffeessssiioonnaall LLeeaarrnniinngg SSeerriieess 11

OOvveerrvviieeww ooff TThhee LLiitteerraaccyy aanndd NNuummeerraaccyy SSeeccrreettaarriiaattPPrrooffeessssiioonnaall LLeeaarrnniinngg SSeerriieessThe effectiveness of traditional professional development seminars and workshops has

increasingly been questioned by both educators and researchers (Fullan, 1995; Guskey &

Huberman, 1995; Wilson & Berne, 1999). Part of the pressure to rethink traditional PD comes

from changes in the teaching profession. The expert panel reports for primary and junior

literacy and numeracy (Ministry of Education, 2003, 2004) raise several key issues for today’s

teachers:

• Teachers are being asked to teach in ways that they themselves may not have experienced

or seen in classroom situations.

• Teachers require a more extensive knowledge of literacy and numeracy than they did

previously as teachers or as students.

• Teachers need to develop a deep knowledge of literacy and numeracy pedagogy in order to

understand and develop a repertoire of ways to work effectively with a range of students.

• Teachers may experience difficulty allocating sufficient time for students to develop

concepts of literacy and numeracy if they themselves do not appreciate the primacy of

conceptual understanding.

For professional learning in literacy and numeracy to be meaningful and classroom-applicable,

these issues must be addressed. Effective professional learning for today’s teachers should

include the following features:

• It must be grounded in inquiry and reflection, be participant-driven, and focus on

improving planning and instruction.

• It must be collaborative, involving the sharing of knowledge and focusing on communities

of practice rather than on individual teachers.

• It must be ongoing, intensive, and supported by a job-embedded professional learning

structure, being focused on the collective solving of specific problems in teaching, so

that teachers can implement their new learning and sustain changes in their practice.

• It must be connected to and derived from teachers’ work with students – teaching,

assessing, observing, and reflecting on the processes of learning and knowledge

production.

Traditionally, teaching has been a very isolated profession. Yet research indicates that the

best learning occurs in collaboration with others (Fullan, 1995; Joyce & Showers, 1995; Staub,

West & Miller, 1998). Research also shows that teachers’ skills, knowledge, beliefs, and under-

standings are key factors in improving the achievement of all students.

Job-embedded professional learning addresses teacher isolation by providing opportunities for

shared teacher inquiry, study, and classroom-based research. Such collaborative professional

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22 FFaacciilliittaattoorr’’ss HHaannddbbooookk

learning motivates teachers to act on issues related to curriculum programming, instruction,

assessment, and student learning. It promotes reflective practice and results in teachers

working smarter, not harder. Overall, job-embedded professional learning builds capacity for

instructional improvement and leadership.

There are numerous approaches to job-embedded professional learning. Some key approaches

include: co-teaching, coaching, mentoring, teacher inquiry, and study.

AAiimmss ooff NNuummeerraaccyy PPrrooffeessssiioonnaall LLeeaarrnniinnggThe Literacy and Numeracy Secretariat developed this professional learning series in order to:

• promote the belief that all students have learned some mathematics through their lived

experiences in the world and that the math classroom should be a place where students

bring that thinking to work;

• build teachers’ expertise in setting classroom conditions in which students can move

from their informal math understandings to generalizations and formal mathematical

representations;

• assist educators working with teachers of students in the junior division to implement the

student-focused instructional methods that are referenced in Number Sense and

Numeration, Grades 4 to 6 to improve student achievement; and

• have teachers experience mathematical problem solving – sharing their thinking and

listening; considering the ideas of others; adapting their thoughts; understanding and

analysing solutions; comparing and contrasting solutions; and discussing, generalizing,

and communicating – as a model of what effective math instruction entails.

TTeeaacchhiinngg MMaatthheemmaattiiccss tthhrroouugghh PPrroobblleemm SSoollvviinnggUntil quite recently, understanding the thinking and learning that the mind makes possible

has remained an elusive quest, in part because of a lack of powerful research tools. In fact,

many of us learned mathematics when little was known about learning or about how the

brain works. We now know that mathematics instruction can be developmentally appropriate

and accessible for today’s learners. Mathematics instruction has to start from contexts that

children can relate to – so that they can “see themselves” in the context of the question.

Most people learned math procedures first and then solved word problems related to the

operations after practising the skills taught to them by the teacher. The idea of teaching

through problem solving turns this process on its head.

By starting with a problem in a context (e.g., situational, inquiry-based) that children can

relate to, we activate their prior knowledge and lived experiences and facilitate their access

to solving mathematical problems. This activation connects children to the problem; when

they can make sense of the details, they can engage in problem solving. Classroom instruction

needs to provoke students to further develop their informal mathematical knowledge by

representing their mathematical thinking in different ways and by adapting their under-

standings after listening to others. As they examine the work of other students and consider

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OOvveerrvviieeww ooff TThhee LLiitteerraaccyy aanndd NNuummeerraaccyy SSeeccrreettaarriiaatt PPrrooffeessssiioonnaall LLeeaarrnniinngg SSeerriieess 33

the teacher’s comments and questions, they begin to: recognize patterns; identify similarities

and differences between and among the solutions; and appreciate more formal methods of

representing their thinking. Through rich mathematical discourse and argument, students

(and the teacher) come to see the mathematical concepts expressed from many points of view.

The consolidation that follows from such dynamic discourse makes the mathematical represen-

tations explicit and lets students see many aspects and properties of math concepts, resulting

in students’ deeper understanding.

LLeeaarrnniinngg GGooaallss ooff tthhee MMoodduulleeThis module is organized to guide facilitators as they engage participants in discussion with

colleagues working in junior classrooms. This discourse will focus on important concepts,

procedures, and appropriate representations of multiplication and division of whole and

decimal numbers.

During these sessions, participants will:

• develop an understanding of the conceptual models of operations with whole and decimal

numbers;

• explore conceptual and algorithmic models of multiplication and division of whole and

decimal numbers through problem solving;

• analyse and discuss the role of student-generated strategies and standard algorithms in the

teaching of multiplication and division with whole and decimal numbers; and

• identify the components of an effective mathematics classroom.

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GGeettttiinngg OOrrggaanniizzeeddPPaarrttiicciippaannttss • Classroom teachers (experienced, new to the grade, new to teaching [NTIP]), resource and

special education teachers, numeracy coaches, system curriculum staff, and school leaders

will bring a range of experiences – and comfort levels – to the teaching and learning of

mathematics. Participants may be organized by grade, division, cross-division, family of

school clusters, superintendency regions, coterminous boards, or boards in regions.

• Adult learners benefit from a teaching and learning approach that recognizes their

mathematics teaching experiences and knowledge and that provides them with learning

experiences that challenge their thinking and introduces them to research-supported

methods for teaching and learning mathematics. For example, if time permits, begin each

session with 10 minutes for participants to share their mathematics teaching and learning

experiences, strategies, dilemmas, and questions.

• Some participants may have prior knowledge through having attended professional

development sessions using The Guide to Effective Instruction in Mathematics, Kindergarten

to Grade 3 or The Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6

through board sessions or Ontario Summer Institutes. These professional learning sessions

are intended to deepen numeracy learning, especially for junior teachers.

FFaacciilliittaattoorrssEffective professional learning happens daily and over time. These professional learning mate-

rials are designed to be used to facilitate teachers’ collaborative study of a particular aspect of

mathematics for teaching to improve their instruction. These materials were not designed as

presentation material. In fact, these sessions are organized so that they can be used flexibly

with teachers (e.g., classroom teachers, coaches, consultants) and school leaders (e.g., vice

principals, principals, program coordinators) to plan and facilitate their own professional

learning at the school, region, and/or board levels.

It is recommended that the use of these materials is facilitated collaboratively by at least two

educators. Co-facilitators have the opportunity to co-plan, co-implement, and make sense of

the audience’s responses together, to adjust their use of the materials, and to improve the

quality of the professional learning for the audience and themselves. Further, to use these

modules, facilitators do not need to be numeracy experts, but facilitators do need to be confi-

dent about learning collaboratively with the participants and have some experience and/or

professional interest in studying mathematics teaching/learning to improve instruction.

Here are a few ways in which facilitators can prepare to use this module effectively:

• Take sufficient time to become familiar with the content and the intended learning process

inherent in these sessions.

• Think about the use of the PowerPoint as a visual aid to present the mathematical prompts

and questions participants will use.

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GGeettttiinngg OOrrggaanniizzeedd 55

• Use the Facilitator’s Handbook to determine ways in which to use the slides to

generate discussion, mathematical thinking and doing, and reflection about classroom

implementation.

• Note specific teaching strategies that are suggested to develop rich mathematical

conversation or discourse.

• Highlight the mathematical vocabulary and symbols that need to be made explicit during

discussions and sharing of mathematical solutions in the Facilitator’s Handbook.

• Try the problems prior to the sessions to anticipate a variety of possible mathematical

solutions.

• As you facilitate the sessions, use the Facilitator’s Handbook to help you and your learning

group make sense of the mathematical ideas, representations (e.g., arrays, number lines),

and symbols.

TTiimmee LLiinneess• This module can be used in different professional learning scenarios: professional learning

team meetings, teacher planning time, teacher inquiry/study, parent/community sessions.

• Though the module is designed to be done in its entirety, so that the continuum of

mathematics learning can be experienced and made explicit, the sessions can be chosen

to meet the specific learning needs of the audience. For example, participants may want to

focus on understanding how students develop conceptual understanding through problem

solving, so the facilitator may choose to implement only Session B in this module.

• As well, the time frame for implementation is flexible. Three examples are provided below.

If you choose to use these materials during:

• One full day – the time line for each session is tight for implementation; monitor the use

of time for mathematical problem solving, discussion, and reflecting.

• Two half days – the time line for each session is tight for implementation; monitor the

MMoodduullee SSeessssiioonnss OOnnee FFuullll DDaayy TTwwoo HHaallff--DDaayyss FFoouurr SSeessssiioonnss

Session A – Modelling and Representing

75 min Day 1

120 – 180 min

90 – 120 min

Session B – Conceptual Development

75 min 90 – 120 min

Session C – Exploring AlternativeAlgorithms

75 min Day 2

120 – 180 min

90 – 120 min

Session D – CommunicatingMathematical Thinking

75 min 90 – 120 min

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use of time for mathematical problem solving, discussion, and reflecting; include time for

participants to share the impact of implementing ideas and strategies from the day one

session.

• Four sessions – the time lines for each session are more generous for implementation;

include time for participants to discuss and choose ideas and strategies to implement

in their classroom at the end of each session; include time for sharing the impact of

implementing ideas and strategies from the previous session at the start of each session.

CCrreeaattiinngg aa PPrrooffeessssiioonnaall LLeeaarrnniinngg EEnnvviirroonnmmeenntt

• Organize participants into small groups – preferably of 4 to 6 people – to facilitate

professional dialogue and problem-solving/thinking experiences.

• Seat participants in same-grade or cross-grade groups, depending on whether you want the

discussion to focus on one grade level or across grade levels.

• Ensure that a blackboard or 3 to 4 metres of wall space is cleared, so that mathematical

work can be posted and clearly seen.

• Provide a container with the learning materials (e.g., writing implements like markers,

paper, sticky notes) on each table before the session. Math manipulatives and materials

should be provided for each pair of participants at each table.

• Provide a copy of the agenda and handouts of the PowerPoint for note-taking purposes or

tell the participants that the PowerPoint will be e-mailed to them after the session so that

they have a record of it.

• Arrange refreshments for breaks and/or lunches, if appropriate.

• If time permits, begin each session with 10 minutes for participants to share their

mathematics teaching and learning experiences, strategies, and dilemmas.

MMaatteerriiaallss NNeeeeddeedd• copy of Number Sense and Numeration, Grades 4 to 6 (Volumes 1, 3, 4, and 6) for each

participant

• Understanding Multiplication and Division of Whole and Decimal Numbers PowerPoint

presentation, slides 1 to 23

• computer, LCD projector, and extension cord

• chart paper (ripped into halves or quarters), markers, sticky notes, highlighters, pencils,

transparencies, and overhead markers (if projector is available), tape for each table of

participants

• base ten blocks, counters, and calculators for every two participants

• for Session D, two BLMs to copy and cut up for use by groups of four in the warm-up game

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SSeessssiioonn AA –– MMooddeelllliinngg aanndd RReepprreesseennttiinngg 77

SSeessssiioonn AA –– MMooddeelllliinngg aanndd RReepprreesseennttiinnggAAiimmss ooff NNuummeerraaccyy PPrrooffeessssiioonnaall LLeeaarrnniinnggBefore Session A begins, work with the session

planning team (administrator, math lead teacher/

division numeracy contact teacher, system numeracy

support personnel, etc.) to inform staff about the

date, time, location, and topic of the session. Remind

participants to bring their copies of Number Sense

and Numeration, Grades 4 to 6:

• Volume 1: The Big Ideas

• Volume 3: Multiplication

• Volume 4: Division

• Volume 6: Decimal Numbers

Display the first and second slides (slides 1 and 2) as

participants enter the learning area. Ask participants

to locate their copies Number Sense and Numeration,

Grades 4 to 6 required for the session (Volumes 1, 3,

4, and 6).

Review the aims of numeracy professional learning

(slides 3 and 4). Emphasize the importance of teach-

ing and learning through problem solving as the

primary teaching approach for mathematics (slide 5)

LLeeaarrnniinngg GGooaallss ooff tthhee MMoodduulleeReview the learning goals (slide 6):

• Develop an understanding of the conceptual

models of whole numbers and decimals.

• Explore conceptual and algorithmic models of

whole number and decimal multiplication through

problem solving.

• Analyse and discuss the role of student-generated

strategies and standard algorithms in the teaching

of multiplication and division with whole and

decimal numbers.

• Identify the components of an effective mathemat-

ics classroom.

Slide 4

Slide 1

Slide 2

Slide 3

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BBooookk WWaallkk:: TTaabbbbiinngg tthhee VVoolluummeess ((11,, 33,, 44,, aanndd 66))Clarify with participants that this professional learn-

ing module is meant to accompany Number Sense and

Numeration, Grades 4 to 6 (in 6 volumes) (slide 7).

Share with the group that a book walk will be used as

a before-reading learning strategy for the use of the

guides, then move to the next slide.

Display the next slide (slide 8) to help participants

begin to understand the organization of the volumes.

Instruct participants at each table to number off 1, 3,

4, and 6 so that each table will have one or two people

to look at each volume. Direct the participants to use

the sticky notes to tab the volume they have been

assigned and then to share the organization of their

volume with the group. This is a tabbing exercise, not

a full reading opportunity. Groups will discover that

Volume 1 is organized by:

• Introduction

• The “Big Ideas” in Number Sense and Numeration

• Quantity (Overview, “Howmuchness”, Meaningful

Contexts, Estimating and Reasoning, Effects of

Operations, Student Learning, Instructional

Strategies)

• Operational Sense (Overview, Properties and

Relationships, Part-Whole Relationships, Flexible

Computation Strategies, Problem Solving and

Models, Student Learning, Instructional

Strategies)

• Relationships (Overview, Number System,

Comparing and Ordering, Operations and

Computations, Fractions/Decimals/Percents,

Student Learning, Instructional Strategies)

• Representation (Overview, Symbols and Placement,

Symbols and Language to Communicate, Student

Learning, Instructional Strategies)

• Proportional Reasoning (Overview, Multiplicative

Comparisons Between Ratios, Using Fractions,

Slide 6

Slide 8

Slide 5

Slide 7

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Decimals, Ratio and Percent, Informal Activities,

Student Learning, Instructional Strategies)

• Glossary

Volumes 3, 4, and 6 are organized by:

• Introduction (Relating to Big Ideas, The Mathe-

matical Processes, Addressing the Needs of Junior

Learners)

• Learning About … in the Junior Grades

(Introduction, Solving a Variety of Problems,

Relating Operations, Concepts, Modelling, Basic

Facts, Developing Computational Strategies,

Developing Estimation Strategies, Decimal

Numbers, Summary)

• Mini-lessons/Models (not in all volumes)

• References

• Learning Activities by Grade (Overview, Big Ideas,

Curriculum Expectations, About the Learning

Activity, About the Math, Getting Started, Working

on It, Reflecting and Connecting, Adaptations/

Extensions, Assessment, Home Connection,

Learning Connections, Blackline Masters)

WWaarrmm UUpp –– WWhhaatt WWaayyss DDoo WWee UUsseeMMaatthh?? Use a think-pair-share process to help participants

access their prior knowledge about mathematical

operations (slide 9). Explicitly describe the think-

pair-share process:

• Allow learners time to think independently before

writing their responses to the question or prompt

(wait time); one idea on each sticky note – each

person should use multiple notes.

• Instruct learners to share their responses with one

other person (pair).

• Instruct learners to share their thinking with the

table (round robin) and to post their notes on

chart paper.

Slide 9

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We often forget that we learn mathematics through

our lived experiences in the world. It is easy for people

to start to think that mathematics only comes into

their lives in the questions posed on the pages of

textbooks and they fail to appreciate how much they

actually do know about math. With questions like this

(How have you used multiplication and division in

your daily life in the past week?), participants will

identify what they know about the two operations.

The activation of prior knowledge will set them up for

studying multiplication and division at a deeper level.

Discuss how this prompt serves to connect mathemat-

ics to “real life” applications and is a useful strategy

to use with students and how it gives students a start-

ing point for instruction. When they have a familiar

context in mind and can imagine the problem they are

solving, they have something to connect to. It sets

them up for learning.

Instruct the table groups to scan all the notes and

discuss a sorting rule for organizing them (slide 10).

They can sort their notes in any way that makes sense

or has meaning to the group. Ask the groups to create

visual representations of their work on chart paper

(e.g., graph, diagram, map) and to establish and

record, with labels, a sorting rule that explains their

work.

Explicitly remind teachers that sorting, classifying,

discovering, and creating rules are key thinking skills

for the teaching and learning of mathematics.

In professional settings, the sharing can be extended

to the whole group by having each table share a

perception, a reflection, or a solution (e.g., their

sorting rule with a few examples).

Slide 10

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MMooddeelllliinngg aanndd RReepprreesseennttiinnggMMuullttiipplliiccaattiioonn –– PPrroobblleemm ##11UUnnddeerrssttaanndd tthhee PPrroobblleemm

Explain to learners that they will now have the

opportunity to experience teaching through problem

solving. Ask participants, working in groups of two or

three, to reflect on how the problem (slide 11) would

be viewed and approached by junior students. Draw

the learners’ attention to the blue box at the bottom

of the slide. Explain that the blue box will appear on

slides to help learners make connections to support-

ing materials in the guide and to establish the focus

for the professional dialogue.

MMaakkee aa PPllaann aanndd CCaarrrryy OOuutt tthhee PPllaann

Display slide 12 and ask participants to think about

each part of Polya’s problem-solving model as they

work through their solutions. They may note that the

process is not a linear one – rather, it is iterative in its

implementation.

Instruct each group of 2 or 3 to solve the problem.

Ask them to use different mathematics to solve the

problem a second way using a second piece of chart

paper. Remind them to model and represent their

thinking with the tools available:

• manipulatives

• chart paper, markers, sticky notes

• diagrams, charts, tables

Ask groups to record the manipulative representations

of their solutions on chart paper with markers to

demonstrate the mathematical thinking. Ask each

group to post their solutions on the wall – grouping

similar solutions together as they hang the chart

paper. If no wall space is available to tape solutions to,

it also works to have participants line their solutions

up on the floor by a wall – a space where the whole

group can gather to talk.

Slide 11

Slide 12

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LLooookk BBaacckk:: RReefflleecctt aanndd CCoonnnneecctt

Ask participants to gather around the display and examine what mathematics they see in the

solutions. No doubt there will be a variety of solutions represented.

For example, you may see solutions like these:

1. (20 x 23) + (9 x 23) = 460 + 207 = 667. It costs $667.00.

2. (10 x 23 ) = 230

(10 x 23 ) = 230

(9 x 23) = 230 – 23 = 207

667

3.

Ask one or two groups to explain their solutions. Discuss the importance of listening and

visualizing while solutions are being shared. It is important for students to learn to follow the

mathematical thinking of others in the classroom. They need to listen and ask questions, and

talk about how their ideas fit into understanding this picture.

Pose the following question: “How does using mathematical models to represent multiplica-

tion promote student learning?” Use a round robin strategy by having each person (or a

sample of people if the group is large) respond to the question.

23

20

9

29

460

207

So, 29 x 23 = $667

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SSeessssiioonn BB –– CCoonncceeppttuuaall DDeevveellooppmmeenntt 1133

SSeessssiioonn BB –– CCoonncceeppttuuaall DDeevveellooppmmeennttPPrroobblleemm SSoollvviinngg ttoo DDeevveelloopp CCoonncceeppttuuaallUUnnddeerrssttaannddiinnggWelcome participants back to the session. Display the

Session B agenda (slide 13). Explain that the focus

will remain on teaching mathematics through problem

solving and that the conversation will now turn to the

role problem solving plays in students’ conceptual

development of operations, in this case, multiplication

of decimal numbers.

WWaarrmm UUpp –– AA MMaatthh CCoonnggrreessss Explain that the debriefing of the problem will be

accomplished through the use of a math congress.

Explain that the term “congress” comes from the

work of Cathy Fosnot from City College of New York,

who has been working successfully with public school

students and teachers for about 15 years to develop

mathematics thinking and learning. The congress

allows the sharing of selected student responses,

analysis of the mathematics used in the solution, and

prompts all students to learn from one another. The

purpose is to debrief the strategies used by students,

unearth multiple representations of mathematical

thinking, and assist in the development of deep under-

standing of concepts. By having students defend and

explain their thinking, teachers give their students an

opportunity to see and hear different perspectives.

This gives students a chance to examine the math very

closely – it is like “laying the concepts and the mis-

conceptions out on the table”. Students have a chance

to learn from the discourse and clarify their thinking

about the math idea. The connection to prior math

knowledge and the development of new knowledge is

made explicit. Enduring learning results.

Slide 13

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TThhee CCoonncceeppttss ooff MMuullttiipplliiccaattiioonn –– PPrroobblleemm ##22 UUnnddeerrssttaanndd tthhee PPrroobblleemm

Draw participants’ attention to the blue box, which

provides specific connections to the guide to support

continued learning. Also note that this is a question

(slide 14) for Grade 6 students who have previously

dealt only with multiplication of whole numbers.

This problem involves multiplication of a 1-digit whole

number and a 3-digit decimal number expressed in

tenths.

MMaakkee aa PPllaann

Instruct participants to solve the problem in more

than one way using different mathematics and then

show their work on chart paper with markers. Remind

participants to make full use of learning tools (e.g.,

manipulatives, diagrams, pictures, words) to represent

and communicate their solutions. Keeping in mind

that this problem would be used prior to the students

having done any multiplication with decimals, ask

teachers to consider which concept or property they

would anticipate students in a junior classroom would

use to solve Problem #2.

CCaarrrryy OOuutt tthhee PPllaann

Among the anticipated solutions, you should expect to

see repeated addition (12.4 + 12.4 + 12.4 + 12.4 =

48 + 1.6 = 49.6), doubling and re-doubling, an open

number line, estimation (resulting in an approximate

answer, such as, four 12s are 48 and three 0.4s are a

little greater than 1, and another 0.4 means I will be

closer to 2, so, the total time is about 50 s), manipu-

latives (where the rod is used to represent ones and

the cubes used to represent tenths). If you are not

seeing these solutions, prompt teachers at a few tables

to consider using different mathematics to make the

concepts explicit and to use manipulatives to model

decimal multiplication.

Slide 14

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SSeessssiioonn BB –– CCoonncceeppttuuaall DDeevveellooppmmeenntt 1155


AA GGaalllleerryy WWaallkkAsk all participants to post their solutions on the wall. There should be at least two from each

table. Conduct a “gallery walk” by asking all learners to travel around the room and read the

solutions from other tables. Learners should each carry several sticky notes and a pen or

pencil. During the gallery walk, encourage participants to write any questions for clarification

they want about a solution and then stick it on the solution on the chart paper. They can

write, “I do not understand.” or “Please explain this step.” and so on.

Following the gallery walk, ask a group to stand beside their solution. Ask them to explain

their thinking and to address and respond to the sticky notes posted on their solution. If time

allows, repeat by having other groups share. All this interaction between a person’s solution

and those of others motivates participants to think more deeply about the mathematics in

their solutions, adapt their thoughts, and learn more about the math related to the problem.

This occurs no matter what understanding they had when they first encountered the problem.

Deep understanding of mathematical ideas, language, and conventions is developed in an

iterative way – a little today and more tomorrow and more next week. The learning continues

to build on what they already know.

Explain to teachers that a typical congress involves a small number of student samples chosen

by the teacher and explained by the student.

Pose the following questions to participants:

• Which four samples would you choose to use for a math congress?

• What concept or property would the samples represent?

• How would you facilitate the discussion in your classroom?

During consolidation, lead a discussion about how they could use a gallery walk to identify the

range of student understandings of decimals. Some of the solutions could, no doubt, be used

to activate analysis of the multiplication of decimals algorithm. This process is also effective

when used for the purpose of improving communication. The sticky notes work to prompt

editing that improves the communication for other audiences.

If you are using this module in a job-embedded format, encourage participants to try the

gallery walk strategy in their own classrooms before the next study session. Ask participants to

bring along samples of student work from congresses they have run. Be sure to start the next

session by revisiting this commitment and making time for debriefing of the process.

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SSeessssiioonn CC –– EExxpplloorriinngg AAlltteerrnnaattiivvee AAllggoorriitthhmmssAAppppllyyiinngg SSttuuddeenntt--GGeenneerraatteedd AAllggoorriitthhmmssaanndd AAnnaallyyssiinngg SSttaannddaarrdd AAllggoorriitthhmmssDisplay the Session C agenda (slide 15) and discuss

the focus of the session: applying student-generated

algorithms and analysing standard algorithms. Explain

that after the problem-solving section of the session,

the group will look back at the ways the problem was

used for instruction.

WWaarrmm UUpp –– PPaarrttiittiivvee aanndd QQuuoottaattiivveeDDiivviissiioonnExplain that the focus of the problem in Session C

will prompt participants to show their many under-

standings of division. Encourage participants to

consider that the teaching of mathematics requires

adults to deconstruct their understanding of mathe-

matics. The two types of division problems on the slide

(slide 16) are examples of this idea. Adults may view

these two questions as the same because both can

be solved with the same operation: division. But, to

students, the two division questions are fundamentally

different.

Instruct participants to briefly discuss at their tables

the differences between the two division questions.

Draw participants’ attention to the connections in

the volumes that can further support teachers’ explo-

ration of partitive (sharing) and quotative (grouping)

division.

Study how the information in each problem differs.

In the first, the number of groups (5 bags) is provided

and the solution will tell the number of things

(apples) in each of the groups when the 30 apples are

shared evenly. In the second question, the number of

things in each group is identified (5 apples), and the

solution will tell the number of groups (bags) created

when the 30 apples are distributed in groups of 5.

Slide 15

Slide 16

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SSeessssiioonn CC –– EExxpplloorriinngg AAlltteerrnnaattiivvee AAllggoorriitthhmmss 1177

Students do not have difficulty sorting division

problems according to these two types – sharing and

grouping. Adults do have difficulty and it is because

our knowledge of division is compressed. We have

generalized these types and forgotten what we were

thinking when we first encountered these problems.

We do not appreciate the differences; so we lump

them together and call them “division problems”.

When students work on representing these questions

with concrete materials, before they have developed

algorithms, they move the manipulatives in very differ-

ent ways. For the sharing (partitive) question, their

hands move and their voices say, “One for you, one for

you, one for you and …” and so on until the counters

are all evenly shared. Then they will count the number

in each group. For the grouping (quotative) questions

they will show the manipulatives and say, “Five for this

group, five for this group, and …” and so on until all

30 apples are in groups. Then they will count the

number of groups. These two fairly standard division

algorithms can be referred to as the “goes into”

algorithm, which shows the division that is represent-

ed in solving sharing problems, and the “repeated

subtraction” algorithm, which represents the process

used in grouping questions.

SSttuuddeenntt--GGeenneerraatteedd aanndd SSttaannddaarrddAAllggoorriitthhmmss –– PPrroobblleemm ##33 UUnnddeerrssttaanndd tthhee PPrroobblleemm

Display Problem #3 (slide 17) and ask participants to

do a Think-Pair-Share on the problem. Have them read

the problem, turn to an elbow partner, and clarify the

problem. Ask, What is the important mathematical

information to pay attention to? and, What do you

anticipate students might do? Ask the pairs to share

their discussions with the table and then with the

large group.


As you guide discussion about this problem, note that

it is appropriate for introducing division of a 2-digit

number by a 1-digit number. So far students will show

Slide 17

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an understanding of division as the opposite of multiplication (7 x 7 = 49 so 49 ÷ 7 = 7).

Likely, students will not have internalized algorithms of the division process and they may not

have explored any methodologies of dividing greater numbers. They need to use their hands

and their knowledge about the other operations to make sense of this new operation called

division.

Direct each group of 2 or 3 to solve the problem and represent their mathematical thinking in

more than one way; each solution should be posted on a separate sheet of chart paper.


BBaannsshhooIn order to make public the mathematical thinking students use to solve a problem, we need

a way of organizing the work so everybody can see the range of student thinking. This allows

students to see their own thinking in the context of the similar thinking of others in the class.

The matching and comparing process promotes learning as students try to understand other

solutions and learn from one another. Japanese educators use a process they call bansho to

organize student work and to lead a conversation that offers everybody a chance to learn more

about the math used in developing solutions to a problem.

Explain that a bansho process will be employed to sort and classify participants’ solutions.

The bansho process uses a visual display of all students’ solutions, organized from least to

most mathematically rich. This is a process of assessment for learning and allows students and

teachers to view the full range of mathematical thinking their classmates and students used

to solve the problem. Students have the opportunity to see and to hear many approaches to

solving the problem and they are able to consider strategies that connect with the next step

in their conceptual understanding of the mathematics. Bansho is NOT about assessment of

learning, so there should be no attempt to classify solutions as level 1, level 2, level 3, or

level 4.

Begin by having representatives from each table post their chart paper solutions on the wall.

Facilitate the group in organizing the bansho by using the following prompts:

• Which solutions show students representing the mathematics using concrete

materials or pictures?

• Which solutions show students working with operations other than division?

• Which solutions show students using an algorithm (effectively or otherwise) that would

have you think they have a deep understanding of the solution? Can they say how and why

the algorithm works?

• How could we organize the solutions to represent a continuum of growth of mathematical

ideas?

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Use the group’s thinking and responses to organize the bansho by posting the charts from left

to right, grouping solutions that show the same mathematics. Later, during discussion, the

mathematics will be named and subsets labelled to describe specific procedures (e.g., count-

ing, array model, repeated subtraction). Ask learners to share how the representations would

benefit different students in a junior classroom.

For example, four sample solutions are shown below and the mathematics that is applied is

named in the paragraphs following the display.

SSaammppllee 11

SSaammppllee 22

96 km is the whole distance

48 km is half way

24 km in each quarter

24 km will be between the starting point,

each refreshment stop, and the end point.

I started with 96 and made 4 groups,

24 km between each.

12

8

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SSaammppllee 33

SSaammppllee 44

96

4

24{96 altogether

I did 96 ÷ 4 on my calculator and got 24.

It looks like there will be 24 km between each stop.

1

24 3

I put 20 in each of 4 groups. I had 16 left over, so I put 4 in for part 1, 4 for part2, 4 for part 3, and 4 for part 4. There will be about 24 km between each stop.

12-5513-Multi 2/12/07 1:54 PM Page 20


The first sample solution shows participants using repeated halving to list half of 96 and

then half of 48 to get 24. There are 4 groups of 24 in 96. Their response, 24 km between

each refreshment stop, shows understanding of the problem, but students may not be able to

generalize the strategy to deal with divisors that are odd (e.g., 95 ÷ 5).

The second sample solution shows the use of concrete materials broken out to show 4 groups.

They show the representation of breaking the total, 96, up into 4 parts. More explanation of

reasoning should be encouraged. It is, again, not clear that these students could deal with a

dividend that does not split evenly into the number of parts named by the divisor.

The third sample solution shows concrete materials being shared among 4 groups. This is a

quotative or sharing kind of division question. This method is generalizable – the students can

show the divisor as the number of circles and the dividend as split equally into groups. Odd

numbered divisors (number of groups) have a place in this strategy.

The fourth sample solution shows a clear understanding of the relationship between division

and multiplication. An open number array is used to show an array with 4 written along the

top to represent 4 columns in the array. The array is filled with 96. The students needed to

find the number of rows that would be needed to provide the other factor, 24, to complete the

equation 4 x ?? = 96. They say they used their calculator to determine 96 ÷ 4 = 24. It is not

clear that the students would know what to do if they got a decimal number answer on the cal-

culator. It may pose a challenge to have students understand the use of the open array for

solving quotative or sharing division problems. The partitive or grouping questions lend them-

selves better to explanations of the open array: 96 separated into groups of 4 by drawing an

array with 4 Xs in each row and then counting the number of rows (96 can be broken into 24

rows/groups of 4).

Use these kinds of questions to connect student thinking to the open array as a model for a

student-generated algorithm.

This presentation strategy is NOT meant for formative (evaluative) assessment. It should not

be used to represent 4 levels of performance of achievement on expectations. It shows the

range of mathematical thinking and knowledge in the class or group. The display, created by

the class, becomes a very powerful tool to help identify the range of understandings among

the students and offers an opportunity to identify starting points for instruction for the

teacher.

The participants examine and discuss the solutions, comparing their own to those of others.

The process helps problem solvers:

• organize their thinking;

• discover new ideas; and

• see connections between parts of the lesson, concepts, solutions, notations, and language.

Ask participants to display a range of student thinking in their classrooms before the next

meeting of the group. They should bring back photos and/or stories of how it worked.

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SSeessssiioonn DD –– CCoommmmuunniiccaattiinngg MMaatthheemmaattiiccaallTThhiinnkkiinnggEEssttiimmaattiinngg DDeecciimmaall DDiivviissiioonnIn learning mathematics, oral communication is as

important as written communication. Words represent

people’s thoughts about the mathematics they are

talking about. Similarly, thought becomes real when

one talks about a topic. Think about your reflection at

the last meeting you attended. As you discussed the

content of the meeting with colleagues, no doubt, you

learned things you didn’t even realize you had learned.

Through talking about it, your thoughts became clear-

er and you gave words to the ideas. This is an essential

part of math class. There has to be time to talk.

In this session, participants will talk with one another,

represent their thoughts on paper, compare their

solutions with those of others, discuss ideas, and

adapt their own thinking. Then they will, together,

analyse solutions presented by a couple of students

(simulation). Through all of this, mathematical com-

munication is strengthened. So, be certain math talk

is highlighted in this part of your collaborative study.

WWaarrmm UUpp –– ““AAllll AAbboouutt PPllaaccee VVaalluuee””GGaammeeDisplay slide 18. Before you start, run one set of cards

(24) for each group. Mix them up before grouping for

play.

Display slide 19. Organize participants into groups of

4. Explain that this session models a three-part lesson

with this game representing the warm-up. To engage

participants and activate their prior knowledge, ask

them to play the game “All About Place Value”. This

game will assist them in understanding the problem.

Slide 18

Slide 19

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SSeessssiioonn DD –– CCoommmmuunniiccaattiinngg MMaatthheemmaattiiccaall TThhiinnkkiinngg 2233

In the second part of the three-part lesson, students

engage in a problem. Students work on the problem

with their peers to uncover the concept or extend

their thinking about mathematics.

Talk about participants’ experiences in using the

problem-solving model, the components of which are:

• Understand the Problem

• Make a Plan

• Carry Out the Plan

• Look Back

MMaakkiinngg tthhee SSttrraatteeggiieess aanndd MMaatthh TTaallkkEExxpplliicciitt –– PPrroobblleemm ##44UUnnddeerrssttaanndd tthhee PPrroobblleemm

Display slide 20. Engage participants in understand-

ing the problem. Ask elbow partners to read the

problem and talk with each other about the important

mathematical elements in the problem.

Ask them to brainstorm questions they would ask

students to prompt them to consider the elements of

the problem.

Some questions might be:

• What do we know from the description of the

problem?

• What numbers are included in the description?

• What does the number in this problem represent?

• Does this problem remind you of another problem?

• If you were going to paraphrase this problem for a

friend, what would you tell them?

• What math vocabulary do you see in the problem?

How can you find out the meaning of the terms if

they are new?

• What will your answer look like (diagrams,

numbers, algorithms, units)?

Slide 20

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MMaakkee aa PPllaann

Have participants work in 2’s or 3’s. Talk about the importance of oral discussion in the

process of solving problems. Discussion is essential to understanding the problem. It lets

everyone clarify their understanding of the problem before starting. In adult workplaces this

is called the “problem definition” stage.

Ask them to consider the usefulness of questions like the following to prompt student

engagement:

• What strategies will you use to solve the problem?

• Is there more than one way to approach the problem?

• Are there tools that will help you to work on the solution or solutions to the problem?

• What sequence of steps will you use to find the solution?


Ask participants to carry out the sequence of steps they planned. Tell them to make modifica-

tions to their plan as new information or understanding comes to them and to be ready to

share information about these adaptations. Remind them to communicate their answers using

numbers, picture, words, and/or tools.


The third part of the lesson is critical to the learning. Reflecting on their own solutions and

the range of solutions from their peers, students are able to connect their prior understanding

of decimal numbers and place value to the operation of division. By thinking about their solu-

tions and expressing them verbally and in writing for their peers, students have access to the

collective learning of the classroom.

After communicating solutions, consider any other possible strategies that could be employed

to solve this or a similar problem. Consider what you learned about mathematics in solving the

problem.

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SSeessssiioonn DD –– CCoommmmuunniiccaattiinngg MMaatthheemmaattiiccaall TThhiinnkkiinngg 2255

Display slide 21. Ask participants to consider the two

solutions to the problem. Have them turn to a partner

and discuss the strategy each student used to attempt

to solve the problem. Consider what the solution

reveals about the understanding of each student.

What question would you ask each student about their

strategy?

Use this slide to lead discussion on strategies used

to divide a decimal number by a whole number.

Participants need to describe what mathematical

thinking they see in the girls’ solutions. After every-

one has shown they can explain the method used by

each girl, ask any participants who answered the

question differently to post their solutions and share

their thinking. Be certain the representation of the

decimal number and the mathematical thinking

involved are made explicit for all participants to

understand. To check for understanding, ask different

groups to restate what one group shared. Thinking,

talking, modelling, and representing are essential

to deep learning about mathematics and its many

applications.

MMaakkiinngg tthhee SSttrraatteeggiieess aanndd MMaatthh TTaallkk EExxpplliicciitt

Wrap up this problem-solving study by discussing

answers to the questions on the next slide (slide 22).

As this professional learning session comes to a close,

remind participants that the whole study has been

about instructional practices that engage students in

learning mathematics through problem solving.

The module has focused on the importance of starting

with children’s informal knowledge and teaching

them ways to represent their mathematical thinking

and to understand the language and conventions of

mathematics.

Slide 22

Slide 21

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The module has also engaged participants in exploring

and discussing their own mathematical thinking and

analysing that of others. As they compared representa-

tions from colleagues, their mathematical understand-

ing has deepened. The module has highlighted the

importance of developing a math-talk community in

the classroom. To communicate mathematically

requires attention be paid to math talk and represen-

tations of math thinking – this is at the heart of learn-

ing mathematics through problem solving.

Ideally, as teachers practise these strategies in their

own classrooms, and discuss and share their learning,

they build confidence in their ability to respond to

students’ ideas and to help them develop ways to

represent and record their own algorithmic processes.

Through talk, deeper understanding of the mathemat-

ics emerges.

One of the most important messages about this kind

of instruction is that it focuses the conversation in the

math classroom on making the strategies explicit so

all students can learn from each other. The classroom

becomes a knowledge-building environment, with

everyone learning through comparing and analysing

the knowledge of the crowd. The goal of instruction

changes from getting the right answer to questions in

the text to focusing on the thinking used to make

decisions and solve problems.

PPrrooffeessssiioonnaall LLeeaarrnniinngg OOppppoorrttuunniittiieessShare the materials available (slide 23) to support

continued professional learning. For more information

and videos of junior students at work, direct partici-

pants to see the materials available on Grades 4–6 at

http://www.eworkshop.on.ca/. Slide 23

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RReeffeerreenncceess 2277

RReeffeerreenncceessFosnot, C.T., & Dolk, M. (2001). Young mathematicians at work: Constructing

multiplication and division. Portsmouth, NH: Heinemann.

Fullan, M. (1995). The school as a learning organization: Distant dreams. Theory into Practice,

34(40), 230–235.

Guskey, T. R., & Huberman, M. (Eds.).(1995). Professional development in education: New

paradigms and practices. New York, NY: Teachers College Press.

Joyce, B., & Showers, B. (1995). Students achievement through staff development:

Fundamentals of school renewal. New York: Longman.

Ministry of Education. (2003). Early math strategy: The report of the expert panel on early

math in Ontario. Toronto: Queen’s Printer for Ontario.

Ministry of Education. (2004). Teaching and learning mathematics: The report of the expert

panel on mathematics in Grades 4 to 6 in Ontario. Toronto: Queen’s Printer for Ontario.

Ministry of Education. (2005). Ontario curriculum, Mathematics: Grades 1 to 8 (Revised).

Toronto: Queen’s Printer for Ontario.

Ministry of Education. (2006). Number sense and numeration, Grades 4 to 6, Volumes 1 to 6.

Toronto: Queen’s Printer for Ontario.

Wilson, S. M., & Berne, J. (1999). Teacher learning and the acquisition of professional

knowledge: An examination of research on contemporary professional development. Review

of Research in Education, 24, 173–209.

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RReessoouurrcceess ttoo IInnvveessttiiggaatteeBurns, M. (2000). About teaching mathematics: A K–8 resource (2nd ed.). Sausalito, CA: Math

Solutions Publications.

Carpenter, T. P., Loef Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating

arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.

Chapin, S., & Johnson, A. (2000). Math matters: Understanding the math that you teach.

Sausalito, CA: Math Solutions Publications.

Coaching Institute for Literacy and Numeracy Leaders. Video on Demand.

www.curriculum.org

Haylock, D., & McDougall, D. (1999). Mathematics every elementary teacher should know.

Toronto: Trifolium Books.

Lester, K., & Charles, R.I. (Eds.). (2003). Teaching mathematics through problem solving:

Prekindergarten–Grade 6. Reston, VA: National Council of Teachers of Mathematics.

Loewenberg Ball, D. (November 2005). Webcast on knowing mathematics for Teaching, avail-

able at www.curriculum.org.

Sullivan, P., & Lilburn, P. (1997). Good questions for math teaching: Why ask them and what to

ask. Sausalito, CA: Math Solutions Publications.

Van de Walle, J., & Folk, S. (2005). Elementary and middle school mathematics: Teaching devel-

opmentally (Canadian ed.). New York: Longman.

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RReessoouurrcceess ttoo IInnvveessttiiggaattee 2299

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00..44 00..4400




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