1 Contents · Overview of The Literacy and Numeracy Secretariat Professional Learning Series ... and Numeracy Secretariat Professional Learning ... co-teaching, coaching, mentoring

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321132CCoonntteennttssOOvveerrvviieeww ooff TThhee LLiitteerraaccyy aanndd NNuummeerraaccyy SSeeccrreettaarriiaatt PPrrooffeessssiioonnaall LLeeaarrnniinngg SSeerriieess .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11

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

SSeessssiioonn AA –– AAccttiivvaattiinngg MMaatthheemmaattiiccaall KKnnoowwlleeddggee .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 77Aims of Numeracy Professional Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Learning Goals of the Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Warm Up – We Are Fractions! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Scavenger Hunt – Volume 1: The Big Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Book Walk – Volume 5: Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

SSeessssiioonn BB –– MMooddeelllliinngg aanndd RReepprreesseennttiinngg .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1111Warm Up – Anticipation Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

What Does It Mean to Model and Represent Mathematical Thinking? . . . . . . . . . . . . . . . 12

Save, Save, Save – Problem #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

A Mini-Gallery Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

SSeessssiioonn CC –– CCoonncceeppttuuaall DDeevveellooppmmeenntt .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1177Warm Up – A KWL Chart – Know, Wonder, Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Quilting – Problem #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

A Gallery Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

SSeessssiioonn DD –– AAlltteerrnnaattiivvee AAllggoorriitthhmmss .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2200Warm Up – The Meaning of Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Best Buy on Juice – Problem #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Engaging in Rich Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Professional Learning Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

RReeffeerreenncceess .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2266

RReessoouurrcceess ttoo IInnvveessttiiggaattee .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2277

BBllaacckk LLiinnee MMaasstteerrss .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2288

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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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3 1 231 321 23 1 32 3 2 22 32learning 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

22 FFaacciilliittaattoorr’’ss HHaannddbbooookk


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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 relationships among fractions, decimals, ratios,

rates, and percents.

During these sessions, participants will:

• develop an understanding of the conceptual models of fractions, decimals, ratios, rates,

and percents;

• explore conceptual and algorithmic models of working with fractions, decimals, ratios,

rates, and percents through problem solving;

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

teaching of the relationships among fractions, decimals, ratios, rates, and percents; and

• identify the components of an effective mathematics classroom.


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3 1 231 321 23 1 32 3 2 22 32GGeettttiinngg 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 that 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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321132• 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

GGeettttiinngg OOrrggaanniizzeedd 55

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

Session A – Activating MathematicalKnowledge

75 min Day 1

120 – 180 min

90 – 120 min

Session B – Modelling and Representing

75 min 90 – 120 min

Session C – Conceptual Development

75 min Day 2

120 – 180 min

90 – 120 min

Session D – Alternative Algorithms

75 min 90 – 120 min


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3 1 231 321 23 1 32 3 2 22 32use of time for mathematical problem solving, discussion, and reflecting; include time for

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

• 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, 5, and 6) for each

participant

• Understanding Relationships Between Fractions, Decimals, Ratios, Rates, and Percents

PowerPoint presentation, slides 1 to 23

• computer, LCD projector, and extension cord

• chart paper (ripped into halves or quarters), markers (6 markers of different colours

for each table group), sticky notes, highlighters, pencils, transparencies, and overhead

markers (if projector is available), tape for each table of participants

• square tiles (at least 100 per table group), base ten blocks, calculators for every two

participants

• BLM1, Scavenger Hunt; BLM2, KWL Chart; BLM3, 10 X 10 grid (about 5 per person)



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SSeessssiioonn AA –– AAccttiivvaattiinngg MMaatthheemmaattiiccaall KKnnoowwlleeddggee 77

SSeessssiioonn AA –– AAccttiivvaattiinnggMMaatthheemmaattiiccaall KKnnoowwlleeddggeeAAiimmss 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 decide how this module

will be implemented (e.g., as 4 separate sessions for

about 90 min each or as a full-day professional learn-

ing day). Include the schedule in the invitations you

send out to inform staff about the date, time, loca-

tion, and topic of the session. Remind participants to

bring their copies of Number Sense and Numeration,

Grades 4 to 6: Volumes 1, 5, and 6.

• Volume 1: The Big Ideas

• Volume 5: Fractions

• Volume 6: Decimal Numbers

Display slide 1 as participants enter the learning area.

Ask participants to locate their copies of Number

Sense and Numeration, Grades 4 to 6, required for the

session (Volumes 1, 5, and 6).

Display Session A agenda (slide 2) and talk with

participants about your plans and expectations for

implementation. That is, give them a sense of the

timing and flow (e.g., 90 min, half day, full day). If

you are using this in a job-embedded format and want

teachers to try out some strategies in their own class-

rooms between gatherings, make that explicit.

Review the aims of numeracy professional learning

(slides 3 and 4). Emphasize the importance of

teaching and learning through problem solving as the

primary teaching approach for mathematics (slide 5).

Discuss how this list of actions is the same as or

different from the goals of a math lesson from 20

years ago. Responses might be organized into a chart

such as the one shown here.

Slide 1

Slide 2

Slide 3

Slide 4


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LLeeaarrnniinngg GGooaallss ooff tthhee MMoodduullee

Review the learning goals (slide 6):

• Develop an understanding of the conceptual

models of fractions, decimals, ratios, rates, and

percents.

• Explore conceptual and algorithmic models of

fractions and decimals through problem solving.

• Analyse and discuss the role of student-generated

strategies and standard algorithms in teaching the

concepts and relationships of fractions, decimals,

ratios, rates, and percents.

• Identify the components of an effective

mathematics classroom.

WWaarrmm UUpp –– WWee AArree FFrraaccttiioonnss!! Show slide 7. Give participants a chance to introduce

themselves to others at their table and then ask them

to generate possibilities for fractions. Ask each group

to share one fraction per table group, and explain how

it fits the criteria for representing “nearly all of us”,

“nearly half of us”, or “nearly none of us”. For exam-

ple, “Almost none of us have seen the film Bon Cop,


Slide 5

2200 yyeeaarrss aaggoo,, mmaatthh ccllaassss wwaass

TTooddaayy,, mmaatthh ccllaassss sshhoouulldd bbee

exclusive inclusive

for some for all

competitive collaborative

about getting the rightanswer to textbook ques-tions

about learning (e.g., thinking, reason-ing, talking, comparing,listening, adaptingthought, and explainingsolutions to problems)

focused on transmittingthe teacher’s thinking

focused on generatingand developing mathe-matics knowledge thatstarts with student’scognition

and so on…

Slide 6

Slide 7


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321132Bad Cop. I asked 23 people and only 1 of us had seen

it. The fraction is close to 0, so it is almost none.”

Discuss the characteristics of the fraction responses

for each case (e.g., for almost none of us, we used

– the numerator is 1 and has a very small value and

the denominator, 23, has a value not very close to 1).

If we had found 10 favourable responses, we might say

is about half of us because half of 23 is 11.5 and

10 is quite close in value to 11.

SSccaavveennggeerr HHuunntt –– VVoolluummee 11:: TThhee BBiigg IIddeeaass Show slide 8. This activity gives participants a chance

to familiarize themselves with Volume 1: The Big Ideas

in Number Sense and Numeration, Grades 4 to 6. The

goal is not for every participant to read the Big Ideas

document, but rather to become cognizant of the

material to which they can refer.

Through a scavenger hunt, the participants will

explore:

• what the 5 big ideas are;

• the importance of learning big ideas;

• characteristics of student learning as they relate

to big ideas; and

• instructional strategies related to big ideas.

Make sure that each participant or pair of participants

has a copy of the document: Volume 1: The Big Ideas

in Number Sense and Numeration, Grades 4 to 6.

Number people at each table from 1 to 5 correspon-

ding to the big ideas, 1 through 5. Together the group

will share the reading and then share their learning

with others at the table.

Give each participant a copy of the Scavenger Hunt

(BLM1). Allow about 20 min for participants to read

the section pertaining to the big idea you have

assigned to them and record their findings on BLM1.

Ask participants to share their findings with others at

their table. This can take up to 10 min.

SSeessssiioonn AA –– AAccttiivvaattiinngg MMaatthheemmaattiiccaall KKnnoowwlleeddggee 99

Slide 8

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3 1 231 321 23 1 32 3 2 22 32BBooookk WWaallkk –– VVoolluummee 55:: FFrraaccttiioonnss The purpose of this segment is to explore Volume 5: Fractions in Number Sense and

Numeration, Grades 4 to 6. Each of the volumes in the series contains an explanation of

mathematical models and instructional strategies that support student understanding of the

topic, in this case, fractions. Each of the volumes also contains sample learning activities for

Grades 4, 5, and 6.

When leading a book walk of the fractions document (Volume 5), spend time on the following

sections, which are listed in the order in which they appear:

• The Mathematical Processes

• Characteristics of Junior Learners (chart)

• Learning About Fractions in the Junior Grades (scope and sequence of expectations in

Grades 4, 5, and 6)

The next section of the Fractions document contains the content knowledge related to the

study of fractions in Grades 4 to 6. Briefly discuss with participants how the illustrations

and text provided help them understand how to teach: modelling fractions as part of a whole;

counting fractional parts beyond one whole; relating fraction symbols to their meaning;

relating fractions to division; establishing part-whole relationships; relating fractions to bench-

marks; comparing and ordering fractions; and determining equivalent fractions. This is only a

brief overview. Entertain questions as time allows.

It is important to discuss the idea that the teaching and learning of mathematics take place

through problem solving. This approach creates conditions in which students generate their

own solutions, share strategies, and struggle to make sense of the math. Through engagement

in this struggle, students construct and deepen their knowledge and understanding of mathe-

matics. A large body of research indicates the power of this approach in ensuring that students

experience enduring learning.

Encourage teachers to read the document and use the sample lessons as models for lessons

they will implement.

If you are wrapping up here and sending teachers to their classrooms to practise what they

have learned, make the expectations for preparation for the next session clear.



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321132SSeessssiioonn BB –– MMooddeelllliinngg aanndd RReepprreesseennttiinnggWWaarrmm UUpp –– AAnnttiicciippaattiioonn GGuuiiddeeIntroduce the agenda for Session B (slide 9).

The use of an anticipation guide (slide 10) enables

teachers to find out what their students know and

what they still need to learn. It also serves to activate

prior knowledge.

As Fullan, Hill, and Crévola say in Breakthrough,

“Our idea is to put the teacher and the student in

the learner’s seat, supported by a surrounding system

that requires and enables focused instruction.” As you

model effective classroom practice throughout this

workshop, you will give teachers an image of what

their mathematics instruction should look like.

Ask participants to decide as a group whether they

agree or disagree with the three statements in the

chart in slide 10. Explain that a group answer means

everyone at the table must be able to explain and

justify the group’s position. Here are some sample

positions:

• The first statement is untrue. A percent of some-

thing like 25% means 25 parts of a whole 100 or a

fraction represented as . The decimal 0.67 has

the value sixty-seven hundredths or , which is a

fraction.

• The second statement is true. If my discount is

0.75, which is or 75%, I am left with paying the

remainder, which is 25%. 75% and 25% make up

the whole price, 100%. 25% means and is equiv-

alent to the fraction . So, if the discount is 75%,

one pays 25% or of the original cost.The third

statement is untrue. Some fraction and decimal

equivalents of are or or 0.50. Some

fraction and decimal equivalents of are or

or 0.80. This is the ascending order: 0.37, , ,

0.93.

SSeessssiioonn BB –– MMooddeelllliinngg aanndd RReepprreesseennttiinngg 1111

Slide 9

Slide 10

25100

67100

75100

25100

14

14

12

510

50100

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WWhhaatt DDooeess IItt MMeeaann ttoo MMooddeell aanndd RReepprreesseenntt MMaatthheemmaattiiccaall TThhiinnkkiinngg??Talk about what it means to model and represent your mathematical thinking. Modelling

means making an image (e.g., with concrete materials, sketches, drawings, graphs, charts).

Models are used to represent an understanding of a situation – the math represented in the

problem you are trying to solve. Notice that represent can also be thought of as re-present.

When you move from a concrete image to a drawn or written one, you are re-presenting (i.e.,

presenting in a different format) your mathematical thinking and on your way to being able

to communicate. That is a major goal of mathematics instruction. We use a great deal of

informal mathematical thinking every day. We may not know the formal math representation

or the standard algorithms, but we can all do math needed to get through daily life. Children

also use informal mathematical thinking as they live their lives. They learn to solve problems

by solving problems, but they do not know how that thinking is named and categorized in the

discipline of mathematics. Math instruction needs to build on this informal knowledge – to

clarify the concepts and build understanding. When connections are made between informal

knowledge, the models that represent that knowledge, the representations that show that

knowledge, and the formal representations using the conventions of math (e.g., language

and symbols), students are enabling themselves to demonstrate and take control of their

mathematical understanding.

The re-presentation of an idea in a second or third way allows students to show what math

they have linked to the problem situation. Math manipulatives can be used by students from

all over the world because the language of the manipulatives is the universal language of

mathematics. If the students can understand the context and relate it to a problem they can

imagine, then they can do the math – represent their solution with concrete materials and

learn to draw and write a related solution. Then they learn to apply that math knowledge to

solve other problems.

SSaavvee,, SSaavvee,, SSaavvee –– PPrroobblleemm ##11Show slide 11. This problem was chosen because it engages students in modelling and

representing their math thinking. Read the problem with the participants and ask a few

people to make the mathematics in the problem explicit and clarify what participants are

expected to do. All students have access to the mathematical thinking when the math

language is explicit and defined. Having students articulate their understanding by explaining

the connections they have made between the numbers and/or symbols (e.g., what ideas from

the problem are represented by makes the math lesson inclusive – for all students.

This problem could pose some confusion because, in reality, it is likely that the discounts

apply to different shirts. If this issue arises in your discussions, ask the participants to make

suggestions to settle possible confusion. For example, everyone could agree to make the

context all about white shirts that all cost, say, $20.00. That way, participants who do not

recognize that the three discounts, as numbers, can be ordered greatest to least have a fixed

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discount. This kind of scaffolding is necessary

occasionally to make the math accessible to all.

After one explanation, ask everyone to show under-

standing with a “thumbs up” or “thumbs down” sign.

Ask someone who shows “thumbs up” to share his

or her understanding. Manage the challenges and

questions as they arise. Make sure the connections

between ideas, pictures, words, numbers, and/or

symbols are shared with all students in the class.

Some problems will have conditions and/or restric-

tions that need to be understood by the participants.

This first step of the problem-solving process – under-

stand the problem – is often omitted, but it is critical

to have all students “on the same page” before the

solving of the problem occurs. Understand the

problem involves more than decoding it – students

must comprehend the situation before they have

access to solving it. Use language strategies for

comprehension to give all students access to engaging

in solving problems.

For mathematicians, often the most time consuming

part of a job is building a detailed understanding of

the way a process works so that he or she can model

it with mathematics. The mathematician needs a

complete and well-defined understanding of the

problem/process before he or she can define and build

a model.

Remind teachers to adapt problems to suit their

students’ needs. Students may need some explanation

of the context, in this case, it may be what having a

shirt sale means. The context must be one in which

students can see or imagine themselves. The context

should be adapted for local variations. (e.g., if the

context is a rural setting that your students who live

in the city will not understand, change the problem

into a context they can understand). Teachers may

need, for example, to reduce extraneous words to

make a problem more accessible to some students.

Other students may need extensions to the problem.


Slide 11


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3 1 231 321 23 1 32 3 2 22 32MMaakkee aa PPllaann aanndd CCaarrrryy OOuutt tthhee PPllaann

Give each group some chart paper and markers and

ask them to record their solutions. Tell them their

solutions will be posted and shared with the large

group. In the planning stage, you would have provided

at least 100 tiles to each group. Encourage teachers

to use the tiles to show their understanding of the

math they used to solve the problem. Explain that

they must show at least two ways to solve the problem

and that they must draw a model of or represent their

mathematical thinking with concrete materials.

Show slide 12. Review or introduce Polya’s problem-

solving model with participants. Throughout these

sessions, the stages of the problem-solving model,

using Polya’s words, are made explicit.

Often adults misunderstand the use of manipulatives,

claiming they don’t need them. Remind them that the

manipulatives are for students, not for teachers whose

knowledge of mathematics is already synthesized –

boiled down and compressed into nice neat packages.

For students to have access to deep understandings,

they need to hear about the concepts from many dif-

ferent points of view and see them represented in a

variety of ways so that during discourse and discus-

sion, all of their questions can be addressed and

resolved. The picture becomes clearer through more

discussion and through discussing your ideas in

relation to the ideas of others. We all think very

differently – we are smarter in crowds. We can help

each other learn. Think about adult professionals

such as engineers, architects, and technicians who

use graphics, visual representations, and concrete

materials to solve problems and display their solutions

all the time.

LLooookk BBaacckk –– RReefflleecctt aanndd CCoonnnneecctt

The third part of a lesson, debriefing and consolidat-

ing, is very important and often overlooked. This

is when the mathematical thinking, language, and

notations are made explicit so that learning is

consolidated or expanded. When new concepts are


Slide 12


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introduced, students need to see and hear others’

solutions to understand the concept(s) being applied

in solving the problem. They need to practise saying

the words so that they clarify their own thinking and

prepare for answering the question, What mathemat-

ics did you learn today?

AA MMiinnii--GGaalllleerryy WWaallkkShow slide 13. In the shirt problem, the expectations

are to explore and deepen understanding of the

relationships among fractions, decimals, and percents.

To model a reflect-and-connect process, ask the small

groups of 2 or 3 to join up with a partner group, lay

out their solutions side by side, and discuss the

similarities and the differences. This is a process of

assessment for learning – for participants to see a

range of understandings about the math in the prob-

lem. No one should try to attach a level of perform-

ance. This is not assessment of learning. The learning

is just starting here.

In preparation for you to lead a discussion of all of

the mathematics represented in the participants’

responses, you need to select a specific set of

responses that show a range of understandings.

Wander around as the initial discussion of same and

different in solutions is occurring between groups of

learners. This process of reflection and connection will

involve your managing a mini-gallery walk (see “Look

Back – Reflect and Connect” in Session C). Ask the

participants who prepared the specific samples you

have chosen to post their responses on the classroom

walls. Some might have used tiles (concrete represen-

tations) and drawn the tiles on their papers, others

may have used a 100 chart, and others may have used

numerical calculations (abstract representations) as

they built their solutions. When this range of 4 to 5

solutions is on display, ask one person to point to

parts of the solution and explain the work. Pose

questions to this person after the presentation to

make all of the thinking explicit. This oral discussion

is invaluable for spreading understanding, but it must

be inclusive.

Slide 13


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3 1 231 321 23 1 32 3 2 22 32At this point, teachers in your session (and hopefully later students in the classroom) will

begin, upon hearing someone else’s strategy, to understand the mathematics that is under

study. Their ideas will bump up against the ideas of others and all learners will come away

with a more solid understanding. Misconceptions will be presented and clarified by and for the

whole group of students. The fact that they need to explain what they did might be what

consolidates their understanding. Oral language is critical to the building of understanding.

So, this form of inclusive instruction gives the responsibility of getting to precision to the

teacher and all students in the classroom. You cannot know what rule or process students have

internalized until they communicate – not parrot – it back to you.

Showing students a range of strategies and solutions empowers them to find many ways to

approach a problem. They begin to believe that math is about thinking and not just about

finding one expected answer.

The consolidation session is also a venue for expanding students’ mathematical language and

their use of mathematical symbols and notations.



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SSeessssiioonn CC –– CCoonncceeppttuuaall DDeevveellooppmmeenntt 1177

SSeessssiioonn CC ––CCoonncceeppttuuaall DDeevveellooppmmeennttDisplay the agenda for Session C (slide 14).

WWaarrmm UUpp –– AA KKWWLL CChhaarrtt –– KKnnooww,,WWoonnddeerr,, LLeeaarrnneedd The use of a KWL chart (slide 15) helps to activate

prior knowledge. Give each participant a copy of a

KWL chart (BLM2). When asking students to fill out

the Know column, the teacher should accept all

responses and not attempt to skew the responses by

accepting those that fit best with the concepts that

will be presented next. What students already know

about a topic serves to scaffold future learning.

KWL charts are often completed in the large group. In

this case, to give all participants an opportunity to

contribute, the charts will be completed in small

groups. Ask participants to answer only the Know and

Wonder columns. It is not always necessary to discuss

responses in the whole group. Do so only if you notice

major differences between what groups of participants

have written. It is not necessary for every group to

have the same responses. The activity is used to get

the thinking started. The only requirements for the

lesson are that the goal of understanding the relation-

ships between fractions and decimal tenths and hun-

dredths is made explicit and that participants learn

many ways to represent that concept.

QQuuiillttiinngg –– PPrroobblleemm ##22 The majority of the time spent on this problem will

be during the reflecting and connecting time. The

concepts and connections between fractions and

decimals and the math language and notation will

be made explicit during this time.

UUnnddeerrssttaanndd tthhee PPrroobblleemm

Show slide 16. Read the problem with the participants

and ask them to turn to an elbow partner and clarify

details in the problem that require their attention.

Slide 14

Slide 15

Slide 16


Ask 1 or 2 two pairs to share their insights with the

group. Be open to all statements and remember, we

all see and hear things differently from one another so

all comments are worthwhile.

The mathematics concepts of the relationships

between fractions and their decimal equivalents will

be provoked in the resolution of this problem. It first

requires students to show their understanding of

hundredths (0.56 expressed as a decimal), then of

tenths on a hundredths grid ( expressed as a

fraction of 100). The question, as posed, will require

teachers to develop a representation and show their

understanding of the relationship between decimals

and fractions. They are asked to use both a 10 x 10

grid and stacked number lines. Their understanding

of the relationships will show in their presentations of

solutions.

Draw participants’ attention to the purple box at

the bottom of some of the slides and make explicit

the instructional strategies being employed to differ-

entiate and deepen the learning.

MMaakkee aa PPllaann aanndd CCaarrrryy OOuutt tthhee PPllaann

Give each group some chart paper and markers and

ask them to record their solutions for sharing with

the large group. Make tiles and 100 charts (BLM3)

available on every table. Ask participants to use

manipulatives, drawings, pictures, words, and symbols

to represent their thinking.


AA GGaalllleerryy WWaallkkShow slide 17. Ask groups to post their solutions.

During Session B, your participants did a mini-form

of a gallery walk. They compared their solutions with

those of a second group and you chose about 4 or 5

solutions that the whole group discussed. During

Session C’s gallery walk, all solutions are posted.

After editing, only 3 or 4 will be discussed publicly.

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Slide 17

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321132Ask your participants to take a gallery walk, in groups of 2 or 3, to view all of the solutions

and discuss their understanding of each representation. They need to think very carefully

about what understandings the solutions show. In mathematics work, we too often identify

what is wrong, rather than focusing on what the solutions show about what students can do

and understand. If there is a part of the solution that the observers cannot make sense of,

they then pose a question for the developers and post a sticky note on the chart paper.

After about 10 min – 15 min, ask the groups to retrieve their initial solutions and answer any

questions that have arisen as noted by the sticky notes. Have them add details or improve

their solution, adapting their thinking according to the discourse and clarification of concepts

they have gained on the gallery walk. This extended discussion builds deeper understanding of

the math concepts and has students practise communicating to an audience.

As the facilitator, circulate while participants are editing their solutions. Select 3 or 4 groups

to discuss theirs publicly. In your selection, look for groups that have made either substantial

changes or additions to their solutions. Choose solutions that shed light on the concepts

identified as the goals of the lesson.

Emphasize that when students share their work, all benefit. Teachers must show appreciation

for a variety of diverse solutions and strategies rather than only evaluating accuracy and

efficiency. The “accuracy” of the answer to a problem is not the single goal of learning mathe-

matics. The mathematical thinking is what needs to endure after the students leave your class-

room. And, remember, it takes a long time and lots of experience to build precision into math-

ematics representation.

The connections students make to their own prior knowledge during the solving of problems

is what informs the teacher about the students’ understanding. Each of us has different prior

knowledge – each of us has our own lived experiences and we cannot anticipate that any two

people share exactly the same prior knowledge. As far as knowing mathematical procedures

or algorithms is concerned, a less efficient or less sophisticated method or strategy that a

student owns is more valuable than a more efficient one that belongs to someone else.

Understanding must reside with the user. Teaching cannot be about zeroing in on pre-

determined conclusions. It can’t be about the replication and perpetuation of a single possible

solution given by one student, in a text, or by a parent, nor can it be about one solution that

resides in a teacher’s head. Rather, mathematics instruction must be more about bringing the

ideas of all students together, making the mathematical thinking explicit and facilitating the

discussion that results in an expansion of knowledge, language, and notation.

Return to the groups’ KWL charts. Ask participants to answer the Learned column. After

the groups have had time to articulate their learning, share in the large group. Talk about

the language that emerged throughout the problem solving and discuss ideas about how this

language can be used effectively on a word wall to make the mathematical language more

explicit for students.

SSeessssiioonn CC –– CCoonncceeppttuuaall DDeevveellooppmmeenntt 1199


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3 1 231 321 23 1 32 3 2 22 32SSeessssiioonn DD ––AAlltteerrnnaattiivvee AAllggoorriitthhmmssWWaarrmm UUpp –– TThhee MMeeaanniinngg ooff RRaattiioo Introduce the agenda for this session (slide 18).

Students are often equipped with formulas and defini-

tions without a true understanding of what they mean

or where they come from. It is important that they be

given an opportunity to reflect on and analyse these

for the purpose of making sense of the mathematics.

Show slide 19. Ask participants to collect personal

definitions of the word “ratio” from 3 other partici-

pants who are not sitting at their table. When all

have returned to their tables, ask them to share their

definitions and collectively write, with examples, their

own group’s definition of “ratio”.

BBeesstt BBuuyy oonn JJuuiiccee –– PPrroobblleemm ##33Show slide 20. This problem was chosen because it

involves participants in activating prior knowledge

about concepts of proportional reasoning and encour-

ages them to represent their thinking using whatever

formal or informal knowledge they have about ratios.

The learning occurs when the math-talk community

engages in reasoning, calculating, proving, and argu-

ing about some of the representations and communi-

cation that relate to the problem solving required by

the problem.

UUnnddeerrssttaanndd tthhee PPrroobblleemm

Read the problem and ask participants to clarify

the details of the problem in groups of 3. Facilitate

conversation that makes public the mathematical

issues in the problem.

There are two pieces of information about prices: 24

boxes cost $27.60 and 18 boxes cost $19.80. A par-

ticipant may offer, “I think the fact that 24 and 18

have many factors in common will help me figure

out the best price.” We are asked to do more than

determine the best price – we must determine if the

best price has a unit price – the price of 1 box – of

less than $1.12.


Slide 18

Slide 19

Slide 20


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SSeessssiioonn DD –– AAlltteerrnnaattiivvee AAllggoorriitthhmmss 2211

Grade 5 students are expected to demonstrate an

understanding of simple multiplicative relationships

involving whole number rates using concrete materials

and drawings. Grade 6 expectations require students

to represent relationships using unit rates. This would

be a Grade 6 question; however, if whole numbers

were used, Grade 5 students could use it to begin

exploring the math in the problem.

Some people may ask, “How many boxes of juice do

the children need for the camping trip?” This may

be an interesting question, but it is not necessary

information for solving this particular question, which

is simply focused on unit rate.

MMaakkee aa PPllaann aanndd CCaarrrryy OOuutt tthhee PPllaann

Give each small group some chart paper and markers.

Ask them to record their solutions in at least two ways

and tell them they will be posted and shared with the

large group.


BBaannsshhoo –– OOrrggaanniizziinngg ttoo SSeeee aa RRaannggee ooff SSttuuddeenntt TThhiinnkkiinngg

Show slide 21. In order to make public the mathe-

matical thinking students used to solve a problem,

we need a way of organizing the work so everybody

can see the range of student thinking. Such an organi-

zation allows students to see their own thinking in the

context of the similar thinking of others. Students are

expected to follow and be able to describe all of the

work represented – not just their own. Students listen

to the explanation of the developers and restate their

solution in their own ways. Mathematical ways of

talking are modelled and practised – resulting in the

creation of a safe, math-talk community. Everybody

has a chance to learn more about the math used in

developing solutions to the problem and to clarify

their understanding of the concepts and/or proce-

dures. Through careful management of discourse,

the mathematics is made explicit.

Slide 21


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3 1 231 321 23 1 32 3 2 22Japanese educators call their process bansho. We will call this process of displaying and

discussing solutions bansho as well. Bansho engages the teacher in selecting the student work,

organizing it, and displaying it to make explicit the goals of the lesson task. For example, if a

goal were to show different addition algorithms, you would organize, from left to right, illus-

trations of the use of concrete materials, addition done by decomposing the numbers into tens

and ones and completed in several steps with regrouping, alternative addition algorithms, and

the standard algorithm.

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 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 con-

ceptual 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.

The matching and comparing conversations focus on the similarities and differences between

the displayed mathematics. The teacher makes the learning explicit by naming the language

and the mathematical concepts and procedures shown in the solutions. Students match their

solutions to the displayed ones and examine those of others to learn more about their own

thinking. When processes in two solutions match, the second solution is taped above the origi-

nal to make a bar graph-like display.

Use a bansho process to sort and classify participants’ solutions. Your goal is to present solu-

tions and then have participants sort and organize the solutions by attending to the mathe-

matical details used to solve the problem. The ideal display is organized to show clusters of

solutions from least to most mathematically sophisticated. Sorting solutions as you walk

around the room is ideal but very difficult to accomplish without practice. A way to start is to

sort solutions that are the same and different mathematically and post these in separate

columns. Then, with the help of participants, stand way back and name the mathematical

strategies that have been applied in the solutions in each column. At that point, your group

may want to rearrange the solutions to show a left to right progression of least to most mathe-

matically sophisticated.

Another way to model this process so teachers can replicate it in the classroom is as follows.

Ask one group whose solution is very concrete to hang their chart paper at the far left. Ask

the next group to compare their solution with the one already posted. If theirs is similar, they

should post it above or below the first (so they are making a bar graph-like display). If theirs

is different, they should post it beside the first. Lead a discussion about the solutions – ask

participants to explain what they understand about others’ solutions.



For example, in the three sample solutions below the mathematics that is applied is named

and underlined in the three paragraphs that follow the display.

The first sample solution shows a solid understanding of the concepts of ratio and propor-

tional reasoning and presents a full solution. The students have used mental math to take

half and one-third of numbers in a systematic way to reach the conclusion that Store 2 has

the better price and satisfies the condition of each box having to cost less than $1.12.

The second sample solution shows students using a calculator to determine unit price by

dividing the total price by the number of boxes. This is an efficient solution but shows little

about student understanding. This is partly because a conclusion is not drawn and the

“answer” is not supported by any explanation that demonstrates understanding. It is close to

completion and could be improved with some editing.

The third sample solution shows the use of a ratio table where the students are doing lots

of calculations and do determine the better price but not the price per box. Again, the

solution is very close to completion but needs some extra work to answer the question with

all its conditions met.

This presentation is NOT meant 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.

First Store:# boxes price

24 27.60

12 13.806 6.90

18 20.60

Second Store:# boxes price

18 20.70

So Store 2’s price is lower. 18 boxesthere only cost $19.80

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First Store:use halves and thirds:24 boxes for $27.6012 boxes for $13.806 boxes for $6.90 and2 boxes for $2.301 box for $1.15

Second Store: use halves and thirdsto calculate18 boxes for $19.809 boxes for $9.903 boxes for $3.301 box for $1.10

The price at the second store is betterat $1.10 per box and satisfiesMother’s instruction that it cost lessthan $1.12.

First Store:(with calculator)27.60 ÷ 24 = 1.15

Second Store:19.80 ÷ 18 = 1.1So the second store is cheaper.

FFiirrsstt SSaammppllee SSeeccoonndd SSaammppllee TThhiirrdd SSaammppllee


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The participants examine and discuss the solutions,

comparing their own to those of others. Bansho helps

problem solvers:

• see what they need to do and think about;

• organize their thinking;

• discover new ideas; and

• see connections between parts of the lesson, con-

cepts, solutions, notations, and language.

The calculations, notations, and use of tools (like a

ratio chart) are all important ideas used in the study

of proportional reasoning that need to be made explic-

it throughout the remainder of instruction in this unit

of study. The way each group used their mathematics

is made explicit and becomes part of the collective

discussion. All the math related to the problem

emerges and each student connects his or her prior

knowledge to some parts of the solutions and the dis-

cussion. Everyone benefits from the discourse on the

mathematics.

EEnnggaaggiinngg iinn RRiicchh PPrroobblleemmssTeaching and learning through problem solving is a

very powerful experience. However, it can only work if

the problems we ask our students to solve are rich

problems. Traditional “word problems” often did not

require our students to think critically and their solu-

tion did not generate new learning. It is important

that teachers reflect on the mathematics that is the

focus of the lesson before designing and assigning

problems.

Show slide 22. The teacher, who is acting as an archi-

tect of rich problems, needs to remember that rich

problems:

• present a context students can relate to and dis-

cuss;

• inherently contain the mathematics that the

teacher wants the students to learn;

• allow every student to engage in solving the prob-

lem using formal or informal strategies;Slide 22


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• have several entry points and are conducive to

extensions, allowing for differentiated instruction;

and

• require students to use high-level thinking skills.

PPrrooffeessssiioonnaall LLeeaarrnniinngg OOppppoorrttuunniittiieess Show slide 23. Discuss the many ways in which

Ontario teachers continue their professional develop-

ment. Ask, What are the next steps in your school? In

your board?

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Fullan, M., Hill, P., & Crévola, C. (2006). Breakthrough. Thousand Oaks, CA: Corwin Press.

Lyons, C.A., & Pinnell, G.S. (2001). Systems for change in literacy education: A guide to

professional development. Portsmouth, NH: Heinemann.

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

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RReessoouurrcceess ttoo IInnvveessttiiggaattee 2277

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www.curriculum.org.

Loewenberg Ball, D. (November 2005). Knowing Mathematics for Teaching (webcast),

available at www.curriculum.org.

Eworkshop.on.ca.

Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics, Grades 3–5.

Boston: Pearson Education.

Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics, Grades 5–8.

Boston: Pearson Education.

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2. For the big idea that you were assigned, read from the “Overview” section up to the

“Characteristics of Student Learning” section. Record five things that best reflect your

big idea.

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3. Read the “Characteristics of Student Learning” section for your big idea. Write down the

three characteristics that you deem most representative.

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4. Read the “Instructional Strategies” for your big idea. Write down the three strategies that

you deem most representative.

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Documents

1 Contents · Overview of The Literacy and Numeracy Secretariat Professional Learning Series ... and Numeracy Secretariat Professional Learning ... co-teaching, coaching, mentoring