1 contents · overview of the literacy and numeracy secretariat professional learning series ......
TRANSCRIPT
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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
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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,
88 FFaacciilliittaattoorr’’ss HHaannddbbooookk
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
123
123
1023
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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
80100
12
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45
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1122 FFaacciilliittaattoorr’’ss HHaannddbbooookk
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
34
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321132price that they can use to calculate the greatest
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.
SSeessssiioonn BB –– MMooddeelllliinngg aanndd RReepprreesseennttiinngg 1133
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
1144 FFaacciilliittaattoorr’’ss HHaannddbbooookk
Slide 12
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SSeessssiioonn BB –– MMooddeelllliinngg aanndd RReepprreesseennttiinngg 1155
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
12-5516 Fractions 2/12/07 11:08 AM Page 17
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.
LLooookk BBaacckk –– RReefflleecctt aanndd CCoonnnneecctt
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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1188 FFaacciilliittaattoorr’’ss HHaannddbbooookk
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.
2200 FFaacciilliittaattoorr’’ss HHaannddbbooookk
Slide 18
Slide 19
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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.
LLooookk BBaacckk –– RReefflleecctt aanndd CCoonnnneecctt
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.
2222 FFaacciilliittaattoorr’’ss HHaannddbbooookk
12-5516 Fractions 2/12/07 11:09 AM Page 22
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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SSeessssiioonn DD –– AAlltteerrnnaattiivvee AAllggoorriitthhmmss 2233312
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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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2244 FFaacciilliittaattoorr’’ss HHaannddbbooookk
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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32
21 13 3232 1
1311 2 31
3211
32 13 3 21 12 21 13
321132
SSeessssiioonn DD –– AAlltteerrnnaattiivvee AAllggoorriitthhmmss 2255
• 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?
312
32
Slide 23
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3
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311 23
1 2 312
23 3 1
132 1
232
32
3 1 231 321 23 1 32 3 2 22
2266 FFaacciilliittaattoorr’’ss HHaannddbbooookk
RReeffeerreenncceessFosnot, C., & Dolk, M. (2002). Young mathematicians at work – Constructing fractions,
decimals, and percents. Portsmouth, NH: Heinemann.
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
Panel on Mathematics in Grades 4–6 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. (2006a). Number sense and numeration, Grades 4 to 6: Volume 1: The
big ideas. Toronto: Queen’s Printer for Ontario.
Ministry of Education. (2006b). Number sense and numeration, Grades 4 to 6: Volume 5,
Fractions. Toronto: Queen’s Printer for Ontario.
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1311 2 31
3211
32 13 3 21 12 21 13
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RReessoouurrcceess ttoo IInnvveessttiiggaattee 2277
RReessoouurrcceess ttoo IInnvveessttiiggaatteeCoaching Institute for Literacy and Numeracy Leaders, Video on Demand, available at
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.
312
3212-5516 Fractions 2/12/07 11:09 AM Page 27
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32
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311 23
1 2 312
23 3 1
132 1
232
32
3 1 231 321 23 1 32 3 2 22
2288 FFaacciilliittaattoorr’’ss HHaannddbbooookk
BBLLMM11SSccaavveennggeerr HHuunntt1. What are the five big ideas in Number Sense and Numeration, Grades 4 to 6?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
3. Read the “Characteristics of Student Learning” section for your big idea. Write down the
three characteristics that you deem most representative.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
4. Read the “Instructional Strategies” for your big idea. Write down the three strategies that
you deem most representative.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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32
31 2
32
21 13 3232 1
1311 2 31
3211
32 13 3 21 12 21 13
321132
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12-5516 Fractions 2/12/07 11:09 AM Page 29
123 12
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232
32
3 1 231 321 23 1 32 3 2 22
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