MATHEMATICS

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MATHEMATICS
as a Teachable Moment
P Meaning P Choice P Diversity P
Trust P Time
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Create Meaning

• Student Projects
• Student Written Problems and Solutions
• Sports / Pets / Cooking
• Date / Special Days / Season / Weather
• Place (Home / Community / School)
• Games

• Discussion in pairs, small groups
• and as a class

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Give Choices
Choices provide meaning through a sense of
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Value Diversity
Diversity should be treated as a positive factor
in the classroom.
We need to
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Create a Climate of Trust
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Ensure There is Adequate Time
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Review of Silent Mouthing
Use the silent mouthing technique
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Student Feedback
to give
to give
When students make errors give them hints,
suggest that they are close, acknowledge that
they are a step ahead or say, That is the answer
to a different question.
Slower processors and complex thinkers the time
they need to do the question.
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Review of Place Value
Place value should be taught at least once a week
but preferably a place value connection should be
made almost every day.
The connections to algebraic thinking should be
made (collecting like terms) as this will pay off
when doing operations with fractions and
algebraic expressions.
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Organization of the CURRICULUM
All four strands (Number Sense, Spatial
Sense, Probability and Data Sense and Pattern
and Relationship Sense) should be covered every
month (every week in Primary).
Problem solving often embeds three of the strands
depending on whether the problem has a focus on
spatial relationships or data relationships.
It is usually preferable to introduce a new
topic through a problem. The Japanese
teachers use this technique effectively.
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Making Meaning with the WEEKLY GRAPH
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Watch for the big ideas in the video.

• What teaching techniques are effective?
• What Mathematical concepts are covered?

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T
Intermediate
eaching
Students
NEW Strategies for OLD Ideas
Where do we find the time to teach this way?
If students are taught this way, how will they
do on the FSA tests?
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Multi-step Division and Decimal Fractions
Placement of the decimal in the quotient should
be done by asking, Where does it make sense to
put the decimal so that the answer makes sense?
The first few times multi-step division is taught
it should be done as a whole class. The errors
made should be used as opportunities to
investigate conceptual understanding.
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Process for Teaching
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If possible, do multi-step division on grid paper
(cm graph paper works well).
If grid paper is not available, use lined paper
turned sideways so that the lines become grids
for keeping the numerals in the correct position.
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.
1 9
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Many algorithms are culture specific time savers
that create accuracy.
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Many algorithms are culture specific time savers
that create accuracy.
In the middle ages we used a box or window method.
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Algorithms in the 21st Century
Algorithms should be developed through discussion
with learners because the purpose of teaching
algorithms is to develop understanding.
The focus should be on accuracy, then on
efficiency.
The most efficient algorithm today is always
based on todays technology.
The most efficient algorithm today is the
calculator or the computer but we do need to
understand the underlying concept or we dont
know if the answer makes sense.
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FRACTIONS are RICH in PATTERNS
Working at your table or in your group, assign
different members of the group to find the
decimal fraction for
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Memorable Fractions and Their Equivalent Buddies
Common Fractions
Simplest Form
Decimal Equivalent
Percentage Equivalent
For example (the first fraction illustrated in
the video)
On your Memorable Fractions sheet please write
in all the fractions studied in the
video. Include some of the equivalent fractions
for these.
There were three other fractions in the problem.
There was one fraction from the graph.
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Have the students draw a bar graph of the results.
In the end, the class will have developed
assessment criteria from a meaningful context by
having students notice what makes a graph a good
communication tool. Self-evaluation is often
the most effective.
Do not give students criteria for creating a good
graph. Discuss the results and focus on the fact
that graphs are supposed to give you a lot of
information at a glance. This means that the
graph should be neat, have a title and a legend
(if necessary).
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Have students discuss (write) what they know
about the class by analyzing the data (graph).
Can they think of any questions or
extensions? Use these for further research.
Use the think/pair/share method to create
discussion, then share as a group (valuing
diversity, creating trust and developing meaning
through choice).
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Collect data.
Decide which fractions (decimals and percents)
you wish to study. If you are worried about
coloring in the hundreds squares for a tricky
fraction, leave this part until the next day and
try it yourself. Enter the fractions on the
Memorable Fraction sheet. Draw a circle graph of
the data. Review the criteria.
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CIRCLE GRAPHS
Can be rich in CURRICULUM Connections
If the number of voters in the class is
12 15 18 20 24 30 36
or
Do the following
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PRINCIPLE of ONE
Find the ones.
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This principle was used in the video to make
equivalent fractions in particular
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PRINCIPLE of EQUIVALENCE
Throughout the video and on the Memorable
Fractions sheet, the students have been making
equivalent fractions and have learned that every
fraction can be expressed as an infinite number
of common fractions, exactly one decimal fraction
and one percentage fraction. It can also be
expressed as a ratio.
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PRINCIPLE of BALANCE
In the video one student noticed that when
equivalent fractions are generated, both the
numerator and denominator have to be multiplied
by the same number. This is also an example of
the Principle of One as
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Please solve the following
Dont forget to show your steps.
2x 5 31
2x 5 5 31 5
2x 26
x 13
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PRINCIPLE of ZERO
This step is necessary for equation solving and
is the only principle that is not generated in
doing the Weekly Graph.
It should have been generated much earlier in the
primary grades when doing the How Many Ways Can
You Make a Number activity during Calendar Time.
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How Many Ways?
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How Many Ways?
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P
How Many Different Ways Can You Make a Number?
Criteria
Criteria
Mark
Mark
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Where any sentence contains the Addition
operation Where any sentence contains
the Subtraction operation Where any sentence
contains the Multiplication operation Where any
sentence contains the Division operation Where
any sentence contains more than two terms (e.g. 2
x 3 5 10) Where any sentence contains more
than two operations (e.g. 2 x 3 4 10) Where
any sentence contains a number more than the goal
number (in this case 10) Where any sentence
contains a number substantially greater than the
goal number (in this case 50 or 100) Where any
group of sentences shows evidence of a
pattern (e.g. 1 9, 2 8, 3 7)
Where any sentence shows knowledge of the power
of zero (e.g. 6 6 10 10 or 10 0
10) Where any sentence uses doubling and halving
to generate new questions (e.g. 4 x 6 24, 2 x
12 24, 1 x 24 24) Where any sentence shows
knowledge of the power of one (e.g. 6 6 9
10 or 10 x 1 10) Where any sentence shows
knowledge of the commutative principle (e.g. 6
4 10 and 4 6 10) Where any sentence
shows knowledge of the number Note this
applies only for numbers greater than 10, such
as 24. In upper intermediate grades, award
marks for exponential notation also. (e.g. 20 4
24 and 2 x 10 4 24) Where any sentence
contains brackets, such as (3 2) (3 2)
(3 2) (3 2) 4 24 Where any sentence
contains exponents, square roots, factorials, or
fractions. Note there should be no expectation
of the demonstration of exponents, square roots
or factorials before grade six, but their use
should be acknowledged and rewarded where a
student chooses to employ such operations in
earlier grades.
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PRINCIPLES of EQUATION SOLVING
Principle of Zero
Principle of One
Principle of Equivalence
Principle of Balance
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to
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PRINCIPLE of ONE
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to
Equivalence is used in all facets of mathematics.
Balance is used in equation solving as well as
multiplication and division of rational
expressions.
The Principle of Zero is extensively in
simplifying rational expressions.
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Probability Makes Meaning
Probability can be introduced during the Weekly
Graph process.
Probability was introduced in the first session
when playing hangman which is an activity
students love to play.
Probability sense is an important skill we use in
everyday life.
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Ten-Frame Probability
In your group, have one person shuffle the red
deck (cards numbered 1 to 10) and a different
person shuffle the blue deck.
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Ten-Frame Probability
Turn over the two decks and find all the
combinations that equal 7.
Was the most common prediction a 7?
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T
Intermediate
eaching
Students
NEW Strategies for OLD Ideas
Which ILOs were covered in the activity?
What are some connected or follow-up activities
that you could use?
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to
Introducing the ten-frame cards this way allows
grade four to eight students to look at numbers
in a new way and learn to add visually without
counting.
The games shown in the video are called
Solitaire 10 and Concentration 10. Some
students in intermediate grades have difficulty
adding, and this is a new way to learn an old
concept of making tens.
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ALL THE
to
FACTS Sheet
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SUBTRACTION
to
FACTS
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All of the fractions generated in the video were
for what you would expect to get. This is
called the Expected Probability.
What we are really interested in is the
Experimental Probability.
The next step is to have each pair or students do
100 trials each and compare the Expected
Probability to the Experimental Probability. The
difference explains why people gamble.
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If each student in the class does 100 trials and
then the data is put on a spreadsheet, it is
clear that while some students will win if they
pick their favourite number, others will lose.
However, the experimental results for the whole
class will usually mirror the expected
probability.
Gambling then is a tax on the under-educated,
often the poor.
Government figures the odds, pays less than the
expected probability, and makes lots of money.
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Do the same activity with six-sided, ten-sided,
or twelve-sided dice.
Probability of getting a specific number or color
of SmartiesTM or other candies on Halloween or
Valentines Day.
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Take out the Decimal Fractions Project sheet.
Enter all the fractions and decimals collected so
far.
Find the prime factorization of the denominator
for each fraction (use fractions in their lowest
terms only.)
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STUDENT FRACTION DECIMAL INVESTIGATION SHEET
5 5
so 4 2 x 2

10 2 x 5
9 3 x 3
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Draw a line from 0 to 2.
0
2
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Sometimes it is important to have the number
lines drawn vertically so that the student makes
the connection to a thermometer.
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Then it is easy to introduce the idea of integers
and negative integers in a natural context.
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MAKING MEMORIES
In the last session the Norman story was
introduced as a way to create a metaphor (based
on scientific theory about the way we create
memories) about how Norman learned to add 8 7
and other numbers by breaking the number up and
using doubles.
Other students were asked if they did the
question in different ways and five responded.
How can this story be used in a classroom when
there is a student who yells out answers or
interrupts with what he or she considers
interesting comments?
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MAKING MEMORIES
What have you mylenized over the course of the
two videos?
Please take 2 minutes of silence to write out a
list.
When the two minutes are up, the facilitator
will ask you to share a strategy or concept you
learned that you feel will be useful.
This writing and then sharing helps
re-mylenize your learning.
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Create a Class List with some or all of the
following headings
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Create Criteria for each Heading
Example Creates a CIRCLE Graph from Raw Data.
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P
Data Heading Legend Neatness
Circle Graph
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P
27 students in the class told how many siblings
they have.
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Number of Siblings
P
Zero siblings
One sibling
Two siblings
Three siblings
P
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EVALUATING Decimals / Fractions / Percentage
Example for Multi-age Grade 6/7 (Grade 6 gets a 4
in the 3 category)
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Probability
EVALUATING
Example for Multi-age Grade 6/7 (Grade 6 gets a 4
in the 3 category)
Given a set of ordinary or special dice or a
spinner, can create a data set and interpret both
the expected and experimental probability.
Creates a data set and interprets but makes some
errors (not fundamental).
Gets a good start and creates a data set but not
both of expected and experimental.
Barely gets started if at all, needs a lot of
help.
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MULTI-AGE
Teaching
Classes
There is some research that shows that students
in multi-age classes demonstrate superior
learning.
This may result from the fact that the teacher
knows she has to individualize more because of
the spread of ability.
I have found it most effective when teaching a
multi-age class to teach to the top grade and
evaluate the lower grade at their own level.
In fact, this is true for all classes even when
they are streamed.
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EVALUATION
Work with someone at your table to create
criteria for at least one of the Intended
Learning Outcomes that you will be evaluating.
Keep in mind that the creation of criteria is
always a process of negotiation between you, the
curriculum and your context (class and school).
If you involve the students in creation of the
criteria, they often create criteria that has a
high standard of expectation for excellence.
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P
A
Good Problem Solvers
T
T
and
RELATIONS
R

• Get started
• Get unstuck
• Persevere
• Can solve problems in more than one way
• Self-correct

N
S
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FACILITATING Problem Solving
Use the think / pair / share method. Give
problems that are multi-step and take note of
student strategies. Record the strategies, slowly
building up a list. Discuss the efficacy and
efficiency of the various strategies that
students use.
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FACILITATING Problem Solving
Use model problems and have students write
problems using the frame as a model. Encourage
the use of mathematical vocabulary by giving
bonus marks. Encourage the use of mathematical
vocabulary by creating a word wall or a glossary
in student workbooks.
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STRATEGIES for Getting Unstuck

• Look for a pattern
• Make a model
• Draw a diagram
• Create a table, chart or list
• Use logic
• Create a simpler related problem
• Work backwards
• Seek help from a peer, the internet, a book

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EVALUATING Problem Solving
Example for Multi-age Grade 4/5 (grade 4 gets a 4
by achieving at the 3 level)
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IMPLEMENTATION
What obstacles do you perceive?
Take the time to make a plan for implementation.
What help do you need?
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Bibliography

• Fuson, Karen C., Kalchman, Mindy and Bransford,
• John D., Chapter 5, Mathematical Understanding
  
• An Introduction in How Students Learn
  Mathematics in the Classroom, Ed. Donovan,
  Susanne and Bransford, John D., National
  Academies Press, Washington, D.C. 2005
• Buschman, Larry E.E... Mythmatics Teaching
  Children Mathematics, Vol.12, No.3, Oct. 2005,
  p136 143
• Calkins, Trevor Mathematics as a Teachable
  Moment Grades K-3, Power of Ten Educational
  Consulting Ltd, Victoria, B.C. 2004
• Calkins, Trevor Mathematics as a Teachable
  Moment Grades 4 - 6, Power of Ten Educational
  Consulting Ltd, Victoria, B.C. 2004
• Silver, Edward A and Cai, Jinfa. Assessing
  Students Mathematical Problem Posing Teaching
  Children Mathematics, Vol.12, No.3, Oct. 2005,
  p129 -135

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Slide presentation created by Trevor
Calkins Power of Ten Educational Consulting 809
Kimberley Place Victoria, B. C. V8X
4R2 www.poweroften.ca
Power Point presentation constructed by Karen
Henderson P. O. Box 18 Shawnigan Lake, B. C. V0R
2W0
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