M ATHEMATICS

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M ATHEMATICS. as a T eachable M oment. P M eaning P C hoice P D iversity P T rust P T ime. 1. C reate M eaning . S tudent P rojects S tudent W ritten P roblems and S olutions S ports / P ets / C ooking D ate / S pecial D ays / Season / W eather

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## M ATHEMATICS

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MATHEMATICS

as a Teachable Moment

PMeaning

PChoice

PDiversity

PTrust

PTime

1

CreateMeaning
• Student Projects
• Student Written Problems andSolutions
• Sports /Pets /Cooking
• Date /Special Days / Season /Weather
• Place (Home / Community / School)
• Games
• Discussion in pairs, small groups

and as a class

2

Control

Commitment

Challenge

Projects,

Student Written Problems,

“How Many Ways” sheets,

and Discussion are all embedded with choice.

GiveChoices

Choices provide meaning through a sense of:

3

Encourage alternative strategies.

Give students choice in the order, methods, strategies and topics.

Make sure all students are involved in creating rules and sharing strategies.

ValueDiversity

Diversity should be treated as a positive factor in the classroom.

We need to:

4

Ask open-ended questions and value diverse strategies for solving problems.

Ask students to explain, discuss and show – especially when their answers are correct.

Value errors as opportunities to investigate conceptual understanding and create

new understandings.

Create aClimate of Trust

5

Teach the curriculum in an integrated manner

so that there are opportunities to review every major theme or

skill set.

Value accuracy

over speed.

Avoid:

Races

Contests

and

Strictly Timed Basic Fact Tests

Integrate the intended learning outcomes (ILOs) into major themes and evaluate over the

whole year.

6

Review ofSilent Mouthing

Use the “silent mouthing technique:

ä

ä

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.

7

Review ofPlace 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.

8

Organization of theCURRICULUM

All four strands

(ŸNumber Sense,

ŸSpatial Sense,

ŸProbability and Data Sense

and

ŸPattern 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.

9

Graphing is a tool for making meaning

if the data is collected from the students.

Eventually the

“Weekly Graph” becomes a day for teaching proportional thinking, decimals, fractions, percents, graphing, patterns and relations, and probability.

The “Weekly Graph” is intended to be student driven

by the fourth week

at the latest.

MakingMeaningwith theWEEKLY GRAPH

10

Watch for the “big ideas”

in the video.

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

11

How do we find the time to teach this way?

How many

ILOS were covered in the previous

video clip?

12

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?

13

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.

14

Process for Teaching

1 ÷ 9

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.

15

1

1

0

)

1

0

0

9

.

-0

1

0

- 9

1

1

0

1

9

9

=

1

1

0

1

1

1

1

.

1

- 9

)

1

9

0

0

0

0

0

0

0

0

.

Ÿ

0.1

0.1

=

or

.

1 ÷ 9

16

The multi-digit regrouping system we use for subtracting is based on the principle of equivalence and is done differently in parts of Europe.

Some Europeans use a system that depends on the principles of balance and equivalence.

Many algorithms are culture specific time savers that create accuracy.

17

The algorithm we use for multi-digit multiplication has changed considerably over the years.

In the fifties we moved the second product over one space which paralleled the way we multiplied using adding machines.

Now we add a zero for the second product.

Many algorithms are culture specific time savers that create accuracy.

In the middle ages we used a box or window method.

18

Algorithms in the 21stCentury

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 today’s technology.

The most efficient algorithm today is the calculator or the computer but we do need to understand the underlying concept or we don’t know if the answer makes sense.

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5

7

7

7

6

7

8

7

3

7

4

7

2

7

1

7

,

,

,

,

,

,

,

,

How many remainders did it take before you achieved a repeating decimal pattern?

Do you notice any patterns?

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:

20

WEEKONE

of the

Weekly Graph

40

100

1

4

2

8

4

16

3

12

5

20

=

=

=

=

=

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.

21

WEEKONE

of the

Weekly Graph

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).

22

WEEKONE

of the

Weekly Graph

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).

23

WEEKTWO

of the

Weekly Graph

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.

24

Have the students find the prime factorization of 360 and the prime factorization of the number who voted (e.g. 30).

Write the equation in fractional form:

360

30

2 x 2 x 2 x 3 x 3 x 5

=

2 x

3 x 5

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:

25

360

30

2 x 2 x 2 x 3 x 3 x

=

2 x

3 x 5

5

2 x 2 x 5

=

=

40

100

x

20

20

5

20

=

2

5

1

4

PRINCIPLE of ONE

Find the ones.

5

This principle was used in the video to make equivalent fractions – in particular:

26

PRINCIPLEof 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.

27

x

=

=

x

5

20

1

4

5

5

PRINCIPLEof 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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2x = 26

2 = 2

Don’t forget to show your steps.

2x + 5 = 31

2x + 5 – 5 = 31 – 5

2x = 26

x = 13

29

PRINCIPLEof 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.

30

P

How Many Different Ways Can You Make a Number?

Criteria

Criteria

Mark

Mark

1

1

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 numberNote: 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.

Where any sentence contains the

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)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

33

These four principles should be generated by and attributed to students. They are all you need to solve most equations and work with rational expressions throughout high school.

PRINCIPLESof

EQUATION SOLVING

PrincipleofZero

PrincipleofOne

PrincipleofEquivalence

PrincipleofBalance

34

ALGEBRAIC

THINKING

Connections

The principle of one and the principle of balance are used in rationalizing radical expressions.

The principles of one and balance can be used to generate an easy to remember algorithm for dividing fractions.

to

35

x

x

=

=

=

1

x

We refer to this as Invert and Multiply which has no other application in mathematics.

x

The Principle of One has many applications.

14

15

5

7

7

5

7

5

2

3

5

7

2

3

2

3

5

7

PRINCIPLE of ONE

36

ALGEBRAIC

THINKING

Connections

All four principles are used in equation solving.

to

Equivalenceis used in all facets of mathematics.

Balanceis used in equation solving as well as multiplication and division of rational expressions.

The Principle of Zerois extensively in simplifying rational expressions.

37

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.

38

Place the decks face down

side-by-side.

Predict the sum if you were to turn over the top two cards.

Collect the predictions from the whole group.

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.

39

How many did you get?

Check with other groups to see how many they got.

Discuss the reason for your answers until you come to a consensus.

What would the probability be for getting 6? 5?

It often is.

What is the probability of turning over two cards whose sum is 7?

Now watch the video.

Ten-Frame

Probability

Turn over the two decks and find all the combinations that equal 7.

Was the most common prediction a 7?

40

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?

41

FACTS

Connections

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.

42

Connections

“ALL THE

to

FACTS” Sheet

43

Visual tools are powerful.

After just this one lesson, which may take two or three days to complete, most students when asked to visualize how to make ’15’ with the cards will say, “Get a ten and a five”.

Connections

cover up or

take away nine.

When they say,

how they see the six.

They should say,

“One and Five”.

This tool works for subtracting 9, 8 and 5, which is almost half of the subtracting facts.

SUBTRACTION

to

FACTS

44

with

PROBABILITY

Connections

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.

45

with

TECHNOLOGY

Connections

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.

46

with

TECHNOLOGY

Connections

47

with

TECHNOLOGY

Probability of a new student in class wanting a specific kind of pizza, liking a certain pop star, or wearing a certain kind of clothing.

Connections

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.

48

DECIMAL FRACTIONS PROJECT

to the

Connections

Do you see any patterns?

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.)

49

5

20

1

4

STUDENTFRACTIONDECIMALINVESTIGATIONSHEET

5 = 5

so 4 = 2 x 2

=

10 = 2 x 5

9 = 3 x 3

50

NUMBER LINES

to

Connections

Place the fraction

on the line.

2

7

Place 83%

on the line.

Place 0.7

on the line.

Place 150%

on the line.

1

9

2

7

5

8

8

5

0.7

83%

150%

1

10

5

8

1

2

1

4

1

2

3

4

1

Place the fraction

on the line.

Place the fraction

on the line.

8

5

The important issue when connecting number lines to rational numbers is to create reference points

(tenths, quarters and halves).

Place the fraction

on the line.

1

9

Draw a line from 0 to 2.

0

2

1

51

NUMBER LINES

to

Connections

Sometimes it is important to have the number lines drawn vertically so that the student makes the connection to a thermometer.

0

Then it is easy to introduce the idea of integers and negative integers in a natural context.

52

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?

53

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

54

EVALUATING

WeeklyGraphActivities

Create a Class List with some or all of the following headings:

55

4

Can create a

circle graph using percentages and includes headings and legend. The graph is easy to interpret

(neat and complete).

Serious errors

and hardly gets started.

1

3

Can create a

circle graph using

a circle graph frame and includes headings and legend.

The graph is easy to interpret

(neat and complete).

2

Can create a

circle graph using

a circle graph frame but is missing headings or legend.

The graph is difficult to interpret

(may not be neat or complete).

Example:

Creates a CIRCLE Graph from Raw Data.

56

3

7

12

P

5

P

Data

Legend

Neatness

Circle Graph

P

P

27 students in the class told how many siblings they have.

P

P

Number of Siblings

P

Zero siblings

One sibling

Two siblings

Three siblings

P

57

Is fluent among the three forms of a rational number, both repeating and terminating common fractions, including most of halves, thirds, quarters, fifths, sixths, eighths, tenths, twentieths, fortieths, fiftieths, hundredths, and thousandths.

4

Is fluent for the three forms for halves, quarters, fifths, tenths and hundredths.

3

2

Can give the three forms for tenths and hundredths.

1

Makes occasional errors with tenths and hundredths.

EVALUATING

Decimals / Fractions /Percentage

(Grade 6 gets a 4 in the 3 category)

58

4

3

2

1

Probability

EVALUATING

(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 createsa data set but not both of expected and experimental.

Barely gets started if at all, needs a lot of help.

59

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.

61

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

62

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.

63

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.

64

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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3

1

Occasionally gets started and perseveres, uses at least one strategy for getting unstuck over the term, occasionally self-corrects, solves one-step problems.

Always gets started, perseveres, uses at least three different strategies for getting unstuck over the term, self-corrects, solves multi-step problems.

4

2

Almost never gets started, gives up easily (demonstrates learned helplessness), never self-corrects, occasionally solves simple one-step problems.

Usually gets started, usually perseveres, uses at least two different strategies for getting unstuck over the term, sometimes self-corrects, occasionally solves multi-step problems.

EVALUATING Problem Solving

(grade 4 gets a 4 by achieving at the 3 level)

66

IMPLEMENTATION

What obstacles

do you

perceive?

Take the time to make a plan for implementation.

What help do you need?

67

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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