Teaching Children Mathematics. Changing Our Thinking Understand math is more than knowing the rules Children form their own understanding of math based

Teaching Children Mathematics

Changing Our Thinking

• Understand math is more than knowing the rules

• Children form their own understanding of math based on personal experiences.

• All children can be successful at math

http://images.google.com/imgres?imgurl=http://www.csos.jhu.edu/P2000/images/tips_images/2007/math.jpg&imgrefurl=http://www.csos.jhu.edu/P2000/tips/math.htm&usg=__sbu--SHPRQpXrRnmZLLEvhmvB6E=&h=168&w=233&sz=26&hl=en&start=8&sig2=xvD2dDpkb_z8xhTs8FwKTw&itbs=1&tbnid=cWbUCnfD5lY1-M:&tbnh=79&tbnw=109&prev=/images?q=children+math&hl=en&gbv=2&tbs=isch:1&ei=R_p-S_HgOIfytQPZuL37Cw

Changing Our Thinking

The needs of students entering the workforce is changing.

Students need to be able to:• Problem solve• Reason• Communicate clearly in a variety of formats• Work with others• Apply their skills/knowledge to new situations

Developing Conceptual Understanding

More than a superficial understanding. Be able to transfer what they know to a another situation. Can recreate their knowledge.

• What is a half? Draw a picture to show what a half looks like?

• Squares activity

Conceptual Understanding

A

BC

D

E

Mental Math and Estimation

Types of Calculations Used in Everyday Life• 200 volunteers recorded all computation over a 24-hour period

•84.6% involved some form of mental math

• Only 11.1% involved a written component

• Almost 60% of all calculations required only an estimate rather than an exact answer

What mathematics do adults really do in everyday life?- Northcote, M., & McIntosh, M. (1999)

Changes in the Mathematics Program

http://www.youtube.com/watch?v=4W7XJ41JPSc

There is an increased emphasis on number sense in the new program.

This is the foundation for success in higher level mathematics.

http://www.youtube.com/watch?v=4W7XJ41JPSc

Teaching Basic Facts

• We teach children strategies for recalling their basic facts.

• They can use these strategies to re-create a number fact if they forget it

• They can use these same strategies for mental math

• Students should recognize which strategy to use depending on the numbers

• Timed fact practice is NOT recommended

Subitizing

• We start to develop number sense through the instant recognition of quantities.

• Students begin to know what quantities such as 6 really mean.

• They begin to see that numbers can be represented in different ways.

Subitizing

• What do you see?

Subitizing


Subitizing


Subitizing

• Five Frames and Ten Frames

• Subitizing provides a foundation for the strategies used in developing basic facts

Subitizing

• Five Frames and Ten Frames

VisualizationVisualization

• 7 + 5 = Visualizing five as three and two rather than as five and zero makes the mental math strategy of making ten and adding 2 more possible.

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Counting on

Doubles

Near Doubles

Names of 10

Nine strategies

Making 10

Skip count by 2

Number in Middle

Commutativity

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

Doubles

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

Doubles

Near Doubles

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

Doubles

Near Doubles

Names of 10

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

Doubles

Near Doubles

Names of 10

Nine strategies

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

Doubles

Near Doubles

Names of 10

Nine strategies

Making 10

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

Doubles

Near Doubles

Names of 10

Nine strategies

Making 10

Skip count by 2

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

Doubles

Near Doubles

Names of 10

Nine strategies

Making 10

Skip count by 2

Number in Middle

+ 1 2 3 4 5 6 7 8 9

1 1+1

1+2

1+3

1+4

1+5

1+6

1+7

1+8

1+9

2 2+1

2+2

2+3

2+4

2+5

2+6

2+7

2+8

2+9

3 3+1

3+2

3+3

3+4

3+5

3+6

3+7

3+8

3+9

4 4+1

4+2

4+3

4+4

4+5

4+6

4+7

4+8

4+9

5 5+1

5+2

5+3

5+4

5+5

5+6

5+7

5+8

5+9

6 6+1

6+2

6+3

6+4

6+5

6+6

6+7

6+8

6+9

7 7+1

7+2

7+3

7+4

7+5

7+6

7+7

7+8

7+9

8 8+1

8+2

8+3

8+4

8+5

8+6

8+7

8+8

8+9

9 9+1

9+2

9+3

9+4

9+5

9+6

9+7

9+8

9+9

Count On

Doubles

Near Doubles

Names of 10

Nine strategies

Making 10

Skip count by 2

Number in Middle

Commutativity

Understanding the Equal sign

7 + 8 = + 12 How do we overcome this?

Number Balances

• Balance-scale inquiries allow students to explore several critical mathematical concepts

Reasoning and Sense Making

We want students to be able to reason and make sense of the mathematics they learn.

Activity sheets

Mental MathMental Math

35 – 16 = 92 – 56 =

1001 – 692 =

Use mental math to find these differences.

As you solve each question, keep track of the processes you are using.

88 + 30 + 15 + 5

Mental Math• Calculating without a paper and pencil

What is 6 X 7?

• (3X7) X 2• (6X6) + 6• (6X5) + 12

Mental Math• Calculating without a paper and pencil

What is 4 X 15?2 x 30 = 60 (halve and double)

What is 18 X 8?20 X 8 – (2 X 8)160 – 16 = 144160 -10 – 6150 – 6 = 144

Division

126 ÷ 3 = (120 ÷ 3) + (6 ÷3)40 + 2 = 42

Halving and Doubling• This strategy is taught in grades 2 -6.12 x 16 = 192 ; 6 x 32 = ?

Array for 12 x 16 becomes array 6 x 32 but area remains the same. So, 6 x 32 = 192

Apply this to finding 3 ½ x 14 = ?

Halving and DoublingApply this to finding

3 ½ x 14 = ?Double x half

7 x 7

Personal Strategies

Do not expect your child to do their calculations and problem solving the same way you did.

They can use any method as long as it is efficient, works all the time and the student can explain it.

Number Line

70 72 100

-30

+2

Another way

28 + 44

20 + 40 = 608 + 4 = 12

72

28 + 44

60 + 12

20 + 8 40 + 4

10 + 2

70 + 2 = 72

Personal Strategies

“When I tried to explain a strategy that all children should use, I felt like I lost the class. When I ask the children to become actively involved, they are excited to explore, to create and modify strategies, and to learn. They have confidence in their own thinking abilities because they build from their own ideas.

“Show and Tell” Linda Dacey and Rebeka Eston

http://images.google.com/imgres?imgurl=http://it.pinellas.k12.fl.us/schools/lealman-es/images/4400870C9C164F7BA273F18BB5C2D9E5.jpg&imgrefurl=http://www.lealman-es.pinellas.k12.fl.us/everydaymath.html&usg=__JtfnKKRCV9Cmb38_THorwOGDUC8=&h=960&w=1280&sz=574&hl=en&start=29&sig2=mmUjY3hzNrqDjAscYgfldw&um=1&itbs=1&tbnid=eW21xUUgoJcoHM:&tbnh=113&tbnw=150&prev=/images?q=children+mathematics&ndsp=20&hl=en&rls=com.microsoft:en-us:IE-SearchBox&rlz=1I7ADSA_en&sa=N&start=20&um=1&ei=rg8oS4_lGZSKtQOq4_ylDA

Problem Solving

In my farmyard I have some chickens and some pigs.

Altogether I can count 25 heads and 78 legs.How many pigs do I have?

Solve it any way you want to. Be ready to share your strategy.

Communicating

Explain your reasoning.How do you know? Convince me that it works all the time.Demonstrate with manipulatives.Use concrete materials, pictures and symbols.

How Can You Support Your Child?

• Discuss how math is used in everyday life• Play games and puzzles that deal with logic,

reasoning, estimation, etc.

http://images.google.com/imgres?imgurl=http://1.bp.blogspot.com/_whk0taV-B5o/SWEMAfU4PyI/AAAAAAAAABU/3D9nc2FD5xg/s320/cribbage15.jpg&imgrefurl=http://thefortuneofe.blogspot.com/2009/01/cribbage-club.html&usg=__8_koeXltRblT-bqwCxtO08C2EUk=&h=225&w=315&sz=23&hl=en&start=3&sig2=G9T7V4uLbHLC17qR4OgEHA&itbs=1&tbnid=d-jXiiFqDMOMyM:&tbnh=84&tbnw=117&prev=/images?q=cribbage&hl=en&gbv=2&tbs=isch:1&ei=5ft-S_aNCYzoswO6mcz8Cw

http://images.google.com/imgres?imgurl=http://images.boardgamegeek.com/images/pic158681_md.jpg&imgrefurl=http://www.boardgamegeek.com/image/158681/othello&usg=__tE35Y3afUAJTpD1q-yFBPaa_Xb4=&h=375&w=500&sz=70&hl=en&start=1&sig2=cp-kU9n70xqM0r0dEIv8cA&itbs=1&tbnid=lFaFn8Zk1YUDKM:&tbnh=98&tbnw=130&prev=/images?q=othello+game&hl=en&gbv=2&tbs=isch:1&ei=Pvx-S63kGIqmtgPQ6_H8Cw

http://images.google.com/imgres?imgurl=http://www.mije.org/files/foldername/sudoku_bowman.png&imgrefurl=http://www.mije.org/bobbibowman/sudoku&usg=__rSOTzAOMSc1q8KM-YBlG_fAWQto=&h=369&w=369&sz=5&hl=en&start=8&sig2=u_1_7XqzsouhY6LwJypWCg&itbs=1&tbnid=aDZKoUBRNHZXsM:&tbnh=122&tbnw=122&prev=/images?q=Sudoko&hl=en&gbv=2&tbs=isch:1&ei=svx-S4TIFZuOtgOw7OD8Cw


Ask questions when helping with homework such as:

• What do you know that might help you?• Can you solve it another way?• How did you get that answer?• Show me what it looks like?• Teach me what you learned about ….


• Allow you child to use strategies that make sense to them. Do not show them the “right way”.

• Encourage your child to explain their thinking• Use correct math vocabulary

http://images.google.com/imgres?imgurl=http://www.mainstaymath.com/images/dictionary.jpg&imgrefurl=http://www.mainstaymath.com/&usg=__ixYkmIRjpGo0edFr_87pj2pWox8=&h=400&w=283&sz=44&hl=en&start=13&sig2=y-X6ZBcDCe9Wh6u54ZwZaA&itbs=1&tbnid=C024o5YAMZcVeM:&tbnh=124&tbnw=88&prev=/images?q=math+vocabulary&hl=en&gbv=2&tbs=isch:1&ei=7_x-S7-AD4fSswOGudX8Cw


• Don’t underestimate your own math capabilities

• Don’t say, “I was never good at math.”• Let your child know he/she can be successful

in math

Documents

Teaching Children Mathematics. Changing Our Thinking Understand math is more than knowing the rules Children form their own understanding of math based