1. Teaching Children Mathematics

2. 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
Procedural knowledge vs. conceptual knowledge ex. 1/3 vs 1/2

3. 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
Procedural knowledge vs. conceptual knowledge ex. 1/3 vs 1/2

4. 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
Identify misconceptions.

5. Conceptual UnderstandingB C D E

6. 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)
Give example: buying a TV $ long division 2. calculator 3 decomposing numbers

7. Changes in the Mathematics Program
There is an increased emphasis on number sense
in the new program.
This is the foundation for success in higher level
mathematics.

8. 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

9. 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.

10. Subitizing
What do you see?

11. Subitizing
What do you see?

12. Subitizing
What do you see?

13. Subitizing Five Frames and Ten Frames
Subitizing provides a foundation for the strategies used in developing basic facts

14. Subitizing
Five Frames and Ten Frames

15. Visualization
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.

16. + 1 2 3 4 5 6 7 8 9 Counting on Doubles Near Doubles Names of 10
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+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

17. + 1 2 3 4 5 6 7 8 9
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+9
9+1
9+2
9+3
9+4
9+5
9+6
9+7
9+8
9+9
Count On 17

18. + 1 2 3 4 5 6 7 8 9 Doubles Count On 1+1 1+2 1+3 1+4 1+5 1+6 1+7 1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+9
9+1
9+2
9+3
9+4
9+5
9+6
9+7
9+8
9+9
Count On
Doubles 18

19. + 1 2 3 4 5 6 7 8 9 Doubles Near Doubles Count On 1+1 1+2 1+3 1+4 1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+9
9+1
9+2
9+3
9+4
9+5
9+6
9+7
9+8
9+9
Count On
Doubles
Near Doubles 19

20. + 1 2 3 4 5 6 7 8 9 Doubles Near Doubles Names of 10 Count On 1+1 1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+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 20

21. + 1 2 3 4 5 6 7 8 9 Doubles Near Doubles Names of 10 Nine strategies
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+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 21

22. + 1 2 3 4 5 6 7 8 9 Doubles Near Doubles Names of 10 Nine strategies
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+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 22

23. + 1 2 3 4 5 6 7 8 9 Doubles Near Doubles Names of 10 Nine strategies
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+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 23

24. + 1 2 3 4 5 6 7 8 9 Doubles Near Doubles Names of 10 Nine strategies
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+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 24

25. + 1 2 3 4 5 6 7 8 9 Doubles Near Doubles Names of 10 Nine strategies
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
3+1
3+2
3+3
3+4
3+5
3+6
3+7
3+8
3+9
4+1
4+2
4+3
4+4
4+5
4+6
4+7
4+8
4+9
5+1
5+2
5+3
5+4
5+5
5+6
5+7
5+8
5+9
6+1
6+2
6+3
6+4
6+5
6+6
6+7
6+8
6+9
7+1
7+2
7+3
7+4
7+5
7+6
7+7
7+8
7+9
8+1
8+2
8+3
8+4
8+5
8+6
8+7
8+8
8+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 25

26. Understanding the Equal sign
7 + 8 = + 12 How do we overcome this?

27. Number Balances
Balance-scale inquiries allow students to explore several critical mathematical concepts
Activities:

28. Reasoning and Sense Making
We want students to be able to reason and make sense of the mathematics they learn. Activity sheets

29. Mental Math
Use mental math to find these differences.
As you solve each question, keep track of the processes you are using.
35 – 16 =
92 – 56 =
1001 – 692 =

30. Mental Math Calculating without a paper and pencil What is 6 X 7?
(3X7) X 2
(6X6) + 6
(6X5) + 12

31. Mental Math What is 4 X 15? What is 18 X 8?
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 = 144
– 6
150 – 6 = 144

32. Division
126 ÷ 3 =
(120 ÷ 3) + (6 ÷3)
= 42

33. 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 = ?

34. Halving and Doubling 3 ½ x 14 = ? Apply this to finding Double x half

35. 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.

36. Number Line
-30 +2 70 72
100

37. Another way 28
+ 44
= 60
8 + 4 = 72

38.
splitting
10 + 2
= 72

39. Personal Strategies

40. “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

41. 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.

44. 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.
1/2
Fraction circle, picture and the symbol ½.

46. 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.

47. How Can You Support Your Child?
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 ….

48. How Can You Support Your Child?
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

49. How Can You Support Your Child?
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
AISI consultants attached to all schools. Schools have lead teachers. Feel free to contact.