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

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

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.

Conceptual Understanding B C 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) Give example: buying a TV $540.43 1. long division 2. calculator 3 decomposing numbers

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. 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 What do you see?

Subitizing What do you see?

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

Subitizing Five Frames and Ten Frames

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.

+ 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

+ 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

+ 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

+ 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

+ 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

+ 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

+ 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

+ 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

+ 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

+ 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

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 Activities:

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

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 = 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 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 = 144 160 -10 – 6 150 – 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 Doubling 3 ½ x 14 = ? Apply this to finding Double x half

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 -30 +2 70 72 100

Another way 28 + 44 20 + 40 = 60 8 + 4 = 12 72

28 + 44 60 + 12 20 + 8 40 + 4 splitting 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

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

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.

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 ….

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

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.