Teaching Math to Young Children

Cognitively Guided Instruction

In a first grade class…

• Three children successfully solved addition and subtraction tasks for two digit numbers.

• Fourteen of the twenty-one children used their fingers to count all or count on as they solved such problems as 6 + 3 = ___ and 8 + 9 = ___.

• Three of the children needed cubes to solve such problems and counted all the cubes.

• One child had difficulty counting more than seven cubes accurately.

Number sense first

• The first priority is to develop early number sense in all children:

• Counting

• Comparing Sets• Composing and Decomposing Numbers

What operation is this?

• Steven had 4 toy cars. He wanted 9. How many more toy cars would Steven need to have 9 altogether?

– Show how a kindergarten or 1st grade student might solve this.

Modeling the Action

• Liz had 8 cookies. She ate 3 of them. How many cookies does Eliz have left?

• Liz has 3 marbles. How many more marbles does she need to buy to have 8 marbles?

• Liz has 3 fish. Tom has 8 fish. How many more fish does Tom have than Eliz?

Rachel’s Problems

• Try each of the problems. Think about how students might model the action in the problem.

• Discuss your solutions with a partner.

– As you watch the video, think about which problems seem harder for Rachel.

Basic assumptions about children’s learning of mathematics

• Very young children know how to solve math problems.

• Children develop mathematical understanding and acquire fluency with whole number computation by solving a variety of problems in any way that they choose.

• Children learn more advanced computational and problem solving strategies by watching their classmates solve problems.

Where is the unknown?1. Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then?

3. Janelle has 7 trolls in her collection. How many more does she have to buy to have 11 trolls?

2 TJ had 13 chocolate chip cookies. At lunch she ate 5 of those cookies. How many cookies did TJ have left?

4. Max had some money. He spent $9 on a video game. Now he has $7 left. How much money did Max have to start with?

Problem Types - Action

Result Change Start

Unknown Unknown Unknown

Join 5 + 2 = 5 + = 7 + 2 = 7

Separate 8 – 3 = 8 – = 5 – 3 = 5

No-action problems

• 6 boys and 4 girls were playing soccer. How many children were playing soccer?

• 10 children were playing soccer. 6 were boys and the rest were girls. How many girls were playing soccer?

• Mark has 3 mice. Joy has 7 mice. Joy has how many more mice than Mark?

Are some more difficult?

• There are 14 hats in the closet. 6 are red and the rest are green. How many green hats are in the closet?

• 14 birds were in a tree. 6 flew away. How many birds were left?

Try this once a week

• Present a problem to the whole class, let them work on it individually, then have several students present their approaches.

• Keep track of their solutions

• Use problems with numbers that are appropriate for your students.

Simple multiplication and division problems

• A bug has 6 legs. How many legs do 5 bugs have?

• Sam found 3 bird nests. Each nest had 5 eggs in it. How many eggs are in all the bird nests?

• You have 20 cookies. You want to share them equally among 4 friends. How many cookies does each friend get?

Solution strategies

• Children learn more advanced computational and problem solving strategies by watching their classmates solve problems.

• How many different strategies so you see in this video?

Solution strategies

• As you watch these children solve simple joining and separating problems, think about who in your class might be at each stage.

Solution Strategies

• Direct modeling of the action in the problem

• Counting strategies

• Derived facts

• Fluency

Solution Strategies

• Try “How Would Children Solve These Problems?” using each of the types of strategies.

• Then try “Finding a Problem for a Strategy” – the Jeopardy Game of CGI.

Solution Strategies

• Use problems like these often in class and record students’ progress through the strategies.

• The packet has so many problems that you shouldn’t run out, but if you do, they’re easy to make up.

• The teacher’s role is to guide student’s learning by knowing each child’s cognition.

Which Problems are Harder?

• The structure of a problem determines how difficult it is for children to solve and determines their initial solution strategies.

Fluency with “math facts”

• The use of manipulatives, counting and derived-fact strategies eventually grows into knowledge of most math facts.

• Explicit instruction on strategies can be helpful for building math facts that haven’t come naturally through problem solving, but that isn’t necessary until 2nd grade.

Number Talks

• A different approach for helping children see number relationships and move toward fluency involves work with dot cards and five- and ten-frame cards.

• Not to take the place of problem-solving, but to supplement it.

Try problems like these in your class.

Record students’ strategies.

You’ll be amazed to see the variety of ways of thinking.

Common Core – K

1. Add and subtract within 10 to solve word problems, using objects or drawings.

2. Decompose numbers less than or equal to 10 into pairs, using objects or drawings. Record with a drawing or equation.

3. Fluently add and subtract within 5.

4. Compose and decompose numbers from 11 to 19 into ten ones and some additional ones.

5. Count by tens to 100.

Common Core – 1st Grade

1. Add and subtract within 20 to solve word problems involving joining, separating and comparing, with unknowns in all positions, using objects, drawings and equations. Use strategies such as counting on, making ten, using doubles, etc.

2. Fluently add and subtract with 10.

3. Find the missing number in addition and subtraction equations.

Common Core – 1st Grade

4. Compare two two-digit numbers based on the number of tens and ones in each.

5. Add within 100 for special cases including adding a two-digit number and a multiple of ten, and a two-digit number and a one-digit number, composing tens as needed. Add or subtract multiples of ten from two-digit numbers without counting. Subtract a multiple of ten from a two-digit number. Use concrete models or drawings and strategies.

Base Ten Concepts

Using objects grouped by ten:

• There are 10 popsicle sticks in each of these 5 bundles, and 3 loose popsicle sticks. How many popsicle sticks are there all together?– Students’ strategies?

• The extension: The teacher puts out one more bundle of ten popsicle sticks and asks students “Now how many popsicle sticks are there all together?” What strategies would students use to answer this?