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 Eliz had 8 cookies. She ate 3 of them. How many cookies does Eliz have left? Eliz has 3 marbles. How many more marbles does she need to buy to have 8 marbles? Eliz 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.
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. • For older children, use some two-digit problems that don’t require regrouping and some that do • Or use simple multiplication and division problems. • Keep track of their solutions
Simple multiplication and division problems • An insect has 6 legs. How many legs do 4 insects have? • A bag of candy has 5 pieces in it. How many pieces of candy are in 7 bags? • You have 20 cookies. You want to share them equally among 4 friends. How many cookies does each friend get?
Solution Strategies • Direct modeling of the action in the problem • Counting strategies • Derived facts • Fluency 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.
Explicit strategy instruction • For addition and subtraction, children should focus on the anchor sums to 10 (6 + 4, 7 + 3, 8 + 2) using ten-frame cards to visualize these combinations. They build on these, as well as “doubles,” to derive other combinations. For example, 7 + 6 can be derived by knowing that 7 + 3 is 10, and 3 more is 13, or that 7 + 7 is 14, so the answer is one less.
Practice with “math facts” • Once these strategies are learned, students can often be induced into remembering combinations by playing competitive games that require quick recall of number combinations, such as “addition war” or “subtraction war” or games where pieces are moved around game boards by rolling dice and calling out sums or differences.
Multidigit Addition and Subtraction • There were 51 geese in the farmer’s field. 28 of the geese flew away. How many geese were left in the field? • There were 28 girls and 35 boys on the playground at recess. How many children were there on the playground at recess? • Misha has 34 dollars. How many dollars does she have to earn to have 47 dollars? • Strategies? Counting single units. Direct modeling with tens. Compensating.
Multiplication (Grouping) • Our class has 8 bags of doughnuts. There are 5 doughnuts in each bag. How many doughnuts do we have all together? • Students’ strategies? Try to generate several.
Partitive & Measurement Division • Charlie has 12 cookies and 3 friends. If each of his friends gets the same number of cookies, how many will each friend get? • Jim picked 54 flowers. He put them into bunches with 10 flowers in each bunch. How many bunches of flowers did Jim make? • Students’ 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?