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Knowledge of Children's Mathematics: A Foundation for

Classroom Discourse

Perspectives from Cognitively Guided Instruction.

Linda Levi

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Summary of Major Cognitively Guided Instruction (CGI) Research Thomas Carpenter, Megan Franke, Elizabeth Fennema,

Penelope Peterson, Linda Levi, Victoria Jacobs, Susan Empson and others.

• 1970s Research on Children’s Thinking• 1984-88: CGI Experimental Study• 1989-95: CGI Longitudinal Study• 1996-05: CGI/Algebra Development and

Research

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1970s: Research on Children’s Mathematical Thinking

• Thomas Carpenter and James Moser research the development of children’s strategies for addition and subtraction problems

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1984 - 1988: CGI Experimental Study

• First grade teachers– 20 participated in CGI professional development

(summer workshop)– 20 participated in general problem solving workshop

• Treatment and Control group were compared– Teachers’ knowledge– Teachers beliefs– Students’ achievement (problem solving and facts)– Classroom practice (amount of problem solving, type

of teacher talk, type of student talk…)

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1989-1995:CGI Longitudinal Study

• Kindergarten through third grade teachers and students in four schools

• All teachers attended CGI professional development

• The following were assessed over three years:– Teachers’ knowledge and beliefs– Students’ achievement and beliefs – Classroom practice

• A study of these teachers’ classroom practice, knowledge and beliefs was done five years after the workshop ended.

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1996-2005: CGI/Algebra Research

• 1996 – 2000 CGI Algebra Professional Development Program was developed by researchers in conjunction with expert CGI teachers.

• 2000 -05 Experimental Research – 90 Kindergarten – Sixth grade teachers attended

CGI/Algebra professional development– These teachers were compared with 90 teachers who

did not attend• Student Achievement• Teacher Knowledge

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Cognitively Guided InstructionSummary of Major Research Results

• CGI Students’ achievement on problem solving tasks is higher than non-CGI students’.

• CGI Students’ performance on computation and facts is not significantly different from that of non-CGI students.

• CGI Teachers have greater knowledge of their own students’ thinking than non-CGI teachers.

• CGI Teachers have greater knowledge of children’s mathematics than non-CGI teachers.

• CGI Classrooms involve a great deal of problem solving and student discussion.

• Most Teachers sustain their practice 5 years after PD ends.

• Some Teachers generated additional growth 5 years after PS ended.

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Outcomes of Classroom Discourse

Discourse

Students learnfrom students

Students’ reflect on their own

ideas

Teachers learn about

their students’ thinking

Teachers learn about

Children’s mathematics

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“I always knew it was important to listen to kids, but I didn’t know what to listen for.”

CGI teacher, 1989

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Foundation for Discourse:

• Teacher’s Knowledge of Children’s Mathematics

• Teacher’s Knowledge of his/her individual students’ thinking.

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Rachel’s Problems

• Rodney is having some kids over for jelly donuts. 7 donuts can fit on one plate. How many plates will Rodney need for 28 donuts?

• Karina had 20 cupcakes. She put them into 4 boxes so that there were the same number of cupcakes in each box. How many cupcakes did Karina put in each box?

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Multiplication, Measurement Division and Partitive Division Problems

• MultiplicationSusan has 8 boxes with 5 marbles in each bucket. How many marbles does Susan have?

• Measurement DivisionSusan has 40 marbles. She wants to put them into boxes with 5 marbles in each box. How many boxes would she need to hold all of her marbles?

• Partitive DivisionSusan has 40 marbles. She has 5 boxes to hold these marbles. How many marbles can she put in each box if she wants to put the same number of marbles in each box?

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Problems to classify:1. I have 21 cents to buy candies with. If each

gum drop costs 3 cents, how many gum drops can I buy?

2. Janelle has 21 beads. She wants to make 3 braids in her hair and put the same number of beads in each braid. How many beads can go in each braid?

3. Kevin earned 89 bonus points when playing his computer game. If it takes 7 bonus points to get an extra life, how many extra lives will he get?

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Multiplication Problems

1. There are 2 bags of soccer balls with 10 balls in each bag. There are also 4 extra balls. How many balls are there altogether?

2. Mia has 7 bags of beads. There are 10 beads in each bag. She also has 6 extra beads. How many beads does she have?

3. Ms. Keith has 6 packages of cookies. There are 10 cookies in each package. She also has 4 other cookies. How many cookies does she have?

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Measurement Division Problems

1. Matt has 36 pennies. He puts 10 pennies into each box. How many boxes can he fill with 10 pennies?

2. David has a rock collection. He has 54 rocks in his collection. He puts them into boxes with 10 rocks in each box. How many boxes does he use?

3. The second graders had 54 balloons for the school carnival. They put balloons into bunches of ten. How many bunches could they make?

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Strategies for Solving Multiplication and Measurement Division Problems with Tens

• Counting by Ones

• Counting by Tens

• Direct Place Value

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• Sam eats 1/10 of a pound of fudge a day. How many days would it take him to eat 2 pounds of fudge?

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• An animal at the zoo eats .1 of a pound of food each day. If the zookeeper has 65.4 pounds of food for this animal, how many days can she feed the animal before the food runs out?

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• How many tens are in 387? How many tenths are in 387?

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• In the early 1900’s, a farmer could pile up stones to construct .1 of a mile of fence a day. If a farmer worked for 35 days building a fence, how long would the fence be?

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• Sam builds .35 mile of fence every day. How long would his fence be after 32 days?