Knowledge of Children's Mathematics: A Foundation for Classroom Discourse

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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 Childrens 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 Childrens Mathematical
Thinking

• Thomas Carpenter and James Moser research the
  development of childrens 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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1989-1995CGI 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 childrens
  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
Teachers learn about Childrens mathematics
Students learn from students
Students reflect on their own ideas
Teachers learn about their students thinking
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• I always knew it was important to listen to
  kids, but I didnt know what to listen for.
• CGI teacher, 1989

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Foundation for Discourse

• Teachers Knowledge of Childrens Mathematics
• Teachers Knowledge of his/her individual
  students thinking.

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Rachels 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

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

• I have 21 cents to buy candies with. If each gum
  drop costs 3 cents, how many gum drops can I buy?
• 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?
• 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

• 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?
• 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?
• 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

• Matt has 36 pennies. He puts 10 pennies into
  each box. How many boxes can he fill with 10
  pennies?
• 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?
• 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 1900s, 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?