Why Explore
Children’s
Mathematical
Thinking?
Vicki Jacobs
San Diego State University
Dinah Brown
San Diego State University &
Oceanside Unified School District
Why Explore Children’s
Mathematical Thinking?
Benefits for the Teacher
 Gain information
 Allow differentiation
 Make math more fun to teach
Benefits for the Child Solving
Benefits for Other Children
Teacher’s Role in Exploring
Children’s Thinking
Solve and Share….
19 children are taking a minibus to the zoo. They will have
to sit either 2 or 3 to a seat.
The bus has 7 seats. How
many children will have to sit 3
to a seat, and how many can
sit 2 to a seat?
What did children do?
19 children are taking a mini-bus to
the zoo. They will have to sit
either 2 or 3 to a seat. The bus has
7 seats. How many children will
have to sit 3 to a seat, and how
many can sit 2 to a seat?
What % had a correct answer?
What did children do?
19 children are taking a mini-bus to
the zoo. They will have to sit
either 2 or 3 to a seat. The bus has
7 seats. How many children will
have to sit 3 to a seat, and how
many can sit 2 to a seat?
What % had a correct answer?
Kindergarten: 51%
What did children do?
19 children are taking a mini-bus to
the zoo. They will have to sit
either 2 or 3 to a seat. The bus has
7 seats. How many children will
have to sit 3 to a seat, and how
many can sit 2 to a seat?
What % had a correct answer?
Kindergarten: 51%
First grade: 33%
What did children do?
19 children are taking a mini-bus to
the zoo. They will have to sit
either 2 or 3 to a seat. The bus has
7 seats. How many children will
have to sit 3 to a seat, and how
many can sit 2 to a seat?
What % had a correct answer?
Kindergarten: 51%
First grade: 33%
Second grade: 26%
Why Explore Children’s
Mathematical Thinking?
Benefits for the Teacher
 Gain information
 Allow differentiation
 Make math more fun to teach
Benefits for the Child Solving
 Encourage sense-making
 Validate child’s thinking
 Consolidate child’s reasoning
Allen’s problems
Grade 1


21 people are going to the zoo. There
are 3 cars to take the people to the
zoo. How many people will go in each
car if the same number of people go in
each car?
I have 21 cents to buy gum drops with.
If each gum drop costs 3 cents, how
many gum drops can I buy?
Solve this problem as
Allen would….
Grade 3 Textbook Problem
An astrophysicist takes 12
photographs of 3 regions of
space. He takes the same
number of photographs for
each region. How many
photographs of each region
does he take?
Textbook Goal:
Show Division as
Repeated Subtraction
An astrophysicist takes 12 photographs
of 3 regions of space. He takes the
same number of photographs for each
region. How many photographs of
each region does he take?
Grade 3 Textbook Lesson
You can use repeated subtraction.
Keep subtracting until there is nothing left.
Then count how many times you subtracted.
12 – 3 = 9 9 – 3 = 6 6 – 3 = 3 3 – 3 = 0
You subtracted 4 times.
So 12 : 3 = 4.
He took 4 photographs of each region.
Why Explore Children’s
Mathematical Thinking?
Benefits for the Teacher

Gain information

Allow differentiation

Make math more fun to teach
Benefits for the Child Solving

Encourage sense-making

Validate child’s thinking

Consolidate child’s reasoning
Benefits for Other Children
 Expose children to new strategies
 Value multiple strategies
 Make math more fun to learn
Kindergarten Class
Mr. Meyer found 3 bird nests.
Each nest had 5 eggs in it.
How many eggs is that?
Mr. Meyer found 3 bird nests.
Each nest had 5 eggs in it.
How many eggs is that?
Jeffrey:
I know a easier way for Shawn
Hart that could have done it.
Another child:
I do too.
Jeffrey:
He just could have gone
5—10—15.
****************************************
=>Ask students to try a second
strategy.
=>Ask students to try on another
child’s strategy.
Mr. Meyer found 3 bird nests.
Each nest had 5 eggs in it.
How many eggs is that?
Shawn: Drew 3 circles with 5 dots
 What did you do?
 What did you put—5 what?
 How did you get 15?
 How did you count? Can you show me
how you counted?
Jeffrey: Knew 3 groups of 5 is 15
 How do you know that? Is there a
slideshow in your head?
Melissa: Drew nests, counted 5–10–15, &
wrote 3 X 5 = 15
 How did you write the number sentence?
 Can you read it for me?
Brianna: Counts from 10 on fingers
 What if there were 5 more?
 If you have 15, what’s 5 more?
Mr. Meyer found 3 bird nests.
Each nest had 5 eggs in it.
How many eggs is that?
Shawn: Drew 3 circles with 5 dots
 What did you do?
 What did you put—5 what?
 How did you get 15?
 How did you count? Can you show me
how you counted?
Jeffrey: Knew 3 groups of 5 is 15
 How do you know that? Is there a
slideshow in your head?
Melissa: Drew nests, counted 5–10–15, &
wrote 3 X 5 = 15
 How did you write the number sentence?
 Can you read it for me?
Brianna: Counts from 10 on fingers
 What if there were 5 more?
 If you have 15, what’s 5 more?
Mr. Meyer found 3 bird nests.
Each nest had 5 eggs in it.
How many eggs is that?
Shawn: Drew 3 circles with 5 dots
 What did you do?
 What did you put—5 what?
 How did you get 15?
 How did you count? Can you show me
how you counted?
Jeffrey: Knew 3 groups of 5 is 15
 How do you know that? Is there a
slideshow in your head?
Melissa: Drew nests, counted 5–10–15, &
wrote 3 X 5 = 15
 How did you write the number sentence?
 Can you read it for me?
Brianna: Counts from 10 on fingers
 What if there were 5 more?
 If you have 15, what’s 5 more?
Mr. Meyer found 3 bird nests.
Each nest had 5 eggs in it.
How many eggs is that?
Shawn: Drew 3 circles with 5 dots
 What did you do?
 What did you put—5 what?
 How did you get 15?
 How did you count? Can you show me
how you counted?
Jeffrey: Knew 3 groups of 5 is 15
 How do you know that? Is there a
slideshow in your head?
Melissa: Drew nests, counted 5–10–15, &
wrote 3 X 5 = 15
 How did you write the number sentence?
 Can you read it for me?
Brianna: Counts from 10 on fingers
 What if there were 5 more?
 If you have 15, what’s 5 more?
What is the Teacher’s
Role in Exploring
Children’s Thinking?


Notice and ask about details
of the child’s strategy
Link strategy to problem
context
What is the Teacher’s
Role in Exploring
Children’s Thinking?



Notice and ask about details of
the child’s strategy
Link strategy to problem context
Ask questions after right &
wrong answers
Javier’s Problem
Grade 5
How many eggs would you
have if you had 6 dozen?
Javier’s Problem
Grade 5
How many eggs would you
have if you had 6 dozen?
Try Javier’s strategy…
5 x 26
5 x 26
10 x 26 = 260
Half of 260 is 130 so
5 x 26 = 130
What is the Teacher’s
Role in Exploring
Children’s Thinking?




Notice and ask about details of
the child’s strategy
Link strategy to problem context
Ask questions after right &
wrong answers
Look for what the child does
understand
Carla’s Problem
Grade 2
Evelyn is reading a book on Ruby
Bridges for her report. She has
been reading for 6 days. Each
day she reads 263 pages. How
many pages has Evelyn read?
Discuss…



What does Carla understand?
What does Carla not understand?
What question(s) would you ask
Carla?
What does Carla
understand?
Multiplication is repeated addition
 Multiplication problems can be
changed into smaller problems
using place value
6 x 263 = (6x200) + (6x60) + (6x3)
 Numbers can be decomposed
60 = 50 +10 so
6 x 60 = (6x50) + (6x10)
 Facility with adding numbers

What does Carla not
understand?

Skip counting over 1000

Commutativity of multiplication
6x3=?
3+3=6
3+3=6
3+3=6
6 + 6 = 12
6 + 12 = 18
What questions would
you ask Carla?

How did you solve this problem?

Where did you start?



How did you know to use
multiplication?
Why did you decide to break up the
60 into 50 + 10?
Why did you decide to break up the
problem into
(6 x 200) + (6 x 60) + (6 x 3)
What is the Teacher’s
Role in Exploring
Children’s Thinking?





Notice and ask about details of
the child’s strategy
Link strategy to problem context
Ask questions after right &
wrong answers
Look for what the child does
understand
Consider what question the child
might be answering correctly
Felicia’s Problem
Grade 4
We have 5 3/4 lbs. of jelly beans
and we’re going to put the jelly
beans in bags so there is a 1/2 lb.
in every bag. How many bags are
you going to need?
Felicia’s Problem
Grade 4
We have 5 3/4 lbs. of jelly beans
and we’re going to put the jelly
beans in bags so there is a 1/2 lb.
in every bag. How many bags are
you going to need?
Felicia’s initial response: 6 bags
Interviewer:
Would that be for a 1/2 lb. in
each bag or a full pound in each
bag?
What is the Teacher’s
Role in Exploring
Children’s Thinking?






Notice and ask about details of
the child’s strategy
Link strategy to problem context
Ask questions after right &
wrong answers
Look for what the child does
understand
Consider what question the child
might be answering correctly
Consider leaving a wrong answer
Gretchen’s Problem
Grade 2
70
– 23
What is the Teacher’s
Role in Exploring
Children’s Thinking?







Notice and ask about details of
the child’s strategy
Link strategy to problem context
Ask questions after right &
wrong answers
Look for what the child does
understand
Consider what question the child
might be answering correctly
Consider leaving a wrong answer
Give children time to think
Kevin’s Problem
Grade 1
One summer TJ saved $451.
At the end of the summer, she
spent $269 on a new bike.
How much money did TJ have
then?
One summer TJ saved $451. At
the end of the summer, she spent
$269 on a new bike. How much
money did TJ have then?
400 – 200 = 200
200 – 60 = 140
140 – 9 = 131
131 + 50 = 181
181 + 1 = 182
Kevin’s Problem
Grade 1
One summer TJ saved $451.
At the end of the summer, she
spent $269 on a new bike.
How much money did TJ have
then?
How long did Kevin think?
Kevin’s Problem
Grade 1
One summer TJ saved $451.
At the end of the summer, she
spent $269 on a new bike.
How much money did TJ have
then?
How long did Kevin think?
70 seconds
Kevin’s Problem
Grade 1
One summer TJ saved $451.
At the end of the summer, she
spent $269 on a new bike.
How much money did TJ have
then?
How long did Kevin think?
70 seconds
What would have happened if
the teacher had interrupted?
What is the Teacher’s
Role in Exploring
Children’s Thinking?







Notice and ask about details of
the child’s strategy
Link strategy to problem context
Ask questions after right &
wrong answers
Look for what the child does
understand
Consider what question the child
might be answering correctly
Consider leaving a wrong answer
Give children time to think
Why Explore Children’s
Mathematical Thinking?
Benefits for the Teacher
 Gain information
 Allow differentiation
 Make math more fun to teach
Benefits for the Child Solving
 Encourage sense-making
 Validate child’s thinking
 Consolidate child’s reasoning
Benefits for Other Children
 Expose children to new strategies
 Value multiple strategies
 Make math more fun to learn
So What?
Discuss…


What big ideas will stay with
you?
What will you try in your
class?