Transcript Teach Primary Number Sense

Teach Primary Number Sense with More
Understanding and Less Counting
Joan A. Cotter, Ph.D.
[email protected]
California Mathematics Council-South
Saturday, October 25, 2014
1:15 – 2:45
Palm Springs, California
© Joan A. Cotter, Ph.D., 2014
Is Counting the Core of Mathematics
• Arithmetic is only one of about 200 branches
© Joan A. Cotter, Ph.D., 2014
Is Counting the Core of Mathematics
• Arithmetic is only one of about 200 branches
• Abacuses made counting unnecessary
© Joan A. Cotter, Ph.D., 2014
Is Counting the Core of Mathematics
• Arithmetic is only one of about 200 branches
• Abacuses made counting unnecessary
• Counting determines quantity, not efficient
© Joan A. Cotter, Ph.D., 2014
Is Counting the Core of Mathematics
• Arithmetic is only one of about 200 branches
• Abacuses made counting unnecessary
• Counting determines quantity, not efficient
• Counting doesn’t work for fractions or decimals
© Joan A. Cotter, Ph.D., 2014
Is Counting the Core of Mathematics
• Arithmetic is only one of about 200 branches
• Abacuses made counting unnecessary
• Counting determines quantity, not efficient
• Counting doesn’t work for fractions or decimals
• Very slow for multiplication
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from the larger number.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from larger number.
7. Subtracting by counting backward.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from larger number.
7. Subtracting by counting backward.
8. Multiplying by skip counting.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
•
•
•
•
•
String level
Unbreakable list
Breakable chain
Numerable chain
Bidirectional chain
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
• Requires stable order for counting words
• Common errors: double counting and
missed count
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
• Unlike anything else in child’s experience
(e.g. in naming family, baby ≠ all others).
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
• Unlike anything else in child’s experience
(e.g. in naming family, baby ≠ all others).
• “How many” not a good test; take n is
better.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
• Focuses more on counting than adding.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on. (Jack and Jill)
• Leads to counting words.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on. (Jack and Jill)
• Leads to counting words.
• No need to learn strategies.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on. (Jack and Jill)
• Leads to counting words.
• No need to learn strategies.
• Very difficult. (article in Nov. 2011, JRME)
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on. (Jack and Jill)
6. Adding by counting from larger number.
• First need to determine larger number.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on. (Jack and Jill)
6. Adding by counting from the larger number.
7. Subtracting by counting backward.
• Extremely difficult. (Easier to go forward.)
© Joan A. Cotter, Ph.D., 2014
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on. (Jack and Jill)
6. Adding by counting from larger number.
7. Subtracting by counting backward.
8. Multiplying by skip counting.
• Tedious for finding multiplication facts.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
Because we're so familiar with 1, 2, 3, we’ll use letters.
A=1
B=2
C=3
D=4
E = 5, and so forth
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
E
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
E
F
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
E
F
A
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
E
F
A
B
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
E
F
A
B
C
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
E
F
A
B
C
D
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
E
F
A
B
C
D
E
What is the sum?
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
A
B
C
D
E
F
A
B
C
D
E
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
F+E=
K
A
B
C
D
E
F
G
H
I
J
K
© Joan A. Cotter, Ph.D., 2014
Adding on a Number Line
D+C=
© Joan A. Cotter, Ph.D., 2014
Adding on a Number Line
D+C=
* A B C D E F G H I J K L M
© Joan A. Cotter, Ph.D., 2014
Adding on a Number Line
D+C=
* A B C D E F G H I J K L M
© Joan A. Cotter, Ph.D., 2014
Adding on a Number Line
D+C=
* A B C D E F G H I J K L M
Are we counting lines or spaces?
© Joan A. Cotter, Ph.D., 2014
Number Line Drawbacks
© Joan A. Cotter, Ph.D., 2014
Number Line Drawbacks
• Quantity of a number not obvious.
© Joan A. Cotter, Ph.D., 2014
Number Line Drawbacks
• Quantity of a number not obvious.
• Requires counting in order to compute.
© Joan A. Cotter, Ph.D., 2014
Number Line Drawbacks
• Quantity of a number not obvious.
• Requires counting in order to compute.
• Ignores place value.
© Joan A. Cotter, Ph.D., 2014
Number Line Drawbacks
• Quantity of a number not obvious.
• Requires counting in order to compute.
• Ignores place value.
• Hard to visualize.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
E+D=
Find the sum without counters.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
G+E=
Find the sum without fingers.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+D
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+D
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+D
D
+C
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+D
D
+C
C
+G
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+D
D
+C
C
+G
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
H–C=
Subtract counting backward by using your fingers.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
J–F=
Subtract by counting backward without fingers.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . , T.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . , T.
What is D × E?
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
“Special cases” of place value (1.NBT.2)
L
is a “bundle” of J A’s
and B A’s.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
“Special cases” of place value (1.NBT.2)
L
is a “bundle” of J A’s
and B A’s.
huh?
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
“Special cases” of place value (1.NBT.2)
L (12)
is a “bundle” of J A’s
and B A’s.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
“Special cases” of place value (1.NBT.2)
L (12)
is a “bundle” of J A’s (ten ones)
and B A’s.
© Joan A. Cotter, Ph.D., 2014
Traditional Counting
“Special cases” of place value (1.NBT.2)
L (12)
is a “bundle” of J A’s (ten ones)
and B A’s. (two ones)
© Joan A. Cotter, Ph.D., 2014
Calendar Math
© Joan A. Cotter, Ph.D., 2014
Calendar Math
August
1
2
3
4
5
6
7
8
9
10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
© Joan A. Cotter, Ph.D., 2014
Calendar Math
August
1
2
3
4
5
6
7
8
9
10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
© Joan A. Cotter, Ph.D., 2014
Calendar Math
August
1
2
3
4
5
6
7
8
9
10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
© Joan A. Cotter, Ph.D., 2014
Calendar Math
August
1
2
3
4
5
6
7
8
9
10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
© Joan A. Cotter, Ph.D., 2014
Calendar Math
August
September
1
2
3
4
5
6
7
8
9
10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
This is ordinal counting.
© Joan A. Cotter, Ph.D., 2014
Calendar Math
August
September
1
2
3
4
5
6
7
8
9
10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
1
2
3
4
5
6
© Joan A. Cotter, Ph.D., 2014
Calendar Math
August
1
2
3
8
9
10
4
5
6
7
© Joan A. Cotter, Ph.D., 2014
Calendar Math
The calendar is not a number line.
• No quantity is involved.
© Joan A. Cotter, Ph.D., 2014
Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines.
© Joan A. Cotter, Ph.D., 2014
Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines.
Children need to see the whole month, not just part.
• Purpose of calendar is to plan ahead.
© Joan A. Cotter, Ph.D., 2014
Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines.
Children need to see the whole month, not just part.
• Purpose of calendar is to plan ahead.
• Many ways to show the current date.
© Joan A. Cotter, Ph.D., 2014
Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines.
Children need to see the whole month, not just part.
• Purpose of calendar is to plan ahead.
• Many ways to show the current date.
Calendars give a narrow view of patterning.
• Patterns do not necessarily involve numbers.
© Joan A. Cotter, Ph.D., 2014
Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines.
Children need to see the whole month, not just part.
• Purpose of calendar is to plan ahead.
• Many ways to show the current date.
Calendars give a narrow view of patterning.
• Patterns do not necessarily involve numbers.
• Patterns rarely proceed row by row.
© Joan A. Cotter, Ph.D., 2014
Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines.
Children need to see the whole month, not just part.
• Purpose of calendar is to plan ahead.
• Many ways to show the current date.
Calendars give a narrow view of patterning.
• Patterns do not necessarily involve numbers.
• Patterns rarely proceed row by row.
• Patterns go on forever; they don’t stop at 31.
© Joan A. Cotter, Ph.D., 2014
Counting Model
Summary
© Joan A. Cotter, Ph.D., 2014
Counting Model
Summary
• Is not natural; it takes years of practice.
© Joan A. Cotter, Ph.D., 2014
Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
© Joan A. Cotter, Ph.D., 2014
Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
© Joan A. Cotter, Ph.D., 2014
Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
© Joan A. Cotter, Ph.D., 2014
Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
• Is tedious and time-consuming.
© Joan A. Cotter, Ph.D., 2014
Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
• Is tedious and time-consuming.
• Does not provide an efficient way to master
the facts.
© Joan A. Cotter, Ph.D., 2014
Problems Learning to Count
• Children with dyslexia or dyscalculia
© Joan A. Cotter, Ph.D., 2014
Problems Learning to Count
• Children with dyslexia or dyscalculia
• Children with SLI (specific language impairment)
© Joan A. Cotter, Ph.D., 2014
Problems Learning to Count
• Children with dyslexia or dyscalculia
• Children with SLI (specific language impairment)
• Children with memory challenges
© Joan A. Cotter, Ph.D., 2014
Problems Learning to Count
• Children with dyslexia or dyscalculia
• Children with SLI (specific language impairment)
• Children with memory challenges
• Children from low SES backgrounds
© Joan A. Cotter, Ph.D., 2014
Research-Based
Solutions
© Joan A. Cotter, Ph.D., 2014
Research from Finland
Riikka Mononen, Pirjo Aunio, Tuire Koponen.
(2014) A pilot study of the effects of RightStart
instruction on early numeracy skills of children
with specific language impairment. Research in
Developmental Disabilities.
Abstract:
....The children with SLI began kindergarten with
significantly weaker early numeracy skills compared to
their peers. Immediately after the instruction phase, there
was no significant difference between the groups in
counting skills….
© Joan A. Cotter, Ph.D., 2014
Earlier Research
• Cotter, Joan. “Using Language and Visualization to
Teach Place Value.” Teaching Children Mathematics
7 (October, 2000): 108-114.
• Also reprinted in NCTM (National Council of
Teachers of Mathematics) On-Math Journal and in
Growing Professionally: Readings from NCTM
Publications for Grades K-8, in 2008.
© Joan A. Cotter, 2010
Grouping in Fives
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Chinese abacus
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Early Roman numerals
1
2
3
4
5
8
I
II
III
IIII
V
VIII
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Musical staff
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Clocks and nickels
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Clocks and nickels
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Tally marks
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Subitizing
• Instant recognition of quantity is called subitizing.
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Subitizing
• Instant recognition of quantity is called subitizing.
• Subitizing 1–4 is called perceptual subitizing.
© Joan A. Cotter, Ph.D., 2014
Grouping in Fives
Subitizing
• Instant recognition of quantity is called subitizing.
• Subitizing 1–4 is called perceptual subitizing.
• Subitizing by grouping in fives is called
conceptual subitizing.
© Joan A. Cotter, Ph.D., 2014
Subitizing
• Five-month-old infants can subitize to 1–3.
© Joan A. Cotter, Ph.D., 2014
Subitizing
• Five-month-old infants can subitize to 1–3.
• Three-year-olds can subitize to 1–5.
© Joan A. Cotter, Ph.D., 2014
Subitizing
• Five-month-old infants can subitize to 1–3.
• Three-year-olds can subitize to 1–5.
• Four-year-olds can subitize 1–10 with
conceptual subitizing.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
• Children learn subitizing up to 5 before
counting.—Starkey & Cooper
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
• Children learn subitizing up to 5 before
counting.—Starkey & Cooper
• Subitizing seems to be a necessary skill for
understanding what the counting process means.
—Glasersfeld
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
• Children learn subitizing up to 5 before
counting.—Starkey & Cooper
• Subitizing seems to be a necessary skill for
understanding what the counting process means.
—Glasersfeld
• Children who can subitize perform better in
mathematics long term.—Butterworth
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
• Adult Pirahã from Amazon region.
Edward Gibson and Michael Frank, MIT, 2008.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
• Adult Pirahã from Amazon region.
Edward Gibson and Michael Frank, MIT, 2008.
• Adults, ages 18–50, from Boston.
Edward Gibson and Michael Frank, MIT, 2008.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
• Adult Pirahã from Amazon region.
Edward Gibson and Michael Frank, MIT, 2008.
• Adults, ages 18–50, from Boston.
Edward Gibson and Michael Frank, MIT, 2008.
• Baby chicks from Italy.
Lucia Regolin, University of Padova, 2009.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
In Japanese schools
• Children are discouraged from using counting
for adding. They are not taught to count on.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
In Japanese schools
• Children are discouraged from using counting
for adding. They are not taught to count on.
• They consistently group in 5s.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Finger gnosia
• Finger gnosia is the ability to know which fingers
can been lightly touched without looking.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Finger gnosia
• Finger gnosia is the ability to know which fingers
can been lightly touched without looking.
• Part of the brain controlling fingers is adjacent to
math part of the brain.
© Joan A. Cotter, Ph.D., 2014
Research on Subitizing
Finger gnosia
• Finger gnosia is the ability to know which fingers
can been lightly touched without looking.
• Part of the brain controlling fingers is adjacent to
math part of the brain.
• Children who use their fingers as representational
tools perform better in mathematics.—Butterworth
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Using fingers
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Using fingers
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Using fingers
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Using fingers
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Using fingers
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Using fingers
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Subitizing 5
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Subitizing 5
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Subitizing 5
5 has a middle; 4 does not.
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Tally sticks
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Tally sticks
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Tally sticks
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Tally sticks
Five as a group.
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Tally sticks
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Tally sticks
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Entering quantities
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Entering quantities
3
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Entering quantities
5
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Entering quantities
7
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Entering quantities
10
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
The stairs
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Adding
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Adding
4+3=
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Adding
4+3=
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Adding
4+3=
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Adding
4+3=
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Adding
4+3=7
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Visualizing
4+3=7
Japanese children learn to do this mentally.
© Joan A. Cotter, Ph.D., 2014
Visualizing
• Visual means to see.
• Visualize means to form a mental image.
© Joan A. Cotter, Ph.D., 2014
Visualizing
“Think in pictures, because the brain
remembers images better than it does
anything else.”
—Ben Pridmore, World Memory Champion, 2009
© Joan A. Cotter, Ph.D., 2014
Visualizing
“The role of physical manipulatives was to
help the child form those visual images and
thus to eliminate the need for the physical
manipulatives.”
—Ginsberg and others
© Joan A. Cotter, Ph.D., 2014
Visualizing
Japanese criteria for manipulatives
© Joan A. Cotter, Ph.D., 2014
Visualizing
Japanese criteria for manipulatives
• Representative of structure of numbers.
© Joan A. Cotter, Ph.D., 2014
Visualizing
Japanese criteria for manipulatives
• Representative of structure of numbers.
• Easily manipulated by children.
© Joan A. Cotter, Ph.D., 2014
Visualizing
Japanese criteria for manipulatives
• Representative of structure of numbers.
• Easily manipulated by children.
• Imaginable mentally.
—Japanese Council of
Mathematics Education
© Joan A. Cotter, Ph.D., 2014
Visualizing
Necessary for:
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
© Joan A. Cotter, Ph.D., 2014
Visualizing
Necessary for:
• Reading
• Architecture
• Sports
• Astronomy
• Creativity
• Archeology
• Geography
• Chemistry
• Engineering
• Physics
• Construction
• Surgery
© Joan A. Cotter, Ph.D., 2014
Visualizing
Try to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2014
Visualizing
Try to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2014
Visualizing
Now try to visualize 8 apples: 5 red and 3 green.
© Joan A. Cotter, Ph.D., 2014
Visualizing
Now try to visualize 8 apples: 5 red and 3 green.
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=
+
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=
+
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=4+1
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=
+
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=3+2
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=
+
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=2+3
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=
+
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=1+4
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=
+
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Partitioning
5=0+5
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Place value
• Place value is the foundation of modern
arithmetic.
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Place value
• Place value is the foundation of modern
arithmetic.
• Critical for understanding algorithms.
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Place value
• Place value is the foundation of modern
arithmetic.
• Critical for understanding algorithms.
• Must be taught, not left for discovery.
© Joan A. Cotter, Ph.D., 2014
Learning 1–10
Place value
• Place value is the foundation of modern
arithmetic.
• Critical for understanding algorithms.
• Must be taught, not left for discovery.
• Children need the big picture, not tiny snapshots.
© Joan A. Cotter, Ph.D., 2014
Place Value
Two aspects
© Joan A. Cotter, Ph.D., 2014
Place Value
Static (Recording)
Two aspects
Dynamic (Trading)
© Joan A. Cotter, Ph.D., 2014
Place Value
Two aspects
Static (Recording)
• Value of a digit is determined by position.
Dynamic (Trading)
© Joan A. Cotter, Ph.D., 2014
Place Value
Two aspects
Static (Recording)
• Value of a digit is determined by position.
• No position may have more than nine.
Dynamic (Trading)
© Joan A. Cotter, Ph.D., 2014
Place Value
Two aspects
Static (Recording)
• Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value of each position
is ten times greater than previous position.
Dynamic (Trading)
© Joan A. Cotter, Ph.D., 2014
Place Value
Two aspects
Static (Recording)
• Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value of each position
is ten times greater than previous position.
• (Shown by the place-value cards.)
Dynamic (Trading)
© Joan A. Cotter, Ph.D., 2014
Place Value
Two aspects
Static (Recording)
• Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value of each position
is ten times greater than previous position.
• (Shown by the place-value cards.)
Dynamic (Trading)
• 10 ones = 1 ten; 10 tens = 1 hundred;
10 hundreds = 1 thousand, ….
© Joan A. Cotter, Ph.D., 2014
Place Value
Two aspects
Static (Recording)
• Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value of each position
is ten times greater than previous position.
• (Shown by the place-value cards.)
Dynamic (Trading)
• 10 ones = 1 ten; 10 tens = 1 hundred;
10 hundreds = 1 thousand, ….
• (Represented on the abacus and other materials.)
© Joan A. Cotter, Ph.D., 2014
Place Value
Transparent number naming
• Asian children do not struggle with the teens.
© Joan A. Cotter, Ph.D., 2014
Place Value
Transparent number naming
• Asian children do not struggle with the teens.
• Their languages are completely “ten-based.”
© Joan A. Cotter, Ph.D., 2014
Place Value
Transparent number naming
• Asian children do not struggle with the teens.
• Their languages are completely “ten-based.”
• Asian countries use the ten-based metric
system.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
12 = ten 2
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
20 = 2-ten
21 = 2-ten 1
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
20 = 2-ten
21 = 2-ten 1
22 = 2-ten 2
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
20 = 2-ten
21 = 2-ten 1
22 = 2-ten 2
23 = 2-ten 3
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
20 = 2-ten
21 = 2-ten 1
22 = 2-ten 2
23 = 2-ten 3
....
....
99 = 9-ten 9
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
137 = 1 hundred 3-ten 7
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
137 = 1 hundred 3-ten 7
or
137 = 1 hundred and 3-ten 7
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Average Highest Number Counted
100
90
80
Chinese
U.S.
Korean formal (transparent)
Korean informal
70
60
50
40
30
20
10
0
4
5
6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Average Highest Number Counted
100
90
80
Chinese
U.S.
Korean formal (transparent)
Korean informal
70
60
50
40
30
20
10
0
4
5
6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Average Highest Number Counted
100
90
80
Chinese
U.S.
Korean formal (transparent)
Korean informal
70
60
50
40
30
20
10
0
4
5
6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Average Highest Number Counted
100
90
80
Chinese
U.S.
Korean formal (transparent)
Korean informal
70
60
50
40
30
20
10
0
4
5
6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Average Highest Number Counted
100
90
80
Chinese
U.S.
Korean formal (transparent)
Korean informal
70
60
50
40
30
20
10
0
4
5
6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
• Only 11 words are needed to count to 100 the
math way, 28 in English. (All Indo-European
languages are non-standard in number naming.)
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
• Only 11 words are needed to count to 100 the
math way, 28 in English. (All Indo-European
languages are non-standard in number naming.)
• Asian children learn mathematics using
transparent number naming.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
• Only 11 words are needed to count to 100 the
math way, 28 in English. (All Indo-European
languages are non-standard in number naming.)
• Asian children learn mathematics using
transparent number naming.
• They understand place value in first grade; only
half of U.S. children understand place value at
the end of fourth grade.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
• Only 11 words are needed to count to 100 the
math way, 28 in English. (All Indo-European
languages are non-standard in number naming.)
• Asian children learn mathematics using
transparent number naming.
• They understand place value in first grade; only
half of U.S. children understand place value at
the end of fourth grade.
• Mathematics is the science of patterns. The
patterned transparent number naming greatly
helps children learn number sense.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Compared to reading
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Compared to reading
• Just as reciting the alphabet doesn’t teach reading,
counting doesn’t teach arithmetic.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Compared to reading
• Just as reciting the alphabet doesn’t teach reading,
counting doesn’t teach arithmetic.
• Just as we first teach the sound of the letters, we must
first teach the transparent name of the quantity.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
4-ten = forty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
4-ten = forty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
6-ten = sixty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
3-ten = thirty
“Thir” also
used in 1/3,
13 and 30.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
5-ten = fifty
“Fif” also
used in 1/5,
15 and 50.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
2-ten = twenty
Two used to be
pronounced
“twoo.”
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
A word game
fireplace
place-fire
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
A word game
fireplace
place-fire
newspaper
paper-news
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
A word game
fireplace
place-fire
newspaper
paper-news
box-mail
mailbox
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
ten 4
Prefix -teen
means ten.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
ten 4
teen 4
Prefix -teen
means ten.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
ten 4
teen 4
fourteen
Prefix -teen
means ten.
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
a one left
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
a one left
a left-one
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
a one left
a left-one
eleven
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
two left
Two said
as “twoo.”
© Joan A. Cotter, Ph.D., 2014
Transparent Number Naming
Regular names
two left
twelve
Two said
as “twoo.”
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten
30
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten
30
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten
30
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten 7
30
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten 7
30
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten 7
30
7
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten 7
30
7
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
3-ten 7
30
7
Note the congruence in the way we say the number,
represent the number, and write the number.
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1-ten
10
Another example.
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1-ten 8
10
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1-ten 8
10
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1-ten 8
10
8
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1-ten 8
18
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
10-ten
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
10-ten
100
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
10-ten
100
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
10-ten
100
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1 hundred
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1 hundred
100
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1 hundred
100
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1 hundred
100
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
1 hundred
100
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
What number comes after 109?
© Joan A. Cotter, Ph.D., 2014
Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading of
numbers and text:
4258
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading of
numbers and text:
4258
268
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading of
numbers and text:
4258
269
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading of
numbers and text:
4258
270
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading of
numbers and text:
4258
The place-value cards encourage reading
numbers in the normal order.
271
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
Scientific notation
3
5000 = 5 ×
10notation, we “stand” on the
In scientific
left digit and note the number of digits to
the right. (That’s why we shouldn’t refer
to the 5 as being in the fourth column.)
272
© Joan A. Cotter, Ph.D., 2012
Learning the Facts
© Joan A. Cotter, Ph.D., 2014
Learning the Facts
Limited success, especially for struggling
children, when learning is:
© Joan A. Cotter, Ph.D., 2014
Learning the Facts
Limited success, especially for struggling
children, when learning is:
• Based on counting: whether dots, fingers,
number lines, or counting words.
© Joan A. Cotter, Ph.D., 2014
Learning the Facts
Limited success, especially for struggling
children, when learning is:
• Based on counting: whether dots, fingers,
number lines, or counting words.
• Based on rote memory: whether flash
cards, timed tests, or computer games.
© Joan A. Cotter, Ph.D., 2014
Learning the Facts
Limited success, especially for struggling
children, when learning is:
• Based on counting: whether dots, fingers,
number lines, or counting words.
• Based on rote memory: whether flash
cards, timed tests, or computer games.
• Based on skip counting: whether fingers or
songs.
© Joan A. Cotter, Ph.D., 2014
Time Needed to Memorize
A study with college students
• To learn 200 nonsense syllables: 93 minutes.
Time Needed to Memorize
A study with college students
• To learn 200 nonsense syllables: 93 minutes.
• To learn 200 words of prose: 24 minutes.
Time Needed to Memorize
A study with college students
• To learn 200 nonsense syllables: 93 minutes.
• To learn 200 words of prose: 24 minutes.
• To learn 200 words of poetry: 10 minutes.
Memorizing Math
Percentage Recall
After 1 day After 4 wks
Rote
Concept
32
69
23
69
8
58
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote
Concept
32
69
23
69
8
58
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote
Concept
32
69
23
69
8
58
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote
Concept
32
69
23
69
8
58
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote
Concept
32
69
23
69
8
58
Math needs to be taught so 95% is
understood and only 5% memorized.
Richard Skemp
© Joan A. Cotter, Ph.D., 2012
Flash Cards
• Often used to teach rote.
© Joan A. Cotter, Ph.D., 2012
Flash Cards
• Often used to teach rote.
• Liked only by are those who don’t need
them.
© Joan A. Cotter, Ph.D., 2012
Flash Cards
• Often used to teach rote.
• Liked only by are those who don’t need
them.
• Give the false impression that math isn’t
about thinking.
© Joan A. Cotter, Ph.D., 2012
Flash Cards
• Often used to teach rote.
• Liked only by are those who don’t need
them.
• Give the false impression that math isn’t
about thinking.
• Often produce stress – children under stress
stop learning.
© Joan A. Cotter, Ph.D., 2012
Flash Cards
• Often used to teach rote.
• Liked only by are those who don’t need
them.
• Give the false impression that math isn’t
about thinking.
• Often produce stress – children under stress
stop learning.
• Not concrete – use abstract symbols.
© Joan A. Cotter, Ph.D., 2012
Fact Strategies
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Complete the Ten
9+5=
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Complete the Ten
9+5=
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Complete the Ten
9+5=
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Complete the Ten
9+5=
Take 1 from the
5 and give it to
the 9.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Complete the Ten
9+5=
Take 1 from the
5 and give it to
the 9.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Complete the Ten
9+5=
Take 1 from the
5 and give it to
the 9.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Complete the Ten
9 + 5 = 14
Take 1 from the
5 and give it to
the 9.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Two Fives
8+6=
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Two Fives
8+6=
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Two Fives
8+6=
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Two Fives
8+6=
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Two Fives
8+6=
10 + 4 = 14
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting Part from Ten
15 – 9 =
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting Part from Ten
15 – 9 =
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting Part from Ten
15 – 9 =
Subtract 5 from
5 and 4 from 10.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting Part from Ten
15 – 9 =
Subtract 5 from
5 and 4 from 10.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting Part from Ten
15 – 9 =
Subtract 5 from 5
and 4 from 10.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting Part from Ten
15 – 9 = 6
Subtract 5 from 5
and 4 from 10.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting All from 10
15 – 9 =
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting All from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting All from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting All from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Subtracting All from 10
15 – 9 = 6
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Going Up
15 – 9 =
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2014
Fact Strategies
Going Up
15 – 9 =
1+5=6
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2014
Money
Penny
© Joan A. Cotter, Ph.D., 2014
Money
Nickel
© Joan A. Cotter, Ph.D., 2014
Money
Dime
© Joan A. Cotter, Ph.D., 2014
Money
Quarter
© Joan A. Cotter, Ph.D., 2014
Money
Quarter
© Joan A. Cotter, Ph.D., 2014
Money
Quarter
© Joan A. Cotter, Ph.D., 2014
Money
Quarter
© Joan A. Cotter, Ph.D., 2014
Trading
1000
100
10
1
© Joan A. Cotter, Ph.D., 2014
Trading
Thousands
1000
100
10
1
© Joan A. Cotter, Ph.D., 2014
Trading
Hundreds
1000
100
10
1
© Joan A. Cotter, Ph.D., 2014
Trading
Tens
1000
100
10
1
© Joan A. Cotter, Ph.D., 2014
Trading
Ones
1000
100
10
1
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
14
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding
1000
100
10
1
8
+6
14
Same answer
before and after
trading.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers
from left to right.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers
from left to right.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers
from left to right.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers
from left to right.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers
from left to right.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers
from left to right.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Add starting at the
right. Write results
after each step.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Add starting at the
right. Write results
after each step.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
6
Write 6.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
6
Write 1 for the
extra ten.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
6
Add the tens.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
6
Add the tens.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
96
Write the tens.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
96
Add the hundreds.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
96
Add the hundreds.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
96
Trade 10 hundreds
for 1 thousand.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
96
Trade 10 hundreds
for 1 thousand.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
396
Write the hundreds.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
1
3658
+ 2738
396
Write the 1 for the
extra thousand.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
1
3658
+ 2738
396
Add the thousands.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
1
3658
+ 2738
396
Add the thousands.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
1
3658
+ 2738
6396
Write the thousands.
© Joan A. Cotter, Ph.D., 2014
Trading
Adding 4-digit numbers
1000
100
10
1
1
1
3658
+ 2738
6396
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Page 5
“These Standards do not dictate curriculum or
teaching methods. For example, just because topic A
appears before topic B in the standards for a given
grade, it does not necessarily mean that topic A must
be taught before topic B. . . . Or, a teacher might
prefer to teach a topic of his or her own choosing
that leads, as a byproduct, to students reaching the
standards for topics A and B.” —CCSS
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Page 5 summary
• Standards do not dictate curriculum or
teaching methods.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Page 5 summary
• Standards do not dictate curriculum or
teaching methods.
• Within a grade, topics may be taught in
any order or taught indirectly.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.NBT)
Know number names and the count sequence.
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number
within the known sequence (instead of having
to begin at 1).
3. Write numbers from 0 to 20. Represent a
number of objects with a written numeral 0-20
(with 0 representing a count of no objects).
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.CC)
3. Write numbers from 0 to 20.
Number Chart
1
6
2
7
3
8
4
9
5
10
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.NBT)
Work with numbers 11–19.
1. Compose and partition numbers from 11 to 19
into ten ones and some further ones.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.NBT)
Work with numbers 11–19.
1. Compose and partition numbers from 11 to 19
into ten ones and some further ones.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.NBT)
Work with numbers 11–19.
1. Compose and partition numbers from 11 to 19
into ten ones and some further ones.
10
6
18
6
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Understand addition and subtraction.
1. Represent addition and subtraction with objects,
fingers, . . . equations.
2. Solve addition and subtraction word problems,
and add and subtract within 10.
3. Partition numbers less than or equal to 10 into
pairs.
4. For any number from 1 to 9, find the number
that makes 10.
5. Fluently add and subtract within 5.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
2. Solve addition and subtraction word problems,
and add and subtract within 10.
Whole
Part-whole circles
Part
Part
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
What is the whole?
3
2
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
What is the whole?
5
3
2
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
Is 3 the whole or a part?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
3
Is 3 the whole or a part?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
5
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
3
Is 5 the whole or a part?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
5
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
3
What is the missing part?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
5
3
2
What is the missing part?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
5
3
2
Write the equation.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
5
3
2
Write the equation.
2+3=5
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
5
3
2
Write the equation.
2+3=5
3+2=5
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
Lee received 3 goldfish as a
gift. Now Lee has 5. How
many goldfish did Lee have
to start with?
5
3
2
Write the equation.
2+3=5
3+2=5
5–3=2
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
• Research shows part-whole
circles help young children
solve problems. Writing
equations do not.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
• Research shows part-whole
circles help young children
solve problems. Writing
equations do not.
• Do not teach “key” words.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
4. For any number from 1 to 9, find the number
that makes 10.
10
7
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Kindergarten (K.OA)
4. For any number from 1 to 9, find the number
that makes 10.
10
7
3
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Understand and apply properties of operations
and the relationship between addition and
subtraction.
1. Use addition and subtraction within 20 to solve
word problems involving situations of adding
to, taking from, putting together, taking apart,
and comparing….
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
larger
set
smaller set
difference
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
larger
set
smaller set
Alex has 2 apples. Morgan
has 5 apples. How many
more apples does Morgan
have?
difference
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Alex has 4 apples. Morgan
has 7 apples. How many
more apples does Morgan
have?
Is 4 the larger, smaller, or the difference?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Alex has 4 apples. Morgan
has 7 apples. How many
more apples does Morgan
have?
4
Is 4 the larger, smaller, or the difference?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Alex has 4 apples. Morgan
has 7 apples. How many
more apples does Morgan
have?
4
Is 7 the larger, smaller, or the difference?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
7
Alex has 4 apples. Morgan
has 7 apples. How many
more apples does Morgan
have?
4
Is 7 the larger, smaller, or the difference?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
7
Alex has 4 apples. Morgan
has 7 apples. How many
more apples does Morgan
have?
4
What is the difference?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Alex has 4 apples. Morgan
has 7 apples. How many
more apples does Morgan
have?
7
4
3
What is the difference?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Understand and apply properties of operations
and the relationship between addition and
subtraction.
3. Apply properties of operations as strategies to
add and subtract, commutative property and
associative property of addition.
4. Understand subtraction as an unknown-addend
problem. [Subtract by going up.]
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
1. Apply properties of operations as strategies to
add and subtract, commutative property and
associative property of addition.
6+3=9
3+6=9
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Work with addition and subtraction equations.
7. Understand the meaning of the equal sign.
8. Determine the unknown whole number in an
addition or subtraction equation.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
Math balance
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
7=7
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
10 = 3 + 7
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
8 + 2 = 10
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
7 + 7 = 14
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
8. Determine the unknown whole number in an
addition or subtraction equation.
8 + _ = 11
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
8. Determine the unknown whole number in an
addition or subtraction equation.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
8 + 3 = 11
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Extend the counting sequence.
1. Count to 120, starting at any number less than
120.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.OA)
Extend the counting sequence.
1. Count to 120, starting at any number less than
120.
100
10
9
10
10
9
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
Understanding place value.
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
6. Subtract multiples of 10 in the range 10-90
from multiples of 10.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
40
6
60
4
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
40
6
46
60
4
64
Put two dots by greater number.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
40
6
60
4
.
46 . 64
Put two dots by greater number.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
40
6
60
4
.
46 . 64
Put two dots by greater number.
Put one dot by lesser number.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
40
6
60
4
.
.
46 . 64
Put two dots by greater number.
Put one dot by lesser number.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
40
6
60
4
.
.
46 . 64
Put two dots by greater number.
Put one dot by lesser number.
Connect the dots.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
40
6
60
4
.
.
46 . 64
Put two dots by greater number.
Put one dot by lesser number.
Connect the dots.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 + 10 = __
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 + 10 = 34
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 – 10 = __
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 – 10 = 14
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
6. Subtract multiples of 10 in the range 10-90
from multiples of 10.
90 – 30 = __
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 1 (1.NBT)
6. Subtract multiples of 10 in the range 10-90
from multiples of 10.
90 – 30 = 60
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.OA)
Work with equal groups of objects to gain
foundations for multiplication.
3. Determine whether a group of objects (up to
20) has an odd or even number of members.
4. Use addition to find the total number of objects
arranged in rectangular arrays.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.OA)
3. Determine whether a group of objects (up to
20) has an odd or even number of members.
Is 17 even
or odd?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.OA)
3. Determine whether a group of objects (up to
20) has an odd or even number of members.
Is 17 even
or odd?
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.OA)
4. Use addition to find the total number of objects
arranged in rectangular arrays.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.OA)
4. Use addition to find the total number of objects
arranged in rectangular arrays.
5 + 5 + 5 + 5 = 20
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.OA)
4. Use addition to find the total number of objects
arranged in rectangular arrays.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.OA)
4. Use addition to find the total number of objects
arranged in rectangular arrays.
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
5 + 5 + 5 + 5 = 20
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
Number and Operations in Base Ten.
2. Count within 1000; skip-count by 2s, 5s, 10s,
and 100s.
3. Read and write numbers to 1000 using base-ten
numerals, number names, and expanded form.
4. Compare two three-digit numbers based on
meanings of the hundreds, tens, and ones digits,
using >, =, and <.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5,
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5, 10,
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5, 10, 15, . . .
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
1000
100
10
1
100, 200, 300, . . .
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
2. Count within 1000.
3. Read and write numbers to 1000 using base-ten
numerals, number names, and expanded form.
300
70
8
378,
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
2. Count within 1000.
3. Read and write numbers to 1000 using base-ten
numerals, number names, and expanded form.
300
70
8
300
70
9
378, 379,
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
2. Count within 1000.
3. Read and write numbers to 1000 using base-ten
numerals, number names, and expanded form.
300
70
8
300
70
9
300
80
378, 379, 380
© Joan A. Cotter, Ph.D., 2014
Meeting the Standards
Grade 2 (2.NBT)
4. Compare two three-digit numbers based on
meanings of the hundreds, tens, and ones digits,
using >, =, and <.
700
6
60
70
706 > 670
© Joan A. Cotter, Ph.D., 2014
Is Counting the Core of Mathematics?
© Joan A. Cotter, Ph.D., 2014
Is Counting the Core of Mathematics?
It doesn’t have to be.
© Joan A. Cotter, Ph.D., 2014
Teach Primary Number Sense with More
Understanding and Less Counting
Joan A. Cotter, Ph.D.
[email protected]
California Mathematics Council-South
Saturday, October 25, 2014
1:15 – 2:45
Palm Springs, California
© Joan A. Cotter, Ph.D., 2014