1. Kindergarten PARTNERS for Mathematics Learning Module 6 Partners 1

2.  An intuition about shapes and 2
Spatial Sense Is…
 An intuition about shapes and
the relationships among shapes
 A feel for the geometric aspects of their
surroundings and the shapes formed by
objects in the environment
(Van de Walle & Lovin, 2006)
 Mental imagery and visualization
Partners
for Mathematics Learning

3. Why is it important?  Students who have well-developed spatial3
Why is it important?
 Students who have well-developed spatial
sense tend to solve tasks in more
meaningful ways,
 They are not just rote learners
(Reynolds & Wheatley, 1997)
Partners
for Mathematics Learning

4.  Students can learn to recognize sets of4
Spatial Relationships
 Students can learn to recognize sets of
objects in patterned arrangements and tell
how many without counting (subitizing)
 For most numbers, there are several patterns
Learned
pattern 5
Combining
two patterns 6
Partners
for Mathematics Learning
6 and 1
more 7

5. How Many Dots?  How many dots do you see in the next slide? Ready…5
How Many Dots?
 How many dots do you see in the next
slide?
Ready…
Partners
for Mathematics Learning

6. 6
Quick Look
Partners
for Mathematics Learning

7. How Many Did You See?  Draw the dots  How did you think about them?7
How Many Did You See?
 Draw the dots
 How did you think about them?
 Compare with your neighbor
 Did you see the same number?
 Did you think about them in the same
way?
Partners
for Mathematics Learning

8. Determining the Number of Dots8
Determining the Number of Dots
Partners
for Mathematics Learning

9. Subitizing  Instantly seeing how many  Recognizing patterns to9
Subitizing
 Instantly seeing how many
 Recognizing patterns to
know how many without
counting
 Identifying small quantities
without counting
Partners
for Mathematics Learning

10. Deeper Understanding  Counting is important, but it is not enough10
Deeper Understanding
 Counting is important, but it is not enough
 A deep understanding involves the use of
imagery
 Students need to think in units
as part of their development
of number sense
Partners
for Mathematics Learning

11.  To conceptualize a number as being made11
Part-Part-Whole Relationships
 To conceptualize a number as being made
up of two or more parts is the most important
relationship that can be developed about
number
Six and three is nine
Partners
for Mathematics Learning

12. Try This…  Count out a set of eight  Separate your counters into12
Try This…
 Count out a set of eight
counters in front of you as if you
were a kindergarten student
 Separate your counters into
2 piles and think of the
combination
Nothing in counting causes
one to focus on the fact that
8 could be made of two parts
Partners
for Mathematics Learning

13. Try This…  Change your 2 piles and say the new13
Try This…
 Change your 2 piles
and say the new
combination to yourself
 Change them again
Thinking about a number in
terms of its parts is a major
milestone in developing
number sense
Partners
for Mathematics Learning

14. Benchmarks of 5 and 10 Five and three more14
Benchmarks of 5 and 10
 Developing number relationships for
numbers 1 to 10 is a major kindergarten
emphasis
 Both five and 10 are important anchors or
benchmarks
Five and three more
Two away from 10
Partners
for Mathematics Learning

15. Fives, Fives, and More Fives15
Fives, Fives, and More Fives
 One challenge for every kindergarten
teacher is having numerous activities that
give children opportunities to learn basic
concepts
 Be prepared to quickly summarize the
activity given to you and your partner
 If time at the end of this “round robin,”
share other activities for composing and
decomposing five and ten
Partners
for Mathematics Learning

16. Under the Rug  Count five objects and place them on the16
Under the Rug
 Count five objects and place them on the
table
 Put some of the objects
under “the rug”
 How many objects are not
under the rug?
Partners
for Mathematics Learning

17. 17
Break Down the Wall
Partners
for Mathematics Learning

18. Five Little Monkeys  Five little monkeys jumping on the bed18
Five Little Monkeys
 Five little monkeys
jumping on the bed
 One fell off and bumped
his head
 Mama called the doctor
and the doctor said, “No
more monkeys jumping
on the bed!”
 Four little monkeys
jumping on the bed……..
Partners
for Mathematics Learning

19. Five in a Hive  Three bees and two  Four bees and one19
Five in a Hive
 Three bees and two
more bees make five
bees in a hive
 Four bees and one
more bee make five
Partners
for Mathematics Learning

20. Five Little Ducks  Five little ducks went out to play20
Five Little Ducks 
Five little ducks went out to play
Over the hill and far away
Mother Duck said, Quack, quack, quack!”
Four little ducks came waddling back
for Mathematics Learning

21. Fun With Five  Use number cards to make different21
Fun With Five
 Use number cards to make different
combinations of five
 Watch out for the frown face so you do not
lose your cards
 You might even have to trade your game
board for another
Partners
for Mathematics Learning

22. Showing Five-ness  Students make models of five using22
Showing Five-ness
 Students make models of five using
wooden craft sticks
 What might the number
combinations be for this
representation?
Partners
for Mathematics Learning

23. Pretty Red Apples Pretty red apples growing on the tree23
Pretty Red Apples
Pretty red apples growing on the tree
Pick three apples - How many do you see?
Do apples grow below or above the ground?
Pick four apples - How many do you see?
What two shapes could you use to make a tree?
Pretty red apples growing on the tree; Pick five
apples - How many do you see?
How many more than 3 is 5?
Partners
for Mathematics Learning

24. Five Little Snowmen  Five little snowmen playing in a band24
Five Little Snowmen
 Five little snowmen playing in a band
 One fell down and hurt his hand
 How many snowmen playing in the
band?
 Four little snowmen…
How many small, medium,
and large circles do you need
to make three snowmen?
Partners
for Mathematics Learning

25. 25
Partners
for Mathematics Learning

26. 26
Partners
for Mathematics Learning

27. 27
Partners
for Mathematics Learning

28.  Have children make books: “My Book 28
Capitalizing on Connections
 Have children make books: “My Book
About 5” or “All About 8”
 What other ways could class book ideas
be used in kindergarten classrooms to
enhance mathematics?
 What themes, formats, and tasks might be
beneficial for students?
Partners
for Mathematics Learning

29. Making Five Frames Mrs. Richmond wants each of her kindergarten29
Making Five Frames
Mrs. Richmond wants each of her kindergarten
students to make a set of five frames from 0 to
5. How many sticky dots will each child need to
make a set? How do you know?
Partners
for Mathematics Learning

30. Why 5 Frames?  Five is an important benchmark number for30
Why 5 Frames?
 Five is an important benchmark number for
young children
 Use of 5 frames greatly reduces the
reliance on counting by ones
 Frames allow students to see combinations
to make five
Partners
for Mathematics Learning

31. 5 Frames Also…  Focus on how far away from 531
5 Frames Also…
 Focus on how far away from 5
 Show numbers greater than 5 as a full
five-frame and “some more” on the mat
Partners
for Mathematics Learning

32. 32
NCTM - Illuminations
Partners
for Mathematics Learning

33. 33
NCTM - Illuminations
Partners
for Mathematics Learning

34. 34
NCTM - Illuminations
Partners
for Mathematics Learning

35. Checking Out Resources35
Checking Out Resources
 Go to to link to the
Illuminations website and many other
resources
 Look at the number strand essential
standards
 What have we discussed?
 What still remains to be addressed
Partners
for Mathematics Learning

36. Let’s Do Equipartitioning36
Let’s Do Equipartitioning
 Equipartitioning refers to dividing a whole
into equal parts, or making fair shares
 Children often begin to think of partitioning
in terms of sharing with friends with each
getting the same amount
Partners
for Mathematics Learning

37. Paper Partitioning – Part 137
Paper Partitioning – Part 1
 How many times do you think you can fold
the patty paper in half?
 How many equal parts would be created
with the number of folds you predicted?
 Explain your reasoning
 Begin folding! Record the total number of
folds and the number of parts created
Partners
for Mathematics Learning

38. Part 2 – Let’s Experiment38
Part 2 – Let’s Experiment
 What would happen if you folded in thirds
each time instead of halves?
 What would happen if you folded the paper
in half and then in thirds?
 How many different ways could you fold a
piece of paper to create 24 equal parts?
 Which way uses the least number of folds?
The greatest?
Partners
for Mathematics Learning

39. Fair “Sharing”  Look at the folds of others at your table39
Fair “Sharing”
 Look at the folds of others at your table
 What similarities and differences do you
notice in your work?
 What problems might children have with
the term “fair shares” (when
defined mathematically as
equal portions)
Partners
for Mathematics Learning

40.  Sharing fairly collections of discrete items40
Equipartitioning in Kindergarten
 Sharing fairly collections of discrete items
between 2 people and reassembling them
 Typically done by dealing
 Look for one-to-one correspondence
 Put parts back together to get the original
collection
 No remainders
 New as an emphasis in K

41. Can We Share? Work with pairs of students41
Can We Share?
Work with pairs of students
 How can we share these objects fairly?
 What would be a fair share for each
person?
 How do you know?
 If you put the two fair shares back
together, how many objects would
we have?
Partners
for Mathematics Learning

42. 42
Take a Quick Look
Partners
for Mathematics Learning

43. What Did You See?  Compare your drawing with your partner  Remember…43
What Did You See?
 Compare your drawing with your partner
 Remember…
 Students with well-developed spatial sense tend
to solve tasks in more meaningful ways
(Reynolds & Wheatley, 1997)
 A network of mental imagery of mathematical
patterns and relationships helps students solve
problems in multiple ways
(Wheatley, 1996)
Partners
for Mathematics Learning

44. 44
Do You Have A Match?
Partners
for Mathematics Learning

45. Focusing on Problem Solving45
Focusing on Problem Solving
 Children come to school with an intuitive
knowledge of mathematics upon which to
build
 The teacher’s role is to act as a facilitator,
connecting new ideas to what students
already know
 CGI is based on
these assumptions
Partners
for Mathematics Learning

46. A Classroom Routine  A problem is presented46
A Classroom Routine
 A problem is presented
 Students solve using their own strategies
 Students explain and justify ideas usually
using informal language rather than
symbols
 Teachers focus on student thinking and
decide what problems to present next
based on that thinking
Partners
for Mathematics Learning

47. Problem Structures  CGI organizes problems into different47
Problem Structures
 CGI organizes problems into different
types
 The Essential Standards promote the use
joining, separating, and part-part-whole
problems
 These problems should be presented in
familiar contexts for students
Partners
for Mathematics Learning

48. How Are These Different?48
How Are These Different?
 Brittany had 5 marbles. Carlos gave her 3 more
marbles. How many marbles does Brittany have
altogether?
 Brittany had 5 marbles. How many more marbles
does she need to have 10 marbles altogether?
 Brittany had some marbles. Carlos gave her 5
more marbles. Now she has 9 marbles. How
many marbles did Brittany have to start with ?
Partners
for Mathematics Learning

49. Joining Problems  Result Unknown - Brittany had 5 marbles.49
Joining Problems
 Result Unknown - Brittany had 5 marbles.
Carlos gave her 3 more marbles. How many
marbles does Brittany have altogether?
 Change Unknown - Brittany had 5 marbles.
How many more marbles does she need to have
10 marbles altogether?
 Start Unknown - Brittany had some marbles.
Carlos gave her 5 more marbles. Now she has 9
marbles. How many marbles did Brittany have to
start with ?
Partners
for Mathematics Learning

50.  Result Unknown – Brittany had 10 marbles. She50
Separating Problems
 Result Unknown – Brittany had 10 marbles. She
gave 4 to Carlos. How many marbles does
Brittany have left?
 Change Unknown – Brittany had 9 marbles.
She gave some to Carlos. Now she has 2
marbles left. How many marbles did Brittany
give to Carlos?
 Start Unknown – Brittany had some marbles.
She gave 5 to Carlos. Now she has 2 marbles
left. How many marbles did Brittany have to start
with?
Partners
for Mathematics Learning

51. Part-Part-Whole Problems51
Part-Part-Whole Problems
 Whole Unknown – Brittany has 5 red
marbles and 4 blue marbles. How many
marbles does she have?
 Part Unknown – Brittany has 7 marbles.
5 are red and the rest are blue. How many
blue marbles does Brittany have?
Partners
for Mathematics Learning

52. Analyzing Problems  Work with a partner to decide the problem52
Analyzing Problems
 Work with a partner to decide the problem
type for each problem listed on the sheet
 When you have categorized the problems,
go back through them and rank them from
easiest to hardest
 Be prepared to explain your rankings to
the group
 Write your own sample problems
for Mathematics Learning

53. Choosing Problems  Problem difficulty can be influenced by:53
Choosing Problems
 Problem difficulty can be influenced by:
 Structure
 Location of the unknown quantity
 Size of the numbers
Partners
for Mathematics Learning

54. In the Classroom  Give students a variety of problem types in54
In the Classroom
 Give students a variety of problem types in
varying situations
 Allow give students the opportunity to
make up their own number stories
Partners
for Mathematics Learning

55.  Talk with your group about things you55
Incorporating CGI
 Talk with your group about things you
already do in your classroom that connect
to the ideas from CGI.
 Take notes on what others in your group
do that you would like to remember.
Partners
for Mathematics Learning

56.  What things are you already doing to56
Reflecting on Essential Standards
 What things are you already doing to
encourage students’ understanding of
number concepts?
 What things might you do differently with
your students as a
result of the revised
math standards?
Partners
for Mathematics Learning

57. Renee Cunningham Kitty Rutherford Robin Barbour Mary H. Russell 57
DPI Mathematics Staff
Everly Broadway, Chief Consultant
Renee Cunningham Kitty Rutherford
Robin Barbour Mary H. Russell
Carmella Fair Johannah Maynor
Amy Smith
Partners for Mathematics Learning is a Mathematics-Science
Partnership Project funded by the NC Department of Public Instruction.
Permission is granted for the use of these materials in professional
development in North Carolina Partners school districts.
Partners
for Mathematics Learning

58. PML Dissemination Consultants58
PML Dissemination Consultants
Susan Allman
Julia Cazin
Ruafika Cobb
Anna Corbett
Gail Cotton
Jeanette Cox
Leanne Daughtry
Lisa Davis
Ryan Dougherty
Shakila Faqih
Patricia Essick
Donna Godley
Cara Gordon
Tery Gunter
Barbara Hardy
Kathy Harris
Julie Kolb
Renee Matney
Tina McSwain
Marilyn Michue
Amanda Northrup
Kayonna Pitchford
Ron Powell
Susan Riddle
Judith Rucker
Shana Runge
Yolanda Sawyer
Penny Shockley
Pat Sickles
Nancy Teague
Michelle Tucker
Kaneka Turner
Bob Vorbroker
Jan Wessell
Daniel Wicks
Carol Williams
Stacy Wozny
Partners
for Mathematics Learning

59. 2009 Writers Partners Staff Kathy Harris Rendy King Tery Gunter59
2009 Writers
Partners Staff
Kathy Harris
Rendy King
Tery Gunter
Judy Rucker
Penny Shockley
Nancy Teague
Jan Wessell
Stacy Wozny
Amanda Baucom
Julie Kolb
Freda Ballard, Webmaster
Anita Bowman, Outside Evaluator
Ana Floyd, Reviewer
Meghan Griffith, Administrative Assistant
Tim Hendrix, Co-PI and Higher Ed
Ben Klein , Higher Education
Katie Mawhinney, Co-PI and Higher Ed
Wendy Rich, Reviewer
Catherine Stein, Higher Education
Please give appropriate credit to the Partners
for Mathematics Learning project when using the
materials.
Jeane Joyner, Co-PI and Project Director
Partners
for Mathematics Learning

60. Kindergarten PARTNERS for Mathematics Learning Module 6 Partners 60