Session 1: Childrens Mathematical Thinking

1
Session 1ChildrensMathematical Thinking
2
This Session

• Describes the development of key math concepts
• Explores the math from the childs perspective
• Encourages teachers observation of childrens
  mathematical behaviors
• Looks at challenges of school math
• Goal To promote teachers understanding of
  childrens mathematical learning as a basis for
  instruction

3
Lets Try Some Math

• Do a math problem in your head

?
4
Our Approach Is

• Drawn from research on childrens mathematical
  development.
• Based on these methods
• Tasks
• Flexible interviews
• Naturalistic observations
• Inspired by Piaget Vygotsky, but presents new
  views
• Video enhanced

5
Everyday Math in the Classroom

• What kind of mathematical thinking do you see?
• Arnie Video

6
What is the Math?

• Many different kinds of mathematical thinking
  occur in this video
• Geometry (shape, symmetry, spatial relations)
• Measurement
• Patterns
• Number concepts the idea of more
• Informal strategies such as estimation
• Math is more than just numeracy

7
Two Stories we will cover today

• The How Many Story
• The Addition Story
• Note Mathematics has many more stories to tell

2
1

8
The How Many Story

• What knowledge does a child need to understand
  the concept of how many?

9
Once upon a time

• The How Many story begins with an attempt to
  understand the concepts of more/less/and same.
• Examples from your experiences?
• Arnies (video) example?

10
Theres more to the concepts of more/less/the
same

• Watch this video clip
• What does Tina understand about the concepts of
  more / less / the same?
• Tina Video

11
What we should know about the more/less/same
concepts

• Spatial relationship is a sensible approach, but
  appearances can be misleading.
• Adults understanding of more is different from
  a childs understanding of more

12
The How Many story continues Enumeration
develops

• Enumeration
• The ability to count to understand how many
• Enumeration can help children
• Further explore the concept of how many
• Build on the concept of more and less
• Children come to school already knowing a lot, as
  youll see

13
Enumeration

• Henry What does he know about enumeration? What
  does he not know?
• Henry Video

14
Enumeration

• Viola What does she know about enumeration? What
  does she not know?
• Viola Video

15
What are the rules for Enumeration?

• Say the number words in the proper order
• Count objects in any order
• One-to-one correspondence
• Count each once and only once
• Match up each number word with a thing

16
Enumeration rules (continued)

• Inclusion of Set/Cardinality
• Last number refers to set as a whole
• Rearrangements do not matter
• You can count any discrete unit

17
Informal strategies

• Subitizing Seeing the amount
• Pushing aside with a finger
• Putting objects in a line
• Counting on fingers
• Grouping Split objects in two convenient groups
  and count
• Unitizing Grouping by 10s

18
How many are here?
19
How many are here?
20
What does each child know about number?
21
Problem 1
22
Problem 2
23
The Addition Story
and 3

• What does a child need to know to be able to
  add?

24
Addition

• Ophelia What does she know about addition?
• Ophelia Video

25
Addition

• Franko What does he know about addition?
• Franko Video

26
What Ophelia and Franko can teach us about the
Addition Story

• Addition is an outgrowth of counting.
• When you enumerate, you add 1.
• Addition can involve some abstract ideas.
• For example, using mental images, or a mental
  number line.
• Children begin kindergarten with some informal
  strategies for addition already in place.

27
The Addition Story goes to school
2 1 3
28
School math
29
School math continued

• School math
• Written, arranged systematically, and has
  explicit rules and procedures
• Good for remembering what was done and
  communicating with others
• Not totally inventible and must be taught
• Jewels of the Culture or Imposition from the
  Outside?

30
Strategies for solving small number problems

• Small number addition problems are
• 1 digit 1 digit
• Why consider them?
• Memorization strategy (4 2 6)
• Counting strategies already used
• Counting all (1,2,3,4 1,2 1,2,3,4,5,6)
• Counting on (5,6)

31
Mathematical Principles for solving small
number problems

• Derived Facts (2 4 ?)
• I know that 2 3 5, so 2 4 is just one more
  than 5. The answer is 6.
• Commutative Property (2 4 6 ? 4 2 6)
• Zero fact (Anything 0 Anything)
• Addition-Subtraction related facts (inverse
  relationship)
• (2 4 6 ? 6 4 ?)

32
Strategies

• What strategy does Joe use?
• Joe Video

33
Who is right?
34
Another issue

• What do these children see when they look at the
  symbols and ?
• Equals Video

35
Symbols A new language

• While learning number facts, children are also
  learning symbols
• Symbols are a new language for children
• Equals sign () can be particularly difficult
• Children think equals means makes
• Adults think equals means equivalence

36
Johnny had 5 apples. Then he picked 2 more
apples. How many apples does he have now?

• Word problems must be interpreted and then
  translated.
• An equation (such as 5 2 7) is a static
  representation of a dynamic operation.

37
Strategies for solving larger number problems

• Invented strategies
• Rearrangements into simpler forms
• child composes and decomposes numbers, showing an
  understanding of the Part-Part-Whole concept
• Counting methods (Laborious, but still done)

38
Algorithms taught in school

• Algorithm
• Conventional and foolproof procedure
• Some children learn to do algorithms properly but
  do not really understand them
• Other children cannot do the algorithm but can
  intuitively understand the problem.
• Bugs (systematic mistakes) may occur

39
Tanya 19 15

• How does Tanya solve this problem? What
  knowledge about addition does she possess?
• What does this clip tell us about childrens use
  of conventional addition algorithms?
• Tanya Video

40
Julie 26 17

• How does Julie solve this problem? What
  knowledge about addition does she possess?
• What does this clip tell us about childrens use
  of conventional addition algorithms?
• Julie Video

41
Examples of bugs
Student 1
Student 2
54 16 610
44 27 611
46 29
21 5 8
33 19 16
70 24
42
To sum up (so to speak)

• Math in K-2 is multifaceted (not just about
  number)
• Everyday math is complex involving BOTH
  concrete and abstract ideas
• Children bring their own ideas into school and
  interpret school math
• We need to help children achieve proficiency with
  understanding
• Understanding how kids think about math will help
  us to teach them better.