Learning Mathematical Problem Solving

1
Learning Mathematical Problem Solving

• David. C. Gary, 1994.
• Childrens Mathematical
• Development Research and Practical Applications.

2
????

• ???????,?????????????
• ??????,?????????????
• ????????????????????????

3
Focuses

• Identify features of the problems themselves that
  made word problems difficult than number problems
• how the child understands and conceptually
  represents word problems and how this affects the
  childs problem-solving processes

4
Arithmetic Word Problem Solving

• Influence of general structural features of the
  problem such as length,
• Influence of semantic structure
• effect of conceptual knowledge
• developmental considerations and sources of errors

5
Problem FeaturesJerman Rees, 1972

• Linguistic features (e.g., no. of words in the
  problem)
• computational demands (e.g., no. of required
  arithmetic operations)
• accounted for 87 of the variability for a group
  of 5th graders

6
Different Features predict problem difficulty for
elementary and junior secondary school
childrenJerman and Mirman, 1974

• Grades 4 to 6 computational demands and no. of
  mathematical terms (diameter, remainder)
• Grades 7 to 9 No. of words between the first
  number and the last number presented in the
  problem (30 of the variability)-- working memory
  demand

7
Semantic Structure

• Meaning of the statements in the problem and
  their interrelationships.
• Most word problems (addition and subtraction) can
  be classified into 4 general categories
• change
• combine
• compare
• equalize

8
Change

• Amy had two candies. Mary gave her three more
  candies. How many candies does Amy have now?
• Then she gave three candies to Mary. How many
  candies does Amy have now?
• Amy had two candies. Mary gave her some more
  candies. Now Amy has five candies. How many
  candies did Mary give her?
• Mary had some candies. Then she gave two candies
  to Amy. Now Mary has three candies. How many
  candies did Mary have in the beginning?

9
Combine

• Amy has two candies. Mary has three candies. How
  many candies do they have altogether?
• Amy has five candies. Amy has two candies. How
  many fewer condies does Amy have then Mary?
• Amy has two candies. Mary has one more candy than
  Amy. How many candies does Mary have?
• Amy has two candies. She has one candy less than
  Mary. How many candies does Mary have?

10
Equalize

• Mary has five candies. Amy has two candies. How
  many candies does Amy have to buy to have as many
  candies as Mary?
• Mary has five candies. Amy has two candies. How
  many candies does Mary have to eat to have as
  many candies as Amy?
• Mary has five candies. If she eats three candies,
  then she will have as many candies as Amy. How
  many candies does Amy have?

11

• Amy has two candies. If she buys one more candy,
  then she will have the same number of candies as
  Mary. How many candies does Mary have?
• Amy has two candies. If Mary eats one of ther
  candies, then she will have the same number of
  candies as Amy. How many candies does Mary have?

12
????

• ??????, ????? ????
• ?????????, ???????
• ??????????????

13
Characteristics of Change problems

• Change-join problem change involves adding to
  the initially defined set
• change-separate problem involves removing a
  subset from the originally defined set.

14
Characteristics of Combine Problems

• Involve the same basic arithmetic
• static relationship rather than the implied
  action
• does not change what each individual has
• involves developing a superordinate

15
Characteristics of Compare Problems

• Quantity of the sets does not change

16
Semantic Structure and Problem-solving Strategies
(Carpenter et al., 1981)

• Little relationship for addition problems since
  most of the problems were solved by means of
  counting-all strategy.
• For subtraction problems, 76 of the change
  problems were solved by using separating-from,
  11 by using matching
• For compare problems, 49 used matching, 23 used
  separating-from

17
Addition Word Problems DeCorte Verschaffel,
1987

• Adding method construction of a set of objects
  to represent the augend, then adds to this set a
  number of objects equal to the addend. Then
  counts all the objects.
• Joining representing the value of the augend and
  addend with separate sets of blocks, physically
  moving these sets together, then counting all the
  blocks
• No-move the sets of blocks are not physically
  moved together.

18

• The adding method method was used on 75 of the
  change-problems trials
• no-move methods was used on 68 of the
  combine-problem trials

19
Problem Solving Process (Mayer, 1985)

• Problem translation
• problem integration,
• solution planning
• solution execution

20
Building of a problem representation

• Understanding of the text
• meaning and mathematical implication of specific
  words (e.g., more implies addition)
• structure of the entire problem
• Order and manner with which the information is
  presented can make the problem more or less
  difficult.

21
Which one is more difficult?
22
Representation of Problem A
1st sentence
2nd sentence
Quantity
who
How many
what
Mary
Candy
?
23
Representation of Problem B
1st sentence
Converted to
2nd sentence
24
Problem Integration
Amy
Mary
Amy
She Mary ?
25
Solution Planning

• Choosing the most appropriate strategy, such as
  counting all, for solving the problem.
• Riley et al. Argued a 3rd type of schema --
  Action schema
• link between the relational schema and the actual
  strategy used.
• Provide implicit knowledge about the results that
  the various strategies produce, and the contexts
  in which they are most typically used.

26
Other Factors

• Reading skills
• understanding of basic counting and arithmetic
  concepts
• working memory capacity
• strategies available

27
Algebra
28
Dimensions that Algebraic Problem Solving differs
from the Solving of Arithmetic Word Problems

• Translation of algebraic word problems into
  equations tends to be more difficult
• the greater complexity of algebraic problems
  results in a larger problem space (all the
  procedures and rules that the person knows about
  a particular type of problem, as well as all of
  the different ways that the problem can be solved)

29
Processes in solving algebraic word problems

• Problem translation transforming the meaning of
  the problem statements into a set of algebraic
  equations, includes
• problem-translation
• problem-integration
• solution planning
• Problem Solution the actual use of algebraic or
  arithmetical procedurs to solve the resulting
  equations.

30
Problem Translation

• Guided by schemas mental representations of
  similarities among categories of problems
• basic structures most problems centred around 4
  types of statements
• assignment statements assign a numerical value
  to a variable
• relational statements specifies a single
  relationship between 2 variables
• questions
• relevant facts information presented that might
  be needed to solve the problem

31

• Translation errors frequently occur during the
  processing of relational statements.
• For example There are six times as many
  students as professors at this university.
  (Clement, 1982)
• S no. of students
• P no. of professors
• equation 6SP? S6P?

32
Another Example (Mayer, 1982)
Laura is 3 times as old as Maria was when Laura
was as old as Maria is now. In 2 years Laura will
be twice as old as Maria was 2 years ago. Find
their present ages.