Transcript Effects of Problem Context on Strategy Use within Functional Thinking

Katherine L. McEldoon, Caroline Cochrane-Braswell
& Bethany Rittle-Johnson
Outline
 Current Study
 Three different problem formats
 Their effect on problem solving strategy use
 What is Functional Thinking
 What makes it challenging
 What facilitates student understanding
2
Functional Thinking
The table shows how the “In” numbers are
related to the “Out” numbers. When a 38 goes
in, what number comes out?
A.41
B.51
C. 54
D. 77
• A type of mathematical thinking which
focuses on the relationship between two (or
more) varying quantities, specifically the
kinds of thinking that lead from specific
relationships to generalizations of that
relationship across instances (Smith, 2008).
77
• Encapsulates important core components of
early algebraic reasoning, such as
generalization and covariation.
Out = (In x 2) + 1
Y = 2X + 1
3
Functional Thinking
Performance – Grade 4
The table shows how the “In” numbers are
related to the “Out” numbers. When a 38
goes in, what number comes out?
A.41
B.51
C.54
D.77
4
National Assessment of
Educational Progress
(NAEP), National
Performance results in
Mathematics at Grade 4;
2007
Functional Thinking in
Elementary School
Two Possible Relationships
 Recursive Relationship
Y1-Y2
 Functional Relationship
X1-Y1
X
Y
2
8
3
12
4
16
5
20
6
24
7
28
20
5
Fostering Functional
Thinking
 Alternate Surface Feature Format
 In novices, surface features compete with deep
X
Y
3
8
5
16
features attentionally (Sloutsky & Yarlas, 2000)
•
Story Problem Format
•
•
6
Depends on linguistic proficency (Kintsch & Greeno,
1985; Koedinger, Alibali & Nathan, 2008)
When readers can handle it, they help support
informal solution methods (Carraher et al., 2003)
9
25
Julia is selling boxes of cookies as a fundraiser for her school. Three
boxes cost $9 and five boxes cost $15. Her aunt wants to buy 6 boxes,
and her neighbor wants to buy 8. How much do they have to pay? Julia
sells 25 boxes in all. How much did she earn?
6
Current Study
 Investigate the effect of problem context on
problem solving strategy within function table
problems
 Contexts (between subjects)
 Baseline
 Alternate Surface
 Story
7
Contexts & Hypotheses
Baseline
Alternate Surface
X
Y
X
Y
2
6
2
6
3
7
4
8
4
8
5
9
5
9
7
11
6
8
14
14
25
41
25
41
Story
Cost of
Present
Cost of
Present w
Gift
Wrapping
2
6
3
7
4
8
5
9
6
14
25
What is the rule for this function table?
41
At a gift shop, you
can pay extra to have
your present giftwrapped, as shown in
the table above. What
is the total cost of the
present with giftwrapping if the cost of
the present is $6?
$14? What about
$41? If the total cost
of a present with giftwrapping is $25, what
was the cost of the
present itself?
Hypotheses
Most recursive
strategy use,
less functional
More functional
strategy use,
less recursive
More functional
strategy use,
less recursive
8
Method
 Participants
 232 students in grades two through six in a
middle class community
 Procedure
 Three different forms of the assessment
(baseline, alternate surface, and story)
were randomly distributed to students
during their normal math class as a whole
class activity.
9
Assessment
 Three isomorphic versions of the assessment
 Baseline (or)
 Alternate Surface (or)
 Story
 Three function table problems
 Additive Y = X + 4 (and)
 Multiplicative Y = 3X (and)
 Combo Y = 3X + 2
10
Strategy Coding
 The student’s strategy code was based on their
function table values and their rule for the table
Functiona
X
lY
Recursive
X
Y
2
6
2
6
3
7
3
7
42%
4
8
4
8
5
9
5
9
Rule: Add 4 to
the X number
to get the Y
number
6
10
6
10
14
18
14
11
21
25
15
25
41
45
41
26
Other
47%
11%
Rule: Add 1 to
the number
before it to get
the next one
We compared the effect of problem context
(baseline, alternate surface, and story problem) on
strategy use (functional or recursive)
11
Strategy Use by
Grade & Condition
Functional
Recursive
75%
30%
50%
Baseline
25%
0%
Baseline
15%
Alt Surface
Alt Surface
Story
Story
0%
2nd & 3rd
5th & 6th
2nd & 3rd
5th & 6th
12
Results- Functional Strategy
Percent Strategy Use
60%
40%
20%
0%
Baseline
Functional
Alt Surface
Stor
y
 Overall, the alternate surface context was the most
conducive to the functional strategy
 significantly more than the story contex
 marginally more than the baseline
 There was no difference in strategy use in the baseline and
story contexts.
13
Results- Recursive Strategy
Percent Startegy Use
25%
13%
0%
Recursive
Baseline Alt Surface
Story
 The baseline context elicited the most recursive strategy
use.
 Significantly more than alternate surface
 No sig. difference compared to story
 There was no difference in the alternate surface and story
contexts.
14
Conclusions
Strategy Use by Context
60%
40%
Baseline
20%
Alternate Numeric
Story
0%
Recursive
Functional
 The baseline context resulted in the most use of the
naïve recursive strategy
 The alternate surface context elicited the most
functional strategy use
 Story context did not increase recursive strategy
use, but it also did not encourage functional strategy
use
15
Thank you