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Heller, P., Ahlgren, A., Post, T., Behr, M., & Lesh, R. (1989, March). Proportional Reasoning: The Effect of Two Context Variables, Rate Type and Problem Setting. Journal for Research in Science Teaching, 26(1), 205-220.

 

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Tables and Figures

 

 
  Figure 2
 
Table I

Some Rate Types Found in Textbook Problems

Definition
Examples
Distribution: divide something equally among people, groups, or objects. cookies per person
acres of land per family
electric charge per electron
   
Packing: spread something evenly over a spatial dimension books per foot of shelf space
mass of aluminum per
cubic centimeter
   
Package Size: count or measure the amount of something in a "package" (when things are not necessarily spread evenly over a spatial dimension). candies per box
ounces per bottle
electrons per neutral iron atom
   
Exchange: exchange or trade one kind of thing for another kind of thing.unit prices (buying goods or services)
money earned per week (salary)
   
Mixture: mix two (or more) things together into some whole, or separate a whole ito its constituent parts. lemonade mix per glass of water
questions right per questions wrong on a test
molar concentration of acids
   
Speed: how fast or slow an object moves or an event takes place. gallons of water emptied from a tank per hour
speed of light
   
Consumption/Production: how efficiently something is consumed (used up) or produced (made). gallons of oil burned by a furnace per hour
electrical energy used per hour (power)
   
Scaling: enlarge or shrink something billions of dollars per mm on a graph of national debt
inches per mile map scale
   
Conversion: convert a quantity from one unit of measurement into a different unit of measurement. centimeters per inch
square feet per acre
pounds per dollar

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Table II
Rate Types and Problem Settings Used in This Study

 
Problem Setting
Rate Type
   
 
More Familiar
Less Familiar
Speed
running laps
driving cars

Variable 1

distance (laps)
distance (miles)

Variable 2

time (minutes)
time (hours)
Exchange
buying gum
buying records

Variable 1

No. pieces gum
No. records

Variable 2

price (cents)
price (dollars)
Consumption
gas mileage of truck
oil burning furnace

Variable 1

distance (miles)
volume (gallons)

Variable 2

volume (gallons)
time (hours)

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Table III
Data Used in Numerical Proportional Reasoning Problems

 
Person 1
Person 2
Problem Format
       
   
V1
V2
V1
V2
Missing Value          
  #1
4
20
12
?
  #2
8
24
?
6
  #3
3
12
?
36
Numerical Comparison          
  #1
8
32
2
10
  #2
3
15
9
45
  #3
6
24
2
6

Note. V1 and V2 are the two variables in the problem setting (for example, number of pieces of gum and price in cents).

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Table IV
Direction of Changes in the Numerator and Denominator of the Rates in the Qualitative Questions

Problem Format
Numerator
Denominator
Qualitative Rate Change    

#1

less
more

#2

less
less

#3

same
more

#4

same
less
#5
more
less
Qualitative Comparisons    

#1

more
more

#2

same
less

#3

more
more
#4
more
less
#5
same
more

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Table V
Means and Standard Deviations for Context Groups on Proportional and Directional Reasoning Scales

 
Score
Context Group
(Ratio type, and Setting)

Proportional Reasoning
(range 0 - 6)

Directional Reasoning
(range 0 - 8)

Exchange    

more familiar

4.23 (1.71)
6.85 (1.48)

less familiar

4.38 (1.64)
6.53 (2.21)
Speed    
more familiar
3.95 (1.89)
6.28 (1.75
less familiar
3.40 (1.81)
5.35 (2.26)
Comsumption    
more familiar
3.54 (1.89)
6.00 (2.19)
less familiar
2.97 (1.44)
4.00 (1.96)

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Table VI
Analysis of Variance for Directional Reasoning Score

Source
MS
df
F
P

Main effects
68.7
5
20.1
<.001
Rational no. level
74.9
2
22.0
<.001
Ratio type
54.6
2
16.0
<.001

Setting

84.3
1
24.7
<.001
2-way interactions
4.2
8
1.2
.29
Rational no. x Ratio type
2.2
4
0.6
.64
Rational no. x Setting
0.6
2
0.2
.84
Ratio type x Setting
13.4
2
3.9
.02
3-way interaction
2.4
4
0.6
.65
Error
3.4
219
 
 


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Table VII
Analysis of Variance for Proportional Reasoning Score

Source
MS
df
F
P

Main effects
41.5
5
17.9
<.001
Rational no. level
75.3
2
32.5
<.001
Ratio type
25.2
2
10.9
<.001

Setting

6.5
1
6.5
.10
2-way interactions
2.7
8
1.2
.32
Rational no. x Ratio type
1.4
4
0.6
.66
Rational no. x Setting
3.6
2
1.6
.21
Ratio type x Setting
4.3
2
1.8
.16
3-way interaction
0.9
4
0.4
.82
Error
2.3
219
 
 

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Fig. 1
   
   
1a. X is similar to Y.
1b. X is necessary, but not sufficient, for Y.
   
   
1c. X is sufficient, but not necessary, for Y.
1d. X is similar to Y, but restricted variance gives appearance of 1c.
   
Fig. 1. Some possible relationships of two scales.

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Fig. 2.

Fig. 2. Context group means for directional and proportional reasong tests.

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