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Heller, P., Post, T., Behr, M., & Lesh, R. (unpublished). The effect of two context variables on qualitative and numerical reasoning about rates.


The Effect of Two Context Variables on Qualitative and
Numerical Reasoning About Rates
 

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Tables and Figures
 
 
 

Table 1
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. (buying goods or services)
money earned per week (salary)
   
Mixture: mix two (or more) things together into some whole, or separate a whole to 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

This table appeared in Heller, P, Ahlgren, A., Post, T., Behr, M., & Lesh, R. (1989). Proportional Reasoning: The effect of two context variables, rate type and problem setting, Journal of Research in Science Teaching, 26(3): 205-220.
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Table 2
Examples of Parallel Items on the Rational Number and Context Test


Rational Number Test

Context Testa


Missing-Value Problems

 

Place a number in the box that will make the two fractions equal.

4/20 = 12/[ ]

Steve and Mark drew the same size maps of their classroom. Steve drew the 20 foot long windows as 4 inches long on his map. How far apart are two desks that Mark drew as 12 inches apart on his map?

Numerical-Comparison Problems

Circle the smaller fraction. If they are equal, circle both fractions.

 

6/24     2/6

Alice and Jenny hammered (equally spaced) nails into different boards. Alice hammered a line of 24 nails into a board 6 ft long. Jenny hammered a line of 6 nails into a board 2 ft long. On which board are the nails hammered closer together?

 

a) Alice

 

b) Jenny

 

c) Their nails are spaced exactly the same.

 

d) There is not enough information to tell.


Fraction Test

Context Testa


Directional Questions

What will happen to the fraction 7/8 if the top number gets smaller and the bottom number gets bigger?

QRb: If Nick mixed less green tint with more white paint than he did yesterday, his green paint would be

a) Fraction gets bigger

a) a darker shade.

b) Fraction gets smaller

b) a lighter shade.

c). Fraction stays the same

c) exactly the same shade.

d) There is not enough information to tell.

d) There is not enough information to tell.

 

What will happen to the fraction 3/4 if the top number gets bigger and the bottom number gets bigger?

QC: Nancy drove more miles than Kathy. Nancy drove for more time than Kathy. Who was the faster driver?

a) Fraction gets bigger

a)Nancy

b) Fraction gets smaller

b)Kathy

c) Fraction stays the same

c)They drove at exactly the same speed.

d) There is not enough information to tell.

d) There is not enough information to tell.


a Each version of the Context Test used only one rate type. For illustrative purposes, four different rate types have been shown on this table.

b QR stands for qualitative-rate-change question and OC qualitative-comparison question.

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Table 3
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)
Mixture
mixing lemonade
mixing paint

Variable 1

concentrate (teaspoons)
1 tint (drops)

Variable 2

water (ounces)
white paint (ounces)
Scaling
making classroom map
reading city map

Variable 1

length (inches)
length (inches)

Variable 2

distance (feet)
distance (miles)
Density
movie lines
nails in board

Variable 1

objects (people)
objects (nails)

Variable 2

line (yards)
board (feet)
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Table 4
Means and Standard Deviations for Context Groups on Qualitative Reasoning and Numerical Reasoning Scales
   
Score
Context Group
(Rate type, Setting and Grade)

Qualitative Reasoning
(range 0 - 8)

Numerical Reasoning
(range 0 - 8)

Speed      
running laps Grade 7
5.86 (1.61)*
4.84 (2.20)
  (n = 56)    
  Grade 8
6.31 (1.70)
6.02 (1.92)
  (n = 58)    
driving cars Grade 7
4.86 (1.77)
4.12 (2.10)
  (n = 57)    
  Grade 8
4.90 (1.59)
5.53 (2.21)
  (n = 61)    
Linear Density      
movie lines Grade 7
5.69 (1.18)
4.25 (2.43)
  (n = 48)    
  Grade 8
6.06 (1.56)
5.83 (2.38)
  (n = 64)    
nails in board Grade 7
5.41 (1.53)
4.50 (2.03)
  (n = 46)    
  Grade 8
6.22 (1.54)
5.62 (2.23)
Mixture      
lemonade Grade 7
5.21 (1.42)
3.70 (2.38)
  (n = 5 6)    
  Grade 8
5.54 (1.45)
5.49 (2.33)
  (n = 57)    
paint Grade 7
5.02 (1.49)
3.80 (2.23)
  (n = 54)    
  Grade 8
5.30 (1.57)
5.26 (2.50)
Scaling      
classroom map Grade 7
2.04 (1.65)
3.06 (2.18)
  (n = 47)    
  Grade 8
2.67 (1.91)
4.39 (2.24)
  (n = 61)    
road maps Grade 7
3.16 (1.84)
3.54 (2.02)
  (n = 57)    
  Grade 8
3.52 (1.74)
4.94 (2.09)
  (n = 62)    

* Standard deviations
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Table 5
Means and Standard Deviations for Rational Number Groups on Qualitative Reasoning and Numerical Reasoning Scales
   
Score
Rational Number Ability and
Rate Type Groups

Directional Scale
(range 0 - 8)

Numerical Scale
(range 0 - 8)

Low Ability      

Speed

n = 67
4.30 (1.95)*
3.46 (1.97)

Density

n = 59
4.91 (1.76)
2.88 (1.87)

Mix

n = 61
4.51 (1.56)
2.33 (1.62)

Scale

n = 58
2.33 (1.51)
2.50 (1.84)
Medium Ability      

Speed

n = 80
5.49 (1.51)
5.23 (1.93)

Density

n = 85
5.72 (1.48)
4.83 (2.08)

Mix

n = 95
5.14 (1.40)
4.54 (2.04)

Scale

n = 75
2.54 (1.74)
3.73 (1.97)
High Ability      

Scale

n = 85
6.39 (1.25)
6.41 (1.60)

Speed

n = 84
6.67 (1.24)
6.89 (1.24)

Density

n = 72
6.08 (1.15)
6.65 (1.58)

Mix

n = 92
3.64 (1.98)
5.45 (1.86)

* Standard deviations
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FIGURE 1

 
Figure 1. Context group means for seventh and eighth grade students on the directional scale.
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FIGURE 2

 
Figure 2. Context group means for seventh and eighth grade students on the numerical scale.
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Figure 3

 
Figure 3. Rate-type group means on the numerical scale for three levels of rational number ability.
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FIGURE 4

 
 
 
Figure 4. Graphs of the percent of students who solved correctly (a) missing-value problems, (b) numerical-comparison problems, and (c) directional questions on the context test versus the percent of students who solved the numerically and/or structurally equivalent problem correctly on the rational number test.
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