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


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Tables
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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):
205220. 
(top) 
Table 2
Examples of Parallel Items on the Rational Number and Context Test


Rational
Number Test

Context
Test^{a}


MissingValue
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?

NumericalComparison
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
Test^{a}


Directional
Questions

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

QR^{b}:
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 qualitativeratechange question and OC qualitativecomparison
question.


(top) 
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)


(top) 
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 

(top) 
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 

(top) 
FIGURE 1


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


(top) 
FIGURE
2

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


(top) 
Figure
3


Figure
3. Ratetype group means on the numerical scale for three levels
of rational number ability.


(top) 
FIGURE
4






Figure
4. Graphs of the percent of students who solved correctly (a) missingvalue
problems, (b) numericalcomparison 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.


(Top) 