Vicky
Kouba, O. Cezikturk, S. Sherwood, and C. Ho
Teaching, learning, and assessing mathematics in context requires careful thought and
planning. This paper will address context in assessment: assessment using tasks or
situations that illustrate the applicability of mathematics in other school subjects, in
the world of work, or in the students’ personal lives.
Context should be employed carefully, however. As we illustrate in the following
examples, context can also obscure the mathematics and divert the intended direction of
the development of mathematical understanding.
We begin with an analysis of students’ responses to tasks designed to test their
understanding of mathematical principles. After illustrating how the contexts in which the
tasks are posed influence students’ performance, we then consider the implications of
that analysis for teaching.
In the current standards-based reform movement, students’ mathematical
understanding often is measured using tests the students’ teachers did not design.
Tests of this sort that "fall from the sky" require students to answer questions
in unfamiliar contexts and without benefit of the knowledge of the teacher’s
expectations. All of the student responses reported here are based on tasks that underwent
careful design and pilot-testing, but that were not developed by the students’
teachers. The tasks are posed in contexts with which the students are not necessarily
familiar. As these tasks illustrate, a student’s familiarity with the context can
have profound influence on the student’s performance.
The Mathematics National Assessment of Education Progress (NAEP) is an example of
an examination that "falls from the sky." Students’ responses to a task
from the1996 NAEP Mathematics Assessment (NAEP 1996) illustrate the influence of context
on students’ interpretation and response to a mathematics task.
DOGHOUSE TASK
Julie wants to fence in an area in her yard for her dog. After paying for the
materials to build her doghouse, she can afford to but only 36 feet of fencing. She is
considering various different shapes for the enclosed area. However, she wants all of her
shapes to have 4 sides that are whole number lengths and contain 4 right angles, All 4
sides are to have fencing. What is the largest area that Julie can enclose with 36 feet of
fencing? Support your answer by showing work that would convince Julie that your area is
the largest.
Response A
In order to achieve maximum area with 36 feet of fencing, the best plan would be to
make a square area. The area inside would be 81 ft2 while in other plans, the
area would decrease.
Response B
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This way the dog has enough room to run and play. Plus Julie has room for the dog
house.
This task was designed to test students’ understanding of the principle that of
all possible rectangles, a square has the maximum area for a given perimeter. Student
A’s answer demonstrates understanding of the principle. The task developers intended
that the maximization of area be the first (and only) condition to be met. Student
B’s response, on the other hand, demonstrates an understanding of the requirements of
confined dogs. That is, Student B apparently decided the primary consideration was that
the enclosure allow the dog enough room to run. Anyone who has boarded a dog at a kennel
knows that dog enclosures are constructed to maximize length. Thus, Student B’s
answer is a reasonable one for the practical context. But is it reasonable in the
context of taking a mathematics test? In most test-taking situations, as was the
case with this particular NAEP item, being alert to the test-maker’s intention is
essential to scoring well. Although we often assume that a familiar context will make
understanding a mathematics principle more easily demonstrated, the dog-enclosure task
illustrates how a student’s personal knowledge about the context obscures the
test-maker’s intent.
Invisibility and Context
Another example illustrates how context can render the mathematics invisible. This
task and the student’s response are data from a research study investigating
students’ understanding of multiplication and division (Kouba, 1991). A second grade
boy, who had responded correctly to multiplication and division tasks presented in
context, was asked to perform the following task:
You have 18 apples that you want to have 3 horses share fairly. You want to use up
all the apples. How many apples does each horse get?
The boy was quiet for a moment and then responded, "One."
The interviewer, a bit nonplused, repeated, "One? Are you sure?"
"Yes," replied the boy, "one, because if you give a horse more than one
apple at a time it could get sick."
The boy’s previous responses were evidence that the boy understood multiplication
and division. However, the context in which this item was presented made the mathematics
invisible to him. The youngster, who lived on a horse farm in rural New York, had a
knowledge base about horses quite different from that held by the item designer. No amount
of coaxing on the interviewer’s part could get the boy to violate the "one apple
to a horse" principle.
Avoidance and Context
The influence of context on performance is observed in older students as well and
may be even more pronounced when the context is intentionally an integral part of the task
as the following example illustrates. Pre-service mathematics teachers in a graduate
methods course were asked to respond to the following task:
SOLVENT TASK
Solvents are used in many industrial and domestic situations to clean food containers,
machines and tools. The use of solvents, even harmless ones such as water, pose serious
environmental problems. The technological problem posed by the use of solvents is how can
the use of solvents be minimized? A solvent-use situation that you have encountered in
your daily life is cleaning a paint brush.
Use the information below about cleaning a paint brush to develop a mathematical model
of the process.
A paintbrush has just been used and the owner wishes to clean it. After the brush
has been scraped against the side of the paint can, it still contains 4 fluid ounces of
paint. The owner dips it into a quart of clean solvent and stirs well until the diluted
paint solution is uniform. After draining, the brush still holds 4 fluid ounces, part of
which is paint and part solvent, since the diluted solution is uniform. The process is
repeated with a fresh quart of solvent. Explain how you went about developing the model.
One pre-service mathematics teacher answered:
I am really not sure how to answer this question. I don’t think solvents should be
used. With the situation given, I think the brush is not really being cleaned but truly
being covered by solvent and eventually the solvent overcomes the paint and breaks it
down, the pain brush is cleaned depending on the amount of solvent used. I feel that water
could be used instead of solvents, If you wash the brush out with water long enough, it
will get cleaned out and it is safer for everyone. (Adapted from the Engineering Concepts
Curriculum Project, 1970).
This response by a pre-service mathematics teacher is typical and illustrates two
important features. First, it demonstrates that the typical mathematics graduate student
does not understand solvents and solutions well enough to be confident with the
interpretation of the situation. Second, it shows how a typical graduate student chooses,
to discuss the environmental issues rather than to develop a mathematical model for the
rinsing process.
Many students ignore even the environmental issues and question the task’s
appropriateness as a measure of mathematical understanding. These students are critical of
the amount of reading, writing, and information about solvents required for successful
performance.
Why do students discuss the environmental issues or critique the task rather than
develop the mathematical model? One plausible explanation is based on the fact that the
majority of pre-service teachers find the mathematical task of developing a model for
rinsing difficult or impossible. Thus, we propose that more sophisticated individuals may
respond to a task using context to avoid revealing that they do not have the mathematics
knowledge required to answer the question posed.
Assumptions and Context
Tasks posed in context often require individuals to make assumptions on which to
base their responses. These assumptions are often implied rather than specifically stated.
When the assumptions of the student are the same as those of the task designer, the
response matches the scoring standard and the response gets the full score. However, as
our next example illustrates, if an individual bases a response on an alternative
assumption, it may be marked wrong.
A group of middle and high school mathematics teachers in a professional development
workshop were developing scoring standards for mathematics tasks. They began the
development process by writing individual responses to the following task:
PIZZA TASK
Your class has decided to have pizza for its end-of-the-year party. You are trying
to decide which pizza store has the cheapest price. The local pizza stores and their
prices are listed below.
Pizza Prices
Sam's Pizza House $8.50 8 slices per pizza
Pizza Palace $10.25 10 slices per pizza
Pizza & Stuff $6.25 6 slices per pizza
Assume that there are 30 students in your class and that each person (including our
teacher) will eat two slices. Also assume that the slices in the pizzas from the different
stores are the same size. Where should you buy the pizza? Please show all your work and
write a brief description of how you decided on your answer.
S0LUTION
Sam's Pizza House offers pizza at $8.50 for 8 slices. That comes to $1.06 per slice.
Pizza Palace offers pizza at $10.25 for 10 slices. That comes to $1.02 per slice.
Pizza & Stuff offers pizza at $6.25 for 6 slices. That comes to $1.04 per slice.
From this analysis it appears that Pizza Palace is the cheapest source of pizza.
However we must also consider the number of pizzas that must be purchased for each person
to have two slices.
The class and the students together will eat 62 slices of pizza. That means they would
have to buy: 8 pizzas from Sam's Pizza House for $68.00; 7 pizzas from Pizza Palace for
$71.75; 11 pizzas from Pizza & Stuff for $68.75
Thus it appears that the cheapest source of pizza considering the number of pizzas that
must be purchased is Sam's Pizza House. (Danielson, 1997, pp. 146-147)
Then the teachers worked in two groups to develop consensus solutions, with the middle
school teachers in one group and the high school teachers in the other. The middle school
teachers assumed that only whole pizzas could be purchased, whereas the high school
teachers assumed that the pizza could be purchased by the slice. The different assumptions
yielded different answers. When the teachers shared answers, they acknowledged that
neither group had even considered the other's approach.
The teachers discussed the task designers’ scoring guide, which is based on just
one of the possible assumptions. The teachers concluded that if they, as adults, were not
aware initially that alternative appropriate interpretations of the task were possible, it
is unfair to expect their students to analyze tasks for alternative interpretations and
weigh the multiple possibilities. At least, it is unfair until students have had an
opportunity to develop skills in exploring multiple assumptions or interpretations of
tasks, and until they learn that presenting multiple perspectives is a performance
expectation,
Teaching about Context
Our examples illustrate several ways in which placing tasks in context makes it
more difficult for students to demonstrate their understanding of mathematics. One might
argue that we just need to design better tests, ones with tasks that are absolutely clear.
However, contexts are rich and complex, and rightly so. Almost any item, no matter how
carefully designed or fully piloted, still is open to alternative interpretation by a
divergent, creative thinker. The challenge posed by setting mathematics in context is
multifaceted and consequently requires diverse teaching approaches to provide students
with the skills necessary to complete the tasks. Two approaches are described below.
The first method addresses the instances where context obscures the task
developers’ intent. As a part of teaching students to see mathematics in practical
situations, teachers need to provide students with the opportunity to become context-wise.
Context-wise students remind themselves of the kind of test they are taking.
"Remember," the student must think, "this is a test of my understanding of
mathematics, not the design of dog enclosures or keeping horses healthy." This
approach can work for older students, but is not as helpful for younger students who have
difficulty some abstract levels of thinking.
In the second method, students learn to develop analytical skills for communicating
solution strategies and making assumptions explicit. Communication such as the kind that
occurred in the mathematics teacher workshop described above makes students and teachers
deeply aware of the interplay of mathematics and context. This kind of communication, this
sharing of why and how tasks are approached can be made part of the day-to-day mathematics
class. Developing the skills required to make assumptions explicit and to test the
assumptions helps meet the challenge of mathematics tasks with multiple defensible
interpretations of mathematics in context. In other words, students learn to identify and
state their assumptions, and students learn to give multiple answers when they see
multiple possibilities.
In a recent Foxtrot cartoon (written and drawn by Bill Amend) a student, Jason,
puzzles on all of the possible assumptions he might make in trying to solve a classic
"Station A to Station B train mileage" problem. He speculates on such variables
as the curvature of the earth, time zones, and planetary motion before constructing his
answer.
This speculation is precisely the nature of thinking about assumptions that can both
impede and/or expand a student’s capacity for mathematical reasoning.
Amend reminds us that the vast majority of the applied mathematics that students
encounter in mathematics classrooms and tests is contrived. The tasks, projects,
assessments, and activities students encounter in academic mathematics rarely come
directly from the students or from problems they pose. Thus, we, as adults, design
learning and testing situations to capture either our vision of daily life or our
approximation of the students’ vision of daily life. Because the contexts are
contrived and because they originate from the adults’ situated cognition, there are
many opportunities for misconceptions or multiple interpretations of a context. We are not
talking about responses with incorrect mathematics, rather we are concerned with response
where the mathematics, per se, is correct, but something else related to context has
"gone a bit awry" from what was intended.
When the miscommunication or multiple interpretation happens in the classroom, the
astute teacher can capitalize extemporaneously on the differences to build a rich
discourse about multiple perspectives, the value of being able to communicate clearly and
convincingly, and the need to listen with an open mind. Assessment environments require a
bit more foresight in terms of expectations and, in high stakes testing, flexible scoring
guides. This latter point, the need for scoring guides to allow for divergent thinking, is
a critical one for test-designers. It also is critical for teachers who must advocate for
students when scoring guides unduly penalize the creative thinker or the students who know
a context from a perspective different from the test designer’s intent. The challenge
is to draw the line between responses that exhibit true divergent thinking from those of
students who deliberately use context to hide a lack of mathematical understanding.
Conclusion
Many teachers avoid using context in instruction, both because they do not feel
comfortable with it, and because the students would have problems interpreting it. The
students, on the other hand, avoid tasks that are put in context because they do not know
how to make the context transparent enough to see the intended mathematics behind it.
We suggest that understanding the context of mathematics in context, as demonstrated by
the ability to state assumptions and consider multiple perspectives, will give a boost of
confidence to teachers using "out of the sky" tasks as an aid in instruction and
to students dealing with high stakes tests.
References
Amend, B. (February 14, 1999). This comic appeared in the Albany Times Union. Taken
from a FOXTROT cartoon by Bill Amend. Distributed by UNIVERSAL PRESS SYNDICATE. Reprinted
with permission. All rights reserved.
Anderson, J. R., Reder, L. M., & Simon, H. A. (No date). Applications and
misapplications of cognitive psychology to mathematics education [Online]. Available:
http://act.psy.cmu.edu/personal/ja/misapplied.html [1999, Feb 24].
Danielson, C. (1997). A collection of performance tasks and rubrics: Middle school
mathematics. Larchmont, NY: Eye on Education.
Caldwell, J. H. (1984). Syntax, content, and context variables in instruction. In G. A.
Goldin & C. E. McClintock (Eds.), Task Variables in Mathematical Problem Solving (pp.
379-414). Philadelphia, PA: The Franklin Press.
Engineering Concepts Curriculum Project (1970). Man Made World. New York: McGraw-Hill.
Kouba, V. L. (1991). Young problem solvers’ attempts at multiplication and
division word problems. In R. Bangert-Drowns (Ed.) Problem solving, critical thinking and
instructional design: Domain specific and generalized approaches to research (pp. 21
– 32). University at Albany, NY: ACRIDAT, Department of Educational Theory and
Practice.
Kulm, G. (1984). The classification of problem-solving research variables. In G. A.
Goldin & C. E. McClintock (Eds.), Task variables in mathematical problem solving (pp.
1-22). Philadelphia, PA: The Franklin Institute Press.
National Assessment of Educational Progress (1996). [Released items and student
responses from materials packet used in training scorers]. Unpublished data from National
Center for Educational Statistics and Educational Testing Service. (Used with permission.)
National Council of Teachers of Mathematics (October 1998). Principles and standards
for school mathematics: Discussion draft. Reston, VA: NCTM.
Roth, W. (1996). Where is the context in contextual word problems? Mathematical
practices and products in grade 8 students’ answers to story problems. Cognition and
Instruction, 14, 487-527.
Schoenfeld, A. H. (1988). Problem solving in context(s). In R. I. Charles & E. A.
Silver (Eds.), The teaching and assessing of mathematical problem solving (pp.82-92).
Reston, VA: NCTM.
Schoenfeld, A. H. (No date) Toward a theory of teaching-in-context [Online]. Feb. 24, 1999.
Webb, N. (1984). Content and context variables in problem tasks. In G. Goldin & C.
E. McClintock (Eds.), Task Variables in Mathematical Problem Solving (pp.69-102).
Philadelphia, PA: Franklin Institute Press.
* Mathematics
Teaching in the Middle School, December 1999.
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English Learning & Achievement 