Setting the Context for Mathematics in Context

Vicky L. Kouba
with
Özlem Çeziktürk
Susan A. Sherwood
Chia-Huan Ho

University at Albany
State University of New York

"Using mathematics in applied situations leads to a deeper understanding."
(NCTM, 1998, p. 93)

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.

The use of context in mathematics helps to:

• Improve students' ability to demonstrate their understanding of mathematics, and
• Facilitate the development of greater understanding of mathematics principles.

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.

Interference and Context
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 illustrate the influence of context on students' interpretation and response to a mathematics task.

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:

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:

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:

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 with 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.

The cartoon that follows demonstrates the spirit, if not the letter, of our thoughts about the importance of communication and explicit statement of assumptions.

(FOXTROT ©1999 Bill Amend. Reprinted with permission of UNIVERSAL PRESS SYNDICATE. All rights reserved.)

Bill Amend, the cartoonist, 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 designers' 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.

The use of context in testing mathematics can be a rich and vital source of information and can more accurately reflect authentic uses of mathematics. However, students should be given ample opportunity to learn how to think about, analyze, and respond to such items, and scoring guides for the items must take into account the possibility of reasonable alternative interpretations.

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.

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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 and E. A. Silver (Eds.), The Teaching and Assessing of Mathematical Problem Solving (pp. 82-92). Reston, VA: National Council of Teachers of Mathematics.

Schoenfeld, A. H. (No date) Toward a theory of teaching-in-context [Online]. Available http://www-gse.berkeley.edu/Faculty/aschoenfeld/ [1999, Feb 24].

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.

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