Improper use of physics-related context in high school mathematics problems: Implications for learning and teaching
This article discusses potential pedagogical difficulties arising from using physics-related contexts in high-school mathematics problems. It is suggested that such problems should not require any external knowledge of physics by the students; meanwhile, the problems should not contradict the physics knowledge that the students may already have. Several examples of recently published textbook problems are presented along with the discussion of the context pitfalls. A simple technique (SCAN list) is proposed for evaluating the physics-related mathematics problems.
The Quest for the Real-life Problems
The quality of the word problems in secondary– level mathematics textbooks is a common subject among educational researchers and teaching professionals. The tone of the discussion found in literature has often been rather critical:
Textbook story problems have a long history of criticism in the literature… Algebra story problems have earned a well-deserved reputation as a threat to student’ achievement in secondary school mathematical courses… [They are] dull, predictable, and routine… (Silverman, Winograd, & Strohauer, 1992, p. 6)
Or as Mayer (1982) said, “[They are] simple story problems.. greeted with moans, fearful faces and incorrect answers…” ( p. 199).
Indeed, a brief look at the “traditional” textbooks, (for instance, Dolciani, Brown, & Cole, 1988; Fair & Bragg, 1990) instantly reveals how abstract most word problems were.
An “abstract” problem does not necessarily have to be a bad one, of course – the mathematicians could bury us in examples. However, research and teaching literature makes a strong case for including the “reallife” word problems as an integral component of the textbook content.
Notably, a proper definition of such problems is hard to come across, but a certain informal consensus appears to arise from the relevant literature. According to that consensus, a real-life problem has at least one of the following features: it is realistic (uses explicitly “real-world” settings); it is amusing to the student; it is relevant to the everyday life of the students. These features will be incorporated in the discussion througout this article. Such problems, it is argued, make the subject of mathematics not only more appealing and inspiring but also more “authentic” (see, for instance, Fairbairn, 1993; Kilman &Richards, 1992; Muth, 1986; Silverman et al., 1992).
It comes as no surprise, then, that the most recent reform movement in mathematics education produced, among other things, quite a few “real-life” mathematics curricula in the last few years – see, for instance, the list of”exemplary” and “promising” math programs on the US Department of Education web site (www.enc.org/ed/exemplary). Moved by the noble desire to improve the pedagogical effectiveness of the problems, the authors, inevitably, went into a largely uncharted territory.
The task of filling a textbook with a few hundred good problems is immense. However, using pedagogically unsound problems can do a lot of harm in the classroom and diminish the intended effect of using the real-life situations. It is important, therefore, to keep pointing out their shortcomings. One of them – the improper use of physics-related situations as the context of the problems – is the focus of this article.
Physics-related situations are, indeed, common in the real-life problems. Physics is, as many physicists say, “the description of the natural world by means of mathematics” – which makes it a context of choice for the problem writers. As an author of several articles on problem-solving and numerous physics problems written for both high-school and college use (Korsunsky, 1995a, 1995b, 1995c, 1997a, 1997b, 1999) who have focused on the use of the real-life context in problems (Korsunsky, 1997a), I was pleasantly surprised by the popularity of physics settings in the problems found in recently published textbooks. However, the number and seriousness of pedagogical “trouble spots” often accompanying such problems was disturbing. The goal of this article is to point out the most common ones and offer some remedies for both teachers and textbook authors.
The issue is not necessarily new. Traditional textbooks also used real-life context improperly (Pollak, 1978). However, the issue has come to bear much greater significance nowadays, as real-life problems have become more fashionable among the textbook writers.
Real-Life Mathematics Problems: A Missing Part in Educational Research
Research literature, while unequivocally favoring the real-life context in mathematics problems, points out several ways in which the context of a mathematics problem may, in fact, impede learning. The context– related pitfalls that attracted most attention of researchers can be identified as follows:
* Genderbias(Chipman, 1988; Chipman, Marshall, & Scott, 1991; Marshall, 1984; Murphy & Ross, 1990);
* Culture- and language-related bias (Schwartz, 1988; Secada & De La Cruz, 1996; Short & Spanos, 1989);
Unfamiliar context (Chipman et al., 1991; Fairbairn, 1993; Ross et al., 1986; Silverman, et al., 1992).
It seems that the researchers have focused primarily on the equity-related issues. Indeed, it is important not to impose an undue cognitive burden on a particular sub-population of problem-solvers (be it females, ethnic minorities, non-native speakers, newcomers to America, etc.). However, researchers, with few exceptions, did not specifically address the ramifications of using arbitrary, inaccurate, and often absurd real-life contexts in mathematics problems – physics contexts in particular.
The poor use of physics in mathematics education hurts all students, regardless of their background except for the students with some prior physics knowledge, which can, ironically, find themselves at a disadvantage. I was able to locate only one article (Pollak, 1978) that offered a serious review of the problems purporting to be real-life applications of mathematics but are, in fact, “anything but.” Two quotes may sum up the spirit of that article:
The statement of such problems rarely questions the honesty and genuineness of the connection to the real world, but the connection is often false in one or more ways. (p. 233)
Some of these problems perhaps contain a kernel of truth, and provide answers of some qualitative validity. If an attempt were made to discuss their relation to reality, and to be honest with the student generally, they might be acceptable.” (p. 234).
These comments are relevant today more than ever, as the barrage of real-life problems appeared in the recent textbooks. Following is a discussion ofa few problems with such “false connection to the real world” and some recommendations for making these problems “acceptable.”
Examples of Inappropriate Physics-related Problems
In this section, examples of pedagogically unsound problems are described, with some brief comments. All of these problems are quoted, or nearly quoted, from several recently published textbooks. Since the context in which they are mentioned is unequivocally critical, I prefer not to name the sources of my examples in this article. Pollak (1978) did the same.
Real-life problems should not assume the students’ external knowledge of physics (problems must be selfcontained)-yet they should not contradict such knowledge, should it exist. Thus, the sample examples that follow are classified into three groups, according to the main pitfall found in each:
* Problems that require external physics knowledge.
* Problems that use oversimplified or incorrect physics contradicting the external physics knowledge.
* Problems that present physics concepts in confusing manner contradicting the external physics knowledge.
Such classification is provided for the convenience of the reader, despite being somewhat loose: some problems, arguably, are “guilty” of more than one pitfall.
Problems That Are Not Self-Contained
Example 1. Ohm’s law, because of its algebraic simplicity, seems to be a favorite among the writers of the real-life problems. Note the sheer length of this problem, overstuffed with some ad hoc physics:
Mr. Sen has a workshop in his garage. There is no electricity in the garage, so he runs a 100-ft extension cord from his house to the workshop when he wants to use the electrically powered tools. He has noticed that his electric drill works fine but his saw does not. An electrician friend suggested he apply Ohm’s law [bold in the text]. This relates the Voltage [heretofore capitalized by the author] drop, or loss, V to the current flowing through the extension cord and the resistance, R, of the cord. V = i*R
His extension cord has a resistance of R = 1 Ohm. If he knows the current, I, through the cord, he can calculate the Voltage drop, V, along the extension cord. For example, a 60-watt light bulb uses about 0.5 amperes of current. Then V =i*R=0.5*1 =0.5 Volts is the voltage drop due to the extension cord. If the voltage in the house is 117 V, the Voltage at the end of his extension cord will be 117-0.5 = 116.5, which will still be enough to light the light bulb. [A series of questions, related to various power tools, follows.]
The context in this problem has several faults. First, the subject is unfamiliar to many students. Second, it requires a great deal of reading, thus is potentially unfair to students with limited English proficiency. Third, the focus on the power tools may well be considered a gender bias. Last but not least, the physics content poses some serious issues.
The problem purports to be self-contained: a simplifled statement of Ohm’s law is included, along with an example. However, the authors still make many assumptions about the students’ knowledge of electricity, which is crucial to understanding the situation discussed in the problem. Here are several questions that a thoughtful student may pose -but not be able to answer:
What is “Voltage” and how is it different from “Voltage drop”?
* What does the “60-watt light bulb” have to do with the rest of the problem? The authors seem to take for granted the students’ understanding of the laws of current in a series connection even though they never mention at all (!) that the light bulb is, in fact, connected to the cord in series.
* Why does multiplying amperes by ohms yields volts? Would milliamps times ohms also give the answer in volts? The textbook does not address the unit issue at all and drops the units altogether when subtracting 0.5 from 117, compounding the confusion. While this appears to be a common practice in mathematics instruction, it creates serious issues in studying science.
* Just what does is meant by “enough to light the light bulb”? The light bulb probably becomes dimmer if the “Voltage” is lower, but one would assume that it would be “lit” somewhat even under a lower voltage.
To help the students make sense of this problem, the mathematics teacher should be a real physics expert- and the explanations would still be too time-consuming. Example 2.
The basic principle underlying the operation of air conditioners and heat pumps involves a relation between pressure, volume, and temperature in a container. Check with someone who knows about heating and cooling and report how those variables change when the machinery is in operation.
In fact, the physics of air-conditioners is a very complex issue; physics maj ors in college often struggle with it. It involves first and second laws of thermodynamics, among other things – rarely studied by high school students. Making this topic accessible to the students without losing coherence and logic is extremely difficult, and few physics textbooks and instructors are successful at it. A casual search for “someone who knows” is likely to be a huge, timewasting disappointment. To expect a high-school student to prepare a meaningful report on such topic is unrealistic, and including such assignment in a mathematics textbook is irresponsible.
The Landsat 5 satellite orbits the earth at an altitude of approximately 700 lan….Landsat 5 completes a survey of the earth every 16 days. If it passes overhead this very minute, on what day will it again pass overhead? How close to New Years’ day?
How close to midnight of your birthday?
This is atypical real-life problem: it mentions a “real thing” (the Landsat satellite), and it has a personal touch (“your birthday”). So what is wrong with it? The satellite on the 700-km orbit takes about 3 hours to orbit the earth, not 16 days! A professional physicist or an astronomer would not assume that “completes a survey” necessarily means “orbits the earth once.” However, this is exactly what the students are likely to assume – and this would be in direct contradiction with what they have learned, or will learn, in their physics class. On the other hand, any attempt to explain why “completing a survey” is not the same as “orbiting the earth” involves a fair amount of sophisticated information that has nothing to do with the (rather primitive) mathematics involved in solving this problem.
Problems Using Oversimplified/Incorrect Physics Example 4.
Karate chops break bricks and boards by applying carefully aimed bursts of energy. Different targets require different amounts of energy. Think about the four target boards pictured here: [The diagram shows four schematically drawn boards of different length and thickness supported at the ends.]
a. Which board do you think would require the greatest energy to break?
b. The target boards differ in length and thickness. How would you expect those two variables to affect required breaking energy?
c. Breaking energy E depends on board length L and thickness T. What sort of equation might be used to express E as a function of L and T?
d. What other variables would you consider in judging the energy required to break a board? How would you expect those variables to be related to each other and to E, L and T?
In this problem, the authors inappropriately mix the terms “energy” (which is not directly relevant here) and “impulse.” But this is not the worst part. In questions c and d, the student must be a physics or material science expert to even guess intelligently- let alone give the correct answer. Of course, the students can come up with some hypotheses. But they would be highly speculative and based on the students’ (limited) knowledge of functions rather than on the actual situation- which is, in reality, highly complex. How are those answers to be judged by the teacher? Are students just expected to assume that the answer must be either direct or inverse proportionality because this is about all they know?
The likelihood of a fatal accident in a car, van, or small truck depends on many conditions. Two key variables are speed and mass of the vehicle ….[The accompanying photograph shows two cars stuck together after a collision.] If A represents the rate of fatal accidents, s represents vehicle speed, and m represents mass of the vehicle, which of the following equations would you expect to best express the relation among those variables?
* A = 200(s + m)
* A = 200(s – m)
* A = 200s/m
* A = 200sm
In a collision between two vehicles (to which the photograph alludes), it is the ratio of masses of the vehicles that matters, not the mass of each individual one. This makes the whole premise rather confusing. In addition, the first two options – A = 200 (s + m) and A = 200(s – m) are dimensionally absurd, which the textbook never stresses.
Also, since none of the equations proposed in this problem is actually correct, one may ask, just what does it mean to “best express”? To give the best numerical estimate for a given mass and velocity? Or, perhaps, to give a physically sensible model? Depending on one’s understanding, there may well be different answers to the question. In fact, one of the “absurd” formulas may happen to give the best numerical estimate – if the units are played right.
Of course, a thoughtful teacher may be able to navigate this rocky terrain and even draw some useful lessons for the students – but how far from mathematics would those lessons be?
Example 6. The next example is too scattered along the textbook page to quote directly; the context of the problem is the path of rays passing through a diamond. The diagram presents two diamonds and the rays oflight passing through them. The law of reflection is used as the central premise of the problem; however, the rays striking surfaces at an angle, pass through, remarkably, with no refraction at all! In all fairness, the authors admit that they “simplify” the situation (although they do not explain how). How would a student figure out what is wrong? The text suggests, “Consult with a science teacher about reflection and refraction of light and the modeling of the diamond cut….”
One big concern here is that “a picture is worth a thousand words.” The textbook’s (incorrect) diagram is likely to stick in the students’ memory so well that even an impromptu lecture by a science teacher may not correct the misconception. Another, even bigger, concern is that the authors of this textbook frequently refer the students to the “science teacher” to obtain the explanations of the concepts casually mentioned in their textbook (Example 2 also comes from it). The collaboration between physics and mathematics teachers should, indeed, be encouraged. However, it is unrealistic to expect the physics teachers to teach the second law of thermodynamics and the laws of refraction to the mathematics students-just to let them know how inaccurate their mathematics textbooks are.
Problems That Present Physics Concepts in Confusing Ways
Example 7. This problem is preceded by a brief introduction of Ohm’s law as I= V/R:
Use Ohm’s law to complete a table like the one below, showing the current that can be expected in circuits with several combinations of voltage and resistance. [The table gives the values ofR “across” and the values of V “down”; the students are supposed to fill the cells of the table with the values of I.]
a. How does the current.. change as voltage increases? As resistance increases?
b. How could the patterns you noted in Part (a) be predicted from the form of algebraic rule relating I to V and R?
This problem obscures the important scientific question related to Ohm’s law: In the formula I= VIR, what quantity remains constant? Ohm’s discovery was that, under a wide range of conditions, the resistance of a metal wire remains constant. This problem, however, gives the impression that every quantity in I= V/R is subject to change – which, if true, would rob Ohm’s law of any meaning at all. An adequate “physics scaffolding” of this problem would require much more time and effort than answering questions a and b which renders the problem useless.
Example 8. Here is another problem based on a physics law, mathematically (but not physically!) very similar to Ohm’s law.
Have you ever noticed that when you use a tire pump on a bicycle tire, the tire warms up as the air pressure inside increases? [Probably not. The temperature change of the tire surface is usually too little to notice.] This illustrates a basic principle of science relating pressure P, volume Vand temperature Tin a container: for any specific system, the value of the expression PV/T remains the same even when the individual variables change. [Several questions follow.]
In physics (and chemistry), the equation PV/T=coast is known as “the ideal gas law.” Most physicists would agree that calling it “a basic principle of science” is quite a stretch: this equation has limited applicability. It describes the portions of gas under low pressures and fairly high temperatures, and it is only an approximation (which makes it similar to Ohm’s law). Applying it to “any specific system” is absolutely inappropriate. The mistake can be easily detected by any thoughtful student, with or without science experience (just try to apply this “basic principle” to a glass of water). The unfamiliarity of most students with the absolute temperature scale would, no doubt, compound the confusion (“What happens to the volume of the water in this glass if its temperature goes up from 35 deg F to 70 deg F?”). Meanwhile, the actual questions posed in the problem can be answered rather easily – but why invoke a reallife context only to twist it so much?
For an astronaut in a space shuttle orbiting the earth, an increase in distance from the earth reduces the effect of gravity. The astronaut’s weight in space is a function of the distance above the earth’s surface with rule:
How to Use Physics-Related Context in a Mathematics Problem
Most troubles in the examples described in this article arise from the authors’ honest attempts to inject some physics into the mathematics problems-yet keep the physics simple enough not to interfere with math. As the reader can see, such “simplifications” sometimes do not .work, causing pedagogical “pitfalls”. (Some of the high-school physics textbooks may be treating the mathematics involved in the physics problems a little too liberally, thus returning the dubious favor to their mathematics counterparts. Perhaps, another article, by a mathematics educator, can address that side of the issue.) Granted, using simplified physics in a mathematics class is no pedagogical crime – it happens in physics textbooks, too. After all, some concepts or applications may not be fully accessible to the students, in which case a simplified version may give a fair, working representation of reality. However, certain checkpoints must be in place when a physics– related problem is used in a mathematics classroom. I propose a short list of such “checkpoints” for textbook authors and teachers. I call it the “SCAN list” (SCAN stands for Self-contained; Concise; Accurate, Noncontradicting).
This list is based on two basic pedagogical principles proposed in the beginning of this article:
First, the problem should not require any outside physics knowledge beyondthe most common-sense facts: it must be self-contained. Meanwhile, the problem should be concise: The explanations must not be so long as to turn a mathematics class into a physics class. If too many explanations within the text are needed to ensure self– containment, the context may overwhelm the matical ideas presented in the problem, thus diminishing its teaching value.
Second, the problem must be accurate: It should not contain erroneous or confusing information, which may be evident to a student who has, in fact, studied physics. Often, a knowledgeable student may be confused and, hence, unfairly penalized because the problem statement or implications contradict the prior knowledge. By the same token, the problem content shouldnot contradict the physics knowledge that will be learned by the students in the near future. If it happens, it would then make the science class difficult to teach and impose a wrong impression that, in mathematics and science classes, things somehow work out differently – hardly making a case for the real-life problems.
To sum up, physics-related context, indeed, makes for an excellent illustration of connections between mathematics and the everyday world. However, the obvious pedagogical advantages of using such context should be carefully weighed against the less obvious disadvantages. To create the problems with a pedagogically sound real-life context, the writers of the textbook problems must be well-versed in both physics and math. If their primary expertise is in mathematics, then the problem authors should not hesitate to consult with physics experts, who can help “SCAN” the problems.
Of course, these ideas can only be implemented when the publishers finally realize the importance of the quality ofthe problem content in the textbooks and begin to put the appropriate intellectual and financial resources in place. Hopefully, this article will bring such state of events a little closer.
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Editors’ Note: I would like to thank Judah Schwartz of Harvard Graduate School of Education and Paul Goldenberg of Education Development Center for useful discussions on the subject.
Correspondence concerning this article should be addressed to Boris Korsunsky, 20 Jaconnet St., Newton, MA 02461.
Electronic mail may be sent via Internet to firstname.lastname@example.org
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