Burning the Candle at Just One End - Scottsdale...

Burning the Candle at Just One EndAuthor(s): Kien H. LimSource: Mathematics Teaching in the Middle School, Vol. 14, No. 8 (APRIL 2009), pp. 492-500Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41182733 .

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that thirty- two out of thirty- three preservice teachers in a mathematics methods class apply the proportion algorithm to solve this problem:

Sue and Julie were running equally fast around a track. Sue started first. When she had run nine laps, Julie had run three laps. When Julie had completed 15 laps, how many laps had Sue run?

These preservice teachers set up a proportion, such as 9/3 = #/15, which is conceptually more difficult than the correct additive approach of 9 - 3 = x - 15, or 15 - 3 = x - 9. Hence, "we cannot define a proportional reasoner as simply one who knows how to set up and solve a proportion" (Cramer, Post, and Currier 1993, p. 160).

To deeply understand proportional reasoning, students must be able to differentiate proportional situations from nonproportional situations. Un- fortunately, the emphasis on the use of proportional strategies in middle- grades mathematics has led students to develop a disposition to associate certain characteristics of problem for- mulation with the use of proportional strategies. Van Dooren et al. (2005) found that the use of proportional strategies to solve a constant-difference running problem, similar to the problem described above, was highest among sixth and seventh graders, as compared with third, fourth, fifth, and eighth graders, in Belgium.

Feinstein (1979) has cautioned about the danger of overgeneralization in mathematics and argued that "to en- able students to discriminate effectively, several 'nonexamples' must be included in their experiences" (p. 22). Discrimi- nating effectively requires students to understand the structural differences between proportional and nonproportional situations. Lamon (2007) proposes that "proportional reasoning means supplying reasons in support of

claims made about the structural relationships among four quantities, (say ay by Cy d) in a context simultaneously involving covariance [italics added] of quantities and invariance [italics added] of ratios or products" (p. 638).

The objective of this article is to promote the use of nonproportional situations to emphasize invariance while students are learning to reason proportionally. This article is divided into four segments:

1. Missing-value problems, to highlight four types of invariance

2. Preservice teachers' responses to

two of these problems, to illustrate their depth of understanding of proportion

3. Activities that foster algebraic reasoning

4. Pedagogical recommendations

DIFFERENT TYPES OF INVARIANT RELATIONSHIPS IN MISSING-VALUE PROBLEMS In mathematics, a proportion is defined as a statement that two ratios are equal to one another. For example, the candle in task la in figure 1 is burning at a rate of 12 mm for every 20 minutes, or 0.6 mm/min. Since the

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494 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL • Vol. 14, No. 8, April 2009

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burning rate is constant, or invariant, the ratio #/50 should also be equal to 0.6 mm/min. Hence, the two ratios should be equal to each other, spe- cifically, a/50 = 12/20. Solving this proportion will determine that x = 30.

If all the missing- value problems that students encounter involve proportional situations, then checking the equivalence of two ratios in a proportion soon becomes redundant and counterproductive. Nonproportional situations must occasionally be incor- porated into students' problem-solving experience to instill the habit of checking whether two quantities are indeed covarying in a proportional manner.

A missing-value problem (part a of each task in fig. 1) can be designed with quantities that vary so that a specific relationship between them remains invariant. For example, the ratio between the two varying quantities in a proportional situation, such as task 1 or task 5, is invariant. Alternatively, the two quantities may covary such that their sum (task 2), their product (task 3), or their difference (task 4) is invariant.

Using a single context for all the tasks in figure 1 highlights the fact that a typical missing-value problem whose quantities vary proportionally can be rewritten so that quantities vary according to a constant sum, product, or difference. In task 2, the original height of the candle is invariant, although its value in unknown. At any moment in time, the sum of the length of candle burned, x, and the height of the candle, b, should equal its original height. Hence, 30 + 75 = h + x for 0 < h < 105 and 0 < x < 105. In task 3, the duration of the festival is invariant. Since 24 candles, lasting 7 hours each, are needed, the festival should last 24 x 7 hrs., which equals 168 hours, or 1 week. In general, n candles x / hours/candle should be equal to 168 hours. Hence, 21x7 = n x /, where n is a natural number.

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^^^^^^^^^^^^^^^^^^^^^^^^^^^H ■ Task 4 and task 5 seem similar, and they both involve 2 burning candles. Since the candles in task 4 are burning at the same rate but are lit at different times, the difference between the burned lengths of candle A and candle B is constant. Because the candles in task 5 are lit at the same time but are burning at different rates, the ratio of the burned length of candle P to the burned length of candle Qjs invariant.

PRESERVICE TEACHERS1 RESPONSES TO TWO MISSING-VALUE PROBLEMS Tasks 4a and 5a were used in a mathematics course for preservice teachers of grades 4-8. The goal was to necessi- tate an analysis of the two problems to understand why a proportional solution is applicable for task 5 but not task 4. The preservice teachers were asked to show their work and submit their solu- tions. Afterward, they discussed their solution to tasks 4a and 5a in small groups before a whole-class discussion.

On the basis of their written responses, few preservice teachers no- ticed the structural difference between the two tasks. Task 4 is an additive situation in which the difference of 8 mm is constant, whereas task 5 is a

proportional situation in which the ratio of 1.6 is constant. In their written response, twenty- three of twenty-eight participants used the same approach for the two tasks. Seventeen students used a proportional strategy for both tasks; thirteen students set up a proportion and solved for the missing value; and four students used a unit ratio strategy, as shown in figure 2. Five students used an additive strategy for both tasks (see fig. 3). One student used an incorrect strategy involving division for both tasks; another used a unit ratio strategy for task 4a and wrote "not enough information to determine how fast candle P is burn-

ing" for task 5a. Only four students

appropriately used an additive strategy


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for task 4a and a proportional strategy for task 5a (see fig. 4).

Although more preservice teachers obtained the correct answer for task 5a than for task 4a, they appeared to have greater difficulty with conceptual understanding of the solution for task 5a than for task 4a. None of the students who used a proportional strategy for task 5a explained the meaning of the ratio 16/10 or the ratio 35/10 in their written response. The lack of explanation among students was pos- sibly a consequence of being asked to show their work; they were not explicitly asked to explain their work. From my experience, preservice teachers generally do not pay close attention to the meaning of ratios when they set up a proportion to solve a missing- value problem.

On the other hand, students who used an additive strategy for task 4a seemed to have a referential meaning for the 18 mm difference (see fig. 3) or the 8 mm difference (see fig. 4). When one student explained her solution for task 4a, most of her classmates could understand why the "difference" method works. How- ever, they could not explain what was wrong with using a proportion to solve task 4a, and why proportions such as 16/10 = */35 or 35/10 = x/16 could be used for task 5a. At that juncture, many students seemed to need further explanation.

Seeking to understand the structural difference between task 4 and task 5 was not a habit of mind that these preservice teachers had. One preservice teacher even inquired about the use of key words, such as "burning at the same rate" and "lighted at the same time," to decide between an additive strategy and a proportional strategy. To understand why the conditions in task 5 make the situation proportional, one needs to visualize the constantly changing heights of the two burning candles and mentally

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coordinate burning rates of the two candles with respect to time.

FOSTERING ALGEBRAIC REASONING The first question of each task in figure 1 is appropriate for sixth- to eighth-grade mathematics students.

Part b of each question, however, must not be used prematurely. Students must first be given ample "opportunities to explore situations in which they have to reason explicitly in terms of quantities and quantitative relationships" (Sowder et al. 1998, p. 131). In undertaking a quantitative


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analysis of the problem situation, students identify the quantities in the problem and the relationships among the quantities. The activity in figure 5 is designed deliberately for students to identify quantities and relationships, as well as to differentiate among givens, unknowns, and variables. This activity helps students build their capacity to visualize covariation of quantities in a problem.

Whenever possible, students, as

well as teachers, should mentally act out the problem situation and draw diagrams to depict the problem situation. In figure 4a, a student drew two number lines to coordinate the burned lengths of candle A and candle B for task 4a. Since the candles in task 5a are burning at different rates, the student had to create a chart to coordinate the burned length of candle P with that of candle Q^The diagrams in figures 6a and 6b were drawn by

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in-service middle school teachers working on task 2a and 3a. They illustrate how diagrams can help students understand the problem situation and determine the invariant, or constant, quantity where the original height of the candle is invariant (in task 2a) and when the duration of the festival is invariant (in task 3a). Another teacher used a diagram (see fig. 7) to highlight two different multiplicative relationships among the four quantities (20 min., 50 min., 12 mm, and the answer of 30 mm) in task 1. These explana- tions were given:

1. The ratio of 3/5 refers to the invariant burning rate: "No mat- ter how long the candle has been burning, it will always burn 3/5 mm per minute until the candle no longer exists."

2. The ratio of 2.5 is a multiplicative comparison: 50 minutes is 2.5 x 20 min., so the answer of 30 mm is also 2.5 x 12 mm, because the candle burns 12 mm every 20 minutes.

Activities can also be designed for students to represent the covariation between two varying quantities graphically and interpret those graphs meaningfully (see figs. 8 and 9). For example, the difference in vertical intercepts in figure 8 can be inter- preted as the 8 mm head start that candle A has over candle B. Students can also practice making connections between the problem situation and its graphic representation. Doing so, they will recognize that the ratio of candle P's burning rate to candle Q]s burning rate corresponds to the ratio of the slopes of the two lines in figure 9.

Students should be challenged to reason algebraically whenever the opportunity arises. For example, students may be asked to show why the ratio of the burning rate of candle P to that of candle QJs 1.6. The proof,




16 candle P s burning rate _ Tx _16_ candle Qjs burning rate 10_ 10

requires students to pay attention to the time quantity. By explicitly identifying the time 7^ between the moment the candles were lit and the first moment, denoted by Tx in figure 9, students can reinforce their understanding of rate as a quantity whose value is equal to the ratio of two other quantities.

Part b of each task in figure 1 requires students to find an equation that relates the two variables. If an equation is formulated with an explicit focus on what relationship, or quantity, remains constant in the problem situation, then students are more likely to develop a deeper understanding of an equation. This understanding involves an equation

as a constraint that forces the values of the quantities to vary in a particular manner according to an invariant relationship that is specific to the problem situation. Some students may even realize that if the equation relating burned length, yy and time, xy

k is the burning rate for candle Q¿ then the equation for candle P is y = '.6kx (see fig. 9), because the candles were burning at different rates but lit at the same time.

These students are likely to understand the association between

for candle B is y = mx, where m is the burning rate for candle B, then the equation for candle A is y = mx + 8 (see fig. 8) because the candles were burning at the same rate but lit at different times. Likewise, if the equation for candle Q_is y = kx, where

the slope of a line and the burning rate of a candle as well as the association between the ̂-intercept and the time at which the candle is lit. Students' algebraic reasoning can be strengthened from investigating the covariation of quantities in these


Discriminating effectively requires students to understand the structural differences between

proportional and nonproportional situations



problems and making connections among characteristics of the graphs, parameters in the equations, and conditions specified in the problem statement.

PEDAGOGICAL RECOMMENDATIONS To recapitulate, we should use nonproportional situations to foster quantitative reasoning among students and minimize their tendency to automatically apply a proportional strategy to solve a missing-value problem. In solving a missing-value problem, students should identify the

covarying quantities and the constant relationship between them. Whenever appropriate, students should be given the opportunity to reason algebraically, such as using a graph or an

equation to represent covariation of

quantities and the invariant relation-

ship for a particular missing-value problem. This approach has helped many of my students (preservice and in-service middle school teachers) improve their understanding of ratio, proportion, and covariation. I am currently conducting a study to investigate the effectiveness of using nonproportional situations to help preservice middle-grades teachers minimize their tendency to overgen- eralize proportionality.

The ideas proposed in this article challenge the myth that learning mathematics should be made as easy as possible for students. Some teachers believe that students should be sheltered from complex problems or situations

that appear contrary to what they are learning. I believe, however, that students should be challenged to think skeptically by reasoning quantitatively.

CONCLUSION This article advocates for includ- ing nonproportional situations when students are learning rates and proportions. Some teachers might be apprehensive about this approach for fear that their students may become confused. This concern is valid. Nevertheless, genuine learning occurs when students resolve their cohfu- sinn and learn frnm their mistakes.

Hence, teachers should strive to create a challenging yet safe environ- ment in which students are intel- lectually challenged. They should be given opportunities to encounter and overcome certain misconceptions and obstacles, such as overgeneralizing proportionality and extending the multiplication-makes-bigger concept from the domain of natural numbers to the domain of rational numbers. Mathematics teachers should reflect honestly on their teaching so as to avoid propagating a learning culture in which "doing mathematics means following rules laid down by the teacher [and] knowing mathematics means remembering and applying the correct rule" (Lampert 1990, p. 31).

REFERENCES Cramer, Kathleen, Thomas Post, and

Sarah Currier. "Learning and Teach-

ing Ratio and Proportion: Research

Implications." In Research Ideas for the Classroom: Middle Grades Mathematics, edited by Douglas Owens, pp. 159-78. New York: Macmillan, 1993.

Feinstein, Irwin. "Dare We Discover or the Dangers of Over-generalization." School Science and Mathematics 79

(1979): 22-33.

Lamon, Susan. "Rational Numbers and

Proportional Reasoning: Toward a Theoretical Framework for Research." In Second Handbook of Research on Mathematics Teaching and Learn-

ing, edited by Frank K. Lester Jr., pp. 629-67. Charlotte, NC: Information

Age Publishing, 2007.

Lampert, Magdalene. "When the Problem Is Not the Question and the Solution Is Not the Answer: Math- ematical Knowing and Teaching." American Educational Research Journal 27 (1990): 29-63.

Sowder, Judith, Barbara Armstrong, Susan

Lamon, Martin Simon, Larry Sowder, and Alba Thompson. "Educating Teachers to Teach Multiplicative Structures in the Middle Grades."

Journal of Mathematics Teacher Educa- tion 1 (1998): 127-55.

Van Dooren, Wim, Dirk De Bock, An

Hesseis, Dirk Janssens, and Lieven Verschaffel. "Not Everything Is Pro-

portional: Effects of Age and Problem

Type on Propensities for Overgener- alization" Cognition and Instruction 23

(2005): 57-86.

The author would like to thank Rick Anderson, the reviewers, and the journal editors for their valuable comments in the

preparation of this article. 9


If an equation has an explicit focus on what quantity remains invariant, then students are more likely to

develop a deeper understanding of an equation



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