144 Mathematics Teacher | Vol. 102, No. 2 • September 2008
connecting research
to teaching
Teaching Trigonometric Function...
Vol. 102, No. 2 • September 2008 | Mathematics Teacher 145
found only if two sides of a right triangle having an
angle of ...
146 Mathematics Teacher | Vol. 102, No. 2 • September 2008
My interviews with a subset of the students
revealed some reaso...
Vol. 102, No. 2 • September 2008 | Mathematics Teacher 147
ing, thirty-four students gave a correct answer, and
thirty-two...
148 Mathematics Teacher | Vol. 102, No. 2 • September 2008
Tall, D. O., M. Thomas, G. Davis, E. Gray, and A.
Simpson. “Wha...
Vol. 102, No. 2 • September 2008 | Mathematics Teacher 149
3.	 Locate the point of intersection between the ray you have j...
150 Mathematics Teacher | Vol. 102, No. 2 • September 2008
3. 	Approximate the sine and cosine of the angles drawn in the ...
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  1. 1. 144 Mathematics Teacher | Vol. 102, No. 2 • September 2008 connecting research to teaching Teaching Trigonometric Functions: Lessons Learned from Research T rigonometry is an important subject in the high school mathematics curriculum. As one of the secondary mathematics topics that are taught early and that link algebraic, geometric, and graphical reasoning, trigonometry can serve as an important precursor to calculus as well as college- level courses relating to Newtonian physics, archi- tecture, surveying, and engineering. Unfortunately, many high school students are not accustomed to these types of reasoning (Blackett and Tall 1991), and learning about trigonometric functions is initially fraught with difficulty. Trigonometry presents many first-time challenges for students: It requires students to relate diagrams of triangles to numerical relationships and manipulate the sym- bols involved in such relationships. Further, trigo- nometric functions are typically among the first functions that students cannot evaluate directly by performing arithmetic operations. Despite the importance of trigonometry and stu- dents’ potential difficulties in learning it, relatively little research has focused on this subject. This arti- cle will present lessons learned from research inves- tigating the learning and teaching of trigonometric functions (Weber 2005). It will report difficulties that students have in understanding trigonometric functions and describe tested instructional strategies to help students overcome their difficulties. TRIGONOMETRIC FUNCTIONS AS RATIOS AND FUNCTIONS What does it mean to understand a trigonometric function? Just as taking a square root or cubing a number can be thought of as operations applied to numbers, the terms sine, cosine, and tangent can be thought of as mathematical operations applied to angles. These trigonometric functions are generally presented in two ways. First, they are presented as ratios that can be applied to labeled right triangles. For instance, students can use a ratio understand- ing of the trigonometric functions to determine that sin A = 3/5, cos A = 4/5, and tan A = 3/4 (see fig. 1). Using calculators, students can use a ratio under- standing of sine and cosine to determine the missing lengths (a and b) of the triangle (see fig. 2). Such an understanding is obviously useful; it is sufficient to solve various types of word problems and perform other tasks. For instance, adding vectors in physics requires this type of trigonometric reasoning. However, a ratio understanding is also limited. Principles and Standards for School Mathematics argues that understanding an operation involves being able to estimate the result of that operation (NCTM 2000, pp. 32–33). For example, under- standing fractions involves knowing that 7/8 + 13/12 should be approximately 2 because each fraction is approximately 1. By itself, a ratio understanding of sine would not permit students to approximate sin 15°, because sin 15° could be This department consists of articles that bring research insights and findings to an audience of teachers and other mathematics educators. Articles must make explicit connections between research and teaching practice. Our conception of research is a broad one; it includes research on student learning, on teacher think- ing, on language in the mathematics classroom, on policy and practice in math- ematics education, on technology in the classroom, on international comparative work, and more. The articles in this department focus on important ideas and include vivid writing that makes research findings come to life for teachers. Our goal is to publish articles that are appropriate for reflection discussions at depart- ment meetings or any other gathering of high school mathematics teachers. For further information, contact the editors. Libby Knott, knott@mso.umt.edu University of Montana, Missoula, MT 59812 Thomas A. Evitts, taevit@ship.edu Shippensburg University, Shippensburg, PA 17257 Keith Weber Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
  2. 2. Vol. 102, No. 2 • September 2008 | Mathematics Teacher 145 found only if two sides of a right triangle having an angle of 15° were given. A ratio understanding of sine would also not permit students to determine in which quadrants sine was increasing or to graph sin 2x. Many calculus tasks, such as determining the derivative of sin x, make little sense with a ratio understanding of sine. To address the latter tasks, students need to have a function-based understanding of trigono- metric operations (Weber 2005). They need to understand operations such as sine as a process that takes an angle as an input and maps this angle to a real number. To understand a trigonometric operation as a function, students need to know a process they can use to evaluate that function for any given angle, and they must be able to anticipate the approximate result of that method and reason about the properties of that result without actually performing the steps of the process. TRADITIONAL INSTRUCTION OF TRIGONO- METRIC OPERATIONS Too often trigonometry instruction emphasizes procedural, paper-and-pencil skills at the expense of deep understanding (Hirsch, Weinhold, and Nichols 1991). An inspection of several popular high school algebra, geometry, and trigonometry textbooks (e.g., Hollowell, Schultz, and Ellis 1997; Schoen 1990; Larson 2004) reveals that students are typically first taught trigonometric operations as ratios (e.g., sin q is defined as y/r in a labeled right triangle or as “opposite over hypotenuse”). Students are asked to use these ratios to accomplish tasks such as those suggested in figures 1 and 2 and then solve word problems. After several sections devoted to these topics, the textbooks introduce the unit circle model of trigonometric functions. At this point, the texts sometimes ask students to imagine applying a pro- cess to find sine and cosine of a particular angle (such as walking r units around the unit circle and locating the y- and x-values, respectively, of the stop- ping point). However, students are not given the opportunity to apply this process. The exercises they are asked to complete rarely require a process under- standing of trigonometric operations; most simply require using a ratio conception of the trigonometric operations or applying algebraic techniques. Other researchers note that traditional instruction empha- sizes understanding trigonometric functions as ratios and does not enable students to understand them as functions (Kendal and Stacey 1997). THE EFFECT OF TRADITIONAL INSTRUCTION ON STUDENTS’ UNDERSTANDING OF TRIGONOMETRIC OPERATIONS To investigate students’ understanding of trigono- metric operations, I conducted a study of thirty-one students who were completing a college trigonom- etry class. I asked them to take a test without using a calculator and then invited four students for an interview. The teacher of the class, who was unaffiliated with my study, described his teaching as traditional; in particular, he claimed to teach primarily by lectures based largely on the course textbook, with an emphasis on students develop- ing procedural skills. The textbook for the course was Lial, Hornsby, and Schneider (2001), which was structured similarly to the textbooks described earlier. The full results and methodology of this study are reported in Weber (2005). The purpose of reporting these results here is to illustrate the limited understanding of trigonometric functions many students have after completing a trigonom- etry course. Two of the test questions were these: 1. Approximate cos 340°. 2. For what values of x is sin x decreasing? Why? Only six of the thirty-one students estimated that cos 340° was between 0.5 and 1.0. Nine stu- dents correctly noted that sin x was decreasing for 90° < x < 270°, and six offered a convincing expla- nation for why this was the case. These results sug- gest that these students had difficulty understand- ing trigonometric functions. Fig. 1 A labeled right triangle Fig. 2 A right triangle with a given angle, a given side, and two missing sides 5 3 A 4 10 a 30° b 40° 40° 70° 160° The intersection point is about 0.75 to the right of the y-axis. 85° 230° (0, 1) 4 10 a 30° b 40° 40° The intersection point is about 0.75 to the right of the y-axis. The intersection point is about 0.65 above the x-axis.
  3. 3. 146 Mathematics Teacher | Vol. 102, No. 2 • September 2008 My interviews with a subset of the students revealed some reasons why. Consider the tran- scripts below (all students’ names are pseudonyms): Interviewer: Describe sin x for me in your own words. Steve: To find sine, it would depend on the problem that was given to me. If I was given a triangle, I would divide y and r. If I were given one of the special angles, like 30°, 45°, or 60°, I would have this number memorized. There are other problems which can be solved by reference angles, or using formulas, like sin q + cos q = 1 [sic]. How you find the answer depends on how the problem is worded. Interviewer: What can you tell me about sin 170°? Can you give me an approximation for this number? Steve: I don’t ... I would need the triangle. Maybe if you told me what some other value of sine would be, or, like, what cos 170° would be, I could find what sine is. Otherwise, I would need to use a calculator. Steve’s responses were representative of those of the four students I interviewed. Steve seemed to view sine and cosine as algorithms (based on ratios or on algebraic manipulations) that could be applied but only if other information, such as a labeled right triangle, was also provided. Without this information, Steve could not conceive of how to apply the sine or cosine operation to an angle, except for special cases. (Because my tested sample consisted of only one class of students, we must not generalize these results inappropriately.) AN ALTERNATIVE APPROACH TO TEACHING TRIGONOMETRY The following suggested instruction is based on the idea that trigonometric operations such as sine can be understood as geometric processes. One pro- cess for computing sine is to construct a unit circle on a Cartesian plane, use a protractor to draw a ray emanating from the origin such that the angle between the positive half of the x-axis and the ray is the input angle, locate the point of intersec- tion between the ray and the circle, and determine the y-value, or height, of that intersection. Recent research in mathematics education indicates that students have difficulty imagining the application of a process without the experience of actually applying it. Instead, students may best develop a deeper understanding of processes by first applying them and then reflecting on their actions (Tall et al. 2000). Here I describe instruction that I designed and then implemented in a college trigonometry class that I taught. (One typical lesson is provided in the appendix.) To involve students kinesthetically in computing sines and cosines, each student was given a protractor and a unit circle drawn on graph paper and marked with Cartesian coordinates such that ten tick marks constituted one unit. I then described and modeled a procedure for computing sines and cosines by using a protractor to draw an angle with vertex at the origin and one ray along the x-axis, marking where the other ray intersected the circle, and using the tick marks to estimate the x- or y-coordinate of that point. This procedure was also fully described in a student handout. Students worked in groups on a series of classroom activities in which they found the sines and cosines of six angles by using the proce- dure. While they worked, I circulated among them to answer any questions they might have and make sure they were applying the procedure correctly. Students were next asked to evaluate the sines and cosines of some angles by anticipating the results of the procedure without applying the procedure itself. For instance, students could find sin 270° by realizing that they needed to examine where the bottom half of the y-axis intersected the unit circle. Students were also asked to make judgments about the procedure’s output without actually applying it. For instance, they were asked to determine which was greater—sin 23° or sin 37°. These tasks helped students reason about the process in general and make judgments about sines and cosines without applying each step. I used lessons of this type throughout the course. For instance, students learned how to find sines, cosines, and tangents by constructing right trian- gles on a Cartesian plane, measuring the lengths of the sides, and computing ratios. Once the students understood this procedure, they could perform tasks like those required in figure 1, because they understood the essentials; it was as if someone had created a triangle and measured the lengths of the sides for them. However, students’ understanding of the functions was not restricted to reasoning from diagrams; they could imagine producing the diagrams themselves, if necessary. When studying reference angles, students were first asked to draw the desired angle, find the appropriate reference angle themselves, and then compute the sines and cosines by looking at it. Students instructed in this geometric approach understood reference angles easily and did not need to rely on the mnemonic strategies that trigonometry students commonly use. A more thorough description of this instruc- tion is provided in Weber (2005). Toward the end of the course, I gave the forty students in my class the same paper-and-pencil test I had given to the students in the lecture-based class described earlier. When asked to approximate cos 340°, thirty-seven of the forty students gave a response of some number between 0.5 and 1. When asked for those values for which sin x was decreas-
  4. 4. Vol. 102, No. 2 • September 2008 | Mathematics Teacher 147 ing, thirty-four students gave a correct answer, and thirty-two provided an adequate justification. On the basis of the students’ test responses, I interviewed four students whose abilities varied (one very good student, two average students, and one struggling student) and whose responses were typical of those of other students. Throughout these interviews, the students were able to explain properties of the sine function by reasoning about the process of computing sines. Two representative excerpts of these interviews follow: Interviewer: Why is sin x a function? John: Because for, uh, each angle, there’s … going back to the unit circle, if you put something in for the sin x, it’s only going to cross at one point. Each angle is going to be one angle, and that one angle is going to cross the unit circle at one point. That one point is going to have a y-value. It will have one and only one y-value. Note that John refers to the process used to com- pute sines to justify why sine has a certain prop- erty. Three of the four students interviewed gave similar responses. It is also worth noting that none of the four interviewed students in the traditionally taught class could justify why sin x was a function, even after being told that an operation was a func- tion if each input had a unique output. In the following excerpt, Erica is able to use her process conception of sine to approximate sin 170°: Interviewer: What can you tell me about sin 170°? Can you give me an approximation for this number? Erica: The answer would, oh, be, I’d say, 0.1. Interviewer: That is a good guess. How did you get that answer? Erica: I pictured making a 170° angle with a protractor and seeing where the angle intersected the circle. Interviewer: I see. And how did you know it would intersect at 0.1? Erica [drawing a diagram]: Well, it would intersect right there [pointing to the point of intersection]. Here Erica is able to show how she can adapt her understanding of a process used to compute sines to approximate sin x quickly and accurately for an arbitrary angle x. These results are from only one class, one that I taught myself, so it is important not to generalize these results inappropriately. These results, however, do show that a geometric approach to trigonometry can be effective in developing students’ understanding of trigonometric operations and suggest that this approach could be applied in other classrooms. DISCUSSION The data from this research suggest that the geo- metric approach to trigonometry can lead students to understand trigonometric operations as func- tions while traditional instruction does not. In this section, I describe the features that distinguish this approach from the lessons that appear in many popular high school textbooks. The first is the emphasis placed on performing a geometric process to compute sines, cosines, and tangents physically. I examined the way in which several high school and college textbooks presented trigonometric functions and found that most mentioned a process (using the unit circle) only in passing. The questions students were asked to complete could almost always be accomplished by treating trigonometric operations as ratios. Second, the alternative approach asks stu- dents to perform this process physically and reflect on their actions. The textbooks provide neither of these two features. It is tempting to say that the emphasis on the unit circle was the reason students instructed in the alter- native approach did better, but in a large-scale study, Kendal and Stacey (1997) found otherwise. They compared students’ learning in classes in which a right-triangle model was used with students’ learn- ing in classes in which a unit-circle model was used. Students who were taught the right-triangle method performed significantly better on a subsequent post- test than students who were given the unit-circle model. So simply teaching about trigonometric oper- ations by using a unit circle model is no guarantee that substantial learning will occur. However, giving students the opportunity to think of sine and cosine as processes is critical, regardless of the model used to teach these operations. If students are given the opportunity to apply and reflect on the constructive geometric processes used in evaluating trigonometric operations, they will understand these operations at a much deeper level than if they are taught that these operations are merely ratios that can be applied to given right triangles. One appealing aspect of the geometric approach is that implementing it does not require a teacher to alter his or her classroom radically. No special technology or training in special teach- ing methods is required. Implementing the ideas expounded here offers practicing teachers the opportunity to create an active, collaborative, hands- on learning environment that has the potential to help students understand trigonometric concepts. REFERENCES Blackett, N., and D. O. Tall. “Gender and the Versa- tile Learning of Trigonometry Using Computer Software.” In Proceedings of the 15th Meeting of the International Group for the Psychology of Math-
  5. 5. 148 Mathematics Teacher | Vol. 102, No. 2 • September 2008 Tall, D. O., M. Thomas, G. Davis, E. Gray, and A. Simpson. “What Is the Object of the Encapsulation of a Process?” Journal of Mathematical Behavior 18, no. 2 (2000): 1–19. Weber, Keith. “Students’ Understanding of Trigo- nometric Functions.” Mathematics Education Research Journal 17, no. 3 (2005): 94–115. ∞ ematics Education 1, edited by F. Furinghetti, pp. 144–51. Assisi, Italy. 1991. Hirsch, Christian R., Marcia Weinhold, and Cameron Nichols. “Trigonometry Today.” Mathematics Teacher 84, no. 2 (1991): 98–106. Hollowell, K. A., J. E. Schultz, and W. Ellis Jr. HRW Geometry. Austin, TX: Holt, Rinehart, and Win- ston, 1997. Kendal, M., and K. Stacey. “Teaching Trigonometry.” Vinculum 34, no. 1 (1997): 4–8. Larson, R. Trigonometry. Boston: Houghton Mifflin, 2004. Lial, M. L., J., Hornsby, and D. I. Schneider. College Algebra and Trigonometry. Menlo Park, CA: Addison Wellesley, 2001. National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. Schoen, H. Trigonometry and Its Applications. Columbus, OH: Glencoe/McGraw Hill, 1990. KEITH WEBER, keith.weber@gse. rutgers.edu, is an assistant professor of mathematics education at Rutgers University in New Brunswick, New Jersey. He is interested in how high school and college students learn mathematics, including trigonometry and proof. Photograph by Erin Maguire; all rights reserved APPENDIX Computing Sines and Cosines by Using the Unit Circle 1. Start with a unit circle drawn on a Cartesian graph. A unit circle is a circle with a radius of 1 whose center is the origin. 5 3 A 4 10 a 30° b 40° (0, 1) 2. Use your protractor to make an angle with respect to the positive part of the x-axis. A 40° angle is shown here. 5 3 A 4 10 a 30° b 40° 70°
  6. 6. Vol. 102, No. 2 • September 2008 | Mathematics Teacher 149 3. Locate the point of intersection between the ray you have just drawn to complete the angle and the unit circle. Using a ruler or your graph paper, find the coordinates of your intersection. The x-value is the cosine of the angle you have constructed, and the y-value is the sine. In this case, sin 40º is about 0.65, and cos 40º is about 0.75. 40° 70° 1 The intersection point is about 0.75 to the right of the y-axis. The intersection point is about 0.65 above the x-axis. 85° Classroom Exercises 1. Compute the following sines and cosines by using a protractor and a unit circle: (a) sin 30° and cos 30° (b) sin 170° and cos 170° (c) sin 120° and cos 120° (d) sin 260° and cos 260° (e) sin 80° and cos 80° (f) sin 325° and cos 325° 2. Without explicitly computing these values (i.e., without using a protractor and a ruler), compute the following sines and cosines. To get you started, I will do the first exercise for you. (a) sin 90° and cos 90° (see diagram below) Explanation: A 90° angle is a right angle. I draw a 90° angle inside the unit circle. It intersects the circle at the top of the circle. This point is (0, 1). So sin 90° = 1 and cos 90° = 0. 5 3 A 4 10 a 30° b 40° 40° 70° 160° The intersection point is about 0.75 to the right of the y-axis. (0, 1) (b) sin 0° and cos 0° (c) sin 180° and cos 180° (d) sin 270° and cos 270° (e) sin 360° and cos 360°
  7. 7. 150 Mathematics Teacher | Vol. 102, No. 2 • September 2008 3. Approximate the sine and cosine of the angles drawn in the diagrams below: 40° 40° 70° 160° The intersection point is about 0.75 to the right of the y-axis. The intersection point is about 0.65 above the x-axis. 85° 230° 4. Without doing the computations, answer the following questions. Justify your answer. (a) Is sin 140° a positive number or a negative number? (Hint: Draw a unit circle and approximate a 140° angle). (b) Is cos 200° a positive number or a negative number? (c) Which is bigger—sin 23° or sin 37°? (d) Which is bigger—cos 300° or cos 330°? Homework Questions 1. Compute the following sines and cosines by using a protractor and a unit circle: (a) sin 50° and cos 50° (b) sin 127° and cos 127° (c) sin 200° and cos 200° (d) sin 300° and cos 300° 2. Approximate the sine and cosine of the angles drawn in the diagrams below: A 4 10 a 30° b 40° 40° 70° 160° The intersection point is about 0.75 to the right of the y-axis. The intersection point is about 0.65 above the x-axis. 85° 230° (0, 1) 3. Without doing the computations, answer the following questions. Justify your answer. (a) Is sin 240° a positive number or a negative number? (Hint: Draw a unit circle and approximate a 240° angle.) (b) Is cos 300° a positive number or a negative number? (c) Which is bigger—sin130° or sin 147°? (d) Which is bigger—cos30° or cos 230°? 4. In what quadrants will sin x be positive? In what quadrants will cos x be positive? 5. Can you find an angle x so that sin x = 2? If so, what angle is it? If you cannot find such an angle, why not?

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