Prototype images in mathematics education: the case of the graphical representation of the definite integral

Abstract

Many mathematical concepts may have prototypical images associated with them. While prototypes can be beneficial for efficient thinking or reasoning, they may also have self-attributes that may impact reasoning about the concept. It is essential that mathematics educators understand these prototype images in order to fully recognize their benefits and limitations. In this paper, I examine prototypes in a context in which they seem to play an important role: graphical representations of the calculus concept of the definite integral. I use student data to empirically describe the makeup of the definite integral prototype image, and I report on the frequency of its appearance among student, instructor, and textbook image data. I end by discussing the possible benefits and drawbacks of this particular prototype, as well as what the results of this study may say about prototypes more generally.

Appendices

Appendix 1: Interview items

The interview data used for this study came from two sets of interviews and the following are the prompts used as sources of the image data for this report.

Interview 1: Ten students received all of the following prompts, and three students received only prompts #4, 5, and 6.

• 1. Let’s look at this general expression, ${\int}_a^bf(x) dx$, without any specific numbers or functions. What does this mean? What does it represent? Please explain as much as you can.

• 2. Let’s say you had a friend in your calculus class who had been sick for the last week or so and missed everything your class learned about integrals. Now they really need you to help them understand what they are. How would you explain integrals to them? What would you say an integral means?

• 3. Let’s say you enrolled in a physics class next semester and your professor wanted to test everybody to see if you understood what integrals are. Suppose your professor gave you a short quiz that asked, “Write down everything you can think of about what an integral is and what it does.” What would you write for your response?

• 4. In class you’ve probably seen both ∫f(x)dx and ${\int}_a^bf(x) dx$. Is there any difference between these two? If so, are they basically the same thing, or are they totally different from each other?

• 5. What do you think is the most important way to understand the integral ${\int}_a^bf(x) dx$?

• 6. You’ve learned about “Riemann sums” in your calculus class. Could you explain what a Riemann sum is? In your opinion, are they useful, or just something you do in a math class?

• 7. From physics, Mass = Density × Volume. If a box has varying density (that is, it’s not the same throughout the box), we would use an integral to calculate it: M = ∫ _Box D dV. Why does this integral end up calculating the total mass of the box?

• 8. From physics, we have Force = Pressure × Area. If the pressure exerted on a surface, S, is not the same at every point, we usually use an integral to calculate it: F = ∫ _S P dA. Why does this integral end up calculating the total force on the surface?

Interview 2: Ten students received all prompts, with no overlap with “Interview 1” students.

• 1. What does ${\int}_a^bf(x) dx$ mean?

• 2. Are there other way you can interpret it or think about?

• 3. What does dx mean? What does it have to do with the integral?

• 4. [This interview also examined other types of integrals, including multivariate and line integrals. If a student, in the course of these prompts, returned to discuss single-variable integrals, that particular portion of the interview was also included.]

Appendix 2: Survey items

This survey was administered to 205 students.

• 1. Explain in detail what ${\int}_a^bf(x) dx$ means. If you think of more than one way to describe it, please describe it in multiple ways. Please use words, or draw pictures, or write formulas, or anything else you want to explain what it means.

• 2. From physics, Force = Pressure × Area. If the pressure on a surface, S, is not the same at every point, we use an integral to calculate it: F = ∫ _S P dA. Why does this integral actually calculate the total force on the surface?

• 3. Why does an integral need a “dx” on it? For example, why can’t it just be ${\int}_0^1{x}^3$ instead of ${\int}_0^1{x}^3 dx$? Explain in as much detail as you can.

• 4. From physics, Mass = Density × Volume. If an object, S, has varying density (i.e. it’s not constant throughout the object), we use an integral to calculate it: M = ∫ _S D dV. Why does this integral actually calculate the total mass of the object?

Appendix 3: Definitions of each code within the coding scheme

This table provides complete definitions for each code used in the coding scheme (see Table 1).

Codes	Definition of code
1a. Linear function	A graph consisting of a single straight line.
1b. Conic function (parabola or circle section)	A graph that visually appears to be a parabola or portion of a circle, even if no explicit function expression is provided.
1c. Other type of function	Any graph not coded into linear or conic.
2a. Continuous and smooth	A graph that appears continuous, with no apparent cusps.
2b. Continuous with cusp	A graph that appears continuous, but with at least one cusp.
2c. Discontinuous	A graph that has any one of a hole, jump, or asymptote.
3a. Positive only	A graph that is only strictly above the horizontal axis.
3b. Positive and 0	A graph that is above AND touching the horizontal axis.
3c. Negative only	A graph that is strictly below the horizontal axis.
3d. Negative and 0	A graph that is below AND touching the horizontal axis.
3e. Positive and negative	A graph that is both above AND below the horizontal axis.
4a. No steep slopes	A graph such that for all slopes, |slope| < 2.
4b. Steep slopes only	A graph such that for all slopes, |slope| ≥ 2.
4c. Both steep and non-steep	A graph with slopes, in magnitude, both above and below 2.
5a. Increasing	A non-decreasing graph with at least some portion that is strictly increasing.
5b. Decreasing	A non-increasing graph with at least some portion that is strictly decreasing.
5c. Pure horizontal	The graph of a horizontal line, with no increase or decrease.
5d. Both increasing and decreasing	A graph with at least one strictly increasing portion and one strictly decreasing portion.
6a. Straight lines only	A graph containing no curvature, but that is a straight line, or collection of line segments.
6b. Concave up only	A graph containing curvature, which is only concave up.
6c. Concave down only	A graph containing curvature, which is only concave down.
6d. Both concave up and down	A graph containing both a concave up portion and a concave down portion.
7a. 0 < a < b	The domain is completely to the right of the vertical axis.
7b. 0 = a < b	The domain begins at the vertical axis.
7c. a < b < 0	The domain is completely to the left of the vertical axis.
7d. a < b = 0	The domain ends at the vertical axis.
7e. a < 0 < b	The domain begins to the left and ends to the right of the vertical axis.
7 f. a = b	The domain consists only of a single point.
7 g. b < a	Any case in which the upper limit is less than the lower limit.
8a. Little variation from average	The graph has no large peaks or valleys, nor long inclines nor declines.
8b. Significant deviation from average	The graph contains at least one large peak, large valley, long incline, or long decline.
9a. Closed boundary	Vertical lines (solid or dashed) are used at a and b to create a closed shape with the graph and horizontal axis.
9b. Open boundary	No vertical lines are used at a and b, making the shape implied, but not physically closed.

Appendix 4: Textbook image counts

The following table lists each of the 126 textbook images, by page number, that were matches for all nine characteristics of the prototype.

Book	“Prototype” images	Images NOT counted as prototype	Images not included in data set
Stewart (2015)	366, 369, 370, 370, 370, 370, 370, 371, 379, 379, 387, 392, 393, 394	366, 366, 366, 367, 367, 367, 368, 369, 369, 369, 369, 369, 369, 371, 373, 373, 374, 379, 379, 382, 382, 383, 383, 384, 384, 385, 385, 386, 386, 387, 388, 392, 393, 393, 393, 393, 393, 398, 405, 407, 408, 417, 418, 418	366, 366, 366, 393, 395, 395, 404, 415
Briggs et al. (2015)	336, 337, 337, 337, 337, 352, 352, 352, 355, 355, 356, 356, 356, 362, 365, 365, 365, 371, 371, 371	334, 335, 335, 335, 335, 338, 338, 339, 341, 348, 349, 349, 349, 349, 350, 350, 350, 350, 352, 353, 353, 353, 354, 354, 356, 356, 362, 362, 362, 363, 363, 364, 367, 367, 368, 368, 369, 370, 370, 370, 370, 370, 377, 377, 379, 380, 390, 390	334, 353, 362, 364, 364, 370, 370, 379, 381