A Graph-Theoretic Analysis of Calculus Textbook Tasks

Motivation

• Over the decades, there have been numerous calls for more conceptually-based Calculus instruction (Dreyfus & Eisenberg, 1990; Thompson et. al., 2013).

• However, in my institutional setting, procedural fluency in the computation of derivatives remains an important foundational skill.

• This analysis was motivated by a curiosity about why, in my experience, students appear to struggle to understand some derivative computation procedures (for example, the Chain Rule) more than others (for example, the Power Rule).

• As someone with a Discrete Mathematics background, I wondered if certain properties of the structure of assigned tasks might play a role.

Research Question

What is the structure of students’ exposure to derivative computation (DC) skills among textbook tasks? 

Background & Theoretical Perspectives

• A graph-theoretic research method is one that leverages the mathematical definition of a graph (Gilroy, in press), a set of points together with the set of pairwise relationships between the points (Chartrand et. al., 2016).

• A derivative computation task is a task of the form:

"Find the derivative of [insert function]."

• An elementary function is a function comprised of a finite collection of field operations (addition, subtraction, division, extracting roots) and simple functions (constant functions, algebraic functions, exponential functions, and the inverses of these types) (Chow, 1999). 

• Let f be an elementary function. Then, I define the Elementary Function Tree of f, EFT(f), as the directed graph (or digraph) whose vertex set is the collection of simple functions and commutative operations contained in f, and, for two vertices u and v, there exists a directed edge (arrow) from u to v if v is contained in u. 

• My design choices for EFTs were informed by Schema Theory and Symbolic Forms.

Schema Theory

• One way to think about Schema Theory is to imagine the brain as a computer whose memory stores and organizes information in a file directory. The file directory of a novice learner, such as a Calculus I student, will contain numerous file folders related to Calculus content. 

• For example, in the case of derivative rules, there might be separate file folders for the Power Rule, the Product Rule, the Quotient Rule, and the Chain Rule. Furthermore, the Power Rule folder might be multiple folders, with one for positive exponents, one for negative exponents, and yet another for rational exponents. 

• However, with adequate exposure over time, the information contained in these separate folders gets compressed into one single folder about Derivative Rules in the fashion of an expert learner, such as a Calculus I instructor. 

• The Elementary Function Trees introduced in this paper consider the structure of elementary functions from a novice learner’s (or maximum file folders) point of view, meaning that each simple function type will contain multiple sub-types. 

• For example, I chose to subdivide power functions according to their exponents.

  • • P+: Positive integer exponents

    • P-: Negative integer exponents

    • PC: Zero exponents

    • PQ+: Positive non-integer exponents

    • PQ-: Negative non-integer exponents

Symbolic Forms

• In physics education research, Sherin poses the notion of Symbolic Forms (Sherin, 2001), which are knowledge elements consisting of a symbol template and conceptual schema. In this case, a derivative rule like “differentiate term-by-term” is associated with the symbol template. 

• Although Symbolic Forms were initially developed in physics education research, because they involve understanding of mathematical content, this theory has also been applied to mathematics education research and across other disciplines (for example, Rodriguez et. al., 2020). 

• Symbolic Forms motivate the treatment of operations as vertices. However, I chose to only use commutative operations (addition and multiplication) as vertices because the position of vertices in a digraph is not significant. 

• Noncommutative operations, such as division, are represented in terms of commutative operations, such as multiplication by a reciprocal, as nontrivial sub-structures in the EFT. 

Task Analysis

• I wanted to simulate a student's realistic exposure to DC tasks in a Calculus course.

• Since "The Odds" are regarded as the proverbial mathematics homework assignment (Bush, 1987; Bill & Jamar, 2009), I chose to analyze the DC tasks contained in 3 popular United States Calculus textbooks.

  • • Calculus: Early Transcendentals, Eighth Edition by J. Stewart

    • Calculus Ninth Edition by R. Larson & B. H. Edwards

    • Calculus Volume 1, Web Version by G. Strang & E. Herman

• For each of the tasks, I drew its EFT using the following set of vertex labels.

Analysis of 1D Skills (Vertices)

• I call the vertices labeled with simple functions one-dimensional skills.

• I began by computing the relative frequency of one-dimensional skills among all the EFTs for each textbook. 

• For visual clarity, I chose to use the larger classes of simple functions: Power, Exponential, Logarithmic, Trigonometric, and Inverse Trigonometric.

Analysis of 2D Skills (Edges)

• In addition to the digraph's vertices, we can also analyze its edges. Any DC skill represented by a single edge in an EFT I will call a two-dimensional skill.

• The Minimum Weighted Common Supergraph (MWCS) of a set of graphs, S, is the smallest graph (in terms of numbers of vertices and edges) that contains all graphs in S as subgraphs (adapted from Bunke et. al., 2000).

• From each textbook's set of EFTs, I constructed the MWCS. In general, this is really hard to do (Bunke et. al., 2000), but in this case, it's the union of the vertex sets and the union of the edge sets.

• Box-and-whisker plots of the edge weights (frequencies) for each MWCS were created for outlier detection. Outliers represent two-dimensional skills that are practiced significantly more than others. 

• For Stewart’s text, 10 outliers were detected, for Strang & Herman’s text, 7 outliers were detected, and for Larson & Edwards’ text, 4 outliers were detected. 

• Finally, to determine if there were any commonalities among the outliers, I visualized the textbooks' sets of outliers in a Venn Diagram.

Conclusions & Future Directions

• Analyses of both the one-dimensional and two-dimensional skills present in the textbooks' DC tasks revealed that Calculus textbooks give students significantly more practice with the power rule than any other DC procedure.

• Specifically, power functions with positive integer exponents were most common among both one- and two-dimensional skills.

• This result has both positive and negative implications, depending on who I talk to.

  • • Disciplinary Experts who are familiar with Calculus (such as Biologists, Engineers, Physicists, Chemists, and Economists) believe that since power functions are used more often than other function classes in models of natural phenomena, students should have a better grasp on these functions than most other function classes. Therefore, it is okay that Calculus textbooks stress power functions in DC tasks. This approach is focused primarily on the utility of mathematics.

    • Mathematics Education Researchers believe that Calculus should be taught conceptually with a minimal focus on complex procedures (Dreyfus & Eisenberg, 1990; Thompson, 2013). Since power functions are regarded as some of the simplest functions, they are useful for students to "get the idea of" computing derivatives procedurally, but not much more is necessary with today's technology (Garfinkel et. al., 2022). Therefore, it is okay that Calculus textbooks stress power functions in DC tasks. This approach is focused on a blend of utility and mathematical problem-solving as a means to determine the reasonableness of solutions.

    • Mathematicians believe a student should be able to determine how to complete a new DC task by drawing on knowledge from previously completed DC tasks. So, students should have an equal understanding of all DC skills. Therefore, it is not okay that Calculus textbooks stress power functions in DC tasks. This approach is focused on training mathematical problem-solving skills.

• In my institutional context, since Mathematicians teach all of the Calculus courses, the results of my analysis indicate a misalignment between the structure of assigned task sets and Mathematicians' expectations of students.

• It would be an interesting Discrete Mathematics problem to construct DC task sets that align with Mathematicians' expectations (i.e. students practicing skills in proportions as equal as possible).

• From a Mathematics Education Research perspective, it would be interesting to determine if assigning students these "balanced" task sets leads to better outcomes for students in Calculus courses whose curricula stress DC skills.

References

Note: References are listed in order of appearance.

1. Dreyfus, T. & Eisenberg, T. (1990). Conceptual Calculus: Fact or Fiction? Teaching Mathematics and its Applications: An International Journal of the IMA, 9(2), 63-67.

2. Thompson, P. W., Byerley, C., & Hatfield, N. (2013). A Conceptual Approach to Calculus Made Possible by Technology. Computers in the Schools, 30, 124-147.

3. Gilroy, H. (in press). The manifestation of graph-theoretic methods in mathematics education research: A metasummary of intercontinental conference proceedings. In Proceedings of the 26th Conference on Research in Undergraduate Mathematics Education.

4. Chartrand, G., Lesniak, L., & Zhang, P. (2016). Graphs & Digraphs Sixth Edition (pp. 3). CRC Press. 

5. Chow, Y. T. (1999). What Is a Closed-Form Number? American Mathematical Monthly, 106(5), 440-448.

6. Sherin, B. L. (2001). How Students Understand Physics Equations. Cognition and Instruction, 19(4), 479-541.

7. Rodriguez, J. G, Bain, K., & Towns, M. H. (2020). Graphical Forms: The Adaptation of Sherin’s Symbolic Forms for the Analysis of Graphical Reasoning Across Disciplines. International Journal of Science and Mathematics Education, 18, 1547-1563.

8. Bush, W. S. (1987). Mathematics Textbooks in Teacher Education. School Science and Mathematics, 87(7), 558-564.

9. Bill, V. L. & Jamar, I. (2009). Disciplinary literacy in the mathematics classroom. Content matters: A disciplinary literacy approach to improving student learning, 63-85.

10. Stewart, J. (2016). Calculus: Early Transcendentals, Eighth Edition. Cengage.

11. Larson, R. & Edwards, B. H. (2009). Calculus, Ninth Edition. Brooks/Cole.

12. Strang, G. & Herman, E. (2023). Calculus Volume 1, Web Version. Openstax.

13. Bunke, H., Jiang, X., & Kandel, A. (2000). On the Minimum Common Supergraph of Two Graphs. Computing, 65(1), 13–25. https://doi.org/10.1007/PL00021410

14. Garfinkel, A., Bennoun, S., Deeds, E., & Van Valkenburgh, B. (2022). Teaching Dynamics to Biology Undergraduates: The UCLA Experience. Bulletin of Mathematical Biology, 84(3), 43. https://doi.org/10.1007/s11538-022-00999-4