In 2015, the International Journal of Research in Undergraduate Mathematics Education (IJRUME) was launched by Springer with editors-in- chief Karen Marrongelle and Chris Rasmussen from the U.S. and Mike Thomas from New Zealand. It was established to “become the central, premier international journal dedicated to university mathematics education research.” While this is a journal by mathematics education researchers for mathematics education researchers, many of the articles are directly relevant to those of us engaged in the teaching of post-secondary mathematics. This then is the first of what I anticipate will be a series of columns abstracting some of the insights that I gather from this journal.
I have chosen for the first of these columns the paper by Marilyn Carlson, Bernie Madison, and Richard West, “A study of students’ readiness to learn calculus.” [1] It is common to point to students’ lack of procedural fluency as the culprit behind their difficulties when they get to post- secondary calculus. Certainly, this is a problem, but not the whole story. Work over the past quarter century by Tall, Vinner, Dubinsky, Monk, Harel, Zandieh, Thompson, Carlson and many others have led the authors to identify major reasoning abilities and understandings that students need for success in calculus. This paper describes a validated diagnostic test that measures foundational reasoning abilities and understandings for learning calculus, the Calculus Concept Readiness (CCR) instrument.
The reasoning abilities and conceptual understandings assessed by CCR require students to move beyond a procedural or action-oriented understanding of mathematics. Whether it is an equation such as 2 + 3 = 5 or a function definition, f(x) = x2 + 3x + 6, students are introduced to these as describing an action to be taken, adding 2 to 3 or plugging in various values for x. To make sense of and use the ideas of calculus, students need to view a function as a process (defined by a function formula, graph, or word description) that characterizes how the values of two varying quantities change together. Listed below are four of the reasoning abilities and understandings assessed by CCR and which the authors highlight in their article.
- Covariational Reasoning. When two variables are linked by an equation or a functional
relationship, students need to understand how changes in one variable are reflected in
changes in the other variable. The classic example considers how the rates of change of
height and volume are related when water is poured into a non-cylindrical container such
as a cone. At an even more basic level, students need to be able to interpret information
on the velocities of two runners to an understanding of which is ahead at what times.
Another example, which involves covariational reasoning as well as understanding rate as
a ratio, considers the height of a ladder and its distance from a wall (Figure 1). When the
authors administered their instrument to 631 students who were starting Calculus I, only
27% were able to select the correct answer (c) to the ladder problem.
Figure 1. The ladder problem. - Understanding the Function Concept. Too many students interpret f(x) as an
unnecessarily long-winded way of saying y. They see a function definition such as f(x) =
x2+ 3x + 6 as simply a prescription for how to take an input x and turn it into an output
f(x). Such a limited view makes it difficult for students to manipulate functional
relationships or to compose function formulas. Carlson et al. asked their 631 students for
the formula for the area of a circle in terms of its circumference and offered the following
list of possible answers:
a. A = C2/4π
b. A = C2/2
c. A = (2πr)2a
d. A = πr2
e. A = π(C2/4)
Only 28% chose the correct answer (a). As the authors learned from interviewing a sample of these students, those who answered correctly were the students who could see the equation C = 2πr as a process relating C and r which could be inverted and then composed with the familiar functional relationship between the area and radius. - Proportional Relationships. Too many students do not understand proportional
reasoning. When Carlson et al. in an earlier study [2] administered the rain-gauge
problem of Piaget et al. (Figure 2) to 1205 students who were finishing a precalculus
course, only 43% identified the correct answer (as presented in Figure 2, it is 4⅔). Many
students preserve the difference rather than the ratio, giving 5 as the answer. Difficulties
with proportional reasoning are known to impede student understanding of constant rate
of change, which in turn underpins average rate of change, which is fundamental to
understanding the meaning of the derivative.
Figure 2. The rain gauge problem (taken from [3], [4]) - Angle Measure and Sine Function. As I described some years ago in an article for The Mathematics Teacher [5], the emphasis in high school trigonometry on the sine as a ratio of the lengths of sides of a triangle—often leading to the misconception that the sine is a function of a triangle rather than an angle—can lead to difficulties when encountering the sine in calculus, where it must be understood as a periodic function expressible in terms of arc length. An example is given in Figure 3, a problem for which only 21% of the Calculus I students chose the correct answer (e). Student interviews revealed that difficulties with this problem most often arose because students did not understand how to represent an angle measure using the length of the arc cut off by the angle’s rays.
What lessons are we to take away from this for our own classes? Last spring, in What we say/What they hear and What we say/What they hear II, I discussed problems of communication between instructors and students. The work of Carlson, Madison, and West illustrates some of the fundamental levels at which miscommunication can occur and identifies the productive ways of thinking that students need to develop.
References
[1] Carlson, M.P., Madison, B., & West, R.D. (2015). A study of students’ readiness to learn Calculus. Int. J. Res. Undergrad. Math. Ed. 1:209–233. DOI 10.1007/s40753-015- 0013-y.
[2] Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment (PCA) instrument: a tool for assessing reasoning patterns, understandings and knowledge of precalculus level students. Cognition and Instruction, 28(2):113–145.
[3] Piaget, J., Blaise-Grize, J., Szeminska, A., & Bang, V. (1977). Epistemology and psychology of functions. Dordrecht: Reidel.
[4] Lawson, A.E. (1978). The development and validation of a classroom test of formal reasoning. Journal of Research in Science Teaching, 15, 11–24. doi:10.1002/tea.3660150103.
[5] Bressoud, D.M. (2010). Historical reflections on teaching trigonometry. The Mathematics Teacher. 104(2):106–112.
great post on proportional reasoning, thanks so much!
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