Calculus around the World
By: David Bressoud @dbressoud
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and Director of the Conference Board of the Mathematical Sciences
The June 2021 issues of ZDM – Mathematics Education is a collection of articles on Calculus in High School around the World, edited by Patrick Thompson and Guershon Harel. My own contribution, “The strange role of calculus in the United States,” is included in this volume. Other articles explore issues around calculus instruction in Singapore, Tunisia, Brazil, Mexico, Quebec, Germany, South Korea, and Israel as well as the United States. When combined with a 2014 paper, “Calculus in European classrooms” by Törner, Potari, and Zachariades that describes calculus instruction in France, Germany, England, Belgium, Italy, Greece, and Cyprus, we see how calculus instruction is approached around the world and come to recognize how truly strange is the way we deal with it in the United States.
The first and most glaring difference is the assumption in the U.S. that calculus is a university-level course. Almost everywhere else it is assumed that students on a more mathematically intensive track through high school will learn the basic tools of calculus before arriving at university, usually beginning in grade 9 or 10. In France and Germany, calculus was brought into the secondary curriculum early in the 20th century, pushed by Poincaré, Borel, and Hadamard in France and Klein in Germany.
Singapore provides a good example of how this works today. About half of all students enter the Advanced Mathematics track in grade 9. Their first two years include basic rules for differentiation—including for transcendental functions—and a study of stationary points with the various tests to determine their type. The next two years expand differentiation to include concavity as well as implicit differentiation and differentiation of parametric functions. Students are taught integration by substitution and by parts. Taylor series are also introduced in these years.
One thing that makes this possible in Singapore and throughout the world is that no one imitates the crazy U.S. system of compartmentalizing Algebra, Geometry, Precalculus, or Calculus by spending an exclusive year on each. Instead, the world uses what the U.S. calls an integrated curriculum, spiraling through all of the secondary topics each year, revisiting and extending student understanding and ability with each pass. Thus in Israel, for examples, students study calculus in each of grades 10 through 12, but only about 25% of the mathematics lessons in these years are devoted to calculus.
In my article, I trace the development of our truly dysfunctional approach to calculus where the same university course is taught—or attempted—in high school, but then retaught in university as though it is fresh and unfamiliar to the students entering Calculus I, even though the overwhelming majority are reviewing material they have already seen. The result is a course that puts the students who have not studied calculus in high school at a tremendous disadvantage. These are primarily students from under-resourced backgrounds. In the high school graduating class of 2013, 38% of graduates from families in the highest socio-economic quartile had completed a calculus course while in high school. Only 7% from the lowest quartile had accomplished this (Champion & Mesa, 2017). Because the U.S. lacks a national curriculum and suffers from local funding and control, university mathematics instructors face widely disparate backgrounds among their students. Students from privileged backgrounds who have benefited from solid preparation are, inevitably, the winners in university.
I have written extensively in these columns on what needs to be and can be done to support all students. What I want to focus on for the remainder of this column is that—despite superior educational systems such as that of Singapore—universities around the world are discovering many of the same problems with student understanding of and facility with calculus that we encounter here in the United States.
Tin Lam Toh reports on an assessment given to 42 Singapore students in the higher mathematics track before the start of grade 11 and again following grade 12. It describes a discouraging lack of progress on even such simple tasks as identifying the number of points of inflexion given the graph of a quartic polynomial with two local minima and one local maximum.
Students in Hartman secondary school in Jerusalem taking their matriculation exams in mathematics. From The Times of Israel, 30 November 2016.
Dreyfus, Kouropatov, and Ron describe their work on a national curricular document for secondary calculus instruction in Israel. This followed a study by Kourapatov that found that “students are much more successful following an algorithm or manipulating symbols than they are when dealing with the concepts of derivative and integral.” (Dreyfus et al., 2021, p. 681). Their curricular document is based on three principles
Reasoning. “The curriculum elicits, fosters, and supports mathematical ways of reasoning as well as fundamental mathematical ideas such as justification.”
An example from Dreyfus et al. of a question that requires reasoning.
Impact. The ways in which the curriculum incorporates the function and position of mathematics in today’s world.
Cultivation. The ways of creating intellectual ground fertile for a new concept to emerge naturally.
This last is encapsulated in a quote from Movshovitz-hadar and Hazzan (2001),
“Don’t define a notion before you show a particular instance worthy of the definition; don’t state a general phenomenon before demonstrating it on a few (sufficiently large number of, yet not too many) particular cases.” (p. 818)
One aspect of cultivation is what Guershon Harel refers to as “intellectual need”.
The authors’ illustrate their work on the national curricular document with a description of the structure of their unit on integration in which the integral is introduced as the accumulation function.
Integration is also the focus of the paper by Greefath, Oldenburg, Siller, Ulm, and Weigand that examines the Basic Mental Models of integration held by German students entering first-year university calculus at the universities of Augsburg, Bayreuth, and Würzburg. The authors identify four such models: integral as area, integral as total variation of a rate of change function, integral as providing average value, and integral as accumulation. This article is particularly interesting for the tool they developed to explore how students’ conceptualize integration. They give several applications or results and four possible justifications, each of which is in line with one of the four mental models. For example, the fact that the integral of sin(x) from 0 to 2 pi is 0 can be justified in terms of areas, total variation, average value, or accumulation. For each explanation, students responded on a 5-point Likert scale that expressed the degree to which the explanation reflected their own thinking, from “not at all” to “exactly how I think”.
Not surprisingly, area was the most popular mental model. Integration as a means of determining average value was the least common. In addition, those students for whom the integral as area was dominant were not likely to think of integration in terms of any of the other three models. On the other hand, the remaining three models showed strong correlations with each other. This provides further evidence that the integral as area is a bad place to start building student understanding of integration.
While the U.S. approach to calculus instruction is particularly dysfunctional, this volume and the recently concluded Topic Study Group on the Teaching and Learning of Calculus of the 14th International Congress for Mathematics Education (ICME14) reveal that student difficulties with calculus are universal. Dealing with them requires the hard work of research in mathematics education to understand the root causes of obstacles and misconceptions and to identify effective strategies for helping students to overcome them.
References
Bressoud, D.M. (2021). The strange role of calculus in the United States. ZDM Mathematics Education 53, 521–533. https://doi.org/10.1007/s11858-020-01188-0
Champion, J. & Mesa, V. (2017). Factors Affecting Calculus Completion among U.S. High School Students. In D. Bressoud (Ed.) The Role of Calculus in the Transition from High School to College Mathematics. Washington, DC: MAA and NCTM.
Dreyfus, T., Kouropatov, A. & Ron, G. Research as a resource in a high-school calculus curriculum. ZDM Mathematics Education 53, 679–693 (2021). https://doi.org/10.1007/s11858-021-01236-3
Greefrath, G., Oldenburg, R., Siller, HS. et al. Basic mental models of integrals: theoretical conception, development of a test instrument, and first results. ZDM Mathematics Education 53, 649–661 (2021). https://doi.org/10.1007/s11858-020-01207-0
Toh, T.L. School calculus curriculum and the Singapore mathematics curriculum framework. ZDM Mathematics Education 53, 535–547 (2021). https://doi.org/10.1007/s11858-021-01225-6
Törner, G., Potari, D., Zachariades, T. (2014). Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts. ZDM Mathematics Education 46:549–560. DOI 10.1007/s11858-014-0612-0
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