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In an answer to another question I asked, Isaac suggested a book that is the standard "reform calculus" book. In a comment, I asked what the phrase "reform calculus" means, and Isaac provided a link to this summary page. It seems to be a new or revised teaching methodology. However, the page linked to doesn't provide a good explanation of what exactly the phrase "reform calculus" means or its history.

The summary describes things like courses being "leaner in terms of the number of topics in the syllabus" and says that "technology alters the relative importance of specific techniques, and methodology because technology offers opportunities for creating new learning environments". But what do these things actually mean for people, especially the calculus student and the professor teaching a calculus course?

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up vote 6 down vote accepted

It is hard to describe faithfully an entire movement or "ism": suppose for instance you had asked for a description of Buddhism, or Marxism, or post-modernism. For every principle that you put forward as a "plank" of the movement, there is someone to say that that's a misunderstanding/oversimplification, or there is a specific submovement formed out of the violation of that principle.

As a result, let me focus on a particular branch of calculus reform which is better defined (and which I have seen more of myself): the Harvard Calculus Consortium. A very clear description, review and critique of this movement is given by Oliver Knill (who is well placed to comment on it, having been involved in the teaching and administration of calculus at Harvard for many years) here:

So I encourage you to read this first and then ask any further questions with at least one specific platform underneath your feet.

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This is particularly on-point as the Harvard Consortium book (Hughes-Hallett, Gleason, et al) was the one I said was perhaps the standard reform calculus book. – Isaac Aug 27 '10 at 21:37

Here's an excerpt from Reviewing Reformed Calculus, by Lisa Murphy, 2006.

The very beginnings of mathematics reform started in the 1960's, but the big push for Calculus reform started in earnest in 1989 with the publication of the National Council of Teachers of Mathematics' (NCTM) Principles and Standards for Mathematics Education. The NCTM published the Principles and Standards in response to the apathy of students towards math and the lack of academic success in the mathematics classroom. To combat these negative trends the NCTM outlined five goals for "the processes of problem solving, reasoning and proof, connections, communication, and representation" [1]. Through these goals it was hoped that students would be equipped with the basic skills and understanding that they would need to be successful. As the Principles and Standards inspired the reform of secondary mathematics education, thoughts of reform began to surface in the collegiate mathematics arena, especially with regard to Calculus. College Calculus courses were experiencing some of the same problems as secondary mathematics. Of the roughly 300,000 college students that are annually enrolled in an engineering-based Calculus course, only 140,000 earn a grade of D or higher [6]. Less than half of the students were performing "well" in their Calculus courses. Armed with statistics such as this, reform-minded professors set out to develop a new curriculum that would help raise the achievement level and stimulate student interest in mathematics.

From the reform movement, numerous curricular designs have been generated. Calculus and Mathematica; Calculus, Concepts, Computers and Cooperative Learning (C4L); and The Calculus Consortium at Harvard (CCH) are a few of the commonly used curriculums. These new curriculums cover the entire spectrum of reform. Some are grounded in traditional techniques but incorporate snippets of reform, while others differ in most aspects from the traditional approach. Despite this vast array, there are some basic elements that are common to all reform curriculums in varying degrees that separate them from the traditional Calculus curriculum. One of the most noticeable differences of reformed Calculus is the use of graphing calculators and/or computers. The graphing calculator is a critical component in the reform classroom. Many reform classes include a weekly lab session where students meet in a computer lab. The students make use of calculators and math computer programs to investigate new topics and to graphically see what they are working on. Most reform textbooks urge students to read through the text with a calculator in hand to see directly what is discussed in the text. The idea behind the incorporation of calculators and computers is to alleviate the heavy algebraic manipulation that students typically do in a traditional Calculus setting. Reform supporters argue that the removal of manipulation allows students to move beyond the drudgery of computation and start learning the fundamental ideas of Calculus. They additionally argue that topics are discussed more fully with the use of graphical representations.

A reformed Calculus class differs from a traditional course in methods of instruction. When walking into a reform classroom it is immediately clear that it is indeed a reform classroom. Most noticeably the teacher is no longer the central focus of the classroom experience. The lecture method of instruction, a standard of traditional curriculum, has a lesser place within a reform setting. The teacher still lectures occasionally and is available to answer questions from the students, but there is greater emphasis placed on cooperative learning. Reform students often work in groups to determine solutions or to explore concepts in a laboratory setting. This idea is rooted in the constructivist learning theory. Each student constructs their own meaning as they learn. Students are given the basic tools and from these discover how the pieces fit together to form the concept that they are studying. One of the primary goals of the C4L curriculum is to "create situations which foster students to make the necessary mental constructions to learn mathematics concepts" [10]. Within the curriculum itself, the reformed method stresses the applications of Calculus. This emphasis hopes to justify the topics of study, which in theory raises interest in the material. In an effort to accomplish this, some of the mathematical rigor is removed from the curriculum. Most reform textbooks are void of a single proof. In the introduction to Calculus from Graphical, Numerical and Symbolic Point of View, the authors state that "proving theorems in full generality is less valuable, we think, than helping students understand concretely what theorems say" [9]. As a result of this change, a common question that arises from students new to reformed Calculus is "Where is the math?" [5].

Accompanying this application heavy curriculum is a different method of assessment. Reformed Calculus courses emphasize the use of writing. Projects, reports and lengthy explanations of problem solutions are common place within the reform classroom. In some cases, the students are graded more on the thoroughness and completeness of written explanations as opposed to correctness of answer. [...]

The emphasis on correct explanation rather than correct answer is seen explicitly in the directions for the midterm. The problem also provides an example of the type of application problems that reformed Calculus students are accustomed to working with. This midterm question additionally exhibits one of the flaws that traditional professors are quick to point out. The problem asks students to determine when the population becomes infinite, which is a misuse of the word infinite. The population may become uncontrollable but it will never become infinite. Traditional professors argue that the misuse of mathematical terms, such as infinite, teaches students the wrong meaning of or concept behind the term, which results in misunderstandings in future math work. More generally, there is a trend in reformed Calculus moving away from individual study and towards a social study of Calculus. The context of learning Calculus is now placed in a more social setting. Students work primarily in groups to gain knowledge both from a textbook and from each other.

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