The presentation accompanies a draft plenary paper. The paper is at this URL
Calculus addresses two foundational problems:
You know how fast a quantity
varies at every moment; you want to know how much of it there is
at every moment.
You know how much of a quantity
there is at every moment; you want to know how fast it varies at every moment.
We recently discovered that Isaac Newton said essentially the same thing.
"Newton expressed the two principal problems of infinitesimal calculus in directly mechanical terms. [Quoting Newton:]
The length of the space traced out being given continually, find the speed of movement at any given time.
The speed of movement being given continually, find the length of the space traced out at any given time."
(Dahan-Dalmédico, A., & Peiffer, J. (2010, p. 234). History of Mathematics: Highways and Byways. Washington, D.C.: MAA.)
DIRACC is rooted in research on students’ mathematical thinking and understanding
In DIRACC, we always refer to values of variables
In DIRACC, variables' values vary
We make strong distinctions among meanings of constant, parameter, and variable. Whether a mathematical notation is a constant, parameter, or variable depends on what you mean by it:
If you intend to represent the value of a quantity whose measure varies within a situation, then you are using that notation as a variable.
If you intend to represent the value of a quantity whose measure is the same within all situations (e.g., $\pi$), then you are using that notation as a constant.
If you intend to represent the value of a quantity that is constant in a particular situation, but which can vary from one situation to another, then you are using that notation as a parameter.
The word "change" can be ambiguous.
Sometimes instructors use it to mean "change in progress"
Sometimes instructors use it to mean "completed change".
At other times instructors use it to mean "become something different".
Students tend to think "change" means either become something different or completed change.
They rarely think it means change in progress.
We use "change" to speak about becoming something different.
We use "vary" to speak about change in progress.
We use "variation" to speak about completed change.
We point out that "change" in "rate of change" always has the connotation of change in progress.
Differentials are at the heart of DIRACC.
A differential is a variable whose value varies through intervals of specified size. The value of x varies by dx through intervals of size $\Delta x$.
We speak of differentials only in the context of constant rate of change.
The differential du is a varying change in u. It is the amount by which u varies within a $\Delta u$-interval.
The symbol $\Delta x$ represents the size of intervals through which dx varies.
We say that dy is related to dx by being proportional to values of dx.
We avoid "slope". Rather we emphasize rate of change.
The special meaning of "at the moment $x=x_0$" is developed in Chapters 3 and 4, and is used throughout the remainder of the book. By a moment of a variable we mean a small interval containing a particular value of the variable.
We develop the concept of an exact rate of change function. Each value of an exact rate of change function gives the rate of change of an accumulation function at a moment of its independent variable. This is the meaning of derivative we develop.
"rf" in the diagram below stands for the function whose values rf(x) give the exact rate of change of an accumulation function f at each value of x. "$\doteq$" means "essentially equal to".
We develop the concept of an approximate rate of change function. An approximate rate of change function is a function whose values approximate the values of an exact rate of change function.
Every value of an approximate rate function is the exact rate of change of its corresponding approximate accumulation function. This concept is used in developing accumulation from rate of change and in developing rate of change from accumulation.
In DIRACC, an integral is not an area. It is an accumulation. If the accumulating quantity is area of a bounded region, then an integral is the area of a bounded region. In most situations, the accumulating quantity is something other than area.
We use different notations for rate of change functions to emphasize different aspects of their meaning. We also make the design of notational systems an explicit topic of discussion.
Antiderivatives are functions.
We do not develop derivatives, then integrals, then bring them together with the FTC. Rather, like Newton, we begin with the FTC in that rate of change and accumulation are two sides of a coin. We use the concept of rate of change to develop accumulation functions and we use the concept of accumulation to develop rate of change functions.
We state the FTC even before developing rate of change from accumulation (derivatives). In fact, we use the FTC as motivation for finding derivatives and antiderivatives.
Structure of the Textbook
Chapters 1-4 can be thought of as pre-calculus. However, they constitute a forward looking precalculus by focusing on ideas central to calculus: infinitesimal amounts of quantities, variation, covariation, function, accumulation, and rate of change. This is in contrast to the backward focus of many precalculus textbooks which emphasize and extend material students (in the US) have already covered in Algebra 1 and Algebra 2, introducig more advanced algebraic topics along the way.
Chapters 5-7 cover standard topics of Calculus 1--integration and differentiation--but in an entirely novel way. They develop the calculus by explicitly addressing the foundational problems stated at the opening of this page.
Chapters 8-12 cover standard topics of Calculus 2 but do so more coherently than is typical. The emphasis in Chapters 4-7 on differentials as variables is leveraged throughout these chapters. It is this idea, together with the symbiosis between rate of change and accumulation, that creates coherence.
Online Homework
DIRACC homework (Exercises, Activities, and Reflection questions) is programmed into an online iMathAS system. iMathAS is a free and open source web-based assessment and grading tool created specifically for mathematics instruction, and can be run on your institution's server.
Most questions are programmed to be automatically graded by the system, and a small portion of questions (essay and file upload questions) require a teacher or TA to use the system to quickly assign points.
If you wish to explore how the DIRACC iMathAS homework system works, please use the following login:
The only available course is "DIRACC - Template" which contains every programmed problem.
Problems through Chapter 7 (except Chapter 2) will be completed by end of Summer 2018.
Problems through Chapter 12 will be completed by end of Summer 2019.
When teaching your own class you can select any subset of these to assign to students, as well as creating your own problems.
Please feel free to try answering questions to see how the system works - this template "course" has been set up as a review only and does not record any data.
To employ iMathAS homework outside of ASU, you will need to install iMathAS on your local server.
Instructional Advice (in progress)
Discuss Preface for Students with students. Revisit the points made there repeatedly during the term.
Use the textbook (especially animations) as a visual aid during class. More technically, use the textbook and animations as didactic objects. Didactic objects are "things to discuss" around which an instructor can conduct reflective conversations about ways students understand important mathematical meanings and ways of thinking.
Pacing: Will give advice in future revision on how to select review material.
Assign Reflections as homework and discuss them with students, either in class or in recitations.
Emphasize to students the importance that they study animations with the aim of understanding what we intend they illustrate. To study an animation does not mean just to watch it repeatedly. Studying an animation should entail reading the surrounding text that explains the animation and relating that text to the animation itself.
This textbook is written to support instructors wishing that students develop a conceptual orientation toward thinking mathematically.