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Q:
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How ``rigorous'' is the O/Z approach to calculus?

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A:
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``Rigor'' is commonly used in 2 different ways: (i) in reference to
a traditional, Bourbaki-style mathematical sense; (ii) as a
rough synonym for ``hard'' or ``challenging''.
The O/Z approach is certainly not rigorous in the first sense. (Nor,
for that matter, are most other beginning texts. For instance,
a rigorous approach to trigonometric functions should be preceded
with a proof of existence of arclength for circular segments.)
Rigor in this sense is, of course, essential to knowing and
doing mathematics---major students should certainly encounter it
seriously in their courses. But we think that a beginning calculus
course is too ``early'' for this encounter to occur in a concerted way.
A pedagogically sounder goal for beginning calculus students (including
prospective mathematics majors) is to become familiar and proficient with
the objects and methods of calculus. This approach seems more
practical for non-major students; it can also help prospective mathematics
majors better understand and use the very objects they'll encounter
later in more theory-based courses.

On the other hand, we certainly regard our approach as rigorous in the sense of being challenging. One source of challenge is the fact that we expect students to grapple seriously with the concepts as well as the manipulations of calculus.

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Q:
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Why are derivatives defined before formal discussion of limits?

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A:
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In a fully formal development, a full treatment of limits would of course
come first. We prefer, though, to first introduce limits (and
other concepts, for that matter) by showing them in action where
the need for limits arises naturally---in finding derivatives.
So we first introduce limits informally, illustrating their
meaning and uses in context. We study limits in their own right
after a section or two of using limits in context. We think that
students benefit from having some sandbox time with limits
before treating them formally.

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Q:
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Why do DEs appear so early?

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A:
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Since they're so important for solving real problems---much more
important than other things. Since they express laws. Since they
explain why log and exp fns and trig fns are so important.

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Q:
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Do students now need to learn DEs as well as calculus?

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A:
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No; we do not pretend to teach a comprehensive course in DEs. What
we do want mathematically is for students to learn is the idea and
language of DEs and how to check whether something is a solution.

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Q:
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(From a first-year student) What is calculus?

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A:
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Good questions. Like all big subjects, calculus is difficult to
encapsulate in a few words, but here's a try.
I'd say that the main theme of calculus is the study of
* continuously changing*
quantities. (I don't mean ``continuous'' in any terribly technical
sense---it's just that we're talking about things like physical position
and velocity, not
* discretely* changing quantities, like the readout on a digital watch.
The most basic problem in calculus has to do with the relation between
a varying quantity (say * f(x)* ) and the rate of change of that quantity
(say *f'(x)* ). Finding *f'* from *f* is the
*derivative* problem; going the other way is
the * antiderivative* problem. There are tons
of applications of these ideas, ranging from physical ones (e.g., relating
position, velocity, acceleration, etc.) to geometric ones (e.g., finding
curves with certain shapes, perhaps for computer graphic applications).

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Q:
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How do you ensure that your students emerge from the course with good
skills in the basic procedures --- e.g. symbolic differentiation, symbolic
integration, and finding limits?

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A:
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Our texts treat all of the usual differentiation rules
(see Chapters 2 and 3) and all of the standard
antidifferentiation methods (substitution, integration by parts,
partial fractions, trigonometric integrals and
substitutions --- see Chapters 5 and 8 in the second edition).
On the other hand, ``covering'' a set of techniques doesn't in
itself ``ensure that students emerge from the course with good
[basic] skills''---perhaps especially in a course that, like
ours, tries to go beyond building basic skills to concentrate
explicitly on conceptual understanding.
The ``gateway test'' or ``proficiency test'' is one
practical strategy we and others have used, we think successfully,
to convince ourselves (and our students) that students
can really carry out symbolic algorithms correctly and efficiently.
I usually give one of these in differentiation and another
in antidifferentiation. My own typical gateway tests include,
say, 10 straightforward but not completely trivial symbolic
derivatives or antiderivatives; students need to get 9 out
of 10 completely correct (no partial credit for
sign errors, etc.) to pass the gateway test. Passing
is worth enough points to make the effort worthwhile.
I retest (more than once, if necessary) those who fail on a
given attempt. In my experience, only a minority pass
on the first try, but very few take more than 3 attempts.
Now and then a student never passes the test, but not
very often---in this event the student loses a
substantial but not catastrophic number of points.

**
Q:
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Are there any such skills that you deliberately downplay in the
course, and if so, why?

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A:
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I wouldn't say that we ``downplay'' specific skills.
In fact, we've been accused of being on the
right wing of the reform movement in this sense---we think
students DO need a reasonable facility with symbols,
partly as a practical tool, partly because the standard elementary
functions of calculus ARE important in their own right, and
and partly because manipulating objects symbolically
can be a useful pedagogical vehicle for learning what these
objects really are and what makes them tick.
What I * would * say is that we try to present a view of calculus
that balances symbolic skills with other desirable outcomes, such
as a good conceptual and graphical understanding of the derivative and
the integral in Calculus I, and some numerical intuition
for such things as integrals and infinite series in Calculus II.
We think that traditional approaches have often been,
although perhaps quite successful at teaching students to
perform symbolic algorithms, not very good at helping
students ``unpack'' the symbols they use to see the mathematical
ideas beneath. For instance, it's quite possible to ``know'' that
* (x^2)'=2x* , or even that
* (x^2 * sin(x))'=2x*sin(x) + x^2*cos(x)* , without really
having any idea what derivatives are or what they tell about
functions.

**
Q:
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Could you briefly describe the motivation for the calculus reform
movement, and for texts like yours?

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A:
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Some of what we are aiming for in our text is
implicit in what's above, especially the bit about
conveying a sense of what calculus is really about---objects
and processes that aren't just pure symbols on which to
perform mysterious symbolic operations, but that
have a geometric and numerical reality as well. (This
doesn't deny that the symbolic view is important, too---but
it's not the only game in town.)
Quite a bit more (but not TOO much, I hope)
on these subjects is in the Preface for Instructors in
the beginning of the book.
What's above is really about our book, not necessarily
about the ``calculus reform movement'' as a whole. My sense of
it is that the phrase ``calculus reform'' has come to mean
so many things to so many people---ranging from real
ideas to pure advertising hype---that it's hard to know
what it really means, if anything. So I personally avoid
the phrase. Still, there are probably some general
common tenets of ``reform'' approaches, including such
things as emphasis on graphs, a willingness to use computing tools,
some focus on conceptual understanding (whatever that means---it's
not the same for different people), and some RELATIVE de-emphasis
on symbolic manipulation as the principal topic of the course.

**
Q:
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There are those that claim that understanding of mathematics comes
through doing many routine problems. How do you respond?

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A:
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I agree that this is one way of learning; indeed, symbolic drill (I
think that's what the question refers to) is * essential*
for learning some things, at some times. But it's
certainly not the only way to learn, and it's a poor way to
learn some things.

**
Q:
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Why do the sections on theory of continuity and differentiability (Sections
4.8 and 4.9 in the second edition) appear so late? Doesn't one need
these properties much earlier?

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A:
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These topics indeed out of place in the sense
of * mathematical development*, but that's not the same thing as
* pedagogical development*. There are many places in our text
(in all calculus texts, for that matter) where ideas are used before
being formally nailed down. (For example, one could argue that
exponential functions can't be rigorously defined until
after the log, which is in turn only rigorously defined by
the integral.) The idea behind the present placement of
theoretical discussion of continuity and differentiability is that
these sections represent brief excursions
into the theoretical side of the subject. Students (so the thinking
goes) are better primed and therefore more ``ready'' for such discussion after
having spent some * informal* time with ideas like
continuity and differentiability.

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