On Assessment, IV

Last week, I had ended with an interpretation for an A–D grading scale, shown below (here is a link to last week’s post for reference).

Day166grades

I remind you that this scale is not ideal; the purpose was to come up with some system of assigning grades which wasn’t punitive, but rather which motivated students to learn concepts rather than to avoid losing points on exams.

We continue with a discussion of how to use such a system in practice.

“Now let’s consider this in the context of an exam.  The first part of an exam is a skills portion, with, say, ten short problems of roughly equal length.  Expectations for this part of the exam [meaning a grade of B] are seven problems “essentially” correct, and four problems completely correct.  These expectations are written on the exam for students to see.

Are these expectations too low?  Perhaps.  But then an A [refer to the chart below] means eight problems “essentially” correct, with at least five completely correct. Of course we must ask what it means for a problem to be “essentially” correct — but when in doubt, err on behalf of the student.  (Students rarely suggest that their scores be lowered.)

Then grading is actually somewhat easier, and grades can be assigned as follows, with the abbreviations EC and CC meaning essentially correct and completely correct, respectively (for simplicity, a grade of C is assigned for all other cases not accounted for):

Day167grades

Now this eliminates the need for partial credit — but does require a judgment as to what “essentially correct” means.

This also makes grading much easier.  I would suggest that each problem be marked as “EC,” “CC,” or left blank.  Few comments, if any are necessary.  This is the approach I have taken in BC Fast-Track, and it encourages further learning as it leaves the student in the position of needing to work through their mistakes.

I would have students keep a section of their notebooks for exams and revisions, and there they can keep their reworked problems, should they choose to do so.  Then — as I did in BC Fast-Track — students could visit me periodically with their notebooks and I can take a look at their ongoing progress.  This “additional” work, if sufficiently well done, could boost their grade at the end of the semester.

I think this could have the same effect it did in BC Fast-Track — exams were easier and more enjoyable to grade.  But there were more discussions in my office about reworked exams and sources of error that were initiated by the students themselves, and these discussions were not about points, but about concepts.

Now what about the part of the exam which is intended to be more conceptual?  Let us suppose that there are three problems, roughly comparable in length, and of various difficulties.  Then grades might be assigned as follows:

Day167grades2

More details about how this would fit in a classroom environment may be found in a later document [I cannot recall which document is being referred to here].  But this system allows for a more qualitative approach to grading.  Performance expectations are also clearer, but such expectations depend critically upon the nature of the problems given.  Moreover, grades are not assigned punitively, but the emphasis is on doing problems completely and correctly.

For an example, below could be a set of ten skills problems and three conceptual questions for a basic assessment on the rules of differentiation.  This would a 70-minute assessment.  Given expectations for completely correct problems, I think this is reasonable.

Skills questions:

1.  Evaluate

\lim_{x\to\pi/4}\dfrac{\tan x-1}{x-\pi/4}.

2.  If f(x)=e^x\cos x, find f^\prime(x).

3.  Find

\dfrac d{dx}\dfrac 7{\sqrt{x^3}}.

4.  Using the quotient rule, find

\dfrac d{dx}\dfrac{x^3}{\sin x},

simplifying as much as possible.

5.  Find the derivative of f(x)=\left(\sin\sqrt x\right)^{\!2}.

6.  Find the equation of the tangent line to h(x)=\sec(2x) at x=\pi/6.

7.  Using a definition of the derivative, find the derivative of p(x)=x^2-x.

8.  Assume that f and g are differentiable functions. Find

\dfrac d{dx}f(g(x^2)).

9.  Find the derivative of q(x)=x^2e^x\cot x.

10.  Let f be the greatest integer function. Using the definition of the derivative, determine whether or not the derivative exists at x=0.

Conceptual questions:

1.  Using the product rule, find

\dfrac d{dx}f(x)(g(x))^{-1}.

Explain your result.

2.  Suppose that the line y=6x+a is tangent to both f(x)=x^2+b and g(x)=x^3+3x. Find a and b.

3.  Suppose that f is a differentiable function. Discuss the following limit:

\lim_{h\to0}\dfrac{f(x+2h)-f(x-h)}{3h}.

Stay tuned for next week’s final installment of this series on assessment….

On Assessment, III

Today, I’ll continue with the discussion of assessment I began a few weeks ago.  Last week, I ended with the observation that “The performance of our students determines our expectations, rather than the other way around.”  Before continuing, I just want to make clear the context for that remark.  When I wrote this essay, IMSA was fully funded by the state of Illinois.

What would it look like to legislators — who approved our funding — if you brought in the top students (relative to their local regions), and then they regularly received low grades, or worse yet, flunked out?  Not good.  And because the admissions process was fairly involved, would this indicate a major flaw in this process?

Yes, some students couldn’t handle the high-pressure academic environment.  But I know that in several of my classes, a student needed to work hard not to earn at least a C.  In other words, showing up, handing in homework on time, and doing a reasonable amount of studying for quizzes and exams would usually guarantee a grade of at least a C.

Now is not the time to dive more deeply into this artificial adjustment of expectations, but I did want to mention that this issue is a significant one, even at presumably elite schools for mathematics and science students.

So to continue with the essay….

“Now there is a natural give-and-take between evaluating student performance and setting expectations.  And, of course, the above remarks are nothing but generalizations.  But they illustrate some of the important issues at hand, and may bear fruitful discussion.

Moving to more concrete issues, I believe that the assignment of letter grades on exams in BC Fast-Track [recall, this was the colloquial name for the Honors Calculus sequence] was, on the whole, successful.  Without going into unnecessary detail, the classroom environment was such that the assigned grades were meaningful to the students.  To give a few examples, I assigned a grade of A+ for truly outstanding work, perhaps only a half-dozen times throughout the entire semester.  The students knew this, and so that accolade truly meant something.

Moreover, an A meant something as well.  It was truly rewarding to see the real pride of a student who, used to earning grades in the B range, began to earn the occasional, or perhaps more frequent, A.  Admittedly, students who made it to the second semester were essentially guaranteed a grade of no lower than a B-.  But this seemed to make an A that much more meaningful.

So student exams had two letter grades on them — one for the skills portion of the exam, and one for the conceptual portion of the exam.  No points were assigned, and few comments were made.  Students were expected to rework problems on which they made errors.

I bring up this point because I think this system of assigning grades really did motivate students to learn calculus rather than accumulate points.  This is the critical issue: I suggest that the way we assign grades does little to disabuse many students that taking a mathematics course is about accumulating sufficiently many — or losing sufficiently few — points.

Let’s take a particular example.  The past few semesters, I stopped assigning half-points on assessments [the usual practice at IMSA].  I might forgive a sign error now and then, but too many on a single assessment would warrant a point or two off.

In the past, I simply considered a sign error as a half-point off.  And so it was.  But consider that without being able to occasionally perform fairly involved calculations, it is not possible to become a successful mathematician.  Attention to detail is as important in mathematics as it is in any number of other disciplines, and we try to develop that skill punitively — you don’t attend to detail, and we will take off points.

Of course one might argue that points are given for work well done — but any of us could, I think, agree that when discussing the grading of an exam, it’s how many points off for a particular type of error that is discussed as much as, or even more than, how many points are given for work correctly done.

And so the idea of “partial credit” is born.  Perhaps now is not the place to begin this discussion, but consider that a student might meet expectations (that is, earn a B) without ever having done an entire complex problem on an assessment completely correctly.  (Some teachers have even gone so far as to give no partial credit.  See On Partial Credit, Letter to the Editor, MAA Focus, February 2002, p. 17.)

Why this system of points and partial credit?  One may speculate as to its origins, and there is controversy even now about its use on standardized exams.  But I cannot help feeling that one function of partial credit is that it allows a teacher to defend the assignment of a particular grade.  “Every sign error is a half-point off.  That’s why you got a B+ instead of an A-.   I have to use the exact same scale for everyone in order to be fair.

But doesn’t this simply shift the responsibility for the grade onto a rubric?  I suggest that many of us would feel competent to take a set of calculus exams — with names removed — and within five or ten minutes, separate out all the A papers.  Of course this is subjective — but no less subjective than saying that this problem is worth six points while another is worth ten, or that sign errors are a half-point off, unless, of course, the derivative of the cosine is taken incorrectly, in which case it’s a whole point.

Thus the assigning of points is no more “objective” than giving a letter grade.  As I’m sure that anyone who has graded a complex word problem based on an assignment of points can attest to.  Consider the student who has the entire procedure correct, but because of a few algebra errors, has no intermediate calculation correct.  The problem is worth ten points.  How many points should the student receive?

Well, of course, you say you’d have to see the problem first.  But I say, no.  The student receives a C.  Having no intermediate calculations correct demonstrates — regardless of what else — that the student has not met expectations.

So is it possible to avoid points altogether?  Perhaps.  Consider the following grading system:

Day166grades.png

The essay continues in the next installment of On Assessment with a discussion of how to implement this system in practice.

On Assessment, II

Last week, I had mentioned finding an essay I wrote about assessment in an Honors Calculus sequence I had designed.  I took some time to set the stage for this essay — so now it’s time to dive right in!

A caveat:  This essay was written in 2011.  I will attempt to keep as true to the original essay as I can, though I’ll edit for clarity and updated information/links, and will also [add commentary in square brackets].  So here goes!

“To educate is to illuminate the power of ideas.

This, of course, is an ideal — certainly not the last to be articulated, nor perhaps the highest. But putting practicality aside for the moment, how might we take on such a view of education?

Consider the example of planetary orbits.  Tying together history, technology, politics, physics, and mathematics through a discussion of Kepler’s laws is — yes — illuminating. Of course there is no single idea at play here, but consider this: Kepler was perhaps the first astrologer to become an astronomer.  He, in contrast to so many of his contemporaries, asked not only where a particular planet should be on some future date, but why it should be there.  In his mind, predicting the positions of celestial bodies for the purpose of casting horoscopes for royal personages was not enough — he wanted to know why. That, however, would have to wait until Newton and the application of calculus.  And the power of that idea!

There is no need to present further historical examples.  What is important, however, is to move beyond historical ideas and address the issue of the power of a student’s own ideas.  Two semesters of teaching Advanced Problem Solving [a course I taught at IMSA which emphasized the writing of original mathematics problems] have shown me that having students write original problems motivates them to learn.  They are timid at first — for how could they come up with an original problem?  But after some success, students are excited to create — and as a result, learn about a particular topic in a more profoundly personal way than they might have otherwise.  (Interestingly, the writing of original problems as a teaching tool is common in eighth-grade Japanese mathematics classrooms.  See The Teaching Gap by Stigler and Hiebert, ISBN 0-684-85274-8, pp. 6–41.)

Other assessments are more routine: students are informed that they need to be able to solve linear and quadratic equations, as well as graph linear and quadratic functions.  This is perhaps more manageable — the task is well-defined within narrow limits.  But it is rather different in nature than creating an original piece of work.

Abraham Maslow articulates a similar difference in a discussion of growth theory (see Toward a Psychology of Being by Abraham Maslow, ISBN 0-442-03805-4, p. 47).  Psychologically, why do children grow?  “We grow forward when the delights of growth and anxieties of safety are greater than the anxieties of growth and the delights of safety.”   From an educator’s point of view, this implies making the classroom a safe environment for creativity and exploration.

“The opposite of the subjective experience of delight (trusting himself), so far as the child is concerned, is the opinion of other people (love, respect, approval, admiration, reward from others, trusting others other than himself).”  (Maslow, same source, p. 51.)  Who comes to mind is the student afraid of enrolling in BC Fast-Track [the colloquial name of the Honors Calculus sequence at IMSA] because he or she might not earn an A.  [I did have a very capable student who was so concerned about grades, he opted to take the more traditional calculus sequence.  I actually met with his father, who wanted a guarantee that his son would earn an A in the course.  Given the nature of the course, there was naturally no way I could give such a guarantee.]  Or perhaps the student who shuns a difficult course and takes an easier one instead.  Grades are just that — opinions of other people.

The assignment of letter grades is not necessary for learning, but is merely practical for other reasons.  The assignment of letter grades does nothing to illuminate the power of ideas.

A quick Internet search reveals that the assignment of A–F letter grades is a fairly recent phenomenon, not making its way into high schools until the mid-twentieth century.  In the early twentieth century, grades of E (Excellent), S (Satisfactory), N (Needs improvement), and U (Unsatisfactory) were also used.

The question of whether a letter-grade system of evaluation is the best option for a school like IMSA is perhaps a worthwhile one (some argue against grades at all [this was the link, but it is now a dead end:  http://www.alfiekohn.org/teaching/fdtd-g.htm ]).  That is a question which cannot be answered in the short-term, if the question is considered relevant. So the question is this: given that letter grades need to be assigned this semester, what approach should be taken?

What do letter grades mean?

The only consensus I have heard, so far, is that earning a grade of A, B, or C means that a student exceeds, meets, or does not meet expectations.  Of course, this is reminiscent of E, S, and N.  Moreover, a grade of C is passing, so that a student receives credit for a course even if they do not meet expectations.

But this is not what the grades currently mean.  Essentially, points are assigned to hundreds of problems given throughout semester, whether on assignments, papers, quizzes, or exams — and an arbitrary weighted sum of these point assignments is converted to a letter.  (Of course letters may need to be converted to numbers so that they may be used by an online grade calculator to compute a number which is then converted back to a letter.)

What does this mean about meeting expectations?

Well, nothing really.  I would venture to suggest that “meeting expectations” currently means “enough A’s, but not too many C’s.”  Perhaps this is politically necessary, but expedient.  The performance of our students determines our expectations, rather than the other way around.”

 

On Assessment, I

While downsizing for a recent move, I decided to sort through dozens of folders (thousands of pages!) of old problems, notes, exams, essays, and other documents generated in the course of twenty-five years of teaching mathematics.  I found an essay written in 2011 which I though might be of some interest.

This essay might be considered a more thorough follow-up to The Problem with Grades, a satire on assessment practices which I’ve shared with you already.  The essential point in that essay was that how we typically assign grades is punitive:  begin with a given number of points (say, 100), punish students for making mistakes (subtract a certain number of points from 100), and then — magically! — the resulting number reflects a student’s understanding.  (If you think that this is in fact a really great way to assess a student’s learning, you probably should stop reading right here….)

The essay I’d like to share here was written in the context of teaching an Honors Calculus sequence I designed while at the Illinois Mathematics and Science Academy (IMSA).  So before diving in, let me take a few moments to set the stage.

We had quite a broad range of students at IMSA — and so when it came to teaching calculus, some students were bored by the “average” pace of the class.  Of course this happens in a typical classroom; the tendency is to teach to the middle, with the necessary consequence that some students fall behind, and others could benefit from a faster pace or more advanced problems.

Now mathematics educators are looking at ways to design classroom environments where students can work more independently, at their own pace, incorporating methodologies such as differentiated learning and/or competency-based learning.  These ideas could be the subject for wide-ranging (future) discussions; but in my case, I was working in an environment where all students did essentially the same work in the classroom, and assessments were fairly traditional.

In addition to creating a faster-paced, more conceptually oriented course, I incorporated two features into this Honors Sequence.  First, I had students write and solve their own original mathematics problems, as well as reflect on the problem-posing process.  I won’t say more about that here, since I’ve written about this type of assessment in a previous blog post.

Second, I experimented with a different way of assessing student understanding on exams.  Each exam had two components:  a Skills portion and a Conceptual portion.  The Skills portion was what you might expect — a set of problems which simply tested whether students could apply routine procedures.  The Conceptual portion was somewhat more challenging, and included nonroutine problems which were different from problems students had seen before, but which did not require any additional/specialized knowledge to solve beyond what was needed to solve the Skills problems.  (As part of the essay I will share, examples of both types of problems will be included.  I also discuss these ideas in more detail in a previous post, On Grading.)

Moreover, I graded these problems in a nonstandard say:  each problem was either Completely Correct (CC), meaning that perhaps aside from a simple arithmetic error or two, the problem was correctly solved; Essentially Correct (EC), meaning that the student had a viable approach to solving the problem, but was not able to use it to make significant progress on the problem; and Not Correct (NC), meaning that the approach taken by the student would not result in significant progress toward solving the problem.

Letter grades (a necessity where I taught, as they are in most schools) were assigned on the basis of how many problems were CC and EC on the Skills/Conceptual portions.  While I won’t go into great detail here, the salient features of this system are, in my mind:

  1.  Earning an A required a fair number of problems CC; in other words, a student couldn’t get an A just by amassing enough partial credit on problems — there needed to be some mastery;
  2.  Earning an A required as least some progress on the Conceptual problems; typically, I would include three such problems, which would be assessed more leniently than the Skills problems.  In order to earn an A, a student would need to make some progress on one or two of these.  I felt that an A student should be able to demonstrate some conceptual understanding of main ideas;
  3. A student could earn a B+ just by performing well on the Skills problems (or perhaps an A- for performing flawlessly), but an A was out of reach without some progress on the Conceptual problems;
  4. The approach was not punitive:  students were assigned CC/EC based on their progress toward the solution to a problem, not how far they fell short of a solution.

Such a system is not new; I have talked with colleagues who used a 0/1/2 system of grading, for example.  Whatever the format, the approach is more holistic.  And given that problems in mathematics typically admit more than one solution, the idea of creating a point-based rubric for all possible solution paths does, in some real sense, border on the insane.

I should also add that our instructional approach at IMSA was essentially inquiry-based.  While such a classroom environment is conducive to having students write original problems and using alternative assessment strategies, it is not strictly necessary.  I have incorporated both the elements described above in a more traditional classroom setting, but with varying degrees of success.  That discussion is necessarily for another time.

So much for a brief introduction!  I would also like to comment that teaching this Honors Sequence was perhaps my most enjoyable and successful teaching experience in the past few decades.  Students who completed the two-semester sequence left thinking about mathematics in a fundamentally different way, and I stay connected with a few of these students many years later.

Next week, I’ll introduce the essay proper, adding commentary as necessary to flesh out details of the course not addressed here, as well as making remarks which reflect my teaching this Honors sequence in the years since the essay was written.  Until then, I’ll leave you with the opening words:

“To educate is to illuminate the power of ideas…..”

The Problem with Calculus Textbooks (Reprise)

Simply put, most calculus textbooks are written in the wrong order.

Unfortunately, this includes the most popular textbooks used in colleges and universities today.

This problem has a long history, and will not be quickly solved for a variety of reasons. I think the solution lies ultimately with high quality, open source e-modules (that is, stand-alone tutorials on all calculus-related topics), but that discussion is for another time. Today, I want to address a more pressing issue: since many of us (including myself) must teach from such textbooks — now, long before the publishing revolution — how might we provide students a more engaging, productive calculus experience?

To be specific, I’ll describe some strategies I’ve used in calculus over the past several years. Once you get the idea, you’ll be able to look through your syllabus and find ways to make similar adaptations. There are so many different versions of calculus taught, there is no “one size fits all” solution. So here goes.

1. I now teach differentiation before limits. The reason is that very little intuition about limits is needed to differentiate quadratics, for example — but the idea of limits is naturally introduced in terms of slopes of secant lines. Once students have the general idea, I give them a list of the usual functions to differentiate. Now they generate the limits we need to study — completely opposite of introducing various limits out of context that “they will need later.”

Students routinely ask, “When am I ever going to use this?” At one time, I dismissed the question as irrelevant — surely students should know that the learning process is not one of immediate gratification. But when I really understood what they were asking — “How do I make sense of what you’re telling me when I have nothing to relate it to except the promise of some unknown future problem?” — I started to rethink how I presented concepts in calculus.

I also didn’t want to write my own calculus textbook from scratch — so I looked for ways to use the resources I already had. Simply doing the introductory section on differentiation before the chapter on limits takes no additional time in the classroom, and not much preparation on the part of the teacher. This point is crucial for the typical teacher — time is precious. What I’m advocating is just a reshuffling of the topics we (have to) teach anyway.

2. I no longer teach the chapter on techniques of integration as a “chapter.” In the typical textbook, nothing in this chapter is sufficiently motivated. So here’s what I do.

I teach the section on integration by parts when I discuss volumes. Finding volumes using cylindrical shells naturally gives rise to using integration by parts, so why wait? Incidentally, I also bring center of mass and Pappus’ theorem into play, as they also fit naturally here. The one-variable formulation of the center of mass gives rise to squares of functions, so I introduce integrating powers of trigonometric functions here. (Though I omit topics such as using integration by parts to integrate unfriendly powers of tangent and secant — I do not feel this is necessary given any mathematician I know would jump to Mathematica or similar software to evaluate such integrals.)

I teach trigonometric substitution (hyperbolic as well — that for another blog post) when I cover arc length and surface area — again, since integrals involving square roots arise naturally here.

Partial fractions can either be introduced when covering telescoping series, or when solving the logistic equation. (A colleague recommended doing series in the middle of the course rather then the end (where it would have naturally have fallen given the order of chapters in our text), since she found that students’ minds were fresher then — so I introduced partial fractions when doing telescoping series. I found this rearrangement to be a good suggestion, by the way. Thanks, Cornelia!)

3. I no longer begin Taylor series by introducing sequences and series in the conventional way. First, I motivate the idea by considering limits like

\displaystyle\lim_{x\to0}\dfrac{\sin x-x}{x^3}=-\dfrac16.

This essentially means that near 0, we can approximate \sin(x) by the cubic polynomial

\sin(x)\approx x-\dfrac{x^3}6.

In other words, the limits we often encounter while studying L’Hopital’s rule provide a good motivation for polynomial approximations. Once the idea is introduced, higher-order — eventually “infinite-order” — approximations can be brought in. Some algorithms approximate transcendental functions with polynomials — this provides food for thought as well. Natural questions arise: How far do we need to go to get a given desired accuracy? Will the process always work?

I won’t say more about this approach here, since I’ve written up a complete set of Taylor series notes. They were written for an Honors-level class, so some sections won’t be appropriate for a typical calculus course. They were also intended for use in an inquiry-based learning environment, and so are not in the usual “text, examples, exercise” order. But I hope they at least convey an approach to the subject, which I have adapted to a more traditional university setting as well. For the interested instructor, I also have compiled a complete Solutions Manual.

I think this is enough to give you the idea of my approach to using a traditional textbook. Every calculus teacher has their own way of thinking about the subject — as it should be. There is no reason to think that every teacher should teach calculus in the same way — but there is every reason to think that calculus teachers should be contemplating how to make this beautiful subject more accessible to their students.

Calculus VIII: Miscellaneous Problems, I

In this post, I’ll continue discussing problems I’ve been encountering in the calculus textbook I’m reading.  Some problems are involved enough to require an entire post devoted to them; others are interesting but relatively short.  Today, I’ll discuss four shorter problems.

The first problem is Exercise 26 on page 33:

If a cylindrical hole be drilled through a solid sphere, the axis of the cylinder passing through the center of the sphere, show that the volume of the portion of the sphere left is equal to the volume of a sphere whose diameter is the length of the hole.

This is not a difficult problem to solve; I’ll leave the simple integral to the reader.  This has always been a favorite volume problem of mine, and this is the earliest reference I’ve seen to it.

Perhaps it was a classic even back then — remember, the book was published in 1954.  The author usually attributes problems he uses to their sources, but this problem has no attribution.  I would be interested to know if anyone knows of an earlier reference to this problem.

The second problem is not from an exercise, but is discussed in Art. 37 on page 48.  It’s one of those “of course!” moments, leaving you to wonder why you never thought to try it yourself….

Why is the antiderivative of y=x^{-1} the natural logarithm?  There are a few different ways this is usually shown, but here’s one I haven’t seen before:  consider the limit

\displaystyle\lim_{n\to-1}\int_a^bx^n\,dx,\quad 0<a<b.

It seems so obvious when you see it written down, but I’ve never thought to take this limit before.  You get

\displaystyle\lim_{n\to-1}\dfrac{b^{n+1}-a^{n+1}}{n+1}.

Now apply L’Hopital’s rule!  And there you have it:

\displaystyle\lim_{n\to-1}\int_a^bx^n\,dx=\log b-\log a.

I think that perhaps when writing x^n, I’m so conditioned to thinking of n as a constant that I never thought of turning it into the variable.  It’s a nice proof.

Next is Art. 68, which begins on page 82.  Again, you’ll agree that it seems pretty obvious after the discussion, but I’ve never seen this diagram drawn before.  This is likely because hyperbolic trigonometry is downplayed in today’s calculus curriculum.  You might recall the comment I made about a colleague once saying they didn’t teach hyperbolic trigonometry since it wasn’t on the AP exam.

Day151Hyperbola.png

So let’s look at the hyperbola x^2-y^2=a^2.  The goal of this exercise is to find a geometrical interpretation of the relationship

\sec\theta=\cosh u,

which is key to connecting circular and hyperbolic trigonometry by means of the gudermannian, as I have discussed earlier.

Draw the auxiliary circle x^2+y^2=a^2, and consider the point

P=(a\cosh u,a\sinh u).

Now drop a perpendicular from P on the x-axis to the point N=(a\cosh u,0).  Next, draw a tangent from N to the auxiliary circle, meeting it at T.  Finally, join T to the origin.

Since NT is tangent to the circle, we know that \Delta NTO is a right triangle.  Therefore ON=a\sec\theta.  But by construction, ON=a\cosh u, and so

\sec\theta=\cosh u.

Yep, that’s all there is to it!  A geometrical illustration of the gudermannian function.  So very simple.  And incidentally, the author goes on to discuss the gudermannian function in the next section.

For the last example, I’ll need to skip ahead a little bit, since my next exploration is a bit too involved and may need an entire post.  As a teaser, I’ll just say that I learned a completely new way to derive Cardan’s formula for solving a cubic equation!  It involves calculus and quite a bit of algebra.  At some point, I’d like to dive in a little deeper and see if I can relate this new proof with the usual one — but again, that for another time.

So this last example (Art. 96 on page 109) is about differentiating

y=e^{ax}\sin(bx).

Of course this is just a simple application of the product rule:

\dfrac{dy}{dx}=e^{ax}(a\sin(bx)+b\cos(bx)).

But why stop here?  We can go further, using an idea very common when working with physics applications.  We seek to write

a\sin(bx)+b\cos(bx)=c\sin(bx+\theta).

Since

c\sin(bx+\theta)=c\sin(bx)\cos(\theta)+c\cos(bx)\sin(\theta),

this amounts to solving

c\cos(\theta)=a,\quad c\sin(\theta)=b.

This is straightforward:

c=\sqrt{a^2+b^2},\quad\theta=\arctan(b/a).

Thus,

\dfrac{dy}{dx}=\sqrt{a^2+b^2}\,e^{ax}\sin(bx+\arctan(b/a)).

This means that taking the derivative of e^{ax}\sin(bx) amounts to multiplying the function by \sqrt{a^2+b^2} and increasing the angle in the sine function by \arctan(b/a).  Therefore

\dfrac{d^n}{dx^n}e^{ax}\sin(bx)=(a^2+b^2)^{n/2}e^{ax}\sin(bx+n\arctan(b/a)).

I actually did the proof by induction to verify this.  It’s pretty cumbersome.

Note that this also implies that

\displaystyle\int e^{ax}\sin(bx)\,dx=\dfrac{e^{ax}\sin(bx-\arctan(b/a))}{\sqrt{a^2+b^2}}.

The same results hold with sine being replaced by cosine.  Such elegant results.

I hope you found these problems as interesting as I did!  There are so many calculus gems in this book.  I’ll continue to keep sharing….

Calculus VII: Approximations

Although I’ll have a very busy summer with consulting, I’ve taken some time to start reading more again.  You know, those books which have been sitting on your shelves for years….

So I’ve started Volume I of A Treatise on the Integral Calculus by Joseph Edwards.

Day150Cover

I include a picture of the cover page, since you can google it and download a copy online.  Between Volumes I and II, there’s about 1800 pages of integral calculus….

Since I’ll likely be working with a calculus curriculum later this year, I thought I’d look at some older books and see what calculus was like back in the day.  I’m continually surprised at how much there is to learn about elementary calculus, despite having taught it for over 25 years.

My approach will be a simple one — I’ll organize my posts by page number.  As I read through the books and solve interesting problems, I’ll share with you things I find novel and interesting.  The more I read books like these and think about calculus, the more I think most current textbooks simply are not up to the task of presenting calculus in any meaningful way.  Sigh.

This is not the time to be on my soapbox — this is the time for some fun!  So here is the first topic:  Weddle’s Rule, found on page 21.

Ever hear of it?  Bonus points if you have — but I never did.  It’s another approximation rule for integrals.  Here it is: given a function f on the interval [a,b], divide the interval into six equal subintervals with points x_0, x_1,\ldots x_6 and corresponding function values y_0=f(x_0),\ldots,y_6=f(x_6).  Then

\displaystyle\int_a^bf(x)\,dx\approx \dfrac{b-a}{20}\left(y_1+5y_2+y_3+6y_4+y_5+5y_6+y_7\right).

Yikes!  Where did that come from?  I’ll present my take on the idea, and offer a theory.  If there are any historians of mathematics out there, I’d be happy to hear if my theory is correct.

One reason most of us haven’t heard of Weddle’s Rule is that approximations aren’t as important as they were before calculators and computers.  So many exercises in this book involve approximation techniques.

So how would you come up with Weddle’s Rule?  I’ll share my (likely mythical) scenario with you.  It’s based on some notes I wrote up a while ago on Taylor series.  So before diving into Weddle’s Rule, I’ll show you how I’d derive Simpson’s Rule — the technique is the same, but the algebra is easier.  And by the way, if anyone has seen this technique before, please let me know!  I’m sure it must have been done before, but I’ve never been able to find a source illustrating it.

Let’s assume we want to approximate

F(x)=\displaystyle\int_a^xf(t)\,dt

by using three equally-spaced points on the interval [a,x].  In other words, we want to find weights p, q, and r such that

S(x)=\left(p f(a)+ q f\left(\dfrac{a+x}2\right)+rf(x)\right)(x-a)\approx F(x).

How might we approach this?  We can create Taylor series for F(x) and S(x) about the point a.  The first is easy using the Fundamental Theorem of Calculus, assuming sufficient differentiability:

F(x)=f(a)(x-a)+\dfrac{f'(a)}{2!}(x-a)^2+\dfrac{f''(a)}{3!}(x-a)^3+\cdots

Now to construct the Taylor series of S(x) about x=a, we need to evaluate several derivatives at a. This is not difficult to do by hand, but it is easy to do using Mathematica and a command such as

Day150Mma

Doing so yields the following:Day150derivs2

Now the problem becomes a simpler algebra problem — to force as many of the coefficients of the derivatives on the right-hand side to be 1 as possible.  This will make the derivatives of F and S match, and the Taylor polynomials will be equal up to some order.

Solving the first three such equations,

Day150eqns

yields, as we expect, p=1/6, q=2/3, and r=1/6. Note that these values also imply that

\dfrac12q+4r=1,

but

\dfrac5{16}q+5r=\dfrac{25}{24}.

This implies that

S(x)-F(x)=\dfrac1{24}\cdot\dfrac{(x-a)^5}{5!}+O((x-a)^6)

on each subinterval, so that

S(x)-F(x)=O((x-a)^5)

on each subinterval, giving that Simpson’s rule is O((x-a)^4).

So how we apply these to derive Weddle’s rule?  We could try to find weights w_1,\ldots w_7 to create an approximation

W(x)=\left(w_1 f(a)+w_2f\left(\dfrac{5a+x}6\right)+\cdots+w_7f(x)\right)(x-a).

If we apply precisely the same procedure as we did with Simpson’s Rule, we get the following as the sequence of weights to create the best approximation:

\dfrac{41}{840},\ \dfrac9{35},\ \dfrac9{280},\ \dfrac{34}{105},\ \dfrac9{280},\ \dfrac9{35},\ \dfrac{41}{480}.

Not exactly easy to work with — remember, no calculators or computers.

So let’s make the approximation a little worse.  Recall how the weights were found — a system of seven equations in seven unknowns was solved, analogous to the three equations in three unknowns for Simpson’s rule.  Instead, we specify w_1, and solve the first six equations in terms of w_1.  This gives us

Day151Weddle.png

Now all weights must be positive; this gives the constraint

0.046\overline6\approx\dfrac7{150}<w_1<\dfrac{13}{200}=0.065.

Let’s put w_1=1/20, which is in the interval just described.  This gives the sequence of weights to be

\dfrac1{20},\ \dfrac5{20},\ \dfrac1{20},\ \dfrac6{20},\ \dfrac1{20},\ \dfrac5{20},\ \dfrac1{20},

where all fractions are written with the same denominator.  Now imagine factoring out the 1/2, and you notice that all divisions are by 10.  Can you see the advantage?  If you have a table of values for your functions, you just need to multiply function values by a single-digit number, and then move the decimal place over one.  An approximators dream!

So Weddle’s approximation is exact for fifth-degree polynomials, even though it is possible to use six subintervals to get weights which are exact for sixth-degree polynomials.  Yes, we lose an order of accuracy — but now our computations are much easier to carry out.

Was this Weddle’s thinking?  I can’t be sure; I wasn’t able to locate the original article online.  But it is a way for me to make sense out of Weddle’s rule.

I will admit that in a traditional calculus class, I don’t address approximations in this way.  There is a time crunch to get “everything” done — that is, everything the student is expected to know for the next course in the calculus sequence.

Should these concepts be taught?  I’ll make a brief observation:  in reading through the first 200 pages of this calculus book, it seems that all that has changed since 1954 is that content was pared down significantly, and more calculator exercises were added.

This is not the solution.  We need to rethink what students need to now know and how that material should be taught in light of emerging technology.  So let’s get started!