# Tag Archives: Calculus

February 20, 2012, 7:32 pm

# The origin of the “nabla” symbol

We’re about to start working with gradient vectors in Calculus 3, and this topic uses a curious mathematical symbol: the nabla, which looks like: $$\nabla$$. This symbol has several mathematical uses, one of which is for gradients; if $$f$$ is a function of two or more variables then $$\nabla f$$ is its gradient. But there does not appear to be a use for the symbol outside mathematics (and mathematical physics).

One of my students asked me about the origin of this symbol, and I had to confess I didn’t know. I always figured it was somehow related to the much more common capital Greek delta, $$\Delta$$, but the real story is a lot more colorful than that.

The nabla is so-called because it looks like a harp; the Greek word for the Hebrew or Egyptian form of a harp is “nabla” . What does a harp have to do with mathematics? The image came up in relation to mathematics…

October 25, 2011, 7:30 am

# Better examples through peer instruction

I just gave midterm evaluations in my classes, and for the item about “What could we be doing differently to make the class better?”, many students put down: Do more examples at the board. I think I’ve seen that request more often than any other in my classes at midterm. This is a legitimate request (it’s not like they’re asking for free points or an extra day in the weekend), but honestly, I’m hesitant to give in to it. Why? Two reasons.

First, doing more examples at the board means more lecturing, therefore less active learning, and therefore more passivity and dependence by students on authority. That’s bad. Second, we can’t add more time to the meetings, so doing more examples means either going through them in less detail or else using examples that are overly simple. In the first case, we have less time for questions and deep thought, and therefore more passivity and dependence….

September 14, 2011, 8:00 am

# Midweek recap, 9.14.2011

Happy Hump Day! Here are some items of interest from the past week:

September 13, 2011, 7:30 am

# Taking the Fundamental Theorem challenge

To all the new readers: Ready for some math? We love math here at Casting Out Nines, and I’ll be taking at least one day a week to talk about a math topic specifically. If you have a math post you’d like to see, email me (robert [dot] talbert [at] gmail [dot] com) or leave a comment.

The Fundamental Theorem of Calculus is central to an understanding of how differential and integral calculus connect. It says that if f is a continuous function on a closed interval [a,b] and x is in the interval, then the function

is an antiderivative for f. That is, F’(x) = f(x). The FTC (technically, this is just one part of that theorem) shows you how to construct antiderivatives for any continuous function. Possibly more importantly, it connects two concepts about change — the rate of change and the amount of accumulated change in a function. It’s a big deal.

I use a lot of technology in my…

March 29, 2011, 4:25 pm

# Five questions I haven't been able to answer yet about the inverted classroom

Between the Salman Khan TED talk I posted yesterday and several talks I saw at the ICTCM a couple of weeks ago, it seems like the inverted classroom idea is picking up some steam. I’m eager myself to do more with it. But I have to admit there are at least five questions that I have about this method, the answers to which I haven’t figured out yet.

1. How do you get students on board with this idea who are convinced that if the teacher isn’t lecturing, the teacher isn’t teaching? For that matter, how do you get ANYBODY on board who are similarly convinced?

Because not all students are convinced the inverted classroom approach is a good idea or that it even makes sense. Like I said before, the single biggest point of resistance to the inverted classroom in my experience is that vocal group of students who think that no lecture = no teaching. You have to convince that group that what’s…

December 16, 2010, 2:30 pm

# A problem with "problems"

I have a bone to pick with problems like the following, which is taken from a major university-level calculus textbook. Read it, and see if you can figure out what I mean.

This is located in the latter one-fourth of a review set for the chapter on integration. Its position in the set suggests it is less routine, less rote than one of the early problems. But what’s wrong with this problem is that it’s not a problem at all. It’s an exercise. The difference between the two is enormous. To risk oversimplifying, in an exercise, the person doing the exercise knows exactly what to do at the very beginning to obtain the information being requested. In a problem, the person doesn’t. What makes an exercise an exercise is its familiarity and congruity with prior exercises. What makes a problem a problem is the lack of these things.

The above is not a problem, it is an exercise. Use the

November 29, 2010, 9:00 am

# What correlates with problem solving skill?

About a year ago, I started partitioning up my Calculus tests into three sections: Concepts, Mechanics, and Problem Solving. The point values for each are 25, 25, and 50 respectively. The Concepts items are intended to be ones where no calculations are to be performed; instead students answer questions, interpret meanings of results, and draw conclusions based only on graphs, tables, or verbal descriptions. The Mechanics items are just straight-up calculations with no context, like “take the derivative of $$y = \sqrt{x^2 + 1}$$”. The Problem-Solving items are a mix of conceptual and mechanical tasks and can be either instances of things the students have seen before (e.g. optimzation or related rates problems) or some novel situation that is related to, but not identical to, the things they’ve done on homework and so on.

I did this to stress to students that the main goal of taking …

November 15, 2010, 9:17 pm

# Technology FAIL day

This morning as I was driving in to work, I got to thinking: Could I teach my courses without all the technology I use? As in, just me, my students, and a chalk/whiteboard with chalk/markers? As I pulled in to the college, I thought: Sure I could. It just wouldn’t be as good or fun without the tech.

Little did I know, today would be centered around living that theory out:

• I planned a Keynote presentation with clicker questions to teach the section on antiderivatives in Calculus. As soon as I tried to get the clickers going, I realized the little USB receiver wasn’t working. Turns out, updating Mac OS X to v10.6.5 breaks the software that runs the receiver. Clicker questions for this morning: Out the window. Hopefully I’ll find a useable laptop for tomorrow, when I’m using even more clicker questions.
• Also in calculus, the laptop inexplicably went into presenter mode when I tried to…

September 5, 2010, 1:37 pm

# This week (and last) in screencasting: Functions!

So we started  back to classes this past week, and getting ready has demanded much of my time and blogging capabilities. But I did get some new screencasts done. I finished the series of screencasts I was making for our calculus students to prepare for Mastery Exams, a series of short untimed quizzes over precalculus material that students have to pass with a 100% score. But then I turned around and did some more for my two sections of calculus on functions. There were three of them. The first one covers what a function is, and how we can work with them as formulas:

The second one continues with functions as graphs, tables, and verbal descriptions:

And this third one is all on domain and range:

The reason I made these was because we were doing the first section of the Stewart calculus book in one day of class. If you know this book, you realize this is impossible be…

August 21, 2010, 6:50 pm

# This week in screencasting: Contour plots in MATLAB

By my count, this past week I produced and posted 22 different screencasts to YouTube! Almost all of those are short instructional videos for our calculus students taking Mastery Exams on precalculus material. But I did make two more MATLAB-oriented screencasts, like last week. These focus on creating contour plots in MATLAB.

Here’s Part 1:

And Part 2:

I found this topic really interesting and fun to screencast about. Contour plots are so useful and simple to understand — anybody who’s ever hiked or camped has probably used one, in the form of a topographical map — and it was fun to explore the eight (!) different commands that MATLAB has for producing them, each command producing a map that fits a different kind of need. There may be even more commands for contour maps that I’m missing.

I probably won’t match this week’s output next week, as I’ll be on the road in

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