# Tag Archives: Calculus

June 5, 2012, 8:00 am

# Taking time and giving time on assessments

I haven’t given many updates lately about, well, anything, but especially about my Calculus 2 class. Freakishly, we are 2/3 of the way through the course now. First of all let me say that there’s something seriously wrong with having a midterm in a class after three weeks, and then a final exam three weeks later. Students should have more time to dread those things.

I kid, but actually the biggest adjustment I’ve made in the class — and teaching a class that’s as compressed as this one is all about paying close attention to everything that happens and being nimble about making adjustments — has been the testing scheme. I know that I posted earlier about my idea of having in-class assessments that were smaller than the usual test, more frequent, and which leveraged student collaboration and the real-life social network of the class. But after a couple of tries with this, I dropped it…

May 14, 2012, 7:30 am

# Turn to your neighbor… and take a test

Week 1 of the 6-week Calculus 2 course is over, and of course it felt like 2.5 weeks of class because that’s the exchange rate between this course and a normal 14-week course. It was challenging for the students, but I feel like they are on board with what we’re doing. I was especially pleased with the outcome of one of the distinctives of this class: the in-class assessments which are called, er, Assessments.

I said at the outset that the key thing with this class was to force the issue on assimilation of material, and part of that was to engage in early, small, and frequent assessment. For formative assessment, we do daily online homework and clicker questions. There’s no requirement to get clicker questions right at all, and WeBWorK sets have no limits on number of attempts or the amount of collaboration or technology used. For summative assessment, we have a midterm exam and a…

May 8, 2012, 12:52 pm

# A screenshot that illustrates what peer instruction can do

I blog a lot about peer instruction, but I think this screenshot from this morning’s Calculus 2 class is worth 1000 of my blog posts about just how effective a teaching technique PI can be. It’s from a question about average value of a function. Just before this question was a short lecture about average value in which I derived the formula and did an example with a graph of data (not as geometrically regular as the one you see below). I used Learning Catalytics to set up the question as Numerical, which means that student see the text and the picture on their devices along with a text box in which to enter what they think is the right answer. (I.e. it’s not multiple choice.) Here are the results of two rounds of voting (click to enlarge):

After the first round of voting, there were 12 different numerical answers for 23 students!   (Some of these would be the same answer if students …

May 7, 2012, 8:58 pm

# How the technology works in Calculus 2

Today we started the spring term, 6-week Calculus 2 class that I’ve been writing about for the last few days. We had a good time today, getting comfortable with each other and doing some review of the basics of the definite integral. Before we get too far into the term, I wanted to outline the technology infrastructure of the course.

For a long time, I’d used the learning management system (LMS) of my institution as the basic technology for the course, and everything else kind of fit around the LMS. At GVSU the default LMS is Blackboard. But I decided after used Blackboard this past year that we have irreconcilable differences. I don’t ask much from my LMS; I mainly use it to archive files, provide a link to a central calendar, post grades, and to make announcements. I don’t need all the dozens of other features Blackboard offers, and the profusion of features in Blackboard tends to…

May 4, 2012, 4:00 pm

# Peer instruction and Calculus 2

Sorry for the boring title and lack of catchy image, but since my first post about the upcoming six-week Calculus 2 course, I’ve expended all my creativity getting the course put together and getting ready for Monday. In the earlier post I laid down some design ground rules for the course. Here, I’m going to say a little more in detail about what we’ll be doing.

It’s especially important on a highly compressed schedule like ours to use the class meetings themselves to jumpstart the assimilation process and then train students on how to carry that process forward as they go to work on the day’s material in the afternoon and evening. This is always an important goal of class meetings in any course — I’d go as far as to say that this is why we have class meetings at all. But when you cram a 14-week course into 6 weeks, it doesn’t take long for one incorrectly-assimilated concept to…

May 1, 2012, 9:18 am

# Building a six-week Calculus 2 course

I took a two-week blogging hiatus while final exams week, and the week before, played themselves out. Now that those fun two weeks are over, it’s time to start focusing on what’s next. Some of those things you’ll read about here on the blog, starting with the most immediate item: my spring Calculus 2 class that starts on Monday.

Terminology note: At GVSU and other Michigan schools, the semester that runs from January through April is called “Winter” semester. The period in between Winter and Fall is split into two six-week terms, the first being “Spring” (May-mid June) and the second “Summer” (mid June-July). It’s quite accurate to the climate here.

Anyway, my Calculus 2 class runs in that 6-week Spring term. If you know anything about Calculus 2, and you have a sense of just how long, or short, a 6-week period is, the first thing you’ll realize is that this is a lot of content…

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…

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