Mandatory Office Hours

Here’s a quick tip for the new school year: make every student in your class visit your office hours within the first week. Make this a grade. I have a quiz every Friday in my classes, and to get full credit on the first quiz, a student needs to have visited my office by then.

Why? The reasons are two-fold. First, the students benefit because now they know where my office is, and when they can find me there, and how to make an appointment with me if my office hours don’t fit their schedule. And they’ve learned that coming to office hours isn’t so scary, so maybe they’re more likely to come back later. Second, I benefit because now I’ve met and talked to all my students one-on-one, and I’ve had an extra opportunity to learn their names early on (I struggle with names, but I make a conscious effort to know them all by the end of the second week). Everyone wins, and it’s a very low-effort assignment.

Try it! Let me know how it works for your class.

Calculus project – real-world data

I recently assigned and graded a take-home project for my Calc I class. Here is the assignment I gave them:

This is a group assignment. You must find a partner in the class to complete this assignment with. If you cannot find a partner, let me know and I will help you find one.
1. Find a real data-set that represents some quantity measured over time. This can be population data, the position of some object/particle, height, weight, temperature, etc. Look at the exercises from section 2.1 for examples of the type of data you’ll need. You should provide the original data and a citation for where you found it.
2. Sketch a plot of the data over time. This can be done by hand or on a computer, but it should be neat and legible.
3. Using the data, pick one of the times. Then find the average rate of change for six different time periods, where the first time you pick is one of the end points for each time period.
For example, if your data uses the times 1900, 1910, 1920, 1930, 1940, 1950, and 1960, you might choose 1920 as your fixed time. Then you’d find the average rate of change for 1900-1920, 1910-1920, 1920-1930, 1920-1940, 1920-1950, and 1920-1960. Show your calculations for at least one of these, and create a table displaying all of the results.
4. Estimate the instantaneous rate of change at your chosen time, and justify where this number came from. You should write a few sentences here explaining what the instantaneous rate of change represents and how you tried to find it.
5. Discuss briefly the difficulties in using discrete data to estimate instantaneous rates of change, and suggest in what ways the data collection could be changed to make your job easier.

If you have any questions about this assignment, please ask me, sooner rather than later.

I think my instructions for part 3 could have been more clear, but the class seemed to understand, and I was happy with most of their submissions. Several groups happened to pick “population of Atlanta” as their data set, but some were more creative, and this helped me get to know their personalities better. For example, one group did “number of YouTube followers PewDiePie has per month, so now I know these two students are gamers.

I was also happy to see different methods for answering part 4. Some students took the left and right estimates and averaged them (e.g. averaging the 1910-1920 and 1920-1930 rates), while others found a middle rate (e.g. calculating the 1910-1930 rate). People who chose an endpoint for their fixed time struggled here, so in the future I might recommend they always choose a middle time to fix, but it was nice seeing how these students justified their answers.

The answers to part 5 were of varied quality, but I generally saw the answer I wanted: having more frequent data points would help get a more accurate instantaneous rate of change. I think that it’s important for students to interpret their answers in words instead of just providing a number, which might be meaningless to them.

I liked this project for several reasons, not least of which being that now every one of my students knows someone in class, has their contact info, and knows where/when is a convenient time to meet with them. Studies show that one of the best indicators of student success in math classes is studying with peers. As I assign more projects throughout the semester, I’ll probably require students to have new partners each time, so they can continue to meet more classmates and hopefully find one they work best with, with whom they can study in the future.

A quick note on board-work when teaching

A simple hint to improve your teaching: when you’re writing on a board and you run out of space, you should take time to slowly and completely erase part of the board. It is natural to try to erase the board quickly so you can get back to writing notes, but this is a mistake in my opinion, for two reasons.

First, students need a moment to process your lesson as you teach. When you take twenty seconds to erase the board (rather than ten seconds), you give them a better opportunity to catch up with what you’ve been saying. They can finish copying down the last thing you’ve written, or can think about whether the lesson makes sense so far. Students appreciate this short break just to catch their breath during a lesson.

Second, quick erasing is often sloppy erasing. Do yourself a favor and clear the next section of the board completely of all stray marks. Now you’ll have a clean slate (literally) on which to write. Your students will have an easier time reading what you’ve written if there are no straggling serifs sprinkled throughout nor ghosts of what was previously written haunting the background.

Also, taking a bit more time to clearly erase the board gives you a second to breathe as well. You can gather your thoughts and prepare for the next section of notes you want to write, so you can give the best lesson possible.

Emphasizing intuition over algebra

I just saw this great YouTube video giving an intuitive explanation of why e^(i*pi)=-1. The creator doesn’t use any trigonometry or calculus, or really any algebra. Just basic arithmetic and lots of great visuals. Give it a watch here:

I often find myself relying on rigorous notation and calculation when trying to explain an idea–both in the classroom and when writing papers. Of course, being correct is a priority. But being convincing is nearly as important. If a student or reader walks away thinking, “I guess I buy it. I mean, all the algebra checked out,” then I’ve given them very little basis for growth.

I want to start including more intuitive explanations in my teaching. I can still give the formal proof of a concept, but let me first try to justify the outcome naturally.

Why does sin(x)/x approach 1 as x approaches 0? Not because the squeeze theorem says it does, using two cleverly chosen other functions. But because sin(x) has values very close to x near the origin.

Why does the formula for Simpson’s Rule work? Again, not because we can do a bunch of arithmetic and change the indices on a summation. It’s because it’s a weighted average of two other approximation rules, one that over-estimates and one that under-estimates.

I can also focus on giving intuition when writing my research. Before I define some new variable that seems to come from nowhere, and before I start some algorithm on a graph that decomposes it for some reason, let me invest a couple sentences saying “Here is the goal. Here’s how we’ll reach it.” Now the reader has some idea where we’re headed and may suddenly recognize the route we’re taking to get there, instead of spending the entire proof scratching their head. It’s not enough that they agree that all the variables add up the way I claim; the reader should walk away feeling like they’ve learned a method they can reuse.