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“What makes calculus so hard?” and Other Questions with Shandelle Henson

“What makes calculus so hard?” and Other Questions with Shandelle Henson

Shandelle Henson, PhD
Professor of Mathematics and Ecology
Interviewed by Scott Moncrieff

 

Sometimes we say, metaphorically speaking, “that’s as hard as doing calculus,” to describe something very challenging. What makes calculus seem so hard?
Conceptually, calculus is not that difficult. But to be able to work calculus problems you have to be really good at algebra. That’s what makes people struggle.

What classes do you need to be ready for calculus?
In high school you need Algebra I and II. If you did that well, then in college you can start with college algebra, or possibly pre-calculus, but pre-calculus is what you have to have prior to calculus. A few students take math for four years in high school and can get calculus-ready.

And at Andrews you have Calculus and Advanced Calculus?
There’s Calculus I, II, and III, and then Differential Equations. We do have a course called Advanced Calculus, aka Real Analysis—that is a senior course for math majors, and it’s extremely abstract stuff. It’s the hardest math class a student can take at Andrews. We might have 10 students finish that in a year.

When did you decide you loved math enough to make it a profession?
I started out as an engineering and physics major—so I had to have math. I took pre-calculus and Calculus I from a man named Art Richert, at Southern (Adventist University), and he was the most fantastic teacher that I’ve ever met. I thought the way he thought, and I understood everything he said, and I was so shocked by that. He proved everything that he wrote on the board, and it was the first time I’d ever been exposed to a mathematical proof. I realized this is the one discipline where you can prove things with 100 percent certainty. That got me really interested from a philosophical perspective, so interested I ended up doing a Master’s in mathematical logic.

Do you have an artsy/musical side, or are you just drawn to the really logical side of things?
I’m very much a Humanities person, even though by profession I’m a mathematician. I like literature, art and music very much. I like to do oil painting. I’m not much of a musician—I play a little piano—but I really wish I could play the violin or the cello. I love serious music, especially J.S. Bach. The music is very intellectual, mathematical. I particularly like fugues.

Did you read the book Gödel, Escher, Bach?
Yes, it’s right here on my bookshelf. There’s something very structured about Bach’s music that can appeal to a mathematician. Math is really the study of patterns, deep structure, so you have that resonance between the two fields. I also like to write. I wanted to be an English major, but math pulled me in a different direction. You will find that mathematicians are usually really plugged into the humanities. I’m an applied mathematician, but pure mathematicians consider themselves part of the humanities almost more than science, because they do mathematics for aesthetic reasons, as opposed to solving scientific problems. We like being here next to physics and chemistry and all, but we’d also like to be over in Nethery Hall. By the way, if you think about the seven traditional classical liberal arts, three of them involve mathematics (logic, arithmetic, geometry). Back then, mathematics was considered essential to general education because of the aesthetic value, because it helped you understand nature, and because it helped you think clearly.

What does calculus do that can’t be done by “regular” math?
Everything in the universe changes over time, so to describe physical and biological systems you have to be able to describe things that change over time, aka “dynamical systems.” The language of that study is calculus, because calculus is the study of rates of change. When you write down an equation to describe how something changes over time, you have to use calculus. That’s why calculus is the language of science.

What’s an example of some exciting event or invention for which calculus played a crucial role?
How about the first moon landing in 1969? The ability to set people on the moon and bring them back is all based on calculus. More recently they set a probe down on a comet.

Maybe a dumb question, but do you ever use calculus around the house, outside of your research and professional contexts?
No. It doesn’t help me mow my lawn. But when we do science…

You recently were appointed editor of the journal Natural Resource Modeling.What is that journal about?
We publish research papers that use mathematics to illuminate how to manage natural resources, such as soil, air, water and wildlife.

Could you give me an example of a paper that would be published in the journal?
Suppose fish and wildlife managers are interested in how many Harbor Seals are in Puget Sound, and where they’re hauling out—coming out on the beach. This is of interest to managers because they don’t want people disturbing the seals. So a researcher might construct a mathematical model, a group of related equations, that describes where those animals are at what time, and how many there are. And the model would be tested against field data to make sure it’s a good model. And then the wildlife manager could use that information. The good models will not only describe, but will predict.

If I see that there are eight seals on the beach today, 12 tomorrow, and 10 the day after that, I can predict that there might be around 10 there next week. How is what you’re doing different from this kind of averaging?
Averaging is fine, as long as the conditions remain constant, but in the Puget Sound you have tides rising and falling, the sun rising and setting at different times, temperature changes, cloud cover changes, all kinds of environmental conditions that are in constant flux. Some you can predict—like tides and solar elevation—and some you can’t, like wind speed. The mathematical model allows you to put together all the predictable variables in a series of equations that can be much more accurate than the kind of averaging you mentioned.

What are your main duties as editor of this journal?
When people submit manuscripts they come to me, and I have an editorial board with experts in different areas of natural resource modeling. So, after checking the threshold suitability of a new manuscript for our journal—which eliminates about half the papers—I assign it to one of those editors, and he or she will send it out for review to two or three experts or “referees” in that area. The referees read the manuscript anonymously—without knowing who wrote it—and return a recommendation to that editor, who collects those reports and returns a recommendation to me, and I make the final decision to accept the manuscript, reject it or ask for a major revision.

How many papers will you be processing in a year?
We get in about a manuscript a day.

What do you like about working at Andrews University?
I was recruited to come here by Don Rhoads (former chair of the mathematics department), in order to build undergraduate research into the curriculum, and to help the department raise its research profile. And with all of us working together in the department, I think we’ve done that. I really like Andrews. I’ve never regretted coming here because my research is appreciated and the student body is wonderful. It’s a diverse student body and I like that. I don’t ever want to go back to a school where everybody looks the same. It’s a small university, but the faculty is doing very interesting research.

You’ve got some down time, and you’re not oil painting or listening to a Bach fugue. Any other things you especially like to do?
I like backpacking with Jim (husband and Research Professor Emeritus of Biology James Hayward), running. I run every day. Back when I was in grad school, my Native American office mate at Duke University got me into running. I think she was the first Native American PhD in math. Anyway, she got me into running, and I’ve run ever since for stress relief and health. Oh, and Jim and I read some big Russian novels recently, like The Brothers Karamazov. Now I’m on to Paradise Lost.

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