Mathematical Mondays

Mistakes vs. Incompetence

As some of you know, I’m entering a transition in the day job. This involves a lot of interviews, some where I’m the interviewee and some where I’m on the panel of interviewers. It makes for an interesting dual perspective.

My current school includes something extra in the interview process—candidates have to teach a brief mock-lesson. For me, that’s the make-or-break portion of the interview. I can forgive a few weak answers on the standard interview questions, but if the math teaching isn’t up to snuff, I’m not recommending.

Since we’re nearing the end of the school year, I’m also leading my classes through reviews to prepare for their final exams. This includes going through problems we haven’t discussed in-depth since last fall. Most of the time, it’s fine. But a couple of times last week (in calculus, naturally), I had some ridiculous cerebral failures.

That’s fine, too. I make a point of emphasizing to my students early on that I can make mistakes, and if they catch me at it, good for them. Seeing me make mistakes without falling apart seems to help them be more willing to take risks even though they might be wrong.

I got to thinking about the two situations. Where’s the line between “Oops, the teacher’s human and makes mistakes” and, “No, this interviewee doesn’t have what it takes”?

My guess is that the line is in awareness. When I screwed up in calculus, I either knew right away or within moments. I immediately ‘fessed up to the students and set about figuring out what I’d done wrong. With interviewees who aren’t cutting it, they generally seem to think what they’re doing is fine. Top interviewees often have more to criticize about themselves. There’s a question in the interview about what they think they need to work on most. It’s always interesting to correlate their answer to this question with their performance in the mock-lesson.

So, everyone, let’s aspire to make mistakes. Own them, learn from them. But never let it cross into incompetence. If we are incompetent in an area, let’s be aware of it, and work to correct it.

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How Does Your World Measure Up?

World-building is a key component of writing fiction, particularly in the genres of sci-fi and fantasy. That means you have to have culture, history, and everything else that comes with a real world underlying your story.

Including … measurement units?

Maybe not. Maybe your world is built enough off of ours that it makes sense to stick with the usual feet and inches, pounds and ounces. Or if your world is in a future where scientific reasonableness is king, so you’re all metric.

But what if that won’t work for your world?

My first novel was largely in an alternate dimension with some shared history, but mostly a huge divergence. And a very science-oriented society. In a particular situation, I needed to make a reference to a measurement of volts.

Volts were named for Alessandro Volta. A dude who didn’t exist in that dimension.

First thought: Oh, crap.

Second thought: Okay, what made-up units would make sense in this word I’ve created?

I considered how the society was fairly practical and straightforward in other naming practices, and I thought about what voltage means. In the end, I came up with a fake unit that seemed to fit both needs.

Have you ever thought about how many units are named after a person? Fahrenheit and Celsius for temperature. Volts, amperes, coulombs, and ohms for various aspects of electricity and charge. Newtons for force, pascals for pressure.

If you don’t need to worry about these things in your stories, you’re a lucky one. For the rest of us, make sure you think about a natural way for units to evolve in your world.

Have you invented units/measurements for one or more of your stories? How did you go about it?

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The Makings of Mathematical Mistakes

In my years of teaching math, I’ve amused myself by taking note of the types of mistakes students make. (Yeah, okay. I’m easily amused.) You can pretty much figure out the type of mistake by watching my reaction.

The “Fell Through the Cracks” Mistake

This usually happens in complex, multi-step problems. The student does all the hard stuff right but overlooks something. More often than not, it’s losing a negative or mistyping something in the calculator.

R.C.’s Reaction: I just point wordlessly at the paper or calculator and wait while the student looks, ponders, then says, “Oh! Oops.”

The “You Know Better” Mistake

Another “careless” variety of mistake. Can I tell you how many times I’ve asked what four-squared is only to hear, “Eight. No—wait! Sorry. Sixteen.”

R.C.’s Reaction: Students often catch those without any help from me. When they don’t, they get my ‘Did you seriously just say that?’ look. If that’s not enough, they get a verbal, “Really?”

The “You’re Still Learning” Mistake

This happens when students are mostly getting a new concept but aren’t quite there yet. OR … when they have to apply something they learned previously that hasn’t quite solidified.

R.C.’s Reaction: Usually I ask them to explain their thinking first, then ask some follow-up questions until they see the wrong turn. Sometimes a neighboring student will try to tease the other about the mistake, at which point I remind them that they made the exact same mistake two minutes ago when I was helping them.

The “Someone in Your Past Failed Both of Us” Mistake

I teach high school math, which naturally relies on concepts learned over several years before arriving in my class. Sometimes we’re working on some complicated algebraic thing and I realize some/all of the students have a problem with an underlying principle. (Fractions, anyone? Or measurement conversions?)

R.C.’s Reaction: What can I do? Go off to the side of our work and make up a simplified example (i.e., non-algebraic addition of fractions), quickly refresh the kids’ memories on that, and parallel it to the problem at hand.

The “Back the Truck Up” Mistake

These mistakes on the part of the student tell me that I screwed something up as the teacher. Didn’t explain clearly, allowed for a massive misconception to take root, etc. Sometimes I even did something just plain wrong.

R.C.’s Reaction: Confess to the class that I made a boo-boo, very clearly indicate where we went wrong, and emphasize the proper way to move forward.

Some people might say it’s a teacher’s job to eliminate mistakes and a student’s job to avoid them. I don’t agree with that. Mistakes are great! They’re how we learn. (Well, so they’re great as long as we learn from them.) And one thing to keep in mind is that I have extremely small classes, and I’ve taught most of my students for more than one year, some for up to five straight. My reactions to the “Fell Through the Cracks” and “You Know Better” varieties are done in an environment where the students and I are able to laugh off mistakes without embarrassment. (And where I’ll accept “It’s calculus on a Monday morning,” as an excuse for the careless mistakes as long as they keep trying.)

I know some people who were always terrified to volunteer information in class, certain they’d make a mistake. I was one of them. Now, I’m okay with making mistakes in the classroom. Still working on being okay with it in the rest of my life.

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Earth: By the Numbers

Yesterday was Earth Day, and today is Mathematical Monday, so I figured I’d put the two together with some stats on this planet of ours.

• The diameter of Earth is about 7,920 miles. That’s nearly 140,000 football fields. If it were possible to drive at highway speeds (say, 75 mph) through the planet, it would take nearly 4.5 days to reach the other side.
• The mass of Earth is about 6 septillion kilograms. (That’s 6 followed by 24 zeroes. I’m going by the U.S. definition of the word—apparently it’s different in the U.K.) That’s about the same as 165 quintillion empty 18-wheelers. Or the Empire State Building 16,000,000,000,000,000,000 times over.
• The volume of water on Earth is about 1.5 billion cubic kilometers. That’s about 600 trillion Olympic swimming pools. (I think. I don’t have my preferred calculators handy.) 97% of that is saltwater in the oceans.
• So, Earth is pretty big. But if our sun were empty, it could fit over 1,000,000 Earths inside.

Now, while you ponder that massiveness, check out this very cool view of the planet from orbit.

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From the Department of Made-Up Statistics

I admit it—I’m a data geek. (Shocking, right?) Give me some data, and I can’t help but analyze it at least a little. I’ve even made graphs to analyze my writing.

I’ve often heard people claim you can make statistics say anything you want. That’s not entirely true, but you can usually frame them in a way that leans in a certain direction, even if that direction is misleading. Some easy ways to do this are asking your question in a particular way, choosing a biased sample, and setting up a graph with an inappropriate axis. (All of these will get you labeled a bad statistician, though.)

Sometimes, it’s easier just to skip all the technical steps and just make up results. So here are some claims that are entirely made up based only on my gut instinct. If anyone finds hard data on any of them, feel free to let me know.

• 47% of cancer survivors will (attempt to) write a memoir on the topic.
• 72% of drivers don’t know how to merge properly.
• 96% of literate Americans don’t know how to use a semicolon.
• 83% of teenagers declare something boring a minimum of twice every school day.
• Chocolate makes everything better 99.9% of the time.
• I remember 74% of the useless minutiae I come across, but only 51% of the information I actually need.
• 2% of the people who protest that a given book should be banned have actually read the book.

Go ahead and make up some of your own statistics, or let me know if you think my percentages are off on any of the above. It’s fun and makes you sound knowledgeable. 😉

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Blowing Students’ Minds

This is one of my favorite parts of teaching—that moment when you tell kids something, and they give you that look.

“Seriously? No way!”

I teach such a wide range of kids, those jaw-dropping moments can come in a variety of ways, especially during the years when I’ve taught physics. Here are a few examples.

• When it’s three o’clock Thursday afternoon for us, it’s eight o’clock Friday morning in Australia.
• How dirt-simple derivatives in calculus can be with shortcuts like the Power Rule, yet still give the same information as the long limit-definition way.
• What happens when you shoot a laser pointer through a piece of diffraction grating (or a prism, or a lens).
• What happens when you look through a piece of diffraction grating at a prism while outside in the sun. (This was accidental, and very cool. We had fun in our color and optics unit.)
• The whole idea of traveling close to the speed of light and all the crazy things that go along with it, like the Twin Paradox.
• The crazy things that just plain sound waves do to cornstarch and water.

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