Education

Demonstrating persistence

Posted on by John

“A college degree shows you can finish something.” I’ve heard this forever, but I don’t believe it. Of course a college degree shows that someone finished one thing, namely a college degree. But I don’t think that’s the best predictor of whether someone will finish something else.

College provides a great deal of support: accountability, frequent feedback, a community of peers, etc. Succeeding in this environment is an accomplishment, but it doesn’t necessarily demonstrate that someone can succeed in a less supportive environment. It also doesn’t necessarily indicate that someone can focus on a project that takes more than a semester to finish.

Here are a few things that might be better indicators of initiative and persistence.

  • Learning a foreign language as an adult
  • Losing 50 pounds
  • Learning to play the oboe
  • Quitting smoking
  • Reading Churchill’s history of WWII
  • Starting a business
  • Running a marathon
  • Writing a book

Employers that use college degrees as their only filter on applicants are missing out. An ideal candidate would have a college degree and some proof of independent achievement. But given a choice between someone with only academic credentials and someone with only independent accomplishments, the latter may be a better hire.

Related post: Picking classes

Advanced or just obscure?

Posted on by John

Sometimes it’s clear what’s meant by one topic being more advanced than another. For example, algebra is more advanced than arithmetic because you need to know arithmetic before you can do algebra. If you can’t learn A until you’ve learned B, then A is more advanced. But often advanced is used in a looser sense.

When I became a software developer, I was surprised how loosely developers use the word advanced. For example, one function might be called more “advanced” than other, even though there was no connection between the two. The supposedly more advanced function might be more specialized or harder to use. In other words, advanced was being used as a synonym for obscure. This is curious since advanced has a positive connotation but obscure has a negative connotation.

I resisted this terminology at first, but eventually I gave in. I’ll say advanced when I’m sure people will understand my meaning, even if I cringe a little inside. For example, I have had a Twitter account SansMouse that posts one keyboard shortcut a day [1]. These are in a cycle, starting with the most well-known and generally useful shortcuts. When I say the shortcuts progress from basic to advanced, people know what I mean and they’re happy with that. But it might be more accurate to say the shortcuts regress from most useful to least useful!

I’m not writing this just to pick at how people use words. My point is that the classification of some things as more advanced than others, particularly in technology, is largely arbitrary. The application of this: don’t assume that ‘advanced’ necessarily comes after ‘basic’.

Maybe A is called more advanced than B because most people find B more accessible. That doesn’t necessarily mean that you will find B more accessible. For example, I’ve often found supposedly advanced books easier to read than introductory books. Whether the author’s style resonates with you may be more important than the level of the book.

Maybe A is called more advanced than B because most people learn B first. That could be a historical accident. Maybe A is actually easier to learn from scratch, but B came first. Teachers and authors tend to present material in the order in which they learned it. They may think of newer material as being more difficult, but a new generation may disagree.

Finally, whether one thing is more advanced than another may depend on how far you intend to pursue it. It may be harder to master A than B, but that doesn’t mean it’s harder to dabble in A than B.

In short, you need to decide for yourself what order to learn things in. Of course if you’re learning something really new, you’re in no position to say what that order should be. The best thing is to start with the conventional order. But experiment with variations. Try getting ahead of the usual path now and then. You may find a different sequence that better fits your ways of thinking and your experience.

* * *

[1] Sometime after this post was written I renamed SansMouse to ShortcutKeyTip. I stopped posting to that account in September 2013, but the account is still online.

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Teaching Bayesian stats backward

Posted on by John

Most presentations of Bayesian statistics I’ve seen start with elementary examples of Bayes’ Theorem. And most of these use the canonical example of testing for rare diseases. But the connection between these examples and Bayesian statistics is not obvious at first. Maybe this isn’t the best approach.

What if we begin with the end in mind? Bayesian calculations produce posterior probability distributions on parameters. An effective way to teach Bayesian statistics might be to start there. Suppose we had probability distributions on our parameters. Never mind where they came from. Never mind classical objections that say you can’t do this. What if you could? If you had such distributions, what could you do with them?

For starters, point estimation and interval estimation become trivial. You could, for example, use the distribution mean as a point estimate and the area between two quantiles as an interval estimate. The distributions tell you far more than  point estimates or interval estimates could; these estimates are simply summaries of the information contained in the distributions.

It makes logical sense to start with Bayes’ Theorem since that’s the tool used to construct posterior distributions. But I think it makes pedagogical sense to start with the posterior distribution and work backward to how one would come up with such a thing.

Bayesian statistics is so named because Bayes’ Theorem is essential to its calculations. But that’s a little like classical statistics Central Limitist statistics because it relies heavily on the Central Limit Theorem.

The key idea of Bayesian statistics is to represent all uncertainty by probability distributions. That idea can be obscured by an early emphasis on calculations.

More posts on Bayesian statistics

Slide rules

Posted on by John

Mike Croucher raises an important point for teachers: Are graphical calculators pointless? I think they are. I resented having to buy my daughter an expensive calculator when I could have bought her a netbook for not much more money.

Calculators are obsolete. I can’t remember the last time I used one. On the other hand, it could be valuable to have students use something really obsolete: a slide rule. Not for long, maybe just for a week or two.

  1. Slide rules are basically strips of log-scale paper. If you play with a slide rule long enough, you might get a tangible feel for logarithms.
  2. Slide rules make you concentrate on orders of magnitude. A slide rule will give you the significant digits, but you have to know what power of ten to use.
  3. Slide rules give you a tangible sense of significant figures. You can’t report more than three significant figures because you can’t see more than three significant figures. Maybe some experience with a slide rule would break students of the habit of reporting ever decimal that comes out of their calculators.

I’m not saying that being able to use a slide rule is a valuable skill. It’s not anymore. But the process of using a slide rule for a little while might teach some skills that are valuable. It would be fine if they forgot how to use a slide rule but retained an intuition for logarithms, orders of magnitude, and significant digits.

I’d recommend using a slide rule in high school for the same reason as using an abacus in elementary school: because it’s tangible, not because it’s practical.

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Picking classes

Posted on by John

Here’s a little advice to students picking electives.

Consider taking classes in those things that would be hardest to learn on your own after you graduate. Taking the most advanced courses available in your major may not be the best choice. Presumably you’ve learned how to learn more about your area of concentration. (If not, your education has failed you.) So the advanced courses might teach you the material you’re best prepared to learn on your own.

Maybe it would be better to take a foundational course in a related area than an advanced course in your main area. For example, I suggested to some statistics graduate students yesterday that they take a really good linear algebra class rather than taking all the statistics they can. If they become professional statisticians, they’ll continue to learn statistics (I hope!) but they may find it harder to take the time to really understand mathematical foundations.

Take chances, make mistakes, and get messy

Posted on by John

Magic School Bus

From Magic School Bus:

Take chances, make mistakes, and get messy.

Magic School Bus is an educational television show for children. The quote above is often repeated by the main character of the show, Ms. Frizzle.

Too many programs that supposedly teach science only teach results from science. Magic School Bus does both. It teaches specific facts, such as the names of the planets, but it also teaches that science is about taking chances, making mistakes, and getting messy.

Related post: Preparing for innovation

Computing days of the week in your head

Posted on by John

Years ago I taught a “math for poets” class. (I don’t remember the actual name of the course. Everyone called it “math for poets” because it was the one math class humanities majors had to take.) I taught the students how to mentally figure out days of the week and they loved it. It was easily the most popular topic in the course. It was satisfying to find any topic that was popular in a course that many had put off as long as possible.

I’d thought about turning my old class notes into a blog post, but there’s one minor complication. I taught this course in the 1990’s and the method was designed to make it easiest to work with dates in the 20th century. You could use it to compute days of the week in the 21st century, but doing so would take one more step than revising the method to make it easier to work with 21st century dates.

Fairy dust on the diploma

Posted on by John

When I was in college, a friend of mine gave me a math book that I found hard to get through. When I complained about it, he told me “You’re going to finish a PhD someday. When you do, do you think there’s going to be fairy dust on the diploma that’s going to enable you to do anything you can’t do now?”

That conversation stuck with me. I realized that I just needed to work hard rather than wait for my intelligence to mysteriously rise at graduation.

Accelerated learning

Posted on by John

Derek Sivers tells how a mentor was able to teach him a semester’s worth of music theory in three hours. His mentor also prepared him to place out of four more classes in four sessions. He gives the details in his blog post There’s no speed limit. It’s an inspiring story.

However, Sivers didn’t go through his entire education this way. He finished his degree in 2.5 years, but at the rate he started he could have finished in under a semester. Obviously he wasn’t able to blow through everything as fast as music theory.

Some classes compress better than others. Theoretical classes condense better than others. A highly motivated student could learn a semester of music theory or physics in a short amount of time. But it would take longer to learn a semester of French or biology no matter how motivated you are because these courses can’t be summarized by a small number of general principles. And while Sivers learned basic music theory in three hours, he says it took him 15 years to learn how to sing.

Did Sivers’ mentor expose him to everything students taking music theory classes are exposed to? Probably not. But apparently Sivers did learn the most important material, both in the opinion of his mentor and in the opinion of the people who created the placement exams. His mentor not only taught him a lot of ideas in a short amount of time, he also told him when it was time to move on to something else.

It’s hard to say when you’ve learned something. Any subject can be explored in infinite detail. But there comes a point when you’ve learned a subject well enough. Maybe you’ve learned it to your personal satisfaction or you’ve learned it well enough for an exam. Maybe you’ve reached diminishing return on your efforts or you’ve learned as much as you need to for now.

One way to greatly speed up learning is to realize when you’ve learned enough. A mentor can say something like “You don’t know everything, but you’ve learned about as much as you’re going to until you get more experience.”

Occasionally I’ll go from feeling I don’t understand something to feeling I do understand it in a moment, and not because I’ve learned anything new. I just realize that maybe I do understand it after all. It’s a feeling like eating a meal quickly and stopping before you feel full. A few minutes later you feel full, not because you’ve eaten any more, but only because your body realizes you’re full.

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