Teaching

On College Rankings

This essay from the President of Reed College discusses what it’s like to live outside (and inside) the world of college rankings, essentially asking “Are these rankings meaningful?”

https://www.reed.edu/apply/college-rankings.html

It’s a familiar story to anyone who has ever contemplated teaching to the test. As rankings/ratings/grades become more and more important, colleges/schools/students (and teachers) tend to focus more and more on those metrics, perhaps at the expense of what’s really important (whatever that might be).

A perfect rating system, presumably, would compel the rated parties to meet and expand the standard of excellence. But in practice, it seems difficult to come to a consensus about what comprises excellence, and even harder, then, to construct an appropriate rating system.

So how should we measure a college or university?

By patrick honner, 14 yearsJune 25, 2012 ago

Teaching

Math-Intensive Majors

The MAA and David Bressoud released a report about the current status of math intensive majors in the U.S. (which you can find here).

Bressoud starts with the encouraging news in the STEM fields–Science, Technology, Engineering, and Mathematics. In the past fifteen years, colleges have seen a 33% increase in students in these majors. However, those numbers may be dominated by particularly large increases in Biology and Psychology.

As a mathematician, Bressoud is interested in math-intensive majors, and so he looks more closely at mathematics, engineering, and physical sciences. As total college enrollments and STEM majors have increased, these math-intensive majors attract a consistent percentage of students. In fact, Bressoud notes that this percentage has been stable for the past 30 years, as math-intensive degrees have shown no growth as a percentage of overall college degrees.

This is curious, given the increasingly quantitative nature of modern society, industry, and academia. Are greater percentages of students in other countries pursuing such degrees? Or do we only need 0.5% of our college students studying math-intensive fields?

By patrick honner, 14 yearsJune 21, 2012 ago

Appreciation Teaching Technology

Fun With Geogebra: Cardioid Tortilla

Somewhere out there a tortilla maker is striving to make a perfect Cardioid Tortilla.

But there is still work to do.

By patrick honner, 14 yearsJune 19, 2012 ago

Resources Teaching

Reflections: Students in Math Class

At the end of the term I ask students to write simple reflections on their experiences from the year: what they learned about math, about the world, about themselves. It’s one of the many ways I get students writing in math class.

It’s a great way to model reflection as part of the learning process, and it’s also a good way for me to get feedback about the student experience.

Mostly, it’s fun! I love sharing and discussing the reflections with students, and it always results in great end-of-year conversations.

Here are some of my favorites.

After learning a little more about math, I think math is created rather than discovered. This makes mathematicians and scientists the creators, not merely the seekers.

I learned a lot of things from my classmates that I wouldn’t have learned if I were to just study on my own.

I have learned that I still have very much to learn about myself.

Mathematics is magical; it can lead you to a dead end, but then it can miraculously open up an exit.

Learning how to think of things in three dimensions completely changed the way I saw math.

By seeing algebraic and geometric interpretations, I learned how to communicate math in more ways.

The process which turns a difficult problem into a relatively easy problem is the beauty of math.

One of the best parts of reflection is how much it gets you thinking about the future. Plenty of food for thought here.

For more resources, see my Writing in Math Class page.

Teaching

A Surprising Integral

I had a fun encounter with an innocuous looking integral.

It all started with a simple directive: evaluate $\int{cos(\sqrt{x}) \thinspace dx}$ .

Integration is often tricky business. Although there is a large body of integration techniques, there isn’t really one guaranteed procedure for evaluating an integral. If you see what the answer is, you write it down; if you don’t, you try a technique in the hope that it makes you see what the answer is. If that technique doesn’t work, you try another.

This particular problem is interesting in that it highlights a strange phenomenon that occasionally pops up in problem-solving: sometimes making a problem look more complicated actually makes it easier to solve.

Let $u = \sqrt{x}$ . Thus, $du = \frac{1}{2\sqrt{x}} dx$ , and so $dx = 2 \sqrt{x} du$ . But since $u = \sqrt{x}$ , we have $dx = 2 \thinspace u \thinspace du$ .

This gives us $\int{cos(\sqrt{x}) \thinspace dx} = \int{2 \thinspace u \thinspace cos(u) \thinspace du}$ .

This actually looks a bit more difficult than the original problem, but now we can easily integrate using Integration by Parts!

After applying this technique, we’ll get $\int{2 \thinspace u \thinspace cos(u) \thinspace du} = 2 \thinspace u \thinspace sin(u) + 2 \thinspace cos(u) + C$ . And so, after un-substituting, we get

$\int{cos(\sqrt{x})} = 2\sqrt{x} \thinspace sin(\sqrt{x}) + 2 \thinspace cos(\sqrt{x}) + C.$

I was surprised that this technique worked, so I actually differentiated to make sure I got the correct answer. You can take my word for it, or you can verify with WolframAlpha.

One of the best parts of being a teacher is learning (or re-learning) something new every day!

By patrick honner, 14 yearsJune 7, 2012 ago

On College Rankings

Math-Intensive Majors

Fun With Geogebra: Cardioid Tortilla

Reflections: Students in Math Class

A Surprising Integral

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