How to Avoid Thinking in Math Class (Part 6)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

I had a surreal moment this year. I’d almost finished a lesson when one boy, usually a hyperkinetic little bundle of enthusiasm, raised his hand.

“So, like, I don’t really understand anything you’re saying,” he informed me, “But I can still get the right answer.”

He smiled, waiting.

“Which part is giving you trouble?” I asked.

“Oh, you were talking about this extra stuff,” he said, “like the ideas behind it and everything. I don’t… you know… do that.”

I blinked. He blinked. We stood in silence.

“So is that okay?” he concluded. “I mean, as long as I can get the right answer?”

Here is Part 6:

How to Avoid Thinking in Math Class (Part 5)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

As for students, it can be frightening to start a math problem. You don’t know quite where it will lead. Will my approach be fruitful? Will it falter? Where do I even begin?

But unlike my desk-perching student, most kids don’t recognize that one rope holding them back is fear of the unknown. They just hesitate: too afraid to leap without a net, but never bothering to go in search of a net for themselves…

In all these cases, students are refusing to engage with their uncertainty. But if you’re uncomfortable with doubt, you’ll never break through to the other side. You’ll never have a “Eureka!” moment or an intellectual “Aha!” You’ll never… well… learn. After all, if you can’t bear to face the unknown, how will you ever come to know it?

I find that my desk-percher has it right. At times like these, the mere presence of an expert can supply the confidence you’re lacking.

Here is Part 5, introducing what happens when students get stuck getting started on a problem:

How to Avoid Thinking in Math Class (Part 4)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

This speaks more to my naiveté as a first-year teacher than anything else, but I was shocked to find how fervently my students despised the things they called “word problems.”

“I hate these! What is this, an English lesson?”

“Can’t we do regular math?”

“Why are there words in math class?”

Their chorus: I’m okay with math, except word problems.

They treated “word problems” as some exotic and poisonous breed. These had nothing to do with the main thrust of mathematics, which was apparently to chug through computations and arrive at clean numerical solutions.

I was mystified—which is to say, clueless. Why all this word-problem hatred?

Here is Part 4, addressing students’ fears of word problems:

How to Avoid Thinking in Math Class (Part 3)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

Every rule – even the craziest, most arbitrary mandate – has a reason rooted in this essential purpose. (Why leave the dishes with big particles? Because the person is still eating!) And so it is in math class. If you understand slope not as “that list of steps I’m supposed to follow” but as “a rate of change,” things start making more sense. (Why is it the ratio of the coefficients? Because, look what happens when x increases by 1!)

You get to work a lot less, and think a lot more.

Now, conceptual understanding alone isn’t enough, any more than procedures alone are enough. You must connect the two, tracing how the rules emerge from the concepts. Only then can you learn to apply procedures flexibly, and to anticipate exceptions. Only then will you get the pat on the back that every robot craves.

With my students on Friday, I garbled the whole analogy. I tend to do that.

But there’s a simple takeaway. Even if you don’t care about understanding for its own sake; even if you’re indifferent to the beauty and deeper logic of mathematics; even if you care only about test results and right answers; even then, you should remember that the “how” is rooted in the “why,” and you’re unlikely to master methods if you disregard their reasons.

Here is Part 3, addressing the importance of both computational proficiency and conceptual understanding:

How to Avoid Thinking in Math Class (Part 2)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

Occasionally, we teachers grow frustrated with our formula-thirsty students. (Okay, more like “often” or “weekly.”) Sometimes, we even denounce formulas altogether, deriding them as “brainless plug-and-chug” or “not real math.”

Of course, that’s going too far. The intelligent use of formulas is an important part of mathematics. But we’re right about one thing: there’s a lot more to formulas than just throwing numbers into a blender.

Here is Part 2, addressing students’ natural desire to mindlessly plug numbers into a formula without conceptual understanding:

How to Avoid Thinking in Math Class (Part 1)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

In teaching math, I’ve come across a whole taxonomy of insidious strategies for avoiding thinking. Albeit for understandable reasons, kids employ an arsenal of time-tested ways to short-circuit the learning process, to jump to right answers and good test scores without putting in the cognitive heavy lifting. I hope to classify and illustrate these academic maladies: their symptoms, their root causes, and (with any luck) their cures.

Here is Part 1, introducing the series:

Preparation for Industrial Careers in the Mathematical Sciences: Creating More Realistic Animation for Movies

The Mathematical Association of America recently published a number of promotional videos showing various mathematics can be used in “the real world.” Here’s the first pair of videos describing how mathematics is used for computer animation. From the YouTube descriptions:

Dr. Alex McAdams, Senior Software Engineer at Walt Disney Animation Studios, talks about how mathematics is used to make realistic, yet art directable, animations.

Prof. Joseph Teran of the Department of Mathematics at UCLA gives an overview of the numerical linear algebra and iterative method techniques that are used to simulate physical phenomena such as water, fire, smoke, and elastic deformations in the movie and gaming industries.

Nutritional Science Isn’t Very Scientific: the research behind dietary recommendations is a lot less certain than you think.

Here’s a nice article about the inherent difficulties faced by nutritional scientists and the perils of observational studies (as opposed to controlled experiments):

Money quotes:

Most of our devout beliefs about nutrition have not been subjected to a robust, experimental, controlled clinical trial, the type of study that shows cause and effect, which may be why Americans are pummeled with contradictory and confounding nutritional advice.

To wit:

Many nutritional studies are observational studies, including massive ones like the Nurses’ Health Study. Researchers like Willett try to suss out how changes in diet affect health by looking at associations between what people report they eat and how long they live. When many observational studies reach the same conclusions, Willett says, there is enough evidence to support dietary recommendations. Even though they only show correlation, not cause and effect, observational studies direct what we eat.

Apart from their inability to determine cause and effect, there’s another problem with observational studies: The data they’re based on—surveys where people report what they ate the day (or week) before—are notoriously unreliable. Researchers have long known that people (even nurses) misreport, intentionally and unintentionally, what they eat. Scientists politely call this “recall bias.”

The coupling of observational studies and self-reported data leads some observers to the conclusion that we know neither how Americans do eat nor how they should eat. A recent PLOS One article even suggests that several national studies use data that is so wildly off base that the self-reported caloric intake is “incompatible with survival.” If people had eaten as little as they reported, in other words, they would be starving.

Peter Attia, a medical researcher and doctor, started questioning the basis of dietary guidelines when he saw that following them didn’t work for his patients. They didn’t lose weight, even when they virtuously stuck with their diets. When he took a look at the research supporting the advice he was giving to his patients, he saw shoddy science. Attia estimates that 16,000 nutritional studies are published each year, but the majority of them are deeply flawed: either poorly controlled clinical trials, observational studies, or animal studies. “Those studies wouldn’t pass muster in another field,” he told me.

Collaborative Mathematics: Challenge 15

I’m a few months late with this, but my colleague Jason Ermer at Collaborative Mathematics has published Challenge 15 on his website:

His website also has some follow-up challenge questions.

Five Ways to Lie with Charts

This is a cute article about ways that people can lie with charts: (1) Puzzling perspective, (2) Swindling shapes, (3) Trendsetters are tricksters (implying a false correlation), (4) Hiding in plain sight, and (5) Changing the scale of the axis.