Significant Digits and Useless Digits

A pet peeve of mine is measuring things to far too many decimal places. For example, notice that the thickness of these trash bags is 0.0009 inches (0.9 mil) but is 22.8 microns in metric. There are two mistakes:

  • While the conversion factor is correct, there’s no way that the thickness is known within only 0.1 microns, or 100 nanometers. That’s significantly that a typical cell nucleus.
  • Less importantly, if they rounded correctly, it should be 22.9 microns, not 22.8.

My favorite example that I’ve personally witnessed — that I wish I had a picture of — is measuring student’s perceptions of a professor’s teaching effectiveness is 13 decimal places.

This webcomic from xkcd illustrates the point both cleverly and perfectly.

Source: https://xkcd.com/2170/

What’s bigger: 1/3 pound burgers or 1/4 pound burgers?

I recently enjoyed reading about an unanticipated failed marketing campaign of the 1980s. Here’s the money quote:

One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it.

Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “¼,” larger than the “3” in “⅓,” led them astray.

Here’s the article: https://gizmodo.com/whats-bigger-1-3-pound-burgers-or-1-4-pound-burgers-1611118517

 

Mathematical Imagery: Snow and Sand Patterns

The American Mathematical Society has a neat page of large mathematical images that were created by merely walking in snow and sand. For example, here’s a time-lapse of the 9-hour construction of the Mandelbrot set on a beach (in between high tides and oblivious passers-by that walked through the artwork).

Using Rubik’s Cubes to Teach Math

I enjoyed this opinion piece about creative ways to use a Rubik’s cube to engage reluctant students in a mathematics class.

As an added bonus, the article provides a link to You Can Do The Cube, which includes complex mosaics that can be built by arranging one side of multiple Rubik’s cubes, suggesting this as a strategy for getting children hooked on Rubik’s cubes (instead of frustrating novices with the complex task of solving the cube completely).

Left-Hand Rule?

Misleading pictures in math textbooks always send 10,000 volts of electricity down my spine. Thanks to the right-hand rule, the cross product should be pointing down, not up. This comes from the 2007 edition of Glencoe’s “Advanced Mathematical Topics,” a high-school Precalculus book.

For what it’s worth, this is the same line of textbooks that, in a supplementary publication, said that the rational numbers are not countable.

Expert mathematicians stumped by simple subtractions

This was an interesting psychological article about how the phrasing of a word problem — in particular, adding extra information that has no bearing on the solution — can affect its perception of difficulty. Money quote:

“Sarah has 14 animals: cats and dogs. Mehdi has two cats fewer than Sarah, and as many dogs. How many animals does Mehdi have?”…

“[I]n the problem with animals, we look to calculate the number of dogs that Sarah has, which is impossible, whereas the calculation 14-2 = 12 provides the solution directly,” explains Jean-Pierre Thibaut, a researcher at the University of Bourgogne Franche-Comté. …

“One out of four times, the [professional mathematicians] thought there was no solution to the problem, even though it was of primary school level. And we even showed that the participants who found the solution to the set problems were still influenced by their set-based outlook, because they were slower to solve these problems than the axis problems,” says Gros.

The results highlight the critical impact that knowledge about the world has on the ability to use mathematical reasoning. They show that it is not easy to change perspective when solving a problem. Thus, the researchers argue that teachers need to take this bias into account in math education.

“We see that the way a mathematical problem is formulated has a real impact on performance, including that of experts, and it follows that we can’t reason in a totally abstract manner,” says professor Sander. Educational initiatives are required based on methods that help pupils learn about mathematical abstraction. “We have to detach ourselves from our non-mathematical intuition by working with students in non-intuitive contexts,” concludes Gros.

 

A Father Transformed Data of his Son’s First Year of Sleep into a Knitted Blanket

This is one of the more creative graphs that I’ve ever seen. From the article:

Seung Lee tracked the first year of his baby’s sleep schedule with the BabyConnect app, which lets you export data to CSV. Choosing to work with six minute intervals, Lee then converted the CSVs into JSON (using Google Apps Script and Python) which created a reliable pattern for knitting. The frenetic lines at the top of the blanket indicate the baby’s unpredictable sleep schedule right after birth. We can see how the child grew into a more reliable schedule as the lines reach more columnar patterns.

Pythagorean Theorem and Social Distancing