Tag Archives: Calculators

Student (mis)understanding of the equals sign

Interesting report here (via Reidar Mosvold) about American students’ misunderstanding of the “equals” sign and how that understanding might feed into a host of mathematical issues from elementary school all the way to calculus. According to researchers Robert M. Capraro and Mary Capraro at Texas A&M,

About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students.

Robert Capraro, in the video at the link above, makes an interesting point about the “=” sign being used as an operator. He makes a passing reference to calculators, and I wonder if calculators are partly to blame here. After all, if you want to calculate 3+5 on a typical modern calculator, what do you do? You hit “3”, then “+”, then “5”… and then hit the “=” button. The “=” key is performing an action — it’s an operator! In fact, I suspect that if you gave students that sequence of calculator keystrokes and asked them which one performs the mathematical operation, most would say “=” rather than the true operator, “+”. The technology they use, handheld calculators, seems to be training them to think in exactly the wrong way about “=”. What we have labelled as the “=” key on a calculator is really better labelled as “Enter” or “Execute”.

In fact, the old-school HP calculators, like this HP 33c, didn’t have “=” buttons at all:

That’s because these calculators used Reverse Polish Notation, in which the 3 + 5 calculation would have been entered “3”, then “5”, “+”, then “Enter” — and then you’d get an answer. What HP calculators label as “Enter”, on a typical modern calculator would be labelled “=”, and in that syntax lies a lot of the problem, it seems.

The biggest problem I seem to encounter with “=” sign use is that students use it to mark a transition between steps in a problem. For example, when solving the equation $3x - 2 = 10$ for x, you might see:

$3x - 2 = 10 = 12 = x = 4$

The thought process can be teased out of this atrocious syntax, but clearly this is not acceptable math — even though the last bit of that line (x=4) is a correct statement. If the student would just put spaces, tabs, or even a semicolon between the steps, it would be a big improvement. But many students are so trained to believe that the right answer — the ending “4” — is all that matters, they have little experience with crafting a good solution, or even realizing that a mathematical solution is supposed to be a form of communication at all.

What are some of the student misconceptions you’ve seen (or perpetrated!) with the “=” sign? If you’re a teacher, how have you approached mending those misconceptions?

Calculator blasts from the past

One of the real treats of the ICTCM was the Saturday 8:00 AM session titled “Three Decades of Handheld Devices: How Mathematics Teaching Changed Along with Them” given by John Kenelly. Prof. Kenelly has a long history of involvement in the development of calculator technology, and he gave a fascinating talk full of good thoughts on the direction of handheld technologies today, war stories from the past, and good jokes. (Example of the latter: “Getting a spreadsheet to work on a calculator is like getting a dog to walk on its hind legs — it can be done, but it ain’t pretty!”)

I will try to say more about Prof. Kenelly’s ideas about the future of handheld technologies in a later post, but for now I wanted to share one of the really cool parts of his talk — the calculators themselves, some of which are now antiques. He had a bag full of these old-school devices (some of which are less than 10 years old but still old-school) which he generously let us paw over.

Here is a Hewlett-Packard HP 35, the world’s first handheld scientific calculator, from 1972. Check out that red LED display and, in contrast with the NSpire, the sheer paucity of keys on the keyboard:

Here’s a rare example of a Casio fx7000, from 1985 — the world’s first handheld graphing calculator.

I was downright startled to learn that sitting right across the aisle from me at this talk was Hideshi Fukaya, the lead engineering on the development team for the Casio fx7000 and the person rightfully considered to be the inventor of the graphing calculator.

Moving ahead up the timeline, here is a Casio Cassiopeia. More of a palmtop computer than a calculator, and it ran Windows CE. Anybody remember good old WinCE and why that abbreviation was particularly apt?

I guess I am just a sucker for old-fashioned calculators.

Aside: I’d love to do a spreadsheet in which one column has the year in which a calculator was made and another column has the number of buttons on the calculator, and run a regression analysis on it.