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 for x, you might see:

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?

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Wednesday morning links

• Walking Randomly has an interesting discovery about the Fibonacci sequence and linear algebra.
• The Productive Student offers up some advice on how to be a leader and conduct killer team sessions. It’s good stuff not only for students who are doing collaborative work but also for anybody who goes to meetings. Are there people who don’t have to go to meetings?
• InsideHigherEd reports on an interesting setup to attract Chinese students to study in the US — the 1+2+1 degree, which involves one year in China, two in the US, and then the final year back in China. (Unfortuately, as the article notes, you can’t Google “1+2+1” because all you get is “4”.)
• Also at IHE and a lot of other places, Rice University is now using an open textbook for its elementary statistics course which is not only free but open for rearrangement and adaptation by any user. A shot across the bow of traditional textbook companies?
• Study Hacks offers advice to students on cutting out the single biggest source of stress (according to them) — the killer course load. There’s something to be said for having an unbalanced course schedule — I found grad school to be easier in some ways than college because I was only taking math courses — but I do remember the worst semester I ever had as an undergrad had me taking three senior-level math courses… plus German, orchestra, and concert choir, with a 20-hour work week at a donut shop to boot. Balance is important.
• Reasonable Deviations writes about a hack of the Boston subway system by three MIT students (for what appear to be purely academic purposes). Predictably, the subway authorities have sought legal action against the students. They ought instead to be thanking them, or hiring them outright, for pointing out a security flaw that eventually could have cost the company millions.
• Java is the most popular programming language in the world, but some are saying that using Java as the language of choice in intro programming courses (as is currently done, for example, in the standard AP Computer Science courses) is hurting students in the long run. To me, this article raises the question of just what computer programming is these days.
• A new Zogby poll is indicating that online university programs (meaning, it seems, online programs offered by existing brick-and-mortar universities as opposed to online universities) are rapidly gaining mainstream acceptance, despite the perception (possibly justified) that they offer less academic rigor than traditional university programming. Unfortunately you have to drill down into the Chronicle article mentioned at the above link to discover that the Zogby poll was administered online! So much for unbiased sampling. But at any rate, the trend seems to be limited mainly to older adults who are looking for college coursework, which makes sense. I think if you restricted the polling to a traditional college population — for example, high school seniors who are looking at colleges to attend — I don’t think you’d see nearly as much of a trend toward online programming.