Readers of this ‘blog know I actively use many forms of technology in my teaching and personal explorations. Yesterday, a thread started on the AP-Calculus community discussion board with some expressing discomfort that most math software accepts sin(x)^2 as an acceptable equivalent to the “traditional” handwritten .
Some AP readers spoke up to declare that sin(x)^2 would always be read as . While I can’t speak to the veracity of that last claim, I found it a bit troubling and missing out on some very real difficulties users face when interpreting between paper- and computer-based versions of math expressions. Following is an edited version of my response to the AP Calculus discussion board.
I believe there’s something at the core of all of this that isn’t being explicitly named: The differences between computer-based 1-dimensional input (left-to-right text-based commands) vs. paper-and-pencil 2-dimensional input (handwritten notation moves vertically–exponents, limits, sigma notation–and horizontally). Two-dimensional traditional math writing simply doesn’t convert directly to computer syntax. Computers are a brilliant tool for mathematics exploration and calculation, but they require a different type of input formatting. To overlook and not explicitly name this for our students leaves them in the unenviable position of trying to “creatively” translate between two types of writing with occasional interpretation differences.
Our students are unintentionally set up for this confusion when they first learn about the order of operations–typically in middle school in the US. They learn the sequencing: parentheses then exponents, then multiplication & division, and finally addition and subtraction. Notice that functions aren’t mentioned here. This thread [on the AP Calculus discussion board] has helped me realize that all or almost all of the sources I routinely reference never explicitly redefine order of operations after the introduction of the function concept and notation. That means our students are left with the insidious and oft-misunderstood PEMDAS (or BIDMAS in the UK) as their sole guide for operation sequencing. When they encounter squaring or reciprocating or any other operations applied to function notation, they’re stuck trying to make sense and creating their own interpretation of this new dissonance in their old notation. This is easily evidenced by the struggles many have when inputting computer expressions requiring lots of nested parentheses or when first trying to code in LaTEX.
While the sin(x)^2 notation is admittedly uncomfortable for traditional “by hand” notation, it is 100% logical from a computer’s perspective: evaluate the function, then square the result.
We also need to recognize that part of the confusion fault here lies in the by-hand notation. What we traditionalists understand by the notational convenience of sin^2(x) on paper is technically incorrect. We know what we MEAN, but the notation implies an incorrect order of computation. The computer notation of sin(x)^2 is actually closer to the truth.
I particularly like the way the TI-Nspire CAS handles this point. As is often the case with this software, it accepts computer input (next image), while its output converts it to the more commonly understood written WYSIWYG formatting (2nd image below).
Further recent (?) development: Students have long struggled with the by-hand notation of sin^2(x) needing to be converted to (sin(x))^2 for computers. Personally, I’ve always liked both because the computer notation emphasizes the squaring of the function output while the by-hand version was a notational convenience. My students pointed out to me recently that Desmos now accepts the sin^2(x) notation while TI Calculators still do not.
The enhancement of WYSIWYG computer input formatting means that while some of the differences in 2-dimensional hand writing and computer inputs are narrowing, common classroom technologies no longer accept the same linear formatting — but then that was possibly always the case….
To rail against the fact that many software packages interpret sin(x)^2 as (sin(x))^2 or sin^2(x) misses the point that 1-dimensional computer input is not necessarily the same as 2-dimensional paper writing. We don’t complain when two human speakers misunderstand each other when they speak different languages or dialects. Instead, we should focus on what each is trying to say and learn how to communicate clearly and efficiently in both venues.
In short, “When in Rome, …”.