Computers vs. People: Writing Math

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 sin^2 (x).

From Desmos:sine1

Some AP readers spoke up to declare that sin(x)^2 would always be read as sin(x^2).  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.

MY THOUGHTS:

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).

sine2

sine3

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.

Desmos: sine4

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, …”.

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6 responses to “Computers vs. People: Writing Math

  1. While I was only an acorn this past year, one thing that seemed abundantly clear to me is that there is no “always”. The way we were instructed to score certain problems seemed arbitrary between problems and sometimes inside a problem. For instance, students were given credit for an answer with only two decimal places when the third was a zero on part (b) of BC#2. They lost points though for the same thing on part (d) of the same problem. While I am sure the Question Leader and Chief Reader have their reasons, it was not made clear to everyone else. As for the notation, the very same problem (BC#2) awarded point for an expression for speed. Students with an expression of (x'(3))^2 + (y'(3))^2 inside the radical were given credit immediately. Students with the expression of x'(3)^2 + y'(3)^2 only earned the point for the expression in the presence of the correct answer. The notation was called “ambiguous” and needed the correct answer to show the student used it as required for distance.

    • Thanks, Dennis. Your note reminds me why I emphasize clear writing and understanding of audience to all students (not just AP).

      For the AP, students need to understand that their graders are absolutely not the same as their teachers. A teacher knows what students have learned and probably knows what was mean when by unconventional notations or phrasings. Unfortunately, that assumption can never be made for AP graders. Those who grade AP responses need to be convinced that a student knows what he or she is doing in a very precise way. They don’t know the students, and they should not be expected to give any credit for even marginally vague explanations. It is the students’ jobs to write clearly. It is our job as teachers to help them learn that skill. It’s a good habit for any writing that will be (or might be) read by someone unknown to the writer.

  2. It seems logical to me that the parentheses indicate what is being put into the sine function. I assume that’s why TI calculators force an open parentheses when pressing the sin button.

    • I agree. I was trying to make the point that both notations are entirely logical within their respective environments.

      For someone steeped in handwritten math to complain about computer syntax is much like an English speaker complaining that a French speaker puts adjectives after a noun (e.g. chapeau blanc) instead of before the before the noun like they’re supposed to be [in English] (e.g. white hat). In this example, the complainer misses the point. Both constructions are correct within their environments.

  3. Chris: GeoGebra also accepts sin^2(x) and displays it the way we math teachers would write it.

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