Tuesday, July 17, 2007

2 z or not 2 z

That is the answer

I'm not the only person who can't read my students' writing. They can't either.

Although their thumbs are hyper-developed from strenuous daily sessions of text messaging, their fingers lack the practiced dexterity to render faithfully the arcane symbols required in math class. A large majority print everything, cursive being a lost art. Among the printers, a few favor ALL CAPS, preferring not to bother with the trivial distinction between upper- and lower-case letters. I can only imagine what will happen when some of them take their first course in genetics and encounter the usual notation for chromosome pairs: AA, Aa, aA, and aa. I think frustration will be dominant.

Indeed, the big problem is ambiguity. I've been trying to persuade my students that it is important to distinguish between similar-looking symbols. This usually means conforming fairly closely to standard orthography. My more creative students, however, give free rein to their imaginations. Bless 'em all! Why shouldn't they choose to write the Greek letter theta with a vertical slash instead of a horizontal one? They know what they mean. Why is their teacher such a party pooper?

Now that they've met phi, they know. It is the Greek letter with the vertical slash. Several recalcitrant students are now trying to school themselves to to draw the two Greek letters distinctly. Some pounced on the one-stroke variant for phi—one I favor myself—in which the symbol is drawn as a single flowing curve instead as a separate circle and slash. Unfortunately, about half of them start too low and shorten the initial downstroke. The result is a phi that looks a lot like a rho. Is this really a problem? Oh, yes. I'm talking about a Calculus III class in which we're using spherical coordinates. The letters used to represent three-dimensional spherical coordinates? None other than rho, phi, and theta. Yikes!

A recent egregious example of a student unable to read her own handwriting occurred in an algebra class. She was attempting to solve a rational equation whose variable was z. Using a least common denominator to clear all the fractions, she had to multiply each term by a factor of (z + 1). She needed to write down that factor three times, once for each term. She succeeded with the first term, but by the second term the (z + 1) has magically morphed into a (2 + 1). When she wrote the third term, the z's transformation was complete, having become a thoroughly unambiguous 2, sporting a cursive loop that no printed z would tolerate.

I had counseled her on the ambiguity of her z's and 2's, suggesting that she use a European z (“the one with a mustache”), but she had resisted my recommendation. “I can tell the difference,” she retorted.

No, she really couldn't.

We all suspect the source of this problem. These days most of us communicate preferentially by phone or e-mail. We certainly don't use handwritten letters. In classes other than mathematics, where the symbology remains technologically challenging, keyboard skills have become paramount. Handwriting has faded and our students grasp writing implements as if they are foreign objects. Indeed, they are.

I wonder if the curriculum committee would entertain a proposal for a one-unit course in mathematical penmanship?


Anonymous said...

Of course, there's always Mathematica, Matlab, Mathcad, Mupad, and a few dozen other software packages out nowadays.

Personally, I always found carefully-formed handwritten symbols and equations aesthetically pleasing.

Zeno said...

That is unquestionably true, anonymous, and I've used some of those packages. However, only one student has ever used a computer to take notes in one of my math classes, and he had an electronic tablet on which he used a stylus, not a keyboard. Handwriting is still a lot more flexible than keyboard entry when it comes to recording steps in a mathematical computation.

Qalmlea said...

I had to laugh at this...because I've seen the exact same z/2 confusion in some of my students.

Mr. Lucchese said...

As a holder of a degree in architecture, I can tell you that lettering is in many ways a lost art. With all the computers and plastic templates out there, it is difficult to convince most people of the need to learn proper lettering. I write in all caps myself, although my lower case letters are half size. There is never any danger of myself or anyone else struggling to read my writing. I will even print when scripting Russian, which no native speakers ever do.

Obviously, the era of texting and typing has limited the amount of time these student practice writing by hand. If it were up to me, rather than spend years in elementary school teaching cursive handwriting, I would instead spend that time perfecting printed lettering. I would certainly include numbers, superscripts and subscripts, Greek characters, and other common math notations. In typing classes, it is common for students to type strings of gibberish to test themselves and I think we ought to try to same thing with hand lettering. A first of second grader doesn't need to understand the complicated string of mathematical notation before them, they only need to be able to accurately replicate it. I think something like this might circumvent the Z/2 problem.

(Historical not: The character Z was commonly used as 2 right up until mechanical printing presses gained popularity.)

Lifewish said...

Hey, you think it's bad trying to distinguish your students' greek letters? Imagine how bad it is for a student if your lecturer does them all the same. Now imagine it's your Vector Calculus lecturer. Now scream quietly.

Zeno said...

Omigawd, Lifewish. That's not cause for quiet screaming. That's cause for loud screaming. I hope you survived the experience.

Anonymous said...

Thanks for another post that had me laughing throughout.

Anonymous said...

I am an European, but I honestly wouldn't see a Z at the places you annotated that the 2 was indeed a Z. And about the Greek letters: I sympathise with you. I'm just a student, but some of my letters have to be terrible, although I doubt if any of my theta's have been interpreted as a phi.

TabAtkins said...

Old thread, but I migrated here from a Sunclipse thread.

I forced myself to use the slashed z specifically because I couldn't tell the difference between my Zs and my 2s. Cal3 taught me the importance of disambiguation.

Similarly, I now do Us as a single curve without a downstroke, ever since being introduced to the du notation for integration by parts. Otherwise I can't tell it from my Ns and Ses. Strangely, my 5s used to fall into the same handwriting equivalence class, until I adopted my wife's practice of drawing them in two strokes, with the body coming first and a strong cap finishing them off.

Anonymous said...

My grad school Algebra professor used to hand-write all his assignments for us. I recall one assignment in which one of the problems featured a matrix which had 3's all down the main diagonal. It took me hours of utter incomprehension (not unusual for me in grad school, sadly, so I didn't think anything of it) before I realized they were all very poorly drawn xi's. Upon which I remarked, "Oh! I get it now," stretching the truth somewhat... I also remember feeling very sad by the further realization that numbers had qualitatively less inherent meaning for me than Greek letters. A strange moment indeed