The old saw, “a picture is worth a thousand words”, applies to teaching mathematics and the other STEM disciplines as much as anything, and I think most STEM teachers are reasonably aware of that. But there’s a discipline called graphic communication that covers far more than that one idea, and teachers seem much less aware of the rest of it. Many principles of graphic communication apply even when there’s nothing like a picture in the normal sense, and that’s what I mean by “one-tenth of a picture”: using graphic elements in text, formulas, etc. Here are a few examples and guidelines from my own experience: some from classroom use, some from other situations. For a great deal more on how to use graphical elements of many kinds to convey information, see Edward Tufte’s well-known books (http://www.edwardtufte.com/tufte/books_vdqi). (However, to my knowledge, the “tenth of a picture” phrase is my own.)
1. Showing Grouping
Show grouping explicitly. As a student teacher, to help my 6th-graders remember the order of operations in evaluating complex expressions, I taught the mnemonic “PEMDAS” (or “Please Excuse My Dear Aunt Sally”), just as my mentor did: first handle all Parentheses, working left to right; then all Exponents, Multiplication, Division, Addition, and Subtraction, in that order, working left to right in each case. But, of course, that’s wrong! Multiplication and division are “on the same level”:
3 * 6 / 4 * 2 = (18 / 4) * 2 , not 18 / (4 * 2)
…and likewise addition and subtraction. I wasn’t surprised to find many students trying to do all multiplication before any division, and all addition before any subtraction. Then it occurred to me to have them use “PE(MD)(AS)” instead; the parentheses helped quite a bit.
2. Presentation of Numbers: Needless Digits, Words, Etc.
Be careful writing numbers out in words. Here’s an excerpt from a newspaper article on the “Obamacare” health care reform law, referring to a poll: “Forty-one percent said they expect it to be fully implemented with minor changes, while 31 percent said they expect to see it take effect with major changes. Only 11 percent said they think it will be implemented as passed.” Saying “41 percent” instead of “forty-one percent” would have made it easier to compare the three numbers given, as well as to see that the total percent (83) falls well short of 100! On the other hand, writing, say, “3 reasons to use this method” instead of “three reasons to use this method” just looks silly. A common and sensible rule is to use the word form only for whole numbers of one digit; but even then, it should be used thoughtfully. In particular, don’t switch back and forth between written-out numbers and digits needlessly.
Use scientific notation thoughtfully. Similarly, don’t switch back and forth between standard and scientific notation needlessly; and…
Use an appropriate number of significant figures. The usual problem here is obscuring important information by showing far more digits than necessary. This is a common mistake, and one reason is undoubtedly that many computer programs and programming languages give numeric values to maximum precision by default.
Here’s some of the output from a Java program that computed pi to 5 decimal places.
pi = 3.1405926 (error=-9.99999749998981E-4) prevPi = 3.1425936 (error=0.001001000750251002) (pi+prevPi)/2 = 3.1415931 (error=5.005001257885056E-7)
It’s not exactly obvious that the magnitude of the error in the second line differs from that in the first line by less than 1 percent! I think you’ll agree that giving the error values in the following way makes them far more readable:
pi = 3.1405926 (error=-0.001) prevPi = 3.1425936 (error=0.0010010) (pi+prevPi)/2 = 3.1415931 (error=0.0000005)
3. Lists and Tables of Anything: Typography
Use any eye-catching typographic distinction—boldface, italic, color, size, etc.—to emphasize extreme or otherwise interesting values. This applies to any kind of data, not just tables where the cells contain numbers. Here’s an example, from a class gradebook of mine. Low scores are in red bold italics; high scores are in green bold italics, underlined. I added underlining to them because green isn’t that eye-catching.
4. Lists and Tables of Anything: Text Graphics and Semi-Histograms
In unformatted text (as in a .txt file or some email, etc.) neither real graphics nor typographic distinctions are available, but there’s still room for a small fraction of a picture! You can often add useful graphics with text characters. Here’s an example. The numbers are an imaginary type-1 diabetic’s blood A1C levels, which are typically checked every few months; a value of 7 or less is very good, one of over 9 not good at all.
Key: * = over 7, ** = over 8, *** = over 9 2007 April 9.1 *** June 7.5 * Sept. 7.8 * Dec. 9.1 *** 2008 March 7.8 * June 8.3 ** Sept. 7.1 * Dec. 7.5 * 2009 Mar. 6.8 June 7.4 * Aug. 7.3 * Dec. 7.1 * 2010 June 7.3 * Sept. 7.7 * Dec. 7.5 * 2011 Mar. 7.8 * June 8.4 **
Another situation is where you want to group similar data values together, normally arranging the groups monotonically by their values. This would ordinarily call for a histogram, but what I call (though only since yesterday!) semi-histograms may be more practical than histograms, or preferable for other reasons. Stem-and-leaf plots are a great example. Here’s a stem-and-leaf plot of the numbers from the third column of the fancy table above, the “Exam 4 (of 30)” column:
0 | 0 1 | 2 8 8 9 2 | 1 1 2 2 3 3 5 8 8 8 9 3 | 0 0 0 0
The lefthand column contains the 10’s digits; to the right of each is a list of the 1’s digits of values with that 10’s digit. Thus, the second row contains the values 12, 18, 18, and 19. This data isn’t ideal for a stem-and-leaf plot. Still, note how it lets you see the most and least populous intervals at a glance (like a histogram), and shows the exact values (which ordinary histograms can’t show) simultaneously. And, of course, stem-and-leaf plots can be used in unformatted text.
(revised 22 August 2013)