The adage states that a picture is worth a thousand words. In math education this old saw translates to “making a diagram should make all this (distributive multiplication, variable binomial multiplication and factoring, etc.) clear as a bell;” however, I agree with the need for caution and care in teaching with arrays as expressed in the collection description above. I’d like to add some anecdotal evidence.
First, I relate a little known tale about Sir Isaac Newton, inventor of the calculus. In 1637 Rene DesCartes published a new system of charting functions using coordinate pairs. The concept was so novel that Newton, at first reading, did not understand it. The matter became clear only upon his reading of a commentary by Van Schooten (let’s call him the teacher here).*
Second, if older students and adults fully understood the relationship between area and multiplication as depicted in continuous arrays, then we would be a nation of algebraists, which we are clearly not.
I think the resources in this collection which come from nrich in the UK and from NCTM’s Illuminations do propound the kind of careful teaching that using these models requires. I take exception to the treatment given by the Wolfram resource, especially the oversimplifications in its sparse commentary, to wit:
“Area is width times height.” [Yes, and our high schoolers apply that formula to triangles and trapezoids too, and ...] “Select from the pull-down list of products to see the tallest corresponding area.” ** [This association of adjective and noun might be a first ever.]
If you too have students who struggle with arrays, what problems have you identified and what have you tried that brings about progress? Lots of people should be interested.
*Dunham, The Mathematical Universe. Wiley, NY, 1994. pp. 274-275.
**Wolfram Demonstrations Project - Publisher, Michael Schreiber - Author.