# Caveat Area Model

This discussion is part of a collection:Rectangular Arrays

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.

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## Replies

I take note of the both techniques vary from the conventional calculation that does not demonstrate fractional items, and that is OK. The kispiration was genuinely great in utilizing right math dialect, while the second vid was somewhat less formal. Neither said that the techniques depend on an expansion of the distributive property, nor did they encode the operations that way. Additionally, these I would group as exhibit models and not zone models. In the case of utilizing region models, make sure to have understudies tell the measurements of every segment rectangle – some experience difficulty exchanging direct measures over the outline.

[This post has been modified by mathadmin]

One way to help students who struggle with the array is to show them various models; they may relate better to some then others.
I found this youtube video that demonstrates how to use kidspiration to model multiplication, this could easily be done with base ten blocks but is nicely done here to find the partial products. http://www.youtube.com/watch?v=mjYYbwuued0
Another youtube video demonstrates how to use the area model without manipulatives, http://www.youtube.com/watch?v=8LkEBiq3Xsg; it may be easier for some students to see the numbers rather than the "picture".
I know that personally when I was assigned to teach in a school that used the lattice method of multiplication I was thrown, because it was not the way I had learned. Through this experience I found that showing students a variety of methods and allowing them to choose the one that makes sense for them is really the best for everyone. As an educator it is my responsibility to be familiar with a variety of methods and to expose the students to each of them, but I should not force them to use a model that they are not comfortable with.

I note the both methods differ from the traditional algorithm that does not show partial products, and that's OK. The kispiration was fairly good in using correct math language, whereas the second vid was a little less formal. Neither mentioned that the methods are based on an extension of the distributive property, nor did they encode the ops that way. Also, these I would classify as array models and not area models. If using area models, be sure to have students tell the dimensions of each component rectangle – some have trouble transferring linear measures across the diagram.