Five Objections to teaching the Area Model in Schools

The "area model" for teaching multiplication to students is a hot discussion topic right now. Here are five deep concerns.

[markjames]

"We adults tend to equate familiarity with understanding." So well said!

[RoomtoDiscover]

Finally someone summed up the responses to objections to Common Core in a handy video! Now instead of writing this all out again and again in math forums, I can just link to this video - thanks!

[tringlewitt]

I teach the box method for multiplying and dividing polynomials. It is able to reach so many more students.

[daphneclose]

Hi James! I just saw you teach Exploding Dots at the Global Math Week Kickoff Symposium. When I first saw this video, I was worried that I shouldn't teach area model. I'm so glad the objections are not yours. FYI, next week I'll be teaching this to my fourth graders and after seeing Exploding Dots, I want to change my future lessons to teach Exploding Dots! Thank you so much!

[lacielittle]

This is a great way to explain the "why" behind how we are teaching students!

[arpentnourricier]

Another interesting side-effect (thinking with an engineer's mind) is that when you draw your sub-rectangles to (rough) scale, it becomes visually obvious which are the big chunks and which are the crumbles that can be left out. In a day and age where computers can do the exact computation, I feel it's the right time to teach students approximations and orders of magnitude and not clog their brains with the minutiae (math is infamous in this respect, at least the way I was taught).

[jerklecirque]

"The goal here is not to just answer integer multiplication problems, but to give us structure that goes beyond." I think you can address many of the complaints leveled at Common Core Math with a variation on this argument.

[christopherhorn]

I'm 40 years old, and you just made multiplication easier for me. Mind blown.

[robbesrh]

I completely agree with you, James. The area model is an extremely useful conceptualization of what is actually going on when we multiply numbers. Isn’t this how the Ancient Greeks regarded numerical operations? If that’s true — and I think it is — that’s an impressive pedigree for the area model. Also, I think it’s helpful to remember that mathematics originally grew out of geometry. I detest memorizing tricks and shortcuts. I want to understand it. In my book, that’s the only thing that counts. The fruits of learning are often transitory and provisional. Depth and understanding are the nectar of Gods.

[patsullivan]

I love the problem he used because it can be easily computed mentally if we see the structure (30 + 7)(30 - 7). 30^2 - 7^2 so it is 900-49=851. Love the connection to difference of squares in algebra.

[waynemv]

A geometric model can help one to intuitively understand how and why the "difference of two squares" identity works:

x² - y² = ( x+y )×( x-y )

I've also found it useful to have memorized perfect squares up to at least 25×25, and as far beyond that as I can.

Here's how one might calculate 37×23 (the first problem from this video) in one's head in a mere couple seconds.

37×23 = (30+7)×(30-7) = 30² - 7² = 900 - 49 = 851

While understanding trumps memorization, both are worthwhile, and mighty powerful in combination.

[KartikSoneji]

Great video. I think that a good way of transitioning from the area method to long multiplication will be to show how impractical it is while multiplying 3 or 4 digit numbers.

[donaldrogers]

I don't mind the area approach, it's a visual representation of what we were taught using the "old" way.  I'm a bit disappointed that he dismisses the old way suggesting that we didn't understand or that it was complicated.  We used to be taught a lot of the proofs for math and where they came from and then would apply the rules from the proofs and alternate methods were still accepted.  Like they still say today, so long as you show your work and it makes sense you still get the marks.

[mynameisZhenyaArt_]

I wish I had You as my second math teacher in school (first was also great, I just needed more math :) )

[samisiddiqi]

I am actually conflicted by whether or not I should teach them the Area Model, the Egyptian Model, or the Scaling Model... All of them are good...

[petedaley]

The area model adds a visual/spacial complication to the problem. Why not simply decompose one of the numbers into tens and ones? Multiply 37 x 20, then 37 x 3, then add the results? You still reinforce place value and don't rely on the standard algorithm. "Hindsight Teaching" (i.e. a teaching method that makes a lot of sense to me know, so darn it why wasn't I taught that way when I was in school?) is very attractive though...

[kurtbrayford]

speed/tradesman quoting a house that needs painting, of course some one like me can just look at an house and knows how much the cost will be, lol

[nadeemleon]

Sir I requested many times and kindly asked you can you help me with calculus make it really easy perhaps

[peterosudar]

If we used Quadrea to solidify are understanding of "area" then these "objections" would be a nice reminder of why pure mathematics helped society to move into the 21at century... where rigour, truth, precision, validity, and compact notation help youngsters(and youngsters at heart) to feel confident in the mathematics they do everyday, every moment, thought by rigorous thoughts.
Check out Norman Wildberger:

Yes I am aware that it can be a heavy task to be open to new ways of thinking but if they are true... they will come to fruition naturally... Why did Exploding Dots work so well? Probably because the spirit of the story enabled many people to share it...

I see the same thing soon to happen with Rational Trigonometry... it is only a matter of time.

Feel free to disagree... I look forward to you convincing me I'm wrong.

[alfiogabriele]

what if the multiplication is 137x221? Cannot find an easy way to use the Area Model, when the old standard mode works well for any number of digit