Republic of Mathematics blog

Reducing cognitive load in algebra

Posted by: Gary Ernest Davis on: January 13, 2010

Many years ago I used to get very annoyed with John Sweller, an Australian academic who aggressively pushed his idea of cognitive load as a panacea for problem solving in mathematics. John upset a lot of people in the Australian mathematics education community because they felt he was talking about routine problems, and not what they saw as problem solving.

I have since come to understand better that John’s theory of cognitive load has to do with working memory, the very real constraints on it, and the means that he described and analyzed to reduce the load on working memory in cognitively difficult situations.

So today I am to sing the praises of John  Sweller and his cognitive load theory.

What stimulated me to do this  is the following recent Tweet:

@Educator Just uploaded 60+ free examples of Algebra 1 onto YouTube! http://ow.ly/Vt9k #TUalg1

I went to the site and looked at a few of the algebra videos advertised.

Folks, I have to say this: I was not happy.

The “follow-me, do as I do” style of presentation of algebra is something I do not like.

I thought about this as I drove my wife to the train at 6:30 this morning, and agonized over whether I should write this post.  After all, who wants to be negative about colleagues who are trying hard to educate kids?

The folks at Educator.com seem to have their hearts in the right place, and they are doing a great job getting this site up and running. More power to them.  On Twitter, @Educator describes Educator.com as a: “Pioneering E-Learning Startup aimed to equalize education. Specializing in Algebra, Biology, Calculus, Chemistry, Java, Physics, Statistics, Trigonometry.”

I had decided to shelve writing about this until I could put my money where my mouth is and show how to do better. After all, criticism is easy; getting out there and do something creative and productive is hard, and that’s what Educator.com is doing. I admire them for that.

But I looked at the video on the distributive property in algebra, thought of John Sweller and cognitive load, and … here we are.

The video that got me thinking about cognitive load is this one:

The video set up has a women explaining a procedure for expanding a polynomial expression (the screen states this is “simplifying” the expression. In my considered mathematical view the final result is less simple than the expression at the beginning).

The task is to simplify (by which they mean “expand”):

-3y^2(4y^3-6y+7)

Looking at this – already simplified – expression, and thinking about cognitive load, what I notice right away is that pesky -3y^2 sitting outside the parentheses.

I know, as an experienced mathematician and educator, that  the combination of the “-” sign, the number “3” and the y^2 term is likely to cause a problem for many students. The “-” sign and the y^2 term are particularly apt, even on their own, to cause difficulties, and together they are much more likely to do so.

For a kid who is just learning this – and therefore watching this explanatory video –  there’s a lot to bring to mind and to keep in working memory.

How might we reduce the cognitive load of this expansion?

First off,  what have we got here? We have two polynomials 3y^2 and 4y^3-6y+7 that we have to multiply out, using the distributive property.  Right away we notice that the first polynomial is a monomial – it is just a number times y^2.

Let’s focus on just the y^2 part first, so reducing the number of things of which we have to keep track.  The index laws – so nicely explained in the video – tell us that multiplying a polynomial in y by y^2 will simply raise the powers of y by 2.

If we do not do this simpler step  first we run the risk of overloading our working memory with the “-” sign and the number 3.  This is what Sweller urges us to avoid. And sure enough the person talking in the video forgets to write the y term in the first multiplication! Dang cognitive load went and got her!

A less cognitively demanding first pass at this expansion would be something like:

-3y^2(4y^3-6y+7) = -3(4y^5-6y^3+7y^2)

Having got that far we can forget all about the y^2 term and focus on the “-3” term. Working memory is feeling relieved!

But we know how often we can get those pesky negative number multiplications wrong, so let’s just distribute the “3” over the new polynomial first:

-3(4y^5-6y^3+7y^2)= -(12y^5-18y^3+21y^2)

OK! that was easy enough. Now to deal with the pesky “-” sign. Well, multiplying by -1 changes all positive coefficients to negatives and negatives to positives, so:

-(12y^5-18y^3+21y^2)=-12y^5+18y^3-21y^2

So, I sing the praises of John Sweller. He was right. I misunderstood him, and I think mathematics educators need to pay real close attention to reducing cognitive load in mathematics problems.

Educator.com is doing a good job. I congratulate them. They could be doing a great job, positively impacting millions of young students.

9 Responses to "Reducing cognitive load in algebra"

Great post. I had to send it to my uni lecturer who taught us about Sweller in Educational Psychology.

Bravo! Great post. Excellent example of how to reduce demands on working memory. I will send links to all our math folks.

Great post, Gary! I can’t wait to read more about Sweller’s approach.

Great post, Gary! I’d like to read more about it . . . future posts please?

I agree completely with your perspective, and would add one organizational step and/or visual simplification to your example. Rewrite your “pesky” -3y^2 factor using parentheses or dots to signify multiplication to break up the monomial into the chunks that will be dealt with one at a time next:
(-1) (3) (y^2) (…)

While adding one step makes the process a bit longer yet, I have often found that rewriting multi-factor terms with each factor separated by dots, or with each factor in parentheses, makes it easier for students to “see” the chunks of the problem they are about to tackle and feel less confused about where/how to start the problem.

Once they have more experience planning out this sort of process, they will be more comfortable doing it all in their heads, and they can drop the rewriting step and do it the way most of us probably do it when we are on autopilot.

Whit, your point is well-taken. I’m reminded of a colleague in Chicago, some of whose students had difficulty with operatons on fractions. One day her black board marker ran dry and she picked up a red marker to make the division sign in the fraction she was writing. The students’ eyes light up and they could do the problems – the visual cues of the division sign had not ben interpreted by their brains in the way she expected. Vision, after all, is intelligence.

So, I agree with you about adding parentheses to further clarify expressions. It is notable, to me, that the great John von Neumann was renowned for his “onions” of parentheses – it was apparently how his brain organized mathematical expressions.

I’ve just found your blog and enjoy it and will read more. I like how you broke down the distributive property into parts for clarity. But, the original expression is ‘factored,’ not simplified. And ‘expand’ refers to expressions raised to a power, not the distributive property. The final expression is indeed simplified.

Thanks for this Neil.

“Expand” is in fact used by many CAS’s to do just what I wrote, and not only (or even) raise to a power.

Writing a polynomial in standard = (expanded) form is not simpler from the point of view of knowing the roots.

Of course “simplify” has an unstated intent, which ought to have been stated!

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