A common complaint–as in “see, math is being taught wrong”– is that students fail to see that equations like 3(x-5)=60 can be solved by dividing both sides by 3 first. Progressives seem to make a big deal about this to the tune of “If students are doing this, they lack ‘deeper understanding’ about equations.

Textbooks that claim alignment to the Common Core now make it a practice to show this.  The problem is that if you have 7(x-5)=60, the process isn’t so neat.

In my experience students ignore the lesson and go ahead and distribute.  I point out that they can do it the short-cut way (since all the problems in this particular lesson are structured so that the short cut can be used), but they still do the “long-way” distribution method. At first I was worried when students were not “getting” it, until I realized I was succumbing to the embedded “deeper understanding” aspect of teaching it this way. I caught myself so now I don’t make a big deal out of it. I tell them to use whichever way they find easiest for them.

Students just learning algebraic rules do not readily see (x-5) as a single entity, and so they also fail to see 3(x-5) as something like 3A, where the three can be divided to undo the multiplication.

Says Robert Craigen, Math Professor at U of Manitoba:

” When students distribute first, is that really a sign of lack of understanding?  Why not instead see it as “showing understanding of the operations”?  Why isn’t cancelling the 3 being a math zombie?  These guys aren’t consistent at all.  Either approach might signal understanding, and either one might be the result of mindless application of mechanical rules.  I would rather focus on whether students are correctly performing the steps and can deal with variant problems. With experience comes fuller understanding.  Initially they only need enough understanding to avoid making obvious errors.  The early goals should be fluency and functional (not deep) levels of understanding.”

To which I say, “Amen”.

1. Chester Draws

The problem is that if you have 7(x-5)=60, the process isn’t so neat.

But at least possible.

When they get x(x – 5) = 50, then they have no option but to expand.

So I teach students, explicitly, that they should always expand brackets and denominators before proceeding. Yes, there are a few problems where it is slightly slower, but given where they are going they’re much better to distribute out every time. Less thinking about mechanics gives more time to think about solving.

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2. SteveH

I never really learned algebra until Algebra II. There were lots of problems where I could not “see” the fastest or alternative approaches. That comes with practice and lots of individual homework. Do they offer any specific curriculum and process (other than slow down and take more time) to do a better job? What is full understanding and how do they test for it? Do they have any proof that what they do works?

Back when I taught college algebra, I came up with a method to get students to “see” an equation by looking for the “=” sign and looking at the rest as a summation of rational terms filled with factors and exponents in the numerator and denominator. I had them create rational terms, circle each term and factor and put in an exponent (1) if it was not there. Identities and rules that seem easy become confusing as the equations become more complex. If they can see the factors and exponents, then math becomes easier.

As problems become more advanced and tricky, like logarithms on the AMC/12 test, it’s very easy to mislead very good math students. The best way for my son to prepare for the AMC test was to do as many of their past test problems as possible. There might be a small amount of general problem solving transference, but in the vast majority of problems, the best help is to see something like that problem before.

Like digit problems such as, “The digits of a two digit number add to 12 and multiply to 12 less than the number. What is the number?”, what transference other than to start creating variables and trying to find ‘N’ equations and ‘N’ unknowns works? We all knew how to do that, and it was often not a big help. What if you didn’t “see” a radius in a problem to create the last equation? We could talk about how some questions are not fair tests of understanding.

As for 3(x-5) = 60, solving it any which way shows understanding. Solve enough problem variations correctly at that course level, then everything is fine.

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• Stephanie Sawyer

So I am older than PEMDAS, and I have always taught my students to “see” terms vs factors. That said, about half of my honors algebra I will “see” that they can divide the 7 first, the other half will distribute.
Sixes basically. In the long run, who cares? As long as they all can successfully apply the rules we are all good, AS long as they get “the rules”.
I always tell them if they are making up “rules” they are probably wrong!

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• BGarelick

Tell that to the progressives.

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