One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”:

1a+b“=”1a+1b1a+b“=”1a+1b

2−3“=”−232−3“=”−23

sin(5x+3y)“=”sin5x+sin3ysin(5x+3y)“=”sin5x+sin3y

and so on. Slightly more precisely, I’d call it the tendency to commute or distribute operations through each other. They don’t notice that they’re doing anything, except for operations where they’ve specifically learned not to do so.

**Does anyone have a good cure for this — a particularly clear and memorable explanation that will stick with students?**

I’ve tried explaining it several ways, but never found an approach that I was really happy with, from a pedagogical point of view.

I want to point out that two issues should be separated when talking about what students know:

None of them implies the other. Native speakers correctly apply grammatical “rules” that they have never heard of to invented words because the brain can extract rules from a huge number of examples. People can memorize the meaning of the letters of another alphabet (Russian, Greek, …) in a very short time, but this does not enable them to read known words in the other alphabet with reasonable speed.

I certainly agree with teaching students, meaning, understanding and context, but if you want them to calculate efficiently and reliably, it cannot be avoided that they do a certain significant amount of computations themselves to give their brains a chance to automatize the routine. (And if they do not care about the results of the computations, it will take much, much longer.)

The mere fact that people over-apply patterns to new situations is not something that I find disturbing at all. It is exactly what I want students to do when I introduce matrix exponentials. The goal is to be able to switch between routine mode and reflection mode.

I would like to be more fancy, since you all seem fancy, but I taught adult literacy for a few years. Adults with 1st – 5th grade math level coming in to try and get their GED.

Cut up a circular pizza into 1/21/2 and 1/31/3 each, and then have them cut up a pizza into 1/51/5. They will then intuitively get that 1/2+1/3=′1/51/2+1/3=′1/5, because that’s way less pizza.

Then you can do the same with the numerator to show that 2/5+3/5=5/52/5+3/5=5/5 a whole pizza.

In two years of teaching that class, my most powerful techniques bar none were pizzas and dollars. Even the most self-proclaimed math illiterate will learn percentages when there’s a sale going on.

The prevailing attitude is “I just need to fudge the numbers around until it looks like the answer”. This can basically be attributed to two causes:

The latter is easily solvable with a few hours of tutoring, but ultimately the former seems more prevalent. To most of these students, it’s all just a list of formulas that they have to memorize for no apparent reason, followed by busywork applying the same formulas mindlessly a few dozen times every other night.

The only reliable way to generate interest in a subject is for it to have immediately obvious benefits to the student.

For things like factoring, commutativity/associativity etc, there is no direct benefit – most of the time, in the real world you can compute the value of an expression exactly as it’s written (if I have a 3×4 and a 2×4 flat of soda cans, why would I bother rearranging it into 4 rows of 5 cans before counting them?).

The benefit to the student lies in being able to use these manipulations to create their own formulas that can be used as shortcuts for boring and repetitive tasks in the future. In other words, it needs to be clear to them that the time invested in learning/memorizing concepts and formulas will be paid off with interest in laziness/time saved in the future.

Once a student is genuinely interested in learning concepts and is able to tie them to real-world examples, they then have a vested interest in sanity checking that what they’re writing makes sense – otherwise they are just shooting themselves in the foot.

In the examples you cited, “numerators” are subject to “linearity” but “denominators” are not.

For instance,

a+bc“=”ac+bca+bc“=”ac+bc

is true, but

1a+b“=”1a+1b1a+b“=”1a+1b

is not.

And

2−3“=”1/232−3“=”1/23

, meaning that once you put

2323

in the denominator, the linear relationship breaks down.

Once I learned that expressions are linear in numerators but not in denominators, it was a big step forward for me.

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”:

1a+b“=”1a+1b1a+b“=”1a+1b

2−3“=”−232−3“=”−23

sin(5x+3y)“=”sin5x+sin3ysin(5x+3y)“=”sin5x+sin3y

and so on.

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