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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”:

$$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$

$$ 2^{-3} \mathrel{\text{“=”}} -2^3 $$

$$ \sin (5x + 3y) \mathrel{\text{“=”}} \sin 5x + \sin 3y$$

and so on. Slightly more precisely, I’d call it: the tendency to commute or distribute operations through each other, without even noticing 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.

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Really looking forward to the responses. My own attempts have involved providing counterexamples and demonstrating the correct "formula" (if applicable). While students can see the mistake in a particular instance once I point it out, it doesn't seem to stick and many repeat similar errors. –  Kelvin Soh Jan 7 at 16:27
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Perhaps the answer in the body of you question: maybe you should introduce them the “Law of Universal Linearity”. In my opinion, it's a funny "law" and students(age dependent) love funny stuff, it helps them, in some sort, remembering things.(Edit: You should explain then, the basic concept of linearity first.) –  Salech Alhasov Jan 7 at 17:07
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Give a set of such questions in exams and weight them >60%. Should cure lots of students. –  Xiaoge Su Jan 7 at 18:07
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I'm curious if you're seeing this in all students, or perhaps mostly in ones that are, say, visual learners, or not visual learners. I mention this because the actual symbols used to describe these equations make the distinction of value very subtle. –  Nathaniel Ford Jan 7 at 18:44
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You left out the famous "freshman's dream", $(a+b)^2 “=” a^2 + b^2$! –  MJD Jan 8 at 2:13

32 Answers 32

I think several answers have touched on this, but haven't stated it directly --

One of the features that defines human intelligence (and which has allowed humans to become the pervasive force on the face of the earth) is the ability to form inferences -- to extrapolate from observation to hypothesis. This is innate, and we would not be humans without it, and it's nonsensical to expect students to not employ the technique.

However, in many scenarios (social, political, economic, etc -- not just mathematical) it's possible to jump too quickly from inference to conclusion, bypassing experiment/analysis, and end up being just plain wrong. This in fact happens all the time. (Heck, it probably happened between you and your wife yesterday.) But this is not a reason to stop using such a powerful tool. Rather, students (and non-students) need to be taught (or learn from sometimes bitter experience) that not all inferences are correct, and that while an inference can "inform" a subsequent experiment or analysis, one needs to tread lightly before jumping from inference to (presumed valid) assumption without the intervening experiment/analysis step.

For the OP's situation I would first say, "Lighten up!" This is human nature, don't take it quite so personally! After that it might actually be worthwhile to discuss this aspect of human intelligence with the class in general terms, without direct linkage to mathematics, giving some examples of good and bad inferences from other aspects of life. This might help the students understand that the issue is not about some rigid oddity of mathematics but is about "life skills" that can be applied everywhere.

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It comes from a fundamental misunderstanding of order of operations and their implication. I would attempt to re-teach order of operations, associativity, distributivity, etc. by using new symbols like ✧ and starting from first principles.

Does a ✧ b = b ✧ a

Does (a ✧ b) ∏ c = a ∏ c ✧ b ∏ c

Lead them down the implications of each decision; the contradictions that will crop up with loose rules; the benefits and shortcuts provided by certain choices. Follow through creating a consistent system of operations until analogs are created for the major operators (+ - * / ^ ()). Once the entire system is constructed, map them to our traditional operators by showing which ones apply.

The problem is people are lazy. If they think they can guess and maybe get it right, many will just go with their gut and will never learn there is a system in place and that there are good reasons for it. If you destroy their comfort zone by dealing with abstractions only, you open up their mind to learning instead of guessing.

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protected by robjohn Jan 8 at 11:46

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