Pedagogy: How to cure students of the "law of universal linearity"?

Question

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. 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.

Answer 1

I think this is a symptom of how students are taught basic algebra. Rather than being told explicit axioms like $a(x+y)= ax+ay$ and theorems like $(x+y)/a = x/a+y/a,$ students are bombarded with examples of how these axioms/theorems are used, without ever being explicitly told: hey, here's a new rule you're allowed to use from now on. So they just kind of wing it. They learn to guess.

So the solution, really, is to teach the material properly. Make it clear that $a(x+y)=ax+ay$ is a truth (perhaps derive it from a geometric argument). Then make it clear how to use such truths: for example, we can deduce that $3 \times (5+1) = (3 \times 5) + (3 \times 1)$. We can also deduce that $x(x^2+1) = xx^2 + x 1$. Then make it clear how to use those truths. For example, if we have an expression possessing $x(x^2+1)$ as a subexpression, we're allowed to replace this subexpression by $x x^2 + x 1.$ The new expression obtained in this way is guaranteed to equal the original, because we replaced a subexpression with an equal subexpression.

Perhaps have a cheat-sheet online, of all the truths students are allowed to use so far, which is updated with more truths as the class progresses.

I think that, if you teach in this way, students will learn to trust that if a rule (truth, whatever) hasn't been explicitly written down, then its either false, or at the very least, not strictly necessary to solve the problems at hand. This should cure most instances of universal linearity.

Answer 2

My interaction with the students (non-mathematically inclined ones, in the United States) has lead me to suspect that for some reason they are not taught the following two crucial ideas.

1. Mathematical expressions have meaning.
2. The validity of a rule for manipulating mathematical expressions is determined by what those expressions mean. In particular, the rules themselves are derived from what the expressions mean.

Understanding of those two ideas seems to me is the key difference between the students who "get it" (i.e., the ones that simply do things correctly and their mistakes usually boil down to not noticing something) versus the one who don't "get it" (which do only as well as they can memorize a bunch of boring, arbitrary-seeming rules).

Consequently, I am of the opinion that searching for

a particularly clear and memorable explanation [of when a particular rule is applicable] that will stick with students

is not at all the correct approach to this issue (unfortunately, given the structure and expectations of the educational systems . The sad reality (as I perceive it) is that for the majority of (U.S.) students, mathematics is the art of manipulating weird, meaningless strings of symbols according to equally weird and exception-filled rules that they barely have the mental capacity to remember. In essence, the kind of content that students appear to be taught seems much more appropriate for simple-minded, inhumanly precise computer, than a human being with the capacity to reason.

This is why no matter how much we illustrate and explain the rules to them, they keep misusing them: what they are missing is not explanations or illustrations, but the ability and mental habit of determining on their own whether the mathematics they are doing is correct or not (which is still hard for a computer: computer proof assistants are still in their infancy).

I personally have no idea how such this skill of doing mathematics right can be cultivated without awareness of the two facts above, and I believe that what separates the students who do demonstrate this skill is that they have (at least an implicit) understanding of those two ideas. Furthermore, I do believe that exposing them to, making them think about, and making them use the meanings of the symbols they write, and doing it again, and again, and again, and again, will have a much more significant effect than reminding them of one-off examples and illustrations of why a particular manipulation they did is not allowed. The one-offs they will forget and not be able to reproduce because of their infrequency, but the repeated insistence on using the meaning of the expressions to establish the validity of the manipulations will hopefully make it habitual for them.

In terms of implementing this in practice, I think that college is way too late, and also quite difficult because college math (and STEM) courses tend to be mostly about transmitting massive amounts of boring technical content and technical skills, leaving little to no room for actual ideas or ways of thinking. Nevertheless, I do think it would be an interesting experiment to have students keep something akin to a "vocabulary notebook" where they record the meaning (as opposed to the formal definition) of the various kinds of expressions they run in to. For example, a fraction $\frac ab$ is supposed to mean "a number which when multiplied by $b$ gives $a$"; it is short and illuminating work to figure out from this (using distributivity of multiplication over addition, which we definitely want numbers to satisfy) that $\frac ab+\frac cd=\frac{ad+bc}{bd}$, that there is no number meant by $\frac a0$, and that $\frac00$ can mean any number). This of course, presupposes that somebody takes the time and makes sure that the language in which these meanings are explained is coherent, so it would be a lot of work to design a course around this method.

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I did in fact once successfully disabuse a(n Honors Calculus) student of "the Law of Universal Linearity" using these ideas. The particular instance concerned manipulating the Fibonacci sequence, and the student had made the error of writing something like $F_x+F_x=F_{2x}$. What I did is explain the stuff above and had the student apply them by analyzing the meaning of the various expressions he had written down was, and then ask whether that equality was justified based on what he knew the expressions meant. That seemed to make an impression on the student, but I personally believe it was an impression made ten years too late...

Answer 3

(This is a rather "soft" answer!)

I don't think there is a solution to this.

In my experience the problem is that math beginners don't understand / assimilate formal laws: they agree that $(a + b)^2 \neq a^2 + b^2$ (because "$2ab$ is missing") but they have no problem writing $(x + 3)^2 = x^2 + 3^2$ two minutes later.

The only "solution" is to take money from them / hit them every time they use the "law of universal linearity", but it takes years to have any effect (and earns you thousands of dollars)

Answer 4

I had a teacher in college who was very fond of repeating phrases like "The Flarn of the Klarp is the Klarp of the Flarn" and "The Flarn of the Klarp is the Twarble of the Flarn." I believe these are from Lewis Carroll. But the way they were incorporated in lecture was like an call-and-response.

For example, the teacher might rapid-fire questions at the student audience such as "The product of the sum is the sum of the product?" followed by "The derivative of the sum is the sum of the derivative?" followed by "The product of the logs is the log of the products?" Just seeing if students would get into a pattern of saying "yes.. yes.." and then whacking them with something to think about. I can imagine this working with trig functions as well.

This teacher would also routinely use small hand-drawn pictures in place of variables like x or y. For instance, I learned about the Taylor series expansion of "e-to-the-doggie" being the sum of "doggie-to-the-n-over-n-factorial". We similarly talked about moment generating functions as "e-to-the-tree-x" with a little tree drawn where the transform variable (usually t or s) would go, and then the moment-generating function's domain was the "tree domain" since that was the independent variable there.

I know this sounds ridiculous, but boy did it work. After a few weeks of acclimating to the sheer bizarreness of it, it really started to make the concept of variables disappear. Rather than fixating on why particular weird non-number symbols like x were showing up, you had to hold onto your butt because it might be a little tulip or a fire hydrant on the test and you were supposed to solve equations and whatnot. It was like there was no time to be confused about symbols because the sheer whimsical arbitrariness of whatever the symbols might be forced you to understand how to manipulate any symbol, which was the whole point.

This was for a first course in calculus-based probability, and eventually we started talking about things like variance, which then naturally became a discussion about how Var(X) = E[X^2] - E[X]^2 is totally a kind of measurement of "non-commuting-ness" between the squaring operation and the expectation operation. So whereas E[X] is linear, (i.e. the flarn of the klarp is the klarp of the flarn), for variance this is not true unless it's a Dirac variable with no variance. For everything else, one measure of central tendency is to say "the flarn of the klarp minus the klarp of the flarn equals ..." so you know just how far off you are from those operations commuting with each other.

I'm not sure if this would work with classes where aptitudes vary considerable, or where there are time constraints to hit materials in time for a standardized test. And it certainly is weird and requires great confidence on the teacher's part (the teacher who taught this to me was a Vietnam veteran who truly didn't give a damn about what students or administration thought of him... he was a bit like the character Walter Sobchak from The Big Lebowski actually). But it seemed to be extremely effective in my class and was one of the big milestones in my own study of mathematics where I went from merely knowing how to compute things when given problem set-ups to really trying to suss out deeper connections, analogies, patterns, etc.

Answer 5

TL;DR: Teach your students that "distribution" over addition only works with multiplication over addition, and nothing else (that matters at this point in their education, at least), and maybe show examples like $(a+b)/c = a/c + b/c$ that mix different operations to make things clear.

Longer answer: I personally have thought a lot about this, and it really has to come down to the usual villain, quick, but confusing notation. We are taught addition as $a+b$ and multiplication as $a\times b$, but soon after primary, we drop the $\times$ and let $ab:=a\times b$. This places addition at a different footing than multiplication at a subconscious level for it is now an implied operation, and this leads to trip ups like the ones you mention.

When one sees distribution of multiplication over addition, like $a(b+c)=ab+ac$, it is easier to pseudo-generalize this rule to anything, like $f(x+y)=f(x)+f(y)$ or $1/(a+b)=1/a+1/b$ since they aren't mindful of the words "multiplication over addition." This is because the implicitness of of multiplication is forgotten and thus it's easy to think that distribution is a property of addition only, and therefore applies wherever there is an addition.

Of course it doesn't. For example, $(a+b)/c=a/c+b/c$ but $a/(b+c) \ne a/b+a/c$ because division over a field is only linear in the first argument, not the second, and of course, division isn't Abelian. You can't tell your students that at this point, so the best way is just to be clear of when it works in their world: multiplication over addition. For the "linear in first argument" for division, and may be use a cheat like $(a+b)/c = 1/c \times (a+b)= a/c+b/c$. At this point, since you can't teach them basic abstract algebra, you'll have to do with just keeping them straight with where distribution works, and if they are so keen, tell them they'll learn why one day.

Answer 6

$1.$ Be brutal ! Give them an F when caught red-handed in the act of perpetrating such unholy and illegal activities ! That should teach them ! :-)

$2.$ Show them nice pictures.

$3.$ Give counterexamples ! $\qquad\qquad\dfrac12=\dfrac1{1+1}\color{red}\neq\dfrac11+\dfrac11=2.$

Or just tell them to “read fractions” : $\dfrac13+\dfrac23=1$ third $+2$ thirds $=3$ thirds $=\dfrac33=1$, for the same reason that $1$ sheep $+2$ sheep $=3$ sheep.

$4.$ Tell them that $2^{-3}\neq(-2)^3$ for the same “reason” that $2^{-3}\neq2-3$.

$5.$ In short, just teach them to think, rather than rely on “magic” formulas.

Answer 7

I can't give you advice what to do against it, but I may help you understand why it is happening.

The point is that the "feeling of knowing", or "being certain", is an emotion, just like feeling sad or happy. It can also be compared to visual perception: instead of perceiving something about the state of the outside world (e.g. a blue mug on your desk), you perceive something about the state of your own cognitive processes: you came up with a piece of knowledge and it feels right.

And just like vision, it is susceptible to illusions which can completely fool your brain. Being convinced that (x+1)^2 = x^2 + 1^2 is right is very similar to being convinced that square A and square B are different shades.

The reason these illusions happen come from the way neuronal based intelligence works. Our brains are specialized at recognized similarity in patterns. If we are exposed to one pattern very frequently, it feels more "right" than other patterns. There are also other details, especially for visual illusions, which are dependent on the particular ways neurons in V1 and other perceptional areas work, but here the analogy between visual-illusion and feeling-of-knowing illusion breaks down. But the point is that feeling certain is not related to factual truth directly; it is related to noticing that the new pattern looks similar to older patterns we have come to believe are true trough repeated observation (or being repeatedly assured that they are true). The reason this works is that if we observe a pattern being true frequently enough, or if most people around us have come to recognize it as being true, it is indeed because it is true. Still, it is a matter of persuasion, not logic. Logic can make us understand something, but not make us believe in it intuitively.

So a person who lives in a world where most visible processes are described by simple linear and proportional relationships will intuitively feel that "linear" or "proportional" explanations for everything are right. This happens on a broad level, where exponential growth is completely counterintuitive and people freshly exposed to it are always surprised by the true magnitude of the calculated results even if they have cognitively understood the underlying principle. I think of myself that I should know better by now, but I still get surprised frequently.

It also happens in some specific ways, like the one you describe with math students. Your pupils have been exposed to linear relationships for years. Their neural networks have learned to react with a "this looks good" signal the way pavlov's dog's neural networks have learned to react with "food comes" signal. When they consider possible solutions, once the linear one comes up, it just feels right. Learning to ignore this inner certainty is possible, but it is a hard and slow process which physically requires rewiring the neurons in their brains. You cannot expect a silver bullet for it. Especially trying to find a way to make it better understandable won't work; they have already understood it in their higher, reasoning processes. It is their affect-level response which has to be overruled, and it responds to repeated training, not to logic.

For a better insight in how the feeling of knowing works, read "On being certain" by R. Burton. It is a great book, and I would recommend it for all pedagogues (and actually for everybody else too, but if you are interested in creating a feeling of knowing in your students, it might be especially helpful).

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Edit A way of thinking about how to solve the problem is using mental models. A mental model is an understanding of how a mechanism works. "A wolf eats the sun each day and it gets reborn the next day" is a mental model of how days and nights work. "The earth is a sphere revolving around its axis with the sun to one side" is another mental model for the same mechanism. *

Humans are capable of solving problems when they don't have a clear mental model of the forces working in the background, but they usually do it haltingly, step by step, and cannot monitor the outcome of their steps for veracity of the solution. It is like trying to cross a labyrinth using some algorithm like taking only right turns and retracing to the left when you run into a blind end. It is possible to do it, but at no point do you actually know the way through the labyrinth, even after you have emerged on the other side. On the other hand, if you have memorized a map of the labyrinth, and the labyrinth is of low enough complexity to fit in your spatial reasoning brain areas, you have a good mental model of the labyrinth and you can easily find a way to the other side, and at each step you can monitor your concrete surroundings and relate them to the mental model of the whole, and it will always feel right when you are on the right way and wrong when you are on the wrong way, because your spatial reasoning "subsystems" will create a feeling of certainty for you. Another example which is probably much more "intuitively right" :) for math teachers would be simple geometry problems about triangles. Read the word description, and you probably could solve it step by step, but it would be hard, and you can't keep all the details in your mind at once. Make a drawing, and everything falls into place; you know the solution before you have calculated it.

What you certainly want is that your pupils get a mental model of nonlinear relationships which can be reasoned about on an intuitive level. Getting exposed to nonlinear relationships written as abstract numbers is not good enough, even if the exposure is very frequent. We humans don't have inborn neural circuits for evaluating rational numbers, this is a learned skill. We have inborn neural circuits for evaluating tangible entities, visual input, smells, language, etc. If you want your pupils to create a mental model at all, instead of running around the numbers blindly, you will have to help them relate the numbers to something. I don't know what this something will be, centuries of teaching math have tried to find such solutions and to my knowledge have not gotten beyond cutting one apple in thirds and one in halfs and then showing that one piece of each together don't make a fifth of an apple. But any working solution, if it exists, will have to work along the lines of creating a good, solid mental model. Then pupils will be able to think properly about the problem at hand, to reason about it on a level which creates the feeling of knowing at the right times except of floating in uncertainty at each step.

I don't have a single good book recommendation on mental models the way I had on the feeling of certainty. They are researched within the context of usability, so a textbook on software usability might contain relevant chapters and/or lead you to better, more specialized literature on mental models.

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• The days and nights provide another nice example of how conviction works against logic and how mental models fit into it all. Note that we as individuals are only convinced that "earth revolves around its axis and around the sun" is the true one because we have been told that it is true. I learned in sixth grade about Foucault's pendulum, and Earth's horizon curvature, and all the other experiments together which prove it; but I have never seen the pendulum or conducted these experiments. It doesn't matter, because when I was four, my father had bought me a globe and told me how it works, and I believed it, long before I knew what a physics experiment is. Had I been constantly told that the Earth is flat up until I started taking sixth grade physics, my teacher describing those experiments wouldn't have convinced me. It was seeing the rotating globe, and hearing the explanation from a person whom I trusted, which helped me create a mental model leading to true convictions, as opposed to mere logical inferences.

Answer 8

My views on this matter differ dramatically from all other current answers. Others seem eager to agree about the prevalence of this "disease", and have many theories about causes and treatments. Instead I believe that you are simply finding a pattern among a disparate variety of errors made by learners of mathematics; it is your own mathematical skill at pattern-matching that connects the dots and gives it a name. However for them it is not one missing skill that a silver bullet will kill, but many puzzles that are missing pieces.

Every learner of mathematics, at every stage, struggles with learning not only the uses of a mathematical skill, but its limitations. This is an iterative process, and mastery is achieved only through repeated efforts. It is difficult to learn that $\frac 25 + \frac 15=\frac 35$, and also difficult to generalize to $\frac ac + \frac bc= \frac{a+b}{c}$. It is also difficult to learn that $\frac 15 + \frac 13 \neq \frac 18$, and still more difficult to generalize this fact. Those of us with math Ph.D.'s may not remember these difficulties, because we have have so many additional layers piled on top, but for precalculus and calculus students, these struggles are still quite fresh.

Consequently I believe that even attempting to impose a single answer, no matter how clever, will be entirely counterproductive. Someone that has not yet mastered $\frac 1a + \frac 1b\neq \frac{1}{a+b}$ is nowhere ready to generalize to $f(a+b)\neq f(a)+f(b)$; on the contrary, such a general approach is likely to intimidate and confuse. Simply identify the specific error they made and state that this is an invalid operation. An explanation should be only given upon request, and should be limited to the context of the error, not a general screed about nonlinear functions and the general deterioration of the human intellect.

Answer 9

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 $1/2$ and $1/3$ each, and then have them cut up a pizza into $1/5$. They will then intuitively get that $1/2 + 1/3 \,=' 1/5$, because that's way less pizza.

Then you can do the same with the numerator to show that $2/5 + 3/5 = 5/5$ 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.

Answer 10

This is one of the things that I find very discouraging as a teacher, and which I have never really understood. However I feel that part of the problem is to do with students' basic attitude towards mathematics. A significant number appear to think that it's all a game: the rules are there not "because they are true" but "because the teacher said so".

And what do you do if you are playing a game in which the rules are complicated and you are not very successful? - simple, you play a different game in which you can make up your own rules!

Sadly, I think that many of the suggestions made to overcome the problem are way too sophisticated. In my experience, counter-examples are of little use. If you show a student a counterexample they will generally nod, smile, agree with you and go away to do exactly the same thing: anyone who has trouble simplifying fractions is scarcely going to appreciate the logic which says that a single counterexample disproves an "all" statement. As Jesse Madnick pointed out, many students will happily (or unhappily, but that doesn't help...) write $\frac{1}{x}+\frac{1}{y}=\frac{1}{x+y}$ when $x,y$ are variables, but will not make this mistake if $x,y$ are specific numbers.

One thing I have noticed is that this error is not "symmetric". It is less common, especially when $x,y$ are specific numbers rather than variables, for students to write $$\frac{1}{x+y}=\frac{1}{x}+\frac{1}{y}$$ than $$\frac{1}{x}+\frac{1}{y}=\frac{1}{x+y}.$$ Perhaps this is because in the first case they look at the left hand side and recognise that the first thing to do is to add $x$ and $y$, which is easy; whereas in the second case they do not know what to do with the left hand side and so, once again, they just make up their own rules.

Answer 11

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

1. Being able to consciously and correctly state some fact. (E.g. the formula for the square of a binom or the correct verb form after "if".)

2. Being able to apply the fact routinely, automatically and with high reliability.

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.

Answer 12

There is no one good answer. Distributive/commutative properties are confusing in large part because they are seemingly arbitrary rules.

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

versus

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

is confusing to a lot of people, because they look the same. You can certainly teach them the rule - but the reason for that isn't the same reason that

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

doesn't work, or that

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

doesn't work (well, sort of in the second case).

The general answer for lower levels (high school non-advanced) is simply to teach each of the cases as they come up, and remind students that commutation/distribution only works in specific instances - in particular, primarily with multiplication.

By the time they get to undergraduate or 'advanced' high school math, then, it would be appropriate to teach them some of the skills of proofs; and then explain that if they want to verify whether distribution works with a particular operator or function, it is fairly simple to prove. That's the only true way that will work in every circumstance (and still requires understanding of how the functions, like sin/etc., work, though in those cases you can always try to disprove it by testing a few example cases first).

Answer 13

I think many of the answers here are giving the students too much credit. In my experience teaching a college algebra course, the basic problem is this:

Students do not understand what they are doing.

(this obviously doesn't apply to all students, but it definitely applies to a nontrivial number of them)

Students don't apply $\log(x + y) = \log(x) + \log(y)$ because they think it is true. They are playing algebra blindfolded. They learned a bunch of tricks early on (for my students, it was in high school), and they are faced with new things that they don't really understand, so they just play it by ear hoping that it will work. Sometimes it does work, because they really are using the rules correctly, and every once in a while by accident their mistakes would "cancel each other out" to give the right answer, but usually it doesn't, leading to frustration.

When I taught logarithms, this was probably the most common blatant mistake (it would be more common except due to the focus on the multiplicative log rules logs with additions are not shown very often). But there were others, like solving expressions without equals signs (the instructions would usually just say to simplify), and "canceling" functions (like $\log$), or otherwise treating them like they were just multiplying.

I don't know the solution to this. One thing that I've found really doesn't work is teaching rules. The reason, it seems, is that such students are really bad at pattern matching. We mathematicians tend to be good at pattern matching, and so we think of this as a good way to impart information, but students can get that $\log(a + b) \neq \log(a) + \log(b)$ and then turn right around and apply $\log(3x + 1) = \log(3x) + \log(1)$. Similarly, even if you can convince them that $\log(1000000 + 100)$ is quite different from $\log(1000000) + \log(100)$, they won't apply it to symbolic versions.

Answer 14

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:

• Not caring about the subject
• Missing some basic knowledge

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 3x4 and a 2x4 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.

Answer 15

It must start early, and it must start by divorcing ourselves from educational approaches that teach students to approach problems algorithmically.

Students write $(x+3)^2 = x^2 + 3^2$ because by the time they start looking at things involving $(x+3)^2$, they've just gotten a hang of the distributive property. And it's taken them a while to get a hang of the distributive property because we insist on teaching it as "first multiply this by the first thing, next multiply by the second thing, now add those two results together," and not as an abstract representation of the product of quantities, or even better, the equivalence between multiplying 5 and 11 and 5 and (10 plus 1).

Students are encumbered with homework (yours is not the only homework they have to do!), laziness, distractions, and life. Of course they're looking for shortcuts, foolproof algorithms to solve the problem, and the like.

Answer 16

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

For instance, $$ \frac{a+b}{c} \mathrel{\text{“=”}} \frac{a}{c} + \frac{b}{c} $$

is true, but $$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$ is not.

And $$ 2^{-3} \mathrel{\text{“=”}} 1/2^3 $$, meaning that once you put $$ 2^{3} $$ 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.

Answer 17

I had problems myself with this when I was starting out. I can't remember what I used to get around your first example. For your second example I got it into my head that the minus sign was the "line in the fraction", so $$ 2^{-3} $$ became $$ \frac{1}{2^{3}} $$ Perhaps not for everyone but I found it an easy trick to remember.

For your example $$ \sin (5x + 3y) $$ I just had to hammer it into my head with examples and the log tables. Essentially starting out with something like what's here http://www.math.com/tables/trig/identities.htm and building slowly on that. I know you've said you tried examples but this was worth a shot.

I would have to agree that a students attitude does contribute greatly to the learning/remembering process with such things like this. Our school teacher broke it down to basics. Students were saying "When will I actually need this in the real world", so she asked us all what we would like to do when we finished school. When she came in the next day she had an example for each of us about how at least one of these laws/examples would be needed in our future career. The overall attitude in the class quickly changed and we got the hang of it. I find this very useful in a tutoring situation as many students are sent to find tutors because their parents want them to do better, thus starting with a bad attitude. It may work on a few of your students and even if it is a small few it is a start.

Answer 18

Go back to the basics!

I've seen this in many (all?) of the students I've tutored. I always attribute it to students being taught 'what' and not 'how' which always leads to a gross lack of understanding of 'what' they're REALLY doing with these operations.

$$\mathbf{2^{−3}\,“=”\,(−2)^3}$$

This is sheer lack of understanding what a negative exponent is. Broken down...

$\dfrac{x^4}{x^2} =\dfrac{x*x*x*x}{x*x}$

so we can cancel out pairs -- something they're good at, and we're left with $x^2$.

So if it's reversed:

$\dfrac{x*x}{x*x*x*x*} {} = \dfrac1{x*x}$

(we're clearly in 'negative territory' in our numerator now...)

$= x^{-2} $

Now they should be able to see why the "premise of equality" makes no sense.

Answer 19

Try once more with an example which really brings the error right in front of Them. I like to say, "Would You:

• Wake up
• Go to school
• Put on clothes
• Shower
• Wipe Your behind
• Poop
• Pull down pants and sit on the toilet

in that order? Of course not because order of operations can be significant."

Answer 20

This is an example of only one kind of "linearity." I don't think it's been mentioned yet.

Whenever I write something like $$\dfrac{2x+3}{2}$$ on the board, somebody will inevitably say, "Cancel the twos!!" And I respond, "Wait! So that means five divided by two is three, right?" $$ \dfrac{5}{2} = 3 $$ "because" $$\dfrac{5}{3}=\dfrac{2+3}{2}=3$$ "Right??"

And then I proceed with showing them the way to factor and cancel in an expression such as

$$\dfrac{4x+6}{2} = \dfrac{2(2x+3)}{2} = 2x+3$$

Answer 21

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

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

In these two cases, the error of the student's method is clearly demonstrated by plugging in some arbitrary values:

$$ \frac{1}{2+3} = \frac{1}{5} \neq \frac{1}{2} + \frac{1}{3} $$

This one should be obvious when you make the student think about it for a second; you can't add two positive numbers and get a number smaller than the ones you started with. The sine equation is a little harder to visualize, but try $x = 9, y = 15$; then $ \sin (5x + 3y) = sin(90) = 1$, while $\sin 5x + \sin 3y = \sin 45 + \sin 45 \approxeq 1.4142 \neq 1$.

Exponent signage is harder because overall it's less intuitive; you have to see the math work to understand why negative exponents are fractions and not negative numbers.

Consider that $5^4 / 5 = 5^4 / 5^1 = 5^{(4-1)} = 5^3 = 225$. Therefore, by the same math, $2^{-3} = 2^{(0-3)} = 2^0/2^3 = 1/(2^3) = 1/8$. However, on the other side of the "equation", $-2^3 = -(2^3) = -8$.

As other answers have said, this is all part of elementary math education, which unfortunately in the U.S. is often taught as a series of "do this, don't do this" without the kind of explanation behind why one transformation is valid and works while the other doesn't.

Answer 22

Well, I think that the problem resides in the comprehension of the definition of respective operations. And we know that some textbooks and teachers said that "linear function" consists in functions of the form "$ax+b$", fatal error of mathematical language.

Answer 23

Much has been said about inappropriate pattern matching, and I agree with much of that opinion. But I really think that what is going on with many of these students is that instead of seeing mathematical patterns (albeit incorrectly), they are actually seeing equations and expressions as lexical patterns. $a(x+y)=ax+ay$ , $\frac{a}{x+y}=\frac{a}{x}+\frac{a}{y}$, $2(-(x+y))=-2(x+y)$, and $2^{-(x+y)}=-2^{x+y}$ become the basic designs that a student can tessellate his or her test paper with. One must certainly admit that $\frac{a}{x}+\frac{a}{y}$ has a much more aesthetic quality than $\displaystyle\frac{1}{\frac{x}{a}+\frac{y}{a}}$. I asked one student why, in the equation $y=a(x-c)^2$, does a change in $a$ stretch every point except the vertex. He replied "because the $a$ is closer to the $x$ than the $c$ is".

I'm not sure what can be done about the lexical problem, except to nullify the tendency by teaching why the rules work the way they do. For example, how many students in high school can show why $\frac{a}{c}\cdot\frac{b}{d}=\frac{a\cdot b}{c\cdot d}$? They should be able to, certainly.

One thing that needs to improve in our schools is following through on the "making connections" blurb that we find in every curriculum document across Canada and the United States. For any arbitrary concept we need to teach that there are multiple interpretations of that concept and that students should learn flip back and forth between those multiple interpretations depending on the situation. For example, some possible interpretations of the fraction $p/q$ may be

• $p/q$ is the solution to this equation $qx=p$
• if we partition something of size $p$ into $q$ partitions, $p/q$ is the magnitude of one partition
• it is $p$ multipied by the magnitude of one of the partitions of $1$ partitioned into $q$ equal pieces
• if $p$ is partitioned, and $q$ is the magnitude of each partition, then $p/q$ is the number of partitions

When trying to evaluate $\frac{x+y}{a}$, a student familiar with the multiple interpretations of the fraction concept would find the 3rd interpretation useful $$\frac{x+y}{a}=(x+y)\cdot\frac{1}{a}=x\cdot\frac{1}{a}+y\cdot\frac{1}{a}=\frac{x}{a}+\frac{y}{a}$$ Of course, no useful interpretations can be found for something like $$\frac{a}{x+y}$$ We can only hope that the student doesn't find any interesting tessellation for such a fraction.

As for other Linearities, perhaps the best thing to do is just to teach fundamental (such as the reasons behind the exponent rules), and perhaps to select a smörgåsbord of exercises that don't repeat the same problem ad nauseam.

Answer 24

Functions, like $\exp$, $\sin$, $x\mapsto 1/x$, $x\mapsto x^2$, or $f$, are laws that assign to an input value $x$ an output value $f(x)$. Only in very special cases this law is additive. An example is the price $f(x)$ of $x$ gallons of gasoline: $f(x+y)=f(x)+f(y)$. When $f(0)\ne0$ we don't even have $f(x)=f(x+0)=f(x)+f(0)$, and when the graph of $f$ is not "linear", i.e., a line, then also some weaker form of "additivity" fails.

Answer 25

I think the best way is to give them counter examples, for instance:

$\dfrac1{a+b}=\dfrac1a+\dfrac1b$

$\dfrac1{2+1}=\dfrac13\,\text{ and }\,\dfrac12+\dfrac11=\dfrac32$

so $1/3$ is not $3/2$ and they will see by themselves that they got it wrong, that's what I do to my students most of the time: counter examples.

Answer 26

Pre 16 I actually made that mistake with the

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

The above mistake was from never being corrected up to 16.

Once at 16, I got a new teacher who corrected me on that, and I only needed to be told once such that I understood what he said. It's a significant thing so I thought about it a lot and remebered it. Alternatively I could have memorized it from drills he assigned, but I was probably smart enough at 16 to write my own drills for something so basic that I understood and just needed to practice a bit without making that mistake. I realised the error came from a)not learning the axioms formally and b)knowing $(a+b)/c$ breaks down like that and assuming that a/(b+c) did. I checked with the teacher that $(a+b)/c$ broke down but $a/(b+c)$ didn't.

Regarding this mistake

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

I would not hae made that mistake because I never invented my own rules, and pre 16 i'd not seen a negative indice. By 16 I had a good teacher that taught us that one surprise, that -2^3 was -(2^3) So in BO DM AS there is a U here BOU DM AS. And he taught us

$$ 2^{-3} \mathrel{\text{=}} \frac{1}{2^{3}} $$

So we learnt indice rules from scratch from him. Nobody would have done the mistake you mention. He made sure that anybody taking Math post 16 got an absolute minimum of an A grade in the exam before that(GCSE), to qualify. If we had made up our own rules and not remembered fundamentals it'd have been a problem.

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

I learnt one or two rules pre 16 regarding sin and cos and even back then i'd have been sure as hell not to do the above. It takes a real idiot to make a mistake like that. Back then I didn't really know f(x) notation that well but still. We wouldn't have got an A in GCSE math if we had done that. There's no way somebosy that did that would have qualified well enough for the teacher to have allowed them to do Math A level (Math post 16).

At 16 our new math book (A level is post GCSE) on Pure Math gave clear axioms all on one or two pages so it got even easier not to do something stupid like that.

The 1/(a+b) though is a classic error no doubt from learning (a+b)/c at a young age and assuming and not neing corrected. The rest, especially the last, no way.

The best one can do is show them the're wrong and when they understand, then give them practice examples, mark them, and remind them and test them and so on and see if they're remembering. Give basic examples that goad them into using their made up rule, see if they do. Make an impression and they should be thinking about their mistake for the rest of the day, and they should remember. Their scores in your drills/tests should improve if you're testing the same thing.

I didn't take Math at degree level. It's questionable whether I could have!

Answer 27

My hypothesis is that all these examples of "suspect algebra" are really examples of "imitative algebra"¹.

Much learning is imitation, something that we are basically "hard-wired" for, and therefore lying largely beyond of the constraints of deliberative/logical reasoning. It takes some training to turn off this tendency to "learn by imitation" (in which reasoning plays no role) in contexts, such as learning math, where it is inappropriate.

My point is: don't be alarmed; IMO, what you're seeing is perfectly normal. "It's just a phase," as they say.

My advice would be, first: don't have a cow over such doozies. (It'd be like despairing over the incomprehensibility of an infant's babbling.) (BTW, I suspect that overreacting to such errors may be the genesis of, or at least contribute significantly to, life-long "math phobia".)

Second: use these mistake as teaching opportunities. For example, when you come across something like

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

ask the student to check the equality by replacing $a$ and $b$ with some actual numbers. Learning how to check one's derivations is a crucial, and extremely general, skill, far more important than any one algebraic "rule", and the sooner such "derivation self-checking" becomes second-nature, the better.

_{¹ Infants and young toddlers babble. The hypothesis that babbling is an imitation of talking is supported by the fact that children (whether hearing or not) of parents who use sign language will display "manual babbling" at the stage when "vocal babbling" normally occurs. Less common than babbling, but in a similar vein, some pre-school children will display "mock reading".}

Answer 28

Provide them with concrete examples, using real numbers (not Real as in the complete ordered field, but "real" as in quantitative).

Next time a student thinks that they can use $$\frac{a}{b+c} "=" \frac{a}{b}+\frac{a}{c}$$ give them an easy problem to check: does $\frac{12}{2+4} = \frac{12}{2}+\frac{12}{4}$? No, so the property must not hold.

The problem most (middle and high school) students have is that they can't yet deal with variables in the same way as numbers. They think it's a whole different world; it's too abstract. So give them problems they can understand, with actual numbers. Then they'll begin to notice patterns, which will lead to the abstract.

Answer 29

for young kids, it might be too early to teach them about linearity. I would prefer to teach them the distribution/commutation instead. That is:
$a+b=b+a$
$a \times b = b \times a$
$(a+b)\times c = a \times c + b\times c$

As @noobermin said, you can stress that the laws work on multiplication and addition only. Hence,

$\frac{1}{a+b}=1:(a+b) \neq 1:a + 1:b$ as this is division, not multiplication. But

$\frac{a+b}{c} = (a+b)\times \frac{1}{c} = a\times \frac{1}{c}+b\times \frac{1}{c}$ since we have multiplication with addition here.

Similarly,
$2^{-3} \neq -2^3$ becuase $2^{-3}$ is not $2\times (-3)$
so the only way to to approach is to apply one of the exponential rules $x^{-n}=\frac{1}{x^n}$

However, for the following expression we can apply either rules and get both correct:
$(-2)^3=-2^3$
Approach1: commutation
$(-2)^3=(-2)\times (-2)\times(-2)=(-1)\times2\times(-1)\times2\times(-1)\times2=\\=(-1)\times(-1)\times(-1)\times2\times2\times2=(-1)\times2^3=-2^3$

Approach2: apply another exponential rule $(a\times b)^n=a^n\times b^n$ with a=-1, b=2, n=3
Of course if the students are eager to learn, you can show them that the above exponential rule can be proved using the commutation property.

The $\sin(a+b) \neq \sin a+\sin b$ can be explained in the same way..

In conclusion, my universal rule is as follows:

• The distribution/commutation apply on multiplication and addition only
• When a new concept is introduced, it comes with its own rules (e.g. exponent, trigonometry, complex number...). Try to learn their rules by heart.

Answer 30

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.

Answer 31

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.

Answer 32

I would say that the main causation, the underlying disease, is the fact that, here in the U.S. we often have too many students, all crammed into a small room, that lack a love for math. Students are worried about grades, not the inner machinations of mathematics. Teach them to love maths and they ought to take care of the rest on their own.