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.

Simon Fraser
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Peter LeFanu Lumsdaine
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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 07 '14 at 16:27
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    Don't give it a name! It will catch on! – Pedro Jan 07 '14 at 16:35
  • @KelvinSoh: yes, that’s exactly my experience! For a particular operation, they get the point — but they keep on falling back to the expectation that linearity/commutativity is the default. – Peter LeFanu Lumsdaine Jan 07 '14 at 16:50
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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 07 '14 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 07 '14 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 07 '14 at 18:44
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    When I help others to cope with such problems, I simply tell them to replace those symbols with some small numbers and compute the values themselves. Most of them could realize the point by themselves after trying this a few times. For example, given a = 2, b = 3; 1/(a+b) -> 1/5 = 0.2; 1/a + 1/b -> 0.5+0.333 = 0.833. (A calculator helps.) – Damkerng T. Jan 07 '14 at 19:22
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    I've never met such a thing, and I sincerely hope that disease does not affect science's students and, more specifically, mathematics students... – DonAntonio Jan 07 '14 at 20:19
  • Pay attention to detail!!! – ebyrob Jan 07 '14 at 20:30
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    @DonAntonio This occurs all the time in freshman calculus courses that engineers and other random majors take. I've considered making an "Bingo Card" for exam grading parties which would include such things... – abnry Jan 07 '14 at 20:34
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    Oh, well: then all is fine. After all engineers aren't neither mathematicians nor scientists...:) – DonAntonio Jan 07 '14 at 20:36
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    This is a great question, refutation of such a law is needed in so many more areas other than math. On Cooking.Stackexchange, we have to explain at least once a week that bacterial growth is not linear, so keeping your food at almost fridge temperature doesn't mean that it will be safe as if you had kept it in the fridge... but the next question will again assume that it is true. Or even worse, the next answer. – rumtscho Jan 07 '14 at 21:09
  • Wouldn't a computer algebra system that handles symbolic manipulation help youth keep their rules straight? It's mighty useful to double check answers against a calculator. – Ehtesh Choudhury Jan 07 '14 at 21:30
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    Excellent question. IMHO the real problem is that students are no longer taught to *think*. I sincerely wish I knew how to fix that. – David Jan 07 '14 at 21:32
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    Make them think about each "=" they write. Why is there equality? They should feel they need an explicit reason why equalitiy holds (because it *is* needed!). In exercises, make them write the law (distributivity, for example) over the "=" and grade *that*. – Reinstate Monica Jan 07 '14 at 23:48
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    I disagree with everyone who is saying "confirm with a calculator." Appealing to a calculator as an absolute authority on mathematics is a *dangerous* state. As someone who may one day *make* the calculator, I hope that no one takes my work to be infallible. On topic: I really don't know why, but I have ***never*** had a problem with this confusion. I'm starting to believe that I take math statements to be false until shown true, rather than vice versa. I think this is why students make that error--they assume it is true until shown otherwise. – apnorton Jan 07 '14 at 23:53
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    I guess the question is do they WANT to be cured? I know many students that make errors like this and even once told they are making errors really don't care too much. I guess it all depends on the consequences that arise from making these mistakes. – fretty Jan 07 '14 at 23:57
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    You left out the famous "freshman's dream", $(a+b)^2 “=” a^2 + b^2$! – MJD Jan 08 '14 at 02:13
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    What about $\frac{a}{b}+\frac{c}{d}``="\frac{a+c}{b+d}$? – Joel Reyes Noche Jan 08 '14 at 02:23
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    I'm with DonAntonio. I have never seen this happen in Russia/Ukraine nor in Germany... That is, everyone makes a mistake occasionally due to lapse of attention, but not by lack of understanding. As a consequence, I suspect this is something particular to your corner of the world. – 3yE Jan 08 '14 at 06:14
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    You should not cure them of the often useful human tendency to overapply patterns to new situations. You should teach them to question their results and check them. – Phira Jan 08 '14 at 07:47
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    Yeah I've never seen students actually *believe* these "equalities" to be true. I'm sure we've all made stupid mistakes like these but those have just been mistakes when writing down the formula... – user541686 Jan 08 '14 at 09:02
  • @JoelReyesNoche Wouldn't any configuration of `a`, `b`, `c`, `d` with `b = 0` and `d != 0` be a counter example? – Frerich Raabe Jan 08 '14 at 10:57
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    Giving students lots of counter examples early on, and including "trick questions" on exams. I had one calc teacher for three terms, and every term she would give us a question on the midterm that involved determining whether a given equation was the equation of a circle, ellipse, or what-not. Every year the answer was "point", and every year when I saw that the equation was equal to zero I thought, "it can't be! I've made a mistake". After getting that question wrong thrice, I'm much more careful about these edge cases. Do that for your students! – Ziggy Jan 08 '14 at 11:11
  • @FrerichRaabe, I suppose. But only if your students understand why division by zero is undefined. – Joel Reyes Noche Jan 08 '14 at 12:49
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    As an aside, this seems like a mathematical analogue of the linguistic concept of [hypercorrection](http://en.wikipedia.org/wiki/Hypercorrection)—applying a perceived rule of grammar, spelling, pronunciation etc. in situations where it does not apply. ("He gave it to John and **I**," "**Sieze** the day!") – Jordan Gray Jan 08 '14 at 13:51
  • I'm curious about the responses that involve teaching this concept using exams and grades. All mathematicians here, but everyone accepts without needing proof that testing is a valid pedagogical model... I believe, based on much research by pedagogs and psychologists, that the main problem with instilling any concept in young learners is the attempt at doing so by means of exams. – GPerez Jan 08 '14 at 14:02
  • @GPerez Any reference for this belief of yours? (For the record, I was also surprised to see exams proposed as ways of *learning* a subject.) – Did Jan 08 '14 at 14:28
  • @Did http://www.alfiekohn.org/books/pbr.htm is kind of a different subject, but same idea. Any of his works address similar issues. As another reference I'd choose the following reasoning: Grading is the assumption that a bijection exists (and we know how to apply it) between a poorly defined space of "intelligences", and the real numbers. Also the relationship is linear, so averages are justified. Where's the proof??? – GPerez Jan 08 '14 at 14:57
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    @JordanGray: Interesting analogy. Could we then exploit that analogy to figure out appropriate pedagogy? In child development, AFAIU, the stage where children learn to apply patterns and apply them everywhere ("I **eated** my cereal") is considered a necessary and normal step on the way to learning *where* to apply each pattern. Maybe the answer to the OP's question then is like the answer to what to do if a 10-yr-old keeps saying "eated" and doesn't notice or care that it's incorrect? I don't know if that takes us anywhere helpful. – LarsH Jan 08 '14 at 15:55
  • I cannot answer this question, and maybe this answer is more fit for a comment anyway. My cure is simple. Use reverse psychology. Make a habit of first demonstrating cases where the rules appears to work out. Then show them how this doesn't hold in the general case. This will likely teach the student to be on the look-out for such mistakes. – nitro2k01 Jan 08 '14 at 19:44
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    Use a cattle prod. – Asaf Karagila Jan 08 '14 at 19:49
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    I think a good cure would be to prove that those equalities do not hold by using an example... – zerosofthezeta Jan 08 '14 at 23:11
  • Related joke: http://pubcrawler.org/images/math.jpg – Vi0 Jan 09 '14 at 01:50
  • What I would do: everybody, take a small sheet of paper! Quick! Put your name on it! 1/8 + 1/7 ! Two minutes! Then the works are collected, someone solves it on the board, everything's explained, and then: ok, everybody, take a small sheet of paper! 1/5 + 1/9 ! – 18446744073709551615 Jan 09 '14 at 10:24
  • Maybe you can find some regular time to present some paradoxes to them as a funny break, maybe even make it like a game when a winner would get one error forgiven on the next test work? Like the theorem of cathenus and hypothenuses equality - it is obviously wrong, but it is tricky to find a faux pas. So if the people would regularly play error investigations like that maybe they would learn "doubt everything" concept? By the way, there is one geometric theorem that I remember exactly after realizing, that recalling it in time would instantly solve the cathenus - hypothenuses hoax :-) – Arioch 'The Jan 09 '14 at 20:13
  • Students? You mean people aged 16 and over? Here in Russia pupils in schools are got rid of such a math heresy in 5th-6th grades. Good old stick and carrot method: you have to prove each equality sign you write, or get downscored. – mbaitoff Jan 10 '14 at 06:14
  • It has now been 3 days since this question was asked and I now mark the 171st upvote on this question, and the $x$th comment, where $x>>100$. Nice job! – Ahaan S. Rungta Jan 10 '14 at 17:37
  • @PeterLeFanuLumsdaine With my upvote, you now have a higher reputation than I do. ;) – Ahaan S. Rungta Jan 10 '14 at 17:39
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    @mbaitoff: in English the word student covers студент and школьник. This disease exists at all levels of math instruction because students (in the US at least) can fail to learn how to add fractions as an automatic reflex from sufficient practice, and then they don't learn algebra properly, but they manage to be passed to higher-level courses. Это кошмар. – KCd Jan 10 '14 at 23:33
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    Maybe "universal distributivity" would be a better name. In my experience, students love to distribute over addition, but it's less common that they want to pull out constants. – asmeurer Jan 11 '14 at 19:12
  • I learned about the "Universal Homomorphism Theorem" in grad school at the University of Texas. 'Tis a funny and wry name for this phenomenon. – ncmathsadist Jan 11 '14 at 22:57
  • You did not mention $\sqrt{35}=3\sqrt{5}$ (no kidding...) – Tom-Tom Jan 13 '14 at 14:13
  • @KCd Not to mention how they consider the problem "insignificant", using "kilowatt per hour" instead of "kilowatt-hour", "Amperes per hour" instead of "Ampere-hour" etc. – mbaitoff Jan 15 '14 at 03:53
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    Related post on matheducators.SE: [Whence the “everything is linear” phenomenon, and what can we do about it?](http://matheducators.stackexchange.com/questions/926/whence-the-everything-is-linear-phenomenon-and-what-can-we-do-about-it) – Martin Sleziak May 18 '15 at 07:44
  • A small number plus another small number does not equal a super small number. – Mateen Ulhaq Sep 26 '19 at 00:04

32 Answers32


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.

goblin GONE
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    Just an observation from years of dealing with it. In regards to distributivity, students who believe (or, understand even) that $a(x+y)=ax+ay$, do not always think to observe that reversing the equality allows one to pull out a common factor in a sum. So this raises another question. Are students being taught that equality is a symmetric relation? – Chris Leary Jan 07 '14 at 22:08
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    I must of had some good teachers, this is the method they used and I've never had this problem. – hildred Jan 07 '14 at 22:49
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    I have had high school teacher that do not think that equality is a symmetric relation (as in, if the question ask for $A=B$, proving $B=A$ and then conclude that $A=B$ is wrong). I think the problem is worse than you suspected. – Gina Jan 08 '14 at 00:56
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    Teach them equality does not mean "press = on the calculator"? – user541686 Jan 08 '14 at 09:03
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    I used two ideas. One was saying, "Is it linear? Does f(a+b) = f(a)+f(b)?" And saying it often. The other was to use the geometry they never had in school. Use lengths and areas and volumes. Draw the squares for $$a^{2}\; and\; b^{2}$$ and $$\left( a+b \right)^{2}$$ and repeat often. Sprinkle with plenty of "Why do we call it 'squared' or 'cubed'? This is high school and it works for some of them. Note that in the U.S. group cooperative activities are the 'best practices' for math. Experts think kids will discover these "relations". It is excruciating to watch. We can still teach physics. – C. Towne Springer Jan 08 '14 at 09:23
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    @hildred you must have had some poor English teachers, I'm afraid. :) – Asya Kamsky Jan 08 '14 at 09:36
  • @AsyaKamsky yup. the dyslexia really confused them and grammar was a 'foren' language. – hildred Jan 08 '14 at 09:45
  • Maybe this is the way algebra is taught in US; from what I know, in many other places algebra is taught from rules, not by examples. – Kochede Jan 08 '14 at 11:14
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    I was taught algebra in the US, and was taught by rules followed by examples. I would be very surprised if it were taught by examples without an explanation of the rules, though certainly one doesn't get as deep into fundamental arithmetic in 7th grade as in college. But despite being taught explicit rules, we all know that intuition is much faster and easier, so if a formal rule seems fairly intuitive, it's very tempting to pay little attention to the rule and stick with intuition. It takes pain (seeing examples where your intuition goes WRONG) to cure that habit. – LarsH Jan 08 '14 at 15:38
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    I hope you're not saying that students should be taught not to guess, or not to develop their intuition. Rather, they need to know that their guesses need to be held humbly, and subjected to formal scrutiny when needed. I think all productive mathematicians take advantage of intuition; otherwise they'd be little better than computers. The trick is learning when formal scrutiny is needed, and when it's not. – LarsH Jan 08 '14 at 15:42
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    The problem is that too many math teachers are not well-trained in Mathematics, and many states require education training rather than Mathematics training. When I took my Math degree, there were Math-Ed students who took 1/2 the Math classes, but took education courses. In Georgia, you are not considered qualified to teach math with a Math degree, but someone with an education degree (even Chemistry-Ed or Physics-Ed) can teach math (especially in counties where math teachers are hard to find). Sigh. – ChuckCottrill Jan 08 '14 at 16:53
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    The US is moving away from teaching Geometry, where the students would learn about axioms and proofs. – ChuckCottrill Jan 08 '14 at 16:55
  • @LarsH, of course you're **completely right,** intuition is so, so important, and any educational program that doesn't recognize this is asking for trouble. And yes, guessing and experimenting are fundamental tools for generating understanding and progress. However I think that when manipulating basic algebraic expressions, you basically just need to know the rules, and there isn't much scope for flair with respect to these basic rules. Of course, you can apply them with flair and creativity; say, to solve the general quadratic equation. But its still those basic rules that are being used. – goblin GONE Jan 08 '14 at 17:40
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    EXACTLY! I had to literally tell my teacher to STOP telling me examples and just tell me what the damn laws are. – Michael J. Calkins Jan 09 '14 at 03:51
  • @ChrisLeary That is an extremely important point you hit. I don't think most students in American Institutions understand the symmetry of the equal sign, and that's why most people are screwing up. – user507974 Dec 18 '14 at 05:31

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.

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...

Ryan Haining
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Vladimir Sotirov
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    Can confirm the same student problems in Europe. Usually, for more students Maths is boring. They learn how to solve things for exams (memorizing different types of problems and how to solve them) but never think about the concepts behind - what is a function, what is an equation. And they never think about actual applications which would render the "linearity" errors obvious. Once they start to be interested in Maths and start learning the concepts, they stop doing similar errors. – Sulthan Jan 07 '14 at 20:27
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    In my experience (with engineering students in the US), the idea that math means something doesn't truly dawn on them until their junior year in college. If you ask, for example, what happens to an apple falling from a tree, they know physically that it falls to the ground. If you ask them to find the answer using math, younger students can conclude, using math, that it goes up and yet not see a problem. By junior year, they start to see the problem and change the sign before turning their work in. – John1024 Jan 07 '14 at 23:27
  • For me, the point at which this really hit home was when my teacher taught us about limits. He started with the real definition $$\lim_{x\to a}f(x)=L\iff\forall_{\varepsilon>0}\;\exists_{\delta>0}:\left|x-a\right|<\delta \implies \left|f(x)-L\right|<\varepsilon$$ but then explain how this is really just saying that "you can make $f$ arbitrarily close to $L$ by picking values for $x$ that are closer to $a$". – AJMansfield Jan 08 '14 at 02:15
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    Elementary questions on this website often display exactly this failure of understanding: just search for questions that ask “Am I allowed to…?” [An example that struck me particularly](http://math.stackexchange.com/q/287572/25554) asks why particular symbols change into particular other symbols when a statement is negated. – MJD Jan 08 '14 at 02:18
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    [Here's another example](http://math.stackexchange.com/q/296867/25554) of someone who doesn't seem to have the slightest idea that mathematical expressions actually express anything, or that there is any way of divining the rules other than copying down what the professor scribbles on the board. – MJD Jan 08 '14 at 02:26
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    As others have said, the meaning behind mathematics isn't taught until college. Before that, it is nothing but mere pattern matching. But, is it needed by everyone? That is what algebra, and all math started out as a calculation aid removing "meaning" from apples and sheep and turns them into symbols that we can easily manipulate: 2 apples plus 3 apples is like 2 sheep plus 3 sheep, ie., a mapping from the set of sheep to the naturals that preserves addition but doesn't preserve their wool... However, once you learn the dry symbols *themselves* have meaning, then you become a mathematician. – noobermin Jan 08 '14 at 05:04
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    You are correct that they are not taught. Current curricula require the students work in groups and discover them on their own and sometimes vote on a new "rule". They jump all around from factoring quadratics to matrix multiply to linear programming just grazing the surface and not knowing why. The teachers don't know why either. These circular curriculums have a revisit to each topic every few months and go a little deeper. In the real world, review takes up the time set for the deeper part. Axiom, theorem, and lemma are words not to be found in the books. It is democratic math, and fails. – C. Towne Springer Jan 08 '14 at 09:36
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    This seems to strike home with the common terminology in logic of syntax and semantics. That high school students are taught mathematics only on a syntactical level, and are missing an understanding of the semantics involved. Bu this has nothing to do with proof assistants (which do purely syntactical manipulations). The lack of proofs for rules also seems akin to adding axioms. One way of showing that their rule is wrong is simply showing that it as an axiom is inconsistent, i.e. deriving $1=0$ from the wrong rule. – HaskellElephant Jan 08 '14 at 17:53
  • I have found that most students who enter graduate school _do_ believe that equations have meaning. Unfortunately, many students quickly loose their faith in this meaning. They often write equations where the right-hand-side is not the same type of object as the left-hand-side, particularly in equations with sums or with random variables. For example, $f(x) = \sum_{x=1}^n x^2$ or $E[X] = XP[X=1]$. That is to say, students who have mastered arithmetic and calculus can still struggle with the basics of "what is an equation" when working on more advanced things. – Michael Jun 30 '16 at 20:27

(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)

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    Interesting proposal. However, I don't understand why you call your solution "soft", since it involves flogging your students. – Dominik Jan 07 '14 at 20:23
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    Your you can just write "No!!!" on their homework in big letters with as many exclamation points as possible. – abnry Jan 07 '14 at 20:38
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    I like the idea of taking money. How about I make a bet with them so they owe me a dollar, erm, I mean 2/2 dollars… so let's see, that's $2/(1+1) = 2/1 + 2/1 = 2 + 2 = 4$ dollars. That should teach them right quick. In all seriousness though, people _are_ good at finding the mistakes when it looks like someone is cheating. See, e.g., the abstract [Wason Selection Task](https://en.wikipedia.org/wiki/Wason_selection_task) and the variant with [concrete domains](https://en.wikipedia.org/wiki/Wason_selection_task#Policing_social_rules). – Joshua Taylor Jan 07 '14 at 22:54
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    I think I've related this story elsewhere on MSE, but this seems like a good place for it. I tell my students that, before they go to use an identity, they should at least check a few numbers to see if it works in some cases. One student wanted, on the final exam, to use the identity $\sqrt{a + b} = \sqrt a + \sqrt b$. He dutifully checked $\sqrt{9 + 16}$ and $\sqrt9 + \sqrt{16}$, found that they were unequal, and then *used the identity anyway*. He then went and left me a bad rating on RMP. – LSpice Jul 14 '17 at 16:54
  • Seub hates to see a math amateur go unbeat – Some Guy Mar 20 '21 at 01:24

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.

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  • This is a method that one of my favorite math teachers of all time used; his favorite was "pineapple". – AJMansfield Jan 08 '14 at 01:59
  • Heh. In my calculus class we once spent a whole hour discussing (d/d plankton). – ApproachingDarknessFish Jan 08 '14 at 07:21
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    And of course the standard calculus teacher's joke (please forgive for the English pun): 'What's the integral of (d cabin / cabin )?' 'Um, natural log cabin?' 'No, it's houseboat.' 'Huh?' 'You forgot the constant of integration. ln(cabin) + C. And a natural log cabin on the sea is obviously a houseboat.' – Scott Sauyet Jan 08 '14 at 12:21
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    I had some experience with something of the opposite effect. When I changed notation from variables that were slightly scary to more familiar ones, suddenly students had breakthroughs. This usually meant replacing `epsilon` and `delta` with `e` and `d`. It's a funny old world! – Scott Sauyet Jan 08 '14 at 12:24
  • @ScottSauyet The other way would work fine, too. Just require that students use only greek letters (or other non-latin glyphs) as variables for a little while. – AJMansfield Jan 09 '14 at 01:41
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    Another point is that Greek symbols still "feel" "mathy" and are seen as "expected math symbols" even by young students. So using Greek symbols might actually be counter-productive: to the extent that a student has anxiety about what a symbol is allowed to mean and whether it has some special meaning, using Greek letters over more "regular" things like x and y might only make the anxiety worse. But using a symbol like a tree, fire hydrant, tulip, pineapple, or anything that's totally not at all possibly confusable with "official mathy type stuff" could have a better chance of working. – ely Jan 09 '14 at 15:38
  • +1. My year 7 teacher (age 12) taught me about complex numbers by saying that it was the square root of -1, and that it was normally called i, but could be called anything. We eventually settling on "lighthouse", and used that for the remainder of our discussions :P – Nico Burns Mar 02 '14 at 19:22

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.

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    not *nothing else*. consider $(ab)^c=a^c b^c$. exponents distribute over multiplication, which is a fact that high school algebra 2 students use. – chharvey Jan 07 '14 at 21:48
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    @TestSubject528491 At that level, though, does the student treat exponentiation as (just) iterated multiplication? If the student does, then the student may treat $(ab)^3$ as an _abbreviation_ for $ababab$, in which case it's not distribution, but rather commutativity of multiplication that matters, since $ababab = aaabbb = a^3b^3$, and no distributive property is needed. – Joshua Taylor Jan 07 '14 at 23:03
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    @TestSubject528491 Well, exponentiation over multiplication is the same thing as multiplication over addition since they are both rings. *Nothing else* is pretty strong, you're right. – noobermin Jan 08 '14 at 00:55
  • I have encountered this as a cause for problems too, and you would be surprised how many people forget about operation order: to that end I use an acronym I was taught a good 3 decades ago: BODMAS (Brackets Orders Divide Multiply Add Subtract). Most people once reminded seem to get on fine, but yes, need to remind them that implicit multiplication is still a multiplication. – GMasucci Jan 08 '14 at 09:28
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    @JoshuaTaylor How is it different from $3(a+b) = a+b+a+b+a+b = a+a+a+b+b+b = 3a+3b$? – Ari Brodsky Jan 08 '14 at 18:39
  • More to the point, you have to be able to deal with non-integer exponents. – Ari Brodsky Jan 08 '14 at 18:40
  • @AriBrodsky In general, we have to be able to handle that, but I don't know at what level high school algebra students have to handle that. You're right, though, that if they view the multiplication in $3(a+b)$ as iterated addition, they'll understand why $3(a+b) = 3a + 3b$. However, students learn about multiplication by non-integers much earlier than they learn about non-integer exponents. – Joshua Taylor Jan 08 '14 at 19:14
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    I had never thought about this, but it is interesting that without thinking about it I parse $f(x+y)$ completely differently than $a(b+c)$. It is understandable that a non-math student might not do so. – Carsten S Jan 09 '14 at 08:35

$1.$ 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.

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    I do not think that standing at the front of a room and writing incorrect rules is a good idea. Students will copy it down. When they look at their notes the incorrect rule is the one rule they will remember. – Jay Jan 07 '14 at 23:41
  • @Jay: Better ? Bottom line, you can't dance around it, and avoid the subject. You have to tell them, explicitly, *that* it's wrong, and *why* it's wrong. – Lucian Jan 07 '14 at 23:50
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    I'm not sure how seriously to take suggestion (1) "Be brutal." Especially in the U.S., where so many students suffer from low confidence (to the point of being "math-phobic"), I can't imagine being brutal will help. If anything, it might just reinforce the image of math being solely about mechanical precision rather than ideas. – Jesse Madnick Jan 08 '14 at 02:23
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    @JesseMadnick: Mathematics is as isolated as any other human inclination (say, music or literature). There's no difference between the US or Romania or any other country. But just as any other teacher of any other discipline will correct its pupils, first by warning them, then, if they persist, by low(er) grades, so also must the math teacher. – Lucian Jan 08 '14 at 02:52
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    I'm not saying teachers shouldn't gently correct students. I'm saying that harshly penalizing struggling students might not be a good approach pedagogically. And yes, like any other human inclination, mathematics requires self-confidence to be done correctly. I don't mean to baby students or regard their egos as precious, but it's naive to think that upon receiving an abysmal grade, all students will react the same: some will take it as an impetus to work harder, and others will sink into despair, convincing themselves that they'll just never get it. – Jesse Madnick Jan 08 '14 at 02:56
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    @JesseMadnick: Every advice should be taken with a grain of salt. But children should be taught to put their confidence into logic, not into oneself, and true confidence is never a synonym for not checking or testing to see whether something is true or false. If they believe a formula to be true, teach them to verify it in their mind or on a piece of paper for some small values of the variables in question, just to make sure (like during an exam, for instance). Truth has nothing to hide, and so it's never afraid of being tested. Only lies and falsehoods shy away from inspection. – Lucian Jan 08 '14 at 03:25
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    +1 For suggesting counter-examples. For students who miss (relatively) simple rules of expression writing, providing theoretical explanations for why might not work. However, when teaching, I would always suggest avoiding "magic" numbers like 1 in counter-examples, and typically would use 2's. – nicholas Jan 08 '14 at 14:01

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).

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.

  • 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.
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  • It's easy to forget that we're talking about brains and biology here. This answer helps explain why replacing variables with trees, doggies, and pineapples is so effective: we've added emotion to the picture, and now the idea behind the notation sticks. – Stephen Bosch Jan 12 '14 at 02:05
  • Actually the fact that Earth is round and gravity pulls towards the center can be explained by general relativity. If you hadn't been told until much later, there still would have been a reason to believe it and you maybe would have figure it out because if you're really smart, you're not too old to learn and believe new surprizing things. The sound delay once would have seemed magic to you but it turns out that the observations that lead you to not observe one for short distances derive from other laws that allow for the observation of a sound delay for long distances. After knowing that the – Timothy Jul 03 '19 at 03:43
  • sound delay is possible, if you then learned at a much older age that Earth is round, maybe you would have been like "Maybe the laws I assumed based on my observations are not the correct laws and those observations derive from other laws. Maybe the round Earth can be explained by general relativity. Maybe we don't observe anything attracting anything else when neither of them possess an electric or magnetic field because the gravitational constant is so small but because Earth is so large, it has plenty of gravity despite the such tiny gravitational constant." – Timothy Jul 03 '19 at 03:48

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.

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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.

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    Sure, I can get algebra students to understand that $1/2 + 1/3 \neq 1/5$, and yet nevertheless write $1/x + 1/y = 1/(x+y)$ on homeworks and exams. – Jesse Madnick Jan 08 '14 at 02:26
  • For sure it's a concept that takes some time to get right. My advice is to have them cut and paste this 'proof' themselves. If they cut and paste a few examples, it might work their way into their mind. Also don't doubt the power of having students write a short essay explaining why something doesn't work. If you can write the equations you should be able to qualitatively explain it as well. – R V Jan 10 '14 at 00:17
  • @RV: what do you mean by the term "cut and paste this proof". Do you mean substitute numbers for variables? The phrase "cut and paste a proof" is not standard. – KCd Jan 11 '14 at 00:11
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    "Even the most self-proclaimed math illiterate will learn percentages when there's a sale going on." Best quote ever! :-) – James Aug 16 '14 at 19:19

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.

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    Your comment about symmetry is interesting: for symbolic $x$ and $y$, I've seen far more cases of $1/(x+y)$ turning into $1/x + 1/y$. I call it "miraculous simplification", which is often observed in students computing indefinite integrals... – Erick Wong Jan 11 '14 at 19:19

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.

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

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


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

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).

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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.

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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.

Kevin Lam
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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.

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    Understanding the distributive property should be helping with this particular case: `(x + 3)(x + 3)` = `x(x + 3) + 3(x + 3)` I think there's something else the students are probably missing. – Izkata Jan 07 '14 at 19:29
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    They're missing the idea that says $(x+3)^2$ is "the quantity $x+3$, all of it squared, now apply distributivity." Instead, they see things outside parenthesis as needing to apply to all the things inside the parenthesis. In my experience, the single most common missed concept is that parenthesis group terms not simply for clarity or convenience, but rather to represent a single quantity. – Emily Jan 07 '14 at 19:35
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    I actually find it easier to explain $(x+3)(x+3) = (x+3)*x + (x+3)*3 $. IE, treating the _first_ $(x+3)$ term as a single unit, it becomes identical to $y(x+3)$, which then is easier to understand as $y*x + y*3$. – Joe Jan 07 '14 at 21:45
  • Perhaps some students initially have difficulty in believing that expanding an expression as simple as $(x+3)^2$ cannot be done in a single step (i.e. it's a multi-step problem, requiring multiple applications of the distributive property). – John Joy Jul 22 '15 at 13:35

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.

Tom Au
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    Yes, that's true, but how do you *teach* this? – miniBill Jan 07 '14 at 20:18
  • @miniBill: I would say, by example. Have students work exercises until they can tell the difference between (true) linearity in numerators, and (false) linearity in denominators. True-false questions are easiest for students. If they can't master those, they can't master anything. – Tom Au Jan 07 '14 at 20:38
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    What the OP said is: "I've tried with examples, and failed, what should I try next?" – miniBill Jan 07 '14 at 20:40
  • @miniBill I don't think that precludes *better* examples. – Matthew Read Jan 08 '14 at 07:26

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.

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  • Could the down voter please tell me why they have done so? I will aim to improve my answer if there is an issue with it. – Ahrz Jan 13 '14 at 11:17
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    I down-voted because I don't think this would be a good way to solve the problem the OP referred, that is; that students don't apply “the law of universal linearity” only in specific cases. "Hammering" those into their heads (or any of the specific cases you referred) wouldn't solved the problem at all. Therefore, I up-voted user18921's, noobermin's and Vladimir's answers. – JMCF125 Jan 19 '14 at 12:25
  • In my experience with European students the trick of the "line in the fraction" works every time so I have seen this to be a successful solution to that particular problem. The second one is always tricky to get across. Using simple examples and counter examples of how the rule works and building on that has also shown success with the majority of the students I have worked with. Perhaps "hammer" was too strong a word. – Ahrz Jan 20 '14 at 09:04
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    Perhaps. And I'm not saying those were bad ideas that wouldn't work. I mean, the examples may be good, but the OP has already tried examples; they don't seem to work generally. I feel the answers I up-voted would more likely take care of that general «universal linearity law» problem. – JMCF125 Jan 20 '14 at 11:15

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.


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.

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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."

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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

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

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

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

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.

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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.

Mario De León
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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.

John Joy
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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.

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


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

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.

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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!

Ahaan S. Rungta
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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".

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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.

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    I've used this technique often – just substitute some concrete values and make sure your last operation makes sense and is valid. That said, though, I've also learned to avoid using 2's and 4's, as you used in your example, because, when x=2, x+x=x*x, and 4-x = 4/x, so you could get a false positive. I feel a little safer using 5's and 3's instead. – J.R. Jan 09 '14 at 23:54

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:
$a \times b = b \times a$
$(a+b)\times c = a \times c + b\times c$

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.

$2^{-3} \neq -2^3$ 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:
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.
Tu Bui
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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.

rich remer
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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.

Daniel R Hicks
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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.

Andrew Graham
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