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$$ \color{red}{(a+b)^2 = a^2+b^2}$$ $$ \color{red}{\sqrt{x^4+y^4} = x^2+y^2} $$ $$ \color{red}{e^{t^2+C} = e^{t^2}+e^C}$$

I've observed this phenomenon -- wherein, implicitly, students say, "Everything is linear! Just pass the operation through!" -- in courses at all levels. High school students. Undergraduates in calculus. It's all over the place.

I'm interested in two things about this phenomenon:

  • Where does it come from? Is there a human tendency to view things linearly? More specifically, is there a reason that this occurs quite frequently with students of mathematics? Do we unknowingly "teach" this behavior, only to have to "unteach" it later?
  • What can we do about it? Is there any evidence that this can be "unlearned" at a later age, and how can this be done effectively? Do we need to just accept that it is part of human learning to view the world linearly, and instead focus on ways of addressing it as it comes up? How can this be done?

This phenomenon was addressed in a recent math.se thread: Pedagogy: How to cure students of the “law of universal linearity”? It is a community wiki thread with many answers, but the main question there was essentially, "Do you have a good explanation for when a student commits an error like this that will make them not do it again?" Many answers are suggestions about showing the student why their reasoning is flawed. A couple start to address the "Where does this come from?" question, but not to the depth that I would like to hear about. Some answers amount to, "There's no good answer", interestingly. And none of them (as of the posting of this current thread) cite any research about this.

So, I'm hoping that this thread, on matheducators.se, can broaden the types of answers seen there. I really want to understand why and how this behavior develops (is it innate, is it a consequence of the way we teach, is it both?) and what we can do to address it effectively (both in the short term, like that other thread, but also in the long term). I'm sincerely hopeful that there is research out there about this phenomenon and that it can be shared here.

Addendum: Please do not duplicate answers/suggestions from that math.se thread. I am looking for significant evidence of effectiveness of techniques, and do not want this thread to become a big list of suggestions.

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Behavioral studies tell me that negative reinforcement always works better... so what about telling them that "you are right, but in mod-2 arithmetic: $a^2 + b^2 = (a+b)^2\, (mod \,2)$? –  wilsonw Mar 29 at 7:10
    
I often see students do $\sqrt{x+y}=\sqrt{x}+\sqrt{y}$. I point out to them that if this were true, the Pythagorean theorem would simplify from $a^2+b^2=c^2$ to $a+b=c$. This seem to work for that particular mistake, but I don't think it fixes the underlying problem, which is probably some combination of lack of technical skill and lack of discipline in reasoning according to well-defined principles. –  Ben Crowell Mar 29 at 22:09
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Linearity/distributivity, when first introduced, may appear to be a property of brackets, rather than a property of multiplication, because the multiplication sign is suppressed. This doesn't cover all cases obviously, but this is an explanation I've gotten for attempts at $(x+y)^{1/2}$ or $\sin(a+b)$ from various students. "Can't you just do that when there are brackets?" Nope, and this should be strongly emphasized early on. –  Robert Mastragostino Mar 30 at 7:22
    
I think a better word choice is "everything commutes". –  TheBluegrassMathematician Apr 28 at 18:09
    
The "everything has a linear/proportional effect" is much, much more widespread. –  vonbrand Apr 29 at 1:28

9 Answers 9

The problem you describe is well-known in mathematics education research. I cite the paper of De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311–334. and give some of the references mentioned:

As the concept of linearity itself, the misuse of linearity has many faces: it has been found at different age levels and in a variety of mathematical domains (see, e.g., De Bock et al., 1999). In elementary arithmetic, the phenomenon of improper proportional reasoning is often related to a ‘lack of sense-making’ in the mathematics classroom (Gagatsis, 1998; Greer, 1993; Nesher, 1996; Verschaffel et al., 1994, 2000; Wyndhamn and Säljö, 1997). When confronted with so-called ‘pseudoproportionality problems’ (such as, e.g. “It takes 15 minutes to dry 1 shirt outside on a clothesline. How long will it take to dry 3 shirts outside?”), many students give answers based on direct proportionality (i.e., tripling the drying time because the number of shirts is tripled). [...]

In secondary education, ‘linearity errors’ are often reported in the fields of algebra and (pre)calculus. Students tend to overgeneralise what has been experienced as ‘true’ for linear functions to non-linear functions (e.g. “the square root of a sum is the sum of the square roots” or “the logarithm of a multiple is the multiple of the logarithm”). This type of systematic errors has been discussed and illustrated by Berté (1987, 1993), Gagatsis and Kyriakides (2000) and Matz (1982). According to Matz (1992), these linearity errors result from students’ overgeneralisation of the distributive law. The immense number of occasions wherein students add and use the distributive law in arithmetic and early algebra is very likely to reinforce students’ acceptance of linearity.

Berté, A. (Réd.): 1987, Enseignement des mathématiques utilisant la ‘réalité’, Tome 1, IREM, Bordeaux.

Berté, A.: 1993, Mathématique dynamique, Nathan, Paris.

De Bock, D., Verschaffel, L. and Janssens, D.: 1999, ‘Some reflections on the illusion of linearity’, Proceedings of the 3rd European Summer University on History and Epistemology in Mathematical Education, Vol. 1, Leuven/Louvain-la-Neuve, Belgium, pp. 153–167.

Gagatsis, A. and Kyriakides, L.: 2000, ‘Teachers’ attitudes towards their pupils’ mathematical errors’, Educational Research and Evaluation 6(1), 24–58.

Matz, M.: 1982, ‘Towards a process model for high school algebra errors’, in D. Sleeman and J.S. Brown (eds.), Intelligent Tutoring Systems, Academic Press, London, pp. 25–50.

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Thank you, this is what I was looking for! I will check out these articles. –  brendansullivan07 Mar 28 at 18:01
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If you have problems accessing some papers, let me know. I am not sure if I can get them. At least, I have the main paper. –  Anschewski Mar 28 at 18:16
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"... overgeneralisation of the distributive law." — this is it in a nutshell and also a hidden explanation for why. There is no such thing as "the distributive law." There's a "distributive property of multiplication over addition" (and potentially one of exponentiation over multiplication, or some such), but that whole "of something over something else" part is important. –  Isaac Mar 29 at 2:34
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Here's a fun game to play. Give your students the example of what to simplify before you teach them how to simplify it. What I notice is quite often the type of simplifications described here are what students give as answers before I teach them, and given that I often see these mistakes after teaching them, it suggests my part approach was wrong. Derek Muller suggests that in science education we should give counters to common misconceptions as well as "the facts." See www.youtube.com/watch?v=eVtCO84MDj8‎. I feel fairly certain that this approach might be successful here as well. –  David Wees Mar 31 at 13:45

I don't view these common mistakes as 'universal linearity' assumptions. The mistake that $(a+b)^2=a^2+b^2$ is just a visually appealing statement. It is mistaken to be correct because it looks nice. Our brains tend to like things that look nice. Similarly, $\sqrt{a+b}=\sqrt a+\sqrt b$ is visually appealing and it resembles the correct formula $\sqrt {ab}=\sqrt a\cdot \sqrt b$. This is a fallacy of the kind "I don't really understand why all these algebraic rules for square root are true, I never really took the time to see the proof and I never really stopped for a second to think about the equalities I was given. Instead I just vaguely tried to remember them, totally devoid of content. Thus, I kinda remember that square root behaves like that, kinda, I hope, and so I'll just compute that way. Moreover, I have so little ability to compute even the simplest things in my head, and I don't feel like pulling out my trusty pocket calculator, that I am completely incapable of detecting the falsehood of this silly claim."

There are plenty of such fallacies, linearly looking or not. The cause, I believe, is simply a fundamental lack of understanding of the formulas coupled with a deep lack in ability to compute very simple things mentally. The way to deal with that is to include true/false questions of this type where the students need to prove or provide a counterexample, and disallow use of calculators.

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Indeed, I agree with your assessment. The simple truth is that these errors are not rooted in logical misapplication of rules. Rather, they are rooted in apathy mixed with poor background in arithmetic. For some, it's just apathy, I speak from my own experience both as a teacher and as a student. –  James S. Cook Mar 28 at 0:30
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The first example, $(a + b)^2 = a^2 + b^2$ in the US is the result of being taught the "distributive method", which is a shortcut method that isn't always true. I fell for the same logic myself, but I can assure you, it wasn't because I thought everything should be or was intended to be linear. –  FizzledOut Mar 28 at 6:59
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Excellent answer. Saved me a lot of typing. I see this (and other similar) a lot with my students, and it's always a lack of understanding of how numbers work, coupled with a desire to just get the question done. –  ChrisA Jul 31 at 17:35

This became to big to be a comment.

Layman's opinion.

Where does it come from?
It comes from the fact that universal linearity is useful to move forward in calculations even if it's wrong. Psychologically this is very attractive. The other option is being stuck. Moving forward has the added incentive that it can be right, that maybe the student can get some points. Another reason why this might happen is that typographically linearity is aesthetically pleasing, so it should be right...

What can we do about it?
A solution is to increase critical thinking overall. This can be done by not slacking off on detail, rigor and logic. Students should be taught to justify every step by going back to the definitions or previously proved properties. But this is school of thought isn't used outside of proof-based courses and this is what happens.

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This hits the nail on the head. The alternatives are often "give up" or "try what looks right". What's described as "universal linearity" looks right to a person without the ability to think critically. –  Jonathan Mar 28 at 1:37

You can encourage your students to check their algebra with random numerical examples.

E.g.: Try setting $a=x=t=2$ and $b=y=C=3$. Then the equations above give $$25=13$$ $$9.85=13$$ $$1096.6=74.7$$ which are vividly wrong.

If one pair of numbers comes out wrong, your algebra was definitely wrong. If a couple pairs of numbers come out right, your algebra was probably right. It's a great heuristic.

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I do this quite frequently. And this is almost identical to the "Give counterexamples" suggest in this answer: math.stackexchange.com/a/630653/37705 I am looking for more significant evidence about whether this works. –  brendansullivan07 Mar 28 at 0:08
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"Random numerical testing" can also be further dignified by calling it "the Monte Carlo method". –  paul garrett Mar 28 at 14:14
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While this sounds like it would be effective, my experience teaching algebra to college students suggests that this method is actually completely useless to anyone who does not already understand what is going on. Since although I believe you've posted this with good intentions, I believe you do not have any directly relevant experience, I've downvoted the answer. Nothing personal; have a good day! –  Chris Cunningham Mar 28 at 21:08
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Interesting -- I wonder what causes the difference. Maybe it works with small groups who are independently interested in understanding what is going on? Sorry for the strongly-worded comments, by the way, but I have strong feelings on this particular issue! –  Chris Cunningham Mar 29 at 14:53
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@MichaelE2: In my (admittedly limited) experience, teaching students to routinely perform these checks is a very effective way to help them unlearn wrongly generalized algebraic manipulation rules, and can even provide a kind of immunity against such overgeneralization in the future. It's not, however, a quick fix -- it takes many hours of active teaching and practice to instill such a routine, and many more for the students to slowly figure out which of the rules they thought they knew are actually wrong, and why. It's even harder if you're trying to teach something new at the same time. –  Ilmari Karonen Mar 30 at 13:30

As a student myself, I'd say that, while I'm not representative of all students, some of it is the intimidating or dismissive way some lecturers or teachers might treat questions relevant to simple things like these, or not teach off the exam specification perse. This leads to a subconscious or even conscious bias against asking these sort of questions which we're unsure of.

For example, if I asked a question such as why you can separate roots with multiplication signs, I might be told to just "accept it and learn it" or if I asked if you could do something incorrect, such as one of the examples you gave, I'm likely to receive a response like "Of course not! Don't be silly!"

I think these two factors especially lead people to just try to memorise as much of the method, but not logic or reasoning, behind the mathematics, and as a result end up getting it wrong, improvising, or guessing.

Personally, I think it's a lot due to teachers not concerning themselves with a student's future, only getting the student past the current exam hurdle, which results in generations of students who are assumed to know something relatively simple, but since the teachers deemed it unnecessary to explain since they don't need the explanation for the exam at hand, never learnt it.

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What can we do about it?

Make them feel bad about it?

But seriously...I don't think it's necessarily a linearity thing. I agree with Ittay Weiss that those "simplifications" look nice, resembling the correct shortcuts they've been shown.

I've tried some things to overcome this: having students numerically verify all simplifications below a certain level (pros - they must learn to use a calculator effectively and they know when they've made a mistake; cons - many of them do not get a handle on using their calculator, this doesn't show them where they went wrong with the algebra anyway), having students focus on identifying errors by having them create multiple-choice tests (pros - they have to name the common errors and put them in context, they get practice answering the tests; cons - the skills are single-step simplifications outside of a larger problem, they may spend too much time thinking about potential errors).

...which results in generations of students who are assumed to know something relatively simple, but since the teachers deemed it unnecessary to explain since they don't need the explanation for the exam at hand, never learnt it.

I think Ashley is right on with this. I think that not enough review material is built into our courses, except for the first day or two. I feel like I have to really hustle to keep my students practicing "old" material so they don't take a vague memory and improvise or guess their way through it when they need it.

My answer: Tell your students that you expect they will correctly use exponent rules (or whatever), diagnose where they are at (even if it is an advanced class where they "should" know how to do it), then give them resources to build them up to where they need to be.

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Welcome to the site! To improve your answer and get it to be more well-received, I'd recommend removing the parts that refer to other answers. The parts of your answer that refer to other answers are mostly "I liked their answer" without really building on them, and I think this extra material is diluting the content of your post. I hope you know I mean this constructively, and thanks for the answer! –  Chris Cunningham Mar 28 at 21:11

In addition to other good answers and comments... I think it should be noted that "linear mathematics" at higher levels is the part of mathematics that we (collectively) understand relatively well, while "non-linear mathematics" is often intractable... except to the extent we can usefully approximate it linearly.

It is both symbol patterns and the mathematical assertion(s) given by the simplest symbol patterns that are both appealing and very handy ... if correct.

Paraphrasing, and as in some of the other answers, if the choice is between "being stuck" and "making progress", often an admittedly dubious assumption of linearity, or some other mildly outrageous optimistic assumption, is necessary to avoid getting stuck. That is, methodologically, linearity assumptions (and other such) are entirely reasonable... at least as a transitional device.

And, after all, "differentiability" of a function is in many regards the assertion that the function can be locally approximated by a linear function. The Newton-Raphson method shows how iteration of a nearly-linear device achieves excellent effects.

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I guess you have already got plenty to read for "Where does it come from?" part of you question. Thus I just shortly introduce my favorite strategy that I use for "What can we do about it?" part, when there is a case to work on:

Encourage your students to find conditions that linearity assumption indeed works!

For examples, for which values of $a$ and $b$, $(a+b)^2=a^2+b^2$. Quite often, it turns to a nice challenge and rewarding in the direction you need.

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Just to steer in a different direction. I think it's the way we learn mathematics (and I personally don't think this is a good thing). From a very young age, we learn to assume that everything is linear. In elementary school, we get problems like:

If John paints one house in five hours and Mary paints one house in three, how long does it take them to paint one house together?

Some kids will say: "Well, I don't know. Maybe they have to spend some time to divide the work between them, and maybe they spend a lot of time bringing their stuff in. You can't really divide this work unless they both bring half their stuff, but then they would have to share their equipment and they may have to wait for eachother...". But no, according to the teacher and the textbook they are wrong, and the answer should be $\frac{15}{8}$ hours. This reinforces the idea that you should just make the assumption of linearity when the problem is complicated.

In this case probably the only way to do an actual computation is to assume linearity. I think the despair of the teachers to show them that mathematics is useful has driven them to this kind of examples. But, it doesn't stop in primary school, linearity is almost always assumed. I think I even saw questions in mathematics and physics courses where you had to assume linearity to solve the question.

(I think the easiest examples that I can think of have to with pressure and the speed of water streaming out.)

Now linearity is not a bad thing (even if the thing we are trying to compute isn't exactly linearly we can still use it as an approximation), but the hidden assumption of it is a bad thing. Students will get confused and assume things are linear when they don't. The sentence 'a manager is someone who expects that two women can deliver a baby in 5 months' comes to mind now (but actually, you can substitute most students for the manager, and all non-linear things for 'delivering the baby').

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