# Whence the “everything is linear” phenomenon, and what can we do about it?

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

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.

• 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.

• 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 '14 at 22:09
• 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 '14 at 7:22
• I think a better word choice is "everything commutes". – TheBluegrassMathematician Apr 28 '14 at 18:09
• The "everything has a linear/proportional effect" is much, much more widespread. – vonbrand Apr 29 '14 at 1:28
• I taught my Calculus I students: "if you think you have an identity like $\sqrt{x + y} = \sqrt x + \sqrt y$, then try plugging in specific numbers. If they don't come out equal, it's not really an identity." On the exam, one student actually wrote: "I want to simplify $\sqrt{x^2 + y^2}$. Try $x = 2$ and $y = 3$: $\sqrt{4 + 9} = 3.605$, but $\sqrt4 + \sqrt9 = 5$." On the next line, he continued: "$\sqrt{x^2 + y^2} = x + y$". (I pass otherwise silently over the issue of believing that $\sqrt{4 + 9} = 3.605$.) – LSpice Apr 2 '15 at 15:47

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.

• Thank you, this is what I was looking for! I will check out these articles. – Brendan W. Sullivan Mar 28 '14 at 18:01
• 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 '14 at 18:16
• "... 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 '14 at 2:34
• 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 '14 at 13:45
• Do you have an electonic copy of of Matz (1982)? I have seen it referenced as discussing "overgeneralised linearity" in a paper of Zazkis (pdf) but found nothing in my various searches. – Benjamin Dickman Jun 8 '15 at 17:10

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.

• 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 '14 at 0:30
• 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. – person27 Mar 28 '14 at 6:59
• 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 '14 at 17:35
• It's worth pointing out that $\sqrt{ab}\neq\sqrt{a}\sqrt{b}$ and this IMO is exactly the sloppy thinking that leads to $(a+b)^2=a^2+b^2$ the fact that noone has mentioned that seems scary. – DRF Mar 22 '17 at 13:46
• Well, the way in which products of square roots need not be square roots of products is qualitatively different, and subtler. I think to first order it is ok to think that they are the same, etc... – paul garrett Jun 27 '17 at 21:40

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.

• 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. – Brendan W. Sullivan Mar 28 '14 at 0:08
• @brendansullivan07: This tweaks that "give counterexamples" suggestion in two ways: 1) students can try random examples themselves, and 2) it helps to reduce everything to decimals: $\frac{1}{2}=2$ may be vivid to you but for most people $0.5=2.0$ is more striking. – user173 Mar 28 '14 at 0:13
• "Random numerical testing" can also be further dignified by calling it "the Monte Carlo method". – paul garrett Mar 28 '14 at 14:14
• 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 '14 at 21:08
• @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 '14 at 13:30

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.

# Edit

I'm now reading Thinking, Fast and Slow by the überpsychologist Danny Kahneman. In the book Kahneman provides a conceptual framework that attempts to describe how people's brains function and he does this by distinguishing 'between two modes of thought: "System 1" is fast, instinctive and emotional; "System 2" is slower, more deliberative, and more logical'. System 1 is what allows us to map the physical world, to react quickly to stimuli (as weather conditions, people's expressions and body language, driving, accidents about to happen, etc). System 2, on the other hand, is what is engaged when we actually think, when we solve a problem, when we perform a difficult task. I believe that my answer above is very much related to these ideas and can be summarized as follows.

Where does it come from?
It comes from System 1 being in the driver's seat.
What can we do about it?
Train System 2 to takeover.

Below is a section of the book which I think is very relevant and exemplificative of the issue at hand.

## The Lazy System 2

One of the main functions of System 2 is to monitor and control thoughts and actions “suggested” by System 1, allowing some to be expressed directly in behavior and suppressing or modifying others. For an example, here is a simple puzzle. Do not try to solve it but listen to your intuition:

A bat and ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?  A number came to your mind. The number, of course, is 10: 10¢. The distinctive mark of this easy puzzle is that it evokes an answer that is intuitive, appealing, and wrong. Do the math, and you will see. If the ball costs 10¢, then the total cost will be \$1.20 (10¢ for the ball and \$1.10 for the bat), not \$1.10. The correct answer is 5¢. It is safe to assume that the intuitive answer also came to the mind of those who ended up with the correct number — they somehow managed to resist the intuition.
Shane Frederick and I worked together on a theory of judgment based on two systems, and he used the bat-and-ball puzzle to study a central question: How closely does System 2 monitor the suggestions of System 1? His reasoning was that we know a significant fact about anyone who says that the ball costs 10¢: that person did not actively check whether the answer was correct, and her System 2 endorsed an intuitive answer that it could have rejected with a small investment of effort. Furthermore, we also know that the people who give the intuitive answer have missed an obvious social cue; they should have wondered why anyone would include in a questionnaire a puzzle with such an obvious answer. A failure to check is remarkable because the cost of checking is so low: a few seconds of mental work (the problem is moderately difficult), with slightly tensed muscles and dilated pupils, could avoid an embarrassing mistake. People who say 10¢ appear to be ardent followers of the law of least effort. People who avoid that answer appear to have more active minds.
Many thousands of university students have answered the bat-and-ball puzzle, and the results are shocking. More than 50% of students at Harvard, MIT, and Princeton ton gave the intuitive—incorrect—answer. At less selective universities, the rate of demonstrable failure to check was in excess of 80%. The bat-and-ball problem is our first encounter with an observation that will be a recurrent theme of this book: many people are overconfident, prone to place too much faith in their intuitions. They apparently find cognitive effort at least mildly unpleasant and avoid it as much as possible.
Now I will show you a logical argument—two premises and a conclusion. Try to determine, as quickly as you can, if the argument is logically valid. Does the conclusion follow from the premises?

 All roses are flowers.


A large majority of college students endorse this syllogism as valid. In fact the argument is flawed, because it is possible that there are no roses among the flowers that fade quickly. Just as in the bat-and-ball problem, a plausible answer comes to mind immediately. Overriding it requires hard work—the insistent idea that “it’s true, it’s true!” makes it difficult to check the logic, and most people do not take the trouble to think through the problem.
This experiment has discouraging implications for reasoning in everyday life. It suggests that when people believe a conclusion is true, they are also very likely to believe arguments that appear to support it, even when these arguments are unsound. If System 1 is involved, the conclusion comes first and the arguments follow.
Next, consider the following question and answer it quickly before reading on:

How many murders occur in the state of Michigan in one year?


The question, which was also devised by Shane Frederick, is again a challenge to System 2. The “trick” is whether the respondent will remember that Detroit, a high-crime city, is in Michigan. College students in the United States know this fact and will correctly identify Detroit as the largest city in Michigan. But knowledge of a fact is not all-or-none. Facts that we know do not always come to mind when we need them. People who remember that Detroit is in Michigan give higher estimates of the murder rate in the state than people who do not, but a majority of Frederick’s respondents did not think of the city when questioned about the state. Indeed, the average guess by people who were asked about Michigan is lower than the guesses of a similar group who were asked about the murder rate in Detroit.
Blame for a failure to think of Detroit can be laid on both System 1 and System 2. Whether the city comes to mind when the state is mentioned depends in part on the automatic function of memory. People differ in this respect. The representation of the state of Michigan is very detailed in some people’s minds: residents of the state are more likely to retrieve many facts about it than people who live elsewhere; geography buffs will retrieve more than others who specialize in baseball statistics; more intelligent individuals are more likely than others to have rich representations of most things. Intelligence is not only the ability to reason; it is also the ability to find relevant material in memory and to deploy attention when needed. Memory function is an attribute of System 1. However, everyone has the option of slowing down to conduct an active search of memory for all possibly relevant facts—just as they could slow down to check the intuitive answer in the bat-and-ball problem. The extent of deliberate checking and search is a characteristic of System 2, which varies among individuals.
The bat-and-ball problem, the flowers syllogism, and the Michigan/Detroit problem have something in common. Failing these minitests appears to be, at least to some extent, a matter of insufficient motivation, not trying hard enough. Anyone who can be admitted to a good university is certainly able to reason through the first two questions and to reflect about Michigan long enough to remember the major city in that state and its crime problem. These students can solve much more difficult problems when they are not tempted to accept a superficially plausible answer that comes readily to mind. The ease with which they are satisfied enough to stop thinking is rather troubling. “Lazy” is a harsh judgment about the self-monitoring of these young people and their System 2, but it does not seem to be unfair. Those who avoid the sin of intellectual sloth could be called “engaged.” They are more alert, more intellectually active, less willing to be satisfied with superficially attractive answers, more skeptical about their intuitions. The psychologist Keith Stanovich would call them more rational.

• 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 '14 at 1:37
• This I think is an extremely essential point. Often students don't know what else to do and have been taught that doing something is better then doing nothing since in the US at least but other places possibly there is nothing to lose by having nonsense on the exam. In my country when I got these types of answers during an oral exam that was the end of the exam. Students soon learn it's better to think for a while or say I don't know instead of saying whatever randomness comes first. – DRF Mar 22 '17 at 13:44
• "It is safe to assume that the intuitive answer also came to the mind of those who ended up with the correct number" That's quite presumptuous. There's $1.10 shared between the bat and the ball. Of that,$1 is unique to the bat. Therefore, the remaining $.10 is shared between the bat and the ball. So "clearly" the ball costs$.05. Maybe most of the people who got the right answer suppressed their intuition, but it's odd to just dismiss out of hand the possibility that some people find the right answer intuitive. – Acccumulation Apr 17 '18 at 15:23

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. 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.

Edit: Glancing back at this, it occurs to me I should mention that I would probably blame government policy more than teachers, who no doubt are under lots of unhelpful pressure and stress which may make it much more difficult to teach "well" - this is just what I have observed from my teachers from an objective standpoint.

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.

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.

I've read all the existing answers long ago but still feel that none have gotten to the heart of the issue. We obtain mathematical results through a process of reasoning. That reasoning must be logical and enough to convince anyone that our results are correct given our initial assumptions. That is the actual purpose of a proof. It does not matter what form the reasoning takes, whether using only words or only mathematical symbols or only a diagram. The requirement is simply to convince the other person. If we cannot do so, then our reasoning is insufficient or incorrect.

This proper attitude must start right from the basics. For example $\frac{1+2}{1+3} \ne \frac{\not{1}+2}{\not{1}+3}$. Explaining to the student that one cannot do that is almost useless. Instead, the student should be asked: "Why do you cancel?" and then "Why does cancelling keep the value the same?".

The problem is that if this is not done from the beginning of arithmetic, it simply causes students to create for themselves a deep quagmire of guesswork in order to heuristically write down things which they believe will get them their grades. If you have seen students who try to mimic their teachers' phrasing but clearly without understanding of the meaning, or students who care only about how to get the answer and not why the method is correct, you know what I mean.

As a result, very few students have a full grasp of even the fundamentals, namely the field of rationals. What I mean by this is that few are able to state all the field axioms correctly and prove results like the uniqueness of inverses (when they exist) and that $0 \times x = 0$ and that $-x \times -y = x \times y$. (Out of these, fewer still can give any explanation as to the rationale for the axioms, but that is another topic.)

It is obvious that with a proper foundation as I briefly described above, no student would ever write $(a+b)^2 = a^2+b^2$. Why? Because they know that "$x^2$" is defined as "$x \times x$" and "$()$" are used to denote what to do first, so $(a+b)^2 = (a+b) \times (a+b)$. Moreover, they also would know the distributivity field axiom that gives first $(a+b) \times (a+b) = a \times (a+b) + b \times (a+b)$ and then after 2 more applications the full expansion, using the commutativity and associativity axioms. Likewise none of the other mistakes that you mentioned would occur.

Furthermore, if students cannot handle the field axioms correctly, one might as well throw the induction axiom out of the window. The way it is taught in most textbooks and curricula is seriously lacking, precisely because it is not based on sufficiently formal reasoning. A simple example that most students who were brought up with textbook induction fail to solve is:

Given a function $f:\mathbb{Z}\to\mathbb{R}$ such that $f(0) = 0$ and $f(1) = 1$ and $f(x+1) + 6 f(x-1) = 5 f(x)$ for any $x \in \mathbb{Z}$, prove that $f(x) = 3^x - 2^x$ for any $x \in \mathbb{Z}$.

It is not hard at all, but only those who understand the logical structure of induction would be able to give a correct proof. In case anyone is wondering what I mean by textbook induction, two examples that I would consider seriously lacking are:

Finally, proper reasoning naturally requires sufficient precision, because one cannot reason logically about statements whose meaning is undefined or unclear. Vagueness in mathematics is one great recipe for confusion. This must start with the teacher. A teacher who is sloppy with mathematical statements or steps in reasoning is simply telling the students that it is alright to be sloppy and by extension it is alright if they do not know what they are doing as long as they get the answer!

One terrible example of sloppiness in most high-school curricula is solving differential equations by "separating variables". Try giving the following to any student:

Solve for $y$ as a function of a real variable $x$ given that the differential equation $\frac{dy}{dx} = 2\sqrt{y}$ holds.

You know what answer to expect, and I hope you know the correct answer. Even Wolfram Alpha gets it wrong. Now for students who give the wrong answer, tell them that it is wrong but do not tell them the correct answer, and ask if they can identify the mistake and fix it. Most will fail to identify the mistake, and fixing the mistake will require the foundation in logic that most students do not have.

Here are the solution sketches for the problems I've given above. I strongly encourage one to thoroughly check one's own work to verify whether each step follows completely logically from the preceding deductions, and merely look at these solutions to confirm.

Problem

Given a function $f:\mathbb{Z}\to\mathbb{R}$ such that $f(0) = 0$ and $f(1) = 1$ and $f(x+1) + 6 f(x-1) = 5 f(x)$ for any $x \in \mathbb{Z}$, prove that $f(x) = 3^x - 2^x$ for any $x \in \mathbb{Z}$.

Hints

Induction only allows you to derive something about the natural numbers. The desired theorem is about integers. Also, if you cannot prove the implication needed for the induction, a key technique that often works is to strengthen the induction hypothesis to include enough information so that you can prove the implication step. Of course that also means that the implication you need to prove has changed!

Solution sketch

First notice that the theorem to be proven is that $f(x) = 3^x - 2^x$ for all integers $x$, and so induction in one direction is not enough! Also, notice that it is impossible to prove that $f(x) = 3^x - 2^x$ implies $f(x+1) = 3^{x+1} - 2^{x+1}$, and hence the induction hypothesis must contain information about at least two 'data points' for $f$. The easiest one would be to let $P(x)$ be "$f(x) = 3^x - 2^x$ and $f(x-1) = 3^{x-1} - 2^{x-1}$". Then one must prove $P(x+1)$, which expands to "$f(x+1) = 3^{x+1} - 2^{x+1}$ and $f(x) = 3^x - 2^x$". I would not accept if the student does not fully prove $P(x+1)$. This would handle the natural numbers, and a similar induction would handle the negative integers. It is of course possible to combine both inductions into one, which it should be explored, although in general it is good to keep a proof as modular as possible.

Problem

Solve for $y$ as a function of a real variable $x$ given that the differential equation $\frac{dy}{dx} = 2\sqrt{y}$ holds.

Hint

The answer is not $y = (x+a)^2$, which you would get by the method of separating variables. What went wrong? Note that the error would still be there if you used the theorem that allows change of variables in an integral. Look carefully at each deduction step. One step cannot be justified based on any axiom. Think basic arithmetic. After you get that, you need to consider cases and use the completeness axiom for reals to extend the open intervals on which the standard solution works.

Solution sketch

The field axioms only give you a multiplicative inverse when it is not zero. Now how to solve the problem? Split into cases. Note that you need to work on intervals since having isolated points where $y$ is nonzero is useless. First prove that for any point where $y \ne 0$, there is an open interval around $x$ for which $y \ne 0$. Then we can use the completeness axiom for reals to extend the interval in both directions as far as $y \ne 0$. Now we can use any method to solve for $y$ on that interval. Note that the method of separating variables is formally invalid, so we should use the change of variables substitution. But the prerequisite for that is that $\frac{dy}{dx}$ is continuous, so we need to prove that! Well, $y$ is differentiable and hence continuous, so $2\sqrt{y}$ is continuous. So we get the solution on the extended interval, and it shows that $y$ becomes zero in exactly one direction in this example. Hence after some checking you will get either $y = 0$ or $y = \cases{ 0 & \text{if } x \le a \\ (x-a)^2 & \text{if } x > a }$ for some real $a$.

Alternative subproof

In fact, the substitution theorem can be completely avoided as follows. On any interval $I$ where $y \ne 0$, we have $y'^2 = 4y$, where "${}'$" denotes the derivative with respect to $x$. Thus $(y'^2)' = (4y)'$, which gives $2y'y'' = 4y'$, and hence $y'' = 2$ since $y' = 2\sqrt{y} \ne 0$. Thus $y' = 2x+c$ on $I$ for some real $c$, and hence $y = x^2+cx+d$ on $I$ for some real $d$. Note that most of the above steps are not reversible and hence we need to check all the solutions we finally obtain with the original differential equation. We would get $c^2 = 4d$. After simple manipulation we obtain the same result for $y$ on $I$ as in the other solution. The other parts of the solution still need to be there.

• I think you're right about the first part: students develop a veritable morass of heuristics and "rules" that somehow "make sense" to them because they get the "right answer" but they cannot explain them at all! – Brendan W. Sullivan May 1 '15 at 18:37
• I don't understand your differential equation example, though. In what sense does Wolfram "get it wrong"? What mistake do you observe students making? I think you should explain it here as opposed to being coy with what you had in mind.. – Brendan W. Sullivan May 1 '15 at 18:38
• Of course. But I think answers on this site should be self-contained and immediately helpful. There's nothing to be gained by being coy/deceptive about it! – Brendan W. Sullivan May 2 '15 at 4:10
• @brendansullivan07: No that's not my purpose. My purpose is to not give an easy way to get the solution, to neither students nor teachers. I want people to spend enough effort to get the answer on their own, before I give any answers. But as I said, give me a while to put the answers in. – user21820 May 2 '15 at 4:20
• @ChrisCunningham: Guess what... I made that mistake in my first version of my answer! For that function when $x < b$ the gradient is negative, but $2 \sqrt{y}$ is positive, so it is not a solution. However, it is a solution to the differential equation I stated 5 comments up. =) – user21820 Sep 10 '15 at 9:11

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').

• I had always been one of those "some kids"; page 30 of this and this remind me of how I typically want to object upon being pushed such an ill-formed question. – Vandermonde Jan 30 '15 at 0:42
• This is an interesting perspective that I hadn't considered before. Somehow I feel this is a more subtle and deliberate form of linearity as simplifying assumption that it's hard for me to connect it to the phenomenon in the question. The latter type of linearity seems more likely to conclude that John and Mary together paint the house in 8 hours! – Erick Wong May 16 '15 at 17:24
• The manager joke is iconic! – Engelsmann Sep 16 '17 at 14:11

I am deeply convinced that these fallacies come from the way we teach maths.

There is some research indicating that for the over reliance on linearity (see e.g. page 51 of the last EMS newsletter, as I commented on another answer giving more references).

But it seems to me that this is a particular case of a more general phenomenon: maths are just thought by many student as black magic. It is about learning and reciting formulas, using precise methods to solve precise exercise, always in the same way. Beware if you misspell a formula, as you might summon an efreet by accident (i.e. the teacher will be mad at you)! In other words, students are lacking the relation between symbols, rules, formulas, theorems in the one hand, and the meaning of them in the other hand. Without such connection, there is simply no way to tell a correct formula from an incorrect one; even checking may be out of the question, as replacing letters with specific values can only be done with a certain understanding of the role of variables, as opposed to the mere ability to reproduce formal manipulations of symbols.

I do not know if the following hypothesis has been tested rigorously, but I also think that this is linked very closely to the way we evaluate our students: the more we ask them to solve standardized exercises, the less sense and reasoning they would put into their maths.

• "Deeply convinced" sounds overstrong for something where you only have "some research" (not spelled out) and a rationale. [As opposed to organized evidence.] There is a concept in critical thinking called multiple hypotheses. You might consider alternate explanations. Rather than misteaching, it could be the result of how many places linearity does work. Or trying simplest first. [And if anything a rules-based computer (person) would be LESS likely to guess linearity as he just crunches an algorithm, versus guessing a simpler model.] – guest Apr 15 '18 at 14:42
• I am deeply convinced because the roots of this conviction are deep in me (reflecting on my experience as a math teacher). I am not asking anyone to trust me on my word, I tried to convey my feeling and I guess other teachers relate to this. (Just as I do not think we should ask students to think a formula is true just because we teachers claimed it.) – Benoît Kloeckner Apr 16 '18 at 20:08

What can we do about it?