What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
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10$\begingroup$ In view of many of the answers to this question, it might help to have in the statement a definition of heuristic as it is applied to mathematics. $\endgroup$– Pete L. ClarkCommented Apr 26, 2010 at 3:56
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11$\begingroup$ In fact, the harmful entity in most answers is not a heuristic at all! $\endgroup$– Victor ProtsakCommented May 22, 2010 at 15:07
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1$\begingroup$ Calculus. In many small Universities (mine included) students have to take Calculus before Real Analysis, and I think that this does some serious damage. $\endgroup$– Nick SCommented Jan 24, 2021 at 0:53
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39 Answers
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The "size" of a finite-dimensional vector space is proportional to its dimension.
In fact, the "size" of a finite-dimensional vector space is almost always better thought of as being exponential in its dimension. This is easiest to see for (finite-dimensional) vector spaces over finite fields, which have finite cardinality. But it's a better heuristic even for vector spaces over infinite fields.
Internalizing the correct intuition makes it clear why forming the (algebraic) direct product of two vector spaces causes their dimensions to add, and not to multiply as you might naively expect based on the fact that taking the direct product of groups multiplies their orders.
Another confusing point to which this misconception leads regards the advantage that quantum computers give over classical ones. The difference is sometimes stated as "quantum computers have a state space that's exponentially large in the number of qubits," but this is highly misleading, because classical computers also have a state space that's exponentially large in the number of bits. The better intuition is: since quantum computers have a state space whose dimension is exponentially large in the number of qubits, the state space itself is actually doubly exponential in the number of qubits, while the state space of a classical computer is only singly exponential in the number of bits.
The reason why this misconception is so widespread is that early courses in linear algebra almost always begin with vectors spaces over infinite fields (usually $\mathbb{R}$ or $\mathbb{C}$), which have infinite cardinality, so the dimension is the only finite number available. This practice leads to misleading intuition for general vector spaces.
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"Mathematical knowledge is contained and communicated primarily by documents."
I'm not sure if this is a heuristic, but in terms of beliefs that inhibit learning, this is definitely the one that hurt my mathematical development the most.
I would say the correct statement is "Mathematical knowledge is contained primarily in the minds of mathematicians and communicated primarily by informal oral communication."
This problematic belief grew out of the way that I (and pretty much everyone else) was taught mathematics at the undergraduate and beginning graduate level. In this setting texts are a central authority and a complete, well-written resource for the knowledge needed to solve any mathematical problem encountered.
In the world of mathematical research, this is no longer the case. I finally figured this out by reading Thurston's essay "On proof and progress in mathematics", which I would strongly recommend for any beginning mathematician.
Maybe it is possible to do research mathematics using papers as a primary resource, but I believe this is highly inefficient. I spent several years trying to learn the noncommutative standard model by reading the available papers on the subject and made no real progress. Looking back, I don't think I ever had a chance of succeeding with this approach.
I would guess that to be successful in mathematics, it is absolutely vital to become regularly involved in conversations with working mathematicians, as awkward and intimidating as that might be.
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5$\begingroup$ This has nothing to do with heuristics, whatsoever! $\endgroup$ Commented Apr 17, 2013 at 20:52
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2$\begingroup$ Sorry but I disagree. There is so much buried knowledge in the unread works of the past that I wouldn't be surprised if it surpasses the knowledge of currently living mathematicians. Take into account that many authors forget their own papers after a couple of decades... $\endgroup$ Commented Apr 17, 2013 at 22:56
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3$\begingroup$ I disagree with the disagree-ers. Sure, this answer is a little more "meta" than the question likely intended, but not overwhelmingly so: if we take "a heuristic in math" to mean "a rule of thumb for how to prove things in math", then this answer is arguably on target, even though it is more methodological and less domain-specific. Besides, I think it's an important message to have out there, a realization that every mathematician will have to come to in order to be successful. Even though it's not the sort of issue you'll find discussed in papers :). $\endgroup$ Commented Nov 14, 2013 at 21:52
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2$\begingroup$ Especially a lot of the intuition is communicated orally and informally. And it is impossible to do mathematics without having an intuition about what you're doing. $\endgroup$ Commented Sep 10, 2014 at 15:13
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Talking about "functions" when we are actually talking about equivalent classes of functions almost everywhere equal
An element of a $L^p(X)$ space is usually called a "function", and is usually denoted by letters that are used typically for functions ($f$, $g$, $h$, etc.).
It seems to be a harmful heuristic to act "as if" $L^p(X)$ is made of functions, as a function is really something that should give you a value for each point $x$ in $X$. I am aware that it is now common practice, but I am sure it would help to actually introduce an actual name besides "function" to call "equivalent classes of functions modulo equality almost-everywhere". This concept is fundamental, and should be given a proper name. I don't have a proposition for such name, but I kind of wish someone in the past did.
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4$\begingroup$ So, what actual harm does it cause? $\endgroup$ Commented Jan 18, 2017 at 11:42
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1$\begingroup$ @GerryMyerson It can lead to errors in the theory of Hilbert spaces. Many people incorrectly think of the Hilbert space $L^2(\mathbb{R})$ as the space $\mathcal{L}^2(\mathbb{R})$ of square-integrable functions. But $\mathcal{L}^2(\mathbb{R})$ isn't an inner product space at all, because the naive "inner product" isn't positive definite. Moreover, the "square-integrable function" intuition often leads people to believe that orthonormal bases of $L^2(\mathbb{R})$ have the cardinality of the continuum, which would imply that the Hilbert space is non-separable, when in fact it is separable. $\endgroup$– tparkerCommented Oct 20, 2018 at 3:35
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7$\begingroup$ @GerryMyerson It can lead people to ask for "values" of elements of $L^p(X)$ at points of $X$. For an individual element of $L^p(X)$ this may actually be made sense of (due to some kind of regularity), but it is surely misleading to think that there is a linear functional like "evaluation at a point". $\endgroup$– KapilCommented Oct 27, 2018 at 5:19
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1$\begingroup$ Why not build the term "ekafunction" from sanskrit eka meaning "one" ? The sound "ek" evocates "Equivalence Class" while the meaning of the prefix suggests that one single equivalence class corresponds to several different genuine functions. $\endgroup$ Commented Oct 27, 2018 at 8:26
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The following refers to the school system in Germany, it may be different in other countries:
In my opinion, one really bad heuristic happens in elementary school, when children learn arithmetics with natural numbers. They learn that addition and subtraction are two entirely different things, because they are taught $a+b=b+a$ but $a-b\neq b-a$. Thus addition is commutative, and subtraction is not. At that level, numbers are solely understood as enumerations of objects.
Then they learn about numbers with units, such as lengths or prices or weights. Also they learn that numbers might have geometric meaning, e.g. as lengths of line segments. But still no concept of negative numbers.
Years later, when they finally get to know negative numbers as well, they have so much incorporated that subtraction is something different from addition that they have difficulties to grasp that $a-b=a+(-b)=(-b)+a$, i.e. that subtraction is nothing else than addition of a negative number.
I think that postponing negative numbers so long is a mistake, and that children in elementary school would very well be capable to understand them.
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1$\begingroup$ Not a heuristic, but my least-favourite mannerism in additive groups is the habit of reading '$-x$' as 'negative $x$'. This is at best meaningless (for example, when $x$ is a non-real complex number) and at worst false (students with this habit cannot understand how $|x| = -x$ can be true, since the left-hand side is positive (not necessarily, but that's not the real issue) and the right-hand side is negative …). $\endgroup$– LSpiceCommented Nov 28, 2017 at 20:08
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"Basic (and useful) mathematics is about calculations and higher (pure) mathematics is about proof."
One reason I think this is harmful is that there is no sharp line between calculations and proofs. Very often a certain calculation is essentially the proof except for a few logical connectives. Conversely, in formal logic, one can create a "calculus" that makes proofs appear to be calculations.
Another reason is that it leads students (and more importantly teachers!) to think that a drastic change of mindset is required to learn higher mathematics.
It is indeed true that analysis is quite different from calculus even though there is strong linkage. However, the former also leads to better techniques to calculate things. Putting too much emphasis on the "proof" aspect of analysis tends to put off a lot students who enjoyed playing with polynomials, trigonometry and calculus. Conversely, many students who like to work with proofs are encouraged to believe that what they are doing is somehow "superior" (higher) to "mere" calculus; they then do not do enough computational drills which ill-serves them if they actually take up mathematics!
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1$\begingroup$ I think a drastic change of mindset is required to learn higher mathematics. That doesn't mean throwing away computation, but embracing proof - partly recognizing the importance of proof, and partly coming to grips with proofs having structure beyond that of just algebraic manipulation (e.g. the first time a student learns about proof by induction). I do agree that devaluing computation is bad, but I think you're underestimating the conceptual change needed to move into higher mathematics. $\endgroup$ Commented Oct 27, 2018 at 15:24
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Division by Zero is Infinity. This was taught in my seventh grade while the teacher was explaining the concept of infinity and it's definition(s).
False: division by zero is Undefined.
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4$\begingroup$ Well, as a heuristic that works sometimes, as long as one remembers it's just a heuristic and takes limits to double-check the conclusions. $\endgroup$– MichaelCommented Nov 16, 2016 at 16:59
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4$\begingroup$ -1. Only $0/0$ should be thought of as being defined. Otherwise, it is usually a good idea to consider $1/0=\infty$ as a heuristic. After all, the mapping $x\mapsto 1/x$ becomes an automorphism of the Riemann sphere if and only if one sets $1/0=\infty,1/\infty=0.$ $\endgroup$ Commented Jan 10, 2017 at 23:39
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8$\begingroup$ $0/0$ should be thought of as defined? $\endgroup$ Commented Jan 11, 2017 at 10:42
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$\begingroup$ I meant to say 0/0 should be undefined. $\endgroup$ Commented Jan 24, 2017 at 22:35
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1$\begingroup$ There are certain contexts where this makes sense, such as the projectively extended real line or the Riemann sphere (extremely useful in complex analysis, since it reveals a symmetry between zeros and poles in various theorems). $\endgroup$ Commented Dec 12, 2020 at 3:27
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In linear algebra, things that are specified by a single number are scalars and things that are specified by a collection of multiple numbers are vectors (or higher-rank tensors).
This is wrong for at least two reasons. First, it blurs the distinction between a one-dimensional vector space over a field and field itself. Second, and perhaps more problematically, it gives the incorrect impression that (e.g.) if $\vec{V}(\vec{r}) = (V_x, V_y, V_z)$ is a vector field, then the individual component $V_x(\vec{r})$ is a scalar field and transforms accordingly under coordinate rotations.
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The excluded middle ( A Law or an Heuristic) .
On a more general level given any closed question: Is it A or B ? , the heuristic says it is one or the other disregarding the option : the question is wrong or stupid or irrelevant or incomplete.
The principle of excluded middle disregards intuitionist logic. And has been harmful in not providing direct (constructive) proofs which are often more clear - yet can be harder to find.
Intuitionism is is also rather natural : being against anti-communists does not means you are a communist.
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6$\begingroup$ Perhaps more proponents of intuitionism should have readily available examples for the glaring question: what are some natural settings where classical logic is faulty next to an (intuitionistic) alternative. Compelling answers to this question are much scarcer than suggestions to consider intuitionism. $\endgroup$ Commented Aug 10, 2011 at 23:51
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7$\begingroup$ A topology is an example of a Heyting algebra, not a Boolean algebra. How's that? $\endgroup$ Commented Aug 25, 2012 at 20:02
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Perhaps one of the worst heuristics is Cramer's rule as a method of computing determinants (a hideous sum over $n!$ signed permutations ...) in linear algebra classes. (I don't know why it is even mentionned.) So often have I seen students (correctly) compute the determinants of $3 \times 3$ matrices and then take only 6 permutations to try to calculate the determinants of $4 \times 4$ matrices.
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3$\begingroup$ I think you mean Leibniz formula, not Cramer's rule $\endgroup$– WojowuCommented Jan 5, 2021 at 1:02
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1$\begingroup$ How is the method for computing $3\times3$ determinants a heuristic? $\endgroup$ Commented Jan 5, 2021 at 3:20