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A friend of mine that teaches math has made many times the following experiment :

  • drawing two circles on the blackboard representing two sets A and B such that A and B are disjoint

  • writing on the blackboard $A \cap B$

  • writing below : " true, or false?"

I let you guess what is the most frequent answer ( this answer being not worse, in fact, that the other, for they are both inappropriate).

The interesting thing, according to me, is not which answer is given but the fact that very few students say: " but, thtis is neither true nor false! this denotes a set ! "

I think this experiment is interesting inasmuch as it puts into light the fact that not grasping the category a symbol belongs to might be one of the major causes of misundertanding in mathematics.Here the confusion is between a symbol denoting an object and a symbol expressing a sentence. I express this as a hypothesis.

Could you give examples of analogous misunderstandings?

Could you imagine analogous experiences that would enable a math teacher to clear up students' confusions?

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    @Namaste. The little experiment I refer to was in my mind a simple example of a more general problem. Do you think "category mistakes as to symbols" is an interesting track to follow in order to analyse misunderstandings in math education? –  May 21 '19 at 20:38
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    @Namaste. The experience you suggest could reveal another thing: the fact that some students believe that when A and B are disjoint, A Inter B does not exist. In other words the confusion between " nothing" and " null set" ( which is parallel to the confusion between "nothing" and " zero"). –  May 21 '19 at 20:42
  • @Namaste. Just a last comment. I'm not interested in showing that students fail to understand. I'm interested in logical processes in mathematical thought and in the fact that math education fails to give students all the tools that are necessary to make understanding possible. To my mind, mathematical language/ symbols analysis is one of the major tools required. –  May 21 '19 at 20:47
  • No, I am suggesting that interpreting $A\cap B$ to mean "Does A intersect B?" is erroneous. Rather, $A\cap B$ is an expression. If you want students to evaluate the expression, ask, "Evaluate $A\cap B$. This is a classic example of the difference between an expression ($2+3$) which can be evaluated, and a proposition/sentence ($2+3 = 5$) which is boolean (true or false). Only in the latter can one ask "true or false". – amWhy May 21 '19 at 20:49
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    @Namaste "Drawing two disjoint sets, A, B, and then writing A ∩ B, and then asking "true or false" (without the option "neither true nor false") sets the students up for failure" — no, it simply proves that the rigid limitations of multiple-choice questions are extremely damaging and stifling, and even a minimal attempt to lead students beyond these limitations shows how thin and fragile their knowledge is, and how their ability to think outside the pattern is inhibited. Multiple-choice questions are evil. – Rusty Core May 21 '19 at 21:31
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    The confusion here seems to belong to your friend, for asking a nonsense question, not to the students for making a good faith effort to understand it. – Henry Towsner May 22 '19 at 07:24
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    On a few occasions, I've explained to a class that, when correcting homework and exams, I use "wrong" and "nonsense" differently: To say that 7 times 5 is 33 is wrong; to say that 7 times 5 is Arizona is nonsense. Perhaps replacing the polite-sounding phrase "category error" with the more brutal "nonsense" helps some students. – Andreas Blass May 23 '19 at 02:23
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    Prior to asking this question, did your friend explain that "mu" is sometimes the best answer to a question? – Jasper May 23 '19 at 02:43
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    @AndreasBlass: You are definitely right about that. I think it is a disservice to students to just say it's a "category error". It is as you said meaningless, and as teachers we can and should both explicitly point out to students any nonsense as well as gently guide them to type-check, so that everything they say next time will be meaningful. Unfortunately, a lot of self-proclaimed teachers criticize me for daring to identify this lack in mathematical pedagogy. – user21820 Jun 13 '19 at 09:32

7 Answers7

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I like the idea here, but I agree that it misleads students, and might have the opposite of intended effect.

Why not hand out a paragraph to the students, and ask them to critique it. Say that the paragraph is a fake student response to an exam question. One sentence in the paragraph could be something like ``Since $A \cap B$, there must be an $x$ so that $x \in A$ and $x \in B$''.

Try to pack in other common errors you notice into this paragraph.

The resulting conversation could be really valuable, and will address the issue you were trying to target in a more natural way.

Steven Gubkin
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$A\cap B$ is not true or false; but it is true that the intersection of two disjoint sets is the empty set, that is, that $A\cap B = \varnothing$ is true. This comes down to a question about whether A\cap B is a boolean statement (a proposition): that which can be answered with "true or false". Asking if $A\cap B$ is true or false is no different than asking if 2+3 is true or false.

I think your presentation to the students invites such confusion. Why not draw two sets, $A, B,$ on the blackboard, disjoint, and ask $A\cap B = ?.$ By your own description, drawing two disjoint sets, A, B, and then writing $A\cap B$, and then asking "true or false" (without the option "neither true nor false") sets the students up for "failure", as you measure it. –

Among sets, we are able to know decisively, when given sets A, B the following boolean statements are such that they are always either true or false:

  • $|A|>|B|, |A|< |B|,$ or $|A| = |B|$
  • $A\subset B, B\subset A, \text{ or } A = B$.

In arithmetic we get boolean statements when we compare of equate numbers:

  • $5+8 = 13$ (true)
  • $8<5$ (false)
  • The set of real numbers is a subset of the set of rational numbers. (false).

On the contrary,

  • $5+8$
  • $3x^2-4$
  • $\frac{d}{dx} (8x-4 - x^2)$

Are not in the category of sentences that be be deemed true or false.

$A=\{1, 3, 5, 7, 9\}; B=\{2, 4, 6, 8, 10\}$. Then $A\cap B = \varnothing$, and is a true statement, whereas $A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, which is also a true statement. But merely writing $A\cap B$, $A\cup B$ yields no sentence with a truth-value, and hence to ask if either is true or false misleads.

I think what is most misleading about the set of your friend is what appears to be an abuse of notation.

Let's partition the set of positive integers, which I refer to as $\mathbb N,$ into two sets:

  • Let $A=\{2n-1\mid n \in \mathbb N\}$;
  • Let $B= \{2n\mid n \in \mathbb N\}$.

Suppose your friend then writes below these sets: $$A\cap B$$

And then asks "True or False ?". The only way such a question has meaning is if the teacher is abusing the notation $A\cap B$ (which denotes the intersection between sets A and B), to mean "Is the intersection between A and B nonempty"? That's not what $A\cap B$ in fact means.

amWhy
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    I think the point is that the students should know enough to realize that the question is non-sensical, but they do not. This is similar to the question "A shepherd has 100 sheep and 5 dogs. How old is the shepherd?". When you give this question to 6th graders, many of them will happily compute $100 \div 5 = 20$ years old or $100 + 5 = 105$ years old. Only a few will respond that not enough information is given. – Steven Gubkin May 22 '19 at 15:01
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    @StevenGubkin: I am surprised at your claim that "many of them will happily" do nonsense. It suggests an even bigger failure of mathematical pedagogy than I thought existed. – user21820 Jun 13 '19 at 09:35
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    And I think it boils down to the same thing Andreas Blass said, which I added to, namely that we should be more explicit in identifying errors, in other words actually requiring students to be precise and logical. It also reminds me of this, which is a related issue of lack of skepticism in one's own hypotheses. Teachers should not only teach students to actively try to disprove their hypotheses, but also to formalize their reasoning so as to be 100% sure about justifications. – user21820 Jun 13 '19 at 09:45
  • @user21820 https://www.youtube.com/watch?v=kibaFBgaPx4 – Steven Gubkin Jun 13 '19 at 14:31
  • @StevenGubkin It's one thing to say "many of them will fail," and another thing to say "many of them will happily" fail. Your comment was a bit over the top. Where to you get "happy"? Your comment to user21820 only supports their comment! – amWhy Jun 13 '19 at 14:43
  • @Namaste: Haha I think he was being hyperbolic. I sometimes catch myself using "happily" in that peculiar sense of "carefreely", though I try not to. – user21820 Jun 13 '19 at 15:45
  • @StevenGubkin: Thanks for the video. Is there any published study? I am a bit wary of videos many of which are fake, though I have no way to estimate the authenticity of the one you linked to. – user21820 Jun 13 '19 at 15:46
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    @user21820 https://eric.ed.gov/?id=ED270327 – Steven Gubkin Jun 13 '19 at 15:53
  • @StevenGubkin: Thanks a lot for that! It was an interesting read. The one in the video was Example 2, and Example 3 (page 26) is even more striking. I am glad to see that I am not alone in my observations: (1) students often solve problems without understanding them; (2) students typically do not critically evaluate solutions presented by the teacher; (3) students often attempt to produce solutions just for the sake of giving an answer... (like those people (teachers included) who try to make sense of the fantastic Example 5.); (4) people self-delude about their logicality (pages 34-36). – user21820 Jun 13 '19 at 16:48
  • @user21820 Most likely, you're correct on this. But sometimes I think "hyperbole" demonizes the group of folks that are "herperbolized". I think it has less to do with "happy" or "carefree" students, and more to do with how they have been taught, and what sorts of answers count (encouraging answers with justifications and the students' testing their justifications vs. "give an answer, or you automatically fail"). – amWhy Jun 14 '19 at 18:05
  • @Namaste: Indeed I agree we should try our best to avoid words that convey the wrong meaning, including "happily" here since the default meaning of "happy" is not the one we want. And yes, I totally agree with encouraging justifications and discouraging just giving an answer, both in teaching and in assessment. =) – user21820 Jun 15 '19 at 07:39
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Another example of this sort of issue:

I have often seen students write things like $\frac84$ when they mean $4\mid 8$ (i.e., $4$ divides $8$.)

It takes a while (for some) to see that the former has a numerical value, and the latter has a truth value.

paw88789
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I'll go with the common " '$=$' is a key on the calculator" misconception.

Many students don't see a problem with and write down things like

$$3 \cdot 4 = 12 - 5 = 7$$ when asked to calculate $3\cdot 4 - 5$. This error is caused (or at least reinforced) by the fact that "$=$" can in almost all cases be read as "calculate the left hand side and write the result on the right hand side" when doing exercises where only numbers are involved. So there is a lot of time in a student's live to memorize this misconception about "$=$".

It of course breaks down when you are dealing with terms and equations which depending on your curriculum is probably around 7th grade. Perhaps one huge obstacle for the students is not only the increasing level of abstraction but also the need to "re-learn" what "equals" actually means.

Things I've tried to deal with this:

  • Emphasize that "$=$" means "is equal to" in my speech and encourage students to spell it out completely: not "three times four is twelve" but "three times four is equal to twelve". If I feel very pedantic, I'd also elaborate that "three times four" is a multiplication while "twelve" is not, so one thing can't possibly be the other.
  • Emphasize transitivity of "$=$" and look at $3 \cdot 4$ and $7$.
  • Have the students actually look at their calculator. TI uses "Enter", Casio uses "EXE" on their advanced models. There are sadly many calculators that support the misconception because they have the dreaded "$=$" key.
  • Forbid multiple equals signs in a single row altogether.
Jasper
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    I think that three times four is twelve. The string "$3*4$'' is not the string "$12$'', but they do represent the same number.

    Also, I wouldn't forbid multiple equals signs in a row, because it can be used to good effect. I recall a proof of the an equality of the form $a = b = c = d $ by actually proving that $a \leq b \leq c \leq d \leq a$ for instance. Wish I remembered which one!

     – Steven Gubkin May 22 '19 at 17:47
  • Re "3*4": "they represent the same number" is exactly what I mean by "is equal to" as opposed to identity (of strings). – Jasper May 22 '19 at 18:32
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    I guess it depends on what "is" is. To me, Superman is Clark Kent. – Steven Gubkin May 22 '19 at 18:58
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  • If $A$ and $B$ are two disjoint sets, writing $A\cap B=0$.
  • If $\ln x=2$ then $x=\dfrac{2}{\ln}$.
  • How many corners a circle have?
  • If $\sin x<\sin\frac{\pi}{4}$ than, after simplifying the sines, we get $x<\frac{\pi}{4}$.
  • If $\frac{\pi}{6}<x<\frac{\pi}{3}$ then $\cos\frac{\pi}{6}<\cos x<\cos\frac{\pi}{3}$. What's wrong? I just added a cosine !
  • The area of a circle is given by $\pi r^2$. Since when does a circle have an area?
  • This is a square not a rectangle ! This is an equilateral triangle not isosceles !
  • Using L'Hôpital's rule to find the limit of a sequence : $\displaystyle \lim \frac{2n^2-5}{n^2+1}=\lim\frac{4n}{2n}=2$.
  • The function $f(x)$ is increasing. The function is $f$ not $f(x)$.
user5402
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    What is wrong with the L'Hôpital's rule examples (perhaps besides omitting $n \to \infty$)? Perhaps your objection is that one cannot differentiate a function with a discrete domain? This is easily fixed: just enlarge the domain in the obvious way. Definitely something I would want students to think about, but it is pretty harmless shorthand. – Steven Gubkin May 22 '19 at 17:52
  • @StevenGubkin It's a harmless shorthand if the student understands that we're finding the limit of a function over $\mathbb{R}$ then restricting its domain. Even if $u_n=f(n)$ than $\lim_{x\to\infty}f(x)$ and $\lim_{n\to\infty}u_n$ are not necessarily equal; for example $f(x)=\sin\left(2\pi x\right)$. – user5402 May 22 '19 at 17:57
  • Some definitions state that an isosceles triangle has exactly two equal sides (with the third side being of a different length). – paw88789 Jun 08 '19 at 13:13
  • @paw88789 I've never seen such definitions. An equilateral triangle has all the properties of an isosceles triangle. – user5402 Jun 09 '19 at 12:31
  • @Paracosmiste see for instance: https://en.m.wikipedia.org/wiki/Isosceles_triangle – paw88789 Jun 09 '19 at 14:01
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    @paw88789 That was probably the definition of Euclid; he also didn't consider a square to be a rectangle. But most (if not all) modern books don't follow Euclid's definition. – user5402 Jun 09 '19 at 14:14
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Among my top frustrated wishes are that there should be much more emphasis around the first year of algebra on directly connecting natural-language grammar to algebraic grammar. My students are extremely hazy on what counts as a statement or non-statement (expression/fragment) throughout the curriculum; including second-year math and computing majors at my community college.

In particular, I wish there was solid drilling on the difference between an "operation", which has the grammatical status of an adjective phrase (e.g.: arithmetic or set operations like intersection), and "relations", which have the grammatical status of verbs. Without the latter we have a fragment, not a propositional statement that can be assessed as true or false.

Daniel R. Collins
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To a computer programmer familiar with the Lisp family of languages, it is the original poster who has the misconception.

Lisp is one of the most ancient computer languages. It is about as old as Fortran and Algol; it is older than Basic, C, Pascal, and all of the object-oriented languages. Lisp was designed by a mathematician to illustrate how set theory could be used as pseudo-code. It was simple enough that his students promptly turned it into an actual programming language. The Logo turtle graphics language is a dialect of Lisp. Lisp is still used as an extension language for the emacs text editor and the AutoCAD drafting system.

In Lisp, everything is an ordered set, a number, or a string. In particular, the concept of false is implemented as the empty set, a.k.a. "nil". So if A and B are disjoint, the intersection of A and B is empty, which evaluates to false.

Jasper
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    There are other places where 0 is identified with false, for example Morse's book "A theory of sets". There are also places where 0 is identified with true, for example in continuous model theory. But such identifications need to be made explicit; otherwise they are, as the OP says, a category error. – Andreas Blass May 23 '19 at 02:18
  • Indeed, from a computer/programming/formal-languages viewpoint, much of standard school mathematics has a great number of implicit, never-discussed assumptions. And some important issues, like "scope" of a variable name, or "return value" as opposed to "side effects", are very rarely mentioned at all, and, if anything, denied to have applicable sense. E.g., in some contexts, the instruction to evaluate 34 returns "true", because it is non-zero... or because the operation succeeded... As opposed to assigning a new variable name to the "outcome" of 34... – paul garrett May 23 '19 at 21:13
  • I think this is an important point. However, I wouldn't go so far as to say that OP has a misconception. Rather, anticipating that some of our students may do computer programming, I would point out that in many computer languages $A\cap B$ will evaluate to False if and only if $A$ and $B$ are disjoint. However, the standard mathematical way to express disjoint-ness is $A\cap B=\emptyset$. – paw88789 Jun 08 '19 at 13:23