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As of last year, I began teaching a course in theoretical computer science, and one of the topics that we cover is sets. In particular, we go over:

  • the actions $\cup$, $\cap$, and $-$,
  • cardinalities
  • Infinite sets in $\aleph_0$, and proving that sets are (or are not) in that set.

I thought that the last part would be tricky (particularly since we not only teach the diagonal graph proof, but also use the proof that hinges an element that points to the set of all elements that don't point to sets that contain themselves... I don't remember the name of that proof, but it is certainly cognitively tricky.)

I hit these last two subjects hard, doing quite a lot of practice and review. In the final examination, students did very well with the second two topics, but struggled mightily over (what I believed to be) simple Venn diagram questions like this:

enter image description here

Write an expression that represents the region in gray.

Some number of kids definitely got it, but for an uncomfortably large number of students, the answers they gave were almost comically convoluted, such as:

$((\mathbb{U} \cap (C \cup A)) - ((A - B) \cup (C - B))) \cup (A \cap B \cap C)$

... or were simply flat-out wrong. So, it's clear that I need to teach this better, and I will in the future.

My question, then, is thus: what are the cognitive traps that a teacher should be aware of when teaching union, intersection, and subtraction in sets?

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    $\begingroup$ There's room for more sophisticated suggestions, but consider this: your students practiced the last two subjects a lot, the first one not so much. As a result, they did well on the last two subjects, not so much on the first one. I think the conclusion is simple. $\endgroup$ – Javier Nov 13 '17 at 22:21
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    $\begingroup$ @Javier, That is unquestionably true, and I'm definitely going to be creating more practice for them. However, looking over the results, it occurred to me that I didn't understand what they don't understand, and historically, that's a sense I get when I'm unaware of the cognitive traps in something that feels simple to me. I'll certainly improve in this regard with experience. I was just hoping for a leg up by learning from the experience of others as well. It doesn't all have to be the School of Hard Knocks :) $\endgroup$ – Ben I. Nov 13 '17 at 23:16
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    $\begingroup$ I can't speak to those who got the question totally wrong, but about those who put comically convoluted answers: is that a bad thing? As long as they got it right, I think a complicated answer shows an in depth understanding of union, intersections, differences. For instance, the student example you listed clearly shows a logical thought process ("Ok, start with the whole thing and intersect it with this part. Subtract this unnecessary area and this unnecessary area. But now I have to add this part back in since I subtracted it twice.") That's quite a bit of understanding. $\endgroup$ – ruferd Nov 14 '17 at 13:47
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    $\begingroup$ @ruferd I awarded full credit for any correct answer, but it means to me that the modeling that they are using in their mind is, at best, terribly inefficient. That means that they will be unlikely to be able to utilize this tool in any reasonable fashion in other contexts. $\endgroup$ – Ben I. Nov 14 '17 at 13:54
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    $\begingroup$ @TheSpartan I wouldn't have any problem with a student giving your answer; it's perfectly reasonable. I'm not looking for a guaranteed minimum set of terms, just something more reasonable than the sample student answer I gave. $\endgroup$ – Ben I. Nov 14 '17 at 20:37
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I think your specific situation with the exam question is similar to a pre-Calc class that spends a good amount of time on solving trig equations, but then asks on the exam for the student to write down the equation to describe a given scenario. These are just different skills. In a similar Discrete Math course I learned the hard way (as you have now) that students have trouble creating mathematical descriptions (a.k.a. modeling) and adjusted subsequent semesters to include more examples like this in lecture and homework.

Another "cognitive trap" they struggle with is drawing degenerate Venn diagrams, e.g. a Venn diagram where $C \subseteq A \cup B$ or some other condition that reduces the number of regions and changes the way the ovals overlap.

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I think the issue here is really more one of problem solving skill or abstraction ability. You could see the same thing in various problems in computer science or in college algebra where some students can solve an apparently complex problem simply and others get caught up in the complexity. Or you see the same thing on problems that contain extraneous information where some are able to ignore it and others think they need to incorporate it.

So my main point is that you are probably thinking of this wrong as some special issue of set theory. It is more of a common human failing. And the way to deal with it is probably practice and discussion. Yes there are some students who can learn the basic rules and then just apply them in a new context. But this is rare(er). Probably very common in the sort of people that teach math. But rare in the student population. So just live with it and train students on the problems you want them to solve and don't worry so much that not everyone is super smart.

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    $\begingroup$ I agree this is a general issue of problem solving skill and abstraction ability and not something specific to set theory. But our job as educators in math is precisely to get students better at problem solving and abstraction, and not to train students on how to solve specific problems. Our job is to make our students super smart. If our students don't learn to take basic rules and apply them in new contexts, we have failed. $\endgroup$ – Alexander Woo Nov 14 '17 at 23:41
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I may be wrong, but from personal experience I think the other answers are missing a crucial point, what it is like to be taking the test itself, and how people attempt to learn material.

If the students come up with convoluted, but correct answers, that typically doesn't mean they are "bad at abstract thinking". Students come up with convoluted answers to simple problems on tests for many reasons, but if many are doing the same thing then its probably one or many of these:

  • Your test is time restrictive in some way, in the same way say some sort of entry exam would be. Its misleading to use the metric of students getting answers correct, when the mental time load for students to actually answer those questions takes a long time. It might simply be a matter of time for them to get the question correct, but now they have less time to look at other problems. When they encounter something like this, they might even see something that looks easy. But when the answer isn't immediately obvious in a situation where it looks obvious, student panic, because they already spent so much time on the other problems and this one should be easy. Students will then take the path of least resistance which means mental brute forcing in some form. If this was one of the few "easy" problems in the test, it might hint further that time restriction was part of the problem, not that the students necessarily lack understanding of intersections and unions.

  • There was no brush up for the students on these concepts, no matter how simple. People don't know what they don't know until they encounter a problem that shows them they don't know something even if that thing was something they used to know. People don't learn by confirming their biases, they learn by seeing challenges to their mental model. They will remember when they were wrong far more often that when they were right. So if you don't provide material to test their knowledge with and have them verify independently what they got wrong, you are taking a large risk in assuming understanding. Don't just assume students know things they give you the impression they do with out first verifying it. Even if they do know how to do this stuff on a proficient level, if not enough practice has been given to them, their lack of confidence will be their downfall on the test. It will always take them longer to complete a question correctly they have less confidence in than one they do.

  • Your test includes many orthogonal concepts, or concepts which can not be applied in a personally meaningful practical cohesive manner until outside the context of the class. When you include orthogonal concepts on a test, the time to switch how you think, or change your mental model increases (for a variety of reasons like definition overloading). If students were faced with previously completely different questions, for example those last two subjects, they would have a harder time adjusting to how to answer the next question you just presented, especially if you didn't touch on recently (or at all!) the kind of reasoning needed for your vendiagram question. Their lack of expectations for such a question will cause a huge slow-down in the ability to answer, or hinder ability to come up with a reasonable answer.

One way to fix it is to give a hint on how to think about the problem, ie how difficult the problem should be to solve. Don't confuse this with how difficult it should be for the student to solve, but how many steps it should take and/or how simple the answer should be. Students may still panic, but will be able to use that as a crutch to understand how they need to think about the problem, with out you actually given them any real help on how to answer vendiagram set question in general. When they start thinking of complicated answers they can quickly short cut all the time wasting bad solutions.

To give a hint you just need to say something a long the lines of "The solution should be simple". You may want to put a number of statements max hint (something large enough so it doesn't actually effect answers for students who don't actually know how to do these), but I'm not sure that is the best precedent to set for your questions at large.

Another way to help is to provide homework for the problem concept. This would be graded (probably recorded for a completion grade, but all wrong solutions actually marked), then the work for those problems is posted on a place they have ready access too. After the first assignment is due, give them a second assignment covering the same topics but with different questions in different order. Finally, grade that assignment and cover what students still got wrong in class, still providing the worked out solutions, but not providing more homework assignments on that subject after that point. These assignments should not be alien to the associated test problems they encounter, or else you have the same orthogonality issues from before. What alien means is subjective, so use your best judgement, they should at least not be surprised by what they see on the test or how its formatted.

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I don't think I can address cognitive traps, but perhaps this tool could be used for student practice. There is a "Wolfram Alpha Widget" for exploring set Venn diagrams. The widget code is available at this link.


                VennW
                Submitted:(A Union B) Intersect (Complement C)
I hope this links to the widget itself for experimentation: link.

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  • $\begingroup$ @BenI.: (A difference B) works as one would expect. E.g., (A difference (B union C)). If one uses 4 or 5 sets, it still works, using ellipses to represent the sets. $\endgroup$ – Joseph O'Rourke Nov 14 '17 at 14:09
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I think it might be helpful to teach for your students to train ways to simplify and transform such longer equations of sets. Since it is a course in computer science, you might even teach them one of the classic algorithms for minimal DNF, which can be easily turned into a problem of sets instead of variables. This should help them to get a feel for things and even the worst students might in the end be able to answer such a question with an overly long but correct union of intersections.

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There seems to be something very wrong here, not in the mind of the student, but in yours.

You describe the mentioned answer as comical, but there's nothing comical about it: it is the answer of a hardworking student, who understands set theory very well and who has given you a correct answer to a question that was undoubtly very difficult for him/her.

When I saw the answer my first reaction was (just like yours): "Jezus, what is this???".

But then I started investigating the answer and after some time I must admit that the answer looked correct and revealed true creativity and problem solving skills. I admit, there are easier ways to solve the question, but there's where calculation rules come in: using calculation rules you can simplify the answer from the student and like this (s)he might understand that another approach might have simplified his/her work.

This latter is also enormously important: when starting to learn mathematics a student starts learning that there is exactly one way to solve a question (which is the way the teacher is doing it).

Only after some time the student might learn that different approaches for the same question might up given a correct answer (this is crucial in the development of the mind of a student!).

Instead of judging that answer as a wrong one, I would advise you to use that answer as a starting point for learning calculation rules about set operations, and from the result of that calculation/simplification you might invite the student to derive from that an easier response to your question, but please, please, never treat the creative result of a hard working student as something comical anymore!

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    $\begingroup$ The sample answer belongs to no specific student, it is purely my own (though it is certainly modeled after what my students gave me back). I would never post a direct student answer to a public forum. Also, while compassion is important, if you don't learn to find some humor in what your students give you back, you're going to have a rough time with grading. ;-) $\endgroup$ – Ben I. Nov 15 '17 at 14:53
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I can not see why someone can not see the simple solution. However their approach may be more scalable.

To solve the over complex problem.

Anyone that produces a solution that complex can not be a good computer programmer. Therefore help them to see it more clearly, or help them to simplify it. Like we simplify other mathematical expressions.

What is it (set theory) like? It is like boolean algebra, which in turn is like high-school algebra. So if they are confident in high-shool algebra, then transform to regular high-school algebra.

  • $\mathbb{U}$ → $1$
  • $\cup$ → $\lor$ → $+$
  • $\cap$ → $\land$ → $\cdot$
  • $A - B$ → $A \cap \bar{B}$ → $A \land \lnot B$ → $A \bar{B}$

And simplify.

  • algebra rules
  • plus additional rules:
    • $1 = 2$ e.g.
      • $2A = A$,
      • $A^2=A$,
      • $A+1=1 or 2 = 1$,
    • $A\bar{A}= 1\times0 = 0$
    • $A+\bar{A}=1+0=1$
    • $\bar{\bar{A}}=A$
    • de Morgan’s Theorem

I got $B(A+C)$ → $B\cap(A\cup C)$

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what are the cognitive traps that a teacher should be aware of when teaching union, intersection, and subtraction in sets?

That's your job to figure out. You should be giving your students exercises during class, seeing what they fail at, and talking with them to understand what they're having trouble with. You should be giving them homework problems, and the homework problems should be designed so that when the students get a problem wrong, you have some idea what concepts they're having trouble with. You should have quizzes every week (if not more frequently) that again are giving you feedback about they are understanding (this is one of the purposes of partial credit: you should make it as clear as possible that the more information your students give you about what they do and don't understand, and WHY they are getting a problem wrong, the more points they get). You shouldn't be getting to the final and being surprised at what your students don't understand.

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    $\begingroup$ This is not an answer. It sounds like that may be a deliberate choice, but it is still a non-answer. It's also not as helpful to me as you presumably imagine that it is, as I'm already redesigning the lessons to encompass more practice. Finally, "that's your job to figure out" is terrible for the kids, and literally what I came here to prevent. That process of discovery is always painful to students. Securing advice from subject-teacher veterans is simply an attempt to minimize that pain. Advising someone not to seek such advice is counterproductive for student and teacher. $\endgroup$ – Ben I. Nov 14 '17 at 19:39
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    $\begingroup$ I'm not advising you to seek such advice. I'm simply saying that you should build your course around minimizing the need for such advice. And you can significantly decrease the pain of discovery by presenting it as YOU trying to learn from THEM, rather than them failing to learn. Maybe there's more than what's what you could fit in your question, but the way you described made it sound like you're treating tests as being the end of the process, rather than the middle. $\endgroup$ – Acccumulation Nov 14 '17 at 20:31

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