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I base this question off where I got my motivation for math and science. Throughout several attempts in my junior years, I was able to design a perpetual motion machine, design a free energy device, prove Einstein $E=mc^2$ wrong. Obviously I was way, way off, and I eventually found that out by myself. However, the amount of interest in math and science and the amount I learned was greatly benefited. So I ask the question: should students be told they're wrong about an idea they have?

The example I will give is a student "proving" $1=2$ by method of division by 0, such as the one described here. I classify this in 4 ways:

  1. They should be told they're right. This makes the student feel smart and makes them have a higher interest in the subject. Eventually they will find out they are wrong. (This obviously being a lie.)

  2. They are told that they came up with good thoughts, not confirming or denying they are right. They may not receive as much interest, but they are still in the process of confirming if they are right and wrong, and if curious enough and believe they are right, perform their own study.

  3. They are told they are wrong, but are not told why. Which still may present some curiosity, maybe enough to find it themselves. In this way they are not risking the trust between the teacher and student.

  4. They are told they are wrong, and explained why. This way they are taught the process themselves, but may not receive as much interest from it.

What's the most effective way for their education?

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  • $\begingroup$ Sounds like you have a good point. If a student has a proof that 1 = 2, let them have their magic trick. Maybe someone will spot the error. Only then interject. $\endgroup$ – djechlin Jan 11 '15 at 2:43
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    $\begingroup$ You could tell them they've done well but are wrong and offer them an incentive to figure out why they're wrong. That way they know they're wrong and still have a reason to go and learn more things. $\endgroup$ – Pharap Jan 11 '15 at 12:55
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All four of your options lead with "They are told..." Consider asking the student questions instead. At the very least, this shows interest, and they may end up catching their own mistakes as they try to explain to you what they had glossed over in their own heads.

When I have the opportunity, I like to challenge my students to explain EVERY step of their work to me, including the steps that are correct. This builds their self-checking reflexes and their communication skills, and when they do catch a mistake, they don't feel like I did the work for them.

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    $\begingroup$ "What if?" questions that suggest a counterexample to be constructed are particularly good to ask...since then a student can "feel smart" by constructing a counterexample AND feel smart if they can explain why such a counterexample could never be constructed. It's the old adage: On Monday you believe your theorem is true, on Tuesday you try to disprove it, on Wednesday you try to prove it again. $\endgroup$ – Jon Bannon Jan 9 '15 at 20:24
  • $\begingroup$ I really like the jist of this answer. I've often found that the best way to teach math is to do less telling them about it and more asking them questions about it. "What do you mean divide by 0? How do you divide by 0?" These kinds of questions inherently make them examine what they've done with a critical mindset in an effort to supply an answer to what you've asked. It's almost like you've tricked them into criticizing their own work, without all of the inherent negative connotations asking for that directly can entail. $\endgroup$ – Justin Benfield Aug 31 '16 at 3:42
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This is obviously a subjective topic, but here's my take:

As an educator, you should see yourself as a resource to your students. You have certain knowledge that they seek to obtain. You should never withhold useful knowledge to a student if they explicitly ask for it. Rather than you attempting to force their actions, it's more about how you react to their actions.

Thus, if a student approaches you and hands you his proof that 1 = 2, you should review it and perhaps say, "I disagree with step 4". You should absolutely say that you think there is an error in their work, otherwise they may have misconceptions that will lead to confusion down the road, or a lack of trust in your knowledge as a teacher, or a feeling that you don't care about their learning.

After you mention that you disagree with step 4, the student will then do one of a few things - they'll (a) review step 4, find their mistake, and that will be that. Or (b) they'll review step 4 and not be able to spot a mistake and ask for you to explain their mistake. Or (c), they won't bother to review their work and will directly ask you to explain their mistake to them. That's o.k.

In either scenario, you should react to the way in which they want to learn. If they specifically ask for you to explain their mistake, don't withhold your knowledge by forcing them to work it out for themselves - that might have the unintended effect of frustrating them, rather than encouraging them. In most cases, you're o.k. to reveal their error gradually though, so that you give them many opportunities to prove themselves. So say, for example, "Did you consider what would happen if b=0 in step 4?" rather than just rewriting step 4 so that it's correct.

Again, they'll react in one of the ways covered above. And you should respond accordingly - reveal more information if they request it until they understand their error, all the while giving them opportunities to make the leap themselves. But if they choose to give up and wish for you to explain their error to them in full, then I believe you should comply. It's either that, or they leave knowing that some part of their knowledge is wrong, without knowing exactly what or why. And you cannot expect for every person to be capable of solving each of their misunderstandings on their own. That is one reason for why we share our knowledge.

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    $\begingroup$ I disagree with the following: "You should never withhold useful knowledge to a student if they explicitly ask for it. Rather than you guiding their actions, it's more about how you react to their actions." When students ask for something explicitly, I believe it is often better to withhold on sharing my own knowledge and help them construct their own understanding. Perhaps this disagreement in orientation extends to your earlier comment on how you conceive of an educator's role: "You have certain knowledge that they seek to obtain." $\endgroup$ – Benjamin Dickman Jan 11 '15 at 11:31
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    $\begingroup$ @Benjamin That might just be a misunderstanding. My wording on "rather than you guiding their actions" was a bit off - I was trying to establish that the teacher should consider themselves a slave (for lack of a better word) to the student, rather than the other way around. It's perfectly acceptable to encourage (guide) a student to think something out themselves. But if they choose to give up and wish for you to explain their error to them in full, then I believe you should comply. It's either that, or they leave knowing that some part of their knowledge is wrong, without knowing what or why. $\endgroup$ – Ponkadoodle Jan 12 '15 at 7:13
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Presumably students who "prove" $1=2$ already know they're wrong. They don't really believe that $1=2$ do they? If a student brought me the "proof" you linked to, I'd first make sure they realized that $1\neq2$. Then, I'd suggest they check the "proof" by choosing a particular number, say $3$, for $a$ and $b$. That way they can see exactly where the equations make the transition from true to false. I'd hope they can also see why that particular step was wrong, but if not then (and only then) I'd explain it.

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    $\begingroup$ Frequently when I make some sort of arithmetic mistake in my research, particularly when the arithmetic has passed through sufficiently many high level concepts (complexification of the jet bundle of the...), I often get a sinking feeling that I have just undermined all of mathematics. Not sure how common this feeling is... $\endgroup$ – Steven Gubkin Jan 9 '15 at 15:49
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    $\begingroup$ @StevenGubkin Fortunately, the example given by the OP is easier to handle than yours, because the former can easily be made very concrete by just plugging in a suitable value for $a$ and $b$, so that the location of the error is easy to pin down. If you've complexified jet bundles, then it's far harder to work through a concrete instance, but, if it can be done at all, it seems worthwhile. $\endgroup$ – Andreas Blass Jan 9 '15 at 16:16
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It depends on the student and the problem at hand, though I'd never suggest one lie to them - when they discover the lie they will suspect everything else you've taught them, and possibly what others have taught.

In most cases I'd suggest that the right answer is to encourage and challenge them.

When I was in High School a teacher indicated that a high school student shouldn't be capable of solving a particular problem. Some overconfident students respond to that type of challenge - I did, and was rewarded with what I learned along the journey. More important than the skill and knowledge gained, was the affirmation that I can accomplish difficult tasks if I set my mind to them.

When a student presents something you already know to be wrong, rather than shooting them down, or praising them falsely, consider challenging them to take the next step. "That's an interesting design for a perpetual motion machine. What is the calculated energy output? How much of that energy will be used in friction in each of 29 bearings and sliding surfaces? Where do you propose to obtain single-pole magnets?" Rather than placing a road block on the path they are traveling, encourage them to continue down that path - they will gain more concrete knowledge with experience than they will by a quick, "It can't work."

If it's dangerous to pursue, or they would learn nothing new by taking something to its logical conclusion, then it might be worthwhile short-circuiting the process and engaging their mind in a more enlightening direction.

I've found, however, that "Fail fast, fail often" is not a bad way to learn things, and with good mentorship it can yield unexpected benefits in a learning environment.

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    $\begingroup$ A good student should suspect that everything he's taught might be false. Skepticism is good. $\endgroup$ – Christian Jan 10 '15 at 21:32
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    $\begingroup$ Skepticism is good, however I would never recommend lying to students on a regular basis as a good teaching style. But that's just me. $\endgroup$ – Adam Davis Jan 10 '15 at 22:48
  • $\begingroup$ Yes, I think that this is a matter about which intelligent teachers disagree. IMHO pedagogically-intended lying and obfuscation are beautiful things. Students know the difference between a teacher hiding math knowledge from them in order to get them to think for themselves, vs. actually being a sneaky and untrustworthy human. I admire teachers who implement deceit consistently enough that their students have learned to think for themselves where it counts. See this lesson, for example: jd2718.org/2009/11/14/… $\endgroup$ – benblumsmith Jan 18 '15 at 18:03
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Let me start by saying that I believe what effectively educates one person will not necessarily be effective for someone else.

With that in mind, I would never personally use number 1 as I feel it is lying. However if I knew the person well enough, I might consider doing it if I feel they learn it better that way (you sound like you benefit from this from your perpetual machine example). However, I think those situations would be quite rare.

Another problem I have with 1 is that sometimes a person may not figure out they are wrong until it is too late! Sometimes thinking something is correct when it is not can be harmless, but what if that person starts telling other people it is true? Sometimes it is surprising how far people can get in life never being told something they believe to be true is wrong. If this person ever became a teacher and started teaching what they believed is correct, when it is not, there could be problem. I would feel it is best not to reassure them they are right when they are in fact incorrect. I feel like this can become more than just a white lie.

Instead, I would probably lean more towards a mixture of 3 and 4. Tell the person they are incorrect but encourage them to try to find out why without giving away the answer (much like your 1=2 link does towards the end by asking: See if you can figure out in which step the fallacy lies.)

But again, a lot of people have a different learning style. For situations like this, I would let the person they know that they are overlooking something within their assumptions, and encourage them to find out what it is.

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My philosophy is that, yes, students should be told when they're wrong. On the one hand, we surely all have self-discovery experiences like the OP's (with the alleged "perpetual motion machine") that were transformative, highly memorable, and likely very dear to each of us. But: We simply don't have the time to self-construct all knowledge that way. Indeed, the whole power of language and writing is to be able to share in others' experiences without re-inventing the wheel individually every time.

Now, the way that this is explained or revealed to the student can of course take many approaches. Requiring a detailed justification of each step is one way. Asking pointed questions around things they've overlooked is another way. But in general, the risk of a student proceeding forward with mistaken conceptions, and the avalanche of problems that can cause down the road, is so great that I wouldn't want someone leaving my classroom thinking that something wrong is right.

As a side note: With my remedial college students, I find it rather stunning the degree to which they will completely tune me out until I specifically say the magic words, "Your answer is wrong". For example: In many cases during in-class exercises, if I start pointing out mistakes in student writing (say: mis-transcriptions, substitutions, etc.), then they'll nod blankly and go "Yeah, I get it" or something, as though I'm being vacantly pedagogical. Not until I say "Your answer is wrong" do they sit up and go "Really?" and then start looking to make corrections.

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Should students be told they're wrong about an idea they have?

Absolutely not. Mathematics is not a matter of right or wrong: according to what/whom?

On the other hand, they should be told, up front, that, in science just like in a court of law,

  1. we always have to make our case,
  2. we always have to submit our case to adversarial proceedings (peer review).

Then, what I tell my students is that, when reading their case, either I agree, will say so and that will be the end of that or, if I disagree, it will then be up to me to make my case.

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