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Background: To cope with online education, I taught linear algebra using a variant of the flipped classroom. I recorded videos and put them up on YouTube and students presented the content in these videos during class - sometimes in pairs, sometimes individually. During the presentation, I used to point out gaps or lack of understanding.

Long story short, the students did not gain much from the course - especially towards the end of the course. My students did terribly in their final exam. As they all did terribly, I feel I failed them. I had a conversation with them and they said that me pointing out their mistakes or lack of understanding demotivated them. Sending them to a vicious cycle and the result of the final exam is a consequence.

I also taught an IBL course and a colleague attended this course. He too told me that I shouldn't say "You are wrong" - it demotivates the students. At the time I had argued, "how else will they learn?".

Question: Is it wrong to say "you are wrong"? If yes, how do we correct mistakes in a way that is not demotivating? My policy has been, give elaborate feedback, but grade leniently. But, feedback often meant I point out their mistakes. Of course, I was also generous with praise. How can I give feedback that is not demotivating?

I didn't use the exact phrase "you are wrong". I used different phrases in different situations. I mostly said the argument was wrong. I do, however, say "I don't think you have understood this concept". Sometimes they reply I understand; I am just unable to explain. So, I generally add the disclaimer - "I do not mean you did not understand anything, but there is some lack of understanding. If you cannot explain, that means you have not understood the concept well enough".

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    $\begingroup$ (+1) for a good question that I don't think has been explicitly asked here before, although someone will probably show that I am wrong shortly after I post this comment. My initial thought is that if you actually said (or wrote) you are wrong, try using wording that is less likely to be seen by some students as a personal attack, such as the answer/reasoning you provided is not correct (or "is not entirely correct" for more distance from "you are wrong"), and if you didn't really say "you are wrong" but instead said something softer, then you should edit your question accordingly. $\endgroup$ Dec 12 '20 at 14:49
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    $\begingroup$ Somewhat related: Should students be told they're wrong $\endgroup$ Dec 12 '20 at 14:53
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    $\begingroup$ A suggestion I have received from my colleagues and was in the answers to the other question is "ask questions that will make them realise their mistakes". However, time constraints make it very difficult. I do tell my students to discuss with me before presenting it to the class, but very few students take up that offer. $\endgroup$ Dec 12 '20 at 15:31
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    $\begingroup$ Thank you for this question. I appreciate the answers, and am grateful to be given a chance to think about this. I am guessing that most of us are struggling right now to find ways to help our students learn. $\endgroup$
    – Sue VanHattum
    Dec 13 '20 at 20:33
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    $\begingroup$ Just noting that the 'flipped classroom' and the research justifying it's use are about class time being spend on what would have classically been the homework so that there is an increased amount of 1:1 time when students have trouble with questions. Having students primarily 'waste' other students time by giving a 'worse than you' presentation can have it's place for sure, but really shouldn't be associated with the concept of a flipped classroom (although it could be one small facet of a flipped classroom). $\endgroup$ Dec 15 '20 at 8:13
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I don't think your students are responding negatively to being told they're wrong. They're responding negatively to being told they're wrong in this specific context. You've set them up in a very high-stress situation where they have to watch your lecture video and then immediately act like little professors. Since the video isn't interactive, they have no opportunity to ask questions if they're confused. They know that their understanding is weak at this stage, so they're being set up for failure when you ask them to do the presentations.

A side note: -- I use the flipped-classroom technique myself, but I continue to be baffled by the custom of implementing this using videos of lectures rather than having students read the book. This seems to me like a ridiculous practice with no sane justification.

If you're going to do the flipped classroom, then you should start with some reason to do that. That reason should be based on evidence, and the practices you implement should be ones that have evidence to show that they work. I think most people who use the technique successfully make sure that once the students are in person with you, they start off with easy tasks and have plenty of opportunities to get help and ask questions. Only later should they be asked to do more difficult things. In linear algebra, an easy task would be something like finding the dot product $\langle 1,0,0 \rangle \cdot \langle 0,1,0 \rangle$. A student who attempts this but makes a mistake will not feel that they have been set up for failure.

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    $\begingroup$ @DivakaranDivakaran They had just 4 days to learn the concepts and then present about them?! $\endgroup$ Dec 12 '20 at 21:54
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    $\begingroup$ Yes. The videos were only 15 min long. Is it too much to expect students to be able to understand a 15 min video and present it in 4 days? $\endgroup$ Dec 12 '20 at 21:56
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    $\begingroup$ @DivakaranDivakaran Absolutely. The reason why the flipped classroom can be so effective is that teaching a topic requires near mastery, and makes gaps in mastery very visible. Learning a topic to some basic level is certainly possible in 4 days, but mastering it is not, especially when students have other courses to balance with their learning process. Students have only a few hours each day to devote to each course. If a student has 8 hours a day to study, and is taking 4 courses, that's 2 hours per course. You're asking students to master a topic in only 8 hours. That's quite unreasonable $\endgroup$ Dec 12 '20 at 23:18
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    $\begingroup$ @DivakaranDivakaran I don't think the choice to "flip" was necessarily a mistake, but the way you did it made the course very difficult for students. (I've never heard of anyone else doing what you did.) My colleagues and I commonly make students go through prepared worksheets in groups for our flipped classes rather than asking them to present lecture material, and sometimes they are asked to present their solutions. $\endgroup$
    – Kimball
    Dec 13 '20 at 14:40
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    $\begingroup$ Just adding more to @TheEnvironmentalist's comment: you're not even factoring in how much time it takes to make a good presentation of material, even for a topic that has been mastered! That's a hard enough thing to do on its own, let alone while simultaneously learning the topic itself. $\endgroup$ Dec 13 '20 at 16:47
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Here are some rhetorical devices I use to deliver corrections in class that are very gentle to students' egos:

  • "You have a slice of something correct there, but maybe we should look more closely at..."
  • "I think you're definitely on the right path here, and what should we do about this part?"
  • "That would definitely be the right answer in [this other situation]. But notice that we have [this current situation]. What should we do with this?"
  • "Hmmm, is that really the definition of that object? Perhaps we should go back and review the exact definition." [Side note: One reason why it's good to have digital slides is to allow a quick jump-back to prior materials like this.]
  • "Interesting. Now, it's always good to double-check our work, so let's check the result by [some validation method]..." [delivered in a case when I know on sight that the answer is incorrect]

That said: I don't think that any of this will be a silver-bullet for the challenges faced by your students. I think that almost all instructors are facing great difficulties in the current emergency-distance-learning environment, a steady dropoff in participation rates, etc. If students start falling behind in any STEM course, and need increased personal responsibility in distance learning, then the falling-behind will snowball and you'll get these kinds of results at the end. Students saying they're discouraged by corrections may be a convenient ego-saving scapegoat, but it's not necessarily the full story.

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    $\begingroup$ Excellent, IMO. $\endgroup$ Dec 13 '20 at 6:40
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An effective educator should not only point out a student's mistake, but also provide the student with the opportunity to apply their newfound knowledge so that it is more likely to stick with them.

The example I'm about to provide relates to high school algebra, but I'm sure it can be applied to college level math such as Linear Algebra.

The other day I was helping a student with solving radical equations. We were working on solving $\sqrt{x^2-4x}=x-4$. I let the student work on solving this independently until they arrived at a solution. Early on, when the student tried to square both sides of the equation, they incorrectly expanded $(x-4)^2$ as $x^2-16$ (this common mistake is known as the "Freshman's Dream" if anyone is curious). I informed the student of their mistake, but I did not fix their work for them. Instead, my goal was to remind the student how to square binomials (I assumed this was a prerequisite for being able to solve radical equations).

Rather than letting them continue solving the original problem, we put it on hold and I computed $(x+3)^2$ for the student, drawing colorful arrows that represented FOIL (that is my username after all $\ddot\smile$). Then I asked the student to expand $(2x+5)^2$ to boost their confidence. Afterwards, the student said they were ready to return to the original problem (if this was not the case, then I would have provided more exercises until they felt ready). With their newfound confidence, the student was able to correctly expand $(x-4)^2$ as $x^2-8x+16$ in the original problem. From there, the student had no trouble finding the solution.

I hope this example shows that constructive feedback is insufficient unless the student is, if time permits, shown what their mistake was. I personally don't mind making up my own problem on the spot and working it out for the student. I also want to mention that if this path is chosen, it's important to not focus on the problem that the student was stuck on. Choose a different problem to solve for the student. Then if they can go back and correctly modify their calculations for the problem they were struggling with, they will not only be learning, but they will also feel much better about their ability to succeed in math.

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    $\begingroup$ I would do the same in one on one interactions. But, as I had mentioned in a previous comment, the biggest constraint is "time" - we also need to do justice to the syllabus. $\endgroup$ Dec 12 '20 at 18:55
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I'm not sure how well this suggestion would fit into your specific course set up, but I want to share something that has been helpful for me in other contexts:

I like to set a general theme/expectation that submitted work from students is supposed to be helpful for a classmate reading along.

So, when I see something on a problem set that is not quite right, or correct but not detailed enough, I get to say something like this:

"I think that I see what you're getting at here, but I worry that a classmate would be confused."

It acknowledges that they have something reasonable, that they're "on the right track". But it also points out that their work could be improved in some way. I think this may help point out errors without "demoralizing" the student.

It also may push them to think this way in the future and in other contexts. In my experience, it tends to lead to conversations about the mathematical ideas themselves and how best to present them, as opposed to how many "points" it's worth.

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Quote, emphasis [and implications] added:

Question: Is it wrong [for the teacher] to say "you are wrong"? If yes, how do we [the teacher] correct mistakes in a way that is not demotivating? My policy has been, [teachers] give elaborate feedback... But, feedback often meant I point out their mistakes. Of course, I [the teacher] was also generous with praise. How can I [the teacher] give feedback that is not demotivating?

Perhaps your students responded productively to elaborate feedback, but most research and casual empiricism say they don't; students overwhelmingly pay attention to grades. [One can like or dislike this, but I see almost no room to debate its veracity.] In this case, they interpreted your feedback as "I'm going to get a low grade whether or not I try hard, so why put time into this class?"

EDIT: OP said that many students received good grades for the presentations and may have inferred that effort was not necessary. Overall, the comments and grading strike me as sending an inconsistent message.

More importantly, though, this quote is all about the teacher doing a lot of mathematical thinking while the students - presenters and audience - doing very little.

I humbly suggest that what you're really looking for is:

  • How do I get students to think and persevere mathematically?

  • How can a student learn to assess the quality of their own math and that of others?

  • How can I give students better odds/opportunities to shine when presenting in front of the class?

  • What motivates students to respond productively to feedback? [Hint: GRADES]

The types of comments from @FoiledIt24 and @Daniel R. Collins are quite good for dealing with richer and advanced content. That's definitely part of the solution.

Here, I suggest an approach that can tackle all four of those questions at once: assessment as learning. [This link is OK. If anybody has a better link please paste it in the comments!]

You could implement assessment as learning as follows:

  1. Discuss the learning outcomes with the students and create criteria for the various concepts and skills to master. Create one big rubric for the course plus smaller rubrics for other critical learning outcomes. Involve students in this as much as possible. Ideally, within a week of the course and/or unit(s) starting, you can say "Here are the scoreboards we will use to track grades and learning." This is a large but worthy investment because without knowing goals and criteria of assessment, nothing below will work. Also, their involvement in creating rubrics will increase intrinsic motivation.

  2. Provide examples of varying quality and have students assess them according to the criteria you've established together, then have students slap an A+ or C- or some other letter grade on them. Discuss as a class and ensure everyone knows the difference between A-quality work vs B-quality work vs C-quality work etc. These varying examples are arguably the single best place to tackle the most common mistakes and misconceptions. You might say something like: "The fictitious students Sue, Al, Bob, and Zach have all attempted their own proofs of the associativity of matrix multiplication. Pretend to be the teacher and mark their work according to the rubric we agreed upon earlier. Give each student specific suggestions. End by giving each a letter grade and prepare to justify it all to other members of your group. Try to reach a consensus with them, then we'll discuss it as a class." And you can do so, knowing it's a safe place to share candid assessments, whether positive or negative.

  3. As students work on proofs and problems on their own, give them feedback in the form of:

  • Reminding them of the criteria. "This is a word problem and one of the criteria is correct interpretation of numbers in context." This is the best type of assistance.

  • "Keep thinking" questions and comments such as those mentioned elsewhere in this thread

  • Descriptive praise [EDIT: I realize I gave this short thrift originally. I should have said that among pedagogical techniques that are easy to implement, descriptive praise is IMHO the most powerful. Other suggestions in this post are much harder.]

  • Hints that reference success but require deep thinking to utilize. "I see you're stuck on exercise 38, but you aced last week's problem set and its last problem is quite relevant here."

  • Highlighting specific progress and excellence in front of the class and celebrating it. Even as little as a short applause or a cheer or giving them a token prize. "Last week, Joe was unhappy with his proof by induction. This week his induction proof got an A- and the grand prize of a $0.25 chocolate bar. Look at both proofs and tell me what improvements he made. Reference the criteria we discussed earlier."

  1. As students become more proficient in assessing their own work, you can start calling on them to present in front of the class. "Who has thoughts on this they'd like to share?" Or, more privately, you can speak to one student and say "Hey, listen, your work here is really elegant even though it's not quite complete. I know you don't like performing, but can I share a screenshot of that with the class?" After this, have the class discuss in terms of the pre-established criteria what makes these such shining examples. When they're done commenting, as the teacher, say "I'd give this a B+. Not bad for 5 minutes work! Here's a similar problem. Now, everyone, gimme some A+ quality work!"

  2. Have students write in PENCIL a practice midterm as if it were the real thing, but do NOT count it for grades. Then, have them mark their own practice midterm in BRIGHT RED PEN, you guessed it, according to the criteria. (They might need a solution key, too.) Then, still in RED, each student writes a letter grade at the top of their test along with a bunch of specific comments to themselves on how to improve for the real midterm. Another student then approves the grade and comments, then submits everything to you and you give them written feedback on the whole thing - but mainly the RED stuff - ideally at least a few days before the real midterm. Trust me - they will respond productively to this feedback!

I hope this provides some useful ideas on how to ensure assessment and feedback are productive and motivating. It really just boils down to clarifying expectations, building on success, aligning incentives and feedback, and profound involvement of students.

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  • $\begingroup$ Thanks for the wonderful suggestions. I will try to incorporate these suggestions, but I guess it will take me years before I can do it perfectly. I wanted to mention one thing though. I graded them very leniently and the grades were revealed soon after their presentation. So, it is impossible for them to think "I'm going to get a low grade whether or not I try hard, so why put time into this class?". Maybe the opposite happened "anyway I will get decent grades, so why to put time into this class?". $\endgroup$ Dec 16 '20 at 7:40
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    $\begingroup$ You're welcome! I don't think anybody implements these things perfectly. Peter Liljedahl said that big improvements to assessment and feedback are both the hardest and best thing you can do for your class. I certainly do NOT execute all of the above perfectly. As for grades, I wonder if there is discrepancy in their minds when they interpret your comments as harsh but get a variety of letter grades afterward. I will correct my post in response to your comment. $\endgroup$ Dec 16 '20 at 8:56
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    $\begingroup$ As someone who will begin student teaching in the spring, I found this answer very useful. $\endgroup$
    – FoiledIt24
    Dec 16 '20 at 15:52
  • $\begingroup$ Glad you liked it! $\endgroup$ Dec 16 '20 at 17:50
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If you meet 3 times weekly, allocate 2 of the 3 days to doing justice to the syllabus and the third to an open, but directed solving of homework problems (beware of undirected, open discussion, which will simply waste time).

Ideally you'll be able to

  • more rapidly cover the material
  • discover what students are missing
  • provide a better opportunity for students to teach in an inverted classroom

The discussion day is a terrific opportunity for your positive feedback; students will listen in a better light, ask better, and try to understand because it's worth "free" points on their homework.

My answer to a similar trouble on Academia SE covers the process and structure in detail

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