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John is trying to learn some field of mathematics. John's teacher Sarah gives constructive criticism to John after seeing John's attempts on exercises related to the field he studies. Later, John tries a different set of exercises on the topic, and yet again, Sarah finds John making the same mistakes as before.

How would the teacher Sarah help John who has "internalized" the mistake? It seems that directly telling the student to do more exercise would only make their condition worse as the "mistakes" would be more drilled in.

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  • $\begingroup$ Are these called internalized mistakes or systematic mistakes (or something else)? $\endgroup$ Nov 7, 2022 at 12:16
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    $\begingroup$ Sorry, but making the same mistake twice doesn't amount to having internalized it in my humble opinion. Rather, your description seems like a completely normal learning process. $\endgroup$ Nov 7, 2022 at 13:54

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Coach them through doing it the right way. Have them repeat it the right way, several times. In front of you. And go very easy, including repeats. Gradually relax the guardrails and keep drilling. They are not going to get it by comprehension, but by exposure. Perhaps comprehension will follow long later. But for now drill it the way you would drill a physical task. I suggest computer drill sites and very easy ones.

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    $\begingroup$ Although this answer sounds like the opposite of mine, I mostly agree with it. I think the student needs both comprehension and exposure. Doing it properly, over and over will build and strengthen the new neural pathway that comprehension has shown the need for. They will need a way to know the difference between the right way and their wrong way for a while. apps that say 'right' or 'wrong' can help. $\endgroup$
    – Sue VanHattum
    Nov 7, 2022 at 1:31
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    $\begingroup$ I find that comprehension usually comes before correct execution in such cases. Bad habits take a lot drill work to suppress as you advise. It's frustrating for the student, who curses and says, "I knew that!", whenever they repeat the mistake. Not only do they have to construct a new pathway and train it to activate in the right context as @Sue says, they have to construct an inhibitory one to block the incorrect one. $\endgroup$
    – Raciquel
    May 3 at 14:13
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The student has a wrong process internalized. Can you explain how their process works? (What I mean here is, could you predict their wrong answer to a new question.) I think that will help you analyze what they are thinking.

If you can find the nugget of right thinking in their wrong thinking, then you can show them how they were right up to this point and then took this one wrong step. They will need to see very clearly why that one step leads them astray.

If you could give an example of the mistakes this student is making, I could perhaps give a more useful answer.

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    $\begingroup$ Also, if their process is predictable, try to make a question where it gives an obviously wrong answer. E.g. zero or a negative number where that makes no sense. $\endgroup$ Nov 7, 2022 at 11:39
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    $\begingroup$ That sounds like a good idea, but I know plenty of students who write down obviously wrong answers. It's obvious to us, but apparently not to them. $\endgroup$
    – Sue VanHattum
    Nov 7, 2022 at 16:06
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In some cases it helps to first try to understand exactly why they might have some misconception. If you can do that, though not always possible, you can more easily find a way to guide them.

An example: I once had a student that evaluated math expressions strictly left to right without considering operator precedence. It was very hard to convince him that he was getting the wrong answers and why. So 2 + 3 * 4 was 20 in his computations.

After some explorations I discovered that there were two things that got him to this situation and reinforced his thinking. The first was bad teaching at some point in his past. He was convinced by some instructor that this was correct. Then, he happened to buy and use a calculator that didn't take account of precedence so his earlier misconception was reinforced daily.

It took a while to convince him that, while he made correct inferences based on his available evidence, it was the evidence that was the cause of his issue, not a personal failing. But until I understood that, and could communicate it, along with the proper rules, he resisted my efforts as somehow undermining him. He hated me for a while, I think, until we worked it out.

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Assuming the student seems to understand your correction, but simply forget or fall back into bad habits, you can teach the student to use metacognition.

Tell them that they seem to have made the same mistake which they made earlier. Ask them questions along the lines of, "What do you think happened? What situations do you think you might find yourself making the same mistake again in the future? Can you think of any reminders for yourself to avoid making the same mistake in those situations?" Then you can teach them any time they are doing this process that they can ask themselves, "I know I tend to make X mistake here; did I make it this time? Let's check." Next, have them do the problem in front of you, not only showing you that they learned the correct process but also the questions to ask themselves to make sure they haven't repeated the same mistakes.

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  • $\begingroup$ That's a big assumption! If it's a tricky subject, making a similar mistake occasionally due to lack of attention is understandable, and cross-checking yourself can find those errors. But consistently making the same mistake in every case is a guaranteed indicator that the student did not actually understand the "correction". In that case the student can't cross-check themselves because they haven't learnt the basic concept. $\endgroup$
    – Graham
    Nov 9, 2022 at 18:06
  • $\begingroup$ I am talking about the situation in which a student fixes their mistake one session and appears to be able to explain it, but then returns to making the same mistake later. Often with a small amount of prompting they can identify their mistake, and the main problem for this student is that they are not self-evaluating enough to recognize that they returned to an old bad habit. $\endgroup$
    – Opal E
    Nov 10, 2022 at 5:39

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