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How do you deal with such errors committed by students in Math classrooms?

Common errors in secondary Math classrooms -

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Not really an answer but thought to recommend John Mason's book 'Developing-thinking-in-algebra.' ISBN:1412911702/83501031 Another of his books I have found most useful is 'The Art of Noticing.'
Have you seen the Classic Mistakes site? As well as the posters, podcasts are available discussing the errors. The posters are good for classroom displays.
Try using the erroneous conclusion to work backwards to see if you get the original problem. For instance, in #1 say something like what if x = 2, or x = 3 -- is the answer still 6?

That way they discover the error themselves, which is more powerful than anything you can say
Surely I agree with Mrs Carolyne Melo - and me also do the same. But I also can see that the origin of these mistakes is in the way of learning - not in pupils. But what a nuisance it is that pupils are subject to double standard of math culture. Why the traffic lights are the same everywhere, and teaching language - not. The language of teaching has the same function as the traffic lights.
I do use this technique at tmes. It is effective and need less explanation. However, I do chip in some theory to help the students be aware of the necessity to follow maths rules and they are there not for fun.


Carolyn Melo said:
Try using the erroneous conclusion to work backwards to see if you get the original problem. For instance, in #1 say something like what if x = 2, or x = 3 -- is the answer still 6?

That way they discover the error themselves, which is more powerful than anything you can say
Discussing common errors in the classroom can be effective way to make aware students the possible mistakes at the time of examination.

Posting articles for students on my class Blog ...Planet Infinity

While solving this question, most of the students used total surface area of the article = CSA of cylinder - 2 (CSA of hemisphere).

Instead of adding the CSA of hemispheres they subtract it.
Most of these seem to be conceptual errors to me, where students are confusing that nature of one operation with another. With 14-18 year-old students, I would address this by revisiting the concepts behind each type of error. I suspect the students have sought to memorize how to do each type of problem, and are getting procedures confused.

For example, order of operations seems a frequent error. Rather than reviewing the order of operations, I would try to go "deeper" and explain why the order of operations is what it is (we do the most "powerful" operations before less powerful ones, so if multiplication can be thought of in some situations as repeated addition, then multiplication is more powerful than addition, etc.). I would then confront the students with more a challenging problem than the one in which they made a mistake, say:
3x -5 + 2x +7 -5x -7x +10

I have often found that a concept or process is grasped much better doing "ugly" problems than in doing very simple ones.

The second approach I would also bring in is visualization: visualize 3x as (x + x + x), or x^3 as (x)(x)(x). Students often need to practice this by writing out the "visualization" on paper for a few problems using each of the arithmetic operators before they get the hang of it. Once they can quickly and habitually "see" the problem in expanded form in their mind's eye, they are less likely to make many of the mistakes you show.
I deal with these type of errors using the "face palm":

Not in front of my students, of course. :)

Seriously though, once I recognized that my students were prone to these type of mistakes, I made sure to repeatedly drive home the concepts that would avoid those kind of mistakes.

For example, #4 was very common in my class. So then I excessively repeat the mantra:

"You CANNOT cancel out terms or parts of terms. You can only cancel out factors." I say it all the time when working with that type of material. Of course, then you need to be certain that the students know the difference between a term and a factor, which can become confusing, because you can have terms inside of factors that are inside of terms (like 3(x+2)+5).

Thanks also to Colleen Young for the Classic Mistakes Web page reference. I may print these out and post them around my classroom.
What if you intend to show the mistake and some students instead of learning to rectify the mistake, remembers it and repeat it. EX #4. I explained and some students did the same mistakes in the test.
I had a challenging issue today in my year 10 class. I had (1+cot^2x)/cotx which I wrote as (1/cotx)(1+cot^2x) and one student completely objected to this! She wanted me to write it as (1/cot x) + cot^2x, which is similar to #4 mistake referred here. It took me nearly 15 minutes to make her understand using numbers like (1+2)/3 is 1/3(1+2), when ultimately she accepted!
I am sure she will have the same doubt if I do it again.
I try to do a lot of error analysis with my students. I find that when they are confronted with correcting their own work they are often more likely to remember the solution. I use James Kaput's work with multiple representations as a foundation for the work students do. If for example my 7th graders miss a problem like 4+3x5 I have them create a model of the problem to find the answer first, describe the problem in writing that confirms their answer, and then work the problem symbolically to confirm the answer a third time. Student's who don't confirm their work through multiple ways of thinking about a problem are more likely to make these kinds of procedural problems. Finally, I have students reflect on the mistake they made, create a plan for remembering the correct way of processing these kinds of problems, and reflect on how comfortable they are with these kinds of problems now. I like the principles outlined in "How Students Learn: Mathematics in the Classroom" by the National Research Council for tying learning together.

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