Constructive refutation of student misconception

Although @Gareth Shepherd recently posted Addressing fundamental math errors close to the issue, I experienced my problem of misunderstanding in class, where two good K10 students were asked to present at blackboard the reduction of $\frac{12-3a^2}{3(2+a)}$.

Self-invented "rule"

The students factored 3 in to denominator, then applied a self-invented rule of "divide numbers by numbers, symbols by symbols", transforming $\frac{12-3a^2}{6+3a}$ into $\frac{12}{6} - \frac{3a\cdot a}{3a}= \underline{\underline{2-a}}$, where - of course - the right hand side (RHS) is correct, but the left hand side (LHS) represents a faulty operation on a fraction in focus here, as the RHS is not a transformation of the LHS (despite the result is that LHS$=$RHS, as pointed out in the blog of Daniel R Collins).

In class, I pose as a proponent for documenting ones (mathematical) actions with rules, and I have engaged in a negotiation with my students, who are used to emphasizing on results only from earlier school experience.

Building knowledge on mistakes

I expect to continue teaching the class after summer, and see the mistake by two relatively strong students asn opportunity to build a faults-are-necessary-culture in he math lessons for this class. Thus, I revisited Lakatos' Proofs and Refutation where both Teacher and Pupils attempt, refute, and improve.

I am looking for suggestions as to how I can use the two students' mistake constructively. Deployment of the misconceptions should meet the following criteria.

• Both the two mistaken students and their classmates must feel encouraged rather than intimidated by me bringing up our discussion in the public of the class (thus exposing their misunderstanding). This can be done by not mentioning the discussion, but then I would run the risk of the two students feeling my approach surreptitious - sneaking in on their weaknesses. Hence, I prefer an approach, where I take responsibility for the discussion, and the entire class sees the mistake as an opportunity.

• The explanation / treatment should simple enough to be briefly presented, yet sufficiently stringent to state the point that the introduced, self-invented rule was based on an invalid conjecture

The 'proof' tag is mainly intended to the students' lack hereof in their deployment of the self-invented rule.

Edit 1

There are no counterexamples, so the comment of @Amir Asghari is taken into account by deleting, what mentioned a counterexample.

Edit 2

Instead of "indicating" that a transformation is not generally valid by the use of $\neq$, I acknowledge the insight of @DRF -- Students may have a hard time interpreting the symbols, including $=$ and $\neq$, in their original, intended meaning. Thus, it is counterproductive to use (any) sign in class to "indicate" anything apart from what these signs are actually meant to signify. I have thus reformulated my "temptation" to use that sign from symbolic to worded.

• I will point out that these exercises of a difference-of-squares canceling one of its factors with a linear denominator are infamous for supporting this sabotage-understanding in every single example. My advice is to make sure you do a bunch of reductions with nonlinear denominators to shake out the broken idea. On my blog: madmath.com/2010/06/stuff-that-shouldnt-work.html – Daniel R. Collins May 25 '17 at 14:42
• @DanielR.Collins - I shall calibrate a bunch of exercises for next class incorporating your experience. The mindboggling Stuff that Shouldn't Work could sure be good fun to work with, then reflect about the meaning of the concept of a rule. – Engelsmann May 25 '17 at 15:23
• Please please do not use $\neq$ "to indicate" anything other than not equal. Students already have no idea what $=$ means and this will just confuse them further. – DRF Sep 16 '17 at 12:53

Mathematics. This seemingly faulty transformation is trickier than it looks. First, notice that it is not completely wrong and in a sense, it is even correct! As Daniel has mentioned in his comment, this is somehow related to a difference-of-squares. Let's have a closer look. $(a-b)(a+b)=a^2-b^2$. A student's version for getting from the LHS to the RHS would be something along this line: a times a (that is $a^2$), minus times plus (that is $-$), b times b (that is $b^2$). Having that version in mind, it would be quite "meaningful" to simplify $\frac{a^2 - b^2}{a-b}$ or $\frac{a^2 - b^2}{a+b}$ by dividing $a^2$ by $a$, sign by sign, and $b^2$ by $b$.

This allows us to link another pair of students' version of doing algebra, one of them being the most infamous algebraic misunderstanding: $(a+b)^2=a^2+b^2$. See how this is supported by the student's way of thinking mentioned above. $(a+b)^2$ is $(a+b)(a+b)$ that "is" $a$ times $a$, plus times plus, $b$ times $b$. "Accordingly", $\frac{a^2+b^2}{a+b}=a+b$!

Pedagogy. This is a great opportunity to build knowledge on a network of mistakes and takes. First of all, noticing the difference between two cases above, we have to qualify "not a transformation" (bolded in your question) to "generally not a transformation". From here you can transform the "mistake" into an investigative task. The question for students is "when that self-invented rule works?" As such, instead of giving them plenty of examples, you encourage them to generate their own examples while they are consciously working with the mistakes rather than being told about the mistakes. (Please also see my answer to a relevant question here)

"How I can use the two students' mistake constructively?"

This warm up routine was shown to me a couple years ago as a way to show common math mistakes in a constructive manner and I used it some of the time (not everyday like the teacher in the video does).

1. Post warm up problems that has common misconceptions or easy to fall into traps.
2. Have students complete the problems anonymously
3. Collect the answers and display one or two of the wrong answers to show on a document cam
4. since the class knows the answer is wrong, have a discussion or Think-Pair-Share about what is wrong in the problem
5. Say something along the lines of "Thank you to whoever wrote this, because otherwise we wouldn't have had a chance to have a great class discussion and now everyone who made the same mistake will learn from your submission."

Perhaps step 5 is the most important as it makes students feel helpful for making a mistake and there is no penalty for doing so, only helping other who had the same mistake. Additionally, this way there is no confrontation with the students who got it wrong (the teacher approaching them and asking to show their mistake to the class can feel intimidating perhaps).

• Note that the teaching Channel video, students hand in named answers, but the names are not "published" to the class, only numbers of yes and no are numbered, and to the class only by the frequency of the teacher pronouncing the "label" words yes and no in response to reading the answers. Of course, if you want to display the answers directly (in the video the teacher rewrites the answer), you may abstain from knowing the identity of the student answering correctly or wrongly. Knowing that you know their identity makes a different contract with your students than the anonymous case. – Engelsmann Jun 28 '18 at 19:20
• I would parallel steps 3 and 5 in importance: Why should one pick this mistake as a 'good' one? What is indeed right about this - wrong - answer? How can you indicate that the answer is right in part? These steps are clearly walked through in the teaching Channel video, which also emphasizes the importance of being respectful, being nice, to the mistake-maker. – Engelsmann Jun 28 '18 at 19:32
• Good points, The other benefit of the teacher copying the mistake instead of directly showing it is that the handwriting can't be identified (and probably also increases the legibility of it!). And the teacher knows who is making these mistakes and who isn't. And being able to decipher what is a "good" mistake and what isn't depends on the context and having taught the class enough times to know what mistakes students typically make for certain problems. – ruferd Jun 28 '18 at 22:00
• I agree with @ruferd as to the video painting a highly edired picture of classroom reality: Teacher will run dry of good, catchy mistake-inviting warm-up exercises, and students will get bored without properly dosing a varied diet of methods. – Engelsmann Jun 28 '18 at 22:43

I love the Ruferd suggestion.

In general, I think that instructors here think too much that students learn from explanation or correction (the way a computer would) versus drill and practice and example and mimicing. I suggest the human animal is not not a completely rule based system. Thus methods to correct back inferences should include some sort of feedback method for getting it wrong, versus getting it right, seeing other students get it right, etc. And being forced to use the proposed general method. Not just "nice" explanation of why they were wrong.

• Fully agree with @guest, in that explanations build no skills. But explanations build cognition, and niceness my overcome motivational barriers. Way back, one of my violoncello teachers practiced the principle that "you have to do it right more times than you have done it wrong before you can say you have learned". Thinking of how many times one has made something (slightly) wrong makes this demand staggering. – Engelsmann Jun 29 '18 at 7:25