This post recently appeared on twitter:
It is indeed a lovely answer. A number of people pointed out the inconsistency between line 2 and line 3, including these responses (click to view):
What actions are available to the teacher upon noticing this?
One approach might be to tell the learner that her answer is wrong, and perhaps explain why. Some teachers might say something along the lines of, 'It's OK to make mistakes', or, 'We learn from mistakes', perhaps even revelling in the 'mistake' as something wonderful. Perhaps some other questions involving indices might then be given, for 'practice'.
Let's suppose that this learner doesn't mind her attempts at solving this problem being described as a 'mistake'. The best case scenario, then, is that given some or all of these interventions, she does not apply this line of reasoning again given other situations 'like this'.
But it is my experience that the chances are that learners will follow similar lines of reasoning in the future unless something else happens, beyond those teacher actions described above. As a wise person once said: "One thing that we learn from experience is that we don't learn from experience alone."
What might the learner take away from this experience? She might remember this particular example, and/or conceivably come away with a rule along the lines of: 'I should not take roots when there are powers that are being added', or perhaps the non-rule 'a^n + b^n = c does not imply a + b = n-th root of c'.
I would conjecture that this learner might well already know this. I suspect she 'knows', at least on some level, that 3^2 + 4^2 equals 5^2, and also that this does not imply that 3 + 4 = 5.
All we can say for certain on looking at this work is that this person took this approach in this context at this time. It is my experience that people do surprising things when faced with cognitive (or emotional) complexity that does not imply a lack of 'understanding' (whatever that might mean).
If this is the case, will correcting her and telling her (again) that 'you can't take a root if there are indices being added', or practising lots of questions involving indices, be enough for her not to do something 'like this' in the future?
It is possible, but certainly not guaranteed. Firstly, there is the difficulty in recognising another situation as being like a previous situation, particularly when embedded in a complex problem.
There is also the matter of learners altering 'local generalisations' they have made, that have proved successful in the past, perhaps in more limited domains. It is almost certainly the case that 'taking roots of expressions with powers' has been a successful strategy in the past for this learner, hence her doing it here. But now she must expand that generalisation to: 'taking roots of expressions only works sometimes, but not if there are two terms being added.'
In my experience, it is very difficult for learners to alter the (valid, at the time) local generalisations they have made, although this at the heart of learning mathematics.
One of the respondents to the post expresses his exasperation at these 'sorts of errors', but I have found this feeling of exasperation to be counter-productive. Expecting all learners to be able to assimilate all of the possible algebraic rules and non-rules, and then be able to apply them in unfamiliar contexts, may lead to disappointment, which may diminish the quality of subsequent interactions with students.
How feasible, or productive, is it for a teacher to try to train learners to avoid making all of the possible 'errors' that they could make? I suspect it is neither feasible, nor desirable. The creation of novel ways of solving unfamiliar problems is exactly the thing that I hope learners will do. As such, this is still, for me, a "lovely answer".
This is not to say that inconsistencies and local generalisations are not to be exposed and challenged; this is the work of learning mathematics. The question is rather how, and by who?
Rather than the exasperation that comes as a teacher from spotting and correcting every 'error', only for the same errors to resurface time and again, I have found it more useful to recognise each piece of inconsistent reasoning as a creative attempt made under uncertainty.
Then what is required is to work towards the development of the learner's internal monitor: the awakening of the desire/need/habit to treat each line of working as a conjecture to be tested.