All doing is knowing, and all knowing is doing.
Humberto Maturana & Fransisco Varela, The Tree of Knowledge
This post contains some further thoughts about the role of articulating before doing when solving mathematical problems, illustrated by some examples from today's Higher maths lesson with J and K.
In my experience, many learners are reticent to spend a few moments articulating before doing, myself included. Sometimes it is clear what I need to do, and there feels no need to talk about what I am going to do before doing it.
In a recent communication, Alf Coles noted that he might be hesitant to invite learners to articulate what they are going to do before doing, as it may stop them beginning. He suggests that sometimes we only know where to go when we get there (or as we are getting there).
Sometimes just doing something is a useful - or the only - way forward. If I am not sure what to do, creating examples or drawing a diagram helps me get a sense of the problem. But there are times when I feel articulating may be useful:
- Taking a few moments to describe/name (to another or inwardly) what I notice and what I want, in the hope that relationships and associations come to mind that had not previously.
- Articulating what I am going to do and why, in order to identify whether I have a rationale for what I am about to do, whether what I am about to do is suitable for this problem.
What follows are two examples of problems in which doing before articulating, and articulating before doing, both proved useful.
Both students could articulate what they needed to do for part (a). K initially conjectured the roots to be 2 and -1, but then spent some time plugging in numbers into the expression x^2+1 to convince himself that the only root was 2. This is an example of getting a sense of by doing. In contrast, J could immediately explain why x^2+1 could never be zero.
The sketch was tricky, in that they only had two points, which is not enough to sketch the cubic accurately. Both students realised, on plotting the two points, that they had to plot more values, another example of getting a sense of by doing.
K then guessed the transformation in part (c)(ii) to be a reflection in the y-axis (as opposed to the x-axis). He sketched the reflection, and was about to move on from this question when I asked him if he could articulate why it was a reflection in the y-axis. He could not. Here is an example in which articulating before acting might have revealed to him the absence of a rationale, signalling that his response was nothing more than a guess, and that more certainty (a rationale, or some means of checking his answer) was required.
This problems with doing in the absence of any rationale was also evident yesterday in at least two examples, firstly when K wanted to use the quadratic formula to find the value of k for the discriminant problem (Dialogue #1), and when J wanted to find the equation of a circle because that is what you normally do with circles (Dialogue #2). These are both instances in which it might have been helpful to articulate before doing.
This is what I would describe as a novel problem for J and K, in the sense that they have not encountered anything that looks like it before. After re-articulating what is a clumsily worded question a few times, we identified that it was about finding the domain and range of this function.
K tried plugging in values in for x into g(x), but was not able to evaluate them as this is from a non-calculator exam. He tried a number of examples but got no closer to identifying the underlying structure. Here is an example of numerous examples not bringing an answer, of doing not leading to knowing. What would have been useful then is to realise he was getting no closer to an answer, and 'return to the problem', but which I mean re-articulate, in the hope that some structure comes to mind by considering the function as it is (i.e. considering the effect of squaring the sine function).
J was not sure what to do for a long time. After eliminating A and B, he came across the idea of substituting values for g(x) instead of x. He was initially not sure what values to choose ("there are not many to choose from!"), but then after discussing with me (a form of articulating) that he could try any value between -1 and 1, he selected -1, creating the equation -1 = sin^2(...). Square rooting both sides of this equation brought the realisation that g(x) could not be -1, leaving D.
So for this example, neither student was able to articulate what they might do to solve this problem. Doing brought J to the solution (although it is possible that he may have stumbled upon the answer). Doing did not bring K any closer to a solution, and there is a chance that returning to the problem (i.e. re-articulating), may have been helpful.
This post contains some examples in which doing before articulating proved successful. There are also examples illustrating that doing before articulating may not be most useful approach, particularly when:
- Doing in the absence of any rationale - and then, crucially, not testing what was done in some way, either by reasoning or checking.
- Doing what is 'normally' done without thinking about why it might be suitable for this problem.
- More doing, when what is needed is a 'return to the problem', i.e. re-articulating in the hope of identifying some structure.
These are instances in which articulating before doing might prove useful.