I have been working with John Mason on some problem solving videos recently, and was struck by his use in one video script of the phrase, "I might recognise this as... but I might not."
I have become sensitised to notice instances of learners recognising-things-as in more and less useful ways. Here are some examples from my lessons this week.
Example 1. A learner solved the equation 9 = x^2, by writing x = √9, and then x = 3.
Example 2. Given the problem below, a learner solved dy/dx = 0, instead of knowing to find where the line and curve meet (by solving simultaneously):
Example 3. A learner used calculus to find the answer to part (b) below - see image - rather than recognising the usefulness of considering k.cos(x+a) as a transformation of cos(x):
Possible reasons for these and other similar occurrences, in which a learner might not recognise the object as something useful, include:
- Switching into an 'automatic' or habitual mode, particular if something seems simple, such as example 1 above.
- Working solely algebraically, without perhaps an underlying visual image. This seemed to be the case with example 2 above. I will give another example of this below.
- Applying something that was learned very recently, rather than using something more suitable from prior knowledge, as in example 3 above.
- Under- or over-familiarity. Under-familiarity may be the result of a seemingly novel or complex looking problem, or may be from lack of practice.
- Not having a useful interpretation of a word (or symbol), such as the word increasing in the example below. What is required here is to interpret the word increasing as finding the values for x where the function has positive gradient.
Note that finding the stationary point is useful for solving this problem, but it was clear that this learner was not sure exactly why it was useful (the notes in black pen are mine, made during discussion).
Part of teaching mathematics is about helping learners make associations: to learn how to interpret words and symbols mathematically, and then to provide opportunities for reflection and re-interpretation . Here are this learner's personal notes after discussing this question:
All of the steps make 'sense', apart from the fact that sin(A) and cos(A) cannot be greater than 1. I asked this learner to re-articulate the question, and then asked: "What visual image does this create, for you?" He immediately drew the image below, and then solved the problem with ease:
One example was a learner solving the equation 2sin(x) - 3cos(x) = 2.5 by choosing to write it in the form k.sin(x+a)=2.5, without being prompted. Another was knowing-to and knowing-how to write complex expressions in index form before differentiating.
The following example was particularly encouraging. I had worked with this learner's awarenesses regarding the discriminant a few months ago; the idea to solve this problem in this way came to his mind in the moment:
This is a good example of recognising-something-as in a way that is useful.