I’ve seen a few blog posts on modelling in Maths this past few weeks so I thought I’d add my thoughts on this absolutely vital skill for teachers…
A perennial problem for Maths teachers is developing students’ ability to ‘see’ their route into the question when it is not immediately apparent which mathematical techniques they need to apply. This week I began teaching the first lesson of the Edexcel C2 Trigonometric Identities and Equations chapter to my Year 12 Class. For those unfamiliar with this chapter, it requires knowledge and application of the following identities:
Normally when selecting the examples I am going to model with the class, I take a bit of time to carefully select the question I am going to use to ensure that the example contains an appropriate level of challenge whilst still keeping the concept that I am trying to teach at its heart. However, this week when modelling for the class I made a point of (somewhat theatrically) picking a question that I had not reviewed beforehand and talking students through my thought process as I looked at the question. By chance the question I had chosen was a great one from Integral Maths:
Obviously I knew that the question would involve the use of one or both of the above identities, however, the students also have this knowledge. What students don’t have is:
- A full mastery of the two identities including the ways in which the identities can be rearranged;
- Exposure to a sufficient number of questions that allows them to spot the ways into the question that examiners tend to use
In order to help expedite the process of spotting some of the commonly used methods required to solve these problems, I channelled my inner James Joyce and gave students a stream of conciousness style account of how I would look at and complete the question.
I started off by pointing out that the first thing I noticed was that the LHS was a difference of two squares, although not in the form students would be used to seeing given that this was their first foray into trigonometric equations. I then said that I looked at the RHS and spotted that there was no cosine squared on the right hand side. I then said that I know that the sine squared theta + cosine squared theta identity can be rearranged to give cosine squared theta in terms of sine squared theta and so a substitution would likely be the way forward with this question once the brackets had been expanded. I then continued to talk students through exactly what I was thinking as I solved the question. In the second example, students had to do a bit more of the ‘leg work’, spotting some of the interesting features of the question and doing more of the talking than in the first question.
Students really appreciated being explicitly told about how I went about looking at this question, especially given that I was doing it ‘live’ and so was, essentially in the same boat as them. Allowing students an insight into how an ‘expert’ thinks is one of the most valuable things any teacher can do and is one reason why didactic teaching is not the evil many seem to have argued. Discovery-based, project-based or inquiry-led learning just isn’t an efficient or effective way to develop students into competent Mathematicians. They need to ‘see’ how an ‘expert’ might see a problem, at least when developing their understanding of a topic.
When modelling all teachers do some work talking through how they ‘see’ a question, however, I contend that most of the time we can do more to ‘make the implicit, explicit’ (to borrow a phrase from David Didau) and really talk students through exactly how we, as Maths teachers, view and solve a the problems presented to us.