To calculate or not to calculate?

How to get the best out of this tool that IB allow at all times for students is a constant challenge. It should most definitely be an advantage to students and almost certainly is on balance, but this does not come without careful consideration of how best to employ it in our teaching. Within that come the following questions;

How do we get students to be come really familiar with this tool so that they use it fluently, know the subtleties of the settings and take full advantage of having it?

How do we make sure students know what they could do with a calculator and what they cant?

How do we discern and how do help students discern when the calculator is the most efficient tool for a job?

Whilst anyone of these questions is worthy of significant discussion and debate, it is the last that is currently occupying my thoughts and as such the subject of this blog entry.

Obviously, the last question relates to the second, in that you must know what it can do before you decide what it should do. The teacher support material for GDCs is a good comprehensive list of calculator skills, but I would suggest that it is a very valid exercise to read through that list and decide which of those skills you would actively encourage students to use. For example, its perfectly possible to ask calculators to round numbers to given degrees of accuracy. Would this be efficient? Would this defeat the object of teaching about accuracy? I understand that philosophy behind allowing the calculator is to remove the emphasis away from calculation and place on it application and generally speaking I subscribe to that philosophy, but I think there examples when it is actually simpler to learn how to do a skill manually than it is to learn how to do it on the calculator and that that later sometimes risks removing all recognition of a concept and thus the ability to judge whether or not an answer 'looks right'. 

I must be careful not to sound like I have a downer on technological aid, when in fact the opposite is true, but, like most things the advantages come when a considered balance is struck in how best to make use of it. Let me try another example; consider a question that asks a student to factorise a given quadratic, say x² + 5x + 6 for example, then asks the student to 'sketch' the curve. How should a student go about that? One possibility is to simply enter the function on a calculator, have it draw the graph, use it to establish the coordinates of the intercepts and the vertex and then copy that all down. The other is to recognise that from the factorised form of the quadratic you can deduce the x - intercepts by solving for zero, the y - intercept can be deduced by substituting x = 0 and the coordinates of the vertex by using x = -b/2a. So which is the best? Which is the most efficient? Which requires the most understanding of mathematics? In true educator style, I will avoid answering the question directly, but say that it comes back to defining the goals of the course. If its worth knowing how to factorise the quadratic, its worth exploring why solving for zero gives you the x - intercepts. In fact this relationship and others like it is what makes the study of quadratics most interesting for me. It is a fundamental concept to understand the significance of either of the variables being zero and one that is great to discover and own. Using -b/2a to get the vertex is no different from asking the calculator unless you do so with experience of knowing why it works at which point is has the same, if slightly more complex value as knowing about the intercepts.

If the goal of teaching quadratics is to use them as modelling tools in context then I would still argue that understanding the significance of either variable being zero was important, but can see that knowing how to use a tool that reads from the model is an important but different skill. In practise I find the Maths Studies syllabus a little caught between the two and often the best way to discern whether to calculate or not is to look at how many marks the question is worth or look for word like 'show' in the question. In a sense, my not always being decided on this issue explains why the syllabus encourages a bit of both and my writing this blog is helping me at least to remind myself of why I ask the questions and why I try to strike the balance!

This is just about scratching the surface of a big debate that is coming the way of curriculum designeres for mathematics and which is being fuelled by people like Conrad Wolfram in the video below. Watch it, think about it and be prepared for a debate another day I reckon.