Why Calculus gets such a bad press!

Most of my blog entries will come about as part of my preparations to teach in the coming week and this one is no different. Tomorrow I start the calculus unit with my 2nd year IB Maths Studies students and I can't decide if I am looking forward to it or not! The teaching of this unit always presents a challenge to me. Whats the best time during the course to do it? What is the required knowledge? (and do my students have it?). It is such a wonderful branch of mathematics that it would be a terrible shame to treat it as skills only and yet we are only three months away exams and have a lot of revsion to do. How much time shall we spend exploring? More to the point, what will the overall benefit of exploring be to the students in terms of being able answer exam questions? Lastly, what is going to get in the way?

Well, as with most things, the more you do it the more experience you have to draw on and the more prepared you are for the answers to these questions (notice how avoided comitting to a how much better I will teach it). So before I even get started with the calculus I am going to spend at least a lesson exploring the principle in the picture above. This is definitley required knowledge, but, in my experience, it is a notorious barrier to progress and if you are not careful students end up thinking calculus is difficult because they dont understand and cant remember this rule about negative powers. This is completely unfair to calculus! I will go right back to basic laws of indices to get students to explore why this rule is so, in an attempt to remove this barrier once and for all. That way, when I get to differentiating functions with negative powers we will glide smoothly past this potential barrier. This is my top tip for the teaching of this maths studies calculus unit. It is assumed knowledge and not covered anywhere in the syllabus. If it is not dealt with then it will put a major spanner in the works just when you think you are getting somewhere. 

Negative powers are one of the reasons why I find this calculus unit hard to teach. The calculus element is actually very small and most of the unit is taken up with the associated supporting algebra. I think its very important to help students understand which specific parts of any problem require calculs to solve them so as not to give calculus a bad press. Quite often a 'calculus' question will follow the following pattern

• A description of a situation with some variables and expressions given. Students are then expected to manipulate this algebra in order to form an expression for a given quantity in terms of just one other variable by using algebraic substitution. This often involves some very sphisticated algebraic skills and reasoning and can scupper a problem before its even begun!
• A question may then ask students to optimise the function - enter the calculus. Students differentiate the function, often worth a couple of marks, then put it equal to zero. This is the calculus done! Now they are solving equations, more algebraic manipulation.
• Students may then be asked to substitute the solutions to these equations back into their original expressions - algebraic substitution.

Of course you cant strip all that algebraic manipulation out of optimisation problems, but it is important to note that much of what students are being tested on here is algebraic manipulation and not calculus. Along with that, this is where the graphical calculators should come in really handy. This is really worth bearing in mind when we teach it!

To answer my other questions, I will be giving time to exploration and discovery. I have made previous attempts to rush this module through and just teach the skills, not focussing on the 'why?' and had little success. Calculus is a rich topic and should be shown to maths studies students. Please see the general section on calculus in the main part of this website and the activities section for examples of what you can do with your classes and how to deal with 'Negative power'!