Further Processes

Thursday 11 February 2016

What counts and what doesn't?

The following is a reply I have found myself giving often regarding the use of 'Further processes' in Internal assessment. the question is 'Is there a comprehensive list of processes that can be considered as 'Further processes' in criterion C of the Internal assessment?' It is a great question and it makes perfect sense to ask it. Unfortunatelty, the short answer is no, but there are some related issues worth considering. Here is my reply.....

'I understand the frustration, but there is not a comprehensive list. In defence of their (IB) position, there is a desire to leave some options open and if there was a comprehensive list then this would rule out anything that wasn't on it. In criterion 4 of the IA markscheme it says 'Examples of further processes are differential calculus, mathematical modelling, optimisation, analysis of exponential functions, statistical tests and distributions, compound probability' Of course, this doesn't tell us what exactly we would expect to see. eg what would we expect to see for compound probability? This is part of the reason so many opt for the statistical test, because it is easy to know what is is expected. Of course, many studies students find stats the more interesting and relevant part of the course for them too. The next level issue is that given the greyness surrounding what we might expect to see for - say modelling - it is open to interpretation and so whilst you may argue well that it should be considered further, a moderator may disagree... BUT, whilst this may all sound a bit inconclusive, I honestly believe that the best approach is to help students follow what they are interested in and do some meaningful mathematics, then try your best to justify the marks you give them. If you think the modelling is good enough (ie not technology only) then go ahead and mark it as further. Happy to hear anyone elses views on this post and my conclusions and hope I have helped a little..... Jim' 

So you can see the problem! What I would like to see in future guidance though is some specific examples of what a moderator will be looking for if students are to get credit for processes other than statistical tests as further processes. I have a few suggestions on that based on the examples on the TSM and years of podering that question! DISCLAIMER - The following are just suggestions based on my opinion. I would give the following credit as further processes if done correctly and relevantly.

Optimisation

The general model for this is to set up a problem like, how to package a given volume with minimum surface area of material and given constraints. Then students collect data by examining particular cases and build towards deducing a model for the surface area and optimising with calculus.

Modelling

This is essentially about collecting information/measurements and trying to find a function that fits the data so that it can be used for forecasting. Of course, like with statistical tests, this can be done with a variety of tech so we have think hard about what manual mathematics we might expect to see....

Example - Say a students is trying to fit an exponential curve to some growth (population?) data. I would expect students to sketch a curve through the points and from tjat curve and the data make estimates or duduce the value of both the y - intercept and the equation of the assymptote. Since...

\(f(x)=k\times { a }^{ x }+c \)

and the y-intercept is at (0, k+c) and the assyptotoe is at y = c

we could reasonably expect that students then deduce the values of k and c. They might then create a dynamic function that allows them to find the best value of a to make that fit. In this sense the student has done better than a tech only solution and better than a random sliding of variables to make it fit. they have understood the key properties of the data and used them to deduce a possible model.

Compound Probability

I have always thought that this should be a problem that involves more than 2 events and conditional probability with probabilities of various combinations being calculated by hand. I am just not confident that a problem with 2 idependent events would really qualify.

Example - If you play with the ideas on the  Fairground Games activity, there are a number of possible avenues. For example, if you get three throws to get the paper ball in the bin, are you more likely to get the second or thrid throw in? A student might determione a relative frequency for the success with the experiment and compare it to more data where people get three throws and so on. thin about it for a while. I think there is lots of possibility.

3D geometry

I think this one is definitely harder, but it often goes hand in hand with optimisation. It is hard to think of examples where this is would be used to solve a problem.

Example - With the  Cuboid Challenge it is possible to make the cuboid from three different pyramids which is nice, for lots of reasons, but mostly because it means they must all have an equal volume because they are all a third of the cuboid they fit in. I think this has potential to be explored in different ways and for some proof, but it would get pretty tricky.

Example - When doing this rice show activity, the rice was falling in some wonderful shapes. I wonder, for example, how many cones of which dimensions could be made with a kilo of rice..... Again, there is thinking to do, but there is potential.

As I said, these are just some suggestions to provoke a little thought on this topic. To finish, I'll re iterate something from my response above. 

'I honestly believe that the best approach is to help students follow what they are interested in and do some meaningful mathematics, then try your best to justify the marks you give them. '

Happy marking!



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