The Kony 2012 video has recently taken the world by storm. The original video itself now has over 84 million youtube hits and it’s been calculated that all videos relating to Kony 2012 have managed to gather over 100 million hits quicker than any other viral video in the history of the internet (Susan Boyle’s Britain’s Got Talent first audition comes in at second place).
Quite recently the IB (International Baccalaureate) decided to change the coursework element of the Maths syllabus. Instead of students answering questions provided by the IB, the students will now be required to devise their own questions to explore. This is an intriguing change in structure and I will be interested to see how well students, and teachers, cope with this freedom.
I envisage that many students will undertake an investigation which requires statistical analysis or one which incorporates an aspect of modelling. Just in case you’re not aware, students are currently required to do one piece of modelling coursework and one piece of coursework which aims at finding patterns and producing a general formula. With this in mind I decided to have a think about some possible projects which the students could embark upon. I quickly realised that the most efficient way to do this is to make a note every time I see something that is simultaneously easy enough for a student to investigate and hard enough for it to be worthwhile. While this defeats the object of the students devising their own questions I feel that I need to prepare for this in some way, just in case it comes to a point in which a student is absolutely stumped on what to investigate. Oh, and just so you know, whilst this sounds easy in theory, in practice it’s easy to forgot to make a note of everything you see!
I have to admit that at first I was a bit sceptical of how much work this would create for me, both in terms of thinking about investigations and marking individual courseworks. However, I do think that it could be beneficial for both students and teachers and it’ll hopefully be something which the students can take a lot away from. I would be interested to hear what other people make of this change?
Anyhow, with this in mind I thought it was a perfect opportunity to mathematically model something which has caused such a huge amount of debate over the last few weeks. Here is a graph of the number of views the original Kony 2012 video has had since March 6th. If you’re wondering if I’m out of my mind by thinking that it is at all worthwhile to model this then please bear with me.
When I use this data to teach modelling, I’m going to start with a simple model and build it up to be more complex. As can be easily seen, the growth rate of this data starts off slowly, then increases rapidly and then slows down again. Hence it’s not far-fetched to immediately think of modelling this data using the logistic function. In terms of y and x, the general form of the logistic function is as follows:
Applying this function to model the data, I ended with the function below. (Must have accidentally put a capital V in there instead of y!?!)
I would say this was a satisfactory model of the data in that it does model the general pattern. However, it does quite drastically over and under estimate some of the data points and as x tends to infinity (which is the number of days since March 6th), the number of hits will not exceed 82.98 million.
Since I want the function to continue to increase slowly after point F I decided to add a little extra to it and ended up with the function below.
This function is a better fit to the data and of course it continues to increase indefinately. As you can see it predicts that the number of hits on Kony 2012 will be at 100 million, 27 days after March 6th which will be on April 2nd. Obviously, the number of hits will not increase at a constant rate indefinately and so this model is an extreme case of what is probably the least amount of time it will take the Kony 2012 video to hit 100 million views.
Since I don’t want the model to increase that quickly forever, I decided to change it once again. The model below increases at a slower and slower rate as x increases.
This model predicts that it will take much longer – around one year – to reach 100 million views.
Having specialised in Pure and Applied Mathematics at University and never coming across mathematical modelling of this sort until having to teach IB, I have never modelled using functions before. However, it is plainly obvious that there are infinitely many functions which could model this data which mean that there is no way of telling when the kony 2012 video will reach 100 million views. There are simply too many variables which affect this data.
This connects to one major problem with an IB modelling task that my students recently did based on modelling population trends in China. There are way too many variables which affect a population and therefore there are infinitely many functions which make different predictions about the future population of China. Indeed, every single one of my students concluded the assignment with a different model.
We surely cannot allow students to believe that they can fit a function to data which is dependent on such a huge number of variables, and then make any type of logical prediction as to what could happen in the future.
Whilst I agree that this is a fun and engaging task which the students can put into context, we should surely be modelling phenomena such as tide heights or concentration of a drug in the bloodstream which are dependant on significantly less variables. Students need to realise this and they shouldn’t therefore model data in their individual exploration which is in essence impossible to make any real predictions about.
Am I completely off the mark here or do other people agree?