All models are wrong!
Saturday 14 September 2024
May 25 - ToK prescribed title 5
You may or may not have seen the prescribed titles for the May 25 ToK essays and so I thought I'd just write a quick post about title number 5. Students are always asked to compare two areas of knowledge in their essays and so mathematics can feature in lots of the titles, but this one specifically asks students to compare mathematics with one other area of knowledge. They have to discuss the extent to which they agree with the statement with reference to mathematics and one other area of knowledge. As a ToK teacher I am often talking to staff about how we need them to help students with examples they can use to make their arguments and I think this question in particular gives us, as mathematics teachers, lots of potential to help. So we better be ready!
A crude summary of ToK essays is that students are expected to demonstrate that they understand the different ways in which knowledge is produced, acquired and communicated across the areas of knowledge, the corresponding difference in the nature of that knowledge and the implications for its application. A favourite little riddle that exemplifies this...
A mathematician and a scientist are held captive on one side of a locked room. They are told that every hour they will be allowed to move half of the distance remaining between them and the locked door on the other side of the room and are given the key.
The mathematician hangs their head in tears and exclaims 'Oh no, we are never getting out of here' because they know they have been set on an infinite path. Meanwhile the scientist grins broadly and comforts the mathematician by telling them that 'Its OK, we will be close enough for all practical purposes'
Although not about models, their is a similar sentiment in the quote above from George Box. As mathematics teachers, we don't have to worry too much about helping students write these essays, but it would be great if we had some examples at the ready for them to work with. These examples are often best when they are things students know well and have worked with, so have a look through your notes and examples and see what you can find. Below are some ideas....
SL Quadratic models for projectiles - This comes up in loads of questions and be anything from an 'Angry Birds simulation' to an exploration of 'the Dam busters'. The theory is quadratic, but in practise the variables involved with the real context might make the model 'Wrong' but 'useful.'
SL Linear correlation and bivariate data - This whole area about looking for correlation between variables is gold for this question. Even a strong positive correlation with a PMCC of 9.7 might actually be 'wrong' but 'useful' and as that value decreases the blend of wrong and useful might change accordingly.
SL Normal Distribution and all the associated probabilities and hypotheses tests and also right in the zone.
These are all areas where mathematical models are created as a means of summarising and explaining patterns that we observe. Showing that they are 'wrong' will be easy and demonstrating their 'use' might take a bit more precision of thought.
As rule, with such a question, it is always necessary to consider different arguments. In this case, what example might you use to argue that you disagree with the statement?
At first thought, and from my obviously conditioned perspective, I feel this would go well with the Human or Natural Sciences but who knows...
Anyway, this is just intended as a provocation and hopefully we will be able help students here with lots of pertinent examples.