So, why do we teach students maths in school? How tempting it is, to say “because it’s useful”. Well, I defy anyone to respond to this post by finding a single example taking from a school maths text book, in which something happens that (a) could be described as useful and (b) happens in the manner that it might do if someone were actually doing it. Even the examples specifically formulated to test functional skills, which is expressly designed to engage students in genuine applications of mathematics, are laughable in their credibility. The fundamental problem is that the real world does not behave in the nice neat clean way that mathematics exercise problems do and we are stuck in a situation in which assessable school maths can brook no argument. The real world has messy tests of validity and continual critique. Jean Lave in ‘Cognition in Practice‘ tracked shoppers in the supermarket and found (a) no relationship between success in school maths and the methods used or the complexity of arithmetic deployed to make shopping calculations and (b) that a successful outcome was always found but regularly for reasons outside of the calculation (notably that you don’t buy the large soap powder even though has a lowed cost per unit, simply because your shelves aren’t tall enough for the box).
We worked on a project for the Bowland trust in which we looked at the cooling of pizzas, engaging participants in the role of consultants to a new pizza business needing to know how long the pizza would remain saleable for in delivery. The real aim is to allow students to develop a mathematical model, validate it, then critique it. I heat a pizza and allow it to cool for ten minutes, getting participants to do two thing (i) estimate the temperature in each successive minute and (ii) say with increasing accuracy and an increasingly specific calculation, the basis on which they are estimating. The participants (who this week were the experienced teacher mentors for our PGCE school partnership) generate a model which says the temperature is the starting temperature (about 90o) minus about 2o per minute, i.e. T=90-2x where x is the elapsed time in minutes. We then test this function by fitting it to the data (which we have generated on a graphing package using a data logging temperature probe as the pizza cooled). The graph fits perfectly and I mean really perfectly. So much so, that everyone is very happy to accept the linear model and calculate the time taken for the pizza to reach 48o (which I have previously tested is the lowest acceptable temperature. Only when we look to see what this model will predict for the temperature after two hours does anyone see the problem. Left on a table for two hours, a pizza does not freeze all by itself! So, we see the model is good in the short term, but cannot account for the medium term behaviour. I have written this up now in some detail (see my chapter here).
The key point is that if we want to develop skills in applications of mathematics we have to recognise that maths can not be applied by wrapping a story around an exercise question. All applications require modelling. This requires developing skills in validation and critique (amongst other things). We have developed a narrative that expressly foregrounds these issues, providing the possibility of movement in the student’s thinking. We say this tentatively because many other factors are involved. However, functional skills exams are not designed to engage learners in credible problems and therefore they cannot possibly achieve this.
Equipment: You can use HP StreamSmart data streamer equipment with HP39/40 and HP50 calculators and emulators. Alternatively TI-nspire with EasyTemp probes. I use Chicago Town mini deep dish pizzas, because you can put the probe between the cheese and the tomato, so it stay hot better! (Truth be told recently I’ve done it with water in an insulated mug, which also works very well).