A couple of years ago I was given the wonderful brief of devising a set of mathematical science tasks to help scientists and engineers in the mathematical aspects of the transition from school to university. Rather than rattle something off in a week or two, I had luxury of thinking about this over an extended period of time, during which I worked with school and university students and lecturers from different faculties.
I think that the concept of the superficial context proved to be the most significant issue arising and the most challenging one to overcome. Superficial contexts are tasks in which there is superficial or artificial reference to science in a maths question or superficial or artificial use of mathematics in a science question. In both of these cases, the importance of either the real-world part or the mathematics part is overstated. And students see straight through this deception.
It might seem clear that there will be many fascinating scientific-mathematics questions out there. After all, fundamental to the process of all (and I really do mean all) scientific activity is quantification of the world. This quantification may be more or less explicit; hidden or overt. But it is there. Mathematics is the study of quantification in its purest state. Thus, mathematics is essential to the activity of the scientists.
In my opinion, a good scientific mathematics question is one in which the level of the mathematics and the science are comparable. Moreover, it is a question in which both the mathematical and scientific parts of the question are meaningful to the solver at their particular stage of education. Ideally, both a scientist and a mathematician would declare the task to be meaningful from the context of their individual disciplines.
Unfortunately, I soon found that good tasks for the classroom which included both scientific and mathematical elements were hard to come by, although there were plenty of tasks which made use of superficial contexts. It is easy to describe the usual suspects in the world of superficial contexts:
1) The cut and shut: A science task at a certain level and a mathematics task at a certain level are welded together because they are, on the surface, on the same topic. These are easy to create, but make meaningless links.
2) The post-script: A mathematics task or a science task is bolted onto the end of a science task or mathematics task respectively, often with a throw-away remark such as ‘something to think about’, ‘extension’ etc. Post-scripts almost always result in a mismatch of levels and styles, such as a hopelessly advanced investigation appended to something rather procedural and routine.
3)The reach for the stars: In the interest of being interesting, tasks refer to ultra-high level mathematics or science, such as string theory or DNA folding. Such tasks revolve around the sort of content that students might encounter should they elect to study for a PhD in a very specialised niche topic. Not only is this entirely irrelevant to the students, the task will necessarily touch the motivating topics in utterly superficial ways.
4) The neither one thing nor another: In the interests of making a tractable problem the resulting task is neither mathematically nor scientifically interesting or relevant. Such tasks are quite easy to make, and will never be used as intended.
5) The mutton dressed as lamb: A simple mathematical task is dressed up in pointless and inappropriate real-world clothing. This degrades the mathematics and fools nobody.
I have no straight-forward resolution of these difficult issues. However, I found that making the simple split between basic quantitative reasoning and mathematical modelling was useful in thinking around the issues:
Basic quantitative reasoning: counting, ratios, proportional reasoning, estimations, approximations, use of numbers and units. You simply cannot have a sensible conversation about science without being able to speak the language of quantification. ‘Basic’ quantitative reasoning might involve many subtleties and actually be very difficult, but is thoroughly grounded in school mathematics.
Mathematical modelling: Using formulae or mathematical techniques to answer scientific questions or to make scientific predictions.
In general the importance of the basic quantitative reasoning was often underplayed by the scientists and the importance of the mathematical modelling often overplayed by the mathematicians: scientists commonly assumed that the mathematicians would find the basic quantitative reasoning, well, basic; mathematicians often lacked appreciation of the scientific complexity underlying the application in which genuine modelling could only really occur at university.
From my work, it seems that the area of basic quantitative reasoning is the best source of many meaningful contexts in which the science and mathematics support each other in a genuine context. Such contexts might well lack the ‘reach for the stars’ aspects but are intellectually meaningful and satisfying to a learner at a particular phase of education.
The area of mathematical modelling is far more challenging. I found few, if any, areas where it was possible to create a task in which the mathematics and the science were genuinely equal partners. Historically, the one area of the school curriculum where mathematical modelling meaningfully took place was in physics, with its essential ties to mathematics. Sadly, many recent approaches to physics have removed the serious parts of mathematics from the subject. Were it to be reinstated students could experience some meaningful understanding of mathematical modelling whilst at school. As it stands, students need to look elsewhere for this stimulation.
So, in conclusion, I found that contexts in which classroom mathematics can be meaningfully linked to classroom science are difficult to come by. Moreover, I have come to the firm conclusion that superficial attempts to dress up mathematics and science in each others’ clothing degrades both subjects. However, I have learned that elevating the intellectual status of the quantitative reasoning in science can be of benefit in both the mathematics and science classrooms and, therefore, the students. I find this both exciting and liberating in an age of targets and subjects silos.
You can see some of my attempts at creating problems in meaningful contexts at http://nrich.maths.org/stemadvanced