One of the unique features of the IB Diploma Mathematics curriculum is its Exploration. This constitutes the Internal Assessment for the subject and is worth 20% of the final grade.
The Exploration is not a project, assignment or prescriptive piece of work but allows the maths student to literally explore one area of maths combined with a personal interest.
The Exploration is assessed against five criteria; Communication, Mathematical presentation, Personal engagement, Reflection and Use of Mathematics. Interestingly, that final criterion, Use of Mathematics is worth just six marks out of a possible 20. This means that the purpose of the Exploration is not to discover (or invent) new mathematics, so neither to confirm Goldbach’s Conjecture nor prove Fermat’s Last Theorem (It’s too late for that one, anyway).
In the past few years our students have tackled a wide variety of topics including; analysing the harmonics of sound waves, modelling excess deaths from the spread of influenza viruses and developing unbreakable ciphers. One student studied Disney’s Tea Cup ride and modelled the displacement, velocity, acceleration and ‘jerk’ on the ride using trigonometric functions and calculus. Some of the patterns formed by the Tea Cups were not only surprising but also revealed a distinct mathematical beauty. Another student designed a suspension bridge for a remote Indonesian village while yet another thought outside the box, and indeed our atmosphere, by planning a trip to Mars using the Hohmann Orbit transfer. Last year, there was an excellent exploration which conclusively determined that the economic welfare loss created by subsidies in the Australian coal industry was less than the welfare loss created by subsidies provided to Chinese coal producers by a total of $188 000 000.
The IB Diploma program also offers a Mathematical Studies course which is aimed at a lower level of Maths content (somewhere between Maths A and B). This also includes an internal assessment similar to the Exploration but with different criteria; Introduction, Information/measurement, Mathematical processes, Interpretation of results, Validity, Structure and communication, Notation and terminology. Again, the emphasis is on a personalized approach to research. Our students have investigated optimal packaging for different shaped containers, modelled the correlation between health spending and life expectancy, and even studied mathematical paradoxes. Did you know that Zeno of Elea, the Greek philosopher actually taught using paradoxes in order to promote a deeper understanding in his followers? His arguments are examples of a method of proof called reductio ad absurdum or proof by contradiction, where a reasonable assumption leads to a contradictory statement showing the proof to be false. Zeno demonstrated that the fleet-footed Achilles could never overtake a tortoise in a race where the tortoise had a head start! Another interesting paradox comes in the shape known as Gabriel's horn. This has an infinite surface area but finite volume which means it could hold a finite amount of paint but the surface could never be completely painted!
The Exploration and Investigation are developed in Semester One of Year 12 and provide yet another dimension to the study of Mathematics. At the end of their journey, the students have become ‘experts’ in one area of Mathematics and produce a document of which they can be proud. I always enjoy supervising students with their IB Internal Assessments, quite often marveling at their curiosity, creativity and commitment.
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