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I often get asked what the Cambridge Mathematics Framework actually is? Now don’t get me wrong – this is one of the most enjoyable questions you could ask me, and I could wax lyrical about our work, going into the minutiae of its design and functionality but sometimes a metaphor does a better job.
In this case I think of the Framework as a map of the mathematical ideas that learners of any age, although aimed at those from 3 to 19, may encounter or visit. As such it is greater than what the vast majority, if not all, will experience in education, in the same way that a geographical map shows more than I will ever visit in my lifetime.
This makes me consider a world map. We often use the term “the world map” but what attributes or features are we mapping?
Are they political?
Used under Creative Commons licence CC0 1.0.
Aliaksei; used under Unsplash Licence.
Courtesy of the University of Texas Libraries, The University of Texas at Austin.
Or some combination of these?
Is it a map highlighting particular objects, places or geographical features? Such as large cities, including capitals?
Yevhen Borysov, via Getty Images.
Or ocean currents?
Public domain image used from Wikimedia Commons.
Or is it a heat map identifying particular characteristics of the population, environment or something else? Such as population density?
© gmnenad; used under Creative Commons Licence CC BY-SA 4.0.
Or child mortality rates?
© Our World Data; used under Creative Commons Licence CC BY 3.0.
If I’m browsing Google Maps (other software is available) – and believe me that is the kind of thing I do – I can change the map type or the perspective I’m taking. In the default view I can see street and place names, businesses and sites. The satellite view gives me a bird’s-eye view showing the type of landscape and geographical features. Finally, the terrain view gives an indication of the topological nature of the map. I can choose to add additional detail like public transport and traffic information. I can zoom in and out; even go down to street view. In each view I can zoom in to look at things in more detail; I can click on businesses and sites to be swiftly directed to their website or additional information. The world is literally at my fingertips.
But there are even more things to take into account when we’re considering a world map. Have you ever tried to wrap a football in a single piece of non-overlapping paper? Well, probably not. It’s not easy – I’d say it’s actually impossible. This leads to even more considerations of trying to represent the world, nominally a sphere, on a 2D surface.
Several different ways of doing this exist and each presents a different representation of the world in which we live.
Mercator projection maps keep longitude and latitude lines at 90° to each other:
© Daniel R Strebe; used under Creative Commons Licence CC BY-SA 4.0.
They distort the shape and relative size of large areas of the globe, especially towards the poles. This is wonderfully demonstrated on the website The True Size; for example, in the picture below the outline of Greenland’s real size has been superposed over Algeria.
Map data © Google Maps; app created by James Talmage & Damon Maneice.
However, the strength in Mercator projection maps is their ability to present accurate compass bearings, which was essential for the seafarers of the 1600s and beyond.
Peters projection maps accurately show the relative sizes of countries around the world:
The resulting distortion of the easily recognised land masses avoids the exaggeration of size of developed nations in Europe and North America when compared to developing equatorial regions, as is seen on Mercator projection maps.
Other representations exist. What characteristics or properties a projection maintains will impact on the way in which you perceive different parts of the world. One issue that sticks in my mind is whether we’re aware of these and the impact they have on our views and beliefs? I doubt it.
The Cambridge Mathematics Framework is a map constructed by a group with a particular perspective, with certain resources, histories, and culture. They choose to highlight certain features or mathematical ideas and draw in particular paths, identifying how mathematical ideas build on each other, rely on shared structures or key concepts and are connected. If others were to construct the map they might choose different characteristics to emphasise or show different routes. This isn’t the map – it is a map. The Framework’s digital nature and design means different features can be explored, highlighted or preferenced. Its functionality means content can be clustered, ordered or connected. Much like the different maps described above identify various features, show different affiliations or characteristics, satisfy a need, or impact on your perception of the world, the Framework has the same potential – but just for maths.
Join the conversation: You can tweet us @CambridgeMaths or comment below.