The aim of this Blog is to provide tasks that might put some spark back into the school geometry curriculum that is currently in force in England and Wales. The focus is on Key Stage 3 (lower secondary school). However, teachers should find that many of the tasks are suitable for other age groups and for curricula in places other than England and Wales.
The visual, tangible, seemingly concrete nature of school geometry would seem to make it an ideal area for students to develop mathematical thinking. However, we have struggled to turn this into reality, at least in the UK. In my secondary school days the focus was on geometric proof, based on the theorems of Euclid. Though the formal, systematic work of Euclid is an astonishing achievement, it seemed dry, legalistic and rather remote to many of my 16 year old contemporaries. When I first started teaching, curriculum development projects like SMP (School Mathematics Project) introduced exciting new geometric elements into the curriculum, such as topology (in the form of 'rubber sheet geometry') and isometric and other affine transformations. However, much of the content was again too ambitious, especially the use of matrices to represent transformations, even though these provided a startling demonstration of how movements in space can be represented just with numbers.
Gradually, and since the introduction of the National Curriculum for England and Wales in the late 1980s, the school geometry curriculum has become ever more fragmented and ever more thin. A few years ago I responded to this by developing a series of diverse geometry tasks for teachers (and their students) which were serialised in the ATM journal Mathematics Teaching (MT229 to MT252). Some of these were put together as the e-book Animating Geometry published by ATM, and the individual tasks, with their numerous animations, can still be found on my Maths Medicine website. My approach in this blog is very different - and very different too, from the approach to the curriculum taken in my blogs on algebra and on multiplicative reasoning (Algebradabra, Algeburble, MultipliXing).
In this blog I have narrowed the scope of the work by basing the tasks on the classification that NCETM has devised for the Key Stage 3 geometry content of the National Curriculum. This classification is described in a series of documents starting with this one: Overview for Theme 6, Geometry. The geometry theme, as NCETM calls it, has been divided into 4 core concepts, which in turn have been divided into a total of 12 statements of knowledge, skills and understanding. The statements are listed here:
This Blog will eventually contain 20 'weekly' sets of 5 'daily' tasks, in common with my other mathematics blogs mentioned above. So for each of the NCETM's 12 statements, there will be 1, 2, 3 or perhaps 4, weekly sets of 5 tasks. This means that teachers will be able to tie the tasks quite closely to the National Curriculum, should they wish to do so. However, there is an important proviso. The geometric content of the tasks given in the NCETM's Theme 6 documents, is, in the main, fairly meagre. Their rationale might be described as Get geometry done, rather than Remain engrossed in geometry. To the extent that the NCETM tasks are typical of what happens in school, teachers will find that many of the tasks in this Blog are rather different, since their aim is to challenge and engage and as such might seem quite demanding for students, at least at first. I hope that some of the tasks will also challenge and engage teachers, especially if they themselves have been deprived of a coherent geometry curriculum while students at school.
As with my other blogs, the notions of weekly and daily should not be taken too literally. The tasks can be done in any order, and teachers might decide to spend just part of a lesson on just one of the tasks, or use a set of tasks over several lessons. Teachers should also feel free to modify the tasks to suit their students.
Important note: A typical format for the tasks in this Blog is to pose a question that requires a specific answer. However, for students to get the most out of the tasks one would want them to explain their answers and to share, discuss and critique each other's methods. I have left this implicit, but would urge teachers to draw this out, so that their students might come to see that this kind of activity is an intrinsic part of doing mathematics.
STOP PRESS: I have produced a revised version of this blog for ATM, who have agreed, in principle, to publish it as a book. I sent the latest revisions to ATM on 16 June 2022. Watch this space!
STOP PRESS 2: The Geometric Sparks book, published by ATM, came out in October 2022.
Dietmar Küchemann
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