19 March 2007
24 February 2011
History of Mathematics – Snezana Lawrence
I often tell the kids in my classroom that mathematics is one of the most creative of the activities that they may, one day, endeavour to dedicate their time to: as creative as any art, and often requiring much less resources – good brains would do!
But how do you actually show this to the kids in order to:
a. persuade them that they can be creative in mathematics and not only through the creation of ‘new mathematics’
b. show them good examples of creativity in the subject
c. encourage them to try some tried pathway themselves before they (hopefully) fly off to some uncharted territory?
There are various methods and techniques one can employ, ranging from the teaching through ‘rich’ tasks, motivating pupils by searching for the role models in modern mathematics, to those employed by other professions which pay much attention to the development of creativity in their educational practices. All of these can be used in everyday teaching by placing mathematics in an historical context. But the most important aspect of this ‘historical context’ aspect is the rule that I always keep to: do it only if it makes you better understand a mathematical concept; use it only if your pupils can see and understand the concept better than if they didn¹t know how it came about.
There is, however, little widely available material that, in a simple and accessible way, introduces primary or secondary school children to the world of creative mathematics in a historical context. This paper is therefore aimed at teachers who would like to do that –
1. by introducing some of the principles that have proved to be useful in constructing a programme of maths teaching in an historical context
2. by pointing to some existing resources, both printed and on-line which can be used in the classroom.
Some points to consider would surely be the following:
- Using the history of mathematics at primary/secondary level raises very many issues and questions. The right material has to be meaningful to the level of mathematical understanding, it needs to relate to the topic being introduced, and it has to achieve some kind of purpose. What exactly would that purpose be? Why does one want to use history of mathematics when teaching children?
- A great many academic sources offer a great many useful answers to this type of question. The problem for a working teacher, who has around twenty hours of teaching to perform each week (apart from marking, setting assignments, testing, coursework moderation, after-school meetings, maths clubs, and many other small things such as mentoring, or perhaps leading a department), is that they don't have much time to spare. The Internet then becomes helpful, although its greatest strength - the uncontrolled and constant growth of resources - may sometimes have a negative effect on teachers by making them unsure of what is appropriate and at the same time effective. Apart from establishing the reliability of the web-based material, it is therefore useful to have in mind the reasons for using historical resources, particularly by teachers who are just beginning to develop techniques for teaching mathematics in such a way.
When looking for resources in the history of mathematics to help teaching and learning of mathematics, one always has at the back of one's mind John Fauvel and Jan van Maanen's twelve questions that were put forward to the history of mathematics education community as part of the ICMI Studies 1997-2000.
You can see the discussion document at www.emis.de/journals/ZDM/zdm974i1.pdf
A shortened list of the questions is directly relevant to every primary or secondary teacher who is considering using history of mathematics resources in their teaching:
- How does the educational level of the learner bear upon the role of history of mathematics?
- At what level does history of mathematics as a taught subject become relevant?
- Should different parts of the curriculum involve history of mathematics in a different way?
- Does the experience of learning and teaching mathematics in different parts of the world, or cultural groups in local contexts, make different demands on the history of mathematics?
- What role can history of mathematics play in supporting special educational needs?
- What are the consequences for classroom organisation and practice?
Some of the principles when deciding what resources to use/make in order to teach maths in historical context should also include the following:
- Adding an international dimension
- Encouraging the desire to understand and compare different cultural approaches to mathematics and to see mathematics as part of the cultural heritage of all societies
- Teaching through an interdisciplinary approach, in particular in relation to visual and literary arts. This may, in a very succinct and inspiring way, demonstrate mathematics as one of the most creative human activities. Examples could be
- Reference to great works of architecture and studying some of their aspects (like symmetry, use of geometrical instruments and samples of geometrical constructions)
- References to mathematics in literature
- Development of a platform for the further study of 'old masters', offering a safe environment for self-discovery and self-identification in the context of the history of mathematics
- Nurturing intellectual fascination with mathematical concepts
- Attaching a meaning to mathematical skills through contextual referencing.
The focus, when first introducing the teaching of maths in historical context, should definitely be on finding out what strategies could be best employed (what works best for you as a teacher, and for your particular classes) I have found that three strategies seemed to work for me:
- Learning a skill through worksheets which explain skills in their historical context
- Story telling about mathematicians, the geography of mathematical discoveries, and about great national and international collaborations or competitions between mathematicians
- Introduction of the research-based activities in whole-class teaching and through themed homeworks, leading to the initiation of students' self-confidence, and their increased ability to take ownership of their learning progress.
Here are two examples which you can use, and are based on the material from the www.mathsisgoodforyou.com site which I developed over the past three years.
1 - Interdisciplinary approach, story-telling and the teaching of maths
Mathematics, as well as any other human activity, is partly dependent on story-telling. This gives mathematicians a context in which to work, as it entails a social as well as a personal platform whereby mathematics is both related to and helps explain other human activities. The more sophisticated the society is, the more complex the story-telling becomes, but the primal need for understanding the simple, archetypal stories remains strong nevertheless. This need is strongest in children as they strive to understand the worlkd around them.
Example: a project to understand the meaning of 'proof' begins by showing the importance Greeks paid to the development of geometry. This leads to exploring famous mathematicians, and famous places. You can start from talking about Euclid and his Elements, show the first edition of the Elements in English (which was the first pop-up book in English), talk about Alexandria and some other famous mathematicians who lived there. Euclid's Elements can then be looked at again, and some theorems investigated,before closer examination of Pythagoras' theorem (and also see again Book I, Proposition 47. This can lead to the life of Pythagoras and the question about reliability of historical sources, his school of mathematics, and his (supposed) travels to Egypt. Pupils are asked to show their understanding of proof by finding more about Euclid's Elements, Alexandria and its famous mathematicians, Pythagoreans, and the origin of Pythagoras' theorem.
2 - Project work, taking ‘possession’ of the search for knowledge and induction of younger pupils
Familiarity with a topic need not finish with the work on the topic. By taking possession of it, students are able to pass the message on to younger pupils, therefore doing the right thing twice over: the first time when they strive to understand the topic themselves through the historical context and then by teaching (and thereby learning more about it) the same topic to the next generation of students. This is most useful and successful when there is a practical task attached to the problem, as the logistical questions provide more opportunities for taking ownership of a particular topic.
Example: a project with Year 7 pupils was recreated in the Master Class for Year 5 pupils who visited St Edmund's in July 2005. Year 7 pupils who had completed the project some months earlier were invited to help Year 5 pupils. This was a project that linked, through an introduction to Egyptian mathematics, three topics: numerals, geometry, and fractions Egyptian fractions and the Eye of Horus fractions were thus linked with experimental building of shapes with paper pyramids, and finding the fraction of the volume of the shape through counting pyramids.
Rather than a large number of projects, you, and the students, are probably much better off with a limited number to cope with during the year. This offers opportunities to explore topics in depth and in relation to all the principles of the project as previously listed. For KS3 pupils projects aim to link more than one topic and lead to an investigative task. KS4 pupils may be given projects as preparation for their GCSE investigative work; for statistics coursework, for example, they can participate in the Cryptography project – make it what you like it to be and for the number coursework you can introduce them to topics such as: Numbers, and The development of algebra and algebraic symbols.
Suggestions for further reading
For various web-based resources which you may use see www.dcs.warwick.ac.uk/bshm/resources.html#education
Other sites which have some resources for the teaching and learning of mathematics in historical context are:
Nrich: www.nrich.maths.org/ (then search for 'history')
Plus Magazine: plus.maths.org/
Famous Problems in the History of Mathematics from Math Forum: mathforum.org/isaac/mathhist.html
Convergence!, an online journal of the MAA: convergence.mathdl.org
The excellent site of David Joyce with interactive Java applets of Euclid’s Elements: aleph0.clarku.edu/~djoyce/java/elements/elements.html .
You may also wish to join in the discussion on the NCETM web portal in the History of Mathematics community.
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