Maths circles – Goals
1st year Maths circles:
- Encourage an appreciation/enjoyment of maths.
- Develop problem solving skills based on an investigative method, pattern guessing, trial-and-error, ...
- Develop independent and team working skills.
- Develop an intuitive understanding of some mathematical concepts (e.g. limits, Combinatorics)
2nd-3rd year Maths circles:
- Encourage an appreciation/enjoyment of maths.
- Develop critical thinking – a wish to understand rather than accept rules blindly – even in, (or especially in?) regards to the school curriculum.
- Teach a structured programme which includes extra-curriculum concepts/methods as well as developing curriculum concepts through problems and examples, in a natural and systematic way, and aligned to requirements of international curricula for that age
- Prepare the grounds for sportive Maths –team contests, Olympiads
- Open up horizons to the larger mathematical context – an intuition for how vast and rich it is
Which aspects of mathematics/types of activities make us enjoy maths?
- Learning something new – a property, a method, a concept that we wouldn’t have thought of ourselves, but which is compelling and memorable once seeing it (Kieran couldn’t explain this precisely but he gave an example: the 1 min proof that the harmonic series is convergent, based on powers of ½, made a 1 hour of lecture seem worthwhile to him)
- Tricks – problems relying on some maths laws or formulas generally unknown to the audience and which, once understood, can be played with friends/family to make one look like a genius : )
- Patterns, symmetries – they’re aesthetically rich and they intrigue us because there seems to be a hidden law behind them. They make us want to discover and understand that law.
- Surprises, e.g. when we don’t see the solution to a problem right away but it is right behind the corner, you just have to look at it the right way.
- The humane factor – intelligent word problems
- Visual elements
Some ideas for developing a lesson plan:
- Start with a rough sketch, list the activities and use “Which aspects of mathematics/types of activities make us enjoy maths” as checklist.
- Preferably there’s a goal/sense of purpose in each section of a lesson which is clear to the students from the beginning of the lesson/section.
- Topics follow naturally from each other.
- Each topic is covered leisurely on its own, with lots and lots of examples emphasizing its various aspects.
- Use one representative common-sense example before the abstract formulation of a law.
- Inasmuch as possible definition of a concept is discussed early on in a common-sense way rather than later after using some technical rules thought in school without proof.
- Each new concept/piece of theory has some applications in the same lesson.
- There is some gradation from simple to complex, with examples at each end.
- General rules may be inferred from simple examples, but beware of encouraging false generalizations and transferring rules into wider contexts before understanding their significance and area of applicability.
1st year lesson plan content:
- List of activities
- Presentation of activities and required materials in ready-to-go (student oriented) format
- Hints
- Solutions
- Take-home problems
2nd- 3rd year lesson plan content:
- The first part models a teacher- classroom discussion on a given theme, developing the theme through examples, questions, hints, solutions, explanations of concepts, practice exercises.
- Second part – more practice exercises. Solutions of these can be discussed in a subsequent hour.
Some electronic resources for 2nd- 3rd year lesson plan:
- http://www.artofproblemsolving.com/
register for Alcumus as a teacher or student and solve their problems
- http://www.cut-the-knot.org/arithmetic.shtml
- http://www.olimpiade.ro/olimpiade.ro/informatii.php?materie=2&editie=11
(use materials from the 4th, 5th and 6th primary school)
- http://euclid.ucc.ie/pages/MATHENR/index.html#Mathematics_Circles
(scroll down for links to Kangaroo, US maths circles etc– some of these also available at http://www.mathscircles.ie/links/
- http://purplecomet.org/welcome/practice
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