Aims

• To give students responsibility for taking control of their learning
• To make school Maths closer to what real Mathematicians do outside school
• To allow a high level of detailed exploration of some complex and deep Mathematics
• To move the emphasis from assessment to evidence for learning
• To kindle enthusiasm for Maths, its complexity, its beauty and its structure
• To replace the learning of mathematical content that has very little value by doing actual mathematics

Course Outline

The course has approximately 20 areas of mathematical study. These are defined loosely, focussing on the high level concept rather than any prescribed list of techniques or tools. These are temporarily called Domains and could include the following:

• Iteration
• Limits
• Types of number
• Polygons
• Conics
• Encryption
• Polynomials
• Game theory …

Students will choose a number of these Domains over a two year period for an in depth study. They will research and explore each Domain leading to a presentation in some format of the work they have done. Ideally the number of domains should be allowed to vary, but should have a maximum of, say, 5. The maximum is to make sure there is time for sufficient depth in each Domain. The lack of minimum allows for a particularly gifted student to spend two years researching, say, Equations and end up with a clear understanding of Galois Theory

In each chosen Domain they will demonstrate a number of types of learning. These are temporarily called Attributes and could include the following:

• Original research
• Historical research
• Explanation of Mathematical ideas
• Maths as a formal language
• Maths as a creative discipline
• Use of Mathematical techniques …

 

The role of the student

The student will choose a unit of study and plan an open ended exploration of it. They will look at different aspects of the unit and research the content and techniques of that area. They will not be expected to be comprehensive in breadth, rather to show as much depth as possible. See example ideas at the end of this paper.

The final product for each exploration could be a written paper, a website, a computer programme, a piece of music, a picture or anything else. It will be accompanied by an explanation of the Maths that led to the final product (this could be in the product in the case of a paper for example) and a journal/workbook./blog giving reflections as they go along. Reflections will include comments from the teacher. The students curate a portfolio of their work with summaries of the Attributes linked to evidence for them.

Working in Groups

Students will be able form research groups to explore a Domain. If they do they will need to demonstrate collaboration as they work together to produce a common final product, or individual final products from a common investigation.

The role of the teacher

The teacher will support students in making wise choices of Domains, act as the sounding board for their ideas and prompt and nudge students into profitable areas of study. They will suggest that students need to show more of a particular mathematical Attribute. They will add comments to the growth of the piece of work to the journal.

Assessment or evidence gathering

It is important that “Students themselves are the measure, not the results of measuring the students”. This means that they choose the evidence from their work that shows their learning and progress in the various Attributes. Their portfolio demonstrates what they have learned by highlighting the best of their learning by Attribute.

Someone is going to ask how all this is graded. The point about evidence over assessment is that it avoids the reduction of a wonderful piece of work to a number. How is a painting by Picasso graded, or to take mathematical examples, Wileys’ proof of Fermat’s theorem or Mandelbrot’s definition of fractal dimension? Let the work be its  own measure.

For more thoughts on assessment look at my general education site www.educhanges.com

Having said that, I am sure some pedant will find a way to reduce this to numbers!

What this curriculum does not have

We are trying to avoid the role of the external quiz master who puts together hurdles that students need to clear to show their learning. This there will be no testing and no predetermined criteria.

There is no externally decided list of techniques to be learned and, for most students, never used again. There is no breadth that is put into the course other than the breadth that students give  themselves.

There are no limits. A student can go into any Domain as far as their ability and interest will take them.

Examples

A piece of study based on the domain of iteration.

The student or students will research from texts what iteration means. They will trace the historical path of iteration as a tool to get to an outcome. They might pick geometric examples such as Archimedes iteration of inscribed polynomials approaching a circle to approximate π. They might pick numerical examples such as the ratios of the Fibonacci series tending to the Golden ratio.

Gradually the Maths will get more complex as they show the definition of a limit. A student interested in the historical development might contrast Newton’s intuitive ideas of infinitesimals with Cauchy and Wierstrauss’ definitions of limits. They may go back to Infinitesimals with Robinson’s non standard analysis.

Having looked at simple limits, a student may look at more complex ones by exploring the logistic mapping, period doubling, strange attractors and the Fiegenbaum number. They may use spreadsheets, programmable calculators or computers to get values more efficiently. They can lead off into original research by exploring whether the Feigenbaum number holds for different iterated functions.

Another student might take this research into the domain of complex numbers and lead to depictions of Julia and Mandelbrot sets, Newton domains of attraction etc. They will demonstrate originality and understanding by creating programmes to do this.

There are a thousand ways this can go. A research group can start together and diverge, or they can converge to their own limit (poor joke!)

Throughout they show the attributes they are using in their Journal.

A study in the Domain of equations

Students could start by looking at historical origins of the solution of quadratics in Babylonian Mathematics. They could demonstrate the equivalence to the quadratic formula. They can then explore the search for higher degree solutions. The heady days of the renaissance are exciting, for students to read about Cardano, Tartaglia and Ferrari. They can demonstrate Cardano’s formula by algorithm or by programme and do the same for Ferrari’s. To take the geometric description in the Ars Magna and demonstrate line by line how this translates to the algebraic formula is a significant piece of Maths.

From here they could follow Bombelli into complex numbers and head for the fundamental theorem of algebra. Can they create cubics for which Cardano’s formula does not work unless we dip into complex numbers and out again?

Or, they can keep it real, learn some group theory and rediscover Galois. Can they generate an unsolvable cubic? What does this mean? What is the difference between types of irrationals?

A third course might take them back into iteration and methods of solving equations without algebraic solutions.

© Chris Binge www.mathsthoughts.com