ReMSTEP Maths – Knots

This video is one is a series of three that were developed as part of  the REMSTEP Project (http://remstep.org.au/). The videos were produced through collaboration between staff in the Faculties of Education and Science at Monash University. The videos are aimed at pre-service teachers but are just as relevant to practicing teachers of secondary school mathematics. The purpose of the videos is to inspire pre-service teachers to think of mathematics as a beautiful, creative and relevant discipline. We aim to challenge preconceived ideas about what maths is, and isn’t and for the beauty of maths to be appreciated, understood and shared with learners of maths.

A key objective is for the video to provide inspiration for classroom activities that deal with the underpinning concepts, rather than the technical and computational aspects, of mathematics. For further information see the Mathematical Association of Victoria’s publication – Common Denominator, term 3, 2017, pg15.

Acknowledgements
These videos were produced as part of the ReMSTEP Project. We are grateful to the following people who made these videos possible; Norman Do, Rebecca Cooper, Jo Burke, Magnolia Lowe Film Communications, Owen Kazula, Chris Nagy, Greg Lancaster and the pre-service and in-service teachers who provided us with feedback.

Dr Norman Do is a self-proclaimed mathematics geek, and lecturer at Monash University. He loves to study and teach mathematics and aspires to engage people in study mathematics and to appreciate the diverse and varied jobs that exist for mathematicians. His main research interests lie in geometry and topology, including knot theory.
Norman believes “We need to encourage people to take on challenges, do hard things and gain a sense of self-worth from learning,” he adds “We need to celebrate our mathematical heroes like Terry Tao and we need to understand that if you’re a mathematician you can work behind the scenes in jobs as varied as biochemistry, animation and finance.” View Dr Do’s profile.

There are place markers in the video labelled ‘Stop & Think’. This gives the viewer opportunities to pause the video and to complete some related activities. These are accessed using the Tabs across the bottom of the video. They are designed to assist pre-service teachers to appreciate the concepts, attitudes and contexts of contemporary mathematics and their relevance to classroom teaching.

Before Viewing the video

1) Consider –

  • What do you consider to be Mathematics?
  • Why is it important to teach it?
  • What is the purpose of Maths in the lives of your students?
  • How might Maths influence what happens in the real world?
  • How do you justify the inclusion of Mathematics in the school curriculum?
  • Can you think of examples of doing Maths which don’t involve numbers?

2) Which of the choices below do you consider to be the best metaphor and why? Can you think of a better one?

How is an excellent teacher of mathematics like a …?

  • Social Worker
  • gardener
  • orchestra conductor
  • actor
  • doctor
  • librarian
  • missionary
  • sculptor
  • games show host

Stop & Think (1)

1. Consider some maths associated with torus, e.g. the surface area or volume. Are there opportunities to investigate these in your maths lessons?  Identify the level(s), strand(s) or substrand(s) of the Australian/Victorian maths curriculum where it might be appropriate. What level of mathematical understanding is required?

Hint: see https://www.mathsisfun.com/geometry/torus.html

2. Dr Do uses the example of PACMAN, a popular video game character in the early 1980s. Is PACMAN culturally relevant to today’s secondary school students? What alternative(s) could you use today which demonstrate this same form of movement?

3. Consider satellite navigation (Sat Nav) using global positioning systems (GPS) as an example where variations in the surface of the earth is important. Research GPS and autonomous transport (driverless cars & trucks) and the key maths ideas that are involved in Sat Nav?

Stop & Think (2)

1. Make a Mobius strip yourself and devise an approach that could convince another person that it has only ‘one’ side and ‘one’ edge.

Hint: see https://brilliant.org/wiki/mobius-strips/ for step by step instructions for creating a Mobius strip.

2. Three manifolds are of great interest to mathematicians and physicists. So what is the shape of our universe?

How could you help students to consider the concept of the size and shape of our universe?
See https://www.youtube.com/watch?v=y7TafCe8NyA

Stop & Think (3)

1. Dr Do talks about knots from a historical perspective but where might students come across representation of knots in their everyday lives?

 

Stop & Think (4)

1. When do students learn about DNA molecules and replication in Science at your school (see Australian or Victorian Curriculum).

2. How could you collaborate with your class’s science teacher to explore the potential learning opportunity for your students?

 

Stop & Think (5)

1. Investigate how ciprofloxacin and knot theory is connected to immunity & cancer treatment? Does this perspective impact on the way you view mathematics contexts and maths education? Explain your answer.

 Stop & Think (6)

1. In everyday life like when sailing, rock climbing or even when tying our shoe laces, we talk about ‘knots’ but are these knots according to a mathematician? What characteristics must a ‘mathematical’ knot have?

For example, the Clove Hitch.This is a very useful knot with a range of purposes.

clove_hitch_with_end-bkgnd

  • Can you tie a clove hitch?
  • If you remove the rod, will it shake out and untangle?
  • Is this an example of a ‘mathematical’ knot?
  • How could you change this to make it a ‘mathematica’l knot?

 

What other knots can you tie?

bow-line-bkgndWinsor knot? Square knot? Rolling hitch & Reef Knot?

For more examples, See: Animated Knots

Stop & Think (7)

1. You have 2 nails in a wall. How can you hang a picture from them in such a way that the picture falls down if either of the two nails is removed, but remains hanging while both nails remain in place?

Stop & Think (8)

1. Brainstorm the ‘beauty of maths’ and how it can simplify or help us understand real life situations.

After viewing the Videos consider;

1. Each video contains a story of human endeavour in mathematics. Whilst watching the series, consider how this might be a useful perspective to adopt in mathematics education.

2. This video series is presented by a Mathematics lecturer and researcher, Dr Norman Do – does his specialist knowledge and expert understanding make a difference when you are watching the video? Do you think it would make a difference to your students?

3. Complete each of the sentences below…

I used to think that a mathematician was … now I think a mathematician is …

I used to think a maths teacher was … now I think a maths teacher is …

I used to think a maths learner was … now I think a maths learner is …

Links

1. Australian Curriculum – mathematics
http://www.australiancurriculum.edu.au/mathematics/rationale

2. Victorian Curriculum – mathematics
http://victoriancurriculum.vcaa.vic.edu.au/mathematics/introduction/rationale-and-aims

3. http://www2.memenet.or.jp/~keizo/0011.htm
Interesting sculptures by Keizo based on knots.

4. http://www.georgehart.com/bagel/bagel.html
A mathematically correct breakfast bagel demonstrates how it is possible to slice a bagel in half and still keep it connected.

5. http://www.georgehart.com/bagel/knot.html
How to slice a bagel into a trefoil knot.

6. http://mathworld.wolfram.com/topics/Topology.html  Resources for teaching many aspects of topology including knot theory.

7. http://www.aussieeducator.org.au/resources/teaching/mathematicsresources.html
A website of resources and links to other websites to support teaching mathematics.

8. https://www.math.hmc.edu/funfacts/ffiles/20006.7.shtml
A couple of hands on activities to try – Rubber bands stuck on a torus.

9. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Mobius.html
Explore where the name mobius strip come from?

10. http://www.thenational.ae/news/uae-news/science/natures-knotty-problem#full
Article explaining how scientists are combining biochemistry and knot theory to understand DNA.

11. http://www.csiro.au/en/Education/DIY-science/Maths/A-Tangled-Web
Try and solve this tangled web math puzzle. A fun activity to help primary school students to learn why knots are important in mathematics.

12. http://www.ics.uci.edu/~eppstein/junkyard/knot.html Lots of links about the geometry of knots

Books & Articles

Adams, C. C. (2004). The knot book: An elementary introduction to the mathematical theory of knots. Providence, RI: American Mathematical Society.

– We use knots to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry.

Crato, N. (2010). Figuring it out: Entertaining encounters with everyday math. Berlin: Springer-Verlag.

– This is a book of mathematical stories — funny and puzzling mathematical stories. Chapter 12 explains how GPS works.

Pickover, C.A. (2006). The mobius strip: Dr. August Mobius’s marvellous band in mathematics, games, literature, art, technology, and cosmology. Thunder’s Mouth Press.

Sumners, D. W. (2011). DNA, Knots and Tangles BT  – The Mathematics of Knots: Theory and Application. In M. Banagl & D. Vogel (Eds.), (pp. 327–353).  Berlin, Heidelberg: Springer Berlin Heidelberg. http://doi.org/10.1007/978-3-642-15637-3_11

– This article describes how the maths of topology has helped understand DNA recombination.