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08 March 2019
Over the last few years I’ve delivered a number of workshops and lectures talking about the place of maths and its importance to me. I regularly start this with a picture of multiple images all depicting the same thing:
Many people – both adults and children – quickly spot that each image shows the digits 1 to 10. Once you know this, it is incredibly easy to make a reasonable translation of many quite complex calculations and even to solve them. It is great fun working with hundred squares in more challenging scripts such as Thai, Egyptian hieroglyphs, or Maya glyphs. Get learners to analyse what’s going on: spot patterns, find repetitions, answer some calculations and make up their own problems. This is even more fun if you include Hebrew (read right to left) and Cantonese (read top to bottom and right to left).
Maths really is a language that traverses countries and continents. My students have often been amazed by the idea that you can go anywhere in the world and the area of a rectangle will still be base x height, that indices rules still apply and that Pythagoras’ formula will still help in finding an unknown side in a right-angled triangle. It seems obvious to us but when you’re learning new ideas about mathematics it is not always clear to novices quite how powerful those new concepts are.
Mathematics not only spans the world but also time. Although disputed by some, mathematicians and historians have conjectured that the notches on the Ishango Bone, dated to the Upper Palaeolithic era (25,000 years ago) and found in 1960 in the Belgian Congo, exhibit mathematical patterns.
The bone shows particular numbers: one below 10 and 20, one above 10 and 20, primes between 10 and 20, possibly doubles and even a lunar cycle (rows a and b total 60). It has been hypothesised that this may have been a tool for a woman to track her fertility: hence one could hypothesise that women were the first mathematicians.
Even though the craftsperson that made this tool lived thousands of years ago, something mathematical has been communicated to us here and now. With this in mind, the next few blogs I’m writing are a celebration of the wondrous maths I’ve come across through my travels and I’d love you to share some of your own international discoveries.
New Zealand, Land of the long white cloud.
I recently visited New Zealand and learnt that the first explorer to reach Aotearoa was Kupe. He used the stars and the ocean to navigate across the Pacific from Polynesia about 1000 years ago. The Māori tribes arrived over the next few hundred years. As hunters, gatherers and growers, many of the tribes were also skilled warriors and excelled at weaving. Tāniko is a traditional weaving technique that doesn’t require a loom and traditionally used flax fibre. The Tāniko designs vary from tribe to tribe and exhibit some wondrous geometrical properties that are used to decorate a range of garments and objects around the home. Visit New Zealand now and you’ll see these designs on clothing, rugs, homeware and a huge range of tourist gifts and memorabilia.
Image source: https://teara.govt.nz/en/diagram/43485/taniko-designs
Identifying the iterated units and how they have been transformed to produce a design and accurately reproducing patterns offers a chance to investigate the geometry of design. Investigating the symmetries of designs such as these supports the idea that two dimensional transformations involve the whole plane.
Drawing designs free hand, on graph paper and then weaving offers the opportunity to explore patterns, symmetries and mathematical communication.
If you fancy trying your hand at Taniko Weaving, have a look at this pdf.
SOMETHING TO TRY:
KS1: Look at the Aronui/Aonui pattern shown here. Can you arrange beads or shells to make a pattern that looks like this? What shapes can you see here?
KS2: How many triangles can you see in the pattern above? How many white stitches or shapes can you count? Could you draw the continuation of the pattern?
KS3: Look at the Patikitiki pattern shown here. In groups of two, one of you describe it while the other sketches it. How close were you? What language did you need?
KS4: Look at the Patikitiki pattern shown. Can you draw it out onto squared paper? (You will need 0.5cm squared or smaller). What do you notice? How would you characterise one ‘unit’ of pattern (motif)?
KS5: Define a unit of transformation in the pattern below and describe the transformation/s necessary to create the pattern. Use a dynamic geometry program to recreate and extend the pattern.
Join the conversation: You can tweet us @CambridgeMaths or comment below.