This is where you’ll find Bev’s latest thoughts, insights and relevant web links. As a Cafe, it’s a place for you to relax and browse. You’ll find interesting mathematics facts and snippets and helpful hints for making maths exciting and relevant. If you are a new subscriber it’s worth making a nice cup of coffee for yourself and checking out as many posts as you can – you never know what you’ll discover. Everyone can access this part of the website, even non-subscribers.
We suggest that your school Maths Improvement Team conducts a resource check for the major substrands every year. So that this task is manageable, we suggest that these surveys are spread out over 4 terms. To make your life easy, we provide a suggested list for each of the MEASUREMENT and GEOMETRY substrands. Its all ready for teachers to just tick and your Maths Improvement Team to summarise and analyse.
TED Talks: Robert Lang The Math of Origami
TED Talk: Alison Gopnik What do babies think?
TED Talk: Carol Dweck: The power of believing that you can improve
What’s wrong with worksheets
Recently I had a heated discussion with colleagues about the word “worksheet”. They argued that these were linked to textbook activities that are boring, mechanical and invariably worksheets are mindless exercises in mathematics. I argued that my colleagues were referring to “textbook” tasks, or something like that. I didn’t win the argument. I didn’t convince them that worksheets could be wonderful examples of activities for our students. I now try to avoid using the term “worksheet” throughout our Maths Matters Resources website, just in case the term is misunderstood in any way.
But the heart of this conversation was really that in primary mathematics we should be providing our students with worthwhile activities, full of interest, maths language, challenge and real-life relevance. What you call these in your classroom doesn’t really matter. A rose by any other name still smells as sweet. So what is it about an activity that makes it effective in primary mathematics?
An activity should be relevant to the needs and interests of your students. It should encourage them to think together with a partner, to use language to explain and explore, to try out and compare different strategies. When a challenge has more than one solution however, the problem for us as teachers is that it becomes more difficult to “mark”, to evaluate, especially if we have more than 30 students in our class and they are working on a variety of activities with a variety of possible solutions. Textbook examples often have just one expected answer, so we know if it is correct or incorrect. But does ‘easy to mark’ mean ‘effective to practise and explore”? Invariably “no”. Simple answer-driven tasks do not generally encourage effective mathematical thinking or sharing of strategies.
I face the same problem when creating activities for Maths Matters Resources. As you will have noticed I try to focus on activities where students have to talk, think, compare, evaluate and explain their thinking, preferably within a real-life context. And I know that as busy teachers you want me to provide “easy to mark” solutions so you can get on with something else in your classroom. In just about all of my recent activities, I try to do this. This can mean providing a few examples of strategic thinking, or ways to solve a particular problem. For example, I provide sample definitions for you in Grades F/1/2 What 3D Object am I PICTURE CARDS ACMMG022 ACMMG043, or an example of responses to a mental warm-up in Grades 3/4 What do I know AREA Mental Warmups F123456. In Grades 5/6/7 Little Town Shopping Centre POSITION Activities (9 pages) ACMMG113, I provide plenty of checklists where your students can cross reference possible data to select a matching correct solution. Phew. It was incredibly complicated to do all this but hopefully they help to make your classroom management a little easier.
So just stop and take a big breath next time you are tempted to hand out a “worksheet” to your class. It may be easy to copy and mark but the quality of learning can never match the challenge of a stimulating, open-ended problem to solve in pairs or small groups. We want Australians of the future who can think for themselves, tackle challenges head-on, work co-operatively to achieve a common goal.
Jennifer Townley Maths Sculptures
Jennifer Townley is a sculptor who creates beautiful mechanical automata with a mathematical zing. You can see plenty of examples at her website, www.jennifertownley.com. How can you use one of these sculptures to inspire your students to create their own sculptures? What is it that they like or dislike about these sculptures?
And there are so many other artists to explore. Have you discovered the Swedish ceramic artist Eva Hild yet?
Or the British sculptor Andy Goldsworthy?
Or the Russian/US sculptor Naum Gabo?
Cartoons are a great way to use humour to teach and develop mathematical ideas. This Mathematicians Food Fight by cartoonist Daniel Reynolds is a good example. It plays with the idea that the mathematical symbol pi is an actual pie that can be used in a food fight. Very silly but very effective. What a great way to start a whole unit talking about and exploring the wonderful relationship between the circumference of every circle and its diameter. Perhaps your students can create their own maths cartoons to inspire students in another class too.
Amazing facts about animals
Make maths come to life in your classroom with a focus on animals. Your students love to discover amazing animal facts. Encourage them to create mathematical lists about what they learn as they work with a partner or in a small group.
This baby stingray, for example, is like a square – but with an alien body attached. So amazing. Are there any other creatures that have a square body shape? If you look under a microscope, or even better an electron microscope, you can discover a whole world of mathematical wonders. Geometric shapes galore. Encourage your students to collect facts about Number, Space/Geometry, Measurement, Chance and Data. e.g there are over 60 different species of stingray, they can live up to 25 years in the wild, they give birth to 2 – 6 young at a time, they grow up to 1.98 m. What else can they discover?
Mathematics has always inspired humans to decorate their belongings and surroundings with mathematical patterns and designs. One of the earliest artefacts discovered is a 40 000 year old green stone bracelet from the Denisovskaya cave, Russia. And prehistoric hand stencils on a cave in Spain were probably made by neanderthals about 35 000 years ago – the first known symbolic art.
Some modern artists are more closely inspired by mathematics than others. MC Escher for example, or in this picture, the Swiss-German artist Paul Klee (1979 – 1940). This is Klee’s painting Castle and Sun, created in 1928. His greatest inspiration came when he visited Tunisia in 1914, where colour dominated his thinking. “I know that it has hold of me forever… Color and I are one. I am a painter.” He produced about 9 000 works of art in his lifetime, an amazing achievement.
A web search will discover some of these works – why not focus on the mathematical ones? Find your favourite Klee painting, for example, then analyse it for colour, shape and size. Try to copy it exactly or use it as an inspiration for your own mathematical work of art.
Using current happenings to inspire your students
Make sure you keep your eyes and ears open for something mathematical happening in the news. For example the New Horizons spacecraft reached Pluto and its moon Charon on Tuesday 14 July after a 9 year journey. How wonderful that scientists and mathematicians can work together to make such things happen. Think of all the mathematical facts about Pluto you can use to inspire your students – making place value, for example, come to life in your classroom.
Pluto is 7.5 billion kilometres from earth. How can we really think about that? What is a billion? (1 000 000 000) 1 billion seconds is about 31.7 years, so 1 billion seconds ago would put us in 1983. 1 billion minutes is approximately 1901 years, so 1 billion minutes ago would land us in the year 114. 1 billion hours is approximately 114 000 years, so 1 billion hours ago would land us in the Lower Paleolithic era or Old Stone Age. 1 billion days is approximately 2.74 million years, so 1 billion days ago would be when the genus Homo appeared in Africa. That means it is a VERY long way away!!!
Pluto was discovered on 18 January 1930 by Clyde Tombaugh when he was only 24 years old. How long ago is that date in 1930? How do you know? What strategy did you use? How close is Pluto to our sun? How heavy is it? (13 050 000 000 000 billion kg. That’s 0.00218 x the mass of our Earth.). What is its diameter? (2 370 km) How long does it take to make a complete orbit? (246.04 Earth years) How far does Pluto travel when it makes one complete orbit? (5 874 000 000 km). How many moons orbit Pluto? (5) Pluto is one third water. So what is the rest of Pluto made from? (rock)
Did you know that the ashes of Clyde Tombaugh, who died in 1997, were sent with New Horizons to Pluto. So dramatically romantic. What an honour to visit Pluto 85 years after his initial discovery.
Maps and Mathematics
Studying and creating maps is a wonderful way to link both 3D Objects, 2D Shapes and Position sub-strands in your maths sessions. The earliest map found so far is from the Czech Republic dated about 25 000 years ago. This showed geographical landmarks in the area. Another early map was created on the walls of Lascaux Caves, France about 16 500 BC. The ancient Babylonian created clay maps of the world as early as 600 BC. The ancient Egyptians also created maps to record property boundaries. And the ancient Chinese created maps on silk. Islamic scholars had world maps in 1154 AD to help Arab merchants and explorers. Records also show Polynesian maps of the Pacific Ocean to help their sailors travel large distances. Sticks were tied in a grid with palm strips to represent wind and wave patterns. Attached shells showed were to find small islands.
Later in the 16th century, of course, the Flemish cartographer, Mercator, worked out a way to make the 3D world look better as a 2D shape and the Mercator’s Projection map was born. Today we have highly detailed digital maps based on aerial photography and satellite imagery. We even have them accessible on our phones.
Maps helped humans define their 3D world as 2D images and as 3D globes. The word ‘cartography” means the study of maps and it comes from a Latin word “carta” (map). All your students, young and all, can enjoy thinking about how to represent the world around them as a map. You’ll find our suggested activities at Geometry – Position. We also have a few maps in Position Photographs and Position Graphics. We are always on the lookout for plenty more!
Singapore Maths Problem
This problem went viral across the world on 14 April 2015. It is from a test given to 14 year old students competing in the Singapore and Asian Schools Maths Olympiad, aimed at the top 40% of students.
We had a fantastic response from our community regarding the correct answer. It’s an interesting maths challenge that relies on logical thinking.
How do you approach this problem? What strategies do you use to solve it? And imagine being in a test situation too. That is an added distraction for your brain.
Do what’s the answer???
To start, Cheryl gives 10 possible dates for her birthday. Albert is then told the month and Bernard the day of Cheryl’s birthday.The challenge is to work out the actual date of Cheryl’s birthday.
Albert and Bernard at first don’t know the actual date from the information they are given. So it can’t be May 19 or June 18. Otherwise Bernard would definitely know if he had 18 or 19 as his day.
And for Albert to say that he definitely knows Bernard doesn’t know, this means it can’t be any date in May or June (when these two numbers 18 and 19 appear) either.
So that leaves July or August.
But then Bernard says he does know when the date of Chery’s birthday.
It can’t be July 14 or August 14 as then Bernard would not know which month is the correct one.
So that leaves July 16, August 15 or August 17.
And then Albert says he knows now too. That means there is no double up.
It can’t be August 15 or 17 as then Albert wouldn’t know which day to select.
It must be July 16.
Watching the moon every month is a great way to think about both time and 3D objects. Twenty-eight days to observe and record, to think about and spot patterns. That’s what Galileo did way back in 1609 when he created a telescope that could magnify 20 x. He observed the moon each day and worked out the surface of the moon has valleys, shadows and peaks like on Earth. It was not perfectly spherical after all. And like Copernicus, Galileo believed the Sun was the centre of our system, not the Earth. Up until then Aristotle’s idea maintained the Earth was the centre of the Universe. Galileo sparked the birth of modern astronomy.
Your students can create their own daily record of the shape of the moon. Compare it with our March calendar which shows Galileo’s own recorded observations.
If you like mathematical games, Monument Valley for iphones or ipads, is pretty spectacular – one of the most beautiful games we have ever seen. If you like MC Escher, and we love his work, the creators have been influenced by his impossible puzzles. At the conclusion of each segment of the game, the little girl returns a mathematical shape that belongs to a tower. Very addictive.
What a wonderful way to celebrate the brilliance of everything! Create giant bubbles and watch them float away, burst, contort. Just try some detergent and water and a large metal ring on a stick. You can bend metal coat hangers, for example. What is the largest bubble you can create? The longest in length? The longest time it stays afloat?
Here are some great bubble video clips to use as a class stimulous:
Bubbles in slow motion – https://www.youtube.com/watch?v=q4BByh4zrWs
Bubbles on a beach – https://www.facebook.com/video.php?v=10153082881424742
Giant bubble net – https://www.youtube.com/watch?v=Q9kBL1cYd-8
IMPROVE your problem-solving skills using these questions
One reason we learn mathematics is so that we can solve problems. The latest educational research shows that the IMPROVE method of metacognitive questioning, developed in Israel by Mevarech and Kramarski, is very effective. Here are 4 IMPROVE questions to help your Stage 2 and 3 students think about their own thinking. These self-directed questions can be used as a scaffold. They’ll be especially useful when tackling complex, unfamiliar and non-routine problems. Problems like these should be available for all students to solve, not just your low blockage students. Most textbooks are not suitable as they include only routine problems using a known algorism. As Andreas Schleicher (Director for Education and Skills, OECD) says, ” … good mathematics education can … foster the innovative capacities of the entire student population. including creative skills, critical thinking, communication, team work and self-confidence”.
You can read more here.
Or view Andreas Schleicher’s TED talk Use Data to build better Schools
MV Road to Mandalay
There are plenty of websites that summarise key facts about ships and boats across our planet – a wonderful source of mathematical discovery. For example, http://www.wildearth-travel.com/ships/.
Bev Dunbar was recently on a river cruise down the Irrawaddy (Ayeyarwady) River in Myanmar when she spotted this magnificent river boat – MV Road to Mandalay. It’s a luxury boat that was originally brought from Germany in the 1990s but it was completely refitted after the devastating Cyclone Nagis in 2008. It is 101.6 metres long, with an 11.7 beam (width), a 1.45 m draft (underwater depth) and 900 tonnes gross tonnage. There are 4 decks which include 43 cabins that can take 82 passengers down the river in style from Mandalay to Bagan and back again. There are 108 crew members. And there are plenty of maths discussions just regarding the size of the cabins on this vessel.
Amazing maps of Australia
Imagining the size of things is a mathematical skill. It’s called spatial visualisation. Maps are one way humans have invented to imagine what the earth looks like broken up into smaller chunks we call countries. Mercator was a Flemish geographer who imagined the earth as a cylinder in 1569. Most maps we see today are based on this cylinder, opened out to create a rectangle. We can imagine the shape of Australia on this cylinder/rectangle, but what does this shape look like in comparison with other countries.
Your students will enjoy seeing these amazing maps of Australia imposed one on top of another. They will help them visualise the huge shape of Australia better than seeing it alone.
Anthony Fisher Maze Maker
Anthony Fisher is a world famous designer of mazes – all shapes large and small. Explore his maze maker website for an amazing (ha, ha …) insight into the wide variety available throughout the world today. Maze exploration can be part of your study of Space: Position. Solving a maze puzzle uses spatial visualisation skills, an essential aspect of coping with our modern world.
David Bailey’s spectacular Escher-type tessellations
David Bailey has created the most amazing set of Escher-type tessellations, many of which will appeal to your primary students. There is a vast range to explore from geometric designs, birds and animals. A tessellation is when you put tiles together on a flat surface so that there are no gaps or overlaps. Decorative tessellating patterns have been found in many ancient cultures, such as Ancient Rome, Egypt and throughout many monuments in the Middle East. A “tessella” is a Latin word for a small square. The Dutch graphic artist MC Escher created magnificent tessellations using irregular interlocking shapes. Escher had travelled extensively through Italy and also Spain where he was particularly inspired by the tilings he saw at The Alhambra.
Why not encourage your students to explore some of David Bailey’s work too? He has created a fantastic website packed full of information about tessellations with his own drawings plus real-life photographs. http://www.tess-elation.co.uk/new-hom