The wonderful world of MC Escher explained by Roger Penrose
If you enjoy exploring MC Escher’s art then this is a wonderful documentary to show Stage 3 and 4 students. Images of infinity that impress scientists, tesselating tiles that never repeat. Lots of inspiration for inking mathematics and art.
NAPLAN TEST Items
NAPLAN tests are with us whether we like them or not (I don’t). And every year teachers in Year 3 and Year 5 spend too much of their valuable maths session time preparing for the test items in Terms 1 and 2. The pressure is huge. Both teachers and parents want their students to demonstrate that they are progressing well on their learning trajectory. And every year we get the results quite late and feel good or bad depending on whether our class scored above or below the rest of the state.
We have been analysing the NAPLAN Numeracy results for primary students in NSW over the last 10 years and find the proportion of questions always includes too many items that test the stage above and too few questions that test the stage below. We argue that to be a valuable data source the proportion of items should be more from the stage below. As a nation, have the students entering Stage 2 achieved the appropriate standard for Stage 1? As a nation, have the students entering Stage 3 achieved the appropriate standard for Stage 2?
The answers to these two questions really matter. Over the last 20 years Australian students have not demonstrated a high enough level of achievement on international tests. Too few secondary students are selecting mathematics and too few students entering tertiary education have the required mathematical skills. Primary schools are the first port of call for all students of mathematics. Are we teaching mathematics as well as we can? Do our maths sessions provide our students with the best environment in which they can learn Mathematics?
In 2016, the NAPLAN numeracy tests will match the proportion of items in the test from each sub-strand with the sub-stand content in the Australian Curriculum: Mathematics. How will the test creators determine this proportion? e.g. there are 2 substrands for Geometry (Shape/Location & Transformation and even a third, Geometric Reasoning, from Grade 3 onwards) and 2 substrands for Statistics & Probability (Chance/Data Representation & Interpretation). Does this mean the each get an identical number of items? And does that mean that one content descriptor is equivalent in value to another? In Grade 4, for example, one Measurement content descriptor says “Use scaled instruments to measure and compare lengths, masses, capacities and temperatures (ACMMG084)”. Yet this covers 4 separate substrands in Measurement, all of which use separate instruments of measurement. Another issue is how do you write a pen and paper test item for something as practical as measuring a real-life object? The Grade 4 content descriptor for Chance, “Describe possible everyday events and order their chances of occurring (ACMSP092)”, requires just one train of thought and more easily transfers to a multiple choice test item.
The next problem is whether each test item is meant for most students, say a core item, or is it to test achievement at a deeper level that perhaps only low block students can succeed in answering? For example, the 2012 Grade 3 NAPLAN test items for numeracy included only 9 items out of 35 that matched core Stage 1 (Stage Below) content descriptors. If we were successful, we’d expect about 80% of our students to successfully demonstrate their understanding of a core Stage Below question. We think it is more important to know if your students demonstrated achievement at core Stage Below questions than non-core questions. Another way to look at this is to say that in Grade 3 2012, only 26% of NAPLAN Numeracy questions enabled your students to demonstrate achievement of core Stage 1 maths. So 74% of questions were potentially too difficult for most students. What’s the point of knowing this?
The Year 5 paper in 2012 included 25 items out of 40 that tested core Stage 1 or Stage 2 content descriptors, still not ideal but much more effective than the Grade 3 paper.
This dilemma also transfers to programming. How do you know as a teacher what proportion of your time to plan for each substrand in Mathematics? Here at Maths Matters Resources we have very strong feelings about this dilemma and give specific advice for how to solve it. Basically about half your time should be planned for Number & Algebra, and half for Measurement & Geometry and Probability. The content for the second half is not insignificant. Number & Algebra should not take up the majority of your time as a teacher. It is not more important than that second half. And Statistics (Data) covers all substrands in Mathematics and potentially could be part of any maths session. We include sample term overviews in The Maths Session – Programming. In the same section, we also include a document called Reflecting on your Term Overviews. And of course just because you decide to focus on Multiplication in Grade 3 doesn’t mean everything is planned for the Grade 3 content descriptor. Some of your students may still be at Grade 2 or Grade 4 levels and above in their progress along the learning trajectory.
We will be fascinated to see the structure for this year’s Numeracy test items for Grades 3 and 5. Our hope is that about half the test items will relate to Number & Algebra and that some of these include Data items. And our hope is also that a large proportion of questions relate to core Stage Below content. Our hope is also that we get a fast turn-around on feedback from the NAPLAN papers to help teachers adjust their programs to match specific student needs.
Anthony Howe’s mathematical metal sculptures
Anthony Howe, an American sculptor, creates amazing metallic sculptures. Could these sculptures inspire your students to create their own using alfoil and silver cardboard?
They use rotation around a series of axes to create fluid movements like a jellyfish.
Maths jokes are a great way to add some spice to your maths sessions. Challenge your students to collect, discuss and evaluate as many different maths jokes as they can. You could develop a whole school collection to share with parents too.
This red T-shirt is one of my favourites: “Too much pi gives you a larger circumference” – so clever, so mathematical.
Some jokes can be obvious, such as: Question: “How many seconds are there in a year?” Answer: “Twelve, January second, February second, March second, …”
And what about this joke about Venn diagrams, based on the famous old Carly Simon song hit from 1972, “You’re so vain, I bet you think this song is about you, don’t you, don’t you?”
Or unexpected jokes such as:
Why was six afraid of seven? Because 7 8 9.
Jokes also play on words:
Question: What do you call a crushed angle?
Answer: A Rectangle (wrecked angle)
Question: Why was the scalene triangle sad?
Answer: Because he would never be right.
Which maths jokes are your favourites? Why?
Looking for mathematics everywhere
One of the benefits of being a Maths Matters Resources subscriber is you have legal access to 1000s of specifically targeted mathematical photographs. Of course you might use these as part of a non-maths lessons but we try to provide a wide selection of photos from across the world that will enhance your maths sessions in particular. Yesterday we visited the Botanical Gardens and even though this was our third visit for the year, it still provided us with a new selection of 2D patterns and 3D objects. Maths can be seen all around us, you only have to look.
Hex colours are all the ways you can combine the 3 primary colours of light, red, green and blue, as a 6-digit number. There are 16 million possibilities. e.g In the number #123456, 12 is a shade of Red, 34 is a shade of Green, and 56 is a shade of Blue. You can see how changing each digit effects the final colour at quite a few internet sites such as this one:
Some people have used this idea creatively. You know we can write the time of day as a 6-digit number. The first two digits are the hour in 24-hour time from 00 to 23. The next two digits are the minutes from 00 to 59. The last two digits are the seconds from 00 to 59. For example 13: 09: 22 is 9 minutes and 22 seconds past 1 o’clock in the afternoon.
Using this information, you can now go to a website that shows what colour these 6 digits represent on the HEX Colour system. They give a specific colour for every second of our 24 hour day. That’s just 86 400 out of the possible 16 million possible colours.
What else can your students do with this system?
Mathematical Sculptors – Anish Kapoor
Anish Kapoor is one of our favourite sculptors. He was born in Bombay India in 1954 but lives now in London. He creates huge geometric structures, often out of highly reflective polished stainless steel. The final effect is often astounding, even though it may be a simple rectangle with a curve, like C-Curve, or a gigantic squashed cylinder, like Sky Mirror. One of his most famous sculptures is his 2006 Cloud Gate, 110 tonnes of mirrored stainless steel in Millennium Park Chicago.
Bev is standing in front of one of his small granite sculptures, Number 8, at the Peggy Guggenheim Museum, Venice.
You can see all of his work at:
Mathematical Artist – Beth Radford
Art and mathematics can be a magical mix. Check out these amazing paintings from local Sydney artist Beth Redford. Beth uses computer graphics to help her design and calculate then she laboriously measures and creates her final paintings by hand. Just beautiful. What can your students create using their own computer graphics?
There is a fantastic new exhibition at the Victoria and Albert Museum, London – The Fabric of India. It is packed full of mathematical information for your students to explore. It is on from 3 October 2015 until 10 January 2016. Wonderful for understanding how spectacular embroidered patterns are created.
There are plenty of accompanying video clips too. Did you know that red dye techniques were known to the Indus Valley civilisation about 45000 years ago? And that cotton has been cultivated for 9000 years? Or that ‘ari” hook embroiderers from Gujurat, India, were highly prized by the Mughal and European courts?
Christmas Tree Data
Christmas time is packed with interesting facts for data analysis.
Here is an American website we like:
For example Americans spend a little more on fake trees than real trees, yet there are twice as many real trees purchased. And 85% of real trees are pre-cut.
You also get a summary of facts from 2008 -2014 so you have useful comparisons and observations to explore.
This site includes some Australian facts.
It has a Christmas countdown clock and simple costs to use with your Stage 2 or 3 class.
Mental Maths Warmups are a wonderful way to start every maths session. They are short, efficient, language-based, targeted, relevant activities that up to the whole class can be engaged in at one time. You can spend about 10 minutes every day warming up your students strategic thinking, problem-solving and fact recall.
The funny thing about them is that there are 3 distinct types of Mental Maths Warmups:
These summarise, recall and practise maths concepts you have tackled with your students in a previous year, term or week.
These target maths concepts that your class is studying today or this week or in this unit.
These investigate maths concepts your class may not have been introduced to yet, They are great as a quick pre-assessment check.
Although sample Mental Warmups are scattered throughout our Maths Resources, you can find a large collection of them sorted into specific grade groups in The Maths Session.
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.