In the earliest days of human civilisation, people would move around on land and know the local landscape well enough to navigate using landmarks. As we spread across the globe however, the sheer size of the Earth led to the necessity of drawing maps as guides for traders and explorers.

The earliest known map dates back 2600 years to Babylon. From this time until 1492, maps of the world only included Europe, Asia and Africa. After the voyages of Columbus and Magellan, maps started appearing that included North and South America. It was nearly another 300 years before Australia, New Zealand and Antarctica were discovered and the full extent of the shape of the world could be visualised.

Over time, the most effective means of creating maps has been using projections. A projection is a systematic method of drawing the Earth’s meridians and parallels on a flat surface, which is very effective but riddled with inaccuracies. This is due to the distortions that occur during the process of transferring a 3-dimensional object onto a 2-dimensional surface. Over time, several approaches have been presented for reducing distortion when transforming a spherical surface into a flat map. These include:

- first mapping the sphere into an intermediate zero-Gaussian curvature surface like a cylinder or a cone, then converting the surface into a plane
- partially cutting the sphere and separately projecting each division in an interrupted map.

**Map projections onto a 2-dimensional surface**

The surface of a sphere cannot be represented as a plane without some form of distortion. Karl Friedrich Gauss mathematically proved this in the 1800’s. Since the 1500’s mathematicians have worked on ways to accurately translate the globe into something flat and they have found that the best way of doing this is to use a process called projection.

Popular rectangular maps use a cylindrical projection as shown below:

In the above case, each point on the sphere is projected onto the surface of the cylinder which is then unrolled creating a flat rectangular map. However, you can also project the map onto other surfaces. In all cases, the maths used by map makers to project the globe will affect the way it looks once it’s flattened out.

There are many, many different types of map projections, but each of these come with various sacrifices to shape, distance, direction and land area. Depending on the projection, every map has advantages and disadvantages in each of these areas.

One of the most popular maps is called the **Mercator** projection. It is the map most commonly used in schools and by Google Maps because it works really well in preserving the general shape of countries. In other words, Australia on the globe looks like Australia on the map. The original purpose of the Mercator projection was for navigation however because it preserves direction exceptionally well, an important aspect if you are trying to navigate the oceans using only a compass. It was originally designed so that a line drawn between two countries on a map gave the exact angle to travel on a compass. This route was not necessarily the shortest, but it was a simple, accurate and reliable way to navigate across the oceans.

Gerardus Mercator created his map in the 15^{th} century and he was able to preserve this direction by varying the distance between the lines of latitude so that they were closer together near the equator and further away at the poles. This created a grid of right angles across the map similar to the one shown below.

The Mercator map was a great rudimentary map for the purposed outlined above. However, this type of mapping created specific problems with the area and size of different countries relative on their position on the globe. For example, countries located closer to the poles appear to be larger in size than those nearer to the equator even though their actual area was vastly different.

Take a look at the size of Greenland compared to Australia on the Mercator map below. It appears larger in area even though Greenland’s area is only 2,166,086 km^{2} compared to Australia’s 7,692,024 km^{2}.

The image below shows Greenland and Australia superimposed, indicating their true size. As can be seen, this displays a vastly different image and representation of their sizes than how they appear on the Mercator.

So why does such a discrepancy occur? The reason for this size disparity soon becomes clear if we place equal sized dots all over the globe at equivalent intervals and project this onto the Mercator map. As can be seen below, the circles retain their round shape throughout the projection, but are enlarged as they get closer to the poles.

The **Gall-Peters **map, also known as an equal area map is another rectangular map projection that more accurately displays land area. Unfortunately, although the area of the countries is preserved in this type of projection, the shape of the countries becomes quite distorted as shown below. Due to this dramatic distortion, the Gall-Peters map is not popular among map enthusiasts who prefer one that is more visually pleasing.

With the development of Global Positioning Systems (GPS), the need for using paper maps has significantly reduced and any modern projections have become more about aesthetics, design and presentation. The Mercator no longer fits the bill with modern cartographers who see it as misleading. However, Google Maps still uses it because of its ability to preserve direction, an important part of the program’s functionality.

Most cartographers these days have settled on maps that split the difference between the distortion of size and shape and in 1998, The National Geographic Society adopted the **Winkel-Tripel** projection because of its balance between size and shape accuracy (below). Prior to this time, the Winkel-Tripel was not considered particularly exceptional and it was a relatively obscure map.

**Examples of other types of projections**

**Polyhedral Projections**

Polyhedral map projections are an interesting solution to the problem of accurately representing curved features on flat surfaces while minimizing the distortion of cartographic properties.

The techniques described in part one are combined when creating polyhedral maps. To create these kinds of maps, cartographers generally inscribe the sphere in a given polyhedron, then separately project regions of the sphere onto each polyhedral face. These polyhedra are then cut along each edge to create a net and in turn, a 2-dimensional map.

The figures below show a globe inscribed in a tetrahedron. The second image shows the net (map) of the projection.

The distortion in polyhedral maps can be quite significant, particularly at the vertices where the polyhedron is further away from the inscribed sphere. However, increasing the number of faces can greatly reduce the distortion. After all, a sphere is just a polyhedron with infinitely many faces.

The five regular or platonic solids are natural candidates for polyhedral maps. These include tetrahedrons, cubes, octahedrons, dodecahedrons and icosahedrons. Tetrahedral maps are generally considered unacceptable however, due to the exaggerated distortion near the vertices.

The most commonly used projections for polyhedral maps is the **Gnomonic** projection. Gnomonic projection maps are created by projecting points on the surface of the sphere, from the sphere’s center, to a point on the plane that is tangent to the surface.

These Gnomonic projections have a greater expansion, away from the origin, than a conformal projection giving it quite a distorted appearance, but it also has some quite useful properties. The way gnomonic maps are constructed means that all great circles on the globe are represented by straight lines on the map. This makes it very useful in plotting great circle routes between arbitrary destinations, which is extremely important for flight paths.

**Acknowledgments**

https://www.youtube.com/watch?v=kIID5FDi2JQ

http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/projPoly.html

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Fantastic example of collaboration and the amazing people I work with occurred today. The school year has started and after two days of presentations down on the Peninsula I am back in the office today, preparing for a full week of school visits next week.

I’m really exciting about working with the teachers and year 9 students at Dromana College next week, over two afternoons. I want to really make an impression with the students, with a growth mindset presentation, followed by a couple of challenging and fun activities.

The presentation I plan on showing comes from the wonderful Jo Boaler’s website https://www.youcubed.org/. She is so inspiring and I talk about her work in all my schools. The presentation is about maths, believing in yourself, the importance of mistakes and the beauty of maths. Now I needed an activity to encourage and enable collaboration.

Going to nrich I found ‘9 colours’ which I believed looked thought-provoking so I asked my colleagues if they were interested in solving it collaboratively, to help determine how successful it could be. Being in a room full of mathematicians and maths teachers, they jumped at it and the conversation flowed.

Initially, as always happens, someone jumped in and started assembling the first level, which, let’s be honest, isn’t particularly difficult requiring 9 cubes and 9 different colours. The second level was assembled just as quickly before someone called a halt to proceedings and insisted we started thinking more carefully about the placement of cubes. This led to discussion around which colour would sit in the very centre and where the two other cubes of that colour needed to be placed.

We would have made it a great deal easier on ourselves if we had started assemble by laying out the bottom level first and building up, but we started from one side and moved across. This resulted in the need to physically hold cubes in place as we set about placing all cubes of each colour, rather than working level by level. This would have saved the mistake we made when we checked the base; misplaced colours. Correcting this led to further discussion around how many faces of each colour were visible in the completed cube.

Having completed the large cube, discussion turned to how we could develop and extend the activity. Depicting the completed cube on isometric paper seemed to next logical step. 3-D drawings are often difficult for students, something that frequently shows up in NAPLAN results at all levels. Enabling students to first draw 3-D objects in 2-D, and then recreate from 2-D back into 3-D, develops their spatial awareness. Redrawing the large cube after a specific colour has been removed develops this even more.

As we worked through the task, more and more ideas emerged on how to take an engaging activity and turn it in a rich task, suitable as a multi-ability, multi-age investigation. Through collaboration and discussion we were able to relate it to the Australian Curriculum at 6 different year levels, extending it in many ways:

- Draw it on isometric paper from different perspectives.
- How many drawings do you need to show all faces?
- Is there more than one way to place each small cube?
- Is yours the same as another group’s cube?
- If you gave another group a drawing could they reconstruct your design?
- What is the minimum amount of information you need to give them?

- Is there a pattern to the placement of your cubes?
- Remove a particular colour (3 blocks), then draw the design with missing blocks.
- What is the surface area and the volume?
- How does the surface area and volume change when you remove one colour?
- Can you predict the colour of the middle block (the one not visible)?
- Is there symmetry in your design?
- Can you determine the total number of combinations?
- What can you determine about the number of faces of each colour you will see in the completed cube?
- Can you explain this using numbers/algebra?
- How many colours would you need for a 4 by 4 by 4 cube?
- How many cubes in a 4 by 4 by 4 cube?
- Do the same rules and patterns exist in a 4 by 4 by 4 cube?
- Is there a point at which the task is impossible? (5 x 5 x 5; 6 x 6 x 6; 7 x 7 x 7…)

**Assumed knowledge:** partitioning, arrays

**Learning Intention:** To understand “groups of” in an organised manner (arrays). To link visual and written representations of times tables.

**What the lesson might look like:**

- Have someone read the question. Ask for questions/clarifications. Allow those who can make a start to go and start.
- Provide one extra step of clarification for others then expect everyone to make a start.
- Rove and answer questions as necessary. Provide enabling or extending prompts as necessary.
- Select some students to put their solution on the board for all to see
*(student writes their solution on whiteboard, photograph solution for projector)* - Pause the class working and allow the students you have chosen to share their solutions to explain their working.
- Highlight arrays and patterns in student models.
- Relate arrays and patterns to times tables algorithms. Record times tables.

- Allow students more time to work on the problem.
- Rove and assist as needed. Redirect any misconceptions.
- Expect students to share their solution with at least one other person.
- Stop the class for final discussions of solutions and strategies. Connect solutions to times tables.
- Reflect on what we’ve learnt about times tables. Students to record own learnings.

**Possible discussion topics:**

- What is an array? How does the apple field show an array? How do arrays show times tables?
- What are good numbers of trees to have in each different field? Why are they good numbers?
- How can you record your working? Show it in more than one way (diagram and number sentence)?
- Find a pattern in your solution. How does the pattern relate to times tables?
- Vary your pattern to show a different times table. Show a different times table in each field.
- How does your solution compare to someone else’s. What is the same, what is different?
- Explain why you think the farmer should plant his fields using your design.
- Relate the solutions to division as well as multiplication.

**Enabling question 1:** how many trees will you plant in each field, remember one is bigger than the other. *(This questions gives a starting point to those who don’t have one)*

**Scaffold 1:** draw a picture – how many trees are there, how are they arranged, how many more are needed to make 72, how could they be arranged?

**Extension 1:** what if the farmer had 108 trees *(takes the number into 3 digits and out of regular times tables for year 3)*

**Extension 2:** You decide how many trees the farmer will plant in three fields. Show how the trees will be planted using a different pattern in each field.

**Going further:** start making patterns with the number of apples on each tree. Create word problems about apple trees and numbers of apples.

Why 72? It’s large enough, but familiar enough, for year 3s to manipulate. It has lots of factors. It partitions into numbers that have lots of factors. Scaffold 1 – the picture has 24 trees, which leaves 48 trees (double 24) still to be planted.

Download a pdf of the **assessment rubric** here.

It’s also important to note up front that no maths lesson exists in isolation. Each lesson you teach will be one component of a broader topic, which will be one component of a broader continuum of mathematical understanding, and so on. A ‘good maths lesson’ will always necessarily be a part of a sequence of lessons or learning experiences which will ideally build mathematical understanding, improve fluency, build problem solving capacity and then develop mathematical reasoning skills.

What follows is a set of observations and suggestions from a classroom teacher who has yet to have delivered that ever-elusive ‘perfect maths lesson’. Some of my own lessons have been spectacular failures – even some of those in which I have been meant to be modelling ‘good practice’ to colleagues!

Nevertheless, it is my intention that the following might at least provide food for thought and conversation on what constitutes effective mathematics teaching and learning, at least within the structure of classrooms and schools that we currently inhabit.

*You* are encouraged to contribute to an ongoing conversation in your school about what works – and doesn’t work – as you continue on the maths learning journey with your students.

**(a) Using ‘Warm-up’ or Ignition Activities**

The purpose of these is simply to start children ‘thinking mathematically’ – to establish a mathematical mindset. Games, loop activities, short open ended problems (for which solutions can be shared or discussed) – anything which will have students ‘lace up their maths boots’.

Keep this brief – about 5 to 10 minutes – and accessible. Choose something all or most students can readily participate in. Inclusiveness is important. Warm-ups also don’t necessarily need to be directly related to that lesson’s content – although it’s handy if your warm-up activity can be used to ‘bridge’ into your explicit teaching session.

If students have access to devices in the classroom, using maths apps to warm up is fine – but *be selective *and “ask three” at the conclusion of the warm-up session about what they did, what they found difficult and what they learned.

**(b) The Role of Explicit Teaching**

There is *absolutely* a place in your maths class for teacher directed instruction (note that by this we are not referring specifically to the teaching methodology called ‘explicit direct instruction’ (EDI), but rather, the general pedagogy of ‘teacher explanation’: “This is how we do it / solve it / plan it…”).

Spend some time in your lesson either introducing new concepts or revising existing concepts. Remember to involve students as you do this. Ask questions unsolicited (try a ‘no hands up’ policy in your classroom!). Ask students to assist you with working out.

Time taken for explicit teaching will vary depending upon the complexity of the concept you are teaching, but usually fifteen minutes is ideal; even adults will have trouble actively concentrating on someone explaining detail for longer than this!

Use stories / analogies / role plays to get your message across. Only use digital instruction clips in active teaching sessions if (a) they are short, and (b) you can pause them easily to check for understanding. Remember – *you* are the most effective resource / piece of technology your students have in their classroom!

**(c) Improving Fluency with Practice**

Generally speaking, we often tend to ‘overdo’ practice in mathematics lessons. This is because most maths textbooks and resources emphasise this through the provision of lots and lots of closed questions and problems for ‘drill and practice’. Whilst there is no doubt that it is important that students reinforce their understanding of mathematical concepts with practice on set exercises, the temptation can be for this to become ‘busy work’ by requiring to students to work through long sets of repetitive, similarly formatted maths exercises.

Sound ‘practice’ (or development of fluency) in mathematics will ensure that student exercises are *scaffolded* appropriately (e.g. similar examples; prompts; one-to-one or small group coaching where available; provision of tactile and/or visual learning resources such as blocks, diagrams, counters…) to make exercises accessible for students still not confident with concepts.

As students become increasingly confident, remove or ‘fade’ these scaffolds and encourage them to tackle exercises using ‘pen and paper’ working and their own mental strategies. Remember that the development of fluency in mathematics can be defined as a continuum along which students are encouraged to move from concrete to increasingly abstract conceptualisations.

**(d) Practising Problem Solving and Using Group Work**

Once a level of mathematical fluency has been established, there is great value in having students work either individually or together on a problem (related to the focus for the lesson). Doing so will develop higher order thinking skills such as problem solving and reasoning, especially when they are pushed just outside their comfort zone or ‘zone of proximal development’. Building in group work and/or individual problem solving sessions allows students to experiment, work through, persist and learn from mistakes.

Productive discussion should be encouraged during, and at the conclusion of, these sessions. This could be structured (e.g. asking groups or pairs to share their working / thinking with the class) or less structured (e.g. small groups working through an open ended problem together, where there will inevitably a degree of discussion).

As a teacher, listen carefully for ‘student voice’ by prompting students with questions which will promote mathematical discourse, e.g.:

*‘Tell me about how you arrived at that answer’**‘Can you convince me that your answer makes sense?’**‘What was easy in this problem? What did you find difficult?’**‘Who arrived at a different answer? Why might there be differences’*…

Better still, choose problems and tasks that are *open ended* and can be easily differentiated (e.g. multiple solutions; students able to set different parameters to make the problem easier or harder).

Provide resources and working materials – blank paper and/or grid paper, counters, MAB blocks, plasticine… whatever the task lends itself to. This will assist students with ‘tinkering’ with possible strategies and solutions.

Choose mixed ability groups if your intention to benefit from peer-on-peer learning / coaching; choose ability differentiated groups if you want to vary the challenge levels in tasks (but do so discretely – perhaps even allowing students the choice regarding the level at which they would like to challenge themselves.) You can also use this session to focus more individualised attention on students needing support.

Remember – the objective here is to move your students beyond understanding and fluency towards ‘problem solving, communicating and reasoning’. Plan problem solving and group work activities carefully and be prepared for a variety of student responses. Developing a culture of openness to experimentation and making mistakes will make these sessions less threatening and more fun as students give themselves and one another permission to explore and play with mathematics.

** (e) ****Remembering Reflection**

Use whatever your learning intentions for the lesson were to recap and reflect at the end of a lesson or learning sequence. These focus questions in turn should be guided by the curriculum focus – e.g. Year 4 Fractions:

**‘Recognise that the place value system can be extended to tenths and hundredths, and make connections between fractions and decimal notation’**

**–> “Who can tell me about some fractions that are easy to convert into decimals? Why is that?”**

It’s easy to run out of time, especially when students are engaged in their work or when packing up takes longer than expected. When this occurs, perhaps take time after the break to quickly recap and refresh on some of the key questions in the lesson.

Again, using questions to generate mathematical discussion is especially useful here.

Try to reflect at least once on context for the maths covered – *how could this mathematics be useful in the world beyond the classroom?*

Some teachers use the idea of a ‘maths journal’ – where students take 5 minutes at the end of each lesson to write one thing down they have discovered, or a mathematical question to be considered later in discussion (e.g. *“What happens with trading when I have to subtract decimal numbers?”* or *“What is the actual use of doing 2D Shapes?”*).

Covering the appropriate content, at the appropriate level of depth, in a reasonable amount of time will *always *present a challenge for teachers. Some tips to make doing so as painless as possible and to avoid getting ‘bogged down’ in unnecessary detail or content are as follows:

**Start with the syllabus document as your guide.**This tells you clearly not only the generalised outcomes for students at each Stage and then Grade level, but also the expected content to be covered (as well and ‘Background Information’ and useful ‘Language’).**Use a good scope and sequence**that has been worked out with your grade buddies and/or curriculum leaders. Following a textbook scope and sequence is not always a good idea as you may find yourself getting bogged down in too much detail in some areas and neglecting others through lack of time. Let your reference text follow your scope and sequence – and not the other way around. Ensure you have some room for ‘give’, to allow for re-covering some content students find difficult and moving more quickly through areas students seem confident with.**Be confident in your own understanding of the content,**whilst also allowing students to see that even you make mistakes and struggle from time to time. Never give the message “Maths is not my thing” – but rather “I may find this a bit challenging but I will keep working at it.”**Always try to find some ways in which the maths you are teaching relates to the real world**– and include this as a feature in your lessons.*For example, if teaching division (sharing between equal groups) to Year 1 or 2, explain how this could be used for sharing out batches of cakes in a bakery to ensure the same number of cakes in each bag.*

Delivery in teaching is what we might otherwise refer to as the ‘art’ of our profession. It is the way through which we engage our audience, maintain their attention, make our teaching and learning experience ‘un-forgettable’ – and importantly, build trusting relationships with our learners.

‘Mindful delivery’ simply means approaching our lessons remembering that getting into the minds of our students usually requires conscious effort. Try to answer for yourself the following question: “If I were (one of my students), what might it take to attract my attention, maintain my enthusiasm and develop my understanding?”

Some ideas that might help you develop mindfulness of your maths lesson delivery:

**Allow mathematics to be playful.**Remember, we remember much more effectively that which we associate with positive emotions (humour, camaraderie, team spirit, playfulness). Where you can, use games, role plays, outdoor activities and creative tasks to reinforce concepts. Remember the old saying: “Teaching is 90% theatre” (I don’t know who first said this, but I will credit it to John Kouimanos, my first Head Teacher and the current Chairman of the Teachers Mutual Bank).**Tell stories to exemplify concepts.**Especially stories about yourself – students love getting to know you! If you’re not comfortable with them knowing too much about your life outside of school… make it up! Using narrative and oral tradition to help our learners to recall important concepts and details is a trick that is as old as human civilisation itself – and something that the First Australians did instinctively and effectively! (See also our recent ‘Maths in the Media’ story on the work of Dr Chris Matthews.).**Use a range of resources.**Don’t just stick to one textbook, one lesson format, one digital maths resource. Provide a variety of materials to assist students that require visual and concrete aids to their learning.**Remember the importance of ‘student voice’.**Having kids verbalise what they’re thinking about is critically important, not just to have them share their ideas but also to have them become consciously aware of their own ideas and learning processes. During ‘teacher talk’ time students are often either passive ‘sponges’ or simply tuning out altogether. Productive student talk encourages your class to engage more actively in their learning, and gives you highly valuable assessment insights.

For further information and insight into developing mathematical discourse, see the Calculate ‘Munch’ articles ‘Learning Conversations in Mathematics’ and ‘Hints for Turning Teacher Talk into Student Talk’.

** **

Context in maths for children and young adults is especially critical. As educators we are all too familiar with the catchcry from our students, “Why are we even learning this stuff, Miss / Sir?” Very often, objections of this nature indicate some underlying difficulty with content. The real question underlying their grumbling: “If I’m going to be bothered persevering with this, is it really going to be worthwhile at the end?”

Giving examples of where maths is used in the world outside of the classroom shows our students that maths is, at its heart, a tool. With that tool we can describe, explore, discover and predict things in the world around about us in ways and at levels of precision that words or pictures alone could never allow.

It also shows us that without a *context* based in the real world, mathematics often appears nonsensical – in the same way as would a foreign language or a word for which we do not yet know the meaning. However, by persevering with mathematics (as with learning a new language), we are unlocking wonderful new ways of understanding our world and solving problems.

Ways to incorporate context into your lessons include:

**Commencing new topics in mathematics**with an anecdote from your own experience, a Youtube clip or film excerpt or a short talk from a friend or community member;**Pick a maths topic focus and have students take a stroll**with you around the school or the local neighbourhood / community, noting down anything they see that might relate back to their understanding of the topic;**‘Make the maths visible’ in other subjects**, by drawing attention to, or providing a verbal explanation of, how mathematics is used across in other academic disciplines as and when you are teaching them;**Consider maths excursions**, class talks or activity days held in partnership with local business people or industry partners; or**Using ‘authentic’ and ‘rich’ tasks**in the maths classroom, drawn from problems or situations in which students might need to apply or use mathematics in the real world.

For further detail on the importance of context in maths teaching and learning, see the Calculate ‘Munch’ article ‘Why Context is Critical…’.

Remember also, however, that as Bertrand Russell said, “Mathematics, when rightly viewed, possesses not only truth, but supreme beauty.” For a proportion of your students, mathematical inquiry and complexity will be appealing in and of itself, without requiring you to make a real-world application visible (indeed, many mathematical phenomena and relationships do not necessarily have a ‘real world’ application – at least not yet!). Such students should be encouraged to continue to enjoy the beauty of maths for its own sake, bearing in mind that such students may well grow into the advanced mathematical thinkers and problem solvers of the future!

**– Marcus Garrett**

© Australian Institute for Teaching and School Leadership (AITSL), 2014

]]>**However, discovering a critical link between mathematics and symbols in algebra became for him an ‘epiphany’, and he soon realised the power of mathematics as a language for discovering the world around him.**

It was then when sitting with his son many years later, adding up with some pecan nuts at the family table, that he realised the power of his own Indigenous cultural heritage – rich in symbolism and storytelling – for helping children gain mathematical understanding.

In particular, Dr Matthews realised that for Aboriginal students, the right cultural frames could even nurture a genuine sense of fun and enjoyment in mathematics – for example, by tapping into Indigenous Australians’ traditional use of symbols, oral storytelling and even dance.

Dr Matthews observes the enormous extent to which navigating and living with the land required immense symbolic – and therefore mathematical understanding for the traditional custodians of our continent, and that today mathematics is relied on heavily for making decisions regarding the protection of Aboriginal land.

Far from being simply ‘white fella knowledge’, he realised that maths and science require the creation and combination of symbols to understand the real world. Through appropriate cultural expressions such as dance, traditional symbols and oral storytelling Indigenous kids can readily identify with mathematical and algebraic thinking.

In a previous AMSI Schools ‘Munch’ post, we discussed how **telling a story to illustrate a point or bring an idea to life** in the human imagination can create those ‘a-ha!’ moments for children learning mathematics. Sadly, constructing a tale as a metaphor for a mathematical idea is an undervalued strategy, often overlooked in teachers’ lesson planners and curriculum programming documents.

To discover more about Chris Matthews’ innovative approach to teaching maths to Aboriginal children, check out the detailed article **‘Maths, story and dance: an Indigenous approach to teaching’ **on ABC online, or listen to his **podcast interview**.

**Chris Matthews is the head of the Aboriginal and Torres Strait Islander Mathematics Alliance (ATSIMA).**

**References:**

Salleh, Anna (2016), ‘Maths, story and dance: an Indigenous approach to teaching’. Article – ABC Online. URL: http://www.abc.net.au/news/2016-08-15/closing-the-maths-gap-with-story-and-dance/7700656. Accessed 1/11/16. Australian Broadcasting Commission : Sydney, Australia.

Sarra, G. (2011), ‘Indigenous mathematics: Creating an equitable learning environment’, Conference Paper – Indigenous Education: Pathways to Success, 2011. Australian Council for Educational research : Canberra, Australia.

Fanning, E and McCullagh, C. (2016), ‘Teaching maths through dance and story’. Podcast – ABC Radio National. URL – http://www.abc.net.au/radionational/programs/lifematters/teaching-maths-through-dance-and-story/7721036. Accessed 1/11/16. Australian Broadcasting Commission : Sydney, Australia.

**Image Credit: **

Australian Paralympic Committeee / Australian Sports Commission (2000), View towards the centrepiece of Opening Ceremony of the 2000 Sydney Paralympic Games. Used with permission. https://commons.wikimedia.org/wiki/File:201000_-_Opening_Ceremony_Aboriginal_Art_centrepiece_view_-_3b_-_2000_Sydney_opening_ceremony_photo.jpg.

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“I want 2C, the class next door, to know what colour eyes we all have so I am going to take a photo and send it”, the teacher said.

The students formed a huddle and their picture was taken. This was then displayed on the whiteboard.

“What do you think? Will the photo let them know what colour eyes we all have?”

Some skilful questioning by the teacher and there was a consensus that no it wouldn’t. It needed a heading to tell them what to look for. A cardboard sign was made and another photo taken and displayed.

“What do you think? Will the photo now let them know what colour eyes we all have?”

“How could we make it clearer”?

Some more skilful questioning from a teacher on top her game and the class decided it would be easier if they got into groups of the same coloured eyes.

Another photo, another display.

“What do you think? Will the photo now let them know what colour eyes we all have?”

“How could we make it clearer”?

The students decided that it was too hard to see how many were in each group so they should line up.

Another photo, another display.

The students said labels under each line would show what colour eyes the group had.

The teacher produced some labels she had prepared earlier including the word purple.

“Mmmm, should we include the purple label”?

There was some debate, but the students decided that it was fine to include the purple label because there this would show that no one had purple eyes.

Another photo, another display.

“This looks pretty good don’t you think”?

It’s still a bit hard to count how many in each group, what if we put a number line up the side so everyone could see how many in a group without counting offered one student.

Under the guidance of a skilled teacher the class had just discovered Column graphs for themselves. The short direct instruction about column graphs that followed had a context, the students were engaged and the lesson flowed from one activity to the next seamlessly.

The use of technology was brilliant, necessary and unobtrusive.

Arthur C Clarke said, “Any teacher that can be replaced by a computer deserves to be.”

This teacher could not be replaced by a computer and the class was lucky to have her.

]]>**A skull has been found by workers during excavation for foundations of a new shopping centre and the police have been called to investigate.**

**The police need to gain as much information from the skull as possible. So they call in the C.S.I. team to conduct tests on the skull.**

**One piece of information that the police would like to know is: What was the height of the person to whom the skull belongs?**

The C.S.I. team need as much data as they can for their forensic investigations. They would like to answer this question:

**“Is there a relationship between the cranium circumference and height of a person.”**

You are about to assist with investigating the answer to this question.

DOWNLOAD BOOKLET **HERE**

DOWNLOAD DATA SPREADSHEET **HERE**

DOWNLOAD TEACHER NOTES** HERE**

Traditional ‘drill and answer’ questions, and closed worded problems such as those used in most textbooks and on online mathematics websites, are usually great for assisting students with understanding of and fluency with maths concepts. However, they don’t necessarily focus students’ learning on higher order mathematical thinking skills.

By contrast, **open-ended**** problems** *do* have greater potential for stimulating higher order mathematical thinking. Open ended problems encourage students to ‘play around with’ different variables in order to generate different solution pathways, to look for new patterns, to trial and error and explore a range of methods, and to apply reasoning to concepts in a range of unfamiliar situations.

In two parts, the attached article puts open-ended problem solving under the spotlight.

**How Long Is a Piece of String Part 1 **focuses on considering how open-ended problem solving is different from ‘regular’ maths questions or problems, and discusses the significant benefits of using this pedagogical approach to mathematics, for both students and for teachers.

**How Long Is a Piece of String Part 2** provides teachers with practical ways to source, develop and implement open-ended problem solving in mathematics lessons.

This alternate assessment task in Financial Mathematics has been developed for Year 10 students, or for Year 9 at challenge level. It is mapped against the Australian Curriculum (Mathematics) and the NSW Mathematics Syllabus (Stage 5.1 & 5.2) and it includes a marking rubric and a teacher’s marking calculator (in Microsoft Excel format) with which to generate marking responses based on selected student variables.

Concepts covered include calculating gross and net wages before and after tax; operating and solving problems using simple and compound interest; and making valid comparisons between total accumulated costs and the initial (principal) amounts of money borrowed or invested.

The Task Workbook (including curriculum outcomes and marking rubric) can be downloaded **here.**

The Teacher Marking Calculator can be downloaded **here.**

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