The English mathematician Grace Chisholm Young was born on 15 March 1868. She received a 1st class degree in mathematics from Cambridge Uni and was the first woman to sit for and then outperform all the male students at Oxford Uni. Grace studied under the famous mathematician Felix Klein at Gottingham Uni in Germany, one of the most prestigious universities at that time. Later, with her husband, Grace wrote “A First Book of Geometry” on paper folding for children (1905). Together with her husband she wrote 220 articles about mathematics. They wanted to share their mathematical ideas with as many people as possible.

What if your students created their own book about geometry? What topics would they select? How would they illustrate it? Would they work on one book for the whole class or perhaps create a book in pairs? How many pages would it have?

Part of working mathematically is recording what you know and presenting it in a way that other people can read about your ideas.

Try to create at least one class book about mathematical concepts each term. You could lend your class book to other classes in the school and ask them for feedback too.

Niels Helge Von Koch, a Swedish mathematician who was born on 25 January 1970 and died on 11 March 1924, was famous for creating a special pattern from repeated equilateral triangles called the Von Koch Snowflake. And it is easy for your students to replicate his pattern.

Start with two identical size large equilateral triangles and place one upside down on top of the other. You might like to glue these together.You now have a 6 pointed star and the points are 6 smaller equilateral triangles.

Cut out 6 more triangles to fit these 6 smaller outer triangles and place them upside down on top of each one. Now you have created 18 tiny equilateral triangles.

Cut out 18 identical small triangles and place these upside down on each small triangle. Got the idea?

Stop whenever you feel the triangles are getting too fiddly to manage.

You have now created one of the most famous FRACTALS, the Von Koch Snowflake.

We love to holiday at Hawks Nest and frequently see dolphins in the surf off Bennett’s Beach, in the Myall River and in the bay at Jimmy’s Beach. Did you know that this area, Port Stephens, is home to 120 bottlenose dolphins, one of the largest pods in the world?

Some interesting mathematical facts about dolphins:

– they are a type of “toothed whale”
– they grow up to 4 m in length
– they can swim for up to 100 km each day
– they get fresh water from their food (squid, fish)
– they can close down one half of their brain to “sleep” for 8 hours a day
– they can travel up to 35 km per hour and breathe through their blowhole
– this breath can help them dive for up to 4 minutes
– they can also hold their breath for up to 20 minutes
– they live for up to 40 years

Imagine all the wonderful classroom discussions with your students about these maths facts.What else can your students discover? Encourage your students to work in pairs to create a mathematical challenge based on these facts.

Blockages to maths understanding are everywhere. Your role as a teacher is not to teach but to unblock as many students each day as you can. Blockages underpin so-called abilities. Your so-called low ability student is one who has multiple blockages. If you can remove these then your student is better able to understand.

Remember it is vital to talk about what something ISN’T as much as about what it is. I just saw this ABC online video clip about Quarters. Everything there was fine. Cute female child’s voice over, clear graphics. All good. BUT … the biggest problem your students have in fractions is understanding far more than that, far more than just one explanation. What is NOT a quarter, what is NOT an equal part? What happens if there are only 3 equal parts? What happens if there are now five equal parts? Talking about “positive” things is only part of the solution. Talking about ‘negative’ things is also a vital component.

This video clip will be a fantastic starting point for a discussion with your class. Try to generate as many alternative questions that need answering. Get your students to explain to a partner what is and is not a quarter. Get them to draw pictures of what they think represents quarters and non-quarters. Try to spot misunderstandings that can then be shared with the whole class.

The annual NAPLAN tests are supposed to assist governments and schools to gauge whether students are meeting key educational outcomes. These results are supposed to help identify strengths or address areas that need to be improved.

Each year for the last 15 years I have been involved with a select group of colleagues in analysing the NSW Year 3 and Year 5 NAPLAN Numeracy results – initially in my role as Numeracy Project Officer for the Archdiocese of Sydney and lately in my role as Director, Maths Matters.

When we look at the NSW NAPLAN Numeracy results for Year 5, we double check every question for the exact curriculum outcome it matches in the NSW Syllabus. Often we find our decisions differ from the official ones which means we have a different proportion of Stage 1, 2 or 3 content outcomes in our summary. Each year we find that there are insufficient Stage below questions in both the Year 3 and Year 5 papers. We are puzzled by this. For example, in Year 3 we would expect more questions to relate to Stage 1 content. And in Year 5 we would expect more questions to relate to Stage 1 and 2 content. Surely governments want to check that most students have achieved the key Stage below content outcomes.

We next examine each Stage below question and determine if it is one which we expect 80% or more Year 5 students to correctly answer. We call these Core Stage questions. Another name could be “key content outcome”. Notice our 80% cut off still allows for one in five students who might be unable to answer correctly. They may be working at an even earlier stage perhaps. These Core Stage below questions focus on the key outcomes we expect students to have understood and also should be able to demonstrate by answering these questions correctly. If there are fewer students answering these Stage below questions correctly, they alert us to blockages that hinder student understanding. When we analyse the percentage selecting an incorrect answer, we are able to see common error patterns in student thinking. These then assist teachers to focus on specific key content outcomes. Teachers will clearly identify student blockages. The ideal result would be that all Core Stage below questions have 80% or more students answering correctly.

Some questions in this Year 5 paper might be Stage below, but they can include twists that we do not expect 80% or more students to answer correctly. We call these Advanced questions.

All this information is then put into a one-page grid. At first you will find the grid a little disorienting due to its unfamiliarity. We put the question number on the left and its matching result on the right. Questions are colour-coded for easy reference so you can see which Stage it is and also if it is a Core or an Advanced question. Question numbers that are underlined mean that question is an open one, not a multiple choice. Questions with an asterisk (*), mean that it is repeated somewhere in the Year 3 paper.

On the matching square on the right, we record the percentage of students who answered correctly.

• If this result has a green background then 80% or more Year 5 students in NSW schools answered this correctly.
• If the number is bold red, that means that it was a Core Stage below question that 80% or more Year 5 students did not answer correctly.
• If this result has a yellow background, that means that fewer than 50% of Year 5 students answered this question correctly.

For example, Question 5 on the left is coloured pink which means it is a Core Stage 2 question. The matching space on the right shows that 90% of Year 5 students answered correctly. Question 24 on the left is coded purple and the number is underlined. This means it is an Advanced Stage 2 question that is repeated somewhere in the Year 3 paper. The third matching space on the right shows that 48% of students answered Question 24 correctly. It is coloured yellow as this result is less than 50%.

We then include a second page summary. This shows each of the open questions (free response) and the results. We also show each of the repeat questions and the results for both Year 3 and Year 5.

We then show an overall summary which enables us to make big picture statements about how Year 5 students in NSW are tracking. Based on this summary we then provide detailed feedback on relevant questions. Your class or your school may be above, at or below these statistics. But what we look at is the biggest picture we can – the total number of Year 5 students in NSW who sat for that test. We show you the major trends in student misunderstanding. These statements help you identify where students need more support.

These are the major results from our analysis of the 2017 NAPLAN Numeracy test for Year 5:

## To see our complete summary for the 2017 Year 5 Numeracy paper, click here. It is in the column marked Term 3/4.

The annual NAPLAN tests are supposed to assist governments and schools to gauge whether students are meeting key educational outcomes. These results are supposed to help identify strengths or address areas that need to be improved.

Each year for the last 15 years I have been involved with a select group of colleagues in analysing the NSW Year 3 and Year 5 NAPLAN Numeracy results – initially in my role as Numeracy Project Officer for the Archdiocese of Sydney and lately in my role as Director, Maths Matters.

When we look at the NSW NAPLAN Numeracy results for Year 3, we double check every question for the exact curriculum outcome it matches in the NSW Syllabus. Often we find our decisions differ from the official ones which means we have a different proportion of Stage 1, 2 or 3 content outcomes in our summary. Each year we find that there are insufficient Stage below questions in both the Year 3 and Year 5 papers. We are puzzled by this. For example, in Year 3 we would expect more questions to relate to Stage 1 content. And in Year 5 we would expect more questions to relate to Stage 1 and 2 content. Surely governments want to check that most students have achieved the key Stage below content outcomes.

We next examine each Stage below question and determine if it is one which we expect 80% or more students to correctly answer. We call these Core Stage questions. Another name could be “key content outcome”. Notice our 80% cut off still allows for one in five students who might be unable to answer correctly. They may be working at an even earlier stage perhaps. These Core Stage below questions focus on the key outcomes we expect students to have understood and also should be able to demonstrate by answering these questions correctly. If there are fewer students answering these Stage below questions correctly, they alert us to blockages that hinder student understanding. When we analyse the percentage selecting an incorrect answer, we are able to see common error patterns in student thinking. These then assist teachers to focus on specific key content outcomes. Teachers will clearly identify student blockages. The ideal result would be that all Core Stage below questions have 80% or more students answering correctly.

Some questions in this Year 3 paper might be Stage below, but they can include twists that we do not expect 80% or more students to answer correctly. We call these Advanced questions.

All this information is then put into a one-page grid. At first you will find the grid a little disorienting due to its unfamiliarity. We put the question number on the left and its matching result on the right. Questions are colour-coded for easy reference so you can see which Stage it is and also if it is a Core or an Advanced question. Question numbers that are underlined mean that question is an open one, not a multiple choice. Questions with an asterisk (*), mean that it is repeated somewhere in the Year 5 paper.

On the matching square on the right, we record the percentage of students who answered correctly.

• If this result has a green background then 80% or more students in NSW schools answered this correctly.
• If the number is bold red, that means that it was a Core Stage below question that did not get 80% or more students correct.
• If this result has a yellow background, that means that fewer than 50% of students answered this question correctly.

For example, Question 11 on the left is coded olive green and the number is underlined. This means it is an Advanced Stage 1 question that is repeated somewhere in the Year 5 paper. The third matching space on the right shows that 67% of students answered Question 11 correctly. Question 15 on the left is coded lime green. This means it is a core Stage 1 question. We expect 80% or more students to answer correctly. If you look at the matching square on the right, you see that only 59% of students answered Question 15 correctly. This alerts us to pay attention to the incorrect responses to see what blockages may be revealed. As Question 15 is also underlined, this means it is repeated somewhere in the Year 5 paper.

We then include a second page summary. This shows each of the open questions (free response) and the results. We also show each of the repeat questions and the results for both Year 3 and Year 5.

We then show an overall summary which enables us to make big picture statements about how Year 3 students in NSW are tracking. Based on this summary we then provide detailed feedback on relevant questions. Your class or your school may be above, at or below these statistics. But what we look at is the biggest picture we can – the total number of Year 3 students in NSW who sat for that test. We show you the major trends in student misunderstanding. These statements help you identify where students need more support.

These are the major results from our analysis of the 2017 NAPLAN Numeracy test for Year 3:

## To see our complete summary for the 2017 Year 3 Numeracy paper, click here. It is in the column marked Term 3/4.

is not always an easy thing to explain to our students. Yet every human culture finds symmetry so attractive, perhaps because we humans are basically symmetrical, with the axis of symmetry travelling from our head to our toes.

An object has mirror symmetry if you can divide it into two matching pieces, like looking at a reflection in a mirror. We use the term “reflection” to describe the matching image. This is also call a “flip” in more colloquial language. An object has rotational symmetry if it can be rotated about a fixed point and parts match the new position exactly. For example, you can trace the outline of a shape on paper then rotate this shape around the centre point of the paper image. If this new position matches the original shape on the paper then this shape has rotational symmetry. It may match more than one time as it turns a full circle.

Not all things have symmetry. It is important to talk about what is NOT symmetrical too,so that your students build a clearer image, remove any blockages.

When I worked at The LEGO Centre in Drummoyne many years ago, we had over 70 000 children visit our LEGO play area. There we were able to observe that almost all the spontaneous LEGO constructions were symmetrical. Without anyone directing the children, they appeared to want their construction to be shape symmetrical, although colours did not always match.

This collection of photographs enables you to discuss a wide variety of objects and images with a special focus on symmetry. Where is the line of symmetry? Is there more than one axis? Are real-life objects always exactly symmetrcal? Do we call something symmetrical even though not every single piece matches exactly?

The National Gallery at Canberra has a magnificent installation by James Turrell, Within Without (Skyspace 2010). Visitors enter via a beautiful walkway across a water moat into a grassy square pyramid, with the top removed so that from  inside you are open to the sky above. You next encounter a large inner cone which you enter. Inside this cone there are seats around the edge and this time there is a large central circle cut from the dome’s ceiling, revealing the open sky above. Just before sunrise and just before sunset a special lightshow reveals a magical world of colour. Your eyes watch the sky which slowly becomes a pure black circle, even though the sky when you go outside is still in its evening shades of grey and blue. A must-see event if you happen to be near the NGA at sunrise or sunset. It is open to the public and is a free event. Students of all ages love it. We met one local family where the children say to their parents on a lovely clear afternoon that it is Turrell Time and they head over to see it in its sunset glory once more.

James Turrell is an American installation artist who works directly with light to create his artworks. He was originally a perceptual psychologist. “Turrell’s over eighty Skyspaces, chambers with an aperture in the ceiling open to the sky … (where) … the simple act of witnessing the sky from within a Turrell Skyspace, notably at dawn and dusk, reveals how we internally create the colors we see and thus, our perceived reality.”

“My work has no object, no image and no focus. With no object, no image and no focus, what are you looking at? You are looking at you looking. What is important to me is to create an experience of wordless thought.” (James Turrell)

Another conundrum.

If you just heft two objects in your hands, like in this picture, it is easy enough to describe the objects as either lighter than, the same mass as or heavier than each other.

But is becomes a more challenging task when you look at a diagram and need to read something important in it. For example, in the diagram below, you need to see that the object on the left pan balances the 5 objects on the right pan. That tells you something. That tells you that 5 tomatoes have the same mass as one bottle. Or it may even tell you that one tomato has the same mass as one fifth of the bottle.

If you analyse what you see in the two diagrams below, this time you see that 5 small yellow balls are heavier than a large yellow cube. But the second diagram tells you that those same 5 small yellow balls are actually lighter than the large yellow sphere. Both diagrams need to be looked at to work out which object is the heaviest – the large cube? The large sphere? Or the 5 small balls?

In a recent numeracy test, 70% of Year 3 students were correctly able to identify the least heavy mass from a sequence of diagrams. Yet last year, in 2016,  only 67% of Year 5 students could correctly identify the heaviest mass from a similar diagram. After two more years of school. ARRGGGHHH!

Does this mean the Year 3 students are dramatically improving their understandings? It’s so difficult to know how to respond to these results.

The test diagrams show a set of balance pans with one only object in each pan. Some show the objects as equal in mass. Some show the objects as lighter or heavier than each other. You need to use your mass logic to look at each diagram, understand which object is heavier and then transfer this realisation to the second or third diagram. In other words. You need to go beyond your first response, your first diagram.

In this year’s test 11% of Year 3 incorrectly selected their first response. What they selected was correct if there was not a second diagram giving you new information. In other words it was their problem-solving skills that let them down. For these one in 10 students, they need to practice perseverence, rereading the question, making time for the problem to become real, making time for them to use their correct understandings of balance pan diagrams. They do NOT need to be taught how to read a balance pan. Their mass thinking is correct but they just didn’t carry it on far enough.

In last year’s test 27% of Year 5 incorrectly selected their first response. Again, it was their problem-solving skills that let them down, not their ability to read meaning into a mass diagram. This is reinforced by the very small number of students who selected totally incorrect answers from the multiple choice. So 67 + 27 = 94. That’s potentially 94% of Year 5 students who could have been correct – a much more reasonable expectation for Year 5.

So, as teachers, we need to think carefully about what test results actually reveal. We should place more emphasis on developing problem-solving skills than reteaching known mass concepts. All students need time to solve a multi-step problem (of course not in a test situation …). They need to reread the question to see that they need to go further than their initial reaction. They need to use logic when they think about their answers. They need to develop pride in being an effective problem-solver.

What’s going on?

In 2016 a large cohort of Year 5 students had lots of trouble correctly identifying how long a length measure was when the cm tape measure didn’t start at 0. Only 42% of students surveyed answered correctly (5 cm). This year, in 2017, Year 3 students are overwhelmingly successful at correctly identifying a similar measure using informal units. Too many swings and roundabouts for me.

When you analyse the incorrect responses, 46% of these Year 5 students selected the answer which was just one more than the correct answer (6 cm). In other words, they counted the 6 marks on the ruler, counting up from the first mark as 1 rather than 0. What on earth do they think length is about? The ruler measures off units of length and the marks in themselves are not the units. Perhaps too many Year 5 students have experienced pen and paper experiences? Or had too strong a focus on the ruler or tape measure marks rather than the actual length of a unit.

So what do we discover when we analyse the Year 3 response to a similar question? This year’s question was open-ended rather than multiple choice. But in itself this doesn’t explain why so many more students were correct. In fact this year’s question was more difficult as it involved far more units (13 units and last year there were only 5 units). We don’t know how many students answered 14 units, but 81% of Year 3 students answered correctly anyway. This is a dramatic change in behaviour and understanding with the 2016 Year 5 cohort.

I believe that there is a glaring explanation. This year’s question talks about measuring with blocks placed end-to-end. In other words, students are virtually told how to interpret the ruler in the diagram. They are told to see it as blocks in a line. So of course they don”t focus on counting up the marks (which would include the mark at the start to get 14).

This explanation has major implications for how we discuss length units with all students. We need to continuously reinforce that length represents units like blocks. We need to talk about the misunderstanding that creates the blockage for the majority of older students.

Length is not just a measure of marks on a ruler. When your feet are together at the start of a number line you have not moved. You are at 0. It is only when you jump a unit of length that you record 1. This concept needs reinforcing with all primary age students. A ruler is conveniently marked to show where these length units reach. Whenever we begin to measure though, we always start at 0. Even if the ruler shows an object starting at 27, that starting point should now represent 0 in a new counting sequence. If an object starts at 108 on a ruler, 108 should now be seen as 0 in a new counting sequence. Even if an object starts at 389 on a tape measure, 389 should now be seen as 0 in a new counting sequence.

Let’s hope that we can make these length understandings part of all our student’s spatial visualisation skills.