A quick read about the ancient mathematicians who searched for pi.

**https://www.exploratorium.edu/pi/history-of-pi**

A video explaining how to calculate pi using a dart board.

Uses 10 chosen notes to play the first 10 000 digits of pi as a musical sequence (requires Flash).

**https://avoision.com/experiments/pi10k**

A video showing real-life instances of pi and fibonacci.

Search the first 200 000 digits of pi for any string of numbers, such as a birthday.

**http://www.angio.net/pi/bigpi.cgi**

An article showing some fascinating pi-inspired data visualisation.

Exploratorium has a list of hands-on activities for learning more about pi.

]]>Little Miss Three Year Old had grabbed a hold of her mother’s purse and emptied the clinking, shiny objects into mum’s lap. Rather than discouraging her from rummaging through her purse, her mother immediately turned her daughter’s curiosity into an imaginative game. Having allocated the roles of customer and cashier to herself and her daughter, she then asked the child which combinations of coins could be used to make up a dollar. What followed was about 10 minutes of happy chatting back and forth over different combinations – which were ‘too much’, which were ‘not enough’ and which were ‘just right’.

In short, this young mum had taken her child’s natural curiosity and spontaneously created a powerful learning opportunity. He daughter provided the imaginative instinct and she provided some structure, simple mathematical language and a willingness to ‘play’ with her child.

The power of playful interactions between adults and children around math concepts in early childhood is often underestimated. As parents, we readily engage with our children in reading and literacy. Less often do we intentionally interact with young kids in mathematical activities.

Yet, a substantial body of research has demonstrated the value of children’s imaginative play for promoting cognitive development as well as social cooperation and interpersonal understanding (Nicolopoulou 2010, p.2). When it comes to the development of early numeracy, this is most effective when children have adults who are willing and able to provide challenges, structured scenarios and appropriate mathematical language. Combining free play with ‘intentional teaching’ (for example, explaining to a child that shapes with three straight sides are called ‘triangles’), and promoting play with mathematical objects and ideas, provide powerful boosts to childrens’ early numeracy skills (Clements and Sarama, 2014).

As adults we also often tend to underestimate the complexity of children’s games. However, playing games – whether structured (such as board or card games with rules) or unstructured (such as imaginative scenarios and role playing) – can require lots of critical thinking and problem solving (Reiber 1996, p.52).

There is research to suggest that children who regularly engage in mathematical play in early childhood (such as construction with blocks and shape puzzles) may also reap longer term benefits (Wolfgang, Stannard & Jones 2001, p.178). However, these effects are more evident in children who have had parents or teachers ‘scaffold’ their learning through intentional teaching and by introducing appropriate mathematical language and concepts (Fisher et al 2013, p.1877).

**A recent article by Daniel Donahoo (ABC News, Australia) provides some practical tips for ‘nailing play’ with your child, each of which are directly applicable to kick-starting their early numerical development.**

Spending time each day in simple mathematical play will introduce your child to a range of valuable number, shape and logical skills and concepts. However, it will also have the powerful advantage of fostering a ‘mathematical mindset’ in your child, in which she understands mathematics to be fun, engaging – and playful.

** – Marcus Garrett**

**References:**

Clements, D.H. and Sarama, N. (2014), *‘Play, Mathematics, and False Dichotomies’*. Blog post on __Preschool Matters Today__, A blog of the National Institute for Early Education Research. Accessed 12/12/2017. URL: https://nieer.wordpress.com/2014/03/03/play-mathematics-and-false-dichotomies/. NIEER, Rutgers University : New Jersey, US.

Donahoo, David (2017), *‘Five Ways to Nail Playing With Your Child’*. Article, ABC News, Sunday 10/12/2017. URL: http://www.abc.net.au/news/2017-12-10/5-ways-to-play-with-your-child/9240134. Accessed 11/12/2017.

Fisher, K.R., Hirsh-Pasek, K., Newcombe, N., Golinkoff, R.M. (2013), *‘Taking Shape: Supporting Preschoolers’ Acquisition of Geometric Knowledge Through Guided Play’*, in __Child Development__, November/December 2013, Volume 84, Number 6, Pages 1872–1878. p.1877. DOI: 10.1111/cdev.12091. Society for Research in Child Development : Washington, US.

Nicolopoulou, A. (2010), *‘The Alarming Disappearance of Play from Early Childhood Education’*, in __Human Development__, 2010:53, pp.1-4. DOI: 10.1159/000268135. Karger : Basel, Switzerland.

Reiber, L. (1996), *‘Seriously Considering Play: Designing Interactive Learning Environments Based on the Blending of Microworlds, Simulations, and Games’.* In __Educational Technology Research & Development__, No. 2, 1996, pp, 43~58. DOI: 128.250.144.144 on Tue, 12 Dec 2017 23:13:46 UTC.

Wolfgang, Charles & L. Stannard, Laura & Jones, Ithel. (2001). *‘Block Play Performance Among Preschoolers As a Predictor of Later School Achievement in Mathematics’*. __Journal of Research in Childhood Education__. 15. 173-180. 10.1080/02568540109594958.

**Why?**

Teaching decimals can sometimes be a difficult concept to teach and like many topics, students can often possess a number of misconceptions in this area. This lesson has been designed to address many of these misconceptions such as whole number thinking, longer is larger and shorter is larger. It introduces students to decimats and is designed for use with middle and upper primary students.

**What?**

The lesson introduces students to the decimat where they begin to make sense of decimal size and decimal place value as well as use fractional language to describe decimals. A decimat is a proportional model that represents the size of decimals as part of a whole. It provides students a visual representation and the way the lesson is designed, allows students to consolidate their values as a decimal, fraction and representation.

The decimat consists of a large rectangle representing one whole. This is then broken up into ten parts representing tenths. Each tenth is also broken up into ten parts representing hundredths and each hundredth into thousandths.

The students then shade various parts of the decimat to show their decimal values.

When using decimats, students are always encouraged to write the value as a decimal as well as a fraction in order to consolidate the understanding of decimal place value.

**How?**

Before introducing students to the decimats game, assign them the decimals comparison test in order to determine what, if any, misconceptions students have about decimals. It is important to iron out the common misconception of longer is larger and both the test and the task will work together to help students overcome this.

You may introduce the decimats to them slowly by only providing them with one representing tenths, or tenths and hundredths and once they have shown competence with these, you can then introduce the full decimat. When students are doing the task, it is important for them to complete each turn with a different colour in order to ensure that they can keep track of each turn. This will also make it easier for both the students and the teacher to check the work for any errors. Ensuring that the values are written both as a decimal and as a fraction is also extremely important because it establishes the link between the decimal and its equivalent fractional value.

After confidence has been established, encourage students to begin further dividing their tenths into hundredths and thousandths to continue to build understanding.

The Decimats game is designed to be used for an entire lesson and not just as a warm up. The more it is played over a decimals unit, the greater the likelihood that they will more easily make sense of decimals. Once the students are familiar with this model, they can then use it to compare the size of decimals by shading blank decimats and comparing the area.

**Dig Deeper**

For more information, the attached lesson plan shows the relevant links to the Australian Curriculum and highlights the related content and outcomes as listed in the Australian Mathematics F–10A syllabus.

**Why?**

This lesson has been designed for students in Upper Primary. It builds on the problem “Four Goodness Sake” found on the NRICH website (https://nrich.maths.org/1081). It is aimed at helping students develop an increased understanding into the order of operations and how the use of brackets in an equation can modify the solution.

**What?**

The lesson activity requires students to use up to four fours and any operation to make the numbers from 0 to 100. For example, 4 + 4 + 4 + 4 = 16.

Initially, students will use the more familiar operations, +, −, ×, ÷, along with brackets to create their equations. For example, (4 × 4 + 4) ÷ 4 = 5. Gradually, finding solutions for the remaining numbers in the list becomes more challenging. Here, it is a good opportunity to introduce some other operations to the students, including √4 = 2 and 4! = 24. In my experience, at least one student in the class will be familiar with the square root operation √*n*, which will make it easier to introduce. The other operation *n*!, known as *factorial*, will be less familiar to students.

*n*! is the product of all positive integers less than or equal to *n; f*or example, 4! = 4 × 3 × 2 × 1 = 24, or 3! = 3 × 2 × 1 = 6

With the introduction of these two less familiar operations, the number of solutions that students can find to the initial problem increases.

**How?**

Recently, in a 5/6 class in NSW the students were quick to understand the initial problem. After naming some of the more obvious solutions, the challenge soon came to find solutions for the numbers from 1 to 10. We decided to use the large classroom blackboard to record the solutions as they were suggested. On the side of the board a “working out” space was included so problems where the “order of operations” was a factor could be tested before being added to the main list.

Although several solutions larger than 10 were also discovered, the number 10 itself became a bit of a sticking point. It was here that I prompted students to think of using different operations. One student suggested , but soon realised that this would not be allowed as it involved using a number other than 4. Another student said that the opposite to “squared” was “square root” and soon a new operation was available to the students. This “new” knowledge was then used by students to provide multiple solutions for the number 10, including:

4 + 4 + 4 – √4

4 + 4 + √4

After another five to ten minutes of working, when the sharing of possible solutions had again slowed down I introduced the factorial operation (. Not only did this help students find solutions to some different numbers, it also helped them to develop less complicated solutions to some of previous numbers. For example:

4! ÷ 4 = 6

With this new information in place, over the course of one hour, the class of students were able to develop solutions to more than 30 numbers. As the lesson was finishing, students were still eagerly sharing potential solutions.

**Dig deeper**

For more information, the attached lesson plan shows the relevant links to the Australian Curriculum and highlights the related content and outcomes as listed in the NSW Mathematics K-10 syllabus.

Are you a good mathematician? How much maths do you use in your daily life?

For many of us, our initial responses to these questions may be “No” and “Not much,” yet when we think about it maths is something that we all use every day, although we may not realise it.

The problem is that the maths we do every day has become so familiar to us that we may not even see it as maths anymore. One of the biggest areas of maths we take for granted is estimation. We use estimation in almost everything we do, from making a cup of tea, to cooking food, to having a shower. These simple daily tasks all include elements of mathematics.

So, what has happened? Why is there a disconnect between the maths we do every day and the perception surrounding the maths that is done inside classroom?

At a recent Kinder Transition meeting at Scone Public School, I spoke to parents about this idea. As a general groan reverberated around the room at the mere mention of maths in school, it seemed that parents present did not see themselves as mathematicians, with many going further to suggest that they “hated maths at school”.

In some ways, this reaction is not surprising. For many of us, the maths that we may remember from school, is no doubt the more challenging maths from our senior secondary years. Complicated formulas and pages of calculations ensured that our final experience of mathematics in school left many of us feeling anxious and confused.

Hope is not lost. The maths we remember from school is not the maths we learnt about in our first years of school. Nor is it the maths currently being explored by students today in the classroom.

Students in their first years of school learn about numbers, counting, shapes and location. They order objects and collections, and create and continue patterns. They investigate problems, make predictions, use trial and error and talk about their discoveries.

As parents we need to be thinking less about the drilling our child on their “timestables facts” and more about exploring the maths in our everyday lives. This is exactly what my talk with parents, “Finding the Maths” was all about. Rather than purchasing a special book, program or application we just need to look for the maths around us.

We can find maths in books, in the kitchen, in the bedroom, in the bathroom, outside, down the park, in the street, in the playground and walking around the shop. Activities that involve time, money, ordering, location words, going for walks, following directions, counting collections, estimating size, looking for patterns, sorting objects into categories, playing card and board games, using construction materials, building with blocks and completing jigsaws; all involve elements of mathematics.

There is a quote, “If we change the way we look at things, the things we look at change”. The challenge for parents is not to share our anxieties about the maths we remember (or are trying to forget) from secondary school. The challenge is to realise that the maths we take for granted, the maths skills we use every day, are the very tasks that can help establish an inquisitive approach to mathematics and lead to a love of maths with our children.

]]>This was a reply I received recently, after running a class on time for a year 3 class. The teacher and I had spoken briefly before the lesson and she mentioned that the class were struggling with the concept of to and past the hour. This had been the concept covered by the text book the day before. As we talked we both agreed that we shelve the planned lesson on addition, and revisit time.

Reaching in to my kit bag I pulled out a packet of circle papers for the students to make clock faces with.

*Folding a circle in half and opening it up again, *“what goes at the top?”

“The o’clock”

“And what does o’clock mean?”

“…”

“Does anyone know?”

*Shaking of heads*

“Ok. It stands for ‘of the clock’. So 12 o’clock is the hour 12 of the clock, and so 1 o’clock is …”

“1 of the clock”

“And 2 o’clock is?”

“2 of the clock”

“Wonderful, so what do we call it when the minute hand is down the bottom?”

“30 minutes”

“Or …?” *Folding the paper along the crease and out again a few times,* “what is this fraction?”

“Half.”

“Half what?”

“Half past.”

“Yes. Or more completely, half past the hour of the clock.”

There were a few “oh”s and “wow”s at this point.

“Now,” *folding the circle to show just the upper right hand quadrant,* “what time is this?”

I should also point out that I was choosing a different student to answer each question and selecting from both those who had raised their hands and those who hadn’t. In each case I did not move on until the chosen student had given me an answer.

“A quarter…” “To.” “Past.”

And here was the threshold, or liminal point, of their confident knowledge. There is only one “o’clock” every hour, and only one “half past” but there are two quarters? Many young children do not discern the difference between the two.

“Hmm,” *raising my arm so that the elbow is at chest height and my hand is in front of my face,* “what is the hand doing?”

“Moving”

“Is it moving to or from?” *The first time I ask this question I don’t provide a reference point. I wait a few seconds and then elaborate,* “Is it moving up to the hour or down past the hour?”

“Down past!”

*Holding up the quarter circle, *“so this is?”

“A quarter past!”

“Past what?”

“The o’clock” and “of the clock” are both heard.

*Quickly flipping the quadrant around,* “and this is?”

“A quarter to”, then less loudly “the o’clock”

“Wonderful! Next then, what else is half past called?”

“…”

“How many minutes in an hour?”

“60”

“So how many minutes is half past?”

“30”

“And a quarter past?”

“15”

“And a quarter to?”

“45” and a couple of unsure “15”s.

“How many to the hour?”

“15” more confidently and supported around the room by nods.

*Moving to the whiteboard and drawing a circle,* “what numbers go between the o’clock and the quarter past?”

At this point I look around and comment to the teacher that there is no clock in the room for the students to use as a reference point. This is neither good nor bad but does help to explain the puzzled looks. A little more prodding elicits the appropriate responses and it is now time for the students to fill in the remainder of the clock faces by themselves. This allows for a couple of minutes to chat with the teacher about where to next.

Where to next is suggesting that smaller circle paper is now used in the same way to model the hour hand. The teacher immediately indicates that she has some and collects it from a storage tub. A few more questions and the students are able to complete the second clock face as well. All that remains is to guide the reflection time by asking the class what they learned during the lesson. The previous sticking points are all gone and a few added pieces of vocabulary and insight.

I do not expect any teacher reading this to be surprised by the materials used. They are standard for primary classrooms. Further the only point of difference between my approach and the regular teacher of this class was to add a second “face” to highlight the different numbering systems for minutes and hours. So why then did she find observing the lesson to be so valuable?

It begins with her own ability to recognise that even though she and the students had already worked through a lesson on the same material they had not reached a point of understanding. Then, having an opportunity to collaborate with another teacher and the confidence to postpone the planned lesson in favour of review and consolidation allowed the class to dig deeper into the meaning of how we measure time.

From this point on, everything about the lesson was centred on the students. Every question stepped out the path from general to specific knowledge and the responses were all genuinely those of the students. Thinking time, both for the students and the teacher asking the questions facilitated dialogue. Silences indicated confusion at one point, concentration at another and dawning awareness at a third. The feedback was in real time and paced to bring the entire class along together. At points where some students were still confused, their classmates chatted with them and retraced the most recent movement until both could continue.

Of course the students will now need to cement their understanding and improve their fluency over coming weeks. One way to do this is to explore the origins of clocks and the word clock. Painting the picture gives greater vibrancy to the image than just sketching it. In this case the painting includes history, etymology and science as well as maths. Go to http://blog.onlineclock.net/clock-word-origins/ to ick out the colours for the palette.

]]>From time to time we share research and articles written by people outside of AMSI. This article is the result of research conducted by the Graduate School of Education at The University of Melbourne. We thought their findings were thought-provoking, and had messages for classroom practice. This is post 3 of 3, about the challenge of recognising algebraic expressions when different letters are substituted.

**Materials:**

- A pack of cards (with face cards left out).

**WALT:** Use inverse operations to find numbers.

**WILF:** Using known elements to find the unknown (algebra).

** Game Objective:** To work out your number.

This game is to be played in groups of three (one dealer and two players rotating the roles).

- The dealer gives each player a card which they hold on their forehead without looking at it.
- The dealer tells the two players the sum, product, or difference of the two numbers.
- Players must work out what their card is, by listening to the total and looking at the other player’s card.

**Variation:**

For more of a challenge, play with three players and a dealer.

**The NSW Numeracy Continuum is a highly useful diagnostic tool which can be used to observe students’ progression of understanding of essential aspects of numeracy, as they move from pre-Kindergarten (Foundation) through to the middle years of schooling.**

Many NSW schools develop ‘I Can’ statements to supplement their use of the Continuum, so that they can work with students and their parents when assessing and developing their skills and moving them forward in each important ‘Aspect’ of numeracy.

AMSI Schools have developed a series of ‘I Can’ statements, mapped against each Aspect on the NSW Numeracy Continuum. These ‘I Can’ Statements have been updated as at November 2017. AMSI Schools will provide revisions in line with the new ACARA ‘National Numeracy Progression Framework’, currently being piloted in NSW schools and due for release to schools in 2018. You can access these here by downloading each Aspect’s ‘I Can’ Statement in PDF format.

We would stress that it is important for teachers to use ‘I Can’ statements in conjunction with the Continuum document itself, and not in place of it. These simply provide a less technical explanation of what students might demonstrate at each level on the Continuum. Teachers and schools should ensure they are directly referring to the more detailed diagnostic descriptors in the Continuum document, as well as referring to the NSW Mathematics syllabus, when formally assessing children’s progress in mathematics.

The NSW Department of Education & Communities also provides an excellent support site for educators using the Numeracy Continuum: **http://www.numeracycontinuum.com/.**

From time to time we share research and articles written by people outside of AMSI. This article is the result of research conducted by the Graduate School of Education at The University of Melbourne. We thought their findings were thought-provoking, and had messages for classroom practice. This is post 2 of 3, dealing with the difficulties students face in interpreting the meanings of symbols they have met before. Let us know what you think!