When I think back over my memories of mathematics in school one of the most prevalent was of sitting in grade six class frantically trying to complete a worksheet full of arithmetic questions while the teacher timed us to see how many we could complete in the given time. I still remember that feeling of panic as I tried my hardest to get through those questions. To say it was traumatic may be overstretching but given what we know today about maths anxiety or as Stanford University Professor Jo Boaler puts it ‘maths trauma’, I am safe in saying that it definitely helped shape my relationship with maths for years to come.

Thankfully today, that focus on rote memorisation is, or should be, a thing of the past. What many mathematics educators know about maths learning is that it can no longer be looked at as a set of rules and procedures to be memorised, but rather a system of meaningful relationships to be investigated and explored.

Luckily for me, I also had some amazing maths teachers and lecturers who both encouraged and fostered my love of the subject in later years. Sadly however, for many kids and adults, this type of rule based learning and memorisation has caused long term damage and has played a large role in creating maths anxious and disaffected maths learners (Boaler, 2015).

The good news is however, that there are ways that we as educators can help our students by building in them something called *Number Sense*.

Number sense is a relatively new term in maths education. It refers to the ability for students to work flexibly and conceptually with numbers. Research has shown that students who experience the greatest success in maths generally have a deeper, more intuitive understanding of numbers and how they relate to each other. They are able to view them conceptually and have developed strategies to adapt them to solve various situations. Simply put, they have good number sense.

Number sense is referred to in a range of mathematics literature. Books such as *Number Sense Routines* by Jessica F. Shumway and Jo Boaler’s *Mathematical Mindsets* talk extensively about how developing a strong ability to work flexibly with number is crucial in maintaining a deeper connection with maths concepts in later years. However, if I was to choose one book that encompasses this idea of developing strong number sense it would have to be Sherry Parrish’s *Number Talks*.

*mental maths skills*, an important facility for the reduction of the cognitive load when tackling heavy maths problems. During a number talk, students are encourage to justify their thinking while communicating the solutions to the problems solved mentally.

- Students sit either on the floor or at their tables generally facing the front of the room.
- There is no calling out or hands up during a number talk. Instead, students make a fist and place it on the front of their chest.
- As the teacher, you present them with a question. You can say something like “Today we will be discussing the solution to 16 x 25”.
- Give the students time to come up with an answer and to think of a way to communicate the solution to the class.
- When students have a solution to the question they put a thumb up. This allows you, the teacher, to see who has an answer and who has not. The aim of the thumbs up instead of hands up concept is to give everyone an opportunity to solve the problem in their own time without the distraction of who has solved it the fastest.
- While they are waiting, encourage the students to find more than one way to get to the result. Once they have discovered another way they can now also put a finger up. The more ways, the more fingers they hold up to their chest.
- Once you are confident that most students have at least one solution, post them on the board for consideration. Incorrect solutions provide a great opportunity for conversations surrounding common misconceptions, so encourage these as well.
- At this stage, you can ask students to share their strategies for arriving at their answer. Place each strategy on the board along with the students name in order to recognise the individual child and identify their strategy.

In her book, Sherry also encourages the use of open arrays when communicating solutions in order to provide a visual understanding of each strategy used. There has been much literature on the importance of arrays in developing multiplicative thinking in students and they are a vital concept in learning multiplication and division. Their repeated use in a number talk helps to consolidate these important skills.

For the junior primary level, number talks can be done in the form of a *dot talk. *Helping students recognise quantity at an early level through subitising is a vital skill and this is encouraged through a dot talk. The basic rules are the same, but instead of presenting students with a numerical problem, a dot card is held up and students are asked ‘how many dots?’, as well as stating the different ways that they saw the collection. Again, these are all presented on the board and open for discussion and communication.

By immediately recognising a collection of numbers or subitising, students start to understand how a number is made up. This understanding of part-whole relationships helps children to separate and combine numbers and accelerates learning in addition and subtraction. An example of a dot talk is shown below.

Five key components to doing a successful number talk are discussed in the book. These are:

- Classroom environment and community
- Classroom discussions
- The teacher’s role
- Role of mental health
- Purposeful computation problems.

In short, when doing a number talk, it is important to foster a cohesive classroom community where students feel safe enough to offer responses for discussion, question themselves and investigate new strategies. Communication is key and this should be encouraged and supported. Once students have practiced a few number talks, they will soon start to discover new experiences in mathematics and so many different ways of solving the same problem. In fact, I have not done a number talk where a student did not make the comment “I never thought of doing it that way” or, “I’m going to try that next time”. Over time, you will find that with regular use, number talks will dramatically improve their ability to access the mental maths strategies necessary to become facile maths learners. They also provide teachers with a great opportunity to establish the level of numerical understanding of each student.

** **

Boaler, Jo ** ***Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts. *2015

Flick Michael and Kuchey Debbie**. **“Contest Corner: Increasing Classroom Discourse and Computational Fluency through Number Talks”. *Ohio Journal of School Mathematics, Spring 2015, Vol. 71. Pp. 38-41 *

Shumway, Jessica F. 2011. *Number Sense Routines*

Boaler, Jo 2016. *Mathematical Mindsets*

Parrish, Sherry 2010. Number Talks: Helping children build mental math and computation strategies.

Number Sense – Jo Boaler: https://www.youcubed.org/resource/number-sense/

https://www.nesacenter.org/uploaded/conferences/SEC/2013/teacher_handouts/CairaFranklin.pdf

http://brownbagteacher.com/number-talks-how-and-why/

https://www.yellow-door.net/blog/what-is-subitising/

http://www.mathcoachscorner.com/2013/07/using-dot-cards-to-build-number-sense/

http://www.mathcoachscorner.com/2013/07/using-dot-cards-to-build-number-sense/

The students began their day with a welcome to country and then a ‘speed dating’ careers session with what AMSI refers to as their ‘regional champions’. These are professionals that live and work in the local area in careers ranging from nurses and electricians, to radiation and hygiene specialists. The students spent ten minutes speaking with each of our eight regional champions about what makes up their career, their maths journeys and all the places where maths is crucial in their professions. It was fantastic to see all the students so engrossed in the stories and pathways that these amazing people followed to get them to where they are today. It was a highly beneficial and informative session by all involved and we thank each of our champions for taking time out of their days to add value to the students in the region.

The two keynote speakers at the event were Aliza Wajih, a graduate mechanical engineer with BHP currently stationed at Olympic Dam and Squadron Leader Darren Prior from the Department of Defense working out of the RAAF Base at Woomera. Both were exceptional speakers and had the students glued to their seats with all the information about their careers and the engaging mathematics and technology that they utilise on a daily basis. It is a telling sign of engagement by the students when we have to wrap up the speakers because all of the questions they had pushed us well over our time allocation. Both Aliza and Darren did a fantastic job, and everyone appreciated their significant efforts on the day.

Along with the speakers, we had a number of activities running. In the first activity, the students were constructing a directional projectile launcher with a clinometer attached so that they could measure the optimum firing angle of a projectile. They then used the finished product to discover what this optimal angle would be. The students had a lot of fun with this activity and made many mathematical discoveries along the way. It was great to see them all engaging in some practical maths and then hearing about how enjoyable they thought it was. As a fun wrap up activity, the outreach officers running the event then held a competition where each group used the information they acquired during the activity to see who could launch their projectile the furthest. Based on the excitement that ensued during this activity, it may be safe to say that this was the highlight of the day.

After lunch all the students were brought together for a reflection and feedback about the day and it was great to hear how positive they thought it was. Many students stated that it had made them rethink their choices for what maths subject they would choose moving forward and that they would consider choosing a harder one which was wonderful to hear.

Overall, it was a very successful event and discussion has already begun about the 2019 CHOOSE**MATHS** Day. AMSI would like to thank all of the staff, students and members of the local community for making the day a triumph and we look forward to continuing the great relationship between CHOOSE**MATHS** and the schools within the local regions.

Not that unusual, you may say; students attended maths camps all over Australia, and you would be right. What is special about this camp is the distance the students and their teachers travel to participate. The group that was running late was from Newman Senior High School (NSHS), the small Pilbara mining town, 1,186km north of Perth and 610km by road from Karratha. A noise started in the bus and became more and more insistent, so the decision was made to pull over to be on the safe side. They hadn’t arrived in Karratha yet and they still had to get home again in time for school on Monday. The driver called for roadside assist. SLIGHT PROBLEM; the call center is in Melbourne, over 3000 km away. Do you know how difficult it is to explain that you are sitting on the side of the road, somewhere between Port Hedland and Whim Creek, with something wrong with the vehicle but you’re not sure what, to someone on the other side of the country, who probably has no idea of where Port Hedland is let alone Whim Creek?

The other schools present at the maths camp, beside the host Karratha Senior High School (KSHS), were Exmouth District High School (EDHS), 550km to the south and Hedland Senior High School (HSHS), a mere 230km to the north. Everyone brings their swags and camps out in classroom on Friday and Saturday nights. I will be honest and admit I didn’t; I slept in a proper bed at a local motel. Fortunately, the NSHS contingent arrived safely, hungry, but no worse for the delay, before bedtime. When I arrived back at KSHS, at 8 am, to set out the first activity for the day, an orienteering activity, bleary-eyed teachers were shepherding students off to breakfast, or organizing their sleep rooms. Bleary-eyed teachers, who had been awake half the night, telling students to go to sleep but of course many students were just too excited to sleep. How come they’re not bleary-eyed?

After the orienteering activity, the students went chocolate-chip mining, which is always fun. Watching groups try and work out the best way to extract as many chocolate nuggets as possible while only using their mining tools and staying within their mining claims, is always interesting; some don’t care about damage and destruction, just get the ‘gold,’ while others are much neater. The second and third round made it even more difficult for the smash and grab groups when they were rewarded for least amount of damage to their mining claim; mind you there was a lot of ‘Quick, sweep the crumbs back into the circle,’ going on. (If you’re interested in trying chocolate chip mining with your class, just do a Google search as there are several resources available on line.)

After lunch, the out-of-towners all got a chance to visit the local shopping mall. May not sound very exciting to you but when the population of your community is only 2500 (Exmouth) or 4600 (Newman), it’s pretty thrilling. The Karratha kids got to have some downtime, which, in today’s ‘keep ‘em busy’ world, was probably enjoyed just as much.

The maths activities continued later in the afternoon and the evening with different teachers running them. Teachers and leadership from KSHS dropped in throughout the weekend and parents and staff were present for the presentation at 9 o’clock on Sunday morning. HSHS’s team LEAP won with the greatest number of points overall, but HSHS Team Standish and EDHS team The Finger Lickin’ Goons were only 2 and 3 points respectively behind. Jess Wolf from EDHS was awarded the Most Valuable Player prize for the weekend.

Then the journeys home began. NSHS was planning to home by 5 pm, hopefully with no delaying incidents along the way.

]]>*Thanks to teachers from Karratha Public School.*

**Learning Objective:**

Give clear directions, follow directions and use coordinates to locate places on a grid.

**Intended Outcome:**

Following directions accurately.

**Materials:**

- If playing option one, masking tape.
- If playing option one, an assortment of random objects.
- If playing option two, an A4 sized 4 x 4 grid.
- If playing option three, an A4 sized 4 x 4 grid for each student or student pair.
- If playing option four or five, an A4 sized 6 x 6 grid for each student.
- If playing option four or five, an A–F die.
- If playing option four or five, a six-sided die.

**Instructions:**

First, mark out a 4 x 4 grid on the classroom floor using the masking tape. Once this is done, there are five game options to play:

**Option 1:**Place some obstacles on various squares on the grid. Students give directions to one player to get from one side of the grid to the other without hitting the obstacles.- e.g. Starting at the green triangle, take one step forward, turn to the right and take one step, turn to the left and take one step, turn right and take two steps, turn left and take one step to finish on the red triangle.

- This can also be done using compass directions (north, south, east and west).

**Option 2:**The teacher has a hidden copy of the grid that shows a path through the grid.- The students must find their way through the grid by guessing where the correct path lies. When the student takes a step on the grid, the teacher must say “yes” if the student is on the correct path and “no” if they are not on the correct path.
- When the student takes a wrong step, their turn is over and the next student has their turn. Other students can watch to learn where the path lies, and give directions to the student whose turn it is.
- See how many tries it takes to get one student through the grid.

**Option 3:**The teacher calls a series of directions that the students must follow to trace the path on their grid. See if they all end up at the same place!- This could be played using directional navigation (left, right, forwards, backwards), compass bearings (north, south, east, west) or grid coordinates.

**Option 4:**Use a 6×6 grid and label the x-axis with the letters A–F and the y-axis with the numbers 1–6.- Students then colour in ten random squares on their grid.
- Using an A–F die and a six-sided number die, call out the coordinates as they appear on the dice when rolled. If a student has the called coordinate, they mark it off. The first student to have five coordinates marked off wins. Or, alternatively, the first student to have all ten of their coordinates marked off wins.

**Option 5:**This game is played similarly to option four. However, as well as labelling the x-axis with letters A–F and the y-axis with numbers 1–6, the student also fills up the entire grid with numbers that belong to up to four fact families (e.g. 2,5,10…. 3,4,12…. 4,8,12…. 5,10,15….). These numbers can be written in the grid randomly.- Again using an A–F die and a six-sided die, students mark off the coordinates called. The student who is first to tick off an entire fact family wins – if they can tell you the four facts for that family. You may like to limit the fact families to addition/subtraction or multiplication/division or leave it open, depending on the students.
- Hint: Strategy may come into play when choosing the numbers to go on the grid – what numbers belong to more than one fact family?

**Variations:**

For something a bit more challenging, try playing battleship!

- For an interactive one player versus a computer game go to: http://www.mathplayground.com/battleship.html (Math Playground, 2015)
- For an interactive player versus player game go to: https://battleship-game.org/en/ (Battleship-Game, 2016)
- You can always draw up your own paper version of the game too!

For something extra challenging, play on a four quadrant Cartesian plane.

*War is usually played with a pack of cards (e.g. http://www.bicyclecards.com/how-to-play/war/). This version can be played with dominoes.*

**Learning Objective:**

Compare fraction values.

**Intended Outcome:**

Accurate identification of the larger fraction.

**Materials:**

- Sets of dominoes

**Game Objective:**

To collect the most dominoes.

**Instructions:**

This game should be played in pairs, with a set of dominoes for each pair.

- Share all the dominoes between the two players, placing them face down.
- Each player must choose one of their dominoes and place it face up in the centre.
- Compare both dominoes as a fraction, with the lower number as the numerator and the higher number as the denominator, e.g.
- The player with the higher value fraction keeps both dominoes.
- If the fractions are equivalent, each player chooses another domino to place in the centre. The player with the higher value fraction takes all the dominoes in the centre.
- Continue playing the game until one player owns all the dominoes. This player wins the game.

**Variations:**

Instead of playing the game until one player owns all the dominoes, apply a time limit to the game. At the end of the time limit, the player with the most dominoes wins.

**Learning Objective:**

Adding ‘make ten’ numbers.

**Intended Outcome:**

Automatic recall of tens number facts (‘friends of ten’).

**Materials:**

- Pack of cards

**Game Objective:**

To collect the most cards by adding pairs which equal 10.

**Instructions:**

This game is to be played in pairs or small groups.

- Lay out 16 cards face up.
- Players take turns to find and collect a pair of cards that add to 10.
- Replace cards taken with cards from the deck.
- Continue playing until all possible ‘friends of ten’ have been picked up.
- Winner is the player who has collected the most 10s (or the most cards in total).

**Variations:**

- Change the number that the players aim to make e.g. make ‘friends of 17’.
- Different players in the same game could have different totals to make, e.g. Player 1 aims to make ‘friends of 15’ while Player 2 aims to make ‘friends of 9’.
- Collect 3 cards rather than card pairs to total your number.
- Lay out 25 cards and use ‘BODMAS’ and up to 4 cards to find two digit numbers.

**Learning Objective:**

Adding place values, ensuring values are lined up correctly and using partitioning to add to a target number.

**Intended Outcome:**

Effective use of place value.

**Materials:**

- Place value chart for each player
- Ten-sided dice

**Game Objective:**

To be first to total the target number.

**Instructions:**

This game is to be played in pairs or small groups.

- Decide a target number which everyone aims to reach, e.g. 459.
- Player rolls the dice and decides which place value column they are going to place that number in (e.g. if a 2 is rolled and they decide to place it in the Hundreds column, it becomes 200).
- The player then decides whether to add or subtract that number on their chart.
- Players take it in turns to roll and add to their own place value chart.
- Play continues until one player reaches the target number. (Example in PDF download below.)

**Variations:**

- Start at a given number and subtract to reach zero.
- Instead of only using addition and subtraction, use ‘BODMAS’ to reach the target number.
*Hint*: Use MAB or calculators to assist calculations.

**Introducing operations with fractions in mathematics can sometimes be confusing for students and frustrating for teachers.**

Providing some context within which students can conceptualise fractions can really help solidify these abstract concepts in students’ minds, especially when adding and subtracting fractions to amounts that are greater than a whole.

In this ‘low threshold, high ceiling’ task, students take the role of the owner of a cake wholesaler, baking and supplying cakes to local café businesses. As café owners order their weekly cakes by the slice, students are required to add unit fractions together to calculate total cake orders. They then solve problems associated with subtracting fractional remainders, using equivalent fractions and converting between improper fractions and mixed numerals.

This downloadable ‘rich task’ lesson resource is designed for teachers and students in Years 3 and 4. It is mapped against Australian Curriculum (Mathematics), with an emphasis on **problem solving** and **reasoning** in **operating with fractions**.

The task comes with a comprehensive grading rubric to assist teachers who wish to use the project for formal assessment purposes.

*This task was developed in consultation with Greta Public School (NSW), a *CHOOSE**MATHS*** Schools Outreach partner school.*

**Main Image Attribution:** By https://www.flickr.com/photos/markusunger/ – https://www.flickr.com/photos/markusunger/16001463889/, CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=52031968

The sustained and long-term success of students in mathematics depends on their ability to develop successful and flexible number strategies from an early age. Three essential foundations of students’ ability to develop such strategies are:

**Numeration**(ie, an understanding of place value and how it allows us to group numbers together so as to organise them);**Conceptual understanding of the operations**(addition, subtraction, multiplication and division); and**A working knowledge of number facts**based on efficient non-counting mental strategies – in addition and subtraction to 20, and in multiplication and division to 100.

(See *Booker, G., Bond, D., Briggs, J. & Davey, G. (1997) ‘Teaching Primary Mathematics’ (2nd Edition), Longman Cheshire: Melbourne*).

‘**MATH**EMONSTER Catch’ is a game that has been designed to help reinforce students’ numeracy in these three foundations of number understanding. The game can be played with students from Year 1 through to Year 6, in pairs or small groups, either competitively or collaboratively.

This game was trialled in partner schools in the Upper Hunter of NSW in 2016–2018 during the CHOOSE**MATHS** Schools Outreach project. The game is designed to be a fun and engaging way in which to help students grasp the bundling of units into groups of ten and then the use of these bundling strategies for operating with large whole numbers, particularly in the addition and subtraction operations.

The downloadable and printable resources provided here include:

- A full set of
**MATH**EMONSTER Catch game instructions; - A poster explaining the decimal values of each
**MATH**EMONSTER character featured in the game; - A set of
**MATH**EMONSTER ‘Catch Cages’ (place value units), ideally to be printed in A3 poster size or larger; - Game record sheets; and
- A full set of
**MATH**EMONSTER character cards (values from ‘ones’ through to ‘millions’).

Depending on time and available resources, teachers or parents may choose to print, laminate and cut out the attached **MATH**EMONSTER character cards or they may simply select to replace the character cards with coloured counters. Dice or digital random number generators can be used to generate the multi-digit whole numbers used to play the game.

We do hope that you and your students or child have lots of fun playing **MATH**EMONSTER Catch together!

**When was the last time you planned a special event? Have you ever stopped to think about just how much mathematics is involved?**

This investigative project gives your students the experience of being a professional ‘event planner’, by organising a special event such as a wedding reception, farewell or special birthday party.

Students are asked to prepare a comprehensive plan that outlines a floor and seating plan, a fully costed menu, a monetary quote and even a timed playlist and event programme.

It provides teachers and learners the opportunity to integrate the use of digital technologies such as spreadsheets and digital calendars into problem solving and mathematical thinking.

This downloadable lesson resource is designed for teachers and students in Years 5 and 6. It is mapped against Australian Curriculum (Mathematics), with an emphasis on problem solving within **operations with whole numbers**, **time** and **financial mathematics**.

The task comes with a comprehensive grading rubric to assist teachers who wish to use the project for formal assessment purposes.

*This task was developed in consultation with Greta Public School (NSW), a *CHOOSE**MATHS** *Schools Outreach partner school.*