**Here’s why, and how, ‘rich tasks’ can help.**

Teachers are often being asked to differentiate teaching and learning to ensure they are catering for a broad range of student abilities, interests and readiness to learn.

In mathematics, this means simultaneously teaching new content to all students, providing ample opportunity for students to master content, supporting students experiencing difficulties with mathematical understanding and ensuring experiences in higher levels that extend and enrich more capable learners.

It can be exhausting to think about, especially when we know that in our classrooms we have such a broad range of students. As teachers, how can we manage this in real-world classrooms, without driving ourselves to the edge of exhaustion by micro-managing separate tasks for each and every student?

Back in 2016, the World Economic Forum predicted that within five years, over one-third of skills that were considered important in the workforce will have changed. They argued that by 2020, artificial intelligence, advanced robotics, autonomous transport, new materials, biotechnology and genomics will have transformed the kind of skills required in the workforce and in the way we live. The three skills that topped the WEF’s ‘list of 21st Century Thinking Skills’ were, in order, complex problem solving, critical thinking and creativity (Grey, A. 2016).

As educators we can’t ignore these realities. We know we need to be preparing as many of our students as possible for the imperatives of life in the 21st century, developing their problem solving, reasoning and creativity proficiencies, especially in mathematics. These skills underpin so many roles in the new economy. However, it’s also important that at the same time we are catering for students who are having difficulty understanding concepts at much more basic levels.

The standard approach to differentiation in mathematics has been to prepare multiple ‘levels’ of content set to different standards, and/or to make available a variety of learning resources, individualised for each learner in the classroom. When teachers are responsible for classes of around 30 students, often within the context of engagement or behavioural issues, this can be time consuming and professionally draining.

Some digital or online learning providers claim to have solved this by developing software that allows students to follow an ‘individualised’ program of learning in mathematics. Such programs use adaptive pathways which respond according to students’ progressive successes and failures against set tasks. Smart, huh?

However, the problem with many such online or digitally based mathematics learning solutions is the same as that facing the traditional ‘textbook only’ based approaches to mathematics that have plagued maths learning since the 1950s. Typical ‘drill and answer’ exercises and closed worded problems – such as those typically used in textbooks and on online mathematics websites – usually focus students’ learning only on developing *understanding* and *fluency* in mathematics.

These two math proficiencies – the skill of grasping a maths concept (understanding) and of using it flexibly and efficiently (fluency) are of course necessary – but by no means sufficient. Tasks in maths that develop only understanding and fluency for students tend to suggest a single linear pathway to ‘working out’ and a single correct answer.

More importantly, they neglect the *problem solving* and *reasoning* proficiencies both prescribed within the Australian mathematics curriculum framework and required for humans to cope with life and work in the 21st century.

**Problem solving, reasoning and creativity: ‘rich’ mathematical thinking**

In mathematical learning, generally ‘closed’ (single answer/single method) tasks tend to stimulate basic conceptual understanding and develop concept fluency. However, ‘open-ended’ types of problems (more than one correct answer, multiple pathways for working out and justifying) have greater potential for stimulating *higher order* mathematical thinking, that is, creative problem solving and complex reasoning capacities.

This is partly because such tasks involve a search for patterns and relationships between elements in the problem. Students must ‘play around with’ different variables in order to generate different solution pathways, use trial and error (much the same as for problems faced in the real world) and explore a range of methods. They must compare the efficiency and accuracy of solution pathways and use reasoning to adapt and apply previously learned concepts to new situations.

Of course, students who have not understood and/or who are not fluent in math concepts will struggle to solve problems and to apply mathematical reasoning. The key issue for teachers is thus how we effectively and efficiently cater for these less fluent students without compromising opportunities for more fluent students to extend and enrich their mathematical thinking.

**‘Rich’ or ‘Low Threshold, High Ceiling’ (LTHC) tasks **

Above we mentioned the desirability of open ended maths tasks, or tasks which provide scope for more capable students to move toward more sophisticated thinking skills.

As an extension to this, ‘rich tasks’ – or ‘low threshold, high ceiling’ tasks in mathematics are structured so that *all* students can make a start to the problem, even if needing support. The ‘low threshold’ of such tasks reinforces understanding and fluency of a given concept and allow less confident learners to experience some level of success.

At the same time, however, further levels or iterations of the same tasks are designed to engage more independent learners in deep or complex problem solving and reasoning. The ‘high ceiling’ in these tasks provide plenty of opportunity for the participants to have a go at much more challenging maths, albeit within the same concept, topic or skill area.

According to Lynne McClure (2011), “A LTHC mathematical activity is one in which pretty well everyone in the group can begin, and then work on at their own level of engagement, but which has lots of possibilities for the participants to do much more challenging mathematics.”

She goes on to explain that rich tasks can go some way toward resolving the dilemma of more efficiently differentiating for a diverse group of learners; a single task designed with low thresholds and high ceilings can provide for the whole class. Teachers often believe that the only way to challenge learners is to offer them *different* content at a higher grade level. However, in rich mathematical tasks the content itself remains quite simple but the *level of thinking* required – such as non-linear problem solving and mathematical reasoning – can become very complex (McClure 2011, p.2).

Rich or LTHC tasks allow teachers to set one problem for all students, provide some explicit whole-class instruction, and then respond to individual and small group strengths and needs as they arise during the problem. An important role for the teacher during such learning is to cultivate a problem solving ‘culture of iteration’ whereby students learn to push themselves into levels of the task that are difficult or ‘problematic’ for them.

Having most students persevering on harder levels of the task (for them) will also often free up the teacher to work more intensively with students struggling with basic concepts. Establishing collaborative problem solving guidelines such as time for quiet thinking and reflection, ‘ask three before you ask me’, pausing periodically to discuss the approaches of different students in various parts of the task (*not* asking for ‘answers’!) and analysing errors as well as ‘correct answers’, will further facilitate the freeing of teacher time to be spent with less confident individuals.

**Developing LTHC or ‘Rich’ tasks in Mathematics**

*Click to here to **d**ownload an exemplar LTHC (rich) task: ‘ catering-canapes’ (approximately Grade 5 or 6 level).*

To write your own rich tasks in maths in a grade and topic suited to your own classroom (which is always much more fun than using someone else’s!), the following development guidelines might prove useful.

**1. Start with a closed version of a problem within a given topic or concept.** Grade level textbooks and standardised tests (such as past NAPLAN papers) are often a good source of closed mathematics problems.

**2. ‘Open up’ the problem by removing or adapting parameters, to allow for a range of solutions.** As an example, a closed problem might read:

‘Sarah takes 4 hours and 55 minutes to complete a 200-page novel, while Derek takes 5 hours and 12 minutes to read the same novel. Assuming they are each reading at the same speeds as they were for the 200 page novel, how much faster is Sarah than Derek, in seconds, if they both read a 300-page novel?’

To open up this question up, we could remove the parameters in the second sentence of the question, such that it instead reads:

‘…

Assuming they are each reading at the same speeds as they were for the 200 page novel, compare the times taken by Sarah and Derek to read one of the novels selected from a book shelf either in your classroom or your library.’

**3. Push further by adding mathematical or procedural ‘problem solving’ complexity.** For example, in the example above, we could add a third reader (with a different reading speed), select a range of different length novels to read and/or ask students to produce line graphs that compare the reading speeds of students against novels at different lengths.

They key concern here is to ensure a few different iterations of the same type of problem, each of which add increasingly complex levels of problem solving for students in the class.

**4. Introduce a requirement for students to demonstrate reasoning and justification** for a version or versions of the problem. Ways in which to do this might be:

- have students challenge one another by setting their own versions of the task (they will need to have reasoned out their own version and to have ‘worked backwards’ in order to do this);
- have students compare several different methods of working out and write reasons for which they think one is ‘better’ than another; or
- have students design a model, experiment or product that applies the concept or topic in the real world. (An example of the above might be to design an experiment that compares any students’ reading speeds and draws conclusions from this experiment that could be useful to students when choosing the type and length of novel to read).

Designing and using rich tasks in mathematics can be a rewarding and motivating experience for teachers as they watch their students engage in problem solving and demonstrate thinking in ways that traditional ‘text book’ questions rarely allow.

They can also provide a new and rich source of assessment information as teachers gain new insights into how their students are working mathematically.

*– Marcus Garrett*

**References**

BBC Active (2010), *‘Methods of Differentiation in the Classroom’. *Online article, in __BBC Active Ideas and Resources__, URL: http://www.bbcactive.com/BBCActiveIdeasandResources/MethodsofDifferentiationintheClassroom.aspx. British Broadcasting Corporation : London, United Kingdom.

EduGains (2015), *‘Knowing and Responding to Learners in Mathematics’. *Online resource available from http://www.edugains.ca/newsite/di/knowing_responding_to_learners.html. Accessed 17/01/18. Ontario Ministry of Education : Ontario, Canada.

Grey, A. (2016), ‘The 10 skills you need to thrive in the fourth industrial revolution’. Article published by World Economic Forum, 19 January, 2016. URL: https://www.weforum.org/agenda/2016/01/the-10-skills-you-need-to-thrive-in-the-fourth-industrial-revolution/.

Herter, R. (2015), ‘Growth Mindset for Math – Mistakes’ (Youtube). URL: https://www.youtube.com/watch?v=LrgpKjiQbQw. Accessed 8/11/2017

McDonald, S. and Watson, A. (2012) What’s in a task? Generating mathematically rich activity. A report commissioned by the United Kingdom Qualifications and Curriculum Development Agency (now the United Kingdom Standards and Testing Agency). London : United Kingdom. Online report – URL: http://xtec.cat/centres/a8005072/articles/rich.pdf. Accessed 17/05/18.

McLeod, S. A. (2012). *Zone of proximal development*. Retrieved from www.simplypsychology.org/Zone-of-Proximal-Development.html

McClure, L. (2011), ‘*Using Low Threshold High Ceiling Tasks in Ordinary Primary Classrooms’*. Online article on nrich.maths.org. URL: http://nrich.maths.org/7701. Accessed 10/1/2018. NRich Maths, Cambridge University : United Kingdom.

Motter, A. (Date uncertain), ‘George Polya’. Online article, in ‘math.wichita.edu’. URL: http://www.math.wichita.edu/history/men/polya.html. Accessed 10/04/16. Wichita State University : Kansas, United States.

New Zealand Ministry of Education (2010 ? – 2017), ‘Problem Solving’ online resource page. URL https://nzmaths.co.nz/level-5-problems. Accessed 3/11/2017. New Zealand Ministry of Education : Dunedin, NZ.

*NRICH Mathematics (2018), ‘What Was In the Box’. URL: *https://nrich.maths.org/7819*. Accessed 17/01/18. Cambridge University, United Kingdom.*

NSW Department of Education (2014), ‘Newman’s Error Analysis’. On Numeracy Skills Framework support website. URL: http://numeracyskills.com.au/newman-s-error-analysis. Accessed 23 October, 2016. Government of NSW : Sydney, Australia.

Piggott, J. (20018) *‘Rich Tasks and Contexts’*. Online article on nrich.maths.org. URL: https://nrich.maths.org/5662. Accessed 10/1/2018. NRich Maths, Cambridge University : United Kingdom.

Piggott, J. (2011), *‘Integrating Rich Tasks’. *Online article on nrich.maths.org. URL: https://nrich.maths.org/6089. Accessed 10/1/2018. NRich Maths, Cambridge University : United Kingdom.

Thanks to Camp Hill Primary School Bendigo, MAV presentation, 2015.

**Learning Objective:**

Use the four processes to calculate numbers below 20.

**Intended Outcome:**

Accurate and strategic use of calculations.

**Materials:**

- 2 × ten-sided dice

**Game Objective:**

To cross off four numbers in a row.

**Instructions:**

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

- Each player writes the numbers 1–20 on their page.
- First player rolls the two dice and uses the two numbers with any process to make an equation. They can then cross that total off their list of numbers.
- Players take it in turns to roll the dice and calculate, and cross off numbers.
- First player to cross off four numbers in a row wins!

**Variations:**

- Use a hundreds chart and three ten-sided dice to create the numbers.
- Use processes with all three numbers on the dice e.g. ‘2 × 6 + 7 = 19’.
- Or, put two numbers together to make a two-digit number and insert an operation (+, –, ×, or ÷) before the third, e.g. ‘67 – 2 = 65’.
- Cross off four numbers in a row horizontally, vertically, or diagonally.

**What is ‘time’ to children?**

As adults, we know how time is measured; but it’s much harder to describe what time is to the average 7 year old. The standard unit of time is the second, and these are organised into aggregated units (minutes, hours, days etc.).

But what of non-standard measures of time, and the fact that time can ‘feel different’ depending upon what we are doing? For example, when we are having fun with friends or family, time seems to ‘fly’; but if we’re in an examination or a doctor’s waiting room, time seems to ‘drag’. Normally, when introducing measurement to children we start with informal units of measure; with time, this is obviously much harder!

Time is different from most other ‘measurement’ attributes that we experience because it cannot be seen or heard or touched (Van de Walle et al. 2010). We can’t ‘see’ time in the same way we can ‘see’ how tall someone is or how full something is. We can only ‘feel’ time as it goes past – and measure it on a device (a clock).

For this reason, time is a very abstract mathematical construct for children – and therefore their understanding needs to be anchored in concrete experiences drawn from the world around them.

Using the much loved Australian children’s book ‘Dog In, Cat Out’ (written by Gillian Rubinstein and beautifully illustrated by Ann James) as a stimulus, the lesson sequences and activities in this unit are a useful way to help children from Foundation to Grade 2 level gain an understanding of the concept of time, its measurement and its numeration.

**Download the 12 lesson Unit, complete with Outline Masters, here.**

*‘Dog In, Cat Out: A Unit in ‘Time’ has been developed in consultation with Greta Public School (New South Wales), and with the book’s author’s and illustrator’s permission granted.*

**Learning Objective:**

Practise solving division facts.

**Intended Outcome:**

Accurate division.

**Materials:**

- Blank grid template (5 x 5, 7 x 7, 5 x 10 are good sizes)
- Dice

**Game Objective:**

To be the first to get 5 in a row.

**Instructions:**

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

- Each player has their own grid. Each player writes random numbers in every square on their grid. It’s a good idea if the numbers are answers in the times tables! (composite numbers).
- First player rolls a die and looks for a number on their grid that is divisible by the number they rolled. They then write the division fact in that square. If they have more than one number on their grid that is divisible by the number they rolled, they may only choose one.
- Players take it in turns to roll the dice and find a number on their grid that divides evenly by the number they rolled.
- First player to have division facts written in 5 squares in a row wins!

Andamooka, located 612km north of Adelaide is a small opal mining town boasting a population of less than 400 people. The local primary school, led by principal Tricia Williams, along with three classroom teachers and a handful of support staff, provides a wonderful nurturing environment for the 27 students who currently attend the school.

Andamooka Primary School have been a part of the CHOOSE**MATHS** program since 2016 and their ongoing commitment to improving student outcomes in mathematics has helped shape a great collaborative partnership with AMSI and the CHOOSE**MATHS** team.

Recently, that partnership extended to the delivery of a CHOOSE**MATHS **Family Night where Andamooka Primary School invited the parents from the local community to join in for some family maths fun along with their kids. The event was held at the Andamooka Community Hall on the 16th of March and was attended by many teachers, parents and students. Nadia Abdelal, the AMSI Schools Outreach Officer for the area, gave an introductory presentation to the parents on ways that they can support their children at home in their mathematics learning, as well and how they can extend this support to the school. The night finished with a variety of maths games and activities all enjoyed by the Andamooka families, many of whom took some great ideas home with them.

**MATHS** Family Night 2019.

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.

]]>**Learning Objective:**

Multiply and divide integers and decimal fractions by 10, 100 and 1000.

**Intended Outcome:**

Improved fluency in mentally multiplying and dividing numbers by 10, 100 and 1000.

**Materials:**

- 2 x ten-sided dice
- 1 x six-sided dice
- ‘Power of Ten’ dice conversion chart (available in PDF download below)

**Game Objective:**

To be the first team with all members sitting down.

**Instructions:**

This game is to be played in two equal teams.

- All students start the game standing up. The teacher rolls the ten-sided dice to give the number (e.g. ‘8’) and then the six-sided dice to give the operation (as per conversion chart – e.g. rolls a ‘5’ which gives a ‘÷ 100’ operation).
- Starting with a member of Team 1, each student must complete the given Power of Ten operation correctly. In the above example, the student selected must answer ‘0.08’, as this is the answer to ‘8 ÷ 100’. Once they have answered correctly, they may sit down.
- This game is to be played quietly, each chosen student answering the question on their own. If there is calling out, a member of that team must stand up again.
- First team to have all members sitting down is the winner!

**Learning Objective:**

Demonstrate understanding of positional language.

**Intended Outcome:**

Listening and following directions.

**Game Objective:**

To listen and follow directions.

**Instructions:**

This game is to be played outside, as a whole class.

- Define some boundaries for the playing space.
- Teacher calls an instruction, based on positional language, e.g. stand under something, sit beside something red, crawl under something low, stand inside a circle.
- Students follow the instructions.
- Repeat for as long as you like!

**Examples of positional language:**

On, under, over, beside, in front of, behind, next to, north of, south of, east of, west of, above, beneath, inside, between, outside.

**Examples of conditions:**

Something green, blue, red.

Something hard, soft, wet, squishy.

**Learning Objective:**

Research to match capital cities and countries, mark places on a map, plan a route.

**Intended Outcome:**

Accurate placing of cities on a map, finding the best path to take.

**Materials:**

- Blank map of the world

*The following sites include templates which you can print off for this game.*- Free World Maps: http://www.free-world-maps.com/printable-white-transparent-political-blank-world-map-c3
- Macs Stuff: http://macsstuff.net/photobov/blank-world-maps-with-countries

- Atlas
- List of capital cities (included below)

**Game Objective:**

To work out the best path for Santa to take on Christmas Eve.

**Instructions:**

- Students use their blank map and mark to mark all of the capital cities that Santa has to visit (cities listed below).
- They then work out the best path for Santa to take making sure he visits all of the cities.

**Capital cities Santa needs to visit:**

- Athens
- Beijing
- Canberra
- Dhaka
- Doha
- Dublin
- Harare
- Kigali
- Kingston
- Kuala Lumpur
- Lima
- Moscow
- Nuku’alofa
- Ottawa
- Port Moresby
- Quito
- Reykjavik
- Seoul
- Tripoli
- Wellington

**Extension:**

- Ensure Santa is at each of the cities during night time so he doesn’t get seen. Where and at what time would he have to start his journey? Where and at what time would he finish his journey? Taking into consideration time differences.
- Work out the distance travelled.

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.