CENTRE FOR EDUCATIONAL DEVELOPMENT Rich Mathematical Tasks SNP Sustainability July 2010 Anne Lawrence

What are rich maths tasks? Why are they important? How can we use them? Rich mathematical tasks (RMTs)

Let’s do some numeracy 1.Think of a number 2.Multiply by 5, add 3 3.Same starting number 4.Add 3, multiply by 5 Why don’t you get the same answer? What do you notice about your answers? Does this always happen? Why?

What is a Rich Math Task?

Accessible and extendable, Allows learners to make decisions, Involves learners in testing, proving, explaining, reflecting and interpreting, Promotes discussion and communication, Encourages originality and invention, Encourages 'what if' and 'what if not' questions, Enjoyable and contains the opportunity for surprise RMT - one definition

Sesame Street: Cookie Monster's Sorting Song One of these things

Three of these things are a lot the same; One of these is not like the others... Which one and WHY? One of these things

Three of these things are a lot the same; One of these is not like the others... Which one and WHY? One of these things /83/4 4/1210/11

Math class needs a makeover

Math class needs a makeover What is Dan’s key message? Do you agree with this? What are the implications?

Levels of demand Lower level demands – Memorisation – Procedures without connections Higher level demands – Procedures with connections – Doing mathematics (Smith & Stein, 1998, p. 348)

Demanding tasks Students of all abilities deserve tasks that demand higher level skills BUT teachers and students conspire to lower the cognitive demand of tasks!

Bloom’s taxonomy - hierarchy of skills

SOLO – framework for understanding SOLO: the Structure of Observed Learning Outcomes. SOLO identifies five stages of understanding. Each stage embraces the previous level but adds something more. Stages progress from surface thinking (lower-order thinking skills) through to deep thinking (higher-order thinking skills involving relational and extended abstract reasoning )

The Stages of SOLO Prestructural –the student acquires bits of unconnected information that have no organisation and make no sense. Unistructural – students make simple and obvious connections between pieces of information Multistructural – a number of connections are made, but not the meta-connections between them Relational – the students sees the significance of how the various pieces of information relate to one another Extended abstract – at this level students can make connections beyond the scope of the problem or question, to generalise or transfer learning into a new situation

Levels of thinking for NCEA AS 1.1 Number Achieved Apply numeric reasoning Merit Apply numeric reasoning with relational thinking Excellence Apply numeric reasoning with extended abstract thinking

Arguing, convincing and proving 1.Always, sometimes, never... 2.Convincing 3.Proving

Always, sometimes or never true? –The square of a number is greater than the number. –If two rectangles have the same perimeter, they have the same area.

Which is the best proof? Explain? Amy is playing a coin turning game. She starts with three heads showing and then turns them over, two at a time. After a while she states: “If I turn them two at a time, it is impossible to get all three showing tails.” Following are three attempts to prove this. Which is the best proof?

First proof Amy can only get four arrangements when she turns two at a time: HHHHTTTTHTHT So it is impossible

Second proof I will score each position. Let H = 1 and T = 0. So HHH = =3, HTH = =2 and so on. Amy’s first position scores 3. Each time she moves, her score will either increase or decrease by 2 or stay the same. So she can only get into positions with odd scores. So it is impossible to get a score of zero So TTT is impossible.

Third proof It is impossible because the most tails Amy can have showing is two. When she turns the head over from that position, you must also turn over one of the tails, so Amy can’t get rid of all the heads whatever she does.

Where would this task fit? What is the level of demand? How can I extend the task? How can I support students who are stuck? Exploring tasks

Where would this task fit? What is the level of demand? How can I extend the task? How can I support students who are stuck? Exploring tasks

Where would this task fit? What is the level of demand? How can I extend the task? How can I support students who are stuck? Exploring tasks

Plan to get the most out of tasks Choose the starting point; Select interactive and intervention questions for when –students get stuck; –students ‘think’ they have the solution; –students are unable to extend the problem further.

Rich tasks encourage students to think creatively, work logically, communicate ideas, synthesise their results, analyse different viewpoints, look for commonalities and evaluate findings. However, what we really need are rich classrooms: communities of enquiry and collaboration, promoting communication and imagination. Rich math environment

Keeping it challenging It's not so much what you do as how you do it!

An abundance of rich task sources

Let’s talk maths! 1.What is the smallest even number? What is your answer and why? 2. = 9. True or false? What is your answer and why? 3.Find the number halfway between 27 and 54 What is your answer and how did you get it?

Oral counting A rich math task?

Rich maths tasks are essential Rich maths tasks are ‘icing on the cake’ What is your view? Why? Importance of RMTs

How do we develop RMTs? Turning around a question Asking for similarities and differences Replacing a number with a blank Degrees of possible variation

Present a single ‘example’ and invite students to tackle their own variations Present several ‘examples’ and use these to highlight important features What is the same and what is different?

Achieved: Apply numeric reasoning Carry out routine procedure/procedures Explain concept/concepts or use a representation/representation in isolation Solve simple (one step) problem and communicate the solution

Merit: Relational thinking selecting and carrying out a logical sequence of steps connecting different concepts and representations demonstrating understanding of concepts forming and using a model relating findings to a context communicating thinking using appropriate mathematical statements

Excellence: Extended abstract thinking devising a strategy to investigate or solve a problem demonstrating understanding of abstract concepts developing a chain of logical reasoning, or proof forming a generalisation using correct mathematical statements communicating mathematical insight

The big day out The Big Day Out is a popular music festival. The festival organizers want to improve the environment by helping festival attendees purchase carbon offsets. The offsets will fund tree planting to reduce the net CO2 emissions of the concert. This assessment activity requires you to determine the round number of trees that will be planted because of the Big Day Out.

Prepare a written estimate for the festival organizers of how many trees will be planted. Use the following information: The organizers expect 13,000 to 21,000 people to attend the festival this year (to the nearest 1,000); Big Day Out attendees can choose to purchase 0, 1, 2, or 3 carbon offsets when they buy their tickets ­ last year 23% of attendees actually purchased offsets; of those who purchase, one quarter of festival attendees purchase three offsets, one third purchase two offsets and the rest purchase one offset; each carbon offset costs NZ$1.34; three trees are planted for every fourteen dollars of carbon offsets purchased.

In your estimate, state any assumptions, explain the sequence of steps you follow, and what you are calculating at each step of your solution. The organizers are worried that a smaller percentage of people might purchase carbon offsets than last year. They would like to put a projected number of trees on their website as tickets are sold. Enhance your estimate so that it explains how to compute the number of trees for different percentages of offset purchasers and/or different number of attendees. Present your method as succinctly as possible, using appropriate mathematical statements.