rich mathematical tasks (rmts)

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

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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. Think of a number Multiply by 5, add 3 - PowerPoint PPT Presentation

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Page 1: Rich mathematical tasks (RMTs)

CENTRE FOR EDUCATIONAL DEVELOPMENT

Rich Mathematical Tasks

SNP Sustainability July 2010Anne Lawrence

Page 2: Rich mathematical tasks (RMTs)

• What are rich maths tasks? • Why are they important? • How can we use them?

Rich mathematical tasks (RMTs)

Page 3: 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?

Page 4: Rich mathematical tasks (RMTs)

What is a Rich Math Task?

Page 5: Rich mathematical tasks (RMTs)

• 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

Page 7: Rich mathematical tasks (RMTs)

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

Page 8: Rich mathematical tasks (RMTs)

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

250 20

58 1250

7/8 3/4

4/12 10/11

Page 9: Rich mathematical tasks (RMTs)

Math class needs a makeover

http://blog.ted.com/2010/05/math_class_need.php

Page 10: Rich mathematical tasks (RMTs)

Math class needs a makeover

•What is Dan’s key message?

•Do you agree with this?

•What are the implications?

Page 11: Rich mathematical tasks (RMTs)

Levels of demand

• Lower level demands– Memorisation– Procedures without connections

• Higher level demands– Procedures with connections– Doing mathematics

(Smith & Stein, 1998, p. 348)

Page 12: Rich mathematical tasks (RMTs)

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!

Page 13: Rich mathematical tasks (RMTs)

Bloom’s taxonomy - hierarchy of skills

Page 14: Rich mathematical tasks (RMTs)

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 )

Page 15: Rich mathematical tasks (RMTs)

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

Page 16: Rich mathematical tasks (RMTs)

Levels of thinking for NCEA

AS 1.1 Number•AchievedApply numeric reasoning

•Merit Apply numeric reasoning with relational thinking

•ExcellenceApply numeric reasoning with extended abstract thinking

Page 17: Rich mathematical tasks (RMTs)

Arguing, convincing and proving

1. Always, sometimes, never ...

2. Convincing

3. Proving

Page 18: Rich mathematical tasks (RMTs)

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.

Page 19: Rich mathematical tasks (RMTs)

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?

Page 20: Rich mathematical tasks (RMTs)

First proof

Amy can only get four arrangements when she turns two at a time:

HHH HTT TTH THT

So it is impossible

Page 21: Rich mathematical tasks (RMTs)

Second proof

I will score each position. Let H = 1 and T = 0. So HHH = 1+1+1 =3, HTH = 1+0+1 =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.

Page 22: Rich mathematical tasks (RMTs)

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.

Page 23: Rich mathematical tasks (RMTs)

• 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

Page 24: Rich mathematical tasks (RMTs)

• 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

Page 25: Rich mathematical tasks (RMTs)

• 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

Page 26: Rich mathematical tasks (RMTs)

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.

Page 27: Rich mathematical tasks (RMTs)

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

Page 28: Rich mathematical tasks (RMTs)

Keeping it challenging

It's not so much what you do as

how you do it!

Page 29: Rich mathematical tasks (RMTs)

http://www.shyamsundergupta.com/amicable.htm

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/

http://www.curiousmath.com/index.php?name=News&file=article&sid=55

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

http://mathbits.com/virtualroberts/spacemath/BottleTop/project.htm

http://www.noao.edu/education/peppercorn/pcmain.html

http://mathbits.com/virtualroberts/spacemath/BottleTop/project.htm

An abundance of rich task sources

Page 30: Rich mathematical tasks (RMTs)
Page 31: Rich mathematical tasks (RMTs)

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?

Page 32: Rich mathematical tasks (RMTs)

• Oral counting

A rich math task?

Page 33: Rich mathematical tasks (RMTs)

Rich maths tasks are essential

Rich maths tasks are ‘icing on the cake’

What is your view? Why?

Importance of RMTs

Page 34: Rich mathematical tasks (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

Page 35: Rich mathematical tasks (RMTs)

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?

Page 36: Rich mathematical tasks (RMTs)

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

Page 37: Rich mathematical tasks (RMTs)

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

Page 38: Rich mathematical tasks (RMTs)

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

Page 39: Rich mathematical tasks (RMTs)

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.

Page 40: Rich mathematical tasks (RMTs)

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.

Page 41: Rich mathematical tasks (RMTs)

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.