Three starter tasks Dish washing If 8 students take 2 hours to wash dishes, how many hours would 12...

Three starter tasks

Dish washingIf 8 students take 2 hours to wash dishes, how many hours would 12 students take to wash the same number of dishes?

Multiples of 4Is 36,924 a multiple of 4? Is there a shortcut for figuring this out?

Always, sometimes or never true? The square of a number is greater than the number.

http://ced-mxteachers-news-site.wikispaces.com

CENTRE FOR EDUCATIONAL DEVELOPMENT

Digging for Gold: Investigating rich tasksAnne Lawrence NCEA National Coordinator Mathematics & Statistics [email protected]



Setting the scene

Lawrence was Otago's first gold-rush town, in the Tuapeka District, it was originally named The Junction, and then later renamed after the British war hero who defended Lucknow during the 1857 Indian Mutiny. At the height of the gold fever, it's population was 11,500; double that of Dunedin, making it one of the largest communities in the country. It is hard to believe that now with a current population of only 550...

Two Classrooms

•Explore tasks that provide the opportunity to develop reasoning and mathematical thinking. This is the gold we are digging for.

• What scaffolding is needed for students to access these tasks?

•How can we scaffold and still maintain levels of cognitive demand?


Outline of this session

Two Classrooms

Learning environments in which teachers: • encourage multiple strategies and ways of

thinking, and• support students to make conjectures and explain

their reasoning

are associated with higher student performance on measures of thinking, reasoning and problem-solving.


Why?

Levels of cognitive demand

Sort into order of increasing levels of cognitive demand:

• Doing mathematics

• Memorisation

• Procedures with connections

• Procedures without connections



Levels of demand Characteristics of tasksVery Low

Memorisation

Low Procedures without connections

High Procedures with connections

Very High

Doing mathematics



Memorisation



Very High

Doing mathematics

AlgorithmicCorrect answer

ExplanationUnderstanding



Memorisation Reproduction of facts, rules, formulae. No explanations


Algorithmic. Focused on producing correct answers. No explanations needed


Algorithmic. Meaningful /real-world context. Explanations required

Very High

Doing mathematics

Non-algorithmic. Requires understanding and application of math concepts. Real-world context or mathematical structure.Explanations required

Rich Tasks


Levels of demand TasksVery Low

Memorisation



Very High

Doing mathematics

• Dishwashing• Multiples of 4• Always/sometimes/never• Flying horseshoes

• Ostrich & seahorse• Fuel for thought• Disc-ness

Rich Tasks

Three starter tasks

Dish washingIf 8 students take 2 hours to was dishes, how many hours would 12 students take to wash the same number of dishes?

Multiples of 4Is 36,924 a multiple of 4? Is there a shortcut for figuring this out?

Always, sometimes or never true? The square of a number is greater than the number.


Flying horseshoeswww.nctm.org

1. Which expression is the most useful for finding the maximum height of the horseshoe, and why is it the most useful expression?

2. What information can you conclude about the horseshoe’s flight from other equivalent expressions? Explain your answers.


Ostrich and seahorse

On the left is a picture of an adult ostrich. It is drawn to a scale of 1:48. On the right is a picture of a seahorse. It is drawn to a scale of 1:1.5.How many seahorses could you fit into the length of an ostrich neck?

http://balanced assessment.concord.org

Which of the following would save more fuel?

a) Replacing a compact car that gets 34 miles per gallon (MPG) with a hybrid that gets 54 MPG

b) Replacing a sport utility vehicle (SUV) that gets 18 MPG with a sedan that gets 28 MPG

c) Both changes save the same amount of fuel.

Fuel for thought


www.nctm.org

Discs-ness

A gold coin is a disc, and an uncooked piece of spaghetti is a cylinder. If you think about it, a coin is also a cylinder and an uncooked piece of spaghetti is also a disc. Clearly the coin is more disc-like and the spaghetti more cylinder-like. Your task is to devise a measure of disc-ness.

http://balanced assessment.concord.org

• What support do students need to tackle these rich tasks?

• What scaffolding is appropriate?• What prompts would you want to give?

Supporting, Scaffolding, Prompting…


Hold the demand

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

We want to prompt students to reason with mathematics and reason through problems rather than giving prompts that require little engagement and reasoning.

How can we provide support without reducing the cognitive demand?


Discs and cylinders

A gold coin is a disc, and an uncooked piece of spaghetti is a cylinder. If you think about it, a coin is also a cylinder and an uncooked piece of spaghetti is also a disc. Clearly the coin is more disc-like and the spaghetti more cylinder-like. Your task is to devise a measure of disc-ness.

How can we scaffold without lowering the demand?

1. Given a coin, a tuna-fish can and a soup can: Devise a definition of disc-ness that allows you to say which object is the most disc-like and which is the least .

2. Given a mailing tube, a straw, and a piece of uncooked spaghetti: Use your definition of disc-ness to determine which object is the most disc-like and which is the least.

3. Write a formula (or algorithm or algebraic sentence) which expresses your measure of disc-ness. You may introduce any labels and definitions you like and use all the mathematical language you care to.

4. Make any measurements you need, and calculate a numerical value of disc-ness for each of the six items.

5. Discuss whether these numbers seem reasonable in light of your notion of disc-ness.

6. How would you change your answers to these questions if you were asked to write a formula for cylinder-ness rather than disc-ness?

Scaffolding learning

1. Problem solving cycle

2. Warm up activity

3. Success criteria

Two Classrooms

• Problem solving involves investigating a problem when the route to a solution is unclear.

• We need to use tasks which allow for student choice about the mathematics and the problem-solving strategies they use to model and investigate situations.

• Learning needs to be scaffolded so students are able to call on their mathematical knowledge and search for a pathway to the answer.

Digging for gold


Rich mathematical tasks are not enough. Students need rich mathematical experiences:

The greatest learning occurs in classrooms in which mathematical tasks with high-level cognitive demand are used and the demand is consistently maintained throughout the teaching.

Digging for gold


http://www.shyamsundergupta.com/amicable.htmhttp://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/http://www.curiousmath.com/index.php?name=News&file=article&sid=55http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.htmlhttp://mathbits.com/virtualroberts/spacemath/BottleTop/project.htmhttp://www.noao.edu/education/peppercorn/pcmain.html http://mathbits.com/virtualroberts/spacemath/BottleTop/project.htm http://gurmeet.net/puzzles/

Finding rich tasks

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

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

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

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

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

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



May the force be with you

as you dig for gold!

Thank you


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