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Mathematics and the NCEA realignment

- AS 1.5
- Feedback on the standard and the task
- Implications for teaching and learning
- Supporting deeper thinking
- Understanding different levels of thinking
- Next steps

Mathematics and NCEA realignment

Angela Jones

Senior adviser

Secondary Outcomes Team

Ministry of Education

Anne Lawrence

Adviser in Numeracy, Mathematics & Statistics

Massey University College of Education

AS 1.5 Apply measurement in solving problems

Achieve:

Apply measurement in solving problems.

Merit:

Apply measurement in solving problems,

using relational thinking.

Excellence:

Apply measurement in solving problems,

using extended abstract thinking.

Measurement includes the use of standard international metric units for length, area, capacity, mass, temperature, and time. Derivedmeasures include density, speed and other rates such as unit cost or fuel consumption.

Students will be expected to

be familiar with perimeter, area and surface area, volume, metric units.

convert between metric units, using decimals

deduce and use formulae to find the perimeters and areas of polygons, and volumes of prisms

find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders

apply the relationships between units in the metric system

calculate volumes, including prisms, pyramids, cones, and spheres, using formulae.

Key skills and knowledge for 1.5

Achievement standard 1.4Solving problems - using a range of methods solving problems, demonstrating knowledge of concepts, solutions usually require only one or two steps.

Relational thinking - one or more of a logical sequence of steps; connecting different concepts and representations; demonstrating understanding of concepts; forming and using a model, and relating findings to a context, or communicating thinking using appropriate mathematical statements.

Extended abstract thinking- one or more of devising a strategy to investigate or solve a problem; identifying relevant concepts in context; developing a chain of logical reasoning; forming a generalisation, and using correct mathematical statements, or communicating mathematical insight.

Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts. The situation will be set in a real-life or mathematical context.

The phrase ‘a range of methods’ indicates that there will be evidence of at least three different methods.

Solvingproblems at A, M and E for 1.5

Achievement standard 1.4What does excellence look like?

Achievement standard 1.4Student B

Student A

Student C

Student D

Student E

Supporting M and E thinking

Students need to develop their own understanding of what A, M and E looks like.

They need to:

Explore examples of A, M and E work

Discuss student work (their own and others’)

Evaluate student work (their own and others’)

Is this at the M standard?

What else is needed to make it to M?

What could you take away and still have it M?

Dan Meyer Ted Talk - recommended viewing for all maths teachers

http://www.youtube.com/watch?v=BlvKW

Questions to ask as you watch Dan’s talk:

What do you see as Dan’s key message(s)?

What are the implications for the classroom?

What are the key message(s) for you from Dan’s talk?

From Dan Meyer (US maths teacher)Levels of thinking, NZC and NCEA

The NZC requires that deeper and more complex thinking are rewarded along with more effective communication of mathematical ideas and outcomes. These are fundamental competencies to mathematics.

NCEA realignment supports this focus.

Students need to engage with activities that provide the opportunity to develop numeric reasoning, relational thinking and abstract thinking in solving problems.

Key questions:

What sorts of activities are appropriate?

How do we support students to access these activities?

What are appropriate levels of scaffolding?

Rich mathematical activitiesLevels of demand

Lower level demands:

Memorisation

Procedures without connections

Higher level demands

Procedures with connections

Doing mathematics

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

Which of the following would save more fuel?

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

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

Both changes save the same amount of fuel.

Fuel for thoughtAlex: I see that the change from 34 to 54 MPG is an increase of 20 MPG, but the 18 to 28 MPG change is only a change of 10 MPG. So, replacing the compact car saves more fuel.

Bo: The change from 34 MPG to 54 MPG is an increase of about 59% while the change from 18 to 28 MPG is an increase of only 56%. So the compact car is a better choice.

Student responsesChloe: I thought about how much gas it would take to make a 100-mile trip.

Compact car:

100 miles/54MPG = 1.85 gallons used

100 miles/34MPG = 2.94 gallons used

SUV:

100 miles/28MPG = 3.57 gallons used

100 miles/18MPG = 5.56 gallons used

The compact car saved 1.09 gallons while the SUV saved 1.99 gallons for every 100 miles. That means you actually save more gasoline by replacing the SUV.

Using technology

A general graph of what occurs with different MPG amounts…

What do you notice? Can you draw a conclusion?

Always, sometimes or never true?

If two rectangles have the same perimeter, they have the same area.

If two cubes have the same volume, they have the same surface area.

Putting your own spin on this

Think about any topic

Recast the content as questions that students can explore

Resist the temptation to tell students the content. Believe that students can investigate and derive relationships and mathematical concepts.

Where would this activity fit?

What is the level of demand?

How can I extend the activity?

How can I support students who are stuck?

Exploring activitiesPlan to get the most out of activities

Use problems that have multiple entry points

students at different levels of mathematical experience and with different interests all need to engage meaningfully in reasoning about a problem.

Plan questions for when

students get stuck;

students ‘think’ they have the solution;

students are unable to extend the problem further.

An abundance of sources for rich tasks

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

- Rich mathematical activities provide the opportunity for students to develop their thinking
- Sharing, examining and discussing student work develops students’ understanding of A, M and E

Discuss in your department

Participate in the online forum

- Feedback
- Discussion, questions and comments
- Ideas for tasks
- Moderating assessment

Look out for what is on offer next year

Discuss in your department

Participate in the online forum

- Feedback
- Discussion, questions and comments
- Ideas for tasks
- Moderating assessment

Look out for what is on offer next year

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