Mathematics and the NCEA realignment
Part three
Webinar
facilitated by
Angela Jones
and
Anne Lawrence
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
Introductions
Mathematics and NCEA realignment
Angela Jones
Senior adviser
Secondary Outcomes Team
Ministry of Education
angela.jones@minedu.govt.nz
Anne Lawrence
Adviser in Numeracy, Mathematics & Statistics
Massey University College of Education
a.lawrence@massey.ac.nz
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.
Achievement
standard
Key
skills and knowledge
for1.4
1.5
Measurement includes the use of standard international metric units for
length, area, capacity, mass, temperature, and time. Derived measures
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.
Achievement
Solving
problemsstandard
at A, M and1.4
E for 1.5
Solving 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.
Achievement
standard
1.4
What
does excellence
look like?
Excellence
Merit
Achieve
Student B
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?
From Dan Meyer (US maths teacher)
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:
1. What do you see as Dan’s key message(s)?
2. What are the implications for the classroom?
3. What are the key message(s) for you from Dan’s talk?
From Dan Meyer (US maths teacher)
• Dan Meyer Ted Talk
• http://www.youtube.com/watch?v=BlvKW
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.
Rich mathematical activities
Key questions:
• What sorts of activities are appropriate?
• How do we support students to access these activities?
• What are appropriate levels of scaffolding?
Levels 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!”
Fuel for thought
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.
Student responses
Alex: 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.
Chloe: 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
Fuel for Thought
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.
Exploring activities
•
•
•
•
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?
Plan 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
Key implications
– 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
Next steps
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
Next steps
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