Mathematics and the NCEA realignment
A webinar
facilitated by
Angela Jones
and
Anne Lawrence
Mathematics and the NCEA realignment
• Introductions
• Background to the changes
• The new achievement standards matrix in
mathematics and statistics
• SOLO taxonomy and Achievement standards
• 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
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
Aligned curriculum and assessment
The New Zealand Curriculum published in 2007 clearly states that
students are to be engaged in thinking mathematically and statistically, and
to be modelling situations and solving problems at all curriculum levels.
The New Zealand Curriculum is asking us to inspire young people to
become successful learners, confident individuals and responsible citizens
A mathematics curriculum should focus on mathematics content and
processes that are important and worth the time and attention of students.
Alignment and coherence of the three elements – curriculum, standards,
and assessment – are critically important foundations of mathematics
education.
Aligned curriculum and assessment (cont)
Achievement standards are aligned to The New Zealand
Curriculum.
Unit Standards with curriculum links have been replaced by
achievement standards.
Assessment tasks are written to reflect good assessment
practices and the type and level of thinking required by the
curriculum and the achievement standards.
The SOLO taxonomy
SOLO stands for the Structure of Observed Learning
Outcomes. It was developed by Biggs and Collis (1982).
Biggs describes SOLO as “a framework for understanding”.
(1999, p.37)
SOLO identifies five stages of thinking. Each stage
embraces the previous level but adds something more in
terms of the level of thinking.
The stages of SOLO
• Prestructural
the student acquires bits of unconnected information that have no
organisation and make no sense.
• Unistructural and Multistructural
students make connections between pieces of information but not the
meta-connections between them
• Relational
the students sees the significance of how various pieces of information
relate to one another
• Extended abstract
students make connections beyond the scope of the problem or
question, to generalise or transfer learning into a new situation
Surface and deep thinking
Unistructural and multistructural questions test
students’ surface thinking (lower-order thinking skills)
Relational and extended abstract questions test deep
thinking (higher-order thinking skills)
Use of SOLO allows us to balance the cognitive
demand of the questions we ask and to scaffold
students into deeper thinking and metacognition
Multistructural questions
Students need to know or
use more than one piece of
given information, fact, or
idea, to answer the
question, but do not
integrate the ideas.
This is fundamentally an
unsorted, unorganised list.
Relational questions
These questions require
students to integrate more
than one piece of given
knowledge, information, fact,
or idea.
At least two separate ideas
are required that, working
together, will solve the
problem.
Extended abstract questions
These questions involve a
higher level of abstraction.
The items require the
student to go beyond the
given information,
knowledge, information, or
ideas and to deduce a more
general rule or proof that
applies to all cases.
A change of focus in mathematics
• The SOLO hierarchy shifts us from the situation in maths
of “doing more” and “ doing longer tasks” to qualify for a
higher grade.
• This change recognises and rewards deeper and more
complex thinking along with more effective
communication of mathematical ideas and outcomes.
• It is these that are fundamental competencies to
mathematics.
.
An example of the levels in algebra
Houses
Sticks
1
5
2
9
3
__
1. How many sticks are needed for 3 houses? (unistructural)
2. How many sticks are there for 5 houses?
(multistructural)
3. If 52 houses require 209 sticks, how many sticks do you need
to be able to make 53 houses?
(relational)
4. Make up a rule to count how many sticks are needed for any
number of houses.
(extended abstract)
Multistructural
This cylinder is part of a
model for a rocket. The
radius is 3cm and the
height is 5cm.
Neil would like to know the
volume of the cylinder
Relational
This solid is a model of a
rocket. The cylinder has a
radius of 3cm. The height of
the cylinder is 5cm and the
height of the cone is 4cm.
James would like to know the
ratio of the volume of the
cylinder to the volume of the
cone
Extended abstract
This solid is a model of a rocket and
consists of a cone placed on top of a
cylinder.
The total height of the solid is 8cm.
Investigate the possible ratios of
volume of cylinder : volume of cone
Achievement standard
standard 1.4
Achievement
Achieve Apply linear algebra in solving problems
Merit Apply linear algebra, involving relational
thinking, in solving problems
Excellence Apply linear algebra, involving extended
abstract thinking, solving problems
Achievement
1.4
AS1.4: Applying standard
linear algebra
in solving problems
What does the SOLO taxonomy suggest that this might
look like for achieve, merit and excellence?
Excellence
Merit
Achieve
Explanatory notes:
1. The following achievement objectives taken from the Equations
and Expressions, and Patterns and Relationships threads of the
Mathematics and Statistics learning area, are related to this
standard:
• form and solve linear equations
• solve linear equations and inequations and simultaneous
equations in two unknowns
• relate graphs, tables, and equations to linear relationships, relate
rate of change to the gradient of a graph.
•
Explanatory note 3:
Problems are situations that provide opportunities to apply knowledge or
understanding of mathematical concepts.
The situation will be set in a real-life context.
The phrase ‘a range of methods’ indicates that there will be evidence of
at least three different methods.
Students will be expected to be familiar with methods related to:
• using formulae
• forming, graphing or manipulating linear models such as C= 8+0.75t
when solving problems
• relating rate of change to the gradient of a graph
• using simultaneous equations, inequations or graphs when solving
problems such as those involving simple linear programming.
What does the standard say for achieve?
Applying linear algebra will involve using a range
of methods in solving problems, demonstrating
knowledge of algebraic concepts and terms, and
communicating solutions which would usually
require only one or two steps.
What does the standard say for merit?
Relational thinking will involve one or more of:
• selecting and carrying out 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.
What does the standard say for excellence?
Extended abstract thinking will involve one or
more of:
• demonstrating understanding of abstract concepts
• developing a chain of logical reasoning, or proof
• forming a generalisation,
and using correct mathematical statements,
or communicating mathematical insight.
Task 1.4A Taxi charges
At Wiaawe airport, there are three different taxi companies
available for hire: Fred’s Taxi Company, P and G Taxi
Company and Flatrate Taxi Company
The Resource page gives the hire charge for each company and
the distances to common destinations in Wiaawe from the
airport. Represent the three taxi companies' charges using the
same representation, for example, three equations or three
graphs with the same variables and scale.
•Recommend which taxi company to use for a trip to two of the common
destinations.
•Recommend distances for which would it be cheapest to use P and G
Taxi Company.
•Fred, who owns Fred’s Taxi Company, wants to be the cheapest taxi
company people can use to travel to any destination. Write and describe
at least two different ways Fred could realistically change his charges to
achieve this goal. Include specific examples of the rates he could use.
Fred’s Taxi
P and G Taxi Company Flatrate Taxi Company
Company
Cheap rates!
Our advantage is clear!
Come with us!
C = 0.65D + 2
$25.00 flat fee to any
Fixed charge: $5.00
destination (up to 60 km)
where
C
is
the
cost
in
Per kilometre: $0.50
Additional charges apply
dollars, and D is the
over 60 km.
distance in kilometres
Destination
Distance (km)
City Centre
17
Port
24
Bethlehem
45
Evidence for achievement
A student uses different methods by:
• using formulae
• Fred’s cost to city = 0.5 x 17 + 5 = $13.50
• forming, graphing or manipulating linear models
Fred’s: y = 0.5x + 5
Or
Fred’s graph
• relating rate of change to the gradient of a graph
• using simultaneous equations, inequations or graphs
Evidence for merit
A student might demonstrate understanding of concepts by solving
Fred and PG equations simultaneously
C = 0.5D + 5 and C = 0.65D + 2 to give distance of 20km, where
their charges are the same, so PG is cheapest for distances less
than 20km
Clearly communicated, with evidence of method and use of mode
An example of selecting and carrying out a logical sequence of steps:
To go to the Port the costs are: Fred = 0.5 x 24 + 5 = $17, PG =
0.65 x 24 + 2 = $17.60 , Flat = $25. Use PG because it is the
cheapest
Clearly communicated, with evidence of method and use of model.
Evidence for excellence
A student might provide a chain of logical reasoning
For all distances over 40km he needs to drop to a flat rate
(like Flatrate Taxi Company). The rate needs to be
anything less than $25 per journey. e.g. Rate = $24 for any
length journey over 40km
And for any journey less than 20 km Fred needs to have a
charge which is lower than P&G.
e.g. His charges could be C = 0.65D + 1.5.
His fixed charge is $1.50 and his rate is the same as P&G.
Evidence for A, M, E
Final grades will be decided using professional
judgement based on a holistic examination of the
evidence provided against the criteria in the
Achievement Standard.
Process from here
Online forum:
• How does this work?
Homework:
• Trial the task,
• Mark student work - keep notes about judgements
but leave student work unmarked (raw)
• Select student work to cover a range of grades
• Submit raw student scripts
Next steps
•
•
•
•
Trial the task
Mark
Select and submit student scripts
Participate in online forum