Identifying and fostering quality
teaching in numeracy
Kaye Stacey
University of Melbourne
k.stacey@unimelb.edu.au
Improving Numeracy in Australia
Learning
standards
Now
The Future
Numeracy ~ Mathematics
Increasing age
PISA “Take the test”
www.oecd.org
• The following three items are publically
released, and available from the PISA “Take
the test” website. This contains items on
scientific literacy, reading literacy and
mathematical literacy (which is essentially
numeracy in the Australian usage).
PISA “Take the test”
www.oecd.org
PISA “Take the test”
www.oecd.org
Q1. .......
Q2. Explain how the graph
shows that on average the
growth rate for girls slows
down after 12 years of age.
6.3 According to this graph,
on average, during which
period in their life are
females taller than males of
the same age?
From Thompson & De Bortoli (2007) PISA in Brief
from Australia’s Perspective, ACER
PISA 2006 international
rankings
PISA 2006 tested 14170
Australian 15-year-olds
Statistically, 8 countries
better than Australia
Australian score range
(5th to 95th percentile)
is narrower than the
OECD average
Australia does well (here
and other studies)
Master of Numeracy –
Numeracy Subject
Master of Numeracy –
Numeracy Subject
mean
PISA 2006 international
rankings
PISA 2006 tested 14170
Australian 15-year-olds
Statistically, 8 countries
better than Australia
Australian score range
(5th to 95th percentile)
is narrower than the
OECD average
Australia does well (here
and other studies)
5th & 95th
percentiles
Master of Numeracy –
Numeracy Subject
Achievement of top 25%
PISA 2006 international
rankings
PISA 2006 tested 14170
Australian 15-year-olds
Statistically, 8 countries
better than Australia
Australian score range
(5th to 95th percentile)
is narrower than the
OECD average
Australia does well (here
and other studies)
Master of Numeracy –
Numeracy Subject
Percentage of students reaching TIMSS 2003
benchmarks
Year 4 Maths: International mean
18
Yr 4 - INT
Yr 4 - Year
AUS
4 Maths: Australia
19
12
Yr 8 - INT
Year 8 Maths: International mean
24
10
0%
20%
9
21
25
26
25
10%
25
37
26
Yr 8 - Year
AUS
8 Maths: Australia
30
16
36
30%
below Low benchmark
40%
Low
50%
5
7
22
60%
Intermediate
70%
High
80%
7
90%
Advanced
About 60 countries participated: Australia, USA, France, England,
Indonesia, Singapore, Japan, Brazil…
Master of Numeracy –
Numeracy Subject
100%
Percentage of students reaching TIMSS 2003
benchmarks
Year 4 Maths: International mean
18
Yr 4 - INT
Yr 4 - Year
AUS
4 Maths: Australia
19
12
Yr 8 - INT
Year 8 Maths: International mean
24
10
0%
20%
9
21
25
26
25
10%
25
37
26
Yr 8 - Year
AUS
8 Maths: Australia
30
16
36
30%
below Low benchmark
40%
Low
50%
5
7
22
60%
Intermediate
70%
High
80%
7
90%
Advanced
About 60 countries participated: Australia, USA, France, England,
Indonesia, Singapore, Japan, Brazil…
Master of Numeracy –
Numeracy Subject
100%
Percentage of students exceeding high TIMSS
2003 benchmarks
Year 4 Maths: International mean
18
Yr 4 - INT
Yr 4 - Year
AUS
4 Maths: Australia
19
12
Yr 8 - INT
Year 8 Maths: International mean
24
10
0%
20%
9
21
25
26
25
10%
25
37
26
Yr 8 - Year
AUS
8 Maths: Australia
30
16
36
30%
below Low benchmark
40%
Low
50%
5
7
22
60%
Intermediate
70%
High
80%
7
90%
Advanced
About 30% of Australian students
achieving “high benchmark”
Master of Numeracy –
Numeracy Subject
100%
Percentage of students exceeding high TIMSS
2003 benchmarks
Year 4 Maths: International mean
18
Yr 4 - INT
Yr 4 - Year
AUS
4 Maths: Australia
19
12
Yr 8 - INT
Year 8 Maths: International mean
24
20%
30%
About 70% of Singapore students
achieving “high benchmark”
40%
Low
50%
5
16
36
below Low benchmark
Master of Numeracy –
Numeracy Subject
26
25
10%
9
21
25
10
0%
25
37
26
Yr 8 - Year
AUS
8 Maths: Australia
30
7
22
60%
Intermediate
70%
High
80%
7
90%
Advanced
About 30% of Australian students
achieving “high benchmark”
100%
Schooling Issues Digest
(Stacey & Stephens)
• Performance by mean scores
• Performance by content area
– Which square has 2/3 shaded? (Int 57%, Aust62%, Singapore 93%)
• State and Territory comparisons
• Performance by student characteristics
–
–
–
–
–
Gender
Home language and migration background
Remote and Rural location
Indigenous students
Socio-economic background
• School, Classroom and Learning Influences
–
–
–
–
School environment and student attitudes
Students’ learning strategies
Teachers and classes
Instruction
Discussion
Time
What does a numerate student need to
know and be able to do?
Australian Curriculum: Mathematics (2010)
Proficiency strands: actions in which
students can engage when learning and
using the content
•
Understanding
–
•
Fluency
–
•
Students develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately,
efficiently and appropriately, and recalling factual knowledge and concepts readily. Students are fluent when
they calculate answers efficiently, when they recognise robust ways of answering questions, when they
choose appropriate methods and approximations, when they recall definitions and regularly use facts, and
when they can manipulate expressions and equations to find solutions.
Problem Solving
–
•
Students build a robust knowledge of adaptable and transferable mathematical concepts. They make
connections between related concepts and progressively apply the familiar to develop new ideas. They
develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students
build understanding when they connect related ideas, when they represent concepts in different ways,
when they identify commonalities and differences between aspects of content, when they describe their
thinking mathematically and when they interpret mathematical information.
Students develop the ability to make choices, interpret, formulate, model and investigate problem
situations, and communicate solutions effectively. Students formulate and solve problems when they use
mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan
their approaches, when they apply their existing strategies to seek solutions, and when they verify that their
answers are reasonable.
Reasoning
–
Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing,
proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically
when they explain their thinking, when they deduce and justify strategies used and conclusions reached,
when they adapt the known to the unknown, when they transfer learning from one context to another,
when they prove that something is true or false and when they compare and contrast related ideas and
explain their choices.
Mathematical Proficiency
• Content strands – knowledge base is essential
• Understanding — comprehension of mathematical
concepts, operations, and relations;
• Fluency — skill in carrying out routine procedures flexibly,
accurately, efficiently, and appropriately;
• Reasoning — capacity for explanation, justification,
logical thought,
• Problem solving — ability to devise methods to solve
problems, within or beyond mathematics
• Productive Disposition — habitual inclination to see
mathematics as sensible, useful, and worthwhile,
coupled with a belief in diligence and one’s own efficacy
• WHAT
• HOW
• WHY
• USE
Quality teaching attends to all aspects
of mathematical proficiency
•
•
•
•
•
Fluency (fashionable to under-rate this!)
Understanding (Box Plots)
Reasoning
Problem solving (PISA, Barbie Bungee)
Productive Disposition (Engagement article)
Practical activity to illustrate aspects of quality
teaching – especially “understanding”
What is a box plot and how is it interpreted?
Building a boxplot
Click here for Box Plot Construction
Discussion point
• What issues of learning maths/numeracy are
highlighted by this example?
• What features of good numeracy teaching were
demonstrated?
• In what ways was the data set well designed for
this ‘lesson’?
• Suggestions for improvement?
• Except for the obvious points – no practice
provided, little attention to meaning of the
statistical indicators,....
Key points about good numeracy teaching
• Obstacle: establishing relevance to individual
– affective: careful choice of context
– cognitive: demonstration of value of concept/technique
• Obstacle: abstractness of representation
– (student – score – point on number line – disappears into
box plot)
– so start with concrete data set, return as needed
• Obstacle – your suggestions?
Quality teaching attends to all aspects
of mathematical proficiency
•
•
•
•
•
Fluency (fashionable to under-rate this!)
Understanding (Box Plots)
Reasoning
Problem solving (PISA, Barbie Bungee)
Productive Disposition (Engagement article)
Mathematical tasks and student cognition:
Classroom-based factors that support and
inhibit high-level mathematical thinking and
reasoning
Henningsen & Stein 1997 JRME
• Engagement in high-level mathematical reasoning and
thinking important as
– To become better problem solvers
– To learn maths better
• The tasks that students do in class are the primary vehicle
for learning
• High-level tasks often lose their high-level character when
used in classrooms
H &S method
• Selected tasks which had desirable task
features to promote mathematical thinking
• Observed classroom implementation
• Asked
– What classroom factors supported high-level cognitive
processes by students?
– What happened when the cognitive demand declined?
• Looked for factors from the literature and
from the data
144
tasks
58
set up to encourage high level
thinking
36 declined
22
7 mixed
maintained
high level
reasons
8 - procedures
11 - unsystematic
without meaning
exploration
10
no math’l focus
Promoting engagement in
high level thinking
Promoting engagement in
high level thinking
• Built on prior knowledge and extended it
• Appropriate amount of time for engaging with task
• Active teaching
– High-level performance modelled
– Scaffolding that doesn’t remove challenge
– Consistently pressing students to explain and justify
• The article goes on to document what caused decline to
– procedures without connections,
– no mathematical activity, and
– unsystematic exploration. (See article for details).
• With the wider use of tasks with potential to engage
students in high level thinking, these ‘declines’ are also
now very eivdent in our schools.
What caused decline to procedures without
connections?
• Removal of challenging aspects, usually by
subtle alteration
• Too much or too little time allocated
• Focus on right answers only (“what do you
write down here”)
Decline to unsystematic exploration
Make a square metre, centimetre etc
• Students allowed too
much time to build sq
metre – too good
• In building, they focussed
on linear not area
measures
• Missed the maths –
relationships between
linear and area
dimensions
Not to scale
What caused decline to “no mathematical
activity?
•
•
•
•
Inappropriate task for those students
Class management problems
Too much or too little time
Sometimes lack of accountability (“didn’t
count” for marks)
Promoting engagement in
high level thinking
• Built on prior knowledge and extended it
• Appropriate time for engaging with task
• Active teaching
– High-level performance modelled
– Scaffolding that doesn’t remove challenge
– Consistently pressing students to explain and justify
• Avoiding the pitfalls that lead to
– Procedures without connections and meanings
– Unsystematic exploration
– No mathematical focus
Barbie Bungee
• Application of linear functions and graphing for Year 9
• Each student’s own Barbie is going to bungee jump
from a high part of the school (e.g. 7m walkway above
asphalt playground) on a rope made from elastic
bands. Students collect in the classroom from trial
(smaller) jumps and predict number of elastic bands for
real jump, then test.
• Google this, or see our RITEMATHS website for Year
9/10 algebra applications with technology for this and
other resources
http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS
160
Maximum drop (cm)
140
120
100
What features
may make this an
excellent learning
activity ?
How might it
decline from
higher order
thinking?
80
60
40
20
0
0
y = 22 + 9x
2
4
6
8
10
Number of bands
12
14
16
Features of Barbie Bungee
• A real world problem, put into classroom-practical form
– formulating math’l problem from real world problem
– Solving math’l problem
– Interpreting and checking in real world
•
•
•
•
High motivation to do well with your own Barbie
Can lose mathematical focus
Can become “procedures without connections”
Some teachers’ valuing of this task were entirely about the
motivational features, not the mathematical.
http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS
Quality teaching attends to all aspects
of mathematical proficiency
•
•
•
•
•
Fluency (fashionable to under-rate this!)
Understanding (Box Plots)
Reasoning
Problem solving (PISA, Barbie Bungee)
Productive Disposition (Engagement article)
Spotlight: Research into Practice VLNS
Max Stephens
Engagement in Mathematics: defining
the challenge and promoting good
practices
Stephens: Introductory pages
• Wide definition of engagement
– ABC Affective Behavioural Cognitive
• Reasons for disengagement
– Not ‘relevant’, difficult, confusing, causing criticism
• Apparent engagement through ‘busy work’
• Deeper aspects of engagement
– Procedural fluency
• individual improvement, personalised through ICT etc
– Conceptual understanding
• deeper connections, made explicit
– Reasoning and problem solving
• opportunities maintained at high level
– Individual and collective engagement , for different purposes
Stephens: Approaches to engagement
• Cognitive dimensions
– Adaptive: Self-efficacy, valuing learning, mastery orientation
– Mal-adaptive : Anxiety, failure avoidance, uncertain control
• Behavioural dimensions
– Adaptive: Persistence, planning, task management
– Mal-adaptive: Self-handicapping, disengagement
• Attend to students’
voice, belonging, choices, responsibility, success
– Article lists associated classroom values and practices, and
specific strategies.
Stephens (citing Baxter and Williams)
Scaffolding techniques to promote
engagement
• Analytic scaffolding
– Teachers supporting better learning
– 7 varieties
Discussion
Time
• Social scaffolding
– Teachers supporting good interactions in class
– 4 varieties
• Scaffolding autonomy and self-regulation
– Teachers supporting ’positive cognitions’
– 7 varieties
• Different students need different scaffolding
Discussion Points
• Select ONE type of scaffolding for your table
• How can each of the varieties of scaffolding be
put into action?
• What experiences are there at your school of
these?
• How can teachers be helped to use such
scaffolding?
Quality teaching attends to all aspects
of mathematical proficiency
•
•
•
•
•
Fluency (fashionable to under-rate this!)
Understanding (Box Plots)
Reasoning
Problem solving (PISA, Barbie Bungee)
Productive Disposition (Engagement article)
Thank you
k.stacey@unimelb.edu.au