Research into practice: What
we can learn from research
into good tasks
Peter Sullivan
AISNSW 2010
What are the challenges
you are experiencing in
teaching mathematics your
school?
AISNSW 2010
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Timeframe (too much curriculum)
Kids did not problem based teaching
Making it relevant, especially for low achievers
Buildinhg confidence
Diverse ability range
Retention
Busy lives of kids
Extension
External influences incluidng naplan, parent expectations n…
Levels of concentration
Gaps in prior knowledge
Disruptions to school routine
AISNSW 2010
Overview
• Findings from research
• The Australian mathematics curriculum
• 5 principles for improving teaching
AISNSW 2010
TASKS AND TEACHER ACTIONS
• We investigated ways that particular types of
mathematics classroom tasks create different
opportunities for students and different
challenges for teachers.
• the type of task influences the nature of the
learning (e.g., Christiansen & Walther, 1986;
Hiebert & Wearne, 1997)
AISNSW 2010
Task processing model
from task to lesson (Stein, 1996)
• Mathematical task as presented in instructional
materials
– which, influenced by the teacher goals, their subject matter
knowledge, and their knowledge of students, informs …
• … mathematical task as set up by the teacher in the
classroom
– which, influenced by classroom norms, task conditions,
teacher instructional habits and dispositions, and students
learning habits and dispositions, influences …
• … mathematical task as implemented by students
– which creates the potential for …
• … students learning.
AISNSW 2010
Teachers transform tasks
• Stein et al. (1996) noted the tendency of
teachers to reduce the level of demand of tasks.
• Doyle (1986) and Desforges and Cockburn
(1988) attribute this to complicity between
teacher and students to reduce risk
• Tzur (2008) … two key ways that teachers
modify tasks:
– at the planning stage;
– if responses are not as intended.
AISNSW 2010
Our goals were to describe
• how the tasks respectively contribute to
mathematics learning
• the features of successful exemplars of each type
• constraints which might be experienced by
teachers
• teacher actions which can best support students’
learning
Teachers

Teacher knowledge, attitudes
Tasks

Cluster meetings

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Classroom implementation


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Student survey
Teacher responses to tasks

Planning units of work

Teaching units of work

Reflection on teaching
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Students
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Student responses to units
Planning 3 lessons

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Teaching 3 lessons


AISNSW 2010
Self ratings of perception of mathematics in % (n = 930)
Rating
0
1
2
3
4
5
6
7
How good are you
at maths? (Q1)
0
1
3
11
21
33
24
8
How happy are you in
maths class? (Q2)
1
4
9
18
22
24
15
8
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While there was not much difference overall
between levels, there was a big difference
(mean 3.5 to 5.5) between classes
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In summary
• At each of these middle years levels there is a
range of satisfaction and confidence, and
teachers should be aware of this
• Teachers make a difference and they need
support to both find out the students levels of
satisfaction and confidence, and to do
something about it if they are low
Qu 9
In this table there are three maths questions that are pretty much the same type of
mathematics content asked in different ways.
Put a 1 next to the type of question you like to do (learn) most, 2 next to the one you
like (learn) next best and 3 next to the type of question you like (learn) least.
We don’t want you to work out the answers.
Movies tickets are $13 for adults and $7 for children. How much does it cost
for 2 adults and 4 children to go to the movies?
(word problem)
2 adults and 4 children went to the movies. They spent $120 on tickets. How
much might the adult and children’s tickets cost?
(open-ended)
(2 ×13) + (4 × 7)=
(number work)
AISNSW 2010
Preferences for liking, and learning from, the task
types as a % (Q9)
Task
Like most
Learn most
Number work
54
40
Word problem
35
23
Open-ended
12
37
AISNSW 2010
11. ... Put a 1 next to the type of question you like to
do (learn from) most, 2 next to the one you like
(learn from) next best, and 3 for the type of
question you like (learn from) the least:
Find the area inside the following shape:
A shape has an area of 10 square units.
What might the shape look like?
A running track has straights that are 100 m
long with half circles at the end. The inside
is all grass. What is the area of the grass?
AISNSW 2010
Preferences for the Task Types as a percentage
Type
Like most
Learn most
Area by counting
39
33
Practical calculation
20
44
Open-ended
42
22
AISNSW 2010
From a different survey after a unit of
work
AISNSW 2010
Student Preferences (%)
Task Type
Task favourite
2nd favourite
Task best for
learning
2nd best for learning
Finding similar graphs
1
0.0%
0.0%
2.1%
2.1%
Clues on cards
1
0.0%
3.1%
18.0%
10.4%
Using excel to present data
1
14.3%
6.0%
12.2%
9.2%
Matching graphs
1
2.0%
0.0%
2.0%
10.4%
Weather project
2
8.0%
6.0%
2.0%
2.1%
Two way tables
2
2.0%
8.0%
6.0%
6.3%
Average height of class
2
0.0%
6.0%
8.0%
2.1%
This goes with this
2
4.0%
2.0%
4.0%
14.6%
Most commonly used letters
2
6.0%
2.0%
8.0%
2.1%
Average height in school
2
26.0%
18.0%
8.0%
17.6%
A sentence with 5 words
3
4.0%
12.0%
8.0%
4.2%
Rock paper scissors
3
22.0%
27.8%
0.0%
4.2%
Seven people went fishing
3
10.0%
4.0%
18.0%
12.5%
Conducting a survey
3
2.0%
2.0%
2.0%
2.1%
Total:
50
50
50
48
Task
AISNSW 2010
Student opinions about their mathematics
classes: Some free format responses
AISNSW 2010
• In summary, it seems that the students were
extraordinarily articulate about what they wanted in their
maths lessons
• In synthesising the responses, students like lessons that
–
–
–
–
–
–
–
–
used materials (although these were not structured materials),
were connected to their lives,
involved games,
were practical with some emphasis on measurement,
in which they worked outside,
The see “like” and “learn” as different
with the method of grouping being important, and
over half of the students claim to like to be challenged.
AISNSW 2010
What does this mean?
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In summary
• The students were extraordinarily articulate
about what they wanted in their maths lessons
• The first characteristic of the responses is the
diversity of aspects on which the individual
students commented, suggesting that there is
no commonly agreed ideal lesson, and there
are many ways to teach well.
AISNSW 2010
• The ways of working in class are clearly
important for students, and however the
teacher intends that the student work, the
reason for this needs to be clarified for the
students..
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A preliminary task
• In 5 words or less, write down an aspect of
teaching mathematics that you would advise
beginning teachers to ensure they think about
in all of their lessons
AISNSW 2010
But first ...
• An underlying assumption is that at least
some learning should come
• from engagement of individuals
• with
– tasks
– each other
AISNSW 2010
Some of the key decisions
• Mathematics success creates opportunities
and all should have access to those
opportunities
• The curriculum should prioritise teacher
decision making
• The curriculum should foster depth and
important ideas rather than breadth
• Students can be challenged within basic
topics, including the advanced students
AISNSW 2010
There are 3 content strands
• Number and algebra
• Measurement and geometry
• Statistics and probability
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… and 4 proficiency strands
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Understanding
Fluency
Problem solving
Reasoning
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Key teaching idea 1:
• Identify big ideas that underpin the
concepts you are seeking to teach, and
communicate to students that these are
the goals of your teaching, including
explaining how you hope they will learn
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An example for us to discuss
• Write a sentence that has5 words, with an
average of four letters per word (no 4 letter
words)
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Some questions
• What is the mathematical point of that task?
• What is the pedagogical point of that task?
• How do you make these points explicit to
students?
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Some questions
• What mathematical actions can be addressed
by working on that task?
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Which card is better value?
Please explain your thinking.
AISNSW 2010
Some questions
• What is the mathematical point of that task?
• What is the pedagogical point of that task?
• How do you make these points explicit to
students?
AISNSW 2010
year 7
• Determine mean, median, and
range and use these measures to
compare data sets explaining
reasoning including use of ICT
AIZ Zone 2 & 3 Day 1
year 7
• to understand and become fluent
with written, mental and
calculator strategies for all four
operations with fractions,
decimals and percentages
AIZ Zone 2 & 3 Day 1
year 8
• Generalise from the formulas for
perimeter and area of triangles and
rectangles to investigate relationships
between the perimeter and area of
special quadrilaterals and volumes of
triangular prisms and use these to solve
problems
AIZ Zone 2 & 3 Day 1
year 9
• Work fluently with index laws in both
numeric and algebraic expressions
and use scientific notation, significant
figures and approximations in
practical situations
AIZ Zone 2 & 3 Day 1
year 9
• Solve problems involving linear
simultaneous equations, using
algebraic and graphical
techniques including using ICT
AIZ Zone 2 & 3 Day 1
year 10
• Understand and use graphical and
analytical methods of finding
distance, midpoint and gradient
of an interval on a number plane
AIZ Zone 2 & 3 Day 1
Key teaching idea 4:
• Interact with students while they engage in the
experiences, and specifically planning to
support students who need it, and challenge
those who are ready
AISNSW 2010
Establish classroom ways of working
• Examples of “norms”
– errors are part of learning
– all students must persist
– all students must be willing to justify their thinking
– working as a community of learners benefits
everyone
AISNSW 2010
An idea we can discuss
• 5 people went fishing. The mean number of
fish caught was 4, and the median was 3. How
many fish might each person have caught?
AISNSW 2010
Some questions
• What mathematical actions can be addressed
by working on that task?
• What might be the challenges in turning this
into a lesson?
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What are enabling prompts?
• Enabling prompts can involve slightly varying an aspect of the
task demand, such as
– the form of representation,
– the size of the numbers, or
– the number of steps,
so that a student experiencing difficulty, if successful, can
proceed with the original task.
• This approach can be contrasted with the more common
requirement that such students
– listen to additional explanations; or
– pursue goals substantially different from the rest of the class.
AISNSW 2010
Factors contributing to difficulty
• It may not be clear which aspects may be contributing to a
particular student’s difficulty, but by anticipating some of the
factors, and preparing prompts that, for example,
–
–
–
–
reduce the required number of steps,
simplify the modes of representing results,
make the task more concrete, or
reduce the size of the numbers involved,
• the teacher can explore ways to give the student access to the
task without the students being directed towards a particular
solution strategy for the original task.
AISNSW 2010
How might you adapt that task for
students experiencing difficulty?
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How might you adapt that task for
students who finish quickly?
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Sample Task 2
• Seven people went fishing. The mean number
of fish caught was 4, and the median was 3.
How many fish might each person have
caught?
Considering the quality of response
• Basic
– “The number of fish could be 1, 2, 2, 3, 4, 5, 11”
• Developed
– “The following are some possibilities:
•
•
•
•
0, 0, 0, 3, 4, 5, 16
0, 0, 1, 3, 4, 5, 15
0, 0, 0, 3, 4, 5, 16
etc
• Advanced
– “The total fish caught is 28. The middle number is
3, so the first 3 numbers of fish caught must have
a numbers that are 3 or less. For each set, the
latter three numbers make up the total. So if the
first 3 numbers total 5, then the latter 3 numbers
must total 20.”
Key teaching idea 5:
• Adopt pedagogies that foster communication, mutual
responsibilities, and encourage students to work in
small groups, and using reporting to the class by
students as a learning opportunity
AISNSW 2010
How do we turn this into lessons?
•
•
•
•
Launch
Explore
Summarise
review
AISNSW 2010
The Japanese have better words
• Hatsumon
– The initial problem
– Kizuki - what you want them to learn
• Kikanjyuski
– Individual or group work on the problem
– Kikan shido – thoughtful walking around the desks
• Nerige
– Carefully managed whole class discussion seeking the
students’ insights
• Matome
– Teacher summary of the key ideas
AISNSW 2010
My suggested words
•
•
•
•
Task introduction
Facilitation of student engagement
Whole class discussion
Teacher summary
AISNSW 2010
Key teaching idea 3
• Engage students by utilising a variety of rich
and challenging tasks, that allow students
opportunities to make decisions, and which
use a variety of forms of representation
AISNSW 2010
A story from Mr T (sent last week)
• In year 9 revision, the students were working
on this problem
– You earn $12 per hour for 22.5 hours. You pay 26%
of your earnings in tax. How much tax will you
pay?
•
•
•
•
•
A girl, Emma, wanted help.
Mr T: Do you have a job?
Emma : Yes
Mr T: How much an hour do they pay you?
Emma : I don’t know I just started
MAWA National issues
Mr T: Let’s say you ear $12 an hour and you
work for 3 hours. How much is that?
Emma : I don’t know. Do you divide?
Mr T: No. Think about earning $12 an hour.
You work one hour, and then another and
then another. How much have you
earned?
Emma : I don’t know
MAWA National issues
• Kylie (sitting nearby): Is it $36?
• Mr T: Yes. Good.
• Emma (to Kylie): God, you’re smart.
MAWA National issues
One of the key issues with the
Australian Curriculum is teacher
decision making
• Some examples of adapting
AISNSW 2010
Some activities
• Relationships
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The average height of 3 people in this
room is 1.7 m. You are one of those
people.
Who are the three people?
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Making questions open
Method 1:
• Write down a question and work out the
answer.
• Make up a new question that includes the
answer as part of the question.
Method 2.
• Write down a complete question including the
answer.
• Remove some of the question parts.
Race to 10
AISNSW 2010
What questions can we answer?
1
1
2
1
1
2
3
1
4
+
1
4
+
3
4
-
1
2
Using empty number lines
• 37 + 45
40
3
2
40
37
80
10
10
37
47
10
10
57
AISNSW 2010
67
82
3
77
80
2
82
• Use an empty number line to work out
1
22
4
-
1
7
2
5
1
2
3
14
4
2
1
15
4
1
20
4
1
22
4
The first
Key teaching idea
2:part of this
• Build on what the students know, both
mathematically and experientially, including
creating and connecting students with stories
that both contextualise and establish a
rationale for the learning
AISNSW 2010
One perspective on building on what
they know
• The idea of the self- contained “lesson”
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A sample lesson
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Using isometric paper, draw the
shapes made by sticking 3 cubes
together (whole faces touching)
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2 cubes might look like
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Draw a shape made from 10 cubes
• Go beyond the straight line
The next task
• A block of city buildings is 3 cubes wide and 3 cubes
long
• It looks like this from the front
On isometric paper, draw what the set of buildings
might look like
The task
• A block of city buildings looks like this from the side
• And like this from the front
• On isometric paper, draw what the set of buildings might
look like
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Side
Front
Front
view
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In what ways was that different
from a conventional
mathematics lesson?
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“Enabling prompts”
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A different destination
• A block of city buildings looks like this from the side
• Draw what the set of buildings might look like
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For students experiencing difficulty:
• Have some isometric paper with part of the
building already drawn.
• Have some cubes. Ask them to model what
the building might look like.
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How might you extend this for
students who finish quickly?
• How many different designs that fit the
directions can you make?
• Draw a set of buildings on the isometric
paper, and draw the front and side view
as well.
AISNSW 2010
• Draw what this representation might mean
1
3
1
2
3
3
1
1
2
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Key teaching idea 2:
the second part
• Build on what the students know, both
mathematically and experientially, including
creating and connecting students with stories
that both contextualise and establish a
rationale for the learning
AISNSW 2010
Getting to work
Josie
Loreto
20 km
How much does it cost Josie to get to work
and back home again?
Assume that it costs
$2 per km for the
full costs
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Getting to work
Josie
Loreto
73 km
How much does it cost Josie to get to work
and back home again?
Assume that it costs
$1.37 per km for
the full costs
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Getting to work
Josie
Loreto
x km
How much does it cost Josie to get to work
and back home again?
Assume that it costs
$z per km for the
full costs
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Getting to work
5 km
Josie
Assume that it
costs $2 per km
for the full
costs
Loreto
Carmen
20 km
How much should Carmen give Josie if she
picks her up and takes her home?
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Getting to work
13 km
Josie
Assume that it
costs $1.37 per
km for the full
costs
Loreto
Carmen
59 km
How much should Carmen give Josie if she
picks her up and takes her home?
AISNSW 2010
Getting to work
x km
Josie
Assume that it
costs $z per km
for the full
costs
Loreto
Carmen
y km
How much should Carmen give Josie if she
picks her up and takes her home?
AISNSW 2010
Assume that it
costs $2 per km
for the full
costs
Getting to work
5 km
5 km
Josie
Carmen
Susan
Loreto
20 km
How much should Susan and Carmen give
Josie if she picks them up and takes them
home?
AISNSW 2010
A preliminary task
• In 5 words or less, write down an aspect of
teaching mathematics that you would advise
beginning teachers to ensure they think about
in all of their lessons
AISNSW 2010