Four different structures for
engaging mathematics lessons
Peter Sullivan
Four lesson structures
What do you see as the characteristics
of a successful lesson?
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Actively engaged in learning (including the teacher)
Where the learning objective is met
Planned
Collaboration
The talk is about the maths
Students know the focus and purpose (learn ing intentions)
Multiple entry and exit points: differentiation (enabling and
extending)
• Challenging struggle
• Aware of success criteria
• Feedback, either self reflection
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Problem solving
Effective and diverse strategies
Various of approaches
Building on what they know
Relevant, real life connections
Conversation led by the children
Experience success
Opportunity for transfer
Deep and good questioning
Using the language of mathematics
Data collection that informs teaching
Confident teacher, willing to have a go, try new things
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Overview
• The basic unit of teaching is the lesson. We know that lessons
need to be varied, to address the needs of all learners
especially supporting students who need it and extending those
who are ready, to challenge students to do their best, to engage
students in interesting mathematics, and to allow teachers
flexibility for their individual decision making.
• Having clear lesson structures helps all teachers who plan in
teams, and especially those who have the opportunity to
observe each other teaching.
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This afternoon we will look at:
• rationale for focusing on lesson structures
• four types of lessons
• drawing on content from the Measurement
and Geometry Strand of the Australian
Curriculum
• connecting each lesson structures to the
proficiencies in the AC
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The current context
The Australian Curriculum: Mathematics offers us an ideal
opportunity to rethink the teaching of mathematics.
We know that many students
forget what they have learnt from one year to the next,
are unwilling to engage with challenging tasks,
develop negative attitudes to mathematics early.
We also know that there are too few students choosing middle
and high levels of mathematics study …
And that many adults are unable to use the mathematics they
have learnt
It is possible that these issues are a result of teachers overemphasising fluency with procedures.
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In other words ...
“More of the same” is not a desirable
option
It is not just improvement but CHANGE
that is needed
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The Curriculum explicitly challenges teachers to
connect understanding, problem solving and
reasoning with the content as well as fluency.
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Which of these are your top two
priorities?
That students:
• Enjoy the mathematics they are learning
• See the usefulness of mathematics to them
• Be able to interpret the world mathematically
• See the connection between mathematics learning
and their future study and career options
• Know that they can learn
• Know that they can learn mathematics
• Know that they can get smarter by trying hard
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Some Principles Underpinning
the Curriculum
• The curriculum seeks to encourage teacher
decision making
• The intention is that teachers will cover fewer
topics in more depth
• The expectation is that all students have
access to the full curriculum for as long as
possible
• Extension of those who are ready should be
within content at current level
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The Structure of the Curriculum
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Australian Curriculum
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Mathematics
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Three Content Strands (nouns)
• Number and Algebra
Number & Place Value, Patterns & Algebra, Fractions & Decimals, etc
• Measurement and Geometry
Using Units of Measurement, Shape, Location & Transformation
• Statistics and Probability
Chance, Data Representation and Interpretation
Year Level Achievement Standards
By the end of Year 1….
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So far there is not much difference from
what you are doing
It is the proficiencies that are different!
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In the Australian Curriculum
• Understanding
– (connecting, representing, identifying, describing,
interpreting, sorting, …)
• Fluency
– (calculating, recognising, choosing, recalling,
manipulating, …)
• Problem solving
– (applying, designing, planning, checking, imagining, …)
• Reasoning
– (explaining, justifying, comparing and contrasting,
inferring, deducing, proving, …)
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The Proficiencies
Why do we change from “working
mathematically”?
• These actions are part of the curriculum, not
add-ons
• Mathematics learning and assessment is more
than fluency
• Problem solving and reasoning are in, on and
for mathematics
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The (brand) new UK National Curriculum …
all pupils:
become fluent in the fundamentals of mathematics,
…, so that pupils have conceptual understanding and
are able to apply their knowledge rapidly and
accurately to problems
reason mathematically by following a line of enquiry,
conjecturing relationships and generalisations, and
developing an argument, justification or proof using
mathematical language
can solve problems by applying their mathematics to
a variety of routine and non routine problems with
increasing sophistication, including breaking down
problems into a series of simpler steps and
persevering in seeking solutions.
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https://www.education.gov.uk/schools/teachingandlea
rning/curriculum/nationalcurriculum2014/a0022061
0/draft-pos-ks4-english-maths-science
Four lesson structures
Choosing tasks and structuring
lessons will be the key
• If we are seeking fluency, then clear
explanations followed by practice will work
• If we are seeking understanding, then very clear
and interactive communication between
teacher and students and between students will
be necessary
• If we want to foster problem solving and
reasoning, then we need to use tasks with
which students can engage, which require them
to make decisions and explain their thinking
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• In other words, the curriculum informs not
only your yearly plans, planning of units, and
assessment, it also informs the structuring of
lessons
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An illustrative lesson
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I watched a mathematics lesson
when I was in Japan
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First the teacher told a story about
tatami mats that emphasised the
notion of area as covering
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Then the teacher posed the task
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The students had a worksheet with
TWO copies of the question on it that
emphasised to the students it was
the method, not the answer, that was
the focus
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How many squares?
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… And that they were meant to go
beyond counting the squares
The students worked individually
but talked with each other while
working
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The teacher selected students
to share their work, giving them
advance notice, an A3 sheet,
and a pointer
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What are the positive and negative
aspects of the lesson?
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My interpretation
• The lesson was connected to
students’ experience
–Relevance, engagement, utility
• It addressed at least one “important
idea” of mathematics
–Area is the number of rows times the
number of columns
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• The clear expectation is that
students learn from each other
–Culture, community, relationships
• The emphasis was on the process
not on the answer
–Quality of thinking, building capacity to
learn
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It was also going somewhere
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Why lesson structures?
• There are 3 levels of planning: yearly, units
and lessons are the most important of these
• Observing others and being observed is an
expectation of Aust Institute for Teaching &
School Leadership so we need a language and
structure for effective dialog about lessons
• Many teachers plan collaboratively (even if
concurrently) and so an understanding of
structures can help
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For the four lesson structures
• experience and activity happen prior to
instruction (or at least in 3 of them)
• representing solutions in different ways is both
engaging and important mathematically
• the classroom community can work to support
the learning of all
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For such lessons …
• the progression of tasks is important
• pedagogies and expectations should be
explicit
• individual work prior to pair or group
collaboration is critical
• few rather than many tasks
• building a classroom community
• a document camera helps
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What these approaches are NOT!
• Asking questions that are so easy that
everyone can do them
• Lessons that are so hard that the students feel
overwhelmed
• Setting up groups that might allow some
students to hide
• Excessive repetition (of course, some is
needed)
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An illustrative lesson
A filmed lesson for year 3/4 mixed ability group,
on the topic of difference.
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The learning task
The time is now 2:45.
The bus leaves at 10
past 4. How long is it
until the bus leaves?
This layout was
intended to
communicate the
need for two
different
methods
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The students were set to work with limited
explanation of the task, and they were not
shown how to do it.
VIDEO: Introducing the lesson
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For some students, a sheet was
provided that prompted particular
methods
Ten
past 4
2:45
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And some slight variations on the task
were prepared
(enabling prompts)
What could these have been?
• Posing questions –start with 3 o’clock…does
that help ?
• How about the way the time was written ?
2.45 and 10 past 4 ?
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Extension was in-built
The requirement to use two methods provided
challenge, and some were asked to find a third
method.
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Also prepared was the following
extension task in case it was needed
Work out how many days it is from June 29th to
September 7st without using a calendar.
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Discussion/Share Time
An important part of the lesson was the
opportunity for students to share their
thinking with the class.
VIDEO: Class discussion on the first task
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The consolidating task
Sammy goes to bed at quarter past 8 in the
evening. He gets up at ten to 7 in the morning.
How long has he been in bed?
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Thinking about the lesson structure
• In this view, the sequence
– Launch (without telling)
– Explore (for themselves)
– Summarise (drawing on the
learning of the students)
Launch
Summarise
Explore
• … is cyclical and might happen more than
once in a lesson (or learning sequence)
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Template One
Considering Options
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“Considering options” template
• Teacher poses and clarifies the purpose and goals of the task. If
necessary, the possibility of multiple responses can be discussed.
• Students work individually, initially, with the possibility of some
group work.
– Based on students’ responses to the task, the teacher poses variations.
– The variations might be a further challenge for some, with some
additional scaffolding for students finding the initial task difficult.
• The teacher leads a discussion of the responses to the initial task.
Students, chosen because of their potential to elaborate key
mathematical issues, can be invited to report the outcomes of
their own additional explorations.
• Tasks can be posed to consolidate the learning
• The teacher summarises the main mathematical ideas.
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For you to work on
• I did a multiplication question correctly for
homework, but my printer ran out of ink. I
remember it looked like
2_x3_=__0
• What might be the digits that did not print?
(Give as many answers as you can)
• (Please write neatly)
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Introduction
• What do they need to engage with the task
NOT
• What might make the task easier
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Consolidating
• A related task that allows students to apply
the earlier learning
• Sometimes this task might be less complicated
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Enabling prompts
• Reducing
– the number of steps
– the complexity of the numbers
– the form of representation
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Extending
32 × 25
20
30
2
Extended Notation
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5
Some examples of measurement
and geometry tasks that might be
used in this structure
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• Find something in the room longer than this string
• Cut a streamer so that it is longer than your hand
but shorter than your foot
• What are some words you could use to describe
parts of a box to show how big it is?
• What are some words to describe a tree to show
how big it is?
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• Find something that is about 2 shoes long.
• Find something that is longer than 2 shoes but
shorter than 3.
• Write your name using 50 matches
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• Michael and Monica measured the basketball court.
Michael said it was 20 rulers long. Monica said it was
19 ½ rulers long. How could this happen?
• Are you a tall rectangle?
• On your page, draw a line that is 1 m long
• Find something that is longer that 30 cm but less
than 40 cm?
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• Write your name using a line 1 metre long
• Find something that is twice as long as it is
wide
• How many different rectangles can you make
using a string that is 1 m long?
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• How would you work out the length of the
string used to tie up this box without untying
the string?
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• A park is enclosed by a fence that has 6 internal
right angles. What might the park look like?
• The total length of the fence of the park is 1
km. Describe some possible lengths of the
various sides.
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• One of the angles of a triangle is 720.
• What can you say about the other angles and
the triangle?
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Four lesson structures
In the Australian Curriculum
• Understanding
– (connecting, representing, identifying, describing,
interpreting, sorting, …)
• Fluency
– (calculating, recognising, choosing, recalling,
manipulating, …)
• Problem solving
– (applying, designing, planning, checking, imagining, …)
• Reasoning
– (explaining, justifying, comparing and contrasting,
inferring, deducing, proving, …)
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Template Two
Purposeful Games
and Puzzles
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Purposeful Games and Puzzles
• After explaining the rules and purpose of the PGP, the teacher demonstrates
the PGP to the class.
• Students engage in the PGP for a short while, after which there is a teacherled class discussion of the strategies and or mathematical point of the PGP
(also rules and possibilities may need to be clarified)
• The students are then offered further opportunity to engage with the PGP.
The teacher or the students can suggest variations, such as making the PGP
more challenging for some, or less complex for others. It is possible to group
students based on their success at the PGP, so that, for example, students
who complete the activity quickly might be grouped together for the next
implementation of the PGP.
• The teacher leads a discussion of the strategies and mathematics of the PGP.
Specific problems can be posed that allow either fluent practice that focuses
on the mathematical point, or extension of thinking.
• The teacher summarises the main mathematical ideas. The teacher has an
active role to find commonalities, patterns, and principles that can form the
basis of the formalisation of the intuitive insights developed during the
engagement with the PGP.
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Just imagining
Which ones of these can be folded to make a cube?
A
B
C
D
E
F
g
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This is the net of a cube.
2.
4
1
2
6
5
3
When it is folded which number will be opposite the 1?
When it is folded which number will be opposite the 2?
When it is folded which number will be opposite the 3?
When it is folded which number will be opposite the 4?
When it is folded which number will be opposite the 5?
When it is folded which number will be opposite the 6?
•
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What are some other
suggestions for such “different
representations” focusing on
geometry or measurement?
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Everyday objects connected with net etc
Angles in shapes
Telling the time
Area and perimeter
A 2D shape is one face of a 3D
Metre, cm. mm etc
Metre ruler and fractions of a metre
Symmetry and translations
Cross sections
Birds eye views
Different representations of parts of a litre
Descriptions of angles and properties
Units of measurement
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There are stacks of great number
games, but …
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Battleships
Treasure hunt
Mastermind
Feely bag (what is my shape)
Human chess
Celebrity heads
Tessellations
Connect 4
Block us
Tetris
Tangrams
Minecraft
Noughts and crosses in teams with one person facing away
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Pentominoes
Stop the clock
Matching digital and analogue
Rush hour
Safari
Tetris
Bee-bots
10 second walk
Where am I (map of school)
Hopscotch
Games need to be a learning experience !
…..
Four lesson structures
In the Australian Curriculum
• Understanding
– (connecting, representing, identifying, describing,
interpreting, sorting, …)
• Fluency
– (calculating, recognising, choosing, recalling,
manipulating, …)
• Problem solving
– (applying, designing, planning, checking, imagining, …)
• Reasoning
– (explaining, justifying, comparing and contrasting,
inferring, deducing, proving, …)
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Four lesson structures
Isometric drawing
• A key life skill is seeing things in 2D (on paper such as
house plans and maps, on screens) and imagining them
in 3D
• The reverse is also true. You need to be able to show
on paper things you have seen.
• One way of representing 3D shapes is by using
isometric drawings
• It is also important to imagine and draw objects
without seeing them
• The goal in this lesson is to draw using isometric paper,
and to imagine what shape might be represented
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Step 1
• Learning to use the isometric paper
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Start with a hexagon
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Draw the internal lines
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2 cubes might look like
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Using isometric paper, draw the
shapes made by sticking 3 cubes
together (whole faces touching)
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Draw a shape made from 10 cubes
• Go beyond the straight line
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Step 2: Imagining
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Describe what this building might look like from the front.
Side
Front
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Side
Front
Front
view
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Step 3
• 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
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The challenge
• 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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Template Three
The active teaching
template
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The active teaching template
• The teacher revises pre-requisite content with students, and
assesses current understandings.
• Teacher uses lively methods to illustrate or model aspects of
mathematics or procedures, building on suggestions from students,
emphasizing relational understanding and making connections with
previous learning. The teacher gives one or two carefully varied
practice examples, which are reviewed.
• Individually or in small groups, students complete further examples,
tasks or problems designed to give practice and consolidation of
the content that is the focus of the lesson. Teacher monitors work
of students, noticing solution strategies, adapting the questions if
necessary.
• The teacher reviews the methods and answers of the students, and
attends to particular problems or responses that assist in
consolidating the purpose of the lesson. Students might propose
questions of their own, and suggest extensions to the technique.
Four lesson structures
Area units
• The metric area units of square metres,
hectares, and square kilometres are in
everyday usage
• But their relationship to each other is not as
straightforward as for units of length, mass,
capacity
• The goal is that you can convert from one
metric area unit to another
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1 m2 is 100 cm × 100 cm
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1 m2 is 100 cm × 100 cm
or 10 000 cm2
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1 hectare is 100 m × 100 m
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1 hectare is 100 m × 100 m
or 10 000 m2
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1 km2 is 1000 m × 1000 m
or 1 000 000 m2
or 100 hectares
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Some exercises
• What do you guess might be the area of the
MCG?
• The MCG has an area of 20,000 m2
• What is that in hectares?
• What is that in km2?
• What do you estimate to be the area of this
room?
• What is that in hectares?
Four lesson structures
In the Australian Curriculum
• Understanding
– (connecting, representing, identifying, describing,
interpreting, sorting, …)
• Fluency
– (calculating, recognising, choosing, recalling,
manipulating, …)
• Problem solving
– (applying, designing, planning, checking, imagining, …)
• Reasoning
– (explaining, justifying, comparing and contrasting,
inferring, deducing, proving, …)
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Four lesson structures
HOW MUCH MONEY?
What is the maximum number of $1 coins that can
fit, lying flat in a single layer without overlapping, in
a shape with an area of 4 m2?
Explain how you worked out your answer
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Template Four
Practical
Representations
template
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Practical Representations template
After posing and clarifying the problem, the teacher asks the students to record
an estimate.
Students are invited to think about what strategies they might use to calculate
an answer, first individually, then brainstorming in a group, and the groups
report their strategies to the class.
Groups choose (or are allocated) a strategy, they implement the strategy to find
an answer, and prepare a report. The teacher monitors the work of the groups,
ensuring that all students are involved in the strategy implementation, and
anticipating groups who might report at the next phase.
The teacher leads a review of responses, including attending to issues such as
efficiency of a strategy, and appropriateness of the degree of accuracy. Ideally
the teacher selects few rather than all groups to report, particularly those that
are likely to contribute to the purpose of the activity.
Students complete more problems or exercises that consolidate the principles
identified in the investigation or prompt transfer to a related context.
The teacher summarises the main mathematical ideas addressed in the activity.
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How might you structure a lesson
based on this task?
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• A chameleon has a
tongue that is half as
long as its body ...
• … how long would your
tongue be if you were a
chameleon?
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What are some other “contexts” that
we could use for this template?
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How much water for the new fish tank?
Height and shoe size
How many dollar coins would cover MCG
1 million minutes
How many plants if they have to be 10 cm apart
Could you carry $1000000 coins
How long is 1km of $1 coins
1000000 now or 10c doubled
Multiply the recipe by 5/4
Compare 7 hands to 2 m
How many cups of flour in a bucket
Rabbit reproduction
Chickens lay 2 eggs per day, how long for an omlette
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• Make a scarf long enough to keep your
teddy warm
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• Compare your scarf with that of a friend
to see who has the longer scarf.
Four lesson structures
In the Australian Curriculum
• Understanding
– (connecting, representing, identifying, describing,
interpreting, sorting, …)
• Fluency
– (calculating, recognising, choosing, recalling,
manipulating, …)
• Problem solving
– (applying, designing, planning, checking, imagining, …)
• Reasoning
– (explaining, justifying, comparing and contrasting,
inferring, deducing, proving, …)
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In review…
Four lesson structures
“Considering options” template
-more than one method in order to provide challenge
• Teacher poses and clarifies the purpose and goals of the task. If
necessary, the possibility of multiple responses can be discussed.
• Students work individually, initially, with the possibility of some
group work.
– Based on students’ responses to the task, the teacher poses variations.
– The variations might be a further challenge for some, with some
additional scaffolding for students finding the initial task difficult.
• The teacher leads a discussion of the responses to the initial task.
Students, chosen because of their potential to elaborate key
mathematical issues, can be invited to report the outcomes of
their own additional explorations.
• Tasks can be posed to consolidate the learning
• The teacher summarises the main mathematical ideas.
Four lesson structures
Purposeful Games and Puzzles
-need to be a learning experience
• After explaining the rules and purpose of the PGP, the teacher demonstrates
the PGP to the class.
• Students engage in the PGP for a short while, after which there is a teacherled class discussion of the strategies and or mathematical point of the PGP
(also rules and possibilities may need to be clarified)
• The students are then offered further opportunity to engage with the PGP.
The teacher or the students can suggest variations, such as making the PGP
more challenging for some, or less complex for others. It is possible to group
students based on their success at the PGP, so that, for example, students
who complete the activity quickly might be grouped together for the next
implementation of the PGP.
• The teacher leads a discussion of the strategies and mathematics of the PGP.
Specific problems can be posed that allow either fluent practice that focuses
on the mathematical point, or extension of thinking.
• The teacher summarises the main mathematical ideas. The teacher has an
active role to find commonalities, patterns, and principles that can form the
basis of the formalisation of the intuitive insights developed during the
engagement with the PGP.
Four lesson structures
The active teaching template
-actively teaching particular ideas
• The teacher revises pre-requisite content with students, and
assesses current understandings.
• Teacher uses lively methods to illustrate or model aspects of
mathematics or procedures, building on suggestions from students,
emphasizing relational understanding and making connections with
previous learning. The teacher gives one or two carefully varied
practice examples, which are reviewed.
• Individually or in small groups, students complete further examples,
tasks or problems designed to give practice and consolidation of
the content that is the focus of the lesson. Teacher monitors work
of students, noticing solution strategies, adapting the questions if
necessary.
• The teacher reviews the methods and answers of the students, and
attends to particular problems or responses that assist in
consolidating the purpose of the lesson. Students might propose
questions of their own, and suggest extensions to the technique.
Four lesson structures
Practical Representations template
-try it out, show how you could do it
After posing and clarifying the problem, the teacher asks the students to record
an estimate.
Students are invited to think about what strategies they might use to calculate
an answer, first individually, then brainstorming in a group, and the groups
report their strategies to the class.
Groups choose (or are allocated) a strategy, they implement the strategy to find
an answer, and prepare a report. The teacher monitors the work of the groups,
ensuring that all students are involved in the strategy implementation, and
anticipating groups who might report at the next phase.
The teacher leads a review of responses, including attending to issues such as
efficiency of a strategy, and appropriateness of the degree of accuracy. Ideally
the teacher selects few rather than all groups to report, particularly those that
are likely to contribute to the purpose of the activity.
Students complete more problems or exercises that consolidate the principles
identified in the investigation or prompt transfer to a related context.
The teacher summarises the main mathematical ideas addressed in the activity.
Four lesson structures
What do you see as the characteristics
of a successful lesson?
•
•
•
•
•
•
•
Actively engaged in learning (including the teacher)
Where the learning objective is met
Planned
Collaboration
The talk is about the maths
Students know the focus and purpose (learning intentions)
Multiple entry and exit points: differentiation (enabling and
extending)
• Challenging struggle
• Aware of success criteria
• Feedback, either self reflection
Four lesson structures
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•
•
•
•
•
•
•
•
•
•
•
Problem solving
Effective and diverse strategies
Various of approaches
Building on what they know
Relevant, real life connections
Conversation led by the children
Experience success
Opportunity for transfer
Deep and good questioning
Using the language of mathematics
Data collection that informs teaching
Confident teacher, willing to have a go, try new things
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How can we improve the experiences of
students when they are learning mathematics?
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