National Partnerships Schools’
Forum
Korumburra Secondary College
Case Study
Algebra, Differentiation and the PLT
South Gippsland Maths Initiative 2009
Ideas that resonated
•
From inference to evidence
• Data led collaborative inquiry
• PLT – sharing
- “Our” students not “my” students
• importance of team leadership
- Purpose
- Process
- Structure
Coaching 2009
Effect/influence of coaching
• focus on the main idea/s for each lesson/unit
• Development of differentiated tasks/activities
• How will I know what students have learnt?
• Reflection with the coach after the lesson
• Model these coaching ideas with members of my staff
• PLT focus – staff to try at least one of the activities with their classes
and provide feedback
Last year 2010
• Development of units which include
- pre-tests
– a move away from text driving teaching and methodology
- data collection at relevant times
- focus on the big ideas and links to the developmental continuum
- one task which allows for differentiation
- starting to listen for the “golden question” and its implications for my
practice.
• link Year 7 units to Year 8 program
• TPL
• Patrick Griffin – Specialist Certificate in Leadership of Assessment
Practices
PLTs at Korumburra SC
•The Mathematics Year 7/8 PLT comprises 5 teachers; one who teaches
two year 7 classes, one who teaches two year 7 classes and one year 8
class, the PLT leader who teaches a year 7 and a year 8 class, and 2
teachers who each teach a year 8 class.
•PLT members vary in teaching experience from 2 to 30 plus years. Two of
the three teachers teaching Year 7 classes have less than 5 years of
Mathematics’ teaching experience. PLT meetings are held every fortnight
for 1 hour after school. Occasionally PLT members have to attend other
PLT meetings scheduled for the same day e.g. Year 7 PLT.
PLT structure
• Ideally the PLT should –
• have between 4 and 6 members
• have a dedicated meeting every fortnight commencing at
the start of period 6 and continuing until 4.30pm. PLT
members are covered for period 6 classroom duties.
• have teachers teaching more than one class per year level.
Teachers can trial and embed their practice.
Structure of fortnightly
PLT meetings
1. Group check in:
• What’s been happening in your classroom in
the last fortnight. What learning have you
observed in the content area?
2. Mathematics content learning
• Semester 1 focus : fractions
• Semester 2 focus : algebra
• Readings, research, VELS
• Classroom observations/ modelled teaching
What is important to consider in the teaching of algebra in Year 7?
Present the power of
algebra as being a
general rule to show
relationships between
variables. The
relationship holds no
matter what numbers
we input.
Students have a
strategy to explore
relationships between
variables.
What is the effect of
the National
Curriculum on
planning in Algebra?
The use of concrete materials
: to model a scenario
: give all students an opportunity
to access a task, predict and
check/verify
: to generate a table of values
: physically manipulate – scaffold
learning from concrete to
abstract- keep a picture in
students’ heads [a visual tool] – a
way to visualise different
strategies /solutions which helps
to see equivalence of methods
e.g. Max’s, Leanne’s and Di’s
solutions to matchstick patterns
When reviewing number develop number
properties
Applying number properties to patterns and
pronumerals
– equivalence
– structure and use these to develop
understanding of number
Provide opportunities for students to explore in a
safe learning environment
Explore strategies that students use, validate and
verify them, explaining their thought processes,
giving students ownership of the direction of the
conversation i.e. look for “golden questions”.
Use data to find out where students are on the
“algebraic continuum” so we can plan our
teaching and learning experiences to meet the
needs of all students and progress their learning.
Big Ideas in Algebra [from the Maths Continuum]
Level 3

Sign equivalence

Number properties

Number sentences with missing numbers

Grids
Level 4

Shapes and objects according to criteria

Rules for sequences
– recursion [additive thinking]
Formula [multiplicative thinking] written or and
symbolic

Equivalence between expressions

Relationships between variables and describe in
words

Inverse operations, words and symbols to form
simple equations
Level 5

Commutative, associative and distributive
properties

Multiplication and division of power terms

Inverses to rearrange formulas

Use linear and other functions to model
situations

Solve simple equations using tables, graphs, and
inverse operations

Identify function and represent it by a table of
values, graph and rule

Describe and specify independent variable of a
function and domain and the dependent
variable. Construct a table of values and graphs
Pre-assessment
task
Algebra story board to develop concepts on Growing Arms
[four arm shapes]/World’s Longest Lunch Scenario
No
No
Yes
Yes
Yes to describe the pattern
Yes – this needs to be stressed
Yes
Yes
Yes
Yes
Yes
yes
Yes
Yes [both written and symbolic]
Yes [Question 5]
Yes [Q2,3,5]
Yes
Yes
Yes [Q4]
yes
Yes – extension into inverse operations depending on the
problem posed
no
Yes
no
yes [Q2,3,4,5]
yes
Yes – build in this exploration of structure
Yes if we can use this type of problem [quadratic]
yes
yes
yes
partly
no
no
Yes
yes – graph allows this exploration
yes
PLT process
• Teachers bring agreed work sample to
meeting (pre-assessment task to begin
cycle)
• Teachers share work samples across all
classes using agreed process (e.g. rubric)
• Select a group of like students and examine
evidence of what students can do, are on
verge of doing
Zone of Proximal Development - zpd
• Vygotsky’s construct of the Zone of Proximal Development - “ a state of
readiness”.
• The zone where success had odds of 50:50 pointed to the location on a
continuum or trait where intervention had the best chance of assisting
development. [George Rasch]
• Intervention is based on a generalised development not on a specific
item-based interpretation of learning (or lack of learning).
• As a result intervention can be linked to appropriate provision of
resources leading to informed curriculum and learning policy.
Patrick Griffin “Studies in Educational Evaluation”
Sixty percent of student achievement is directly attributable to
the teaching practice in the classroom. (Leithwood, 2004)
Step 1 or 5
Gathering Data and
Analysis
CAT (eg. NOYCE Task)
Step 4
Student Work Samples
Step 2
Identifying Students
Understandings.
Learning monitored
(eg. PLT Log using mode)
PLT Log analysis
Step 3
Setting Student Learning Goals
New Knowledge
Algebra Task Analysis
Year Level
7E
Range: 7
Mode:
4
Median: 4
Mean:
4
Mairead
Tamara
0
Ashley
1
Noyce Task: Necklaces Date July,2010 School: Korumburra SC
[maximum 11]
Jaiden
Troy
Bayley
Nathan
Izaak
Shane
Gemma
Andrew
Bailey Natasha Maddie
2
3
4
Ashlea
Brooke
Mitch
5
Amber
Cooper
Justine
6
Tenae
Sophie
7
8
Algebra Necklaces Task - Pre-test
score
11
10
9
8
7
6
5
4
3
2
1
0
7E
7A
7B
7C
____
____
____
____
____
7D
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
Sixty percent of student achievement is directly attributable to
the teaching practice in the classroom. (Leithwood, 2004)
Step 1 or 5
Gathering Data and
Analysis
CAT (eg. NOYCE Task)
Step 4
Student Work Samples
Step 2
Identifying Students
Understandings.
Learning monitored
(eg. PLT Log using mode)
PLT Log analysis
Step 3
Setting Student Learning Goals
New Knowledge
Necklaces Task Yr 7 What students can do and implications for teaching.
Score
0–1
Attempted the table
Use concrete materials
Make a pattern/complete
Draw the pattern
Copy and continue patterns
Form an array from a pattern
Work from a graphic / picture / diagram
2-3
Completed part of table that relates to
picture
Use concrete materials
Make predictions about the pattern and trial predictions
Start to verbalise how they have made their patterns
What’s The Rule? Making links between related numbers
Transfer patterns into table formats
Read information in tables
Focus on differences as part of a relationship in patterns
4–5
Completed the table with most correct Experiences using algebra storyboard
Use difference patterns
Get answers using draw all / count all
Transfer information from a table to a general formula
Verbalise information in a table to be able to find a rule
Continue counting patterns (oral, visual, symbolic)
Be able to verbalise and utilise mathematical operations/expression
6–7
Completed the pattern
Rule verbalised using additive thinking
Predominately draw all strategy for Q4
8 – 11
Rule verbalised
Guess my rules, using rules
Used a table or division strategy to find Formulate number patterns into rules e.g. a general rule/formula
Q4
Graph results
Use more complicated patterns and represent solutions in a variety of ways
Create new patterns
Verbalise and represent rules from patterns in a number of ways
Explore connections between additive and multiplicative thinking
Move to arithmetic reasoning strategy – the structure behind a problem
What are we drawing when we add a square? What’s the mathematical way to explain that?
If we have 4+3+3+3 … to get to 37 what are some strategies we might use to work out how many?
Sixty percent of student achievement is directly attributable to
the teaching practice in the classroom. (Leithwood, 2004)
Step 1 or 5
Gathering Data and
Analysis
CAT (eg. NOYCE Task)
Step 4
Student Work Samples
Step 2
Identifying Students
Understandings.
Learning monitored
(eg. PLT Log using mode)
PLT Log analysis
Step 3
Setting Student Learning Goals
New Knowledge
• Set a learning goal for the students
• Team members suggest teaching
strategies to support those students
• Agree on what would show that the
students reach the goal
• Bring back agreed evidence to next
meeting in a fortnight and look at next
group of students
Student Learning Log
What can this/these student/s write and do?
Student: Mitch, Brooke, Amber, Maddie, Shane
Level: 4- 5
Review date: 7th September
Completed the table with most correct
Get answers using draw all / count all
What is this student indicating that What teaching strategies What resources are needed?
they can nearly do? [verge]
could you investigate?
Story board, counters, blocks
Transfer information from a table to a general
Use of the algebra story What evidence would show that the
formula
student is moving on? [learning]
Verbalise information in a table to be able to find board
a rule
Continue counting patterns (oral, visual,
symbolic)
Be able to verbalise and utilise mathematical
operations/expression
Emphasis on completing
the table correctly and
looking for a rule
Can verbalise and /or write a rule in
words/ or as a picture
Can write a rule in symbols
Can use a rule to predict patterns
What are the links to other dimensions [Number, Space, Measurement, Chance and Data] or
domains?
Reviewing number and develop number properties
Applying number properties to patterns and pronumerals
– equivalence
– structure and use these to develop understanding of number
Sixty percent of student achievement is directly attributable to
the teaching practice in the classroom. (Leithwood, 2004)
Step 1 or 5
Gathering Data and
Analysis
CAT (eg. NOYCE Task)
Step 4
Student Work Samples
Step 2
Identifying Students
Understandings.
Learning monitored
(eg. PLT Log using mode)
PLT Log analysis
Step 3
Setting Student Learning Goals
New Knowledge
• Questions/Areas to investigate which may arise during
the use of the Algebra Story Board [Longest Lunch
context]
• clarification of information e.g. tables are side to side
• interpretation of the pattern given in words into
symbols e.g.
“the number of chairs is the number of tables times two
plus grandma and grandpa” means
c= tx2+2
“the number of chairs is equal to two times the number of
tables plus two” means
c=2xt+2
or
c=2x(t+2)
Are these three equations the same? How can you check?
Number of
tables
t
1
2
3
4
10
Number of
chairs c
4
6
8
10
22
c= t x 2 + 2
c= 1x2 + 2
=4
c= 2 x t + 2
c= 2x1 + 2
=4
c= 2 x (t +
2)
c= 2x(2+2)
= 2x(4) =
8
If there were 20 tables, how many chairs would be needed?
Can you explain why the answer is not 44 [i.e. 2 x 22 as there
were 22 chairs for 10 tables]?
How many tables are needed to seat 122 people?
How many triangular table would seat 122people?
From the point/line graph the connection between gradient
and y intercept and the rule/equation can be investigated.
Sixty percent of student achievement is directly attributable to
the teaching practice in the classroom. (Leithwood, 2004)
Step 1 or 5
Gathering Data and
Analysis
CAT (eg. NOYCE Task)
Step 4
Student Work Samples
Step 2
Identifying Students
Understandings.
Learning monitored
(eg. PLT Log using mode)
PLT Log analysis
Step 3
Setting Student Learning Goals
New Knowledge
9
Class 7E Pre and post test data for the
Beaded Necklaces Algebra task
8
7
6
5
4
3
2
1
0
Question 4 /2
Question 3 /3
Questions 2&3
/3
Question 1
/3
Two points in time data
Year 7 KSC 2010
Two Points in Time Histogram for the Algebra
Beaded Necklace task
25
F
r 20
e
15
q
u 10
e
5
n
c 0
y
Frequency Pre-test
Frequency Post-test
0
1
2
3
4
5
6
Score
7
8
9 10 11
Implications of cohort report
• Significant shift from
lower end to higher end
Score 0 – 3 from
35 students to 14 students
Score 4 – 7 from 61 students to 63 students
Score 8 – 11 from
2 students to 19 students
Mode from 4 to 7
Algebra Necklaces Task - Pre-test – Post-test comparison
score
11
10
9
8
7
6
5
4
3
2
1
0
7E
7A
____
7B
7C
____
7D
____
____
____ ____
____
____
____ ____
____
____
____
Pre Post
7A & 7B
7C & 7D
7E
____
____
____
____ ____
____
____
____
Pre
Post
Teacher A
Teacher B
Teacher C
____
____ ____
____
____
____ ____
____
____
Pre
Post
____
____ ____
____ ____
____
____
____ ____
Pre
Post
____ ____
____
____
____ ____
____
____ ____
Pre
Post
Necklaces Task Yr 7 What students can do and implications for teaching.
Score
0–1
Attempted the table
Use concrete materials
Make a pattern/complete
Draw the pattern
Copy and continue patterns
Form an array from a pattern
Work from a graphic / picture / diagram
2-3
Completed part of table that relates to
picture
Use concrete materials
Make predictions about the pattern and trial predictions
Start to verbalise how they have made their patterns
What’s The Rule? Making links between related numbers
Transfer patterns into table formats
Read information in tables
Focus on differences as part of a relationship in patterns
4–5
Completed the table with most correct Experiences using algebra storyboard
Use difference patterns
Get answers using draw all / count all
Transfer information from a table to a general formula
Verbalise information in a table to be able to find a rule
Continue counting patterns (oral, visual, symbolic)
Be able to verbalise and utilise mathematical operations/expression
6–7
Completed the pattern
Rule verbalised using additive thinking
Predominately draw all strategy for Q4
8 – 11
Rule verbalised
Guess my rules, using rules
Used a table or division strategy to find Formulate number patterns into rules e.g. a general rule/formula
Q4
Graph results
Use more complicated patterns and represent solutions in a variety of ways
Create new patterns
Verbalise and represent rules from patterns in a number of ways
Explore connections between additive and multiplicative thinking
Move to arithmetic reasoning strategy – the structure behind a problem
What are we drawing when we add a square? What’s the mathematical way to explain that?
If we have 4+3+3+3 … to get to 37 what are some strategies we might use to work out how many?
Teams owning the work
• This PLT model will work if it
• is supported by school leadership teams and
resourced
• is owned by the teachers involved (their
data, their students, their classrooms)
• has an effective team leader who is
committed to the work (team leader may
not be the domain leader).
Implementing the Action Plan –
How do we know it is working?
• Discussion
as a starting point for a unit of work
“What are the big ideas of a topic?”
• Mapping the content in VELS
• Knowledge and use of the mathematics continuum to
selecting suitable tasks to identify students’ ZPD
• Use of data and evidence to have substantive
conversations about student learning
• Use of data and evidence to challenge current practice
Implementing the Action Plan
•Use of common language when discussing our students
• Ongoing and opportune sampling of students’ work
• Teachers suggest strategies to support like students
• Identifying gaps in teacher knowledge and strategies for
addressing these gaps
• development and use of “rich tasks” by teacher/s
A sustainable approach to PLTs
in South Gippsland Secondary
Schools
• Having a common approach across all schools in the
South Gippsland Region is a powerful strategy
•Team leaders meeting regularly across schools
helps to develop leadership skills and promote
collaborative learning and provides opportunity to
practise PLT skills
Where to in 2011?
• Students directing the learning/ collaboration/ follow up
on “golden questions “
• What is the learning from our students’ perspectives ?
• Development of a Year 9/10 PLT
• Have a dedicated meeting every fortnight commencing at
the start of period 6 and continuing until 4.30pm.
• Development of a PLT folder
• The PLT leader uses a network of other PLT leaders to
continue their own skill development
CONTENT
‘You don’t change performance without
changing the instructional core. The
relationship of the teacher and student in
the presence of content must be at the
centre of all efforts to improve
performance.’
(Elmore 2007)
STUDENT
TEACHER
(Diagram from Cohen and Loewenberg, 2001)