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 2009Effect/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 fortnightlyPLT 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

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 propertiesApplying number properties to patterns and pronumerals – equivalence– structure and use these to develop understanding of numberProvide 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.

What is important to consider in the teaching of algebra in Year 7?

Big Ideas in Algebra [from the Maths Continuum] Pre-assessment task

Algebra story board to develop concepts on Growing Arms [four arm shapes]/World’s Longest Lunch Scenario

Level 3 Sign equivalenceNumber propertiesNumber sentences with missing numbersGrids

NoNoYesYes

Yes to describe the patternYes – this needs to be stressedYesYes

Level 4Shapes and objects according to criteriaRules for sequences – recursion [additive thinking]-Formula [multiplicative thinking] written or and symbolicEquivalence between expressionsRelationships between variables and describe in wordsInverse operations, words and symbols to form simple equations

Yes

Yesyes

Yes [Question 5]Yes [Q2,3,5]

Yes [Q4]yes

Yes

YesYes [both written and symbolic]

YesYes

Yes – extension into inverse operations depending on the problem posed

Level 5 Commutative, associative and distributive propertiesMultiplication and division of power termsInverses to rearrange formulasUse linear and other functions to model situationsSolve simple equations using tables, graphs, and inverse operationsIdentify function and represent it by a table of values, graph and ruleDescribe and specify independent variable of a function and domain and the dependent variable. Construct a table of values and graphs

no

noyes [Q2,3,4,5]yes

yes

partlynono

Yes

Yes – build in this exploration of structureYes if we can use this type of problem [quadratic]yes

yes

Yesyes – graph allows this explorationyes

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 5Gathering Data and

Analysis

CAT (eg. NOYCE Task)

Step 2Identifying Students

Understandings.

(eg. PLT Log using mode)

Step 3Setting Student Learning Goals

New Knowledge

Step 4Student Work Samples

Learning monitored

PLT Log analysis

Algebra Task Analysis Year Level 7E Noyce Task: Necklaces Date July,2010 School: Korumburra SC  [maximum 11] Range: 7 Mode: 4 Median: 4 Mean: 4 Jaiden Troy Bayley Nathan Izaak Shane Ashlea Amber Mairead Gemma Andrew Brooke Cooper Tenae Tamara Ashley Bailey Natasha Maddie Mitch Justine Sophie  0 1 2 3 4 5 6 7 8  

Algebra Necklaces Task - Pre-test

score 7E 7A 7B 7C 7D11109 ____87 ____ ____ ____ ____ ____6 ____ ____5 ____ ____ ____4 ____ ____ ____3 ____ ____ ____2 ____ ____ ____1 ____ ____ ____0 ____

Sixty percent of student achievement is directly attributable to the teaching practice in the classroom. (Leithwood, 2004)

Step 1 or 5Gathering Data and

Analysis

CAT (eg. NOYCE Task)

Step 2Identifying Students

Understandings.

(eg. PLT Log using mode)

Step 3Setting Student Learning Goals

New Knowledge

Step 4Student Work Samples

Learning monitored

PLT Log analysis

Score0 – 1

Attempted the table Use concrete materials Copy and continue patterns Make a pattern/complete Form an array from a patternDraw the 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 predictionsStart to verbalise how they have made their patterns What’s The Rule? Making links between related numbersTransfer 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 correctGet answers using draw all / count all

Experiences using algebra storyboard Use difference patternsTransfer information from a table to a general formula Verbalise information in a table to be able to find a ruleContinue counting patterns (oral, visual, symbolic) Be able to verbalise and utilise mathematical operations/expression

6 – 7 Completed the pattern Rule verbalised using additive thinkingPredominately draw all strategy for Q4

Explore connections between additive and multiplicative thinkingMove to arithmetic reasoning strategy – the structure behind a problemWhat 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?

8 – 11 Rule verbalised Used a table or division strategy to find Q4

Guess my rules, using rules Formulate number patterns into rules e.g. a general rule/formula Graph results Use more complicated patterns and represent solutions in a variety of waysCreate new patterns Verbalise and represent rules from patterns in a number of ways

Necklaces Task Yr 7 What students can do and implications for teaching.

Sixty percent of student achievement is directly attributable to the teaching practice in the classroom. (Leithwood, 2004)

Step 1 or 5Gathering Data and

Analysis

CAT (eg. NOYCE Task)

Step 2Identifying Students

Understandings.

(eg. PLT Log using mode)

Step 3Setting Student Learning Goals

New Knowledge

Step 4Student Work Samples

Learning monitored

PLT Log analysis

• 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: Mitch, Brooke, Amber, Maddie, Shane

Level: 4- 5

Review date: 7th September

What can this/these student/s write and do?

Completed the table with most correctGet answers using draw all / count all

What is this student indicating that they can nearly do? [verge]

Transfer information from a table to a generalformula Verbalise information in a table to be able to find a ruleContinue counting patterns (oral, visual, symbolic) Be able to verbalise and utilise mathematical operations/expression

What teaching strategies could you investigate?

Use of the algebra story board

Emphasis on completing the table correctly and looking for a rule

What resources are needed?

Story board, counters, blocksWhat evidence would show that the student is moving on? [learning]

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

Student Learning Log

Sixty percent of student achievement is directly attributable to the teaching practice in the classroom. (Leithwood, 2004)

Step 1 or 5Gathering Data and

Analysis

CAT (eg. NOYCE Task)

Step 2Identifying Students

Understandings.

(eg. PLT Log using mode)

Step 3Setting Student Learning Goals

New Knowledge

Step 4Student Work Samples

Learning monitored

PLT Log analysis

• 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 = t x 2 + 2 “the number of chairs is equal to two times the number of

tables plus two” means c = 2 x t + 2 or c = 2 x ( 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 5Gathering Data and

Analysis

CAT (eg. NOYCE Task)

Step 2Identifying Students

Understandings.

(eg. PLT Log using mode)

Step 3Setting Student Learning Goals

New Knowledge

Step 4Student Work Samples

Learning monitored

PLT Log analysis

Two points in time data

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 7E 7A 7B 7C 7D11 ____ ____10 9 ____ ____8 ____ ____ ____ ____ 7 ____ ____ ____ ____ ____ ____ ____ ____ ____6 ____ ____ ____ ____ ____5 ____ ____ ____ ____ ____4 ____ ____ ____ ____ ____ ____3 ____ ____ ____ ____ ____2 ____ ____ ____1 ____ ____ ____ ____ ____ ____0 ____ Pre Post Pre Post Pre Post Pre Post Pre Post

7A & 7B Teacher A7C & 7D Teacher B7E Teacher C

Score0 – 1

Attempted the table Use concrete materials Copy and continue patterns Make a pattern/complete Form an array from a patternDraw the 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 predictionsStart to verbalise how they have made their patterns What’s The Rule? Making links between related numbersTransfer 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 correctGet answers using draw all / count all

Experiences using algebra storyboard Use difference patternsTransfer information from a table to a general formula Verbalise information in a table to be able to find a ruleContinue counting patterns (oral, visual, symbolic) Be able to verbalise and utilise mathematical operations/expression

6 – 7 Completed the pattern Rule verbalised using additive thinkingPredominately draw all strategy for Q4

Explore connections between additive and multiplicative thinkingMove to arithmetic reasoning strategy – the structure behind a problemWhat 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?

8 – 11 Rule verbalised Used a table or division strategy to find Q4

Guess my rules, using rules Formulate number patterns into rules e.g. a general rule/formula Graph results Use more complicated patterns and represent solutions in a variety of waysCreate new patterns Verbalise and represent rules from patterns in a number of ways

Necklaces Task Yr 7 What students can do and implications for teaching.

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).

• 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 – How do we know it is working?

•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

Implementing the Action Plan

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

TEACHERSTUDENT

‘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)

(Diagram from Cohen and Loewenberg, 2001)