Key TeacherIn NumeracyDay Three

KTN Training Day 3 Agenda

• Welcome and Prayer• Session 1: Effective teachers of Numeracy,

shoulder-to-shoulder professional learning and optimising teaching opportunities

• Session 2: Assessment – the ENI/MAI and Growth Points to inform teaching.

• Session 2: The KTN Role, co-teaching.• Session 3: Fractions, Decimals and

Percentages

How best to teach mathematics?

What we often teach students is an impoverished mathematics, one that focuses

on detailed facts and procedures while neglecting the fundamental nature of the field.

If we taught music as we teach mathematics, students would practice musical scales for years without ever getting to play a song.

If we taught art as we teach mathematics, students would practice drawing lines and shapes for years without ever getting to

create a picture.

If we taught writing as we teach mathematics, students would practice spelling, punctuation, and penmanship without ever getting to use

writing to express what they have to say.Glenn M.Kleiman

A problem to get you thinking!

• My telephone number has eight digits.• When I add up the digits I get a two digit number .• When I add these two digits together I get a

number less than 4.• What could my telephone number be?

Enabling prompt – try it with your own telephone number or with the school

number.

A deep knowledge of:

General classroom pedagogy

Specific pedagogical content knowledge

Pedagogy

A deep knowledge of:Assessment toolsAssessment analysis

and diagnosisHow students learn

(including learning styles, multiple intelligences and brain function)

Students

A deep knowledge of:Subject matterHow students

progress through the desired content

Potential difficulties students may experience with the content

Content

• Build teacher knowledge (expand the circles)

• Increase the teachers’ capacity to integrate the knowledge (increase the overlap)

Content

Students

Pedagogy

Building Concepts

• Pre-concepts – ideas students have before any learning has taken place

• Partial Concepts – parts of concepts on the way to learning the whole concept

• Misconceptions – rules students construct and apply as a result of pre- or partial concepts.

Beliefs about teaching and learning of maths

• Long term outcomes improves clarity of focus: A common understanding of long term outcomes helps when deciding how best to aid student’s learning.

• All students can learn Mathematics: Teachers are responsible for ensuring that each student achieves outcomes to the best of his/her ability.

• Learning Mathematics is an active and productive process: Students need to construct their own mathematical knowledge in their own way and at a pace that enables them to make sense of the ideas situations they encounter.

• ‘Risk’ relates to future Mathematics learning: Students should be encouraged to ‘take risks’ at classroom level to ensure future mathematical progress is not ‘at risk’.

• Successful maths learning is robust learning: A focus on short term performance or procedural knowledge at the expense of robust knowledge places students ‘at risk’ of lessened progress throughout schooling.

Things for you to think about…

• The principles of teaching and learning• Numeracy Dedicated Time• Professional learning community• Data informed instruction• Culture for individual learning and group

work• Your role in all of this.

Effective Teachers of Numeracy

• Read Doug and Barbara Clarke’s article.

• In your journal jot down any of the strategies listed which you implemented in your classroom last week.

• Think about what you are planning to teach this week and jot down the strategies you plan to use/try.

Teacher’s establish and nurture an environment conducive to learning mathematics through the decisions they make, the conversations they orchestrate and the physical setting they create….more than a physical setting with desks, bulletin boards and posters, the classroom environment communicates subtle messages about what is being valued in learning and doing maths.” George Booker 2004

Vision: Where do you want to journey to?

Fuel: What is motivating you to make this journey?

People: Who can help you make the journey?

Luggage: What are you going to take along?

Exhaust: What are you going to leave behind?

• The Physical Environment conveys strong messages to children from the first time they enter the room.

• What messages do students receive from the physical environment of their classroom?

- A message that:- COMMUNCIATES LEARNING

- CELEBRATES LEARNING- SCAFFOLDS LEARNING- ORGANSIES LEARNING- MOTIVATES LEARNING

- What messages do you want your students to create and receive from their classroom environment?

How may we be more effective?

• Implementing classroom strategies such a Numeracy Dedicated Time.

• Allowing children to reflect on their learning• Providing time for reflection (for you and for

your students)• Providing challenge and motivation• Allowing for multiple entry and exit points• Professional learning, especially with our

peers

One on One Assessment

The Early Numeracy Research Project and the Interview

One on One Interview

• The definition of Numeracy used for the Early Numeracy Research Project was

• [Numeracy is] the effective use of mathematics to meet the general

demands of life at school and at home, in paid work, and for participation in

community and civic life.MCEETYA Benchmarking Task Force 1997

Aims of the Early Numeracy Research Project

• To assist schools to implement the key design elements as part of the school’s mathematics program

• To challenge teachers to explore their beliefs and understandings about how children develop their understandings of mathematics, and how this can be supported through the teaching program

• To evaluate the effect of the key design elements and the professional development program on student numeracy outcomes.

Key Differences between Literacy and Numeracy

• Teachers’ personal confidence with mathematics

• The need to improve the perceptions of children teachers and parents regarding mathematics

• The lack of shared understanding of the ‘big ideas’ of mathematics in the early years

• The lack of comprehensive assessment instruments and processes for early years

Why use the Interview?

• It enables children to showcase their skills and understandings

• Allows teachers to discover how their students think and feel during maths tasks

• It provides insights into the child’s thinking• It generates detailed individual and class profiles

which show student achievement relative to growth points

• It allows schools to track student progress over time• It informs focussed planning for teaching at the point

of need

Language and numeracy?

• There is a range of assessment possibilities available but language is an essential element of all of them.

• Did the child understand the language of the task?

• Did the person marking the assessment understand the child’s language?

Growth Points

• A series of Growth Points was developed as part of the ENRP and provided the basis for the construction of a number of tasks for the one-on-one interview.

• Growth Points describe the typical ‘stepping stones’ along the path to mathematical understanding and allow teachers to focus in fine detail on the child’s mathematical learning in a specific aspect e.g. counting.

‘Learning mathematics involves a dialogue between people – a

complex two-way communication in which meanings are negotiated and

ambiguities are navigated.’• Nick Connolly EQ 2008

Growth Points help teachers :

• Understand how children learn• Assess and monitor children’s growth in

understanding• Identifying children who are at risk• Identify the zone of proximal development

for children’s learning• Plan and target teaching so that we can

identify the experiences that will most effectively help children to reach the next growth point in their mathematical learning

Interview Task - Counting

1 Teddy TaskShow the child the teddies and get the cup.

Please take a big scoop of teddies…..Please put a few more teddies in to fill up the cup (You will need at least 20).

a) Hold them in front of you…Tell me how many teddies you think are in the cup.

b) Please check to find out. (If the child takes the teddies out of the cup one by one say ….Please tip the teddies out to check.)

c) How many teddies are there?d) Please put one teddy back into the container. How

many teddies are on the table now?

Your task -

• Working with a partner, study the Growth Point profile you have been given.

• Describe what your student can do.

• Decide what he or she needs to learn next.

• Develop a set of activities to help move the child to the next Growth Point.

Profiles

• Table 1 1,2,2,0• Table 2 2,2,2,1• Table 3 1,2,1,1• Table 4 1,0,1,1• Table 5 4,3,2,2

• Table 6 3,3,2,1• Table 7 2,2,2,2• Table 8 3,2,2,2• Table 9 4.4.3.2• Table 10 3,2,2,1

Approximate Growth Points

Year Level Counting Place Value Addition and Subtraction Strategies

Multiplication and Division Strategies

2 3 2 2 2

3 4 3 4 3

4 5 3 5 3

5 5 4 5 4

6 5 5 5 4

7 6 5 6 5

Co-Teaching

One teach,one drift

StationTeaching

ParallelTeaching

Alternativeteaching

Teamteaching

One teach, one observe

Co-TeachingPossibilities

One teach, one observe

• One teacher teaches the whole group, the other teacher observes the students.

• This observation needs to be focussed eg which students are engaged, contributing.

• A good opportunity to gather students data, monitor and support student behaviour, evaluate individual students’ participation.

Team Teaching

• Teachers share equally in the planning and delivering of the lesson/s.

• Can involve ‘jig-sawing’ – students and teachers put together smaller pieces of learning until they have helped each other to construct the whole.

• Teachers support one another.

Alternative Teaching

• One teacher teaches a small group while the other teacher takes the rest of the class.

Parallel Teaching

• Teachers plan collaboratively and simultaneously teach the same content to two different groups of students eg half the class each.

Station Teaching

• Teachers divide responsibility for content.

One teach, one drift

• Similar to ‘one teach, one observe’ but in this scenario one teacher drifts around the classroom checking for understanding.

• Provides opportunities for 1 on 1 teaching at the point of need.

What might work for you?

• These six options might be useful for you to try at your school.

• Choose the option you are going to try out next week and plan your approach.

• Share your thoughts with the person next you and provide constructive feedback for each other.

Set yourself a goal for this week.

• Decide upon one strategy you would like to put into place this week – with your class.

• Decide upon one strategy you will try to put into place with a peer.

• Plan how you will set up the necessary conditions for success.

Decimals

Some common misconceptions

• Decimals are decoration or punctuation• Numbers on the right of the decimal point are

the small whole numbers• The decimal point separates units of money

or measurement• The numbers on the right of the decimal point

are different in some way eg reverse place values; more digits – smaller number; less digits – smaller numbers; are about parts or fractions eg 0.5=1/2

• When students apply pre-conceptions, partial concepts or misconceptions and get the right answer they think this must be the whole concept. Teachers need to provider activities which provoke students to challenge their own ideas.

• Numbers on the right of the decimal point are tenths and hundredths of the whole.

• Decimal fractions extend whole number place value to represent numbers between the whole numbers

Development of Decimal Understanding

• Students’ understanding of decimal fractions depends on what they understand about whole numbers and fractions.

• We can extend the patterns in the way we write whole numbers to the way we write decimals.

• There are numbers between consecutive whole numbers.• The place value system can be extended to the right of

the units place to show numbers between two whole numbers.

• To represent a number between two consecutive whole numbers, record the smaller whole followed by the part, separated by a decimal point (eg a number between 4 and 5 is 4.67)

• The digits to the right of the units place have decreasing values in powers of ten…and can represent infinitely small numbers. decimal fractions can be partitioned just as whole numbers can.

Relationshipe between the Places

• Students need to move from whole number place value and unit fraction ideas to seeing the link between fractions and place value, and the relationships between the places, not just the value of each place.

Linking decimal and Common Fraction Notation

1 ÷ 10 =

Students use fraction notation to represent the decimal models they

make.

Students use the calculator to see the decimal place value notation.

Consecutive Numbers – a problem to solve

Three consecutive even numbers add to give a number between 100 and 120.

Enabling Prompts

• What does consecutive mean? Even?

• Name two consecutive numbers. What do they add to?

• How many numbers are there between 100 and 120?

Extending Prompts

• What three consecutive even numbers would give a sum between 560 and 580?

• Find an efficient way to find three consecutive numbers that will ‘work’ for any given number.

Decimal Activity

• Write a number with at least two decimal places.

• Use the materials on your table to make a model of the number to show its quantity.

• Turn your number card upside down beside your model.

• Look at the models of others and decide what number each model represents.

Think about -

• What did you need to know in order to make your model?

• What did you need to know to recognise what other peoples’ models represented?

Fractions

“No area of mathematics is as mathematically rich, cognitively

complicated, and difficult to teach as fractions, ratios and

proportions” Smith (2002)

A fractions challenge

Greedy Frogs

Three chairs

Three Freddos

Ten volunteers

Lots of thinking!

Six BIG messages about fractions

Fractions are a challenge to teach and learn

Fractions are an important part of the mathematics curriculum

Common fraction understanding is based on the part/whole concept (partitioning) and fractions are about equal parts

The bigger the denominator the smaller the parts

Developing the idea that the size of the unit/whole determines what the fraction looks like is essential

Multiple representations of fractions are required

Language Difficulties

There are some language difficulties inherent in talking about fractions:

Use of ordinal numbers – we talk about thirds, fifths, sixths

Lack of consistency in the use of ordinal numbers – ‘halves’ not ‘twoths’ and quarters not fourths

The variety of names we have for fractions – one half is also five tenths, zero point five, (0.5) and fifty percent (50%) as well as two quarters, three sixths and so on

A ‘fraction’ usually refers to san little bit but 9/2 is also a fraction

The Fractions Rap

A Fractions task

Write down 10 fractions between 1/3 and 2/3.How many different ways can you show 2/3?(Note if the children understand 2/3 as part of a whole, a part

of a length, as part of a collection so they use a range of materials.

Draw some shapes that are divided into 2 parts but the parts are not halves.

Draw some groups that are divided into two parts but the parts are not halves.

Ask the children to justify why the parts they drew are not halves.

Fractions with Bloom’s Revised Taxonomy

Remembering

List the fractions you know and can show.

Understanding

Find items that you can use to show the fractions.

Applying

Draw a diagram which shows these fractions or take photographs of the fractions.

Analysing

Design a survey to find out which fractions are easy and which are hard. Graph your results.

Evaluating

Choose a diagram or picture to represent the hardest fractions to use in a game.

Creating

Create a PowerPoint presentation game for others to play

Pie fraction

Key Understandings – FSIM fractions

KU 1 – when we split something into two equal-sized parts, we say we have halved it and that each part is half of the original thing.

KU 2 – We can partition objects and collections into two or more equal-sized parts and the partitioning can be done in different ways.

KU 3 – we use fraction words and symbols to describe parts of a whole. The whole can be an object, a collection or a quantity.

Key Understandings – FSIM Fractions

KU 4 - The same fractional quantity can be represented with a lot of different fractions. We say fractions are equivalent when they represent the same number or quantity.

KU 5 – We can compare and order fractional numbers and place them on a number line.

KU 6 – A fractional number can be written as a division or as a decimal.

KU 7 – A fraction symbol may show a ratio relationship between two quantities. Percentages are a special kind of ratio we use to make comparisons easier.

Suggest how we can help students to:

• develop the appropriate images, actions and language to accompany formal fraction work

• conceptualise fractions as quantities before being introduced to fraction algorithms

• construct the procedures of the operations, rules and language of fractions

In your table group develop some ideas for achieving these things. Write them on your A3 sheet.

Remedies

Develop part/whole/partitioning relationships– Sharing discrete objects to find the part– Using knowledge of parts to find the whole– Partitioning continuous quantities into parts– Using knowledge of partitions to construct wholes or multiples

from parts

Develop models for equivalence– Paper folding– Fraction Walls– Number lines

Represent fraction operations using the above models

Develop language of fractions– symbols to words and words to symbols

Partitioning continuous quantities:

Paper Folding – cut five streamer strips the same length.

• Fold one strip into half, quarters, eighths

• Fold another strip into thirds, sixths, ninths

• Fold the third strip into fifths & tenths

• Fold the last strip strip into sevenths

Fractions - folding paper.

• Folding paper helps develop the vocabulary needed.

• Connects to the number line.• Start by folding paper strips eg streamers.• Make connections :

Halves, quarters, eighthsThirds, sixths, twelfthsfifths, tenths

Moving from Fraction Walls to Number Lines

The Fraction Wall uses area as a measure.– Each partition represents the same fractional

part of the whole – The whole is one row of the Fraction Wall.– Combinations of parts can be used to represent

larger fractions e.g. three-eighths or five-eighths – There is no unique representation of three-

eighths on the Fraction wall. – Three-eighths can be represented in many

different ways.

Fraction Wall Activity

• Work in pairs or threes.• You need a fraction wall sheet, coloured

pencils, two dice (numerator, denominator)• Take it in turns to throw the dice and colour in

the appropriate sections of the wall.• Talk within your group about what you are

doing.• Keep a record on the bottom of your sheet.

Fractions on the Number Line

Introducing the Number LineStep through the introduction slowly – don’t assume prior

knowledge.

Keep reminding the children ‘Where is the one?’• Mark in zero and one other reference point.• Discuss convention of negative numbers to the left, zero in

the middle, and positive numbers to the right.• Move towards children drawing their own number lines.• Try it out with others at your table!

Number lines could be -

• Masking tape on the floor.• A fabric strip.• String across the room.• Chalk in the playground.• Magnetized numbers on the whiteboard

or blackboard.• Cash register/adding maching rolls.• Number ladder.

Research shows the importance of:

– teaching how fractions can be represented especially on a number line (building on paper folding & Fraction Wall)

– introducing students to fractions as numbers on the number line as early as possible

– showing how to check results and to estimate answers– inter-relating procedures and making connections between

decimals, fractions, and percentages.– checking strategies– estimating answers– showing students how to represent fractions – assisting students to identify and correct their misconceptions

Fractions

Magic Beans(Lima beans sprayed gold on one side)

Take a handful. Throw them down in front of you. Talk about the number of gold beans out of

the total number of beans. Link to ‘numerator’ and ‘denominator’.

Word Splash for Fractions