Counting On Ray MacArthur. State Numeracy Programs Conference 20102 Assessment For Learning 1.45 + 28 73 How did you calculate this? Algorithm 45 + 20

Counting On

Ray MacArthur

State Numeracy Programs Conference 20102

Assessment For Learning1. 45 + 28

73

How did you calculate this?

Algorithm45 + 20 + 8(40 + 20) + (5 + 8)45 + 30 – 250 + 23


Assessment For Learning2. How many chocolate bars costing

75 cents each can be bought for $5?6

3. How much change should you get from $5?50 cents

How did you do these?


Assessment For Learning4. 216 +

14324

How did the student get this answer?

2732383


A Short History of Counting On• LFIN initially developed for the CMIT project in

1996 by Professor Bob Wright.– Outlines how students move from naïve strategies

to sophisticated strategies to solve problems.– Provides a basis for observing, understanding and

developing students strategies.– Frameworks for Place Value and Multiplication

and Division developed for the Counting On project which was piloted in 1999 and implemented in 2000.


Counting On 2007 -09• Simplified assessment instrument• Newman’s Error Analysis• Revised Counting On CD• Funded by AGQTP• Learning communities of schools• Facilitator conference


Independent Evaluation• Assoc Prof Allan White – UWS• “There is strong evidence that the Counting On 2007

program is a very successful program for assisting the development of the mathematical skills and knowledge of students who have struggled with mathematics in the middle years of school with students improving by one or more levels their understanding of place value (66%) and multiplication/division (65%) ... There was further evidence that this improvement was retained over time.”


Level Changes from Initial to Final Assessment

Place Value Multiplication and Division


Level Changes Across Year Cohorts

Place Value Multiplication and Division


Why Negative Movement?• “The spread of results suggests a mix of reasons where some

can be remedied by further experience with Counting On.• An examination of the student learning outcomes show less

improvement as the year group gets older. Possible explanations could include that the students do not have the capacity to handle the mathematics required, or they have become very resistant due to negative feedback and a poor self image, and so no amount of teaching and experience will change the results.

• It is possible that the students guess when they do not know an answer which would lead to unstable results.”


Counting On 2010• AGQTP funding not available• NSWIT registered course

– Counting On in the Middle Years– 151CUK110– 10 hours– Regional delivery model– Events created using My PL@DET

• Revised Counting On website


Revised Counting On Website• Only available through the intranet• Framework

– Video snippets

• Resources– Assessment instrument– Teaching activities– Learning objects

https://detwww.det.nsw.edu.au/curr_support/maths/counting_on/


The Revised Assessment


Assessment Instrument• Whole class assessment

Question If you need to use pen and paper use the space next to each question.

1. I have $53 and I spend $27. How much money do I have left?

Answer ____________________

2. What number is 152 more than 498?

Answer ____________________

Explain an easy way of working this out.

3. The first international cricket team to tour England was an Aboriginal team. The team won 14 matches, drew 19 matches and lost 14 matches.

How many matches were played?

Answer ____________________

Question If you need to use pen and paper use the space next to each question.

4. If there are 42 biscuits and each child at a party gets 7 biscuits, how many children are at the party?

Answer ____________________

5. Altogether there are 28 cards to be placed in 4 equal rows. How many cards are in each row?

Answer ____________________

6. Natalie paddled 402 km of the Murray River in her canoe over 6 days. She paddled the same distance each day.

How far did Natalie paddle each day?

Answer ____________________


Apparent Expert Intermediate Target Group


Sort CriteriaCategory Criteria

Apparent Expert

Intermediate

Target Student

• Correct working and answers to 5 or 6 items• Clear understanding of correct number

concepts needed to solve the problems.

• Some correct working and answers.• Some understanding of number concepts

needed to solve the problems but still not fully developed or consistent.

• Few or no correct working or answers.• Evidence of misconceptions in working and

answers.


Assessment Instrument• Whole class assessment• Newman’s error analysis


Row, row, row your boatNatalie paddled 402 km of the Murray River in her canoe over 6 days. She paddled the same distance each day.How far did Natalie paddle each day?

What skills are needed to answer this question?


Newman’s Error Analysis• 5 prompts

– Read the problem

– What is the problem asking

– What has to be done to solve the problem

– Solve the problem

– What is the answer to the problem

- Reading

- Comprehension

- Transformation

- Encoding

- Processing


Assessment Instrument• Whole class assessment• Newman’s error analysis• Covered items tasks

– Uncovering dots task– Circles task


Learning Framework In Number


Place Value Framework



Place Value Framework Summary

Level Characteristic Syllabus

0Ten as a Count

Counting by ones

1Ten as a Unit

Ten is a countable unit. Visual materials

2Tens and Ones

Two digit mental addition and subtraction

3Hundreds, Tens and

Ones

Three digit mental addition and subtraction

4Decimal PV

Decimal place value

5System PV

Understands place value

NS1.2

NS1.2

NS2.2

NS2.2

NS2.4

NS3.2


Multiplication and Division Framework



Multiplication and Division Framework Summary

Level Characteristic Syllabus

1Forming Equal Groups

Counts the visible items in each group by ones

2Perceptual Multiples

Counts using groups with visible items

3Figurative Units

Counts using markers for each group

4Repeated Abstract Composite Units

Counts without group markers

5Multiplication and

Division as Operations

Uses multiplication and division as inverse operations

NES1.3

NS1.3

NS1.3

NS2.3

NS2.3


Implementing Counting On



Implementing Counting On• Retracing old learning paths – doing more of

the same techniques that failed initially for these students – just frustrate Counting On students.

• Counting On activities:– offer a novel teaching approach that is working

mathematically– identify the concept being addressed as well as

the context or model being used– are fun


Implementing Counting On• Exploring one or two models in depth is

preferable to using many different models for the same idea.e.g. Clothesline empty number line ribbon

maker is a natural sequence within one model• Choose the activities for a purpose and explain

the purpose to the students.– “Today we are going to play Make 100. This will

help us to group tens and ones and add tens and ones.”


Implementing Counting On• “But what about the syllabus and the normal

teaching program?”• If you were just learning Japanese and I put

you in the Advanced Japanese class you would understand how these Counting On students feel.


Implementing Counting On• Depends on class/school structure• Mixed ability – differentiate the activities• Streamed – have a Counting On class• Write the CO activities into the existing

teaching program• Have a number focus at the beginning of every

lesson then move to another strand• Lesson study


Implementing Counting On• 10 minute activity at the beginning of each

lesson• Whole lessons of Counting On• Differentiated activities eg double number

lines (0-100, 0-1000, 0-1, 0-0.1)


The Final Word• The mathematics syllabus relies on students

having a well developed understanding of the place value system, addition, subtraction, multiplication and division.

• Counting On helps students to develop their understanding

Documents

Counting On Ray MacArthur. State Numeracy Programs Conference 20102 Assessment For Learning 1.45 + 28 73 How did you calculate this? Algorithm 45 + 20