counting on
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Counting On. Ray MacArthur. Assessment For Learning. 45 + 28 73. How did you calculate this? Algorithm 45 + 20 + 8 (40 + 20) + (5 + 8) 45 + 30 – 2 50 + 23. Assessment For Learning. How many chocolate bars costing 75 cents each can be bought for $5? 6 - PowerPoint PPT PresentationTRANSCRIPT
Counting On
Ray MacArthur
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Assessment For Learning1. 45 + 28
73How did you calculate this?
Algorithm45 + 20 + 8(40 + 20) + (5 + 8)45 + 30 – 250 + 23
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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?
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Assessment For Learning4. 216 +
14324
How did the student get this answer?
2732383
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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.
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Counting On 2007 -09• Simplified assessment instrument• Newman’s Error Analysis• Revised Counting On CD• Funded by AGQTP• Learning communities of schools• Facilitator conference
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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.”
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Level Changes from Initial to Final Assessment
Place Value Multiplication and Division
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Level Changes Across Year Cohorts
Place Value Multiplication and Division
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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.”
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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
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Revised Counting On Website• Only available through the intranet• Framework
– Video snippets
• Resources– Assessment instrument– Teaching activities– Learning objects
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The Revised Assessment
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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 ____________________
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Apparent Expert Intermediate Target Group
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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.
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Assessment Instrument• Whole class assessment• Newman’s error analysis
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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?
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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
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Assessment Instrument• Whole class assessment• Newman’s error analysis• Covered items tasks
– Uncovering dots task– Circles task
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Learning Framework In Number
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Place Value Framework
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Place Value Framework Summary
Level Characteristic Syllabus0
Ten as a CountCounting 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
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Multiplication and Division Framework
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Multiplication and Division Framework Summary
Level Characteristic Syllabus1
Forming Equal GroupsCounts 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
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Implementing Counting On
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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
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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.”
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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.
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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
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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)
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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