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Counting On

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

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Counting On

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  1. Counting On Ray MacArthur

  2. Assessment For Learning • 45 + 28 73 How did you calculate this? Algorithm 45 + 20 + 8 (40 + 20) + (5 + 8) 45 + 30 – 2 50 + 23

  3. Assessment For Learning • How many chocolate bars costing 75 cents each can be bought for $5? 6 • How much change should you get from $5? 50 cents How did you do these?

  4. Assessment For Learning • 216 + 143 24 How did the student get this answer? 2732 383

  5. 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.

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

  7. 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.”

  8. Level Changes from Initial to Final Assessment Place Value Multiplication and Division

  9. Level Changes Across Year Cohorts Place Value Multiplication and Division

  10. 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.”

  11. 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

  12. Revised Counting On Website • Only available through the intranet • Framework • Video snippets • Resources • Assessment instrument • Teaching activities • Learning objects

  13. The Revised Assessment

  14. Assessment Instrument • Whole class assessment

  15. Apparent Expert Intermediate Target Group

  16. Sort Criteria • 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.

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

  18. Row, row, row your boat 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? What skills are needed to answer this question?

  19. - Reading - Comprehension - Transformation - Processing - Encoding 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

  20. Assessment Instrument • Whole class assessment • Newman’s error analysis • Covered items tasks • Uncovering dots task • Circles task

  21. Learning Framework In Number

  22. Place Value Framework

  23. Place Value Framework Summary NS1.2 NS1.2 NS2.2 NS2.2 NS2.4 NS3.2

  24. Multiplication and Division Framework

  25. Multiplication and Division Framework Summary NES1.3 NS1.3 NS1.3 NS2.3 NS2.3

  26. Implementing Counting On

  27. 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

  28. 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 makeris 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.”

  29. 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.

  30. 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

  31. 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)

  32. The Final Word • The mathematicssyllabus 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

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