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The Story of Standards: What We Know About Quality, Coherence, and Progression

The Story of Standards: What We Know About Quality, Coherence, and Progression. Joan Ferrini-Mundy, Peter Bates, and the PROM/SE Associates PROM/SE Mathematics Associates Summer Institute, AUGUST 9-12, 2004. DAY 2, TASK 1.

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The Story of Standards: What We Know About Quality, Coherence, and Progression

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  1. The Story of Standards: What We Know About Quality, Coherence, and Progression Joan Ferrini-Mundy, Peter Bates, and the PROM/SE Associates PROM/SE Mathematics Associates Summer Institute, AUGUST 9-12, 2004

  2. DAY 2, TASK 1 Penny had a bag of marbles. She gave one-third of them to Rebecca and one-fourth of the remaining marbles to Aman. Penny then had 24 marbles left in her bag. How many marbles were in the bag to start with?

  3. The PROM/SE Process Gather Data Analyze Data Implement Changes Interpret Data Design Solutions Identify Challenges Conjecture Reasons

  4. The PROM/SE Process Gather Data Analyze Data Implement Changes Day 2 Interpret Data Design Solutions Identify Challenges Conjecture Reasons

  5. The story …. • What Gets Taught: WHO DECIDES? • The Case of Fractions: WHAT ARE THEY? • Big Ideas and Trajectories: HOW CAN WE ACHIEVE COHERENCE?

  6. What Gets Taught: WHO DECIDES?

  7. The Many Aspects of Curriculum …. intended standards, benchmarks, indicators accessed student enrollments what teachers do in classrooms implemented assessed MEAP, Ohio Proficiency and Achievement Tests achieved student learning

  8. The Many Aspects of Curriculum …. intended standards, benchmarks, indicators accessed student enrollments what teachers do in classrooms IMPLEMENTED CURRICULUM assessed MEAP, Ohio Proficiency and Achievement Tests achieved student learning

  9. The intended curriculum ……

  10. High Achieving Countries’ Mathematics Standards

  11. NCTM PSSM with Top Achieving Countries’ Profile

  12. Michigan’s (New) Mathematics GLCEs

  13. Composite Mathematics Standards for MI PROM/SE

  14. The “accessed” and “achieved” curriculum …..

  15. National data, from CCSSO State Indicators of Science and Mathematics Education 2003

  16. Differing expectations leadto differing results Math scores: Minority 12th graders vs. white 8th graders

  17. THE IMPLEMENTED CURRICULUM

  18. MI GLCEs NCTM standards pacing guides professional development textbooks MEAP AP syllabus

  19. PROM/SE data research findings MI GLCEs NCTM standards pacing guides professional development textbooks MEAP AP syllabus national data

  20. Ultimately, teachers determine the implemented curriculum -- with lots of inputs -- and part of what PROM/SE is about is helping to build capacity to do this in a way that supports mathematics learning and growth for all.

  21. COMMENTS ON MARBLES PROBLEM, PETER BATES

  22. PROM/SE ITEM, GRADES 6,7,8 Penny had a bag of marbles. She gave one-third of them to Rebecca, and then one fourth of the remaining marbles to John. Penny then had 24 marbles left in the bag. How many marbles were in the bag to start with? A. 36 B. 48 C. 60 D. 96 Grade 6 35.2% Grade 7 43.0% Grade 8 43.6%

  23. Fraction issues that arise in the Marble Problem: • Meaning of fraction • Equivalence of fractions • Adding and subtracting fractions • Multiplying and dividing by fractions • Representing fractions …….. 1/4 = 8 marbles

  24. The Case of Fractions: WHAT ARE THEY?

  25. What does mean?

  26. What does mean?

  27. What does mean?

  28. What does mean?

  29. Multiple Meanings and Uses of Fractions part of a collection part of a whole point on a number line measurement ratio/rate probability division of two numbers abstract number

  30. What meanings of fraction came up in the marble problem? • Fraction as division • Number meaning (we calculated with fractions) • Part of a collection • Part of a whole

  31. Find the value of each of the following expressions when m = 4, p = 2, and q = 3. NUMBER Grade 6 From Singapore Mathematics Curriculum

  32. RATES Grade 7 From Connected Mathematics

  33. Advantages & Limitations:part of a whole • Advantages: • emphasizes the unit (1 whole) • familiar to children (sharing, splitting, pizza) • Limitations: • hard to calculate with fractions represented as part of a whole (1/3 + 1/5) in some representations (e.g., circles) • Different size “wholes”

  34. Advantages & Limitations:part of a collection • Advantages: • counting is easier • leads to division meaning • Limitations: • the “whole” or unit is arbitrary • computation is hard • difficult to use to show fraction greater than 1

  35. Advantages & Limitations:number line • Advantages: Leads to number meaning Allows for fractions greater than one • Limitations: • More abstract than part of a whole 0 1

  36. Performance on a fractions problem 1/4 1/2 PROM/SE RESULTS: Grade 3 35.2% Grade 4 43.0% Grade 5 43.6% 8/8 10/8

  37. Advantages & Limitations: division is the same as 23 • Advantages: • builds on part of a collection idea • relates to contexts • Limitations: • difficult to justify

  38. WHY IS the same as 23 ?

  39. Review of Research Literature review yielded 73 research studies of student understanding and teaching of fractions Some of the authors: Behr, Bright, Borasi, Michaelsen, Davis, Kerslake, Mack, Middleton, Steffe, Olive, Wearne, Hiebert, and many others…. key findings….

  40. Number line at 4th grade is difficult • Translations between various representations • Need a lengthy readiness period for fractions • Fraction/ratio confusion • It helps to encourage children to talk about their interpretation of fraction • Greater emphasis is needed on division interpretation • Need to recognize the limitations of the “part to whole” model • Need to transition better from realm of counting numbers to rational numbers • Knowing different meanings and interpretations strengthens understanding

  41. Students bring experience with part-to-whole interpretation, and fair shares • Need variety of models for the “whole” (circles, rectangles, irregular shapes) • In part-to-whole, not only do partitioning, but complete the whole

  42. Big Ideas and Trajectories: HOW CAN WE ACHIEVE COHERENCE?

  43. Coherence: A Worthy Goal Content standards are coherent if they reflect a sequence of topics and performances organize along the logical and hierarchical structure of mathematics Schmidt, Wang, and McKnight, in press.

  44. From NCTM’s Principles and Standards for School Mathematics The Curriculum Principle: A curriculum is more than a collection of activities; it must be coherent, focused on important mathematics, and well articulated across the grades.

  45. From NCTM’s Principles and Standards for School Mathematics The Table of Standards and Expectations in the appendix highlights the growth of expectations across the grades. It is not expected that every topic will be addressed every year. Rather, students will reach a certain depth of understanding of the concepts and acquire certain levels of fluency with the procedures by prescribed points in the curriculum, so further instruction can assume and build on this understanding. (p. 30)

  46. Number Emphasis Across the Grades Pre-K–2 3–5 6–8 9–12 Algebra Geometry Measurement Data Analysis and Probability

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