The Quantile Framework ® for Mathematics & West Virginia

1. The Quantile Framework® for Mathematics & West Virginia A Practical Approach: Linking Assessment to Instruction Presented by: Jason Turner, Manager, Professional Services jturner@lexile.com

2. Today’s Session What is the Quantile Framework and why do I need to know about it?

3. Today’s Session Where can Quantile measures be found?

4. Today’s Session What can be done with the Quantile Framework?

5. Today’s Session How will the Quantile Framework help students?

6. Today’s Session How can Quantiles be used in the schools and in the classroom?

7. The Quantile Framework for Mathematics: Question #1 What is the Quantile Framework for Mathematics? Why do I need to know about it?

8. An indicator of mastery of specific skills. A list of mathematical skills. A curriculum for mathematics. A mathematics program. A grade equivalent. Common MisconceptionsThe Quantile Framework is NOT:

9. The QuantileFramework… …is like a “yardstick” ---it’s the way to measure the development of: • Students’ mathematical knowledge and • Difficulty of math skills and topics

10. The Quantile Framework Developmental scale that locates mathematics skills and concepts within a taxonomy. Mathematical Skill/Topic/Concept Quantile Measure (Q) Characterizes the problem-solver with a measure on the same developmental scale. Student Quantile Measure (Q) The Quantile measure uses a common metric for both

11. Vocabulary Break: Let’s define an essential concept

12. What is a Quantile Measure (QMeasure)? • QMeasures are applied to both students and math skills and concepts. • Students: The Quantile measure for an individual or student is the level at which he or she is ready for instruction and has knowledge of the prerequisite mathematical concepts and skills necessary to succeed. When applied to students, the QMeasure indicates the student’s readiness for instruction.

13. What is a Quantile Measure (QMeasure)? • QMeasures are applied to both students and math skills and concepts. • Skills & Concepts: When applied to mathematical skills and concepts, QMeasures indicate the task difficulty associated with a skill or concept relative to other mathematical skills and concepts. For example, the skill of rounding whole numbers to a given place value has a QMeasure of 410Q.

14. The Quantile Measure… Forecasts the student’s readiness to be successful with the skill after introductory instruction.

15. The Quantile Measure… Also refers to the difficulty of a mathematical skill or concept, or how difficult the skill or concept (QTaxon) is compared to others.

16. Vocabulary Break: Let’s define another essential concept

17. What is a QTaxon? • Derived from taxonomy (a category or group) • A specific mathematical skill or concept, e.g., to perform long division with remainders is a specific mathematical skill or concept, or QTaxon. • Within the Quantile Framework, QTaxon is the formal term for skill or concept.

18. Why Do I Need To Know About It? • Developmental Scale: Allows districts and schools to track growth. • Common Scale: Places task difficulty and student readiness on the same scale. • Differentiation & Targeting: Provides math teachers the means to differentiate and target math instruction based on the student’s status.

19. The Quantile Framework for Mathematics Where can Quantile measures be found?

20. Where can I find the Quantile measure? State End-of-Grade Tests (Texas, West Virginia, North Carolina, & Wyoming) Scholastic Math Inventory (SMI) Voyager V-Math Pearson PASeries

21. Where can I find the Lexile measure? • A student’s Lexile measure can be found on the student report for the WESTEST 2 for grades 3 -8 and 10. • More information on a student’s Lexile measure based on the WESTEST 2 score is available on the West Virginia Department of Education’s Web site at: • http://wvde.state.wv.us.oaa

22. West Virginia Educational Standards Test 2 (WESTEST 2)

23. West Virginia Educational Standards Test 2 (WESTEST 2)

24. The Quantile Framework for Mathematics What can be done with the Quantile Framework?

25. Two Issues for Math Educators: First, how does a teacher determine student readiness? OR How does a teacher forecast student preparedness to learn mathematical skills or concepts?

26. Two Issues for Math Educators: Second, how does a teacher establish the measure of difficulty for various skills in mathematics?

27. Sally & Jim Raul Jacob, Lyn, Victor Quantile Scale 1000Q Skill Difficulty + Student Readiness Strengthens Instructional Strategies 900Q +50 800Q - 50 700Q

28. Success Rates for Students of Different Mathematics Ability with a Skill at a Defined Quantile Measure

29. Success Rates for Student of Similar Ability with Math Skills of Different Skill Demands

30. Benefits of the Quantile Framework • DETERMINE: student readiness • ESTABLISH: task difficulty • STRENGTHEN: classroom teaching practices and strategies

31. Benefits of the Quantile Framework LINK: student ability to skill difficulty (using a common measure) FORECAST: student success rate (after introductory lessons) TRACK: growth over time

32. The Quantile Framework for Mathematics How will the framework help the students and teachers in my district?

33. Picturing Mathematics with the Quantile Map

34. Gaps in learning: Student Ability 350Q Skill Difficulty 850Q Student: 350Q Qtaxon: 850Q What must happen to close this gap? What must happen to narrow this gap?

35. The Quantile Framework: Practical Benefits Helps teachers… • Differentiate instruction • by linking students with instruction at their level • by knowing the Quantile level of the skill or concept and matching it with the Quantile measure of the student • Determine student readiness • by knowing each student’s Quantile measure • Forecast student success with specific skills and concepts

36. The Quantile Framework: Practical Benefits Helps teachers… • Monitor student math progress • by tracking student growth using the Quantile developmental scale • Describe pre-requisite skills needed to learn more advanced mathematical concepts • Tailor teaching methods in the classroom to ensure a greater level of understanding and application • Determine supplementary and upcoming skills for students in need of enrichment

37. The Quantile Framework for Mathematics How can Quantiles be used in the classroom?

38. Vocabulary Break: Let’s define some more essential concepts

39. What is a Prerequisite QTaxon? • For any mathematical skill or concept there are prerequisite skills and concepts necessary to succeed. • When we refer to Prerequisite QTaxons, we mean those skills that are necessary to be successful with the skill in question. • For example, to successfully perform division with remainders, one must be able to successfully perform simple division without remainders.

40. What’s an Impending QTaxon? • Every mathematical skill or concept is building toward another skill or concept. • For example, the multiplication of single digit numbers comes after the addition of double digit numbers, and is, therefore, an impending skill… • Impending QTaxons are the skills that ‘come next’ or are the next steps in the Quantile taxonomy

41. What’s a Supplemental QTaxon? • A Supplemental QTaxon is a mathematical skill or concept which broadens, enhances or deepens knowledge of a skill or concept, but is not necessary for success with that skill or concept. • For example, a Supplemental QTaxon of “Calculate the areas of triangles, parallelograms, trapezoids, circles and composite figures”, is “Use models to develop formulas for finding areas of triangles, parallelograms, trapezoids, and circles” This skill enhances understanding of the QTaxon in question, but is not necessary for success.

42. An Essential Feature of the Quantile Framework:The Knowledge Cluster! • Precursory skills to ensure prior knowledge • Supporting skills to connect and add depth across content strands • Quantile measures to address the degree of difficulty for each skill

43. The Power of the Knowledge Cluster To inform instruction and address the needs and abilities of each student To add depth and breadth to concepts in mathematics To adjust Individual Education Plans according to Quantile measures

44. The Knowledge Cluster Accessing the knowledge cluster

45. Knowledge Clusters

46. Knowledge Clusters Q Taxon ID numbers

47. A QTaxon and It’s Knowledge Cluster QT-M-254 Investigate and determine the relationship between the diameter and the circumference of a circle and the value of pi; calculate the circumference of a circle.

50. Subtract 2- and 3-digit numbers with regrouping.