1. The New CIA: Curriculum, Instruction and Assessment in Mathematics Education for the 21 st Century Mathematics Education Colloquia University of Kentucky October 18, 2005 Linda Jensen Sheffield Regents Professor Northern Kentucky University Sheffield@nku.edu http://www.nku.edu/mathed Mathematics Education Colloquia University of Kentucky October 18, 2005 Linda Jensen Sheffield Regents Professor Northern Kentucky University Sheffield@nku.edu http://www.nku.edu/mathed

2. Curriculum: What does it mean for Kentucky to have a world-class mathematics curriculum?

3. What is curriculum?  A body of knowledge to be transmitted (core curriculum)  A product - an attempt to achieve certain ends in students (outcomes based)  A process - the interactions of teachers, students and knowledge (Paideia)  A body of knowledge to be transmitted (core curriculum)  A product - an attempt to achieve certain ends in students (outcomes based)  A process - the interactions of teachers, students and knowledge (Paideia)

4. What is mathematics?  A body of knowledge involving numbers and their manipulations  A set of relationships between quantities and symbols  The study of patterns  A body of knowledge involving numbers and their manipulations  A set of relationships between quantities and symbols  The study of patterns

5. In Kentucky, we have:  Academic Expectations  Communication, reasoning, and problem solving  Pattern and Structure  Quantity - Number concepts  Mathematical procedures  Shape and dimensionality  Uncertainty - Statistics and probability  Change (adapted from Lynn Steen - On the Shoulders of Giants)  Program of Studies and  Core Content  Number properties and operations  Measurement  Geometry  Data Analysis and Probability  Algebraic Thinking  Academic Expectations  Communication, reasoning, and problem solving  Pattern and Structure  Quantity - Number concepts  Mathematical procedures  Shape and dimensionality  Uncertainty - Statistics and probability  Change (adapted from Lynn Steen - On the Shoulders of Giants)  Program of Studies and  Core Content  Number properties and operations  Measurement  Geometry  Data Analysis and Probability  Algebraic Thinking

6. Depth of Knowledge Norman Webb  Level 1 - Recall - Facts, definitions, terms, simple procedures and algorithms or formula applications. This includes one-step, well-defined, and straight algorithmic procedures.  Level 2 - Skills and Concepts - Engagement of mental processing beyond habitual response. This might include classifying, organizing, collecting, displaying, comparing and interpreting data - operations involving more than one step.  Level 3 -Strategic Thinking - Reasoning, planning, using evidence and a higher level of thinking.  Level 4 - Extended thinking - Complex reasoning, planning, developing and thinking most likely over an extended period of time.  Level 1 - Recall - Facts, definitions, terms, simple procedures and algorithms or formula applications. This includes one-step, well-defined, and straight algorithmic procedures.  Level 2 - Skills and Concepts - Engagement of mental processing beyond habitual response. This might include classifying, organizing, collecting, displaying, comparing and interpreting data - operations involving more than one step.  Level 3 -Strategic Thinking - Reasoning, planning, using evidence and a higher level of thinking.  Level 4 - Extended thinking - Complex reasoning, planning, developing and thinking most likely over an extended period of time.

7. Five Components of Mathematical Proficiency from Adding it Up: Helping Children Learn Mathematics National Research Council  Conceptual understanding - comprehension of concepts, operations and relations  Procedural fluency - skill in carrying out procedures flexibly, accurately, efficiently and appropriately  Strategic competence - ability to formulate, represent and solve problems  Adaptive reasoning - capacity for logical thought, reflection, explanation, and justification  Productive disposition - habitual inclination to see mathematics as sensible, useful,a nd worthwhile, coupled with a belief in diligence and one’s own efficacy.  Conceptual understanding - comprehension of concepts, operations and relations  Procedural fluency - skill in carrying out procedures flexibly, accurately, efficiently and appropriately  Strategic competence - ability to formulate, represent and solve problems  Adaptive reasoning - capacity for logical thought, reflection, explanation, and justification  Productive disposition - habitual inclination to see mathematics as sensible, useful,a nd worthwhile, coupled with a belief in diligence and one’s own efficacy.

8. Finding from Trends in International Mathematics and Science (TIMSS) http://ustimss.msu.edu/republican.html Features of the U. S. Educational System:  Curricula that are a mile wide and an inch deep  lacking any real focus  State frameworks that are like long laundry lists lacking coherence across the topics  A static view of what is basic that keeps repeating the same topics from grade to grade  A middle school curriculum that is not intellectually challenging or world class in standard  A system of tracking students in mathematics that gives a different curriculum to different groups of students  Classroom instruction that mirrors the lack of focus and coherence in the curriculum Features of the U. S. Educational System:  Curricula that are a mile wide and an inch deep  lacking any real focus  State frameworks that are like long laundry lists lacking coherence across the topics  A static view of what is basic that keeps repeating the same topics from grade to grade  A middle school curriculum that is not intellectually challenging or world class in standard  A system of tracking students in mathematics that gives a different curriculum to different groups of students  Classroom instruction that mirrors the lack of focus and coherence in the curriculum

9. Recommendations from Trends in International Mathematics and Science (TIMSS) http://ustimss.msu.edu/republican.html  The US needs to develop focus in its curriculum. Covering many topics each year leads to a lack of depth in any one area. This lack of focus is seen in state frameworks, textbooks and in what teachers actually teach.  US 8th grade textbooks (700 pages and encyclopedic) stand in stark contrast with the 200 page paperbacks in other countries and play a major role in the unfocused nature of the curriculum. The decision to include all topics in the book so that it sells in all states might be good marketing but bad educational policy.  State Frameworks need to reflect the coherence of mathematics and science and not just be long lists of topics.  The US curriculum needs a dynamic view of what is basic at each grade level, one that changes over the grades and provides an intellectually challenging middle school curriculum that includes worldwide basics such as algebra, geometry, chemistry and physics.  The US needs to develop focus in its curriculum. Covering many topics each year leads to a lack of depth in any one area. This lack of focus is seen in state frameworks, textbooks and in what teachers actually teach.  US 8th grade textbooks (700 pages and encyclopedic) stand in stark contrast with the 200 page paperbacks in other countries and play a major role in the unfocused nature of the curriculum. The decision to include all topics in the book so that it sells in all states might be good marketing but bad educational policy.  State Frameworks need to reflect the coherence of mathematics and science and not just be long lists of topics.  The US curriculum needs a dynamic view of what is basic at each grade level, one that changes over the grades and provides an intellectually challenging middle school curriculum that includes worldwide basics such as algebra, geometry, chemistry and physics.

10. Conceptual Structure of “Traditional” Math Teaching Sharon Griffin and Robbie Case

11. Conceptual Structure of “Reform” Math Teaching Sharon Griffin and Robbie Case

12. Let’s look at an example of math as a fixed body of knowledge: MA-06-4.2.1 Students will determine and apply the mean, median, mode, and range of a set of data. DOK - 2 Jack had quiz scores of 75, 85, and 80. After the fourth quiz, his average was 85. All quizzes were weighted equally. What score did Jack receive on the fourth quiz?

13. Let’s look at an example of math as a set of conceptual relations: MA-06-4.2.1 Students will determine and apply the mean, median, mode, and range of a set of data. DOK - 2  The sum of five consecutive counting numbers is 90. What is the largest of these five numbers?  The sum of a set of consecutive counting numbers is 90.What is the least number that could be part of that set of numbers?  How many different ways can you get a sum of 90 with consecutive counting numbers?  What if the sum changes to 84? 30? N? How many different ways can you get a sum of N with consecutive counting numbers?  The sum of five consecutive counting numbers is 90. What is the largest of these five numbers?  The sum of a set of consecutive counting numbers is 90.What is the least number that could be part of that set of numbers?  How many different ways can you get a sum of 90 with consecutive counting numbers?  What if the sum changes to 84? 30? N? How many different ways can you get a sum of N with consecutive counting numbers?

14. The number of US citizens training to become scientists and engineers are declining while the need for them is increasing  These trends threaten the economic welfare and security of our country.  Even if action is taken today, the reversal of this trend is 10 to 20 years away. Middle schoolers today won’t complete advanced training for science and engineering occupations until 2018 or later.

15. China now graduates about 200,000 engineers a year; India and Japan, 100,000 each; the United States about 50,000. In the field of mathematics in the United States, based on current trends, one must begin with 3500 ninth-graders in 2005 to produce 300 freshmen qualified to pursue a degree in mathematics. Of these, about 10 will actually receive a bachelors’ degree in the field. Finally, one Ph. D. in mathematics will emerge in about 2019. Norm Augustine, CEO of Lockheed Martin In the field of mathematics in the United States, based on current trends, one must begin with 3500 ninth-graders in 2005 to produce 300 freshmen qualified to pursue a degree in mathematics. Of these, about 10 will actually receive a bachelors’ degree in the field. Finally, one Ph. D. in mathematics will emerge in about 2019. Norm Augustine, CEO of Lockheed Martin

16. Of all pre-college curricula, the highest level of mathematics one studies in secondary school has the strongest continuing influence on bachelor's degree completion. Finishing a course beyond the level of Algebra 2 (for example, trigonometry or pre-calculus) more than doubles the odds that a student who enters postsecondary education will complete a bachelor's degree. This is much more important than SES! Answers in the Tool Box: Academic Intensity, Attendance Patterns, and Bachelor's Degree Attainment, Adelman, Clifford. 1999c

17. Success in the traditional academic curriculum, especially the math curriculum, is the most powerful predictor of wage advantages from increased postsecondary attainment, and improvements in mathematics skills account for most of the growth in wage premium from increased postsecondary educational attainment since the early 1980s. ( Murnane et al., 1995; Grogger and Eide, 1995)

18. Highest Math Studied in HS % of HS Grads Earning BA Calculus79.8 Pre-Calc74.3 Trig62.2 Algebra 239.5 Geometry23.1 Algebra I7.8 Pre-algebra2.3 Answers in the Tool Box: Academic Intensity, Attendance Patterns, and Bachelor's Degree Attainment — June 1999

19. Did you know that?  College students who have not taken an AP class have a 33% chance of completing a Bachelor’s Degree;  College students who have completed one AP course have a 59% chance of completing a Bachelor’s Degree; and  College students who have completed two or more AP courses increase to 76% their chances of attaining a Bachelor’s Degree? Answers in the Toolbox ; http://www.ed.gov/pubs/Toolbox  College students who have not taken an AP class have a 33% chance of completing a Bachelor’s Degree;  College students who have completed one AP course have a 59% chance of completing a Bachelor’s Degree; and  College students who have completed two or more AP courses increase to 76% their chances of attaining a Bachelor’s Degree? Answers in the Toolbox ; http://www.ed.gov/pubs/Toolbox

20. What indicators of teacher quality are related to students’ performance? Harold Wenglinsky, 2000 http://www.edweek.net/context/topics/issuespage.cfm?id=16 Aspect of Teacher QualityDifference in NAEP Math Scores Major/minor in math/math education 39% PD in working with different student populations 107% PD in higher-order thinking skills 40% Hands-on Learning72%