2. The New CIA: Curriculum, Instruction and Assessment in Mathematics Education for the 21 st Century Mathematics Education Colloquia University of Kentucky November 15, 2005 Linda Jensen Sheffield Regents Professor Northern Kentucky University Sheffield@nku.edu http://www.nku.edu/mathed Mathematics Education Colloquia University of Kentucky November 15, 2005 Linda Jensen Sheffield Regents Professor Northern Kentucky University Sheffield@nku.edu http://www.nku.edu/mathed

3. Curriculum: What does it mean for Kentucky to have world-class mathematics curriculum and instruction?

4. 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)

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

6. What is instruction?  A command and explanation for how an action, behavior, method, or task is to be begun, completed, conducted or executed. (Wikipedia)  TEACHING IS A CONSCIOUS STREAM OF DECISIONS, MADE BEFORE, DURING, AND AFTER INSTRUCTION, THE IMPLEMENTATION OF WHICH, WILL INCREASE THE PROBABILITY THAT LEARNING WILL TAKE PLACE. ( Madeline Hunter, Modified)  Academic guidance and intellectual motivation (Ithaca College)  A command and explanation for how an action, behavior, method, or task is to be begun, completed, conducted or executed. (Wikipedia)  TEACHING IS A CONSCIOUS STREAM OF DECISIONS, MADE BEFORE, DURING, AND AFTER INSTRUCTION, THE IMPLEMENTATION OF WHICH, WILL INCREASE THE PROBABILITY THAT LEARNING WILL TAKE PLACE. ( Madeline Hunter, Modified)  Academic guidance and intellectual motivation (Ithaca College)

7. 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.

8. Whenever you read a book or have a conversation, the experience causes physical changes in your brain. It's a little frightening to think that every time you walk away from an encounter, your brain has been altered, sometimes permanently. George Johnson (from In the Palaces of Memory. How We Build the Worlds Inside Our Heads, 1991)

9. What can we learn from brain functioning research?  Construct your own knowledge and make connections between new knowledge and old. (Grow dendrites and synapses.)  Plan ahead and organize your learning. (Use your frontal lobes.)  Use all the senses through active, hands-on learning. (Use the parietal lobes.)  Look for patterns. (Strengthen the P - 300 wave.)  Use novel situations and have fun when you are learning. (Create endorphins and avoid “flight or fight reaction”.)  Challenge yourself; take harder classes, and really try to do well. (Grow and strengthen neurons.)  Construct your own knowledge and make connections between new knowledge and old. (Grow dendrites and synapses.)  Plan ahead and organize your learning. (Use your frontal lobes.)  Use all the senses through active, hands-on learning. (Use the parietal lobes.)  Look for patterns. (Strengthen the P - 300 wave.)  Use novel situations and have fun when you are learning. (Create endorphins and avoid “flight or fight reaction”.)  Challenge yourself; take harder classes, and really try to do well. (Grow and strengthen neurons.)

10. A Heuristic for Constructing Mathematics

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

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

13. Questions for differentiating the mathematics curricula Who? What? When? Where? Why? and How? Who should learn rich mathematics? – Everyone! What or what if? What patterns do I see? What generalizations might I make from the patterns? What proof do I have? What are the chances? What is the best answer, the best method of solution, the best strategy to begin with … ? What if I change one or more parts of the problem? When? When does this work? When does this not work? Where? Where did that come from? Where should I start? Where might I go for help? Why or why not? Why does that work? If it does not work, why not? How? How is this like other problems or patterns that I have seen? How does it differ? How does this relate to "real-life" situations or models? How many solutions are possible? How many ways might I use to represent, simulate, model, or visualize these ideas? How many ways might I sort, organize, and present this information?

14. Your brain never stops developing and changing. It's been doing it from the time your were an embryo, and will keep on doing it all your life. And this ability, perhaps, represents it greatest strength. James Trefil (from Are We Unique?, 1997) Your brain never stops developing and changing. It's been doing it from the time your were an embryo, and will keep on doing it all your life. And this ability, perhaps, represents it greatest strength. James Trefil (from Are We Unique?, 1997)

15. Role of a Student Creator ⇛ Repeat/rephrase ⇛ Agree/disagree...and tell why ⇛ Add on to... ⇛ Wait, think, and go deeper ⇛ Talk to a partner ⇛ Record reasoning and questions Role of a Teacher/Mentor:  Ask questions that encourage creativity and reasoning  Elicit, engage and challenge each student’s thinking  Listen carefully to students’ ideas  Ask students to clarify and justify their ideas  Attach notation and language to students’ ideas  Decide when to provide information, clarify, model, lead or let students struggle  Monitor and encourage participation Adapted from Project M 3 : Mentoring Mathematical Minds

16. The arrows in the diagram stand for +3. What number belongs in the final square? 5 What number would go in the 87th square?

17. Question the answers; don ’ t just answer the questions.

18. Dig more deeply. The number in the final square is 160. What do the arrows stand for? 5 Is another answer possible? The number in the final square is 160. What do the arrows stand for? 5 Is another answer possible?

19. Think deeply about simple things.

20. The number in the final square is 41. What do the arrows stand for? (An arrow above the squares might stand for something different than an arrow below the squares.) How many different answers can you find? 5 The number in the final square is 41. What do the arrows stand for? (An arrow above the squares might stand for something different than an arrow below the squares.) How many different answers can you find? 5

21. What if …?  you change the number of steps?  you change the starting number?  you change the operations? The creative mathematics begins after the original problem has been solved.

22. 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%

23. The principal activities of brains are making changes in themselves. And we as teachers have the power to impact the creation of new minds. Use your power wisely. And we as teachers have the power to impact the creation of new minds. Use your power wisely. Marvin L. Minsky (from Society of the Mind, 1986)