1. Teaching to the “Big Ideas”: Moving beyond the standards Terry P. Vendlinski UCLA Graduate School of Education & Information Studies National Center for Research on Evaluation, Standards, and Student Testing (CRESST) Annual CRESST Conference September 8, 2005

2. Overview Strengths and weaknesses of teaching to standards Integration of cognitive science and educational assessment research Ontological schema and Bayesian networks

3. Strengths and Weaknesses Improving instruction and learning by linking to standards (NCLB) Instruction needs to be focused but…. Organization and Sequence may be lacking Regular monitoring of student progress AND feedback about progress may be lacking

4. Example of Instruction e i + 1 = 0

5. Assessment e i + 1 =

6. Answer e i + 1 = 0

7. What Would You Have to Learn in Order for This to Be Useful? What do the symbols mean? What is this equation about? What can you do with it? How can you use it? What’s important about it? Why has it been called the most beautiful equation in all of mathematics? Someone who can answer these questions understands the meaning of what they’ve learned. Niemi, 2005

8. The integration of cognitive science and educational assessment research Our model of how students learn affects our assessment practice and our inferences (KWSK) These, in turn, affect our instruction

9. Cognitive Science Humans look for organizing features We try to apply prior knowledge The individual constructs meaning and understanding

10. Expert Knowledge Structure

11. Advanced Novice Knowledge Structure

12. CRESST Models Determine the cognitive demand required to master a task or concept Place the task or concept in context Measure that against what is expected

14. Inverses Additive Laws of Arithmetic Expression Attaching NCTM Standards to the Ontology Recognize and use inverse properties (6 – 8) Use properties of zero.. in operations (4 – 5) Understand and use inverse relationships … within the operations of addition and subtraction ( 6 – 8)

15. Inverses Additive Laws of Arithmetic Expression Attaching California Standards to the Ontology Simplify numerical expressions by applying … inverse (7) Use the inverse relationship between addition and subtraction (1 & 2)

16. Inverses Additive Laws of Arithmetic Expression Attaching Items to the Ontology This is how a student did the problem 2 – (-7). If the student asked you to make sure it was correct, what would you say? Give an explanation WHY you think each step is either correct or incorrect: 2 – (-7) 2 + 7 + (-7) – (-7) 2 + 7 9

17. Inverses Additive Laws of Arithmetic Expression Using student work to infer understanding Given: 2 – (-7) Explain the step: 2 + 7 + (-7) – (-7) This is correct since 7 + (-7) is zero and you don’t change anything by adding zero

18. Inverses Additive Laws of Arithmetic Expression Using student work to infer understanding Given: 2 – (-7) Explain the step: 2 + 7 + (-7) – (-7) This is correct because (-7) – (-7) cancel each other out making it 2 + 7 + 0

19. Inverses Additive Laws of Arithmetic Expression Using student work to infer understanding Given: 2 – (-7) Explain the step: 2 + 7 + (-7) – (-7) This is incorrect because the 7 in 2 + 7 should be a -7 Number line

20. Bayesian Inference The probability of event 1 GIVEN that event 2 has occurred is the product of the probability of event 2 given the probability of event 1 and the probability of event 1 divided by the probability of event 2

21. Bayesian Inference II The probability that a student understands GIVEN that they’ve passed a test is the product of the probability that they pass a test given they understand and the general probability of understanding divided by the probability that students pass the test

22. Using the Ontology

24. Bayesian Ontologies We can… probabilistically infer student understanding select assessments that are likely to be most informative see new ways to organize instruction

25. Questions? Terry Vendlinski Vendlins@ucla.edu