Tuesday, 7/26: Week 2 Conceptual vs. Procedural Understanding

1. CI 512: Teaching and Learning Tuesday, 7/26: Week 2 Conceptual vs. Procedural Understanding

2. Class OutlineConceptual and Procedural Understanding Note Taker: Derek Baily Observer: Karen Czarnecki • Logistics (9:00-9:05) • Theory vs. Method (9:05-9:10) • Conceptual and Procedural Understanding • Small Group Discussions (9:10-9:40) • Whole Class Discussion (9:40-10:10) • Break (10:10-10:20) • TIMSS Video Study • Video Viewing (10:20-11:20) • Whole Class Discussion (11:20-11:40) • Observer Observations (11:40-11:45) • Conclusions and Exit Cards (11:45-11:50)

3. Logistics: Readings for Thursday Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (1999). How People Learn: Brain, Mind, Experience, and School. Washington D.C.: National Academy Press. Lobato, J. (2003). How design research can inform a rethinking of transfer and vice versa. Educational Researcher, 32(1), 17-20. Carraher, T., Carraher, D., & Schliemann, A. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3, 21-29 • Optional Reading: Robertson, W. (1990). Detection of cognitive structure with protocol data: Prediction performance on physics transfer problems. Cognitive Science 14, 253-280.

4. Reading Responses • Keep it brief and to the point • No longer a 350 word limit

5. Exit Card Reflections • Readings and discussion don’t focus on my specific teaching context • I/we need to do a better job at bringing conversation back to how these examples relate to our specific foci • Request for more reflection time • Small Group Discussions • Some small groups had successful discussions • Other small groups were dominated by a single speaker • Written questions for the small group time

6. Theory vs. Method • Behaviorism as a theory of learning • Learning is exhibited only through external behaviors • Classical Conditioning: Subjects can be conditioned to have involuntary responses to artificial stimulus • Operant Conditioning: Positive (or negative) reinforcement strengths the bonds between stimulus and response • Drill and practice as a method for instruction

7. What should we learn? • Conceptual and procedural understandings do not form a theory for learning: instead they seek to answer the question “What should one learn?” • Orientations toward conceptual and procedural understandings have implications for teaching practices

9. Small Groups • * Group Moderator: • Help encourage all group members to participate in the discussion • Ensure that all of the discussion questions are covered

10. Small Group Questions • What is procedural/instrumental understanding? Why is it important? • What is conceptual/relational understanding? Why is it important? • Give a description of both conceptual and procedural understanding for (one or more) of the following topics: • The vertex of a parabola • The human circulatory system • Gravity • A topic of your group’s choosing • For one of the above topics, design an activity/questions that help to assess or develop: (a) conceptual understanding (b) procedural understanding

11. Strands of Mathematical Proficiency • Conceptual understanding—comprehension of mathematical concepts, operations, and relations • Procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately • Strategic competence—ability to formulate, represent, and solve mathematical problems • Adaptive reasoning—capacity for logical thought, reflection, explanation, and justification • Productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding It Up: Helping Children Learn Mathematics. download.nap.edu/catalog.php?record_id=9822

12. 10 minute Break

13. TIMSS Study • Trends in International Math and Science Study • Collected data on 4th and 8th grade students in more than 60 countries over a 12 year period (1995, 1999, 2003, 2007) • Video data was collected in three countries: America, Japan and Germany • More information at nces.ed.gov/timss/or google TIMSS

14. TIMSS Methodology • A subset of schools in each nation were randomly selected, and a random class period from each school was selected to be filmed. No substitutions were allowed. • 100 classes in Germany, 50 classes in Japan, and 81 classes in the U.S. • Teachers were asked to teach a typical lesson, and self-reported data on the previous lesson was collected • Although the presence of video cameras might affect classroom behavior, it is unlikely to change the types of follow-up questions a teacher would ask students

15. TIMSS Video Questions • What types of (a) procedural (b) conceptual understandings are evidenced in this video? • What do you think the teachers’ goals are in terms of conceptual and procedural understanding? • How does the teacher encourage (a) procedural and (b) conceptual understanding in this video? • How would you characterize the teachers’ beliefs with respect to conceptual and procedural understanding?

16. Analysis of TIMSS Stiegler & Heibert (1999) The Teaching Gap • International group of teachers and teacher educators reviewed the videos • Complex analytic process of repeated video viewing

17. Analysis of TIMSS Mottos for Teaching Mathematics • Japanese: “Structured problem solving” • German: “Developing advance procedures” • American: “Learning terms and practicing procedures”

18. Analysis of TIMMS (Stiegler & Hiebert) • “In the Japanese lessons, there is the mathematics on the one hand, and the students on the other. The students engage with the mathematics, and the teacher mediates the relationship between the two.”

19. Analysis of TIMMS (Stiegler & Hiebert) • “In Germany, there is the mathematics as well, but the teacher owns the mathematics and parcels it out to students as he sees fit, giving facts and information at just the right time.”

20. Analysis of TIMMS (Stiegler & Hiebert) • “In U.S. lessons, there are the students and there is the teacher. I have trouble finding the mathematics. I just see interactions between students and teachers.”

21. Observer Observations

22. Exit Cards • Rate your level of participation for today (0-3) • What was one thing you learned during today’s class? • What is one thing you would like to know more about from today’s class?