Mathematics Learning for All: How Can It Be Done?

1. Mathematics Learning for All: How Can It Be Done? Delta ‘99, Laguna Quays, Australia November 22, 1999 David A. Smith Duke University There are lots of good fish in the sea. W. S. Gilbert, The Mikado

2. Outline • Educational Problems and Proposals • Educational Myths and Realities • Learning from Research • Neurobiology • Cognitive Psychology • Research-based Curricular Materials • Summary and Marching Orders

3. Educational Problems and Proposals

4. Problems with Mathematics Education • Changing demographics • Watered-down courses • Bored, alienated students • Low success rates • Growing “remedial” enterprise • Frustrated faculty • Disproportional filtering

5. Reform is not a new idea • “It is to be hoped that the near future will bring reforms in the mathematical teaching in this country. We are in sad need of them. From nearly all of our colleges and universities comes the loud complaint of inefficient preparation on the part of students applying for admission; from the high schools comes the same doleful cry. Educators who have studied the work of German schools declare that our results in elementary instruction are far inferior.” • Florian Cajori, 1890 The Teaching and History of Mathematics in the United States

6. Responses a century later • NCTM Standards for School Mathematics • Curriculum and Evaluation, 1989 • Teaching, 1991 • Assessment, 1995 • NSF Calculus Reform Initiative, 1987-94

7. Equity and Expectations • “We have inherited a mathematics curriculum conforming to the past, blind to the future, and bound by a tradition of minimum expectations.” • “Equity for all requires excellence for all; both thrive when expectations are high.” Everybody Counts, 1989

8. Educational Myths and Realities

9. Transmission Myth • Knowledge can be transmitted from knower to learner.

10. Classroom reality: Constructing knowledge together

11. Darwinian Myth Competition between students • builds character, • weeds out misfits. Reality: Cooperation generates better learning gains for all.

12. Cooperation at work

13. Engineering Math at Duke • 1980’s: Few made it to Eng Math II, mostly white males • Now: 2-4 sections of EM I, 2 of EM II every semester, diverse population • High success rate in hands-on sections, low in lecture sections

14. Standards Myth • The quality of our work is measured by the spread of our grades. • (If everyone “gets it,” our standards are too low.) • Reality: A high level of success is both possible and desirable.

15. Elitist Myth • Only special people (like us) can understand mathematics. • Realities: • Students who work hard at meaningful tasks can understand mathematics. • Students will work hard at meaningful and rewarding tasks.

16. Teaching Myth • Is this true or false? • Is it an axiom? • a theorem? • a definition? Good teaching engenders good learning.

17. Teaching vs. Learning • “In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. … This happens most readily when students work in groups, engage in discussion, make presentations, and in other ways take charge of their own learning.” Everybody Counts, 1989

18. Learning from Research

19. Messages from Neurobiology • The human brain has not evolved significantly in the last 10,000 years. • We all have the same basic equipment. • Deep learning is whole-brain activity. • Mind and body are one system, not two.

20. How People Learn • Students’ initial understandings must be engaged. • To develop competence, students must • have a deep knowledge base, • understand in a conceptual framework, • organize for retrieval and application. • Students must monitor progress toward goals. National Research Council, 1999

21. Good Practice in Undergraduate Education • Encourages student-faculty contact • Encourages cooperation among students • Encourages active learning • Gives prompt feedback • Emphasizes time on task • Communicates high expectations • Respects diverse talents, ways of learning Chickering and Gamson, 1989

22. Diverse ways of learning • Learning approach is more important than learning style. • Deep learning approaches are different from surface learning approaches. • A student may exhibit different approaches in different courses. Ference Marton, Noel Entwhistle, Paul Ramsden, and others

23. What encourages surface approaches? • Excessive amount of material • Lack of opportunity to pursue subjects in depth • Lack of choice over subjects and/or method of study • Threatening assessment system

24. What encourages deep approaches? • Interaction -- peers working in groups • Well-structured knowledge base -- connecting new concepts to prior experience and knowledge • Motivational context -- choice of control, sense of ownership • Learner activity plus faculty connecting activity to abstract concept

25. Kolb Learning Cycle

26. Kolb Learning Cycle • Concrete Experience: input to the sensory cortex -- hearing, seeing, touching, movement • Reflection/Observation: internal, right-brain, produces context, needed for understanding • Abstract Conceptualization: left-brain, develops interpretations of experiences and reflections • Active Experimentation: external action, use of the motor brain

27. Research-based Curricular Materials

28. Connected Curriculum Project • Materials for labs and projects • Web pages with text, hyperlinks, graphics, Java applets, problems • Downloadable CAS files in which students respond to challenges, control the interaction, write a report • Content from precalculus through engineering mathematics and mathematical finance

29. http://www.math.duke.edu/education/

30. Summary and Marching Orders

31. Summary Research results in cognitive psychology, neurobiology, and education all lead to the same conclusions about • who can learn, • how students learn, • how to design curricula and pedagogies to engender learning.

32. Marching Orders College and University Faculty: • Make introductory courses attractive and effective. • Restore integrity to the undergraduate program. • Lecture less; try other teaching methods. • Link scholarship to teaching. Everybody Counts, 1989