Mathematics Learning for Mathematics Learning for All: How Can It Be Done?All: How Can It Be Done?

David A. SmithDuke University

Delta ‘99, Laguna Quays, Australia

November 22, 1999

There are lots of good fish in the sea.

W. S. Gilbert, The Mikado

OutlineOutline

Educational Problems and Proposals

Educational Myths and Realities Learning from Research

Neurobiology Cognitive Psychology

Research-based Curricular Materials

Summary and Marching Orders

Educational Problems Educational Problems and Proposalsand Proposals

Problems with Problems with Mathematics Education Mathematics Education

Changing demographics Watered-down courses Bored, alienated students Low success rates Growing “remedial” enterprise Frustrated faculty Disproportional filtering

Reform is not a new ideaReform 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

Responses a century laterResponses a century later

NCTM Standards for School Mathematics Curriculum and Evaluation, 1989 Teaching, 1991 Assessment, 1995

NSF Calculus Reform Initiative, 1987-94

Equity and ExpectationsEquity 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

Educational Myths Educational Myths and Realitiesand Realities

Transmission MythTransmission Myth

Knowledge can be transmitted from knower to learner.

Classroom reality: Classroom reality: Constructing knowledge togetherConstructing knowledge together

Darwinian MythDarwinian Myth

Competition between students builds character, weeds out misfits.

Reality: Cooperation generates better learning gains for all.

Cooperation at workCooperation at work

Engineering Math at Engineering Math at DukeDuke

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

Standards MythStandards 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.

Elitist MythElitist 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.

Teaching MythTeaching Myth

Is this true or false? Is it an axiom? a theorem? a definition?

Good teaching engenders good learning.

Teaching vs. LearningTeaching 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

Learning from ResearchLearning from Research

Messages from NeurobiologyMessages 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.

How People LearnHow 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

Good Practice in Good Practice in Undergraduate EducationUndergraduate 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

Diverse ways of Diverse ways of learninglearning

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

What encourages What encourages surface approaches?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

What encourages What encourages deep approaches?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

Kolb Learning CycleKolb Learning Cycle

Kolb Learning CycleKolb 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

Research-based Research-based Curricular MaterialsCurricular Materials

Connected Curriculum Connected Curriculum ProjectProject

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

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

Summary and Summary and Marching OrdersMarching Orders

SummarySummary

who can learn, how students learn, how to design curricula and

pedagogies to engender learning.

Research results in cognitive psychology, neurobiology, and education all lead to the same conclusions about

Marching OrdersMarching 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