A History of Research in Mathematics Education

1. A History of Research in Mathematics Education By Jeremy Kilpatrick

2. The history of education…. • Education was not considered a “discipline” to be studied. • Germany (1779) and Sweden (1804) were the first countries to chair departments of education. • In the U.S., New York University (1832), Brown University (1850) and University of Michigan (1860) led the way. • Mathematics education programs began to develop at the end of the 19th century.

3. Influences from the Disciplines • Mathematics • Mathematicians were curious about how math was created. • Observations of their children’s and grandchildren’s mathematical helped develop and improve programs. • Psychology • Empirical studies were used to observe patterns in cognitive development of school aged children.

4. Pioneer in Mathematics • Feliz Klein • Professor of mathematics at Erlanger (1872) • Famous for his Erlanger Programm • Believed that mathematics should fall between humanistic and scientific education • Served as president of the International Commission on the Teaching of Mathematics, which reported on state of mathematics teaching around the world. • The comparisons were descriptive rather than analytical. • Reports were generated from data collected from surveys

5. Pioneer in Psychology • Alfred Binet • Director of the first French psychological laboratory • Sought to gather precise data by scientifically training teachers and allowing them to practice experimental pedogogy in their classrooms • Believed that questionnaires, observation and experiment were the best means for gathering this information • Originator of intelligence testing • Believed that intelligence testing should be used for diagnosis rather than ranking • Basis for IQ test

6. Research on Thinking • Jean Piaget studied the processes children used to obtain their answers. His assessment of reasoning processes led to the “clinical method”. • Psychological laboratories were established all over the world, including one at Johns Hopkins University in 1883. • At JHU, G. Stanley Hall brought experimental pedagogy to the U.S. by launching a child study movement.

7. More research on thinking…. • University of Wurzburg’s Otto Selz concentrated his studies on problem solving and influenced a generation of psychologists. • Karl Dunker authored and influential monograph that analyzed the processes of solving problems • Lev Vygotsky formulated a thoery of mental growth where development is guided by instruction

8. Studies of Teaching and Learning • Teaching is taken as a treatment and learning as an effect. • The basic technique for analyzing such effects within the compass of a single investigation is analysis of variance, developed by Ronald A. Fisher.

9. Connectionism • Phrase coined by Edward Thorndike to describe his behaviorist studies in 1900. • Thorndike and Robert Woodward attempted to show the limitations of transfer of training. • For example, they found that practicing estimating the sizes of rectangles did not improve one’s ability to estimate the sizes of triangles. • However, Thorndike did not say that transfer of training was impossible “but only that transfer cannot be assumed to occur, that it is rarely automatics, and that direct teaching for desired outcomes is usually more efficient and economical than are hoped-for, spill-over effects”

10. Charles H Judd • His studies disproved connectionism. • Fifth and sixth graders that were taught the principle of refraction did better shooting darts at a moving underwater target than pupil who were not familiar with refraction. • Believed that arithmetic is a “general mode of thinking”.

11. The Testing Movement • Started in Boston in 1845 to prove to the state board how well their students were performing (backfired) • Detailed work done by reformer Joseph Mayer Rice in the 1890’s. • His studies concluded that by setting standards and administering exams, school districts could get the results they wanted.

12. Child Study • G. Stanley Hall and Maria Montessori published findings on the use of activities and materials in teaching mathematics to young children. • Because child study usually involves pre-school children, it has not produced much worthwhile in the area of mathematic education.

13. The Measurement Movement • Cliff W. Stone conducted a study on 3000 students to measure achievement in reasoning and fundamentals. • Stone wanted to standardize the administration and scoring of the test in order to extract relationships between other factors. • Introduction of the concept of efficiency of instruction-ratio linking achievement to time spent in instruction

14. The Social Utility Movement • What does business want? • Mental calculations • Does the curriculum have too much “fluff”? • Yes. Too many “excessive requirements” and too much time spent on arithmetic. • What arithmetic do people use? • Simple calculations, using numbers under 1000 • Calculations for buying and selling • Simple fractions with numerator of 1

15. A Reaction to Reductionism • Critics argue that curriculum cannot be based on frequency of adult usage • “Shall we say that 60% of the teaching of the schools in spelling and language should be devoted to the 100 word of the most frequent occurrence—to the, and, but, to, be, etc?”

16. Incidental Learning and Readiness • Incidental learning-children learn arithmetic better if it is not systematically taught • IL was the precursor for readiness theory-concepts are better taught to a child once he/she is mature enough to understand it. • Louis Benezet, a superintendent in New Hampshire, conducted a study that didn’t teach arithmetic until after grade 7. After a year’s instruction, these students tested at the same level of students that had been taught in a traditional manner.

17. Responses to Curriculum Issues • Though most agreed on the importance of knowing arithmetic, required math courses in high school was challenges in the 1920’s and 1930’s. • Educators started evaluating individual differences and whether or not all students would benefit from studying math. • Other studies dealt with unified mathematics. • Eight Year Study-allowed secondary schools to experiment with innovative curricula without risking graduates’ chances of admission to college

18. The search continues…. • Math educators still didn’t know where they fit in. • Mathematicians were busy arguing mathematical minutiae. • Psychologist often debated issues that were irrelevant to math education.

19. The Golden Age-early to mid 1950’s • American school were under attack for graduating young adults that were not prepared for college or the working world. • Three gatherings of mathematicians and educators to analyze and/or solve the problem: • 1962-conference that brought together psychologists, mathematicians nad mathematics educators to discuss the problems of mathematical learning. • 1967-more diverse and marked the beginning of true interdisciplinary and community among researchers in math education • 1968-mainly mathematicians and educators called together to identify topics for projects, thesis and postdoctoral research in mathematics education.

20. Organized and progressive research • Journal-Vol.1 No.1 Journal for Research in Mathematics Education (January 1970) • About 85% of the research was being conducted in the U.S. • New doctoral programs in math education

21. Realistic approach • European researchers explored activity theory developed in the Soviet Union. • Mathematics is viewed as a human activity arising out of real situations that ask the students to learn from investigating problems they’ve formulated • What does this sound like???

22. Conclusion • Has the lack of depth in mathematics education research contributed to the disputes over math reform? • Are TERC’s and Everyday Math considered studies for research in math education? • What’s next??