A History of Research A History of Research in Mathematics in Mathematics

EducationEducation

By Jeremy KilpatrickBy Jeremy Kilpatrick

The history of education….The history of education….

Education was not considered a Education was not considered a “discipline” to be studied.“discipline” to be studied.

Germany (1779) and Sweden (1804) were Germany (1779) and Sweden (1804) were the first countries to chair departments of the first countries to chair departments of education.education.

In the U.S., New York University (1832), In the U.S., New York University (1832), Brown University (1850) and University of Brown University (1850) and University of Michigan (1860) led the way.Michigan (1860) led the way.

Mathematics education programs began to Mathematics education programs began to develop at the end of the 19develop at the end of the 19thth century. century.

Influences from the Influences from the DisciplinesDisciplines

MathematicsMathematics– Mathematicians were curious about how Mathematicians were curious about how

math was math was created.created.– Observations of their children’s and Observations of their children’s and

grandchildren’s mathematical helped grandchildren’s mathematical helped develop and improve programs.develop and improve programs.

PsychologyPsychology– Empirical studies were used to observe Empirical studies were used to observe

patterns in cognitive development of patterns in cognitive development of school aged children.school aged children.

Pioneer in MathematicsPioneer in Mathematics

Feliz KleinFeliz Klein– Professor of mathematics at Erlanger (1872)Professor of mathematics at Erlanger (1872)– Famous for his Erlanger ProgrammFamous for his Erlanger Programm– Believed that mathematics should fall between Believed that mathematics should fall between

humanistic and scientific educationhumanistic and scientific education– Served as president of the International Served as president of the International

Commission on the Teaching of Mathematics, Commission on the Teaching of Mathematics, which reported on state of mathematics teaching which reported on state of mathematics teaching around the world.around the world. The comparisons were descriptive rather than analytical.The comparisons were descriptive rather than analytical. Reports were generated from data collected from surveysReports were generated from data collected from surveys

Pioneer in PsychologyPioneer in Psychology Alfred BinetAlfred Binet

– Director of the first French psychological Director of the first French psychological laboratorylaboratory

– Sought to gather precise data by scientifically Sought to gather precise data by scientifically training teachers and allowing them to practice training teachers and allowing them to practice experimental pedogogy in their classroomsexperimental pedogogy in their classrooms

– Believed that questionnaires, observation and Believed that questionnaires, observation and experiment were the best means for gathering experiment were the best means for gathering this informationthis information

– Originator of intelligence testingOriginator of intelligence testing Believed that intelligence testing should be used for Believed that intelligence testing should be used for

diagnosis rather than rankingdiagnosis rather than ranking Basis for IQ testBasis for IQ test

Research on ThinkingResearch on Thinking

Jean Piaget studied the processes children Jean Piaget studied the processes children used to obtain their answers. His used to obtain their answers. His assessment of reasoning processes led to assessment of reasoning processes led to the “clinical method”.the “clinical method”.

Psychological laboratories were Psychological laboratories were established all over the world, including established all over the world, including one at Johns Hopkins University in 1883.one at Johns Hopkins University in 1883.

At JHU, G. Stanley Hall brought At JHU, G. Stanley Hall brought experimental pedagogy to the U.S. by experimental pedagogy to the U.S. by launching a child study movement.launching a child study movement.

More research on More research on thinking….thinking….

University of Wurzburg’s Otto Selz University of Wurzburg’s Otto Selz concentrated his studies on problem concentrated his studies on problem solving and influenced a generation solving and influenced a generation of psychologists.of psychologists.– Karl Dunker authored and influential Karl Dunker authored and influential

monograph that analyzed the processes monograph that analyzed the processes of solving problemsof solving problems

– Lev Vygotsky formulated a thoery of Lev Vygotsky formulated a thoery of mental growth where development is mental growth where development is guided by instructionguided by instruction

Studies of Teaching and Studies of Teaching and LearningLearning

Teaching is taken as a treatment and Teaching is taken as a treatment and learning as an effect.learning as an effect.

The basic technique for analyzing The basic technique for analyzing such effects within the compass of a such effects within the compass of a single investigation is analysis of single investigation is analysis of variance, developed by Ronald A. variance, developed by Ronald A. Fisher.Fisher.

ConnectionismConnectionism Phrase coined by Edward Thorndike to describe Phrase coined by Edward Thorndike to describe

his behaviorist studies in 1900.his behaviorist studies in 1900. Thorndike and Robert Woodward attempted to Thorndike and Robert Woodward attempted to

show the limitations of transfer of training.show the limitations of transfer of training. For example, they found that practicing For example, they found that practicing

estimating the sizes of rectangles did not improve estimating the sizes of rectangles did not improve one’s ability to estimate the sizes of triangles.one’s ability to estimate the sizes of triangles.

However, Thorndike did not say that transfer of However, Thorndike did not say that transfer of training was impossible “but only that transfer training was impossible “but only that transfer cannot be assumed to occur, that it is rarely cannot be assumed to occur, that it is rarely automatics, and that direct teaching for desired automatics, and that direct teaching for desired outcomes is usually more efficient and outcomes is usually more efficient and economical than are hoped-for, spill-over effects”economical than are hoped-for, spill-over effects”

Charles H JuddCharles H Judd

His studies disproved connectionism.His studies disproved connectionism. Fifth and sixth graders that were Fifth and sixth graders that were

taught the principle of refraction did taught the principle of refraction did better shooting darts at a moving better shooting darts at a moving underwater target than pupil who underwater target than pupil who were not familiar with refraction.were not familiar with refraction.

Believed that arithmetic is a “general Believed that arithmetic is a “general mode of thinking”.mode of thinking”.

The Testing MovementThe Testing Movement

Started in Boston in 1845 to prove to Started in Boston in 1845 to prove to the state board how well their the state board how well their students were performing (backfired)students were performing (backfired)

Detailed work done by reformer Detailed work done by reformer Joseph Mayer Rice in the 1890’s.Joseph Mayer Rice in the 1890’s.

His studies concluded that by setting His studies concluded that by setting standards and administering exams, standards and administering exams, school districts could get the results school districts could get the results they wanted.they wanted.

Child StudyChild Study

G. Stanley Hall and Maria Montessori G. Stanley Hall and Maria Montessori published findings on the use of published findings on the use of activities and materials in teaching activities and materials in teaching mathematics to young children.mathematics to young children.

Because child study usually involves Because child study usually involves pre-school children, it has not pre-school children, it has not produced much worthwhile in the produced much worthwhile in the area of mathematic education.area of mathematic education.

The Measurement The Measurement MovementMovement

Cliff W. Stone conducted a study on 3000 Cliff W. Stone conducted a study on 3000 students to measure achievement in students to measure achievement in reasoning and fundamentals.reasoning and fundamentals.

Stone wanted to standardize the Stone wanted to standardize the administration and scoring of the test in administration and scoring of the test in order to extract relationships between order to extract relationships between other factors.other factors.

Introduction of the concept of Introduction of the concept of efficiency of efficiency of instruction-instruction-ratio linking achievement to ratio linking achievement to time spent in instructiontime spent in instruction

The Social Utility MovementThe Social Utility Movement

What does business want?What does business want?– Mental calculationsMental calculations

Does the curriculum have too much Does the curriculum have too much “fluff”?“fluff”?– Yes. Too many “excessive requirements” and Yes. Too many “excessive requirements” and

too much time spent on arithmetic.too much time spent on arithmetic. What arithmetic do people use?What arithmetic do people use?

– Simple calculations, using numbers under 1000Simple calculations, using numbers under 1000– Calculations for buying and sellingCalculations for buying and selling– Simple fractions with numerator of 1Simple fractions with numerator of 1

A Reaction to ReductionismA Reaction to Reductionism

Critics argue that curriculum cannot Critics argue that curriculum cannot be based on frequency of adult be based on frequency of adult usageusage– ““Shall we say that 60% of the teaching Shall we say that 60% of the teaching

of the schools in spelling and language of the schools in spelling and language should be devoted to the 100 word of should be devoted to the 100 word of the most frequent occurrence—to the, the most frequent occurrence—to the, and, but, to, be, etc?”and, but, to, be, etc?”

Incidental Learning and Incidental Learning and ReadinessReadiness

Incidental learning-children learn arithmetic Incidental learning-children learn arithmetic better if it is not systematically taughtbetter if it is not systematically taught

IL was the precursor for readiness theory-IL was the precursor for readiness theory-concepts are better taught to a child once concepts are better taught to a child once he/she is mature enough to understand it.he/she is mature enough to understand it.

Louis Benezet, a superintendent in New Louis Benezet, a superintendent in New Hampshire, conducted a study that didn’t Hampshire, conducted a study that didn’t teach arithmetic until after grade 7. After a teach arithmetic until after grade 7. After a year’s instruction, these students tested at year’s instruction, these students tested at the same level of students that had been the same level of students that had been taught in a traditional manner.taught in a traditional manner.

Responses to Curriculum Responses to Curriculum IssuesIssues

Though most agreed on the importance of Though most agreed on the importance of knowing arithmetic, required math courses knowing arithmetic, required math courses in high school was challenges in the 1920’s in high school was challenges in the 1920’s and 1930’s.and 1930’s.

Educators started evaluating individual Educators started evaluating individual differences and whether or not all students differences and whether or not all students would benefit from studying math.would benefit from studying math.

Other studies dealt with unified Other studies dealt with unified mathematics.mathematics.

Eight Year Study-allowed secondary schools Eight Year Study-allowed secondary schools to experiment with innovative curricula to experiment with innovative curricula without risking graduates’ chances of without risking graduates’ chances of admission to collegeadmission to college

The search continues….The search continues….

Math educators still didn’t know Math educators still didn’t know where they fit in.where they fit in.– Mathematicians were busy arguing Mathematicians were busy arguing

mathematical minutiae.mathematical minutiae.– Psychologist often debated issues that Psychologist often debated issues that

were irrelevant to math education.were irrelevant to math education.

The Golden Age-early to mid The Golden Age-early to mid 1950’s1950’s

American school were under attack for graduating American school were under attack for graduating young adults that were not prepared for college young adults that were not prepared for college or the working world.or the working world.

Three gatherings of mathematicians and Three gatherings of mathematicians and educators to analyze and/or solve the problem:educators to analyze and/or solve the problem:– 1962-conference that brought together psychologists, 1962-conference that brought together psychologists,

mathematicians nad mathematics educators to discuss mathematicians nad mathematics educators to discuss the problems of mathematical learning.the problems of mathematical learning.

– 1967-more diverse and marked the beginning of true 1967-more diverse and marked the beginning of true interdisciplinary and community among researchers in interdisciplinary and community among researchers in math educationmath education

– 1968-mainly mathematicians and educators called 1968-mainly mathematicians and educators called together to identify topics for projects, thesis and together to identify topics for projects, thesis and postdoctoral research in mathematics education.postdoctoral research in mathematics education.

Organized and progressive Organized and progressive researchresearch

Journal-Vol.1 No.1 Journal for Journal-Vol.1 No.1 Journal for Research in Mathematics Education Research in Mathematics Education (January 1970)(January 1970)

About 85% of the research was being About 85% of the research was being conducted in the U.S.conducted in the U.S.

New doctoral programs in math New doctoral programs in math educationeducation

Realistic approachRealistic approach

European researchers explored European researchers explored activity theory developed in the activity theory developed in the Soviet Union.Soviet Union.

Mathematics is viewed as a human Mathematics is viewed as a human activity arising out of real situations activity arising out of real situations that ask the students to learn from that ask the students to learn from investigating problems they’ve investigating problems they’ve formulatedformulated

What does this sound like???What does this sound like???

ConclusionConclusion

Has the lack of depth in mathematics Has the lack of depth in mathematics education research contributed to education research contributed to the disputes over math reform?the disputes over math reform?

Are TERC’s and Everyday Math Are TERC’s and Everyday Math considered studies for research in considered studies for research in math education?math education?

What’s next??What’s next??