Soviet Mathematics Soviet Mathematics Education ResearchEducation Research

A few interesting A few interesting morsels...morsels...

The Soviet PeriodThe Soviet Period

1914 – 19911914 – 1991– The Russian Revolution to GorbachevThe Russian Revolution to Gorbachev– Many circumstances create a special Many circumstances create a special

environment...environment...

– Communism/Revolution/ConformismCommunism/Revolution/ConformismTotalitarianism and good fruits of intellectual Totalitarianism and good fruits of intellectual laborlabor

Special conditions for education researchSpecial conditions for education research– AccessAccess– Philosophy – racial/ethnic equalityPhilosophy – racial/ethnic equality– IQ testing bannedIQ testing banned– Research in rapidly developing areas (agrarian to Research in rapidly developing areas (agrarian to

industrial)industrial)

Important FiguresImportant FiguresLev Semenovich Vygotsky (1896-1934) Soviet Lev Semenovich Vygotsky (1896-1934) Soviet developmental psychologist and the founder of developmental psychologist and the founder of cultural-historical psychology cultural-historical psychology

Alexander Romanovich Luria (1902-1977) Soviet Alexander Romanovich Luria (1902-1977) Soviet neuropsychologist and developmental psychologist, neuropsychologist and developmental psychologist, student of Vygotsky, one of the founders of cultural-student of Vygotsky, one of the founders of cultural-historical psychology and psychological activity theory historical psychology and psychological activity theory

Vasily Davydov (1930-1988) ‘Vygotskian’ psychologist Vasily Davydov (1930-1988) ‘Vygotskian’ psychologist and educationalist, developer of mathematics and educationalist, developer of mathematics curricula aimed at a fundamentally different approach curricula aimed at a fundamentally different approach to learning to learning

(they all got in some trouble for their rather liberal thinking...)(they all got in some trouble for their rather liberal thinking...)

Different Quality of Soviet Math Different Quality of Soviet Math Ed ResearchEd Research

Environment of repression and Environment of repression and freedom demands different thinking freedom demands different thinking and solutionsand solutionsRejection of standardized IQ testingRejection of standardized IQ testingDetailed studies of a small number of Detailed studies of a small number of subjectssubjects– Uzbeki peasants (colors, objects) LuriaUzbeki peasants (colors, objects) Luria– Twins (block building) Gal’perinTwins (block building) Gal’perin

Science and technology highly prizedScience and technology highly prized

Math Education Research, a Math Education Research, a sample result:sample result:

‘‘The difference between capable, The difference between capable, average, and incapable pupils, as our average, and incapable pupils, as our research permits us to conclude, research permits us to conclude, comes down to the following. In comes down to the following. In capable pupils these associations capable pupils these associations (mathematical generalizations) can (mathematical generalizations) can be formed “on the spot”: in this be formed “on the spot”: in this sense they are “born”, if one can so sense they are “born”, if one can so express it, already generalized, with express it, already generalized, with a minimal number of exercises.a minimal number of exercises.

Math Education Research, a Math Education Research, a sample result:sample result:

In average pupils these associations In average pupils these associations are established and reinforced are established and reinforced gradually, as a result of a whole gradually, as a result of a whole series of exercises. They form series of exercises. They form isolated, concrete associations, isolated, concrete associations, related only to a given problem, “on related only to a given problem, “on the spot”. Through single-type the spot”. Through single-type exercises these associations are exercises these associations are gradually transformed into gradually transformed into generalized associations. generalized associations.

Math Education Research, a Math Education Research, a sample result:sample result:

In incapable pupils, even the In incapable pupils, even the isolated, concrete associations are isolated, concrete associations are formed with difficulty, their formed with difficulty, their generalizations are still more generalizations are still more difficult, and sometimes such difficult, and sometimes such generalizations do not occur at all.’generalizations do not occur at all.’

(V. A. Krutetskii, 1976, The Psychology of Mathematical (V. A. Krutetskii, 1976, The Psychology of Mathematical Abilities in Schoolchildren, p.262.)Abilities in Schoolchildren, p.262.)

V V DavydovV V Davydov

So, can generalization, the rapid making of general mathematical associations be taught?Can we teach all students to become ‘capable pupils’?Davydov math curriculum for grades 1 – 3 attempts just that– "ascending from the abstract to the concrete"

(A/C) teaching format – the project still continuesthe project still continues

Philosophy of the Davydov Philosophy of the Davydov CurriculumCurriculum

When we designed a mathematics course, we When we designed a mathematics course, we proceeded from the fact that the students’ proceeded from the fact that the students’ creation of a detailed and thorough creation of a detailed and thorough conception of a conception of a real numberreal number, underlying , underlying which is the concept of which is the concept of quantityquantity, is currently , is currently the end purpose of this entire instructional the end purpose of this entire instructional subject from grade 1 to grade 10….In our subject from grade 1 to grade 10….In our course the teacher, relying on the knowledge course the teacher, relying on the knowledge previously acquired by the children, previously acquired by the children, introduces number as a particular case of the introduces number as a particular case of the representation of a representation of a general relationshipgeneral relationship of of quantities, where one of the numbers is taken quantities, where one of the numbers is taken as a measure and is computing the other.as a measure and is computing the other. (Davydov, 1990, pp. 358, 352)(Davydov, 1990, pp. 358, 352)

Davydov CurriculumDavydov Curriculum

AlgebraicAlgebraicDevelops the concept of equality and inequality before Develops the concept of equality and inequality before numbernumberDevelops part-whole relationships first, thenDevelops part-whole relationships first, thenDevelops the concept of measure,Develops the concept of measure,Then unit,Then unit,Then numberThen numberNot your typical arithmeticNot your typical arithmeticBase 3 numbers developed before base 10Base 3 numbers developed before base 10‘‘Arithmetic’ at the end of 1Arithmetic’ at the end of 1stst grade gradeCurriculum is a series of problems – no expository Curriculum is a series of problems – no expository material.material.

Example 1Example 1

Example 2Example 2

What thinking do you think is What thinking do you think is elicited by these problems?elicited by these problems?

Children’s thinking:Children’s thinking:

Parts Parts ≠≠ whole if there are too many parts. whole if there are too many parts.Parts Parts ≠ whole if one of the parts is the wrong ≠ whole if one of the parts is the wrong type or size.type or size.Parts ≠ whole if there are too few parts.Parts ≠ whole if there are too few parts.Parts = whole if and only if the number, types, Parts = whole if and only if the number, types, and sizes of the parts exactly match the whole.and sizes of the parts exactly match the whole.This series of problems developed the part-whole This series of problems developed the part-whole language with the children.language with the children.

Example 3Example 3

Consider this problem:Consider this problem:

How would you solve it? What do you think the letters ‘C’ and ‘P’ represent? What do you think the letters ‘C’ and ‘P’ should represent?

Children’s thinking:Children’s thinking:

P is for place and C is for the city where the cups P is for place and C is for the city where the cups were made.were made.P is for portion and C is for cup.P is for portion and C is for cup.P and C are for the colors of the cups.P and C are for the colors of the cups.How much they can hold is what is equal.How much they can hold is what is equal.Volume is like three books that go together.Volume is like three books that go together.Volume is like really loud music.Volume is like really loud music.Volume and space are different.Volume and space are different.The perimeters of the cups are the same.The perimeters of the cups are the same.The company that made the cups is the same.The company that made the cups is the same.The heights of the cups are equal.The heights of the cups are equal.The sizes of the cups are equal.The sizes of the cups are equal.The shapes of the cups are equal.The shapes of the cups are equal.

(PAUSE)(PAUSE)

(for exercise)(for exercise)

What did we argue about?What did we argue about?

Is it:Is it:T + T + B B = H + B= H + B

oror

T + T + (some other letter) (some other letter) = H + B?= H + B?

Which answer is correct?Which answer is correct?

Could both answers be correct?Could both answers be correct?

What is the nature of a variable?What is the nature of a variable?

How do we get our students How do we get our students to form the ‘right’ concepts, to form the ‘right’ concepts, make the correct make the correct generalizations?generalizations?

Some things to think about Some things to think about from before....from before....

A variable is a number.A variable is a number.

An equation is an action.An equation is an action.

A variable is a number.A variable is a number.

AA22 – B – B22 = (A + B)(A – B) = (A + B)(A – B)equation?, formula?, expression?equation?, formula?, expression?variables?, unknowns?, something to solve?variables?, unknowns?, something to solve?

Completely factor: Completely factor: 5522 – X – X22 YY22 – X – X22 49 – T49 – T22 4S4S22– 16– 16(m(m22 – n – n22))66 – (m – (m22 + 25) + 25)88

same?, different?same?, different?

An equation is an action.An equation is an action.

Have you seen this before?Have you seen this before?– Simplify/Evaluate the expression: Simplify/Evaluate the expression:

(10(1022)+(12 x 2 – 20 )+(12 x 2 – 20 4 + 3 x 2) = 4 + 3 x 2) =

– And a student writes the following:And a student writes the following:(10(1022)+(12 x 2 – 20 )+(12 x 2 – 20 4 + 3 x 2) 4 + 3 x 2) = 100 = 100 = 100 = 100

+ 24 – 5 + 6 = 124 = 125+ 24 – 5 + 6 = 124 = 125

– What is happening here???What is happening here???

Some thoughts...Some thoughts...

Students make RATIONAL errors Students make RATIONAL errors based on their concepts, the based on their concepts, the generalizations that they have made generalizations that they have made from their own experiences.from their own experiences.

These issues are not just vital for These issues are not just vital for mathematics education, but all fields.mathematics education, but all fields.

These issues are POLITICAL and These issues are POLITICAL and CULTURAL.CULTURAL.

Some (conceptual) thoughts...Some (conceptual) thoughts...

What is quantifiable and what is not?What is quantifiable and what is not?

Why are ‘we’ arguing over Why are ‘we’ arguing over ‘embryonic’ stem cell research and ‘embryonic’ stem cell research and not over ‘zygote’ stem cell research?not over ‘zygote’ stem cell research?

What is the difference between What is the difference between Religious and Scientific explanations Religious and Scientific explanations of the world? Is belief the same as of the world? Is belief the same as theory?theory?

The EndThe End

Or is it?????Or is it?????

It’s up to you.It’s up to you.

““I know why that means equal! Equal I know why that means equal! Equal is like that because it’s like two equal is like that because it’s like two equal lines.”lines.”

““A letter could stand for anything. A letter could stand for anything. Doesn’t matter what, when or how.”Doesn’t matter what, when or how.”