Soviet Mathematics Education Research A few interesting morsels...

The Soviet Period 1914 – 1991 –The Russian Revolution to Gorbachev –Many circumstances create a special environment...

–Communism/Revolution/Conformism Totalitarianism and good fruits of intellectual labor Special conditions for education research –Access –Philosophy – racial/ethnic equality –IQ testing banned –Research in rapidly developing areas (agrarian to industrial)

Important Figures Lev Semenovich Vygotsky ( ) Soviet developmental psychologist and the founder of cultural-historical psychology Alexander Romanovich Luria ( ) Soviet neuropsychologist and developmental psychologist, student of Vygotsky, one of the founders of cultural-historical psychology and psychological activity theory Vasily Davydov ( ) ‘Vygotskian’ psychologist and educationalist, developer of mathematics curricula aimed at a fundamentally different approach to learning (they all got in some trouble for their rather liberal thinking...)

Different Quality of Soviet Math Ed Research Environment of repression and freedom demands different thinking and solutions Rejection of standardized IQ testing Detailed studies of a small number of subjects –Uzbeki peasants (colors, objects) Luria –Twins (block building) Gal’perin Science and technology highly prized

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

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

Math Education Research, a sample result: In incapable pupils, even the isolated, concrete associations are formed with difficulty, their generalizations are still more difficult, and sometimes such generalizations do not occur at all.’ (V. A. Krutetskii, 1976, The Psychology of Mathematical Abilities in Schoolchildren, p.262.)

V 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 continues

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

Davydov Curriculum Algebraic Develops the concept of equality and inequality before number Develops part-whole relationships first, then Develops the concept of measure, Then unit, Then number Not your typical arithmetic Base 3 numbers developed before base 10 ‘Arithmetic’ at the end of 1 st grade Curriculum is a series of problems – no expository material.

Example 1

Example 2

What thinking do you think is elicited by these problems?

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

Example 3

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

(PAUSE) (for exercise)

What did we argue about? Is it: T + B = H + B or T + (some other letter) = H + B? Which answer is correct? Could both answers be correct? What is the nature of a variable?

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

Some things to think about from before.... A variable is a number. An equation is an action.

A variable is a number. A 2 – B 2 = (A + B)(A – B) equation?, formula?, expression? variables?, unknowns?, something to solve? Completely factor: 5 2 – X 2 Y 2 – X 2 49 – T 2 4S 2 – 16 (m 2 – n 2 ) 6 – (m ) 8 same?, different?

An equation is an action. Have you seen this before? –Simplify/Evaluate the expression: (10 2 )+(12 x 2 – 20  x 2) = –And a student writes the following: (10 2 )+(12 x 2 – 20  x 2) = 100 = – = 124 = 125 –What is happening here???

Some thoughts... Students make RATIONAL errors based on their concepts, the generalizations that they have made from their own experiences. These issues are not just vital for mathematics education, but all fields. These issues are POLITICAL and CULTURAL.

Some (conceptual) thoughts... What is quantifiable and what is not? Why are ‘we’ arguing over ‘embryonic’ stem cell research and not over ‘zygote’ stem cell research? What is the difference between Religious and Scientific explanations of the world? Is belief the same as theory?

The End Or is it?????

It’s up to you.

“I know why that means equal! Equal is like that because it’s like two equal lines.” “A letter could stand for anything. Doesn’t matter what, when or how.”