Transcript Concepts and College Math

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 (1896-1934) Soviet
developmental psychologist and the founder of
cultural-historical psychology
Alexander Romanovich Luria (1902-1977) Soviet
neuropsychologist and developmental
psychologist, student of Vygotsky, one of the
founders of cultural-historical psychology and
psychological activity theory
Vasily Davydov (1930-1988) ‘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 1st 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.
A2 – B2 = (A + B)(A – B)
equation?, formula?, expression?
variables?, unknowns?, something to solve?
Completely factor:
52 – X2
Y 2 – X2
49 – T2
4S2– 16
(m2 – n2)6 – (m2 + 25)8
same?, different?
An equation is an action.
Have you seen this before?
– Simplify/Evaluate the expression:
(102)+(12 x 2 – 20  4 + 3 x 2) =
– And a student writes the following:
(102)+(12 x 2 – 20  4 + 3 x 2) = 100 = 100 + 24 – 5
+ 6 = 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.”