Transcript Dedicated to the memory of Graham Littler The long standing chairman of the IPC of SEMT.

Dedicated to the memory of
Graham Littler
The long standing chairman of the IPC
of SEMT
No man is an island, entire of itself;
Any man's death diminishes us,
because we are involved in mankind,
and therefore never send to know for
whom the bell tolls;
It tolls for us.
{ John Donne, meditation XVII (with
some changes )}
Prologue
The knowledge needed for teaching mathematics
at the elementary schools.
What should we expect from
somebody who teaches mathematics
in elementary schools?
Content knowledge and pedagogical content
knowledge
Ball, Hill and Bass (2005)
35

25
175
70
875
Suggest a representation for the result:
A rectangle the vertical side of which is 35 and its horizontal side 25.
20
5
30
600
150
5
100
25
The areas of the four rectangles (which is
30x20+30x5+20x5 +5x5).

35
25
175
70
245
As to the content knowledge
Should we or should we not teach combinatorics or
probability at the elementary level?
At what grade are we supposed to teach fractions or
negative numbers?
Zone of proximal development (ZPD, Vygotsky)
L.P. Benezet, Superintendent of Schools, in
Manchester, New Hampshire, USA.
"we are constantly being asked to add new subjects
to the curriculum … but no one ever suggests that
we eliminate anything."
"Formal arithmetic was not introduced until the
seventh grade. In tests given to both the
traditionally and the experimentally taught
groups, it was found that the latter had been able
in one year to attain the level of accomplishment
which the traditionally taught children had
reached after three and one-half years of
arithmetic drill."
"It would be interesting thing to call some of the
leading citizens in your community around the
table and read the articles to them and to see what
their attitude would be."
As to pedagogical content knowledge
3:(2/5)
How many two fifths are in 3?
(3x5)/2
In order to calculate the result of 3:(2/5) you
should multiply by 5/2.
p:(m/n), where p,m,n are small enough to keep the
drawing clear and simple.
"A generalization":
In order to divide by a fraction m/n one
should multiply by n/m.
Why (4/3): (2/5) is equal to (4/3)x (5/2)?
Mathematical integrity
Question: Are there any good websites or other resources
to help explain neg x neg numbers?
Reactions:
(I) Ds is going through The Key to Algebra, Book 1,
and it uses a football field explanation…He does not
know much about football and it is confusing.
(II) Dd said her math book used football as well (Scott
Foresman). She knows very little about football and feels
his pain. I think she just memorizes.
As to children's mathematical thinking
I) Why do we teach mathematics?
II) What is mathematics?
III) In what ways the teaching of mathematics
serve the ultimate goal of education?
IV) To what extent the elementary teachers have
the necessary background to study what we
expect them to know so that they will be able to
implement the tasks that the educational system
presents to them.
The typical profile of the elementary teachers
The human interaction with little children and
being involved with their intellectual and
emotional development gives them a lot of
satisfaction.
The Mismeasure of Man (S.J. Gould, 1981).
The Immeasurable Man.
Emotional intelligence, social intelligence, and
more (Golman, 1995; Gardner, 1993).
1. David holds 5/8 of the shares of a certain
factory. He gives his son Daniel 2/3 of his shares.
What part of the factory shares is owned by Daniel
after this transaction?
2. Barbecuing meat causes it to lose 1/5 of its
weight. What was the original weight of a piece of
meat, if after barbecuing it, its weight was 300
grams?
"Many U.S. teachers lack sound mathematical
understanding and skills…Mathematical
knowledge of most adult Americans is often as
weak, and often weaker" (Ball, Hill&Bass; 2005).
What should we expect…
"What should we expect… - What are the
demands?
"What should we expect…" - what can be
expected ?
Some recommendations:
i) Ausubel's leading principle: “If I had to reduce all of
educational psychology to just one principle, I would say
this: The most important single factor influencing
learning is what the learner already knows. Ascertain
this and teach him accordingly” (Ausubel, 1960).
ii) The zone of proximal development principle
(Vygotsky, 1986)
iii) The suitable pace of teaching.
Meaningless learning expresses itself very often
in pseudo conceptual and pseudo analytical
behaviors
As to mathematical content knowledge:
Three principles which can help us to determine a
list of mathematical topics required from preservice and in-service elementary teachers in
different social and cultural settings.
These principles will not lead to a uniform
universal curriculum.
Because of the comparative international surveys
in science and mathematics, education has become
an international competition.
As to pedagogical content knowledge:
Concrete models and representations should be
used only if they are simple and clear.
As to children's mathematical thinking:
Prefer clear, simple and straight forward texts to
more sophisticated and complicated studies.
Why do we teach mathematics?
"We live in a mathematical world, whenever we decide
on a purchase, choose insurance or health plan, or use a
spreadsheet, we rely on mathematical understanding…
The level of mathematical thinking and problem solving
needed in the workplace has increased dramatically…
Mathematical competence opens doors to productive
future. A lack of mathematical competence closes those
doors" (Principles and Standards for School
Mathematics; NCTM, 2000)
Underwood Dudley (2010)
In eight categories of work places which were
sampled randomly from the yellow pages no
evidence has been found that algebra is required
there, "even for training or license."
People should study mathematics in order to
train their mind.
In the vast majority of countries around the
world, mathematics acts as a draconian filter to
the pursuit of further technical and quantitative
studies... (Confrey, 1995).
What is Mathematics?
Courant and Robins (1948)
Hersh (1998)
Mathematics is the science of numbers and their
operations, interrelation…and of space
configurations and their structure…
Mathematics is a collection of procedures to be
used in order to solve some typical questions given
in some crucial exams (final course exams,
psychometric exams, SAT etc.)
(I) It is not only for exams, it also for real life
situations.
(II) Mathematics teaches you to think.
In what ways the teaching of mathematics serve
the ultimate goal of education?
An educated person is a thoughtful person.
a) characterized by careful reasoned thinking…
b) given to heedful anticipation of the needs and
wants of others.
"thoughtful" also means "considerate."
The golden rule.
What you hate, do not do to other people.
"careful reasoned thinking."
Procedure is a sequence of actions which
should be carried out one after another in a
certain order.
Respecting procedures, as well as carrying
them out precisely and carefully, can be
recommended as an educational value.
Many procedures in everyday life were formed in
order to serve the golden rule.
Procedures related to behavior on lines,
Procedures related to pedestrians and drivers,
Procedures related to littering and recycling.
Within the traditional setting, the teacher is a tool
of the syllabus.
By adding educational discussions to the syllabus,
the syllabus becomes an educational tool.
The pseudo- analytical and pseudo-conceptual
behaviors as a reaction to exaggerated intellectual
demands.
There are 16 cards in a box. Each card is in an
envelope. All the numbers between 1 and 16
(included 1 and 16) appear on the cards (one number
per card). Describe an event the probability of which
is 1/2.
To pull out of the box the card which has 8 on it.
Because 8/16 is 1/2.
Schor & Alston (1999)
3-(-4)
a) Sandy got squares for positive and negative numbers.
-1= a square in red color.
1= a square in blue color
-(-1)= a square in blue color
She took 3 red squares, and then subtracted 4 in blue.
How many squares in what color did she have?
(b) Sharifa had $3 negative (out of her pocket)
and she gave Maria negative one times minus $4.
How much did they have together?
The ability of prospective teachers to prove or
refute arithmetic statements (Tirosh, H.; 2002).
Checking some examples is not enough to
establish the validity of a universal statement
(for every x, F(x) is true, where F(x) is a quantifier
free statement)
A typical feature of being general is the use of
letters
In order to refute a universal statement, is it
enough to point at one example for which the
statement is not true?
Checking one example is not enough to refute the
universal statement.
In order to establish the validity of an
existential statement, is it enough to point at
one example for which the statement is
true?
Pointing at one example for which the
statement is true is not enough to establish
the validity of an existential statement.
These students are trying to identify proofs by
their superficial form.
They fail to rely on meaning.
Epilogue