Dedicated to the memory ofGraham 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)

875

70175

2535

Suggest a representation for the result:

A rectangle the vertical side of which is 35 and its horizontal side 25.

20 5

30

5

600 150

100 25

The areas of the four rectangles (which is 30x20+30x5+20x5 +5x5).

245

701752535

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 pre-service 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