Mathematical Dialog as a method of teaching and learning mathematics

Mathematical Dialogue as a method of

teaching and learning mathematics

in high school

Boris Tsoniff

Research Proposal for Ph.D

( Mathematics education )

Supervisor: Dr Margot Berger

Somebody in my presence asked prof. Gelfand how to excite and develop in pupils interest to maths. Without hesitation he answered: they have to be given good problems . When I asked which problems he thinks are good he considered for 5 seconds and said: good problems are interesting and simple ones .

From memoirs about prof. Gelfand

In my thesis I am going to investigate and propose one of possible solutions for two paradoxical

situations which any maths teacher working in the high school come across .

Paradox 1.

A teacher and pupils speak different languages. They use the same words but with different sense. Besides they use different logics. Revising congruent triangles in grade 10, I rather frequently came across the typical situation. After finding out that my pupils remember what congruency is and understand the idea of the construction of triangles by three sides I ask them to solve the harmless problem: construct the angle which is equal to the given one. After brief silence the best advanced students absolutely correctly suggest me to measure the angle by a protractor. My pupils logic is different. They are guided by the logic of the common sense experience in measuring of angles. The word construct they miss subconsciously because it does not belong to their active maths vocabulary. For them principal words are angle and equal, and for me they are angle and to construct .

Paradox 2.

A teacher introduces maths to pupils explaining some theoretical conceptions ( rules, definitions, theorems and so on ) in the language and the logic which pupils understand badly ( see Paradox 1 ) and therefore absorb badly. To control the understanding a teacher asks pupils to solve some problems. Now besides language and logic which pupils do not understand they have

to accomplish operations ( to solve, to prove ) which nobody taught them earnestly. One cannot take seriously the opuses which are displayed on the dozens American educational sites with pompous titles like ( for example )

HOW DO PEOPLE SOLVE PROBLEMS?

There is the sponge part: when you soak up all the information you can discover (and a lot of misinformation).

There is the shake part: when you shake out the facts and question the problem itself and start to imagine all sorts of things.

There is the squeeze part: when you wring out the sponge and scribble down the most promising splashes and driblets.

There is the bounce part: when you and another concerned with the problem toss embryonic ideas back and forth until only the fittest survive.

There is the scratch part: like the above, but now you scratch brain against brain hoping to spark a new notion.

There is the once-again-please part: when you examine the survivors in the cold light of reason, abandon most, and incubate a few in the warm darkness of imagination.

There is the dry part: when you quit thinking about the damned problem and turn your mind to pleasure or routine. (You only think you've stopped thinking.)

There is the yahoo part: when things connect and an idea pops into your head that turns out to be the key to the solution. Often this happens when you least expect it and aren't even thinking about the big problem.

There is the do part: when you use your particular talents and learned skills and those of others concerned to shape and form the raw idea into a proper solution.

Then there is the itch part: which maybe should come first instead of last. The drive to solve problems creatively -- with a new and original solution -- stems from some chronic itch; dissatisfaction with all existing solutions. Even when the latest may be your own.

Now try to apply these guidelines to solve simple 2000 years old Chinese problem: The first goose flies from the North Sea to the South Sea for 7 hours and the second goose flies from the South Sea to the North Sea for 9 hours. If they start at the same time, when they meet?

Judge by the facts that for 2000 years Chinese invented gun powder, compass, porcelain, paper ( the list can be continued ) and Chinese students firmly take high positions in different maths competitions, Chinese educators 2000 years ago wrote more deep manuals and Chinese pupils solved this problem successfully.

The attempt ( deliberate or non-deliberate ) to resolve those paradoxes in US by reducing the teachers language and logic to the pupils ones brought to the pitiable results. In according to the survey, only two percents of teachers able to divide by in their minds ( V.Arnold ). Test for university entry includes now the demand to divide 111 by 3 without a calculator.

In 2003 Boing Corporation used the following problem for staff selection ( graduate and post-graduate students ): There are 100kg of cucumbers in the bag. It is known that 99% of a cucumber made of water. After drying a little, only 98% of a cucumber made of water. What is the weight of the bag now? Absolute majority won the 98kg answer ( correct answer is 50kg ).

I think that the way out from this paradoxical situation can be rather found in bringing the pupils language and logic up to the teachers ones. This is not only my personal opinion. In front of you fragment of the project of the base concept of mathematical education in Russian schools

Project of the educational standard in mathematics for the high school Base level Language and logicMathematical language as a part of a natural language, having its own distinctive features. Culture of the mathematical speech. Mutual influence of the mathematical and natural languages, origin of the mathematical terms and symbols. Combination of stringency and subordination to the rules of the natural language as the most important property of the mathematic language. Ability to formulate mathematical clauses in oral and written speech. Equations and inequalities as predicative clauses with variables. Constant and variable in mathematics and their analogues in natural language. Universal and existential quantifiers as the logical equivalents for the corresponding words of the natural language. Base laws of logic and rules of logical deduction as a reflection of the natural language thinking. Universality of the laws of logic and logical deduction rules. Logical paradoxes in mathematics and in natural language. Inductive conclusion made on the base of intuition. Setting up a hypothesis and it testing

As you can see , the base of this concept is the notion that Mathematics is a language and ought to be teaching and learning as a language. I completely share this notion. Judging the numerous articles ( especially in Russian educational magazines, where serious discussion about future of Russian mathematical education is taking place ), with the same notion are agreed majority of educators and practical teachers. A brilliant sentence Mathematics is a language too

is to be considered as a postulate for a long time.

What did Gibbs mean? Analogy is evident. As a natural language is not communication only

but also a repository of the collective social experience saved by generations as a mathematical language is a repository of our knowledge about structure and logic of the Universe.

But is Mathematics a language in the linguistic sense? How does mathematical language correlate with generative language? Why in spite of different types of generative languages (analytic and synthetic ) does mathematical language remain an universal one? Is it a subset (contraction ) or expansion of the generative language?

It is obvious that without answers to these questions the theoretical base for mathematical language acquisition cannot be developed.

There are still more, crucial for my thesis, questions exist. They are questions about interrelation between mathematical language and inner speech. Can inner speech be enriched by the mathematical terminology ? Does inner mathematical speech exist? What is a predicativity of inner mathematical speech in Vigotsky sense?

As an illustration I would like to demonstrate an attempt to record my own inner speech during the solution of the arithmetical problem of average complexity. Please keep in mind that original inner speech was recorded in Russian. The problem was found in the one of Internet forums.

I quote the problem as I got it from the Web-side:

I have no idea how to disprove

500

21016

+

is NOT a perfect square.

Inner speech solution

Prove negativeFactorize? Chap tried

EMBED Equation.DSMT4

n

108

+

sum of cubesrubbishCommon case n=1 36n=2216

2

6

...

3

6

n=3 2016dont know36 not working

162

16200

=

-

interestingcan try

210

2

16200

n

+

-

no

2102

00

216

n

-

+

200 common factor

216827

=

8 common factoraha!

200(101)983

n

-+

wonderfulnow

200(999...)983

+

how many

100199

-=

... nin our case? 498idiotic numberany way common factor 9try:

89[25(11...)3)]

+

111 498 familiardivisible by 37of course

111373

=

498 divide by 3evenly! 1663 - common factorhave what?

893[2537(001001...)1)]

+

27 and 82 we havenumber evenneed 3not divisible by 3proveeasy to say

2537

? 925 three digitsbring in

925925...1

+

...last digit 6...166 times.. that's it 9+2+516166 times2656plus 12657add20not divisiblecant take ...

Solution of the problem

Write down the given number

500

21016

N

=+

in the form

500

210200216

N

=-+

and factorize it:

498

200(101)983

N

=-+

2589(1...........1)983

N

=+

498

times

89[25(1...........1)3)]

N

=+

498

times

Number (1..1) can be represented in the form :

( 111 111 111) =

111

(001 001 001) =

373

(001 001 001)

166

times

Now

893[2537(001...........001)1)]

N

=+

166

times

or

827[925(001...........001)1)]

N

=+

.

Notice that square root of N

66[925(001...........001)1)]

N

=+

can be a natural number, only if the expression in brackets ( which is obviously even ) is divisible by 3.

Prove that

1

925(001...........001)1

N

=+

is not divisible by 3.

166

times

Number is divisible by 3 only if the sum of its digits is divisible by 3.

To find the sum of digits of

1

N

rewrite it in the form:

1

925...........9251

N

=+

166

times

Then the sum of digits

166(925)1

+++

=

2657

. It is easy to check that the result is not divisible evenly into 3 and therefore number

N

cannot be a perfect square.

If I manage to answer all those questions I will come to the core question of my thesis:

How to introduce mathematical language to the pupils?

L.Vigotsky wrote in his main book Thinking and Speech: What was once the dialogue between different people becomes now the dialogue inside the brain Ideas of Vigotsky and

his scientific school show clearly in which direction I have to move. Language acquisition as any

knowledge acquisition is always a dialogue between the competent and zealous for knowledge, because the inner speech is dialogic itself.

Acquisition by means of a dialogue doubtless is the most ancient form of education, and the sole one existed before written language had been invented. Cabbalists successfully used the acquisition dialogue right up to XIV century. Theory of dialogue was developed by Plato, Socrat,

M.Buber, M.Bakhtin, L.Vigotsky, V.Bibler. All of them emphasized cultural significance of a dialogue and V.Bibler was an ideological inspirer of the group of educators and teachers, created a school in which different subjects were taught on the base of the so named dialog of cultures (I.Kurganov).

Accepting that Mathematics is a language and including it in the structure of the dialogue of cultures we return to Mathematics the classical cultural significance which had been lost by modern civilization.

Mathematical language is a dynamic structure which becomes more and more complicated as

our conceptions about Universe change. Not only semantic but logic of the language changes dramatically. The gravitation theory of Newton as valid as the gravitation theory of Einstein. But each one is valid in its own space geometry, with its own language ( space metric, imaginary time, curvature of space) and logic ( dimension, observer, velocities addition ). Scientists now consider mathematics as a hierarchy of structures and with corresponding hierarchy of the mathematical languages of increasing complexity. Hierarchic formalization of the mathematical languages creates selective barriers for those who studies mathematics. For any human being there is a limit of mathematical language abilities. For me, for example, language of non-Abelian groups was a limit. I simply stopped to understand the language.

I think that for my thesis it will make sense to separate and work with only part of the mathematical language which corresponds to our knowledge on the level of the end of XIX century. I would name that part The school mathematical language . It covers all school maths and approximately first two years of university maths education. In that way my work proceeds on the school level with scientific research of my supervisor Dr. M.Berger ( The appropriation of mathematical objects by undergraduate mathematics students: a study ). The language of the school mathematics is still very close to the generative language, and its logic is close enough to the logic of common sense.

Coming to the last group of questions I would like to note that Vigotsky school considers education through activity as the only effective method of education. Activity in mathematics is mostly solution of problems. Therefore solutions of problems ought to form the basis of the mathematical dialogue. In mathematical dialogue pupils in natural way acquaint with mathematical terminology and learn to solve problems step by step forming logic different from the logic of common sense. So, what problems have a teacher to select for the dialogue?

I am going to use ideas of famous mathematician V.Arnold who thinks that the student graduated from the university must know the solution of the certain minimum of the maths problems (200 approximately). Drawing an analogy we can say that at the end of each year at school starting from Grade 10 pupil must know the solution of the certain number of so called frame problems. To the end of the Grade 12 there will be the same 200 problems ( 50 for each year ).

In my thesis I hope to describe some general principles of selection of the frame problems and to find the quantitative criteria for such a selection.

Method of the Mathematical Dialogue cannot be the sole method of mathematical education. Normal educational process made of three information streams:

extra- active ( from a teacher to pupils );

- intro- active ( from a teacher to pupils );

- inter- active ( information stream with changing directions ).

Investigation carried out in Moscow in 1985-91 ( 900 maths lessons ) showed that these three components are in 5:4:1 ratio. But for so called strong maths teachers the same ratio is 4:4:2. So strong teachers spend 20% of the time in dialogue with their pupils, which is twice more than ordinary teachers do. It gives us some food for thought.

Prospective structure of work

Mathematical Dialog as a method of

teaching and learning mathematics

in high school

Introduction

Chapter 1.Dialog and dialogical speech.

1.1Dialogical education: history and traditions.

1.2Dialogue idea development:

-Martin Buber;

- M.Bakhtin;

- L.Bygotsky;

- Dialog of cultures of V.Bibler.

Chapter 2.Inner speech.

2.1Inner speech: structure and language;

2.2Inner speech as a dialogue;

2.3Age - specific features of the inner speech;

2.4 Solution of problems in the inner speech;

2.5 Example of self-observation.

Chapter 3.Mathematical language.

3.1Semantic, grammar and logic of the natural language;

3.2 Semantic, grammar and logic of the mathematical language;

3.3 Universalism of the mathematical language;

3.4 Mathematical language and inner speech;

3.5 School mathematical language;

3.6 Mathematical language acquisition;

3.7Method of mathematic dialog (MMD).

Chapter 4.Frame mathematical problem practical realization

of the method of mathematic dialog.

4.1Selection criteria for the frame problems;

4.2 Cluster problems and their connection with the frame problem;

4.3Selection criteria for the cluster problems;

4.4Quantitative assessment of the problems complexity.

Conclusion

Literature

Research questions

1. Is Mathematics at least particularly a language in the linguistic sense?

2. How does Mathematical language differ from a natural language?

3. Can Mathematical language be considered as increasingly complicative

hierarchy of logical structures?

4. What place in this hierarchy does occupy the school mathematical language?

5. Can inner speech be enriched by mathematical terminology ?

6. Does inner Mathematical speech exist?

7. What is a predicatively of inner Mathematical speech in Vigotsky sense?

8. Is it possible to unite Mathematical language acquisition and methods of

solution problems teaching?

9. Are there any criteria for selection the set of problems?

10. How to estimate the depth of understanding on the quantitative basis?

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