Zoltán Pál Dienes (anglicized

as Zoltan Paul Dienes)

(born 1916) is a Hungarian mathematician whose

ideas on the education (especially of small

children) have been popular in some

countries.[1] He is a world-famous theorist and

tireless practitioner of the "new mathematics" -

an approach to mathematics learning that uses

games, songs and dance to make it more

appealing to children.

Dienes' life and ideas are

described in his autobiography,

Memoirs of a Maverick Mathematician,

and his book of mathematical

games, I Will Tell You Algebra

Stories You've Never Heard Before.

He has also published a book of poetry, Calls from

the Past .

His latter life contributions have been chronicled

by Bharath Sriraman in The Mathematics Enthusiast.

Dienes (1960) originally postulated four principles of mathematical learning through which educators could foster mathematics experiences resulting in students discovering mathematical structures.

This theory relates specifically

to teaching and learning of

mathematics rather than teaching and

learning in general. It consists of four

principles:

Preliminary, structured activitiesusing concrete materials should beprovided to give necessary experiencesfrom which mathematical concepts canbe built eventually. Later on, mentalactivities can be used in the same way.

The first principle, namely theconstruction principle suggests thatreflective abstraction on physical andmental actions on concrete(manipulative) materials result in theformation of mathematical relations.

- In structuring activities, construction

of concepts should always precede

analysis

- Concept involving variable should be

learnt by experiences involving the

largest possible number of variables

In order to allow as much scope as possible

for individual variations in concept-formation

and induce children to gather the

mathematical essence of an abstraction, the

same conceptual structure should be

presented in the form of as many perceptual

equivalent as possible

Most people, when confronted with asituation which they are not sure how tohandle, will engage in what is usuallydescribed as “trial and error”. In trying tosolve a puzzle, most people will randomly trythis and that and the other until some formof regularity in the situation begins toemerge, after which a more systematicproblem solving behavior becomes possible.

Beginning of all learning, how the would-belearner becomes familiar with the situationwith which he/she confronted

• After some free experimenting, it usually

happens that regularities appear in the

situation, which can be formulated as “rules

of a game”. Once it is realized that interesting

activities can be brought into play by means

of rules, it is a small step towards inventing

the rules in order to create a “game”.

• Every game has some rules, which need to

be observed in order to pass from a starting

state of things to the end of the game, which

is determined by certain conditions being

satisfied. It is an extremely useful educational

“trick” to invent games with rules which

match the rules that are inherent in some

piece of mathematics which the educator

wishes the learners to learn.

• This can be or should be the essential

aspect of this part of the learning cycle.

Once we have got children to play a number

of mathematical games, there comes a

moment when these games can be

discussed, compared with each other.

It is good to teach several games with very

similar rule structures, but using different

materials, so that it should become apparent

that there is a common core to a number of

different looking games, which can later be

identified as the mathematical content of

those games that are similar to each other in

structure, even though they might be totally

different from the point of view of the

elements used for playing them.

It is even desirable, at one point, to establish

“dictionaries” between games that have the

same structure, so to each element and to

each operation in one game, should

correspond a unique element or operation in

the other game. This will encourage learners

to realize that the external material used for

playing the games is less important than the

rule structure which each material embodies.

It is even desirable, at one point, to establish

“dictionaries” between games that have the

same structure, so to each element and to

each operation in one game, should

correspond a unique element or operation in

the other game. This will encourage learners

to realize that the external material used for

playing the games is less important than the

rule structure which each material embodies.

So learners will be encouraged to take the

first halting steps towards abstraction, which

becoming aware of that which is common

to all the games with the same rule

structure, while the actual physical

“playthings” can gradually become “noise”.

There comes a time when the learner has

identified the abstract content of a number of

different games and is practically crying out for

some sort of picture by means of which to

represent that which has been gleaned as the

common core of the various activities.

At this point it is time to suggest some

diagrammatic representation such as an arrow

diagram, table, a coordinate system or any other

vehicle which would help fix in the learner’s mind

what this common core is. We cannot ever hope

to see an abstraction, as such things do not exist in

the real world of objects and events, but we can

invent a representation which would in some

succinct way give the learner a snapshot of the

essence that he has extracted or abstracted

through the various game activities.

Each one of the learned games can then be

“mapped” on to this representation, which will

pinpoint the communality of the games.

After students get the similarity about the

number of each polygon, in this stage student

can represent the figure

It will now be possible to study the

representation or “map” and glean some

properties that all the games naturally must have.

For example it could be checked whether a

certain series of operations yields the same result

as another series of operations. Such a

“discovery” could then be checked by playing it

out in one or more of the games whose

representation yielded the “discovery”.

An elementary language can then be developed

to describe such properties of the map. Such a

language can approximate to the conventional

symbolic language conventionally used by

mathematicians or freedom can be exercised in

inventing quite new and different symbol

systems. Be it one way or another, a symbol

system can now be developed which can be used

to describe the properties of the system being

learned, as the information is gathered by

studying the map.

symbolize the formula of pentagon’s diagonal

number from the pattern that they get.

5(5 − 3)

2= 5

3(3 − 3)

2= 0

6(6 − 3)

2= 9

4(4 − 3)

2= 2

3 4 5 6

𝑛(𝑛 − 3)

2= ?

n

The descriptions of the symbolization stage

can get very lengthy and often quite redundant.

There comes a time when it becomes desirable

to establish some order in the maze of

descriptions. This is the time to suggest that

possibly just a few initial descriptions would

suffice, as long as we appended ways of

deducing other properties of the map,

determining certain definite rules that would

be allowed to be used in such “deductions”.

In such a case we are making the first steps

towards realizing that the first few descriptions

can be our AXIOMS, and the other properties

that we have deduced can be our

THEOREMS, the ways of getting from the

initial axioms to the theorems being the

PROOFS.

Student can apply the general formula to

another case, such as they can determine the

number of polygon with side 23.