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.
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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