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Dialogic Mathematical Method

The project is about the development and research of an innovative method for the instruction of mathematics in preschool, called “the dialogic-explorative method”. The project will develop and validate it with preschool teachers and children.

From preliminary explorations it appears that our method helps children develop a high level of mathematical intuition. Children with learning difficulties become talented, and the share of children defined as gifted rises. The method is based on an original mathematical concept we developed called Soft logic, combined with a dialogic approach to the instruction of preschoolers.

We intend to train a group of about 20 preschool teachers in the dialogic-explorative method for the instruction of mathematics in preschool. The teachers will participate in a 60-hours, year-long professional development program (the curriculum is attached in the appendix of this request). In addition, we will provide guidance and assistance for the preschools themselves.

The dialogic-explorative method that guides us is based on the idea of empowering the educational act, so that the educator and the child both experience a process of shared investigation and discovery. For this purpose, we created a fundamental distinction between the concepts of demonstration and experiment. In a demonstration, the results are known ahead of time to the adult, while in an experiment the adult does not know what will occur, and in fact the adult investigates the phenomenon together with the children, and like them is in a state of uncertainty. We applied this investigation and uncertainty approach to the concept of numbers through defining a new type of number called “Soft numbers”. These new numbers correspond to our conception that there is more than one possible solution to a problem, and that contradictions are not necessarily a problem of decision making but can be a possible state of affairs (as in oxymoron in poems and in quantum physics).

Based on previous observations, we reached the hypothesis that in the dialogic-explorative approach one can achieve above and beyond what is expected from preschoolers. Our method of engagement in the dialogic-explorative approach was to research and develop a new way of thinking that is more appropriate for the numerous abilities of preschoolers. Our challenge in the proposed project is to show that it is possible to develop a mathematical language with a Soft logic that is richer than the regular logic, with its dichotomy of truth and falsehood. Furthermore, we will attempt, accordingly, to validate our hypotheses.

We developed an axis that is called the zero axis and it contains all the various multiples of the number zero, which are called Soft zeroes. Absolute zero is also on this axis. Soft numbers are created by adding Soft zeroes to real numbers as follows:  a0+b1. These numbers are located on a new, non-Cartesian system of coordinates. The Soft system of coordinates has an inherent twist that naturally creates a Möbius strip.

Soft numbers create an algebraic structure: they can be added, subtracted, multiplied and divided just like complex numbers, which were created by defining an imaginary root for the number –1 (and later became the basis of Electrical Engineering analysis). The difference between Soft numbers and complex numbers is that instead of multiplying by an imaginary number i, one multiplies by the number zero. For many years the discussion of complex numbers was solely theoretical, until finally they were extensively applied in the exact sciences and in engineering, and we believe that this will be the case with Soft numbers as well, which already have an important potential for pedagogical use.


The two main research questions are:

  1. How does the dialogic-explorative approach expand the range of gifted and genius children?

  2. ​Does the dialogic-explorative approach affect curiosity, creativity and flexibility of thinking? In what ways?

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