Andreas Vohns in his article Fundamental Ideas as a Guiding Category in Mathematics Education—Early Understandings, Developments in German- Speaking Countries and Relations to Subject Matter Didactics says: "In its day-to-day regime the mathematics classroom is mainly focused on students’ mastery of specific knowledge and skills currently at hand. But do they see the bigger picture? Do they get an appropriate idea of what mathematics is essentially about? Fundamental ideas have been a regularly proposed way to outline the bigger picture.".

The second version of the Brazilian Common Core (BNCC) puts the following pairs of Fundamental Ideas in Mathematics: variation and constancy; certainty and uncertainty; movement and position; relationships and interrelationships.

Nilson José Machado, a Brazilian scholar, in turn, proposes the following Fundamental Ideas in Mathematics: equivalence; order; proportionality; interdependence; measurement; invariance; variance; periodicity; randomness; problematicity; optimization.

I'm looking for other authors who have addressed the subject of Fundamental Ideas in Mathematics (including countries other than Germany and Brazil).

In particular, shouldn't "recursion" be included in the list of Fundamental Ideas?

  • $\begingroup$ What does Nilson José mean by "problematicity"? $\endgroup$ Commented Apr 27, 2020 at 17:06
  • $\begingroup$ @MarkFantini; He means "mathematical language; informal and formal logic; equations and inequalities as questions, ...". $\endgroup$ Commented Apr 28, 2020 at 11:58
  • $\begingroup$ Just as an observation, many of the fundamentals become rabbit holes rather quickly. The more something is deemed a "fundamental" the more trick it seems to be to pin down. $\endgroup$
    – Cort Ammon
    Commented Nov 3, 2020 at 2:10
  • $\begingroup$ @CortAmmon, do you have examples of your point of view? $\endgroup$ Commented Nov 4, 2020 at 8:38

1 Answer 1


Researchers Sebastian Rezat, Mathias Hattermann and Andrea Peter-Koop have published a book "Transformation - A Fundamental Idea of Mathematics Education". You can find the link here.

This is a description of the book:

The diversity of research domains and theories in the field of mathematics education has been a permanent subject of discussions from the origins of the discipline up to the present. On the one hand the diversity is regarded as a resource for rich scientific development on the other hand it gives rise to the often repeated criticism of the discipline’s lack of focus and identity. As one way of focusing on core issues of the discipline the book seeks to open up a discussion about fundamental ideas in the field of mathematics education that permeate different research domains and perspectives. The book addresses transformation as one fundamental idea in mathematics education and examines it from different perspectives. Transformations are related to knowledge, related to signs and representations of mathematics, related to concepts and ideas, and related to instruments for the learning of mathematics. The book seeks to answer the following questions: What do we know about transformations in the different domains? What kinds of transformations are crucial? How is transformation in each case conceptualized?

It mentions transformation as a fundamental idea to mathematics education but transformations are very much relevant to mathematics in general.


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