### Abstract

Practices within research mathematics can and do serve as models for mathematics education. However, typically such inspirations impose a devastating narrowness in relation to reflections on mathematics. This narrowness I refer to as the “banality of mathematical expertise”. Reflections on mathematics can be expressed through a philosophy of mathematics that goes beyond the traditional emphasis on ontological and epistemological dimensions, to become four-dimensional by also addressing social and ethical issues. Many working philosophies of mathematics operate within a narrow scope of reflections, seemingly located within an ethical vacuum. The consequence is a cultivation of a banality, manifest in many university studies in mathematics as well as in dominant research paradigms in mathematics. This constitutes a serious limitation in providing models for mathematics education. By contrast, there exist examples of practices of mathematics education that demonstrate a richness of reflections on mathematics. Accordingly, I address the extent to which such practices of critical mathematics education could serve as models for research mathematics and mathematics education at the university level.

There does not exist any single well-defined practice of mathematics; rather we are dealing with a multitude of practices in both research and applications.Footnote1 Neither does it make sense to talk about the practice of mathematics education; also in this case we are dealing with a multitude of practices. Despite all these ambiguities, in this paper I address the relationships between, on the one hand, the practices of mathematics as found at universities and research institutions, and on the other, the practices of mathematics education as found in school contexts.

The relationships between the two set of practices are opaque, which can be illustrated by an observation made by Burton (2004). She completed an extensive study of how mathematicians were doing their research. She interviewed more than 70 research mathematicians working in different areas of mathematics, both pure and applied, and she identified a variety of approaches being used. Some mathematicians were thinking in images and movements, while others engaged in extensive manipulations of symbols and formulas. Some created new ideas by writing out formulas, not on paper but with a hand moving in the air. For some, the work could be undertaken after going to bed—and I wonder if such writing would work equally well in the dark. Some were grasping new connections while taking a shower. Through her study, Burton revealed the existence of a huge variety of creative practices in mathematics research.

Burton also asked the mathematicians what they found to be the best way of teaching mathematics, and here she observed an astonishing agreement: the appropriate way to make students learn mathematics was through lecturing. How the mathematicians got to this shared idea was not revealed through Burton’s research, but the observation highlights that practices in mathematics and practices in mathematics education are not connected in straightforward ways.Footnote2

Mathematics could inspire many innovations and new approaches in mathematics education. In this paper, however, I problematize the very idea that mathematics could serve as a role model for mathematics education in general. The reason for this is that practices of mathematics research often operate within a narrow scope of reflections on mathematics. These practices seem to be located in an ethical vacuum, and this phenomenon I refer to as the banality of mathematical expertise. In order to clarify what this banality does include, I will proceed along the following steps:

First, I refer to an example of the impact of mathematics on mathematics education, and in this way I try to clarify what it could mean to consider mathematics as a role model for mathematics education. Second, I outline a four-dimensional philosophy of mathematics through which I point out what I consider important when reflecting on mathematics. Third, I pay particular attention to the ethical dimension of this philosophy and specify issues that it could include. Fourth, I highlight that ignoring ethical issues brings about a narrowness in reflections on mathematics, which I refer to as a banality of mathematical expertise. This banality makes me question the relevance of research in mathematics being a general role model for mathematics education. Fifth, I present two examples from critical mathematics education that include profound reflections on mathematics. These considerations bring me to observe that there exist practices in mathematics education that might serve as role models for the formation of mathematical expertise. I conclude the paper with a short summing up.

There does not exist any single well-defined practice of mathematics; rather we are dealing with a multitude of practices in both research and applications.Footnote1 Neither does it make sense to talk about the practice of mathematics education; also in this case we are dealing with a multitude of practices. Despite all these ambiguities, in this paper I address the relationships between, on the one hand, the practices of mathematics as found at universities and research institutions, and on the other, the practices of mathematics education as found in school contexts.

The relationships between the two set of practices are opaque, which can be illustrated by an observation made by Burton (2004). She completed an extensive study of how mathematicians were doing their research. She interviewed more than 70 research mathematicians working in different areas of mathematics, both pure and applied, and she identified a variety of approaches being used. Some mathematicians were thinking in images and movements, while others engaged in extensive manipulations of symbols and formulas. Some created new ideas by writing out formulas, not on paper but with a hand moving in the air. For some, the work could be undertaken after going to bed—and I wonder if such writing would work equally well in the dark. Some were grasping new connections while taking a shower. Through her study, Burton revealed the existence of a huge variety of creative practices in mathematics research.

Burton also asked the mathematicians what they found to be the best way of teaching mathematics, and here she observed an astonishing agreement: the appropriate way to make students learn mathematics was through lecturing. How the mathematicians got to this shared idea was not revealed through Burton’s research, but the observation highlights that practices in mathematics and practices in mathematics education are not connected in straightforward ways.Footnote2

Mathematics could inspire many innovations and new approaches in mathematics education. In this paper, however, I problematize the very idea that mathematics could serve as a role model for mathematics education in general. The reason for this is that practices of mathematics research often operate within a narrow scope of reflections on mathematics. These practices seem to be located in an ethical vacuum, and this phenomenon I refer to as the banality of mathematical expertise. In order to clarify what this banality does include, I will proceed along the following steps:

First, I refer to an example of the impact of mathematics on mathematics education, and in this way I try to clarify what it could mean to consider mathematics as a role model for mathematics education. Second, I outline a four-dimensional philosophy of mathematics through which I point out what I consider important when reflecting on mathematics. Third, I pay particular attention to the ethical dimension of this philosophy and specify issues that it could include. Fourth, I highlight that ignoring ethical issues brings about a narrowness in reflections on mathematics, which I refer to as a banality of mathematical expertise. This banality makes me question the relevance of research in mathematics being a general role model for mathematics education. Fifth, I present two examples from critical mathematics education that include profound reflections on mathematics. These considerations bring me to observe that there exist practices in mathematics education that might serve as role models for the formation of mathematical expertise. I conclude the paper with a short summing up.

Original language | English |
---|---|

Journal | ZDM |

Volume | 52 |

Issue number | 6 |

Pages (from-to) | 1187-1197 |

Number of pages | 11 |

ISSN | 1863-9690 |

DOIs | |

Publication status | Published - Nov 2020 |

### Keywords

- Role model
- Philosophy of mathematics
- Ethics
- Banality of mathematical expertise
- Critical mathematics education

## Fingerprint Dive into the research topics of 'Banality of Mathematical Expertise'. Together they form a unique fingerprint.

## Cite this

Skovsmose, O. (2020). Banality of Mathematical Expertise.

*ZDM*,*52*(6), 1187-1197. https://doi.org/10.1007/s11858-020-01168-4