“Mathematics is something more than a subject based on the data”

Travelling through different places allow for observing rich fauna and flora, describing the landscape and recording the most beautiful views. But this is not all. When walking through the wildest jungle, sooner or later do we come across indigenous inhabitants who feel good in their reality which is sometimes totally detached from others. As in every tribe, they have their sages and teachers, and guides who guide them. And a similar phenomenon can be observed at schools across the world. Schools establish their own tribes; belonging to them can be either totally coincidental, considering directive, administrative division into school districts, or meticulously selected, underpinned by either prices of properties around or the tuition fee.

The internal school division of local tribes is definitely far more advanced. And this is not about social relationships. Thinking about learning, especially the school or education as an extensive area, is strictly linked to the division into specific disciplines. The most basic division is into the sciences and the humanities. Mathematics is definitely included in the former.

And maybe the school world is not that simple? Maybe it’s not like people are divided into humanists and those who can’t see the world outside the sciences oscillating around mathematics, physics or chemistry. The world of school and school subjects is divided into these two tribes: the humanists and the scientists; sometimes, there are additional subcategories, “artistic” and “sport” tribes.

Belonging to any of these does not condition the participation in specific classes; the latter is conditioned by the organisation of work of a specific entity. Despite the fact that sometimes artistic classes are voluntary, mathematics is always obligatory.

Ethnomathematics

In our first texts about education, we assumed the perspective of a traveller-scientist who learns about some not-fully-discovered world in the educational space. When deriving from the work of humanity, we develop some new perspective on the not entirely obvious issues related to the functioning of specific disciplines.

When discovering the area of mathematics, a little unexpectedly, I’ve come across the notion of “ethnomathematics.” It deals with issues related to culturally-linked understanding and teaching of mathematics (see D’Ambrosio. 1999, see Powell, Frankenstein, 1997). Sometimes it is related to ways of recording, functioning in different, mainly historical, cultural circles.

We can’t forget that mathematics, despite it seems to be a consistent and objective area of science, was developed in distant corners of the world by difference civilisations, which is why the contemporary mathematics seems to be a creative compilation of the civilisational work of the Ancient Egypt, Greece, China, India or the Middle East (see Seah, Bishop, 2000).

Out of its nature, mathematics is something more than a subject based on the data. The fact that mathematics tends to be treated as a pure and objective science makes that it is hard to find many works that analyse values that can be transferred through dfferent contents, as opposed to teaching history and languages which, in their nature, are full values and contents transferred together with an entire set of specific facts. What is more, currently, when talking about a combination of values and mathematics, we mostly pay attention to inequalities stemming from gender. These inequalities are based on convictions that devaluate girls as those less competent and having poorer skills of understanding and learning mathematics. Interestingly enough, we can obseve measures often taken to change these convictions in western societies through extensive actions aimed at including girls and women into the area of Stem (Science, Technology and Engineering, Mathematics).

Although this is not the objecive of this text, it is hard not to mention STEM teaching. This area is mentioned as the key one from the perspective of creating the society of the future and an economics based on know-how. Preparing to engagement in work in this respect constitutes an aim of functioning of many global educational systems which are the subject of numerous international scientific works (see Ho, 2019).

Bearing in mind the above, the ethnomathematical spirit will now guide us through the world of mathematics as a science, taking us, for a while, to the metalevel of thinking about a subject that is taught, in different versions, in all schools across the world.

Above calculations and figures

Despite the fact that the school is obligatory for almost all the people who were lucky enough to be born in a rich country that ensures universal access to education, sometimes the impressions about the function of the school or what is taught may differ. It is also underlined by scientists who analyse the school curriculum. And it is not only the one that is expressed directly as a set of specific contents, but also the hidden one which reproduces some kind of a knowledge and values expressed indirectly.

This knowledge which is not expressed directly in the curriculum depends on the cultural context in which the school operates. This context includes the freedom to decide about the material passed onto the operational (school) level. Around the world, we operate based on a specified knowledge corpus which forms grounds for teaching mathematics, although the stress sometimes is put on different things. Teaching mathematics seems to be limited to mastering the skill of making calculations and logical thinking, assessed both formally, through validation mechanisms, such as external or standardized examinations, international research done at different stages of education, and entirely non-formally, in the form of using mathematical skills in everyday life, which is difficult, if not impossible, to reliably assess and examine.

Apart from reproducing the knowledge, the school plays a crucial role in the transfer of values. We need to differentiate the values represented by an institution which materialize through the way we organize our work or space from the values displayed by people who work for a given organisation. In the case of school, values that are taught to students are mainly transferred by teachers.

Looking from the perspective of the school as an area that manifests the culture of a given community, it is values that allow for a real continuity of a given social group, based on the historically developed activities that are morally acceptable and lead to keeping their own, tribal identity (see Nixon 1995). Values that are taught at school are remembered better than the knowledge itself (see Bishop 1996), which is why it needs to be taken care of what contents are consciously or subconsciously taught when teaching mathematics. Importantly enough: values may relate to different areas, as described below.

Different authors categorize values taught through mathematics in different manners. Seah and Bishop divide mathematical values into three groups of complementary pairs: rationalism and objectivism, control and progress, openness and mystery (see Seah, Bishop, 2000). They also mention: the formalistic-activist view continuum (Dormolen, 1986); instrumental vs relational understanding/learning (Skemp, 1979); evaluating – reasoning continuum (Robitaille, 1993). The values listed above can be found in manuals used by students. The analysis of student books from the perspective of values was the main subject of the study carried out by Seah and Bishop, described in their article (see Seah, Bishop 2000).

Another perspective (though still within ethnomathematics) was taken by Leslie Dietiker. According to her research, the contents of mathematics student books is some kind of an art; it is mostly about telling a story that contains an aesthetic element related to the language used, composition, etc. (see Dietiker 2015). The author also points out at some element of metaphorical understanding of a mathematical text as an obligatory part of every such text within teaching it as a school subject. Considering this aspect in thinking about mathematical tasks allows for analysing the contents of student books both from the perspective of logical consistency and aesthetic values. Mathematics does not need to only teach efficient calculations, but may also inspire to go beyond and arise interest in students, like the adventures of superheroes.

Although we can assume that the main objective of teaching mathematics is to develop certain skills related to efficient calculations, we should not forget about the abovementioned additional values of teaching mathematics. This stems from the time spent by children on text tasks and mathematical texts in the entire educational cycle. It turns out that using student manuals and working with text is the main form of activity during math classes in primary school (see Hudson, Henderson, Hudson 2015). Naturally, this is not the most efficient manner to acquire and refresh knowledge, but for some, potentially culturally-conditioned reasons, it is common across many educational circles around the world.

Is it for everybody?

Although this question may seem absurd, literature also gives examples of the context of teaching mathematics. The main question in this respect is whether mathematics should be obligatory to the same extent for all students. Interestingly, even math teachers tend to talk about the relation between the intelligence and mathematical skills and define their role as those who divide people into those that can understand math and those who are totally unable to do that (see Boaler 2009). Assuming, however, that all should learn mathematics, how long should it take? What contents should be taught? Should they be culturally-diverse? Or maybe the contents should be more context-specific? (see Clements et.al.).

Since mathematics is sometimes treated in an utilitarian way, every culture or even a sub-culture or a smaller social group tends to create and develop its own, particular mathematics (see Gerdes 1998). Maybe it is more justified to develop a curriculum based on these nuances, retaining some basic corpus considered a minimum that is necessary for correct functioning in the world. Maybe nowadays what is more important than mathematical skills is numeracy, i.e. the skill to apply mathematics in solving everyday problems instead of operating on abstract creations of the math world. Numeracy is of course grounded on basic mathematical skills, but does not introduce the unnecessary knowledge which may not be used in practice in everyday life. This understanding of needs in teaching mathematics may be controversial, but the matter of real skills seems to be of more importance, bearing in mind the possibility of putting these skills into practice in everyday life better than in the case of algebra or geometry. Similarly, Trends in International Mathematics and Science Study (TIMSS) or research that considers mathematical components (PISA) rather focus on the correct application of knowledge instead of operations on abstract models introduced by mathematics as a school subject in primary school.

Maybe mathematics that is taught at school is totally different from mathematics as such and both disciplines do not match the needs of the real world that requires a different type of mathematical agility. The issue of a inadequacy of contents taught at school is well known by education researchers. Inadequacy itself was defined as a so-called didactic transposition which assumes breaking the continuity between the school life and the real life (see Hudson, Henderson, Hudson 2015) in any discipline. This is how it may be assumed that the school should rather teach the “real” mathematics, i.e. a subject that would develop problem-solving skills, developing ideas, discussion and presenting arguments for using specific methods (see Boaler 2009) in the context of usefullness in everyday life.

Summary

Schooling may also take place out of the building, in the circumstances that are called “school” only symbolically. The main part of “schooling” is about passing the knowledge and the culturally-related values. As it turns out, also thinking about mathematics cannot be detached from culture and contents that are not necessarily objective from the scientific point of view. Although math, as a school subject, directly prepares students to enter into real life, a lot of contents on which teachers focus could be presented in a different way. Based on the above, one could say that what is taught on math classes depends on the teacher and their understanding of usefulness of particular things. The usefulness of what is taught may be viewed from at least three perspectives: exam requirements, the imposed curriculum and the needs of everyday life. What knowledge should be passed depends also on a given cultural cycle and the needs that a particular teacher finds justified. However, in every context there will be a focus on the student welfare, although it may be understood totally differently. The way it is understood will be the subject of the next analysis during the journey through the education landscape of the contemporary world.