**‘Mathematical anxiety’ and understanding of mathematics**
A kind of ‘mathematics anxiety’ has always been detected and reported in the empirical realm of mathematics education by all participants in the phenomenon – students, parents, teachers, policy makers, and researchers. Studies within mathematics education have confirmed the existence of this so-called “mathematics anxiety” among secondary and high school students as a global phenomenon and have related it to various social and psychological aspects of the processes of teaching and learning mathematics. The concept of _understanding_ of mathematics has been employed in the analysis of both the causes and the effects of this anxiety: a poor or improper understanding of the taught mathematics leads to anxiety and conversely, anxiety affects understanding. \[Stodolsky (1985), Passolunghi et al. (2016), Foley et al. (2017), Núñez-Peña & Bono (2019), Holmes & Hwang (2016), Guita & Tan (2018), and Choi-Koh & Ryoo (2019)\].
In general, past research on the relationship between understanding of mathematics and ‘mathematics anxiety’ has been conducted with the concepts and methods of educational sciences and psychology; the causes, the effects, and the proposed methodologies for the enhancement of mathematics teaching with respect to this relationship have been analyzed relative to various social and psychological aspects of the processes of teaching and learning mathematics, that is, in a theoretical framework specific to social sciences.
**Conceptual and procedural mathematical knowledge**
There is a consensus that mathematical knowledge is of two distinct kinds: conceptual and procedural knowledge and the relations between conceptual and procedural knowledge are often bi-directional and iterative (Rittle-Johnson & Schneider, 2015). Accordingly, we have conceptual and procedural understanding of mathematics and conceptual and procedural approaches toward teaching mathematics. This distinction maintains a tension between the two teaching approaches, not that they cannot collaborate or be unified, but rather, relative to the practical circumstances of the teaching practice, which actually impose instructors to teach in one way or another.
In practice, the inclination toward the procedural approach is largely the result of policy making, which imposes a “socially useful” mathematics to be taught in schools. This approach is somehow justified by the constraints of the school setup: the limited time frame of a class, the pragmatic goal of passing tests and exams (which are based on problem solving), and the applicative nature of mathematics itself.
**Intensional and extensional conceptual mathematical knowledge**
There is general consensus that conceptual knowledge in mathematics should be defined as knowledge of mathematical concepts, the ways they can be known through their relationships (Hiebert, 1986, pp. 3-4; Star, 2005), knowledge about facts, generalizations and principles (Baroody, Feil, & Johnson, 2007, p.107). However, this view does not extend knowledge much beyond the mere content of mathematics as a discipline, its formalism, principles, and methods. In this intensional view, conceptual knowledge is just a _mathematical_ perquisite of procedural knowledge. Knowledge about mathematics as a whole, its nature and specificity among other disciplines, and methods of acquiring knowledge do not fall within this view: further, the complex epistemology of mathematics does include this kind of knowledge, both intensional and extensional (we are talking here about language, semantics, symbolism, structure, epistemic virtues, truths, motivations and goals, empirical influence, applicability, and cognitive and anthropocentric aspects, among others).
**Philosophical disciplines and mathematics education**
Given the special place mathematics and its nature hold as an object of study in an intersecting zone of epistemology, philosophy of mathematics, and philosophy of science as theoretical-philosophy disciplines, and adjacently, fundamentals of mathematics and history of mathematics, theoretical contributions for an epistemological framework of enhancing mathematics education through a holistic approach are expected to come from these domains. Their potential in this respect has already been established in the works of Ernest (1989, 1991, 1994), Ernest et al. (2016, pp. 3-17), (Godino & Batanero, 1998), Skovsmose (2013/1994), Kitcher (1983), and others, but it was not explored from a practical-applicative perspective in mathematics education, nor applied in the policies of education.
It is not surprising that we place philosophy of science in the list of the potential contributors, since _applicability_ of mathematics and its constitutive role in sciences is a field investigated within this discipline and accounts for the nature of mathematics. Besides, both philosophy of mathematics and philosophy of science have an inner educational dimension that potentially can be exploited in mathematics and science education.
It should be noted that fundamentals and history of mathematics, although having tight connections with philosophy of mathematics and of science, cannot _alone_ provide the extensional knowledge that the philosophical disciplines can provide in order to enrich traditional conceptual knowledge. From a historical perspective, fundamentals of mathematics provide the _formalisms_ of the various theories as attempts to overcome contradictions, paradoxes, or inadequacies, as a foundation of mathematics. But the content of this sub-discipline can be assimilated with the traditional mathematical content taught in schools, since it reflects how mathematics is constructed and has the same formal nature; however, it does not provide the _whole_ picture of the nature and features of mathematics in relation to general and scientific knowledge. Insights from fundamentals and history of mathematics are present to some extent in the curriculum of several systems of high school education worldwide, mostly as notes or optional content; they are taught as disciplines in many mathematics departments of colleges or universities instead.
Unexpectedly, the line of argument of the classical works in 1980s ands 1990s, showing the potential of the mentioned philosophical disciplines to enhance mathematics education, has been ignored at large in further research in mathematics education, practice, and policies.
None focused on arguing for and making applicable the general principle that teaching _about_ mathematics along with traditionally teaching formal mathematics and ‘doing’ mathematics does not render the process more difficult; rather, it enhances mathematics education, with effects on decreasing mathematics anxiety and stimulating performance. We can fairly hypothesize that a philosophically enhanced conceptual understanding of mathematics makes one a better problem solver eventually. This hypothesis is already supported by past research on the relations between the conceptual and procedural learning of mathematics (see for instance Baroody, 2003; Rittle-Johnson & Siegler, 1998; Rittle-Johnson et al., 2001).
**Two additional arguments for employing philosophy**
The necessity of turning to philosophy for mathematics-education needs was motivated in the works of Ernest, Godino & Batanero, Skovsmose, Kitcher (cited in the previous section) and others, who provided arguments based on conceptual clarification, and the cultural and social dimension of mathematics. A similar though differently motivated necessity was advanced for the case of science education (Hills, 1992; Matthews, 1994; Mellado et al., 2006; Höttecke and Silva, 2011; and others), which can stand as an adjacent argument for the mathematical case. Call this as a whole _the disciplinary argument_.
I will also propose two new arguments, related this time to mathematics itself as an epistemic construct: First, _the ‘foundational-dependence’ argument_, based on the premise that philosophy has influenced dramatically the foundations and development of mathematics through its critical questions, conceptual debates, analysis of paradoxes, consistency, and adequacy. Further, contribution to the foundation of mathematics will be shown to imply the potential for contribution in understanding of mathematics. Second, _the ‘nature-of-mathematics’ argument_, based on philosophical investigations into the nature of mathematics (including open questions), which runs on the line of thought that philosophical inquiry on the mathematical concepts, and the concept of mathematics as a whole, adds knowledge on the subject even through its open questions, while stimulating critical thinking, which goes hand in hand with both conceptual understanding and the analytical skills required in mathematical learning and practice (Barboianu, 2021).
**Mathematical understanding**
In the research of mathematical education, the problem of defining a concept of _mathematical_ understanding has been given a central role. Skemp (1976) distinguished between relational and instrumental understanding in mathematics; Nickerson (1985, pp. 229 - 236) identified understanding through its results and the relationships of these results to established knowledge. Hiebert & Carpenter (1992, p. 67) advanced a similar definition for understanding as involving the building of a conceptual structure based on mental representations. Sierpinska (1994, Ch. 2 - 3) distinguished between the ‘act of understanding’, ‘understanding’ as the result of that act, and ‘processes’ of understanding as further cognitive activity. More recent views (such as Dreyfus & Eisenberg, 2012; Pino-Fan et al., 2015; Greeno, 2017; Newton & Sword, 2018) either offer refined versions of these definitions or advance other relationships between the same basic constituent concepts. Let us note that all of these views are more intensional than extensional and draw on the idea that mathematical understanding should be defined structurally.
In theoretical philosophy, understanding is tightly related to the epistemic concepts of _meaning_ and _context,_ not only in what concerns language, but as a general epistemological concept that involves valuable and distinguishable knowledge (see Kvanvig, 2003, 2009; De Regt & Dieks, 2005; Grimm, 2014). But the meaning of mathematical concepts, statements and theories, and context of the creation, development and application of mathematics – all these are investigated within epistemology, fundamentals of mathematics, philosophy of mathematics and of science, and even of language. Importance of meaning has already been established in research as concerns prospecting a theory of mathematical education, still in relation to the philosophical aspects of mathematics (see Godino, 1996; Godino & Batanero, 1998); context, on the other hand, has not been sufficiently researched and one task would be to identify exhaustively the contexts of mathematical understanding (among which are the logical, theoretical, applicative, epistemic, historical, and perhaps the anthropocentric) and incorporate them into the new notion of holistic mathematical understanding that we propose, which is intended to be both intensional and extensional, and to reflect the _entire_ specificity of mathematics. It is the context component of understanding that can reflect this specificity more than the meaning can reflect: Mathematics is by its nature _both_ abstract and concrete, theoretical and applicative, discipline and language, discipline and method, mental activity and propositional construct; mathematics is self-generative and self-applicative, and includes second- and higher-order predications.
**Foreseen outcomes of the epistemic holistic approach**
In such an epistemological framework, we can investigate the hypothesis that many teaching methods and strategies used for and assumed to enhance explanation of mathematical concepts are not enough in ensuring a _full_ understanding (in particular, example-based and metaphorical explanation – that is, making mathematics look “friendlier” or “fun”, including through new video and web technologies, is hypothesized to provide only mental-cognitive comfort for the student and not to add new active knowledge on the topic).
The classical definitions of mathematical understanding put forward a plurality of “understandings”, suggesting that we cannot talk about understanding _per se_. But such separations make difficult any attempt to unify the conceptual and procedural approaches of teaching, and maintain a tension between them. A _holistic_ concept of understanding defined on epistemological grounds complies with the aim of unifying the conceptual and procedural approaches of teaching and would also accommodate with the alleged psychological nature of understanding as a ‘mental state’ (which is meant as singular).
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