# Critical Thinking in Mathematics Education

**DOI:**https://doi.org/10.1007/978-3-319-77487-9_35-4

## Keywords

Logical thinking Argumentation Deductive reasoning Mathematical problem solving Critique Mathematical literacy Critical judgment Goals of mathematics education## Definition

Mainstream educational psychologists view critical thinking (CT) as the strategic use of a set of reasoning skills for developing a form of reflective thinking that ultimately optimizes itself, including a commitment to using its outcomes as a basis for decision-making and problem solving. In such descriptions, CT is established as a general methodological standard for making judgments and decisions. Accordingly, some authors also include a sense for fairness and the assessment of practical consequences of decisions as characteristics (e.g., Paul and Elder 2001). This conception assumes rational, autonomous subjects who share a common frame of reference for representation of facts and ideas, for their communication, as well as for appropriate (morally “good”) action. Important is the difference as to what extent a critical examination of the criteria for CT is included in the definition: If education for CT is conceptualized as instilling a belief in a more or less fixed and shared system of skills and criteria for judgment and associated values, then it seems to contradict its very goal. If, on the other hand, education for CT aims at overcoming potentially limiting frames of reference, then it needs to allow for transcending the very criteria assumed for legitimate “critical” judgment. The dimension of not following rules and developing a fantasy for alternatives connects CT with creativity and change. In Asian traditions derived from the Mãdhyamika Buddhist philosophy, critical deconstruction is a method of examining possible alternative standpoints on an issue, which might amount to finding self-contradictions in all of them (Fenner 1994). When combined with meditation, the deconstruction provides for the student a path toward spiritual insight as it amounts to a freeing from any form of dogmatism. This position coincides with some postmodern critiques of purely intellectual perspectives that lack contact with experience and is echoed in some European traditions of skepticism (Garfield 1990). Hence, paradoxical deconstruction appears more radical than CT as it includes overcoming the methods and frames of reference of previous thinking and of purely intellectual plausibility.

## Introduction

The role assigned to CT in mathematics education includes CT as a by-product of mathematics learning, as an explicit goal of mathematics education, as a condition for mathematical problem solving, as well as critical engagement with issues of social, political, and environmental relevance by means of mathematical modeling and statistics. Such engagement can include a critique of the very role mathematics plays in these contexts. In the mathematics education literature, explicit reference to CT as defined in educational psychology or philosophy is not very widespread, but general mathematical problem-solving and mathematical reasoning are commonly associated with critical thinking, even though such association remains under-theorized. On the other hand, the notion of critique, rather than CT, is employed in the mathematics education literature in various programs related to critical mathematics education. In these programs, the adjective “critical” is used to modify “mathematics (education)” rather than “thinking.”

### Critical Thinking and Mathematical Reasoning

Mathematical argumentation features prominently as an example of disciplined reasoning based on clear and concise language, questioning of assumptions, and appreciation of logical inference for deriving conclusions. These features of mathematical reasoning have been contrasted with intuition, associative reasoning, justification by example, or induction from observation. While the latter are also important aspects of mathematical inquiry, a focus on logic is directed toward extinguishing subjective elements from judgments, and it is the essence of deductive reasoning. Underpinned by the values of rationalism and objectivity, reasoning with an emphasis on logical inference is opposed to intuition and epiphany as a source of knowledge and viewed as the counterinsurance against blind habit, dogmatism, and opportunism.

The enhancement of students’ general reasoning capacity has for quite some time been seen as a by-product of engagement with mathematics. Francis Bacon (1605), for example, wrote that it would “remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they [the mathematics] sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it” (VIII (2)). Even though this promotion of mathematics education is based on its alleged value for developing generic thinking or reasoning skills, these skills are in fact not called “critical thinking.” Historically, the notion of critique was tied to the tradition of historic, esthetic, and rhetoric interpretation and evaluation of texts. Only through the expansion of the function of critique toward general enlightenment, critique became a generic figure of thinking, arguing, and reasoning. This more general notion, however, transcends what is usually associated with accuracy and rigor in mathematical reasoning. Accordingly, CT in mathematics education not only is conceptualized as evaluating rigor in definitions and logical consistency of arguments but also includes attention to informal logic and heuristics, to the point of identifying problem-solving skills with CT (e.g., O’Daffer and Thomquist 1993). Applebaum and Leikin (2007), for example, see the faculty of recognizing contradictory information and inconsistent data in mathematics tasks as a demonstration of CT. However, as most notions of CT include an awareness of the subject doing it, neither a mere application of logical inference nor successful application of mathematical problem-solving skills would reasonably be labeled as CT. But as a consequence of often identifying CT with general mathematical reasoning processes embedded in mathematical problem solving, there is a large overlap of literature on mathematical reasoning, problem solving, and CT.

There is agreement that CT does not automatically emerge as a by-product of any mathematics curriculum but only with a pedagogy that draws on students’ contributions and affords processes of reasoning and questioning when students collectively engage in intellectually challenging tasks. Fawcett (1938), for example, suggested that teachers (in geometry instruction) should make use of students’ disposition for critical thinking and that this capacity can be harnessed and cultivated by an appropriate choice of pedagogy. Reflective thinking practices could be enacted when drawing the students’ attention to the need for clear definition of key terms in statements, for examination of alleged evidence, for exposition of assumptions behind their beliefs, and for evaluation of arguments and conclusions. Fawcett’s teaching experiments included the critical examination of everyday notions. A more recent example of a pedagogical approach with a focus on argumentation is the organization of a “scientific debate” in the mathematics classroom (Legrand 2001), where students in an open discussion defend their own ideas about a conjecture, which may be prepared by the teacher or emerge spontaneously during class work. Notably, in these examples CT in mathematics education is developed as a social activity.

While cultivating some form of discipline-transcending CT has long been promoted by mathematics educators, explicit reference to CT is not very common in official mathematics curriculum documents internationally. For example, “critical thinking” is not mentioned in the US Common Core Standards for Mathematics (Common Core State Standards Initiative 2010). However, in older recommendations from the US National Council of Teachers of Mathematics, mention of “critical thinking” is made in relation to creating a classroom atmosphere that fosters it (NCTM 1989). A comparative analysis of associations made between mathematics education and CT in international curriculum documents remains a research desideratum.

Notions of CT in mathematics education with a focus on argumentation and reasoning skills have in common that the critical competence they promote is directed toward claims, statements, hypotheses, or theories (“texts”) but do include neither a critique of the social realities, in which these texts are produced, nor a critique of the categories, in which these texts describe realities. As it is about learning how to think, but not what to think about, this notion of CT can be taken to implicate a form of thinking without emotional or moral commitment. However, the perspective includes the idea that the same principles that guide critical scientific inquiry could also guide successful problem solving in social and moral matters and this would lead to improvement of society, an idea that was, for example, shared by Dewey (Stallman 2003). Education for CT is then by its nature emancipatory.

### Critical Thinking and Applications of Mathematics

For those who see dogmatic adherence to the standards of hypothetical-deductive reasoning as limiting, the enculturation of students into a form of CT derived from these standards alone cannot be emancipatory. Such a view is based on a critique of Enlightenment’s scientific image of the world. The critique provided by the philosophers of the Frankfurt School is taken up in various projects of critical mathematics education and critical mathematical literacy. This critique is based on the argument that useful things are conflated with calculable things and thus formal reasoning based on quantification, which is made possible through the use of mathematics, is purely instrumental reasoning. Mathematics educators have pointed out that reliance on mathematical models implicates a particular worldview and mathematics education should widen its perspective and take critically into account ethical and social dimensions (e.g., Steiner 1988). In order to cultivate CT in the mathematics classroom, reflection not only of methodological standards of mathematical models but also of the nature of these standards themselves, as well as of the larger social contexts within which mathematical models are used, has been suggested (e.g., Skovsmose 1989; Keitel et al. 1993; Jablonka 1997; Appelbaum and Davila 2009; Fish and Persaud 2012). Such a view is based on acknowledging the interested nature of any application of mathematics. This is not to dismiss rational inquiry; it rather aims at expanding rationality beyond instrumentality through inclusion of moral and political thought. Such an expansion is seen as necessary by those who see purely formally defined CT as ultimately self-destructive and hence not emancipatory.

### Limitations of Developing CT Through Mathematics Education

The take-up of poststructuralist and psychoanalytic theories by mathematics educators has afforded contributions that hold CT up for scrutiny. Based on the postmodern acknowledgment that all forms of reasoning are only legitimized through the power of some groups in society and in line with critics who see applied mathematics as the essence of instrumental reason, an enculturation of students into a form of CT embedded in mathematical reasoning must be seen as disempowering. As it excludes imagination, fantasy, emotion, and the particular and metaphoric content of problems, this form of CT is seen as antithetical to political thinking or social commitment (Walkerdine 1988; Pimm 1990; Walshaw 2003; Ernest 2010; see also Straehler-Pohl et al. 2017). Hence, the point has been made that mathematics education, if conceptualized as enculturation into dispassionate reason and analysis, limits critique rather than affording it and might lead to political apathy.

### Further Unresolved Issues

Engaging students in collaborative CT and reasoning in mathematics classrooms assumes some kind of an ideal democratic classroom environment, in which students are communicating freely. However, classrooms can hardly be seen as ideal speech communities. Depending on their backgrounds and educational biographies, students will not be equally able to express their thoughts and not all will be guaranteed an audience. Further, the teacher usually has the authority to phrase the questions for discussion and, as a representative of the institution, has the obligation to assess students’ contributions. Thus, even if a will to cultivate some form of critical reasoning in the mathematics classroom might be shared among mathematics educators, more attention to the social, cultural, and institutional conditions under which this is supposed to take place needs to be provided by those who frame CT as an offshoot of mathematical reasoning. Further, taxonomies of CT skills, phrased as metacognitive activities, run the risk of suggesting to treat these explicitly as learning objectives, including the assessment of the extent to which individual students use them. Such a didactical reification of CT into measurable learning outcomes implicates a form of dogmatism and contradicts the very notion of CT.

The antithetical character of the views of what it means to be critical held by those who see CT as a mere habit of thought that can be cultivated through mathematical problem solving, on the one hand, and mathematics educators inspired by critical theory and critical pedagogy, on the other hand, needs further exploration.

Attempts to describe universal elements of critical reasoning, which are neither domain nor context specific, reflect the idea of rationality itself, the standards of which are viewed by many as best modeled by mathematical and scientific inquiry. The extent to which this conception of rationality is culturally biased and implicitly devalues other “rationalities” has been discussed by mathematics educators, but the implications for mathematics education remain under-theorized.

## Cross-References

## References

- Appelbaum P, Davila E (2009) Math education and social justice: gatekeepers, politics and teacher agency. In: Ernest P, Greer B, Sriraman B (eds) Critical issues in mathematics education. Information Age, Charlotte, pp 375–394Google Scholar
- Applebaum M, Leikin R (2007) Looking back at the beginning: critical thinking in solving unrealistic problems. Mont Math Enthus 4(2):258–265Google Scholar
- Bacon F (1605) Of the proficience and advancement of learning, divine and human. Second Book (transcribed from the 1893 Cassell & Company edition by David Price. Available at: http://www.gutenberg.org/dirs/etext04/adlr10h.htm
- Common Core State Standards Initiative (2010) Mathematics standards. http://www.corestandards.org/Math. Accessed 20 July 2013
- Ernest P (2010) The scope and limits of critical mathematics education. In: Alrø H, Ravn O, Valero P (eds) Critical mathematics education: past, present and future. Sense Publishers, Rotterdam, pp 65–87Google Scholar
- Fawcett HP (1938) The nature of proof. Bureau of Publications, Columbia/New York City. University (Re-printed by the National Council of Teachers of Mathematics in 1995)Google Scholar
- Fenner P (1994) Spiritual inquiry in Buddhism. ReVision 17(2):13–24Google Scholar
- Fish M, Persaud A (2012) (Re)presenting critical mathematical thinking through sociopolitical narratives as mathematics texts. In: Hickman H, Porfilio BJ (eds) The new politics of the textbook. Sense Publishers, Rotterdam, pp 89–110CrossRefGoogle Scholar
- Garfield JL (1990) Epoche and śūnyatā: skepticism east and west. Philos East West 40(3):285–307CrossRefGoogle Scholar
- Jablonka E (1997) What makes a model effective and useful (or not)? In: Blum W, Huntley I, Houston SK, Neill N (eds) Teaching and learning mathematical modelling: innovation, investigation and applications. Albion Publishing, Chichester, pp 39–50Google Scholar
- Keitel C, Kotzmann E, Skovsmose O (1993) Beyond the tunnel vision: analyzing the relationship between mathematics, society and technology. In: Keitel C, Ruthven K (eds) Learning from computers: mathematics education and technology. Springer, New York, pp 243–279CrossRefGoogle Scholar
- Legrand M (2001) Scientific debate in mathematics courses. In: Holton D (ed) The teaching and learning of mathematics at university level: an ICMI study. Kluwer, Dordrect, pp 127–137Google Scholar
- National Council of Teachers of Mathematics (NCTM) (1989) Curriculum and evaluation standards for school mathematics. National Council of Teachers of Mathematics (NCTM), RestonGoogle Scholar
- O’Daffer PG, Thomquist B (1993) Critical thinking, mathematical reasoning, and proof. In: Wilson PS (ed) Research ideas for the classroom: high school mathematics. MacMillan/National Council of Teachers of Mathematics, New York, pp 31–40Google Scholar
- Paul R, Elder L (2001) The miniature guide to critical thinking concepts and tools. Foundation for Critical Thinking Press, Dillon BeachGoogle Scholar
- Pimm D (1990) Mathematical versus political awareness: some political dangers inherent in the teaching of mathematics. In: Noss R, Brown A, Dowling P, Drake P, Harris M, Hoyles C et al (eds) Political dimensions of mathematics education: action and critique. Institute of Education, University of London, LondonGoogle Scholar
- Skovsmose O (1989) Models and reflective knowledge. Zentralblatt für Didaktik der Mathematik 89(1):3–8Google Scholar
- Stallman J (2003) John Dewey’s new humanism and liberal education for the 21st century. Educ Cult 20(2):18–22Google Scholar
- Steiner H-G (1988) Theory of mathematics education and implications for scholarship. In: Steiner H-G, Vermandel A (eds) Foundations and methodology of the discipline mathematics education, didactics of mathematics. In: Proceedings of the second tme conference, Bielefeld-Antwerpen, pp 5–20Google Scholar
- Straehler-Pohl H, Bohlmann N, Pais A (eds) (2017) The disorder of mathematics education: challenging the socio-political dimensions of research. Springer, BerlinGoogle Scholar
- Walkerdine V (1988) The mastery of reason: cognitive development and the production of rationality. Routledge, LondonGoogle Scholar
- Walshaw M (2003) Democratic education under scrutiny: connections between mathematics education and feminist political discourses. Philos Math Educ J 17. http://people.exeter.ac.uk/PErnest/pome17/contents.htm