KeywordsNumeracy Quantitative literacy Critical mathematical literacy Mathemacy Matheracy Statistical literacy
The neologism “mathematical literacy” belongs to an array of related terms that have been used in English language mathematics education research and policy discourses in the context of suggestions for the improvement of mathematics teaching and learning. While diagnosis of some apparent shortcomings seems to coexist with formal mathematics education since its inception in the USA, “mathematical literacy” is linked to the reform narratives of 1980s (Craig 2018). One of the first written occurrences of the term in the USA was in 1944, when a Commission of the National Council of Teachers of Mathematics (NCTM) on Post-War Plans (NCTM 1970/2002, p. 244) required that the school should ensure mathematical literacy for all who can possibly achieve it. Shortly after (in 1950), the term was used again in the Canadian Hope Report (NCTM 1970/2002, p. 401). In more recent times, the NCTM 1989 Standards (NCTM 1989, p. 5) in the USA spoke about mathematical literacy and mathematically literate students. Apparently, no definition of the term was offered in any of these texts. The 1989 Standards did, however, put forward five general goals serving the pursuit of mathematical literacy for all students: “(1) That they learn to value mathematics, (2) that they become confident with their ability to do mathematics, (3) that they become mathematical problem solvers, (4) that they learn to communicate mathematically, and (5) that they learn to reason mathematically” (op. cit., p. 5).
Mathematical literacy is an individual’s capacity to formulate, employ and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts, and tools to describe, explain and predict phenomena. It assists individuals to recognise the role that mathematics plays in the world and to make well-founded judgments and decisions needed by constructive, engaged and reflective citizens.
In mathematics education research and policy texts, one finds an array of related terms, such as “numeracy,” “quantitative literacy,” “critical mathematical literacy,” “mathemacy,” “matheracy,” and “statistical literacy.” While some of these notions more clearly differ in extension and intension, some authors use “numeracy,” “quantitative literacy,” and “mathematical literacy” synonymously, whereas others distinguish also between these. While the term “mathematical literacy” appears to be of US descent, the term “numeracy” was coined in the UK, although the neologism “innumeracy” spread through a popular science publication in the USA (Paulos 1989). According to Brown et al. (1998, p. 363), “numeracy” appeared for the first time in the so-called Crowther Report in 1959, meaning scientific literacy in a broad sense, and later obtained wide dissemination through the Cockcroft Report (DES/WO 1982), which stated that its meaning had considerably narrowed by then. There have been further shifts in interpretation since then. A recent, rather wide, definition of “numeracy” can be found in OECD’s PIAAC (Programme for the International Assessment of Adult Competencies) “numeracy” framework: “Numeracy is the knowledge and skills required to effectively manage and respond to the mathematical demands of diverse situations” (PIAAC Numeracy Expert Group 2009, p. 20).
The term “quantitative literacy” is yet another term of US descent, going back to the work of Steen (e.g., Madison and Steen 2003). As to countries where English is an official language, Geiger et al. (2015) observe that “numeracy” is still more commonly used in the UK, Canada, South Africa, Australia, and New Zealand, while in the USA, “mathematical literacy” appears to be the privileged term. In South Africa, the pursuit of mathematical literacy has motivated the introduction of a new stand-alone school mathematics subject area available for learners in grades 10–12, which aims at allowing “individuals to make sense of, participate in and contribute to the twenty-first century world – a world characterized by numbers, numerically based arguments and data represented and misrepresented in a number of different ways. Such competencies include the ability to reason, make decisions, solve problems, manage resources, interpret information, schedule events and use and apply technology” (DoBE 2011, p. 8). One motivation for introducing this mathematical subject was to increase student engagement with mathematics.
Characteristics and Delimitation
Even though the notions above are interpreted differently by different authors (which suggests a need to pay serious attention to clear terminology), they do have in common that they stress awareness of the usefulness of and the ability to use mathematics in a range of different areas as an important goal of mathematics education. Furthermore, these notions are associated with education for the general public rather than with specialized academic training while at the same time stressing the connection between “mathematical literacy” and democratic participation. As in other combined phrases, such as “statistical literacy” or “computer literacy,” the addition of “literacy” may suggest some level of critical understanding.
While “mathematical literacy,” “quantitative literacy,” and “numeracy” focus on mathematics as a tool for solving nonmathematical problems, the “mathematical competence” (and “competencies”) and “mathematical proficiency” focus on what it means to master mathematics at large, including the capacity to solve mathematical as well as nonmathematical problems. The notion of “mathematical proficiency” (Kilpatrick et al. 2001) is meant to capture what successful mathematics learning means for everyone and is defined indirectly through five strands (conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition). Furthermore, by referring to individuals’ mental capacities, dispositions, and attitudes, the last two of these strands go beyond mastery of mathematics and include personal characteristics.
The notion of “mathematical competence” has been developed, explored, and utilized in the Danish KOM Project (KOM is an abbreviation for “competencies and mathematics learning” in Danish) and elsewhere since the late 1990s (Niss and Højgaard 2011). Mathematical competence is an individual’s capability and readiness to act appropriately, and in a knowledge-based manner, in situations and contexts in which mathematics actually plays or potentially could play a role. While mathematical competence is the overarching concept, its constituent components are, perhaps, the most important features. There are eight such constituents (“mathematical competencies”): mathematical thinking, problem posing and solving, mathematical modeling, mathematical reasoning, handling mathematical representations, dealing with symbolism and formalism, communicating mathematically, and handling mathematical aids and tools. The description of mathematical competencies does not specifically focus on learners of mathematics nor on mathematics teaching. Also, no personal characteristics such as capacities, dispositions, and attitudes are implicated in these notions.
Motivations for Introducing Mathematical Literacy
We recognize as valid and genuine the concern expressed by many segments of society that basic skills be part of the education of every child. However, the full scope of what is basic must include those things that are essential to meaningful and productive citizenship, both immediate and future. (p. 5)
2.1. The full scope of what is basic should contain at least the ten basic skill areas [. . .]. These areas are problem solving; applying mathematics in everyday situations; alertness to the reasonableness of results; estimation and approximation; appropriate computational skills; geometry; measurement; reading, interpreting, and constructing tables, charts, and graphs; using mathematics to predict; and computer literacy. (pp. 6–7)
2.6. The higher-order mental processes of logical reasoning, information processing, and decision making should be considered basic to the application of mathematics. Mathematics curricula and teachers should set as objectives the development of logical processes, concepts, and language [. . .]. (p. 8)
These examples show that mathematics educators have been concerned with capturing “something more” (in addition to knowledge and skills regarding mathematical concepts, terms, conventions, rules, procedures, methods, theories, and results), which resembles what is indicated by the notion of mathematical literacy as it is, for example, used in the PISA. On the one hand, the arguments for broadening the scope of school mathematics have been utility oriented, based on the observation of students’ lack of ability to use their mathematical knowledge for solving problems that are contextualized in extra-mathematical contexts, in school as well as out of school, an observation corroborated by a huge body of research. On the other hand, the constitution of mathematics as a school discipline in terms of “products” – concepts (definitions and terminology), results (theorems, methods, and algorithms), and techniques (for solving sets of similar tasks) – became challenged. Product-oriented curricula were complemented by, or contrasted with, a conception of mathematics that includes mathematical processes, such as heuristics for mathematical problem solving, mathematical argumentation, constructive and critical mathematical reasoning, and communicating mathematical matters.
There are different views about the amount of mathematical knowledge and basic skills needed for engagement in everyday practices and nonmathematically specialized professions, although it has been stressed that a certain level of proficiency in mathematics is necessary for developing mathematical literacy. The role of general mathematical competencies that transcend school mathematical subareas also has been stressed in the newer versions of conceptualizing mathematical literacy, most prominently in the versions promoted by the OECD-PISA (see above).
Critique and Further Research
Even though the notion of “mathematical literacy” has gained momentum and is now widely invoked and used in various contexts, it has also attracted different sorts of conceptual and politico-educational criticism.
Some reservations against using the very term “mathematical literacy” concern the fact that it lacks counterparts in several languages. No suitable translation exists, for example, into German and Scandinavian languages, where there are only words for “illiteracy,” which stands for the fundamental inability to read or write any text. Indeed, the term “literacy” (both mathematical and quantitative literacy) has been interpreted by some to connote the most basic and elementary aspects of arithmetic and mathematics, in the same way as linguistic literacy is often taken to mean the very ability to read and write, an ability that is seen to transcend the social contexts and associated values, in which reading and writing occurs. However, the demands for reading and writing substantially vary across a spectrum of texts and contexts, as do the social positions of the speakers or readers. The same is true for a range of contexts and situations in which mathematics is used. People’s private, professional, social, occupational, political, and economic lives represent a multitude of different mathematical demands. So, today, for most mathematics educators, the term mathematical literacy signifies a competency far beyond a set of basic skills.
Another critique, going against attempts at capturing mathematical literacy in terms of transferable general competencies or process skills, consists in the observation that such a conception tends to ignore the interests and values involved in posing and solving particular problems by means of mathematics. Jablonka (2003) sees mathematical literacy as a socially and culturally embedded practice and argues that conceptions of mathematical literacy vary with respect to the culture and values of the stakeholders who promote it. Also, de Lange (2003) acknowledges the need to take into account cultural differences in conceptualizing mathematical literacy. There is no general agreement among mathematics educators as to the type of contexts with which a mathematically literate citizen will or should engage and to what ends. However, there is agreement that mathematical literate citizens include nonexperts and that mathematical literacy is based on knowledge that is/should be accessible to all.
In the same vein, mathematics educators have empirically and theoretically identified a variety of intentions for pursuing mathematical literacy. For example, Venkat and Graven (2007) investigated pedagogic practice and learners’ experiences in the contexts of South African classrooms, in which the subject mathematical literacy is taught. They identified four different pedagogic agendas (related to different pedagogic challenges) that teachers pursued in teaching the subject. Jablonka (2003), through a review of literature, identified five agendas on which conceptions of mathematical literacy are based. These are as follows: developing human capital (exemplified by the conception used in the OECD-PISA), maintaining cultural identity, pursuing social change, creating environmental awareness, and evaluating mathematical applications. Some terms have been introduced as alternatives to “mathematical literacy” in order to make the agenda visible. Frankenstein (e.g., 2010) uses critical “mathematical numeracy,” D’Ambrosio (2003) writes about “matheracy,” and Skovsmose (2002) refers to “mathemacy.”
Relations of mathematical literacy to scientific and technological literacy have also been discussed (e.g., Keitel et al. 1993). Challenging questions include the role of mathematics in digital technology and the implications for the development of critical competence to counterbalance the demathematizing effect of mathematics-based technologies that operate as black boxes (e.g., Gellert and Jablonka 2009). This question becomes particularly relevant if the question of interpretability is not based on the lack of expertise of the user of such a black box, but rather is a consequence of the complexity or flexibility of the underlying mathematical model (such as in the context of machine learning).
As to the role of mathematical literacy in assessment, discrepancies between actual assessment modes and the intentions of mathematical literacy have been pointed out by researchers in different contexts (Jahnke and Meyerhöfer 2007; North 2010; Jablonka 2015). In the assessment literature, the contexts in which mathematically literate individuals are meant to engage are often referred to in vague or general terms, such as the “real-world,” “everyday life,” “personal life,” “society,” and attempts to categorize contexts often lack a theoretical foundation. Identifying the demands and knowledge bases for mathematically literate behavior in different contexts remains a major research agenda.
As far as the teaching of mathematical literacy is concerned, the transition between unspecialized context-based considerations and problem solutions that employ specialized mathematical knowledge is a continuing concern. Ethnographic studies of how use of (school-)mathematical notions and techniques is made within other practices (e.g., workplaces) show that (school-)mathematics becomes subordinated to the motives or objects characteristic of these practices. Conversely, out-of-school experiences and knowledge often become a mere springboard for developing school mathematical notions and techniques. Studies of curricula associated with teaching mathematics through and for exploring everyday practices have, for example, usefully drawn on theories of knowledge recontextualization.
These observations suggest that the meanings and usages associated with the notion of mathematical literacy and its relatives have not yet reached a stage of universally accepted conceptual clarification nor of general agreement about their place and role. Future theoretical and empirical research and development are needed for that to happen.
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