“Mathematical literacy” is an individual’s ability to identify, understand, exert well-founded judgment about, and act toward the roles that mathematics plays in dealing with the world (i.e. nature, society, and culture)—not only as needed for that individual’s current and future private life, occupational life, and social life with peers and relatives but also for that individual’s life as a constructive, concerned, and reflective citizen.
Some explanatory remarks are in order for this definition to become transparent.
1. In using the term “literacy,” we want to emphasize that mathematical knowledge and skills that have been defined and are definable within the context of a mathematics curriculum do not constitute our primary focus here. Instead, what we have in mind is mathematical knowledge put into functional use in a multitude of contexts in varied, reflective, and insight-based ways. Of course for such use to be possible and viable, a great deal of intra-curricular knowledge and skills are needed. Literacy in the linguistic sense cannot be reduced to—but certainly presupposes—a rich vocabulary and a substantial knowledge of grammatical rules, phonetics, orthography, and so forth. In the same way, mathematical literacy cannot be reduced to—but certainly presupposes— knowledge of mathematical terminology, facts, and procedures as well as numerous skills in performing certain operations, carrying out certain methods, and so forth. Also, we want to emphasize that the term “literacy” is not confined to indicating a basic, minimum level of functionality only. On the contrary, we think of literacy as a continuous, multidimensional spectrum ranging from aspects of basic functionality to high-level mastery. In the same vein when we use the word “needed” we do not restrict ourselves to what might be thought of as a minimum requirement for coping with life in the spheres that are at issue. We also include what is “helpful,” “worthwhile,” or “desirable” for that endeavor.
2. The term “act” is not meant to cover only physical or social acts in a narrow sense. Thus the term includes also “communicating,” “taking positions toward,” “relating to,” and even “appreciating” or “assessing.”
3. A crucial capacity implied by our notion of mathematical literacy is the ability to pose, formulate and solve intra- and extra-mathematical problems within a variety of domains and settings. These range from purely mathematical ones to ones in which no mathematical structure is present from the outset but may be successfully introduced by the problem poser, problem solver, or both.
4. Attitudes and emotions (e.g., self-confidence, curiosity, feelings of interest and relevance, desire to do or understand things) are not components of the definition of mathematical literacy. Nevertheless they are important prerequisites for it. In principle it is possible to possess mathematical literacy without possessing such attitudes and emotions at the same time. In practice, however, it is not likely that such literacy will be exerted and put into practice by someone who does not have some degree of self-confidence, curiosity, feeling of interest and relevance, and desire to do or understand things that contain mathematical components.
Again, in defining Mathematical Competencies we follow the Mathematics Literacy framework published by the OECD Program for International Student Assessment (PISA). Here is a nonhierarchical list of general mathematical competencies that are meant to be relevant and pertinent to all education levels.
• Mathematical thinking
♦ Posing questions characteristic of mathematics—Does there exist...? If so, how many? How do we find...?
♦ Knowing the kinds of answers that mathematics offers to such questions.
♦ Distinguishing between different kinds of statements (e.g., definitions, theorems, conjectures, hypotheses, examples, conditioned assertions).
♦ Understanding and handling the extent and limits of given mathematical concepts.
• Mathematical argumentation
♦ Knowing what mathematical proof is and how it differs from other kinds of mathematical reasoning.
♦ Following and assessing chains of mathematical arguments of different types.
♦ Possessing a feel for heuristics (what can happen, what cannot happen, and why).
♦ Creating mathematical arguments.
• Modelling
♦ Structuring the field or situation to be modelled
♦ Mathematizing (i.e., translating from “reality” to “mathematics”).
♦ De-mathematizing (i.e., interpreting mathematical models in terms of “reality”).
♦ Tackling the model (working within the mathematics domain).
♦ Validating the model.
♦ Reflecting, analyzing, offering critique of models and model results.
♦ Communicating about the model and its results (including the limitations of such results).
♦ Monitoring and control of the modelling process.
• Problem posing and solving
♦ Posing, formulating, and making precise different kinds of mathematical problems (e.g., pure, applied, open-ended, closed).
♦ Solving different kinds of mathematical problems in a variety of ways.
• Representation
♦ Decoding, interpreting, and distinguishing between different forms of presentations of mathematical objects and situations, and the interrelations between the various representations.
♦ Choosing and switching between different forms of representation according to situation and purpose.
• Symbols and formal language
♦ Decoding and interpreting symbolic and formal language and understanding its relations to natural language.
♦ Translating from natural language to symbolic or formal language.
♦ Handling statements and expressions that contain symbols and formulas.
♦ Using variables, solving equations, and performing calculations.
• Communication
♦ Expressing oneself in a variety of ways on matters with mathematical components, in oral as well as in written form.
♦ Understanding others’ written or oral statements about such matters.
• Aids and tools
♦ Knowing about and being able to make use of various aids and tools (including information technology tools) that may assist mathematical activity.
♦ Knowing about the limitations of such aids and tools.
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