Mathematics in Human Learning
Mathematics Education is the teaching and learning of mathematics, as well research on these matters. The researchers in this field seek conceptual and methodological tools and implement approaches which facilitate their study and use. In Europe, research into Mathematics in Human Learning is known as the Didactics of Mathematics (a common denomination, at least, in France and Germany) or Mathematics Education (denomination in United Kingdom).
The creation of the chair of Mathematics Education at the University of Göttingen, in 1893 coordinated by Felix Klein.
The contributions of Felix Klein, David Eugene Smith, Max Simon, Hans Freudenthal, André Revuz, George Polya, and other mathematicians who encouraged Mathematics Education.
The foundation of the International Commission on Mathematical Instruction (ICMI) in 1908, which was responsible for promoting scientific work in the area based on the collaboration of mathematicians, psychologists, mathematics teachers, and mathematics professors.
The training of an international scientific community in the area of Mathematics Education as a result of the First International Congress on Mathematical Education (ICME), held in Lyon in 1969.
From the seventies in the last century, the research area began to be organized. In France, the IREMs (Instituts de Recherche en Didactique des Mathématiques) were created. These were institutes dependent on the Mathematics Departments of the universities and were directed by a mathematician. In Italy and Hungary, Mathematics Education was institutionalized through the university Mathematics Departments, which led to close collaboration between researchers in Mathematics Education and mathematics professors, as in France.
The constitution of Mathematics Education in the Anglo-American area took a different route. In the United Kingdom, the Association of Teachers of Mathematics (ATM) has served as the main support for the articulation of national research projects in Mathematics Education, such as the Low Achievers in Mathematics Project (LAMP) in 1987 or the Shell Centre for Mathematical Education which was set up in Nottingham in 1968. In North America, the researchers in Mathematics Education may be members of Mathematics Departments or professors ascribed to the Education Science faculties (e.g., the National Center for Research in Mathematical Science Education in the USA).
As stated above, if we compare it with other sciences (physics, philosophy, or pure mathematics), Mathematics Education is a very young discipline; however, it does have social relevance as can be judged by the proliferation of journals (Educational Studies in Mathematics, Journal for Research in Mathematics Education, Journal For the Learning of Mathematics, Journal of Mathematics Teacher Education, etc.) doctoral programs, scientific associations and organizations (e.g., the National Council of Teachers of Mathematics (USA), National Centre for Excellence in Teaching Mathematics (England), SEIM (Spain), Homi Bhabha Centre for Science Education (India), etc.), and congresses (ICME, PME, …).
Mathematics Education is identified as a science and as a scientific discipline.
If an attempt is made to formulate to what extent this discipline self-identifies among the professionals in the area of Mathematics Education, it will be seen that basic epistemological options constituted by beliefs and conceptions formed from collective and individual experiences related to mathematical knowledge and how the teaching and learning of mathematics comes about and should come about come into play. The underlying mathematical epistemologies in these professionals configure a horizon for the understanding of the identity of a varied discipline (Sierpinska and Kilpatrick 1998) which is at the base of the structuring of their didactic proposals. Thus, we will obtain different characterizations depending on whether we subscribe to objectivist epistemologies with affiliations to Platonism, based on a view of mathematics as a static unified body of knowledge, in which mathematics is only discovered, or a dynamic views of mathematics, heirs of a critical view of science (e.g., constructivist positions, perspective on the resolution of problems), which understands mathematics as a field of human creation in continual expansion, and for which mathematics is an open science, in construction, and its results are provisional and remain open to revision. In this chapter, dynamic perspective is adopted, in the terms explained above. We consider mathematics as a science with an object of specific study closely linked to historical and sociocultural human development and for which the actions of creating and discovering are the motor forces.
Research into Mathematics Education has two main objectives, one pure (basic science) and the other applied (engineering). The purpose of basic research is to understand the nature of mathematical thought and its teaching and learning. In these studies, work is done on theoretical perspectives in order to understand this nature, and to do so, descriptions of aspects of cognition are developed (e.g., thinking mathematically, what the students understand or fail to understand as regard the concepts of function, limit, etc.) or descriptions of the influence of the emotional, social, and cultural dimensions in mathematical knowledge.
Applied research seeks the applicability of the knowledge achieved in basic research in order to improve instruction in mathematics. Some of these studies are conducted based on proofs of existence (evidence of cases in which the students can learn to resolve problems, induction, the group theory; evidence of the feasibility of diverse types of instruction) and descriptions of the consequences (positive and negative) of different forms of instruction.
In order to ensure a determined state of opinion, as our starting point we will adopt the definition which the Spanish Royal Academy Dictionary gives of the term religion, “the set of beliefs or dogmas regarding the divinity, feelings of veneration and fear of the divinity, moral norms for individual and social conduct, and ritual practices, principally prayer and sacrifice in order to worship the divinity.” According to this definition, religion is an element of human activity, whose components refer to different environments of individuals and of collectives: comprehension and understanding (beliefs), affective existential experience (feelings, the supernatural dimension), and conduct (ethics and morality of behavior).
If we apply this definition literally to the concept of Mathematics Education, we can affirm that this knowledge does not comply with the parameters established for a religion and it should be added that the dialogue with religion does not become a necessity for mathematics nor for Mathematics Education. Nevertheless, as was noted in section “Religion,” Mathematics Education as a scientific discipline is structured based on a determined epistemological foundation; therefore, it participates in a view and ways of accessing knowledge marked by systems of beliefs of the collectives which work on these. Beliefs do not only refer to the human construction but also, very especially, to mathematical production and the view of the world. Therefore, we could state that religion and Mathematics Education share similar grounds (mathematical metaphysics, philosophical presuppositions of what knowledge is), whose foundations are supported by systems of beliefs and ways of conduct founded on ethical attitudes. This common substratum makes it advisable not to reduce the environment of mathematics and of Mathematics Education to web of systems and results which, however rich and complex they may be, seem to us to be insufficient.
It may be pertinent to point out here that, throughout the history of mathematics, we find mathematicians and collectives of mathematicians who have been led to perceive mathematics as a model and mirror of what human conduct must be and as the object inspiring astonishment and mystery (so linked to the religious phenomenon) by the “certainty” of mathematical knowledge, the order and harmony of the universe discovered through the contemplation of the structures and rhythms of nature. In the case of the Pythagorean school, the revelation of the harmony of the universe expressed in the harmony of the numbers provided a path to union with the divine. It could be said that, on contemplating the independent force and autonomy of the relationships which are created or discovered in mathematics, mathematics can reasonably suspect the presence of something superior to it, which precedes it in intelligence. This conviction crosses the history of mathematics, and we can trace it to Hermite (and Gödel), in its affirmation of the divine origin of the world of mathematical ideas and on the construction of a rational religion based on its logical-philosophical thought or in the formal proofs of the existence of God (Anselm, Leibniz, Gödel).
Mathematics Education is not a particular case in general education. It is important to point out that, although the object of Mathematics Education is mathematics, the epistemological status of this knowledge is different from the mathematics investigated by mathematicians; the nature of the evidence and of the argumentation is different in both disciplines. Mathematics Education must necessarily refer to the person in his global nature, a person still to be conformed, integrated into a specific society, and modeled by the culture in evolution of the society he belongs to; Mathematics Education must unfailingly have the human and material resources required at a specific time and must be aimed at the finalities assigned to education by a determined society, through its political authorities, finalities which can be extraordinarily varied.
Mathematics is characterized by the fact that its studies mental creations, it does not work with objects or physical phenomena, and by the form in which its declarations are justified. Mathematics has its own characteristics such as abstraction, induction, hierarchy, globalization, rigor, which entails an epistemology of its knowledge. Mathematics is a discipline which requires a certain effort for its assimilation and the use of cognitive strategies of a superior order. Mathematics Education is characterized by the application of forms, methods, and strategies which inform the conscious action of the mathematical work in the person learning. To achieve this, it is necessary to have an interdisciplinary approach, which takes into consideration a variety of theoretical frameworks: epistemology, psychology, the sociology of mathematical knowledge.
As the necessary interdisciplinary condition is not sufficient, Brousseau and Artigue in the ICMI Study Conference (p. e., Artigue 1998) pointed out that the questions of meaning and genesis of the mathematical notions are a meeting point for researchers into Mathematics Education and researchers into mathematics. Without this specific contribution, Mathematics Education would run the risk of becoming didactics of a general type.
Relevance to Science and Religion
In relation to the subject Science and Religion, we tend toward an approach which makes it possible to have elements of differentiation, dialogue, and integration between both disciplines and we consider that a pertinent relationship between Science and Religion can be established based on complementariness. In the first place, the differentiation is sustained because Mathematics Education and Religion conserve their legitimate identities and autonomy in the domains recognized for each of them by the respective scientific communities. Each of these disciplines is autonomous when defining their universes and each is free to advance along its own path.
It should be asked whether, despite being disciplines with such different identities and languages, a space, a common territory can be established in which both can dialogue and interact. The response would be affirmative. On the one hand, science (Mathematics Education) is an instrument which forms the subject as regard his perception of the world and the nature of the person who it humanizes and perfects (the human mind, system of values, etc.); on the other hand, religion needs consistent subjects, who provide sense to human action considered globally (the holistic dimension).
From this point of view, it could be stated that Mathematics Education incorporates elements which are related to the metaphysical dimension. In the mathematization process, the human mind finds indications, clues which make him suspect the existence of something in the universe beyond him; this opens up the question on reality, on existence, on being, on the how and the when, on where from and where to; all these questions involve interrogations on axiology and meaning. In this opening of the mind to intellectual mathematical knowledge, being is present as a horizon and a possibility in its infiniteness and the perception of an infinite horizon in knowledge, which stimulates the search for its foundations. Thus, the mathematical mind is opened up to the possibility of moving with no difficulty from specific thought on a specific problem to what is mysterious and transcendent, entering a dynamic search for what the mysterious and the inexpressible/ineffable represents in the mind.
The opening up of our minds is not only the opening up and the dynamism of intelligence but that the person intervene holistically (the whole person; with his willingness; with his capacity for desire, love, freedom, structure of values, etc.).
The teaching and learning of mathematics which is rooted in humanist approaches, undoubtedly, contributes to the fact that the person who learns gradually discovers human values with all their nuances (including absolute values). This discovery acquires more significant qualities to the extent in which it is supported by person to person and person to reality interaction (including mathematical reality). Educational action from mathematics can help to identify and discover dimensions of value within the framework of these interactions. The learners not only discover values but also live them; they transform these into conduct, thoughts, actions, and all kinds of experiences.
Among pure mathematics and technological applications, there is a wide spectrum of mathematical activities which have the search for coherence, clarity, certainty, and effectiveness in the development of knowledge in common. During the learning of mathematics, coherence and the needs of mathematical work do not enclose the learner in a mechanical and automatic control of reality but bring him close to the unexplored and the unknown, to creativity and intuition, to the passion to discover new worlds. It also opens them up to the nonformal world of metaphysics and theology.
If this process is to be feasible, a determined type of mathematical epistemology must govern the learning. An epistemological approach of an ahistorical, amoral nature, isolated from the scientific disciplines, deprived of cultural emotions and values, only formed by a body of facts and truths, a conception which has dominated many of our syllabuses (curriculum) for a long time, would not be feasible. It would be advisable to propose mathematics and education of a humanist type. In the humanist approach, the perspective of the professor, of the educator as a mediator in the learning process, is vital. The professor and his conceptions of mathematics are irreplaceable as regard the revelation and the development of values which can make the learner ask questions on the nature of certainty, truth, existence, limits.
Finally, it should be mentioned that, in some countries, in the Science Religion dialogue, materials have been published on the learning of mathematics at secondary school level, taking the social responsibilities which this science assumes in each context (the axiological structure of the science) as the reference and production framework. We refer to the Charis materials, drafted by members of the Association of Christian Teachers (Scortt et al. 1996). These decide for the option to establish connections between spirituality, charity, and social justice, connections which lead the subject to a search for an explanation of the world in terms of religion, philosophy, science, and mathematics.
Sources of Authority
As in other disciplines, one of the sources of authority of Mathematics Education lies in the methods which make the validation and feasibility of research possible. The results of the research in education are not “proved” in the sense that mathematics is proved. Frequently, experimental methods or direct statistics (of the type used in the physical sciences) are inadequate or insufficient due to the complexity of deciding what is meant by educational conditions being “replicated.” Thus, in Mathematics Education, it is necessary to have recourse to a wide variety of methods of a heuristic or interpretational nature. For example, a look at the study of research into the teaching of calculus in university Mathematics Education in recent decades shows this variety (Schoenfeld 2000). Along the same lines, we find reports of detailed interviews with students, comparisons of calculus courses under the reform with traditional courses and a broad study on the development of the comprehension of calculus in students through the use of learning technologies. These studies, which use anthropological and sociological techniques of a qualitative nature, are increasingly more common in Mathematics Education.
It is also usual that the use of these types of techniques gives rise to questions on their validity and feasibility. In order to neutralize the suspicion, Mathematics Education has a number of criteria for evaluating models, theories, and, in general terms, any empirical or theoretical work; these criteria are the following: descriptive power, explanatory power, scope, predictive power, rigor and specificity, the possibility of refutation, replicability, availability of multiple sources of evidence (“triangulation”). The findings achieved are rarely definitive. With these types of techniques, the results always become provisional and, frequently, they are tentative, like human existence. The evidence, in these cases, does not adopt the form of mathematical proof but is accumulative, progressing toward conclusions which can be considered to be tendencies or reasonable models with explanatory power for the reality they study.
Mathematics as a science is one of the fundamental axes of culture and contracts a singular responsibility with culture for its proper development; a development which could be said to be at human scale. Mathematics, as a formal science, cannot assume the pretension of neutrality; together with other disciplines, it is unfailingly linked to the improvement of the standard of living of the citizens. In this same regard, numerous aspects of mathematical work and its teaching strongly involve our sense of responsibility as members of a global society. This orientation is included in declarations of institutions and the contributions of mathematicians (American Mathematical Society Ethical Guidelines (http://www.ams.org/secretary/ethics.html); interview with Reuben Hersh, What Kind Of Thing Is A Number? A Talk With Reuben Hersh [2.10.97] (http://www.edge.org/3rd_culture/hersh/hersh_p1.html; Moslehian 2005; Guzmán 2000). Among the ethical principles we can point out the submittal to reality; the joyful acceptance of truth and beauty; professional integrity; the deep sense of humility in the search for knowledge; the sense of freedom, of community, and cooperation with others; the respect for mathematical capacity wherever it may be, regardless of race, gender, ethnicity, age, religious or political orientation, etc.
In the last two decades, much effort has been made to identify the set of values associated to mathematical knowledge and its teaching (Brown 1996). As a suggestion, and with the provisional nature of all contextualized knowledge, we propose the categorization of Bishop (1988) which, based on the four components of culture of White (1959): sentimental, ideological, sociological, and technological, organizes six sets of values grouped in three binomials: rationalism– empiricism, control–progress, and opening–mystery.
Rationalism is related to the discussions, reasoning, logical analysis, and explanations. This value is revealed by the teacher when he evaluates the development of the capacities of the students for argumentation, logical reasoning, and mathematical proofs. Empiricism is related to the processes of conversion into objects, specification, and the application of ideas in mathematics. The teacher will show this in his classes when he estimates the development of the practical capacities of the students for the use of the mathematical ideas and symbolism in modeling through diagrams and in the collection of experimental data.
Control relates to the potentiality inherent to mathematical knowledge for the use of established rules, procedures, and criteria; of facts; and of predictions. This value is inculcated by the teacher when he appreciates the capacities of the students when presenting routines and algorithms, when working with mathematical precision, and when exploring mathematical ideas which predict events. Progress is related to the development of mathematical ideas, of individual freedom and creativity; the teacher can foster this in teaching through the stimulation of creative explanations.
The opening up refers to the democratization of knowledge through proofs and through individual explanations. The teacher can foster this value when he stimulates the development of the capacities of the students to structure their own ideas in proofs, verifications, discussions, and in the freedom to express different points of view.
Finally, the mystery and the search for truth, this value is related to the fascination for scientific mathematical ideas and with submittal to the truth and to reality, so rooted in the scientific. This value, undoubtedly, constitutes one of the important features which we should appreciate and encourage in the teaching off mathematics. The teacher can stimulate the imagination of the students in discussions on the nature of the object knowledge and the meaning of the scientific ideas.
Therefore, besides being a determined class of symbolic technology (rules, concepts, algorithms, etc.), mathematics is the carrier and, at the same time, the product of determined values. If we only wish to understand mathematics as a specific symbolic technology, we will only understand a small part of these, perhaps the least relevant for education and for our future as human beings.
Some of the concepts which are defined below are not the objects of study, not of definition in Mathematics Education. In the event that this is done, it should be pointed out that it is not possible to establish a univocal definition because this will depend on the theoretical framework of reference which is adopted. Therefore, we will define these, taking this into account.
Mathematics is an exploration of certain omnipresent and more or less complex structures which appear in nature, in the world; which admit this rational approach, which can be manipulated through symbols; which provide us with a certain domination of the reality they refer to; and which we call mathematization. For example, in the mathematics for a realist education focus, mathematics begins and remains in reality. Mathematical activity commences in processes of mathematization of the real and is expressed in rules, in structures, which in turn become the base material for superior abstraction, generating a hierarchy which is distant from the original common sense until it is converted into the reality most distant from this.
In Mathematics Education, focuses such as the resolution of problems and the constructivist perspective place special stress on the human being as the person who creates and constructs mathematics. The human being is not a concluded being but is a being who becomes within the evolutionary framework.
The transmission of mathematical knowledge is carried out by paying attention to specific persons under a holistic perspective of the integration of the cognitive, affective, and social dimensions of the subject.
Life and Death
These are not the objects of definition by the discipline.
In relation to this concept, it seems to us to be significant to point out one of the facets which is most worked on in Mathematics Education, the objective view of reality. Mathematics fosters a view of reality which is more objective than subjective. The ideas are essentially ideas on objects. Mathematics deals with abstractions, and Mathematics Education puts substantial efforts into developing what is usually called abstract thought. The power to convert these abstractions into objects is what makes it possible to handle these with such precision. In addition, the network of logical connections developed with mathematical ideas through proofs, extensions, examples, counter examples, generalizations, and abstractions helps to give them objective meaning, and consequently, it makes it possible to address these as if they were objects. The language of if; we suppose and the conditional tense of the verb also force an imagined reality at conscious level and, thus, it permits this to be manipulated as if it were an objective reality. Therefore, when encouraging the students to develop their capacity of abstraction, we encourage the ways to specify and “convert abstract ideas into objects” through a wide range of symbols.
Mathematics, like any other type of knowledge is created, applied, and taught within the framework of institutions whose use and values have significant repercussions on the way to focus learning and on the approaches of Mathematics Education. For example, in the constructivist positions on Mathematics Education, knowledge refers to the ordering and organization of our experiences and not to the understanding of an objective ontological reality. In order to provide grounds for their proposals, the constructivisms found several psychological referents, Piaget and Vygotsky, whose theories of knowledge are wider than the epistemology of mathematical knowledge. There are philosophical presuppositions on what knowledge is, and there is an option to decline any type of correspondence with the world in order to define truth and objectivity. The proposals of radical and social constructivism are important examples of these types of conceptions which give rise to beliefs which influence the teaching and learning of mathematics.
From its beginnings, mathematical knowledge has been considered to be sure, absolute, and eternal knowledge. However, in the twentieth century, epistemological approaches appeared and, currently, these have an important influence on the teaching of mathematics and present mathematical work as fallible. The mathematical truths, propositions, and ideas in general are open to the examination of any properly trained person. According to this focus, the education process must be centered on a heuristic focus which is set against the deductivist focus of the formalist school (e.g., Lakatos 1976). The deductivist focus starts from the acceptance of primary truths and definitions, without explaining how these arose, and from these, the theorems and proofs are presented. On the contrary, the heuristic focus shows the counter examples; it does not separate the definitions from the proofs which have generated them; it places emphasis on problematic situations and stresses the logic of the discovery which has given rise to new concepts. The perspective opened up by the approach of heuristics endeavors to show the side of mathematics hidden below the Euclidean rigor and presents it in its process of genesis toward a new level (Polya 1945). The creative phase in mathematics is not governed by logical analyses, but by inquiry which must opt for new views, to relate concepts or properties and create new ones. The consequences of this approach for Mathematics Education are substantial, and, in recent decades, their effects have begun to be seen. Among its more disseminated tendencies is the accent on the transmission of processes of mathematical thought and, in particular, on the mental processes for the resolution of problems.
Perception and action are bases of mathematical thought. We understand perception to be the exercise of a first level of awareness faced with contact with reality. The development of perception implies extending the capacity to be aware, to note, to highlight elements and aspects which configure what surrounds us until it is configuring the subject which it constructs more and more actively. In Mathematics Education, attention is paid to the blocks which more specifically affect our cognitive functioning, the perceptive blocks. These are obstacles which prevent anyone from solving a problem from clearly perceiving the problem in itself or the information required to solve it.
In some subdisciplines such as the didactic of mathematical analysis, the changes and transformations in space and time are analyzed.
Education basically consists of educating the attention, producing changes in the locus, in the focus, and in the structure of attention, and these changes can be increased through others, working in the awareness itself. The act of awareness is an internal individual activity which a student carries out but which is more easily achieved if the educational activity fosters it. Several strategies can favor this. For example, one strategy is that the teacher work with the awareness that the students have of mathematics instead of expressing his own awareness of mathematics. Another strategy is the development of an internal mathematical monitor which presents us with questions such as, “Why am I doing this particular operation?” “Is it supposedly more complicated?” and “Am I doing this wrong?” Developing a monitor which corrects errors is an important part of the metacognition involved in becoming an expert in mathematics.
In this discipline, the mission of the human mind consists of rationally interpreting realities, facts, which are presented as given, as previous, to the best of its ability.
The work of mathematics is necessarily an open activity, inexhaustible, in the sense that it can never be considered to be concluded; the mystery is linked to the fascination for new mathematical ideas and the perception of the infinite.
Finally, we should point out something which we have shown in this document, which is that in the development of the discipline of Mathematics Education, there are different focuses. The conceptions of the knowledge of the world included in each of these focuses have repercussions in the education of the students and can become indirect ways of familiarizing them with different ways to see the world and to conceive objectivity and truth, and, as a consequence, to place the bases for their conception of moral responsibility.
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