Posted at 11.29.2018

A necessary idea to the question What's mathematics pedagogy. is Why do we teach mathematics. If basic number skills are clearly needed in every days life, can we say the same about geometry? Mathematics trains us to believe also to think rationally, rigorously. They can be an important part of individuals culture and a heritage of days gone by and anyone who analyzed mathematics really found serious beauty in them. More important mathematics helps us understand, since Galileo we know they help us understand the technology of dynamics, but also the relationship between the vertices, ends and faces of an polyhedron or that any integer higher than 1 can be expressed as a unique (up to buying) product of perfect numbers

To the question why doing mathematics David Hilbert replied: "The situation is here, you must solve it!" However the answer that would probably convince most is merely because mathematics is useful: in sciences (physics, chemistry, biology, computer knowledge), in executive, in economics and money And unless of course one needs to suppress every form of teaching, we have to teach mathematics. Stable mathematics education is crucial for broadening post-school opportunities. Regrettably, many students progress are hampered by insufficient self-confidence, or low numeracy skills and it is for teachers to boost their practice to crack the pattern. According to the International Academy of Education, a multinational organism dealing with the UNESCO, any mathematical pedagogy should abide to the next principles:

be grounded in the general premise that students have right to access education and the precise premise that all have the right to access mathematical culture;

acknowledge that students, irrespective of age, can develop positive mathematical identities and be powerful numerical learners;

be based on interpersonal value and sensitivity and be attentive to the multiplicity of social heritages, thinking operations, and realities typically within our classrooms;

be centered on optimising a range of desirable academic outcomes which include conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning;

be focused on enhancing a variety of social results within the mathematics school room that will donate to the alternative development of students for fruitful citizenship.

(International Academy of Education, p6)

This article is divided into three parts. First we give some factors about mathematics, what they are and what are the distinctions from other sciences. The second part is a wide look at the notion of pedagogy and the distinctions between traditional and progressive pedagogy. Finally in the 3rd we will concentrate on numerical pedagogy and the two ways of considering mathematical concepts, thing or tool.

Mathematic comes from the Greek јёј± (math"ma) whose first meaning was "science or knowledge", and only later became "mathematics". From Math"ma came up ј±ёј±№є (mathematikos), which first designed "related to learning", and then later "related to the numerical science". In Latin it became mathematicus then mathematique in Romance dialects. Finally mathematic then mathematics.

The Ishango bone is the oldest trace of mathematical activity discovered; going out with back to the prehistoric age groups, it is believed to become more than 20000 years old (Brooks and Smith, 1987). In comparison writing is much more recent, dated between 6000 BC and 8000 BC.

However mathematics once we understand it today started out with the Ancient Greeks between 600 and 300 BC. If all culture and civilization always developed their knowledge of mathematics, they all achieved it through inductive reasoning: one reaches conclusions or rules through repeated observations. On the other hand, the Greek mathematicians used deductive reasoning: they used definitions, axioms and logic to demonstrate their statements (Bernal, 2000). One of the most influential Mathematic wording ever, the Elements was written in 300 BC in Alexandria by Euclid (Boyer and Merzbach, 1991) and was used as a textbook before end of the nineteenth century. This shows one of the most characteristic feature of Mathematics, the permanence of its results. The annals of Mathematics or perhaps a summary of it is very good beyond the range of this article, we will soon present two occasions of the history of Mathematics, that are of relevance because of this essay. WITHIN THE Assayer, Galileo composed "Philosophy is written in this grand booklet, the universe. . . It really is written in the words of mathematics, and its own character types are triangles, circles, and other geometric numbers;. . . . " (Drake and Stillman, 1957). Galileo stemmed the mathematization of technology that gave labor and birth to the knowledge we realize today. The need for present Mathematics, the actual fact that Mathematics support most sciences and techniques, that good education in Mathematics is of primordial importance in composing autonomous individuals roots out of this momentous time.

Modern Mathematics ushered in the early 1920s from the foundational turmoil of mathematics. David Hilbert (1862-1943) proposed a refundation of Mathematics, known as the Hilbert's program. Mathematics was to be written in a formal dialect and rooted in a couple of axioms. Already in the Component, Euclid based his work on a few axioms and postulates. Those were viewed as clear propositions on existing objects. However Hilbert's axioms aren't supported by existing or abstract objects, they are a set of relations and operations. This new procedure shaped the modern mathematics, and connectively how they are taught.

Mathematics is a knowledge in the traditional sense, i. e. an prepared body of knowledge characterized by the decision of the items researched and the internal rules of development and validation of its knowledge. With Galileo emerged the idea that the law of aspect could be written mathematically and due to its phenomenal success, modern research thoroughly use mathematics; frequent efforts are made to write any medical guideline in the numerical terms. However Mathematics is not a modern research nor a knowledge of characteristics.

Defining mathematics is an elaborate job and a work in progress. Mathematicians had and still have heated up debates and there is no satisfying description (Mura, 1993).

One of the first explanation was presented with by Aristotle: "the research of volume" (Franklin, 2009). Where arithmetic was the field of discrete variety and geometry the field of the continuous ones. A far more modern definition was presented with by the North american mathematician Benjamin Peirce, the first line of his Linear Associative Algebra is "Mathematics is the knowledge that draws necessary conclusions". Logicism (i. e. adequacy of mathematics and logic) challenged this definition, for example Bertrand Russel wrote "All Mathematics is Symbolic Logic". (Russell, 1903). Subsequently this view was criticized by the intuitionism mathematicians who observed mathematics as a mental activity creating items: "The reality of a mathematical statement can only be conceived with a mental building that proves it to be true, and the communication between mathematicians only functions as a way to generate the same mental process in different minds"(Iemhoff, 2012).

In order to provide a definition of mathematics one must first correct its field of analysis and second explain its device.

One of the very most talented mathematician of the twentieth century, Alexander Grothendieck composed in Recoltes et semailles "mathematical activity includes essentially three things: learning numbers, studying forms and measuring distances. " Arithmetic studies volumes, geometry patterns and analysis distances. However if Mathematics is an individual science rather than three distinctive ones, it is because there are tight connections between statistics, shapes and ranges and learning those connections requires a great area of the mathematics.

The rules of mathematics will be the most constraining of most sciences. For a concept to be mathematic, its explanation needs to own it entirely. In every day vocabulary, words are understood because they relate with multiple personal experience. In some sense the definition of a term only encapsulate part of its meaning, words don't fully support the notions they specify. In mathematics this facet of words is prohibited, a word can only be utilized if its meaning determines its whole meaning, and this, only using words whose definition are already established. Definitions must hold by themselves without the exterior references. Additionally the rules determining the connection between words, i. e. , the guidelines of logic, must also be completely explicit. A numerical text is about well-defined objects and shows some necessary repercussions of their explanations through a mixture of logical steps.

"Let this variety of ideas be placed before him; he'll choose if he is able to; if not, he will remain in question. Only the fools are certain and guaranteed. For if he embraces Xenophon's and Plato's viewpoints by his own reasoning, they'll no more be theirs, they will be his. He who practices another follows little or nothing. He finds nothing at all; indeed he looks for nothing. We aren't under a ruler; let each one case his own independence [Seneca]. Tell him that he understands, at least. He must imbibe their means of pondering, not learn their precepts. And let him boldly forget, if he needs, where he got them, but let him know how to make them his own. Truth and reason are normal to everyone, no more belong to the person who first spoke them than to the man who says them later. It really is no more according to Plato than relating if you ask me, since he and I understand and see it the same manner. The bees plunder the bouquets here and there, but afterward they make of them honey, which is all theirs; it is no more thyme or marjoram. However with the pieces borrowed from others; he'll transform and combine them to make a work of his own, to wit, his common sense. His education, work, and review purpose only at creating this. " (Montaigne, Essays).

Pedagogy originates from the greek ±№ґ±Ї± (paidagge), where ±№ґ (pais, genitive ±№ґ, paidos) means "child" and (ag) "lead"; hence one can understand pedagogy as "to lead the child". The slave who escorted the child to the institution, but also transported his package and was responsible for his homework was the pedagogue.

In ancient greek language, paideia (±№ґЇ±) imply children education. In Athens children were taught sentence structure, rhetoric, mathematics, music, philosophy, geography, natural history and gymnastic. The paideia was likely to elevate the child to change him into an completed man.

One can define pedagogy as the method and practice of teaching, especially as an academics subject matter or theoretical theory (Oxford dictionary). Those methods and tactics have considerably advanced over time and remain changing and growing. Today, many challenging philosophies and ideas are rivalling. Debates over education are often heated, and always were (two centuries ago Rousseau's Emile: Or on Education was used up in Geneva).

Teaching and learning was always a domain of analysis and reflection. In every society and civilization, children are educated. For that goal, institutions are created and philosophies of education developed. If some scholars consider the particular date 1657 as the beginning of modern pedagogy, it is merely because John Amos Comenius (28 March 1592 - 15 November 1670) (Murphy, 1995) teaching viewpoint is very modern on many accounts. In some influential text messages, Comenius developed the bedrock of pedagogy: 1. Education is made for everybody without variation of class, religious beliefs or sex. It's important to notice that although at the same time that considers women to be inferior compared to men, Comenius argue that ladies have the same intellectual ability as young boys; 2. Instruction should not be limited by intellectual and logical activities, but also must include manual training; 3. Learning sensible content like geography, record or science alternatively than Latin (Comenius argued for less Latin, not to control it); 4. Supplying importance to a creative education, because artwork should be accessible to all or any; 5. Acquisition of knowledge should be pleasurable, not a chore. Comenius remarks that teachers need to help the learners to build inner motivation. To the purpose they have to use images. In his Orbis Sensualium Pictus (The Visible World in Pictures), notions are always combined with woodcuts. In a natural way he opposed to any form of punishment, as the determination for learning must result from the learner himself. In his Didactica magna Comenius stimulates the creation of tutor training companies. This school of thought was revolutionary at that time and its diffusion took a long time.

The progressive education

One can track back some of its rules to the Renaissance humanists, to Rabelais and Rousseau. In Emile: or On Education, Rousseau exposes his pedagogical project: Education should respect the type of the child rather than constraining his physical, intellectual and moral development; allowing him to fully develop in his natural do it yourself, but capable to live in modern culture. It really is still one of the very most read and important word on education. In Japan, it is just a essential reading for principal teacher. Nevertheless the birth of intensifying education is at the finish of the nineteenth hundred years with the idea of John Dewey (1859-1952). The essential principles are in fact very close to Comenius's: 1. Drive should come from the learner, rather than being externally stimulated; 2. Learning should be straight related to the learner; 3. Learners need positive learning experience (no disappointment); 4. The learners must be active participant to the training experience. His method is based on "hands-on learning", the teacher is a guide and the student actively learns by "doing". Like Rousseau, Dewey presumed that education is not about acquiring a catalogue of pre-determined skills, but instead realizing one's full probable. He was very critic to the original methods of instructions, where the learner passively have the knowledge from the tutor, like an unfilled vase that could slowly be filled up. Concomitently, he was very reserved on "child centered method". Although antagonist, both methods show the same flaw: they assume a duality between the learner's experience and the knowledge taught (Westbrook, 2000). Although there have been always many pedagogical philosophy in rupture with the original way (e. g. Johann Heinrich Pestalozzi (1746-1827) founded academic institutions in Switzerland after Rousseau's precepts), Dewey sometimes appears as the "dad" of most progressive education philosophies.

Another cornerstone in intensifying education came with the work of Jean Piaget creator of the genetic epistemology (although the term is older and from Wayne Mark Baldwin). The main idea is to see the process of learning as a construction. Piaget supposes the living of deep emotional structure which control the training process. In addition after meeting Jean Dieudonne in Paris, Piaget would find immediate romance between Bourbaki's three mother structures (order structure, algebraic composition, topological structure) and the three buildings he observed in children considering (Behavioural schemata, Symbolic schemata, Operational schemata) (Wells, 2010). This id would lead to the brand new Mathematics reform of the sixtees whose basis was that children's functions of acquiring (or constructing) mathematical strategy were isomorphic to the way mathematics were built.

The traditional education

By traditional education, usually one refers to methods that are knowledge focused. The teacher is going to pass the science to the learner. The professor would expose the knowledge and gets the students practice through exercises. There is no differentiation and each learner cared for the same. If one considers learning as making connections, traditional training supposes that all learners make the same contacts. The school is conceived "as a place where certain information is to be given, where certain lessons should be discovered, or where certain practices are to be produced" (Dewey, 1897).

In fact the term traditional education is mainly used by practitioners of different pedagogical methods. When Dewey and his pupil William Heard Kilpatrick (1871-1965) identified intensifying education it became necessary to make the differentiation from former means of teaching. So traditional education doesn't make reference to any particular doctrine, but from what was regarded as common practice at that time. Actually you can bring many parallels between Dewey's progressive education and past philosophies on education like Comenius's or even the Greek paideia. An essential difference between traditional and intensifying pedagogies is to be found in the goals that they established and in the role that universities have. Traditional pedagogy goals are encapsulated by Montaigne's quote given above; it is ideal for the learner to acquire knowledge and make it his own. Alternatively, intensifying pedagogy is far more ambitious: " in the meantime there are the enslaved humans who must attain their own liberation. To develop their conscience and consciousness, to make sure they are aware of the proceedings, to get ready the precarious earth for the future alternatives - this is our activity. " (Marcuse, 1967). The target set is infinitely higher and preferring a kind of pedagogy is not any longer only a selection of the most effective practice. It becomes a political choice, which ramified to the role of schools in the culture and the way the modern culture itself is structured.

English doesn't make a definite distinction between pedagogy and didactic. However many languages do. Didactic comes from the Greek didaktikos, signifying to 'instruct'. If one considers the etymologies of both words, it would appear that pedagogy concerns the kid or the learner while didactic would be about the coaching itself. Jean Houssaye defines a pedagogical triangle where in fact the vertices are respectively knowledge, professor and learner (Houssaye, 2000). Houssaye clarifies each of the side represent a process, the relationship between two vertices. The medial side knowledge-teacher signifies the didactic process, concerns with the data, its corporation and delivery. The teacher-learner aspect is approximately pedagogy, the connection between the instructor and the learner. The knowledge-learner part symbolizes the training activity and the appropriation of knowledge. For Houssaye, traditional pedagogy only rests on the knowledge-teacher side while a full "child focused" approach would mostly rest on the teacher-learner area. A desirable pedagogical method would consider the complete triangle; the educator would only purposely and temporarily highlight one aspect of the triangle and would balance it at a later stage of the training process.

As previously stated one of the main feature of mathematics is the vocabulary in which it is written. One of the difficulties in learning mathematics is the discrepancy between the formal explanations and the representations that one can have in his mind. If we consider an object as familiar as a upright line on the aircraft, what proper classification can we give than it? Mathematicians would identify a straight range as an affine space of dimensions 1. But applying this definition suppose understanding of linear algebra which one reflected over the partnership between an affine space and a line drawn on a paper or panel. A school meaning could be an algebraic one: the group of items that satisfies some affine equation. With this example one can understanding the duality in doing mathematics. What one reads or writes differs from what's in ones mind. How can the tutor help the learner to produce and connect an effective but personal group of images with the meanings?

One common method is to bring in new notions or concepts from concrete experience and problems. In doing this the teacher desires to connect the abstract notions to concrete images and representations. Although smart, this approach forgets a duality common to every mathematical notion; what Regine Douady telephone calls the "tool-object dialectic" (R. Douady, 1984). It really is in fact a very simple idea and a very old one; but structuring all teaching in mathematics. It is illustrated in the starting of '2001: A Space Odyssey' (Kubrick, 1968) when an ape runs on the bone as tool and weapon. The world shows the invention of the first tool and hence of the "dialectic" relationship between the thing and the function allocated to it. Another illustration of this concept sometimes appears in modern skill where items familiar to us are shown in complete different setting (e. g. Juan Miro using discarded cans to conceive sculpture). Regine Douady contribution was to transpose this notion to mathematical objects. It is a typical misconception to believe mathematical items are just bits of a vast rational construct without finality or purpose. Actually mathematical items are nothing if one neglect their function, their tool part. It could have been that a particular theory or object have been created to solve a challenge in auto mechanic or optic, or even to solve a particular equation. Some of those objects were developed over many years, sometimes centuries. If they seem to possess lost their original goal and acquired some kind of transcendental living it is only because unexpected utilization have been found, they added dealing with unseen problems during their conception. But also because, new things acting on them have been created to solve increasingly harder problems. Hence, any number, function or condition bears this dual mother nature. They can be regarded in a structural way considering their relationships with other similar things, but they can be also studied independently. In the end many mathematicians began their trip contemplating the strange agreements of decimals in lots, or observing the countless dazzling features contain in a form This causes a fundamental question for any professor: given an idea, should it be launched from its thing or tool's aspect? Should children in most important college learn addition and division by 2 before being exposed to concrete amounts or sharing's problems? How to add algebraic equations? Again through concretes problems or could it be better to first gain familiarity with them? Is it feasible for pupils to understand the concept of function through the study of one particular example?

None of those questions have an obvious answer. Whoever believes the answer to be evident should transfer the dilemma to other problems whose natures are not really different. Should we study the architecture of an bridge only through its function of crossing a river? Can we understand the beauty of a painting through the brushes of the painter? Regine Douady conclusions are extremely mindful: "From our experience, we can draw the next hypothesis; so long as 'enough' notions are unveiled through the tool-object pedagogy, others can be straight provided by the teacher or the reading of any textbook. An important pedagogical problem is about the options criterion, the business and the articulation of the notions in line with the way they are introduced. Concerning this, we are not making any assumption, but giving examples of realization. " (Douady, R, 1984, translation mine).

This essay focused on the didactical part of the coaching process explaining some of the difficulties in coaching mathematics. Most of the properties and laws and regulations of mathematics, even though they seem clear and clear have been developed over decades (because none are obvious) and all carry implicit challenges that the tutor is often unaware of. The tool/thing dialectic is important to consider as it irrigates every mathematics pedagogy; from the abstract teaching of the sixties through the new math reform to the constructivist methods. However there is way out of the tool/object dialectic and since a bottom line I would

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