Modern math teaching methodology offers various possibilities for solving the problem of involving students in independent and research work, it develops their problem solving skills and develops their creative thinking processes and skills. One of those possibilities is in the area of scientific framework. The foundation of a scientific framework is the principle of science and scientific research methods. The article describes science in various segments of math teaching starting with the nature of math to mathematical tasks as an important method in shaping the system of basic mathematical knowledge, abilities and habits in students. In the end, some drawbacks in math teaching are mentioned which occur due to the inappropriate treatment of science in the teaching process.
Math teaching today primarily takes place within a professional framework. However, teaching math is a complex and demanding process. Even though being professional is a condition for its success, it is not sufficient. The complexity is successfully resolved by relating math to other sciences. That way we get a process which has to take place harmoniously within several frameworks. The main frameworks are language frameworks, professional frameworks, methodology frameworks, scientific frameworks, pedagogical frameworks and psychological frameworks.
As it is not easy to achieve harmony, occasional slips and weaknesses occur in math teaching which significantly influence the quality of math education. That reflects negatively on the aims of modern math teaching which emphasizes involvement of students in independent and research work, developing skills for problem solving and the development of creative thinking and creative skills. Modern math teaching methodology offers various possibilities for solving the above mentioned problem. A teacher can find many possibilities within the scientific frameworks. The foundation of scientific frameworks is the science principle and scientific research methods. These concepts often cause a dilemma. What does a scientific approach mean in math teaching? The aim of this article is to describe that meaning and to give a few postulates and issues which arise in scientific frameworks of math teaching. A math teacher does not have to be a scientist in order to appropriately and correctly apply the science principle and research methods in math teaching.
Didactic principles are the founding ideas and guidelines based on which teaching takes place. The basic characteristic of each principle is contained in the name of the principle itself which math teachers mostly understand. The same applies for the science principle. Nevertheless, the principle should be described in detail. The science principle in math teaching consists of the appropriate harmony of teaching content and teaching methods on the one hand and the demands and regularities of math as a science on the other hand. That means that a math teacher should introduce students to those facts and form in their thought processes those mathematical occurrences which are scientifically founded today. Math teaching has to be such to enable further broadening and enrichment of content and a natural continuation of math education at a higher level. It is evident that from the description the principle of science makes a connection between math as a teaching subject and math as a science.
In the process of learning and becoming involved with the law of nature, scientists apply special methods – scientific research methods. Basic methods of scientific thinking and research are: analyses and synthesis, analogy, abstraction, and concretization, generalization and specialization, induction, and deduction. The work of a math teacher in a classroom differs in many respects from the work of a math scientist, but there are also these common characteristics:
In the process of learning the scientist applies the mentioned methods since they are necessary for obtaining new statements, their proof and their link with already known facts and theories. The shortest overview of some mathematical theory has four steps:
A) Stating basic concepts
B) Axiom formulation
C) Introduction of new concepts
D) Deriving and proving a theorem.
In other words, some scientific math area is a formation of axioms, basic concepts, derived concepts and theorems. In the teaching process, a math teacher helps students to discover and learn new mathematical truths. That knowledge can be obtained in various ways and the bases of all those methods are also concepts and theorems.
From the comparison mentioned we can easily conclude that scientific methods are important for modern math teaching. That is why they are the subject of research in modern math teaching methodology. Through the selection of appropriate problems and through the application of that method a creative teacher can prepare students for work which is very similar to research work, work of a scientist. Plenty of math teaching content can undergo such application thus meeting the science principle in its extent. What does our teaching practice show in that respect? During the lesson, the math teacher often says: “the analysis shows”, “let’s have a look at some concrete examples”, “analogous it is proven”, “this set of facts induce the conclusion”, “the result of these observations is a generalization”, “through specialization we get the formula”, “mathematical concepts are abstract” etc. Do the students understand these words? How do we check their understanding? Knowledge of the procedures mentioned is often implied and therefore lack an explanation. That is not good.
Students should gradually and appropriately be taught how to analyze, synthesize, abstract, induce, deduce, generalize, specialize, observe analogies, regardless of whether they will be seriously involved in math at a later stage. As opposed to the usual acquisition of content, this is a higher level of mathematical education. Mathematical way of thinking is a valuable gain of mathematical education, applicable in many other activities. The words gradual and appropriate are emphasized. If scientific procedures are appropriately and correctly applied, with a necessary feeling for the difficulty of math content and mathematical way of thinking, taking into consideration mathematical abilities of each student, it can be expected that math teaching will be successful. On the contrary, students will have significant difficulties in acquiring the teaching content and with time they can get the wrong impression that math is a more difficult subject than it actually is. Sadly, math books, and consequently the teaching process do not pay sufficient attention to the regularities of the application of scientific procedures. In teaching some math content it can be established that they are wrong from that point of view. The science principle is therefore neglected.
Students’ failures in math and the inadequate knowledge which is displays upon the completion of their education are for the majority part a consequence of the fact that teaching is mostly done at a lower level, where acquisition of content is overemphasized, while the higher level is neglected. The reason for this neglect lies in the fact that for higher level math teaching one needs more demanding scientific methods based on teaching which is heuristic and problem solving. On the other hand, the need for (appropriate) use of scientific methods in math teaching can be explained with the following facts:
Developing math is a concrete and inductive science, and math itself is an abstract and deductive science.
What is teaching math in that respect? Teaching math in primary school is also mostly concrete and inductive. Math teachers arrive at abstract postulations, generalizations by observing concrete objects and concrete examples and through inductive conclusions. This method is familiar and appropriate for students of that age. The inductive procedure is made up of a chain of inductive steps which lead to the understanding of the general. We begin with concrete objects and special cases, inductive conclusions are sequenced by analogy, and the observed facts are generalized. We observe a tight link between induction with concretization, specialization, analogy and generalization. The advantage of applying induction: implementation of the easier to more difficult principle, simpler to complex, studying new abstract concepts and phrases through observation and assessment, guiding students to new concepts, expression of new theorems, etc. The inductive approach is important in the development of a student’s thought process which on the other hand is necessary for acquiring a lot of content in school math. Among such content are various rules, regularities, formulas, theorems, especially if they are not strictly derived or proven. The opposite of induction is deduction. The deductive process of thinking and proving, takes place after induction, at a higher level of math teaching and math education.
Concept is a form of thought which reflect important characteristics of the objects studied. The process of formulating a concept is a gradual process. We can roughly describe the process in the following way: The initial and most simple step of being aware of the concept is observation and introduction to concrete objects and their concrete characteristics related to the concept and sensory awareness – observation. The second step is observing something general and common to elements in the observed group of objects – having an idea about the concept. The third step is pointing out the important characteristic of such objects – formulation and acquisition of the concept. It is not difficult to recognize some important scientific procedures in the described process: analysis, synthesis, abstraction and generalization. That means that any concept, including mathematical concepts, after careful analysis develop through abstracting characteristics of objects which exist in nature and through generalization. In that way mathematical concepts, although abstract concepts, reflect some characteristics of the real world and in that way contribute to their awareness.
According to that, in teaching mathematical concepts, the teacher realizes the science principle if the process of formulating concepts is appropriately implemented (observation, the idea about the concept, formulating the concept) and if he adheres to the rules which must satisfy the definition of a concept (appropriateness, content minimum, conciseness, naturalists, applicability, and contemporariness). At first glance it can seem that the need for content minimum in the definition is rather rigorous, even when it can easily be accomplished in teaching. That is not the case. A demand has its methodological explanation. Redundant definitions on the one hand burden the student’s memory, and on the other hand cause confusion in differentiating definitions and theorems. The critical place for working on a concept is the transition to that level where the abstraction procedure begins, since the transfer from concrete to abstract is rather difficult for some students.
What a theorem is we know. A theorem is a mathematical judgment whose truth is established by proof. A theorem is one of the most important mathematical concepts and its analysis demands special attention of every math teacher. Appropriate teaching of that concept enables faster development of mathematical thinking of a student and better understanding of math itself. In teaching a theorem the teacher realizes the science principle if he teaches his students to appropriately and precisely formulate a theorem, clearly differentiate assumptions from a theorem statement, formulate a theorem twist, formulate an opposite statement, and if he achieves understanding of the methodology in proving a theorem. Indirect theorem proofs, especially forms such as proof of contraposition and contradiction (reductio ad absurdum) create great difficulties for students.
The question posed here is: should a student who will not deal with mathematics in everyday life at a later stage in life, or for whom math will not be of essential importance, know and understand these theorems? The answer can be portended from the following irrefutable truth: learning how to prove means learning how to judge (reason), and that is one of the basic tasks in teaching math. Every person should know how to judge (reason) in life. How else can two different statements be compared, or extract from several statements those that are true, check the correctness of a suspicious proof, disprove someone’s opinion, come to the appropriate conclusion about something, etc.? Yes, every student should learn how to prove. That is why education is not complete if a student throughout schooling has not encountered and understood proof for several standard mathematical theorems.
Teaching how to prove presents a great challenge for a math teacher, since it obviously is neither simple nor easy. Especially since a teacher must keep in mind an important fact:
Although math is a deductive science, school math is not developed at any teaching level as a strictly deductive system, but remains within the framework model. This especially applies for math teaching in primary school since it is inductive for the majority part. Many theorems are taught without proof. A critical part for carrying out generalizations through inductive sequences of concrete cases is the transfer to the level where the abstraction procedure begins, since the transfer from concrete to abstract is even at this point quite difficult for some students.
Contemporary math teaching presupposes different knowledge activities than traditional. Emphasis is given to the development of the ability to work independently with a creative approach to math, and on developing conditions for successful application of acquired mathematical knowledge and abilities. Students’ independent work on acquiring knowledge of math is achieved largely through the possibility of appropriately choosing and using teaching tasks. In that way tasks become an important means in forming students systems for basic mathematical knowledge, abilities and habits and aid to the development of their mathematical skills and creative thinking. A task is a complex mathematical object and its composition is not always easy to analyze. However, in a broader sense we can isolate five of its basic constituents: conditions, aim, theoretical basis, solution, overview.
For the topic discussed, the most important constituent is the last one overview. It offers possibilities of testing new ideas and further directions of students’ thoughts. Particular directing can be accomplished by using some of these questions:
Can the manner for solving the problem be made simpler? Can the problem be solved in another way? Have we used the described procedure for finding a solution in some other problem? Can the problem be made simpler? Can the problem be generalized? Can you come up with a similar problem? What is the opposite
statement? Is the opposite statement valid?
The questions obviously point to analysis, synthesis, analogy, specialization and generalization. In seeking answers to those questions particular mathematical skills of students are developed and nourished, and their creativity is lifted to a higher level.
We have already mentioned that a math teacher need not be a scientist in order to appropriately and adequately apply the science principle and scientific methods in teaching. This occurs in math teaching without much interference. Solving a math problem implies some research and development. That is why the
teacher has to create the spirit of curiosity in his students, the inclination for independent mental work and to show them ways to new discoveries. A creative math teacher using creative teaching methods has great chances to develop in his students creative characteristics.
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