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The Semantic Approach in Teaching and Learning Mathematics in Secondary Schools
M. Rodionovα & Z. Dedovetsσ
A clear understanding of the structure and meaning of language is a necessary condition for the effective teaching of mathematics, both at an abstract level and also to equip students with the skills associated with solving problem situations in everyday life. Work on the formation of the ability to solve problems begins with the first days of schooling. Solutions to simple problems usually present no difficulty. But many students have progressively greater difficulty in solving more complex problems. This is particularly noticeable at times of transition between classes. One cause of this is that struggling students do not have sufficient ability to analyze the text of a given problem, and thus to correctly identify both the distinction and the interrelation between the known and the unknown, which is the basis for choosing the way to solve it. In this article, the authors demonstrate the importance of semantic analysis, highlighting the levels of understanding of any given problem, the stages of semantic analysis of a mathematical text, and establishing the relationship between levels and stages. Particular reference is given to geometric and algebraic problems.
Keywords: mathematical text, word problem, semantic analysis, secondary students, mathematics.
Author α: Department of Mathematics and Computer Science Teaching, Penza State University, Penza, Russia.
σ : Department of School of Education, The University of West Indies, Trinidad and Tobago.
In solving various problems in mathematics lessons, students not only acquire mathematical knowledge, but also develop wider thinking skills in the general process of reasoning, analysis, synthesis, comparison, generalization, etc. Any solution of a particular problem must begin with a deep and comprehensive analysis. The purpose of semantic analysis is to understand the content of a text.
It involves a sequence of specific actions. 1) Students must identify and comprehend words, terms and concepts used both in real life and in mathematics-for example sequential grammatical constructions ("if ... then", "after ...", etc.) and words with quantitative characteristics ("everyone", "any", "some", "all", "almost everything", etc). 2) They must be able to apply such language to the given task or problem, using, discriminately, only the textual information essential for the problem solving process. 3) They must then be able to highlight the key purpose of the task, identifying the object or the value to be found, counted, measured or drawn.
Proper semantic analysis will enable students to discern what is required from the text of the task, and to establish the connections between the data and the as yet unknown. To be of use, each mathematical task should give the student “food” for intensive mental activity. The task should not be obvious, but should be have some challenges, requiring effort to overcome them. Simple direct problems (where the choice of action is too clearly determined either by the very situation of the problem or by signal words) make only negligible contribution to the development of the ability to analyze [1, 12]. In such situations, the process of analysis takes place quickly and unconsciously. This can have a harmful effect in the future when the students face more complex tasks in which structured analysis becomes necessary.
THE SEMANTIC CHARACTERISTIC OF MATHEMATICAL TEXT
One of the most important conditions for increasing students motivation to learn mathematics is their understanding of the meaning of the mathematical content. There are a number of requirements for authors of textbooks and the teachers who use them [4,5].
Essential concepts, and their interrelationships, which together determine the sequence of reasoning, actions and operations should be highlighted and explained. With regard to school courses on mathematics, this can be bedeviled by the fact that words can have different meanings. For example, the meaning of the same arithmetic expression can be associated in one case with its result, in the other with the calculation process, in the third with the features of its structure. The reason for this situation is the existence of certain differences between the language of teaching mathematics and the actual pure mathematical language. The language of instruction uses a number of terms and sentences that are not precise enough to be considered strictly mathematical. Key examples are: expression, theorem, proof, equation, and equality. These terms are used by teachers as “working language” rather than as designating certain specific and discrete mathematical objects.
Such terms do not have a strictly logical and unchanging definition, which makes it more difficult to elucidate their meaning. These difficulties are exacerbated by the phenomenon of "polysemy", where one of a number of ostensibly similar terms arises as a development or reinterpretation of another. This can disorient students. For example, it is possible to refer to “equation” in the generic sense and also as the “equation” of a figure in a plane and space; to “polygon” both as a one-dimensional and as a two-dimensional image; and to “angle” as a geometric figure and as an “angle” of rotation. Students need to understand both that there can be such differences and also how they apply to specific situations and problems. Then there will be minimal danger of demotivation [6,7,19].
Any explanation of the mathematical context which gives a correct perception of meaning must address both the solution to a specific problem and its general and future applicability. This transferability is based on the idea of substitution. If in the identity, the variable is replaced by an expression, then the resulting equality will again be the identity. For example, the known formula of the difference of two squares can be used to simplify a trigonometric expression of the form: cosх – sinх).
Practice shows that students often approach assimilable algebraic constructions as relatively isolated phenomena and, accordingly, do not show any tendency to associate them with known algebraic identities. The reason for this situation is the didactic mistake of the teacher in reinforcing an automaton like approach to the application of algebraic language. This can be prevented by careful verbalization and schematization of algebraic expressions, which makes it possible in the future to update the corresponding set of interrelations (for example, instead a - b = (a - b)(a + b), use - = (-)(+) where and may be symbol constructions.
In the process of child development, a child's thinking changes from concrete to abstract, passing through the stages of development corresponding to visual-practical, visual- figurative and verbal-logical thinking. This sequence determines the steps for studying mathematical concepts and theorems: from the tangible to the figurative to the verbal/symbolic form. Thus, the multi-contextual awareness of the meaning of a mathematical text is provided by the availability of a consistent and interchangeable translation of the content from the "language of everyday" to visual-geometric, and then to verbal-symbolic [2,3].
To teach mathematics properly, it is essential to use stimulating questions. Much of the mathematical material contained in school textbooks is descriptively flawed and imbalanced. For example, in many textbooks there are no explanations why the study of planimetry concentrates on triangles and to a lesser extent on quadrangles, but not pentagons. Where is the explanation of the significance of studying combinations of polygons and circles (or polyhedra and rotation bodies)? Why in the textbooks are there descriptions of the law of sines or cosines, but not the law of tangents? In algebra, what are the reasons for not fully outlining the analogy between actions with numbers and actions with polynomials (division of polynomials is not considered)? Such limitations can adversely affect understanding and thus motivation.
SEMANTIC UNDERSTANDING OF THE TEXT
Clearly the effectiveness of the work of students using a particular mathematical text is directly related to their degree of understanding of the text. Psychological and educational literature differentiates three main levels of understanding [9, 12, 17].
4.1 The first level
This level is characterized by an uncritical perception of information. Students connect the meaning of the text with interesting facts, usually represented in the form of an image. They use false analogies to expand the scope of a particular rule. Another feature of this level of understanding is that the students have no desire to 1) identify the essential features of mathematical concepts; 2) consider the implementable limitations of the particular method of mathematical activity being applied; 3) analyze a variety of potentially applicable situations, 4) safeguard the original semantic meaning of the text when it is translated into mathematical language and vice versa. This can be shown by the high number of errors made by students when using mathematical terminology in oral responses.
4.2 The second level
At this level, students master the techniques of mathematical action and realize the limitations of their application. Here, students are interested in the origin of mathematical objects, the emergence of certain contradictions and in the spectrum of diverse relationships between them. When analyzing and solving a problem, students use different mathematical symbols [15,19]. There remains the difficulty of over generalized and insufficiently precise language when applied to various aspects of mathematical sections (arithmetic, algebra, analysis, synthetic and analytical geometry, vector calculus, etc.). Often one mathematical construction has several equivalent forms of descriptions. For example, the concept of "parallel straight lines" can be defined in terms of different methods considered in the school course of geometry: "in the coordinate language" (as lines given by equations y = kx + b1 and y = kx + b2); in " the vector language" (through the collinearity of the vectors corresponding lines); in " the language of geometric transformations" (as straight lines obtained by central symmetry, parallel translation, or homothety); in "the language of synthetic geometry" (as parallel lines formed with the transversal line equal interior and exterior angles), etc.
4.3 The third level
Students identify for themselves the possibility or impossibility of applying a corresponding mathematical rule or method in a particular situation and can assess the nature of the language in a given text (accurately - inaccurately, expediently - inexpediently, heuristically - algorithmically, economically – uneconomically and so on). Such an assessment implies recognition of a particular perspective - that is, of recognizing yourself as an active participant in the discussion. At this stage, the student is able to present information in the form that best reflects individual characteristics [4, 18].
This description of the various levels allows us to identify the main requirements to enable a transition from the lower to a higher level. In the transition from the first to the second level of understanding of the content of a mathematical text, the main requirement is an understanding of its meaning. The student’s overall internal cognitive perspective does not significantly change. The goal of the second stage is a deeper understanding of the content. This can take the form of a dialogue-either explicitly external with another classmate and or implicitly within the student. Through such dialogue the student assimilates the outcome of the interaction between various semantic positions, enriching and at the same time refining personal ideas about the content being studied.
V. METHODOLOGICAL APPROACHES FOR UNDERSTANDING OF MATHEMATICAL TEXT BY STUDENTS
A methodological approach involves the implementation of two intertwining directions: an immediate and active comprehension of the text as it is perceived and an independent production of a meaningful text. In the first case, the main means is the semantic analysis of a mathematical text consisting of three main stages [8,11,13].
5.1 The Anticipation stage
The student becomes absorbed in the text, deciphers scientific terminology, analyses the underlying meaning of key concepts and begins to discern not only immediate but also potential applicability.
5.2 The Introspection stage
Here the student concentrates on practicing self- analysis and acquiring self-control in comprehending the mathematical text. The main didactic means here are -writing a supporting summary; practical verification of the statements; clarifying and responding to questions asked by the teacher and the students; and the use of relevant examples and counterexamples.
5.3 The Retrospective stage
The purpose of this stage is to rethink the text and determine its further development and application. A key manifestation of this stage is an equal exchange of opinions between the participants of the dialogue. This helps to discover alternative meanings, show the ways of establishing new connections, and to reveal to the students the generalizable meaning of the material in the structure of the entire section, topic or course.
Table 1 shows the main components of the semantic analysis of a mathematical text in relation to the transition from the first to the second and from the second to the third levels of understanding of the content of this text. The process can be organic.
Components of semantic analysis
Transition from the first to the second level
Transition from the second to the third level
Result of analysis
Identifying the meaning of terms and phrases
Explanation of terms,
giving examples, demonstrating drawings,
Discernment of essential and nonessential attributes of concepts and patterns of related theorems
Awareness of the content of
concepts and the connections between conditions and conclusions of theorems
Identification of key moments in the text and its semantic microthemes
Division of the text into parts, logical identifying keywords
Systematization and structuring of material
and final conclusions
Determination of the applicability limits of the material
Recognizing situations that correspond to the content of the material
Bringing examples and counterexamples. Awareness of potential for error
Identifying the general
formulas and algorithms
Definition of the language paradigm of a text
Modeling problem situations
Translating the material into mathematical language whilst preserving the meaning of the original language construct
Describing the opportunities and expediency of presenting a text in terms of a particular language
Reflection on the content of the text
Discerning and explaining what is
clear / unclear and simple /complex.
of content of the text, including analysis of the author's position
Wide ranging and differentiated approach to
Profound understanding of the content of text
Inter and intrasubject communication with the teacher
Deepening development of perspectives
Locating the place and meaning of a text within the perspective of the whole topic, section or course
Any semantic analysis of school mathematics material must include syntax. So, the teacher should pay attention to all components of the text (tasks, concepts, theorems, rules and algorithms). During work on and with theorems, an analysis of the logical structure of the theorem has a unique linguistic flavour. It contains the totality and sequence of logical connections between the sentences represented in its formulation. The clarification of the logical structure of the mathematical statements considered in a school mathematics course is a key methodological requirement. It is connected, in particular, with revealing the truth of these statements when applying the known rules for their transformation. Therefore, one of the general approaches to the study of the theorem is work with its structure, which includes the following features:
- A statement of the theorem in the implicative form;
- The identification of the conditional statement, conclusion statement and explanatory part of the theory;
- The clarification of the nature of the relations between the conditions and conclusions of the theorem;
- The transformation of the theorem (the formulation of the converse, the inverse, and the contrapositive theorem or their cycles).
With regard to any motivational plan, the last point is the most interesting since it implies creating opportunities for students to participate in the construction of new theorems, which then acquire a personal and deeper meaning. Thus, for example, in considering the condition of a complex theorem (or a task) of the type: pp …pgg…g, students can "switch" statements from the condition to the conclusion of theorems (and vice versa), replacing them with negatives, and obtain the resulting cycles of theorems or tasks that are equivalent in their semantic meaning to the "mother theorem" . The created task cycle is solved either sequentially (using the solution of the original problem), or it is divided into variants for the purpose of differentiation. For example, of such a cycle.
Problem 1 (a).
The initial problem: if three lines intersect in pairs, then they either lie in the same plane or have a one common point.
New problems (b):
- If three lines intersect in pairs and do not have a common point, then they lie in the same plane.
- If three lines do not lie in the same plane and do not have a common point, then they do not intersect in pairs.
- If three lines intersect in pairs and do not lie in the same plane, then they have a common point.
- If three lines do not lie in the same plane then either intersect in pairs or have a common point.
- If three lines do not have a common point, then they either lie in the same plane not intersect in pairs.
The kinds of transformations of the formulation of the theorem can be extended, but in this case the resulting statement can be false.
The relationships between textual and mathematical languages can be quite complex. There can be an ambiguous correspondence between language constructs and the mathematical operations that model these constructions. For example, the word "more", depending on the context, can be translated into the mathematical language (+) and (-), or the symbol ‘>’. The text can also contain hidden meanings, especially quantifier words such as “all”, “everyone”, “any” or “some”. Often a problem as described within a textbook or as presented by a teacher does not make explicit the characteristics of objects and the relationships that connect them, because it is wrongly assumed that these are intuitively obvious to students . This often leads to misunderstanding by the students and is one of the main causes of a negative attitude to textual descriptions of mathematical constructions.
To overcome these difficulties, the teacher must undertake a rigorous semantic analysis of the problem. This includes the following components:
- Elucidation of the meaning of terms and language constructions used in the description of the task-adding or deleting words as necessary to ensure clarity and to avoid ambiguity or redundancy.
- Reformulation of the task. For example, creating a text problem using a mathematical model whilst preserving the basic meaning.
- Reframing mathematical facts and ideas into a word form. For example, with regard to the practice of reasoning, hypotheses construction and extrapolation.
- Making full use of the information contained within the task, including additional, related questions.
Consider the implementation of the first of these points.
Problem 2: A tourist plans to travel the 360km from city A to city B over 12 days. How much earlier can he arrive at city B if each day he travels 8km further than the day before ?
In the text of the problem, we can distinguish the following significant points for the semantic analysis:
- The word "way" is understood here as a fixed route of a certain length.
- "Plans" means he made a plan that assumes travelling a path length of 360 km in 12 days.
- "Earlier" corresponds to a subtraction operation (if the minuend is already specified in the condition of the problem, then the subtrahend should be found).
- "Further" corresponds to an addition operation (one of the addend is known, and the second one must be found).
- "Arrive" here carries an additional meaning of assumed outcome.
- "Over 12 days" carries an additional meaning, that the tourist passes a certain part of the way on each of the 12 days (doesn't stop and always goes in one direction).
- "Every day" has a quantifier value of "each" and means the distance traveled by the tourist on every day.
- "Plans" also means a plan for one day or 1/12 of a general plan.
Such an analysis enables the teacher to reformulate the problem without changing the meaning.
In practice, it is not always expedient to make a detailed semantic analysis of the text. But in all cases the teacher must pay rigorous attention to and root out ambiguous or hidden meaning.
The ability to distinguish the essential features of mathematical concepts and the interrelationships between them contribute to the development of new skills, such as the ability to analyze the mathematical text and to correctly present the task to others, either orally or in written form. The motivational tone of linguistic activity may change. Initially, the main motivational factor for the student is the clarification and understanding of the meaning of the text. But when conveying this to other students and the teacher the main motivation is to ensure they understand. This requires clarity and succinctness, the ability to convey purpose and content and key aspects of process.
The content of a school mathematics course should contain enough opportunities to involve students in discussion on how to translate (often implicit) language constructs into mathematical language and how to become aware of the applicability and limitations of specific mathematical methods .This discussion can be one to one but can be greatly enhanced in group work .
When studying the topic "Cone", as a initial for the further presentation of the material, the following problem can be chosen.
The height of a cone is 10 cm. Find the area of a cross section passing through the vertex of the cone and the chord of the base. The angle formed between the base and cross section is 300. The chord forms a sector of 600.
In dialogue together, the teacher and students are looking for ways to solve this problem. Having then found a solution, students are encouraged to continue one or more "unfinished" segments at the base of the cone until they intersect with the base circle (the student can have the triangle or the rectangle inscribed in the circle). Then the teacher can discuss the following questions:
- What figure was constructed? (Triangle or rectangle)
- What is the location the center of the circle relative to the obtained figure?
- Is it possible to define elements of this figure using the task data?
- What figure do we get if we connect those new points on the circle with the vertex of the cone? (Pyramid)
- What is the location of this pyramid relative to the cone? (The pyramid is inscribed in the cone)
- What are the common elements of a cone and a pyramid?
- Is it possible to draw a pyramid circumscribed about a cone? How can this be done?
- How to find the volume of a pyramid inscribed and circumscribed about a cone?
Then the students continue work in groups. Each of the groups is invited to create their own task using the combining of a cone and a pyramid. They should use one of the known methods of transforming the problem: replacing the requirement of the task with a new requirement; expansion of the drawing through additional construction of lines; a replacement of the condition of the problem by some new condition.
Such joint work often produces a more integrated and less fragmented understanding than can be obtained when a teacher uses the traditional approach to teaching.
Other methods of involving students in the linguistic aspects of mathematical problems could include being asked to individually or collectively prepare a brief report on the proof of theorems in a way different from that proposed in textbooks or creating entertaining and therefore stimulating problems which illustrate the possibility of applying a particular rule. A menu of prepared tasks can be offered to the attention of the class and discussed together with the teacher.
Another option is to demonstrate how a problem can be solved in different ways – not simultaneously, but as students get to know different mathematical methods. The list of tasks is placed in advance on a special stand or in any other place accessible to schoolchildren. As a method is mastered, students can use it to solve selected problems. Once all the methods of solution have been applied, a concluding systematizing lesson is organized. The prepared solutions are presented for discussion. During the discussion, students are asked to evaluate tasks by their complexity, universality, correctness, creativity and other parameters.
This can take the form of a yearlong project. The list of potential topics is determined by the teacher at the beginning of the school year. Students choose interesting topics for them which are then approved by the teacher. A plan is then drawn up, which includes relevant source material. Regular consultations are held during the year on the content of the issues being developed. The final stage is a student conference. Such an approach is highly motivating for students–both because of the element of ownership from personal selection and also from the individual learning journey required by their research.
The motivational effect of linguistic activity in understanding and producing mathematical text primarily depends on the optimal interaction between the esoteric subject language and students’ “native general” language used when working with this text or in live communication with the teacher or with classmates during lessons. This includes paying proper attention to the general laws of language and the particular features of each students’ language repertoire. The teaching of mathematics is less effective when it is solely conducted in words that carry specific mathematical information.
The degree of understanding of the problem and, consequently, the success of further actions is influenced by an understanding of its content and context. Teachers need to develop the following skills: careful listening to the nature of the task; a similar careful attention to the potential and situational meaning and intonation of words, sentences and punctuation; consequential clear presentation of the task and the methodologies for solution and the imaginative use of various strategies and questions to widen and deepen students’ understanding.
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