CHAPTER TWO
TEACHING AND LEARNING OF MATHEMATICS
2.1 Objectives
By the end of this chapter, you should be able to:
i. Discuss the educational applications of the works of some mathematics psychologists such as: Piaget, Bruner, Gagne and Dienes.
ii. Discuss methods of teaching of concepts, principle, skill and proofs
iii. deductive, analytic and synthetic approaches in mathematics teaching
2.2 Mathematics Psychologists
2.2.1 Jean Piaget's Theory of Cognitive Development
According to the idea of cognitive development proposed by Jean Piaget, the intellect of children increases with age. Children progress through a series of cognitive stages that are impacted by both intrinsic abilities and environmental influences. According to Jean Piaget's theory of cognitive development, children go through four stages of intellectual development that correlate to the increasing complexity of their thinking with age. Each kid goes through the stages in the same order, despite the fact that biological maturation and interactions with the environment determine how children develop. Each stage of development entails a separate intellect, with each child's thinking at each level being qualitatively distinct from the others.
Stages of Cognitive Development
Although no phase may be skipped, individual differences in the rate of a child's growth imply that some children may never reach the later phases. Despite the fact that descriptions of the phases occasionally provide an indication of the age at which the average kid would reach each stage, Piaget did not claim that a certain stage was reached at a specific age.
The Sensory motor Stage (Birth to 2 Years)
Children gain knowledge of the world through their perceptions and actions. During this period, a number of cognitive talents arise. These include self-recognition, delayed imitation, object persistence, and representational play. By the end of this stage, children will have proved via play that they can substitute one object for another. Because people discovered that words could convey both things and feelings, language began to develop. The youngster can now store, retrieve, and categorize information about the outside world.
The Concrete Operational Stage (7 - 11 Years)
At this age, children begin to think rationally about genuine events. Children begin to comprehend the concept of conservation, which asserts that while the appearance of something may change, certain properties stay constant. Children are also capable of psychological turnarounds (e.g. picture a ball of plasticine returning to its original shape). During this stage, children grow less egocentric and begin to consider the feelings and thoughts of others.
The Formal Operational Stage (12 and above)
Physical and perceptual restrictions have no bearing on formal operational reasoning. During this period, teenagers may comprehend abstract concepts (e.g. no longer needing to think about slicing up cakes or sharing sweets to understand division and fractions). They may follow the structure of an argument without considering particular instances. Teenagers are capable of addressing hypothetical situations with several potential answers. For example, if asked, "What would happen if money disappeared in one hour? They may guess on several potential outcomes. Children of around 12 years old can follow the structure of a logical argument without regard to its substance. People gain the ability to think about abstract concepts and test hypotheses rationally throughout this stage. This stage saw the birth of scientific thought, with problem-solvers proposing abstract theories and hypotheses.
2.2.2 Jerome Seymour Bruner
Jerome Seymour Bruner (1 October 1915 – 5 June 2016) was a renowned American developmental and cognitive psychologist. He blends psychological research and classroom practice in his work. Bruner contends, according to Febrianti and Purwaningrum (2021), that mathematics is a science that can be studied through the ideas and structures that already exist in mathematics. Using these mathematical concepts and structures, a search may be conducted for the material's relationships. Bruner claimed that the most crucial aspect of learning is how individuals actively select, retain, and modify knowledge.
Bruner believed that there are three parallel processes involved in learning. The three stages are as follows: (1) acquiring new information, (2) altering information, and (3) assessing the relevance and validity of knowledge. Bruner identifies three stages of child development: enactive, iconic, and symbolic. In the enactive period, toddlers learn through physically interacting with objects. During the iconic stage, children's activities evolve and lead to more abstract concepts. At this level, there is a conceptual process of imagining an item, but no actual manipulation. In the third stage of symbolic development, the kid manipulates the symbol without reference to things. Bruner develops four learning theories: construction, notation, contrast and variation, and connectedness (Tampubolon, 2018).
Construction:
According to this notion, the best approach for a learner to begin learning mathematical concepts and principles is to create them. To create an idea or principle is to reduce the complexity of the concept or principle by analyzing its constituent pieces. Bruner contends that "any idea, topic, or body of information may be given in a way that is simple enough for any student to comprehend" (Bruner, 1996). According to Bruner, the concept of a concept is based on actions using concrete objects in the early phases of a student's learning. The consequences of this approach for mathematics education are that the logical presentation of new concepts is improper. This is plausible if considering Bruner's proposed indicator phases, enactive, iconic, and symbolic.
Notation:
According to the Notation theory, the initial construction is cognitively simplified and better understood by pupils if it is constructed according to a notation that corresponds to the degree of intellectual development of the learner. The impact of this theory on the teaching of mathematics is the use of developmentally appropriate notation for both ideas and principles. A idea's notation should at least indicate one concept and not another. The use of notation that is inconsistent with the degree of intellectual development of the learners will disturb their comprehension.
Contrast and variation
The theory of contradiction and variation implies that the process of learning from tangible to abstract mathematical concepts must be incorporated into the contradictions and variations (Alamian & Kazemi, 2020). Consequently, as students learn mathematical concepts, the examples must change so that their comprehension is enhanced. The application of this principle to mathematics education is that while introducing a concept, its opposite must be introduced. Moreover, examples and non-examples used to illustrate a concept or idea must differ.
Connectivity
According to the connectedness hypothesis, every mathematical notion, structure, and talent is related to other concepts, structures, and skills. Although the explanation of the idea or principle must be related to the preceding concept or principle, it need not be linked to notions that are too distant. The application of this idea to the teaching of mathematics is to illustrate the previous notion or principle prior to explaining the new concept or principle. In addition, the explanation or evidence of the notion or principle is provided. The notion or principle may be illustrated by both examples and non-examples.
Bruner established three cognitive representation phases.
The first level is enactive, which is the expression of knowledge through actions. This phase includes the encoding and storing of data. There is no internal representation of the objects involved in the direct handling of the objects. Learning should begin with direct manipulation of items. For example, in mathematics class, Bruner supported the use of algebra tiles, money, and other materials that might be controlled.
Iconic: Which is the visual summary of pictures that occur between the ages of one and six. This level comprises an internal visual representation of external objects as a mental picture or icon. After a student has had the chance to actually manipulate the items, they should be encouraged to create visual representations, such as a form or a diagram.
Here, words and other symbols are used to depict experiences symbolically. Beginning around the age of seven, information is stored in the form of a code or symbol, such as language. Each symbol has a definite relationship to the item it symbolizes, and a learner eventually comprehends the relationship between symbols and the things they stand for. For example, a student in mathematics learns that the plus symbol (+) indicates to add two numbers together while the negative sign ( - ) means to subtract.
Bruner asserts that children actively engage in learning at a level commensurate with their cognitive development. To enhance the learning experience, educators should optimize the style of presentation rather than the material being taught.
Applications of Brunner's Theory to Education
Bruner felt that by adjusting educational techniques to the cognitive functioning level of children, every topic that can be taught to adolescents and adults can be taught to children. Any topic may be taught to any child at any developmental level.
1. Bruner considered the kid as a minor authority. According to him, a youngster may comprehend the action of disciplines at nearly any age. According to him, the primary difference between a child's and an adult's cognitive processes is quantity, not kind or quality.
2. Every domain of knowledge may be represented as one of three motor, visual and symbolic systems. Teaching every specific topic to any specific individual necessitates one of the three approaches. Bruner stated that, wherever feasible, a teacher should deliver his or her instruction in all three practical, visual, and symbolic systems.
3. Bruner defends the guided discovery technique as opposed to the independent discovery method. Guided discovery is a manner in which learners are encouraged to grasp and realize the questions with the help of the teacher.
4. Encouraging pupils to grasp the subject's essence: According to Bruner, discovery is an interior concept and process. Bruner argued that this internal process requires a rearrangement of the thinking system. To do this, students should be encouraged to acquire a subject's or field of knowledge's principles.
5. The learning atmosphere should be devoid of worry and tension. The educational environment should be structured so that students may freely express their thoughts, consider a variety of subjects, and arrange their mental conceptions in order to enhance their cognitive ability.
Bruner had the following opinions regarding education and learning:
§ He felt that education should encourage the growth of problem-solving abilities via inquiry and discovery.
§ He felt that subject matter should be given in terms of the child's worldview; • Curriculum should be structured so that the mastery of one skill leads to the acquisition of a more advanced skill.
§ He also promoted teaching through the organization of concepts and learning through exploration.
§ Lastly, he felt that culture should affect the ideas that individuals use to arrange their perspectives of themselves, others, and the world in which they live.
2.2.3 Zoltan Dienes
Zoltan Dienes (1916-2014) was a world-renowned Hungarian mathematician and education psychologist who felt that mathematical structures could be effectively taught to children in the primary grades via the use of manipulatives, games, and tales. Dienes discovered that creating game rules that correspond to the rules found in mathematical systems capitalizes on children's innate propensity for game-based learning. Dienes also discovered that manipulatives were an efficient technique to explain complicated mathematical topics. He created Base 10 blocks (commonly referred to as Dienes blocks) to assist children understand the mathematical foundations in an appealing manner. Base ten blocks, also known as multi base arithmetic blocks (MAB) or Dienes, are mathematical manipulatives (wooden or plastic cubes, rods, and flats) used to teach pupils basic mathematical concepts such as addition, subtraction, number sense, place value, counting, and number bases.
Here are three reasons why Dienes regarded games to be an excellent educational tool.
1. Games enhance the enjoyment and motivation of learners.
Dienes emphasizes in his work that children do not need to reach a particular developmental stage in order to feel the joy of mathematics. The most important thing is that youngsters learn how to think.
Teaching mathematics with games helps engage students who find the subject intimidating and dull. Understanding mathematical patterns and relationships can be an exciting and inspiring experience for youngsters when they learn via games.
2. Games increase learners' problem-solving skills
Fun board games like Snakes and Ladders, Cards, Monopoly, and Scrabble hide mathematical structures and rules that even young children can use to develop higher-level problem-solving abilities like trial-and-error techniques, task simplification, pattern-finding, hypothesis formation and testing, reasoning, and proving and disproving. Games may also be a helpful evaluation tool. It is feasible to more accurately gauge the learners' present level of understanding by paying attention to the strategies they choose throughout the game.
3. Games facilitate the practice and reinforcement of mathematical abilities
Games not only aid in the development of learners' problem-solving skills, but also provide them opportunities to practice and reinforce their mathematical abilities. Learners flourish when they have a specific objective to achieve, even if the route to success is difficult. Children are allowed to consider all possible tactics, and if one fails, they are encouraged to try again. When children receive positive comments for their hard work, they obtain a sense of hope and the realization that they can learn and develop as they face new problems. Mathematical games combined with the use of manipulatives promote and deepen students' thinking without requiring tedious memorization.
2.3 Dienes' Six Learning Phases
Dienes reconstructed Piaget's well-known four-stage process of conceptual development into a six-stage process pertaining to the production of mathematical concepts.
1. Free Play: a requirement or beginning point for higher abstractions, involving the acquisition of concrete experiences about the world, its objects, and their relationships through "trial and error"
2. Rule-based Games: methodical finding of regularities, rule-invention, learning to play by the rules, making differences between the beginning state and the final state, the rules to be satisfied, and the conditions to be met.
3. Comparative Structuring: a discussion of the games, a comparison of the rules, and a search for commonalities among rule-based game structures, excluding specific elements. Searching for the "common core" of comparable games as their structural (mathematical) content and presenting dictionaries for shared characteristics.
4. Representation: diagrammatic, visual, or multimodal portrayal of the abstracted characteristics of the games, extracting the essence of the communalities and mapping the rules and regularities of the actual games to the representations.
5. Symbolization: analysis of the representation, study of the cleanable (not comparable) properties of the games as classes of regularities, verbal description of the extracted rules by introducing symbols for the "map" that represent abstract components, and verification of the outcomes of abstract rules in concrete games.
6. Formalization: discovery of relationships between the described and symbolically represented qualities of the representation, attaching methods for deducing further properties, finding of descriptions that imply other descriptions, and determination of the rules of deduction. Beginning the process of picking axioms, locating theorems, and developing proofs.
2.3.1 Inductive Method
The Inductive Method is founded on the induction principle. Induction is the process of establishing a universal truth by demonstrating that if it is true for a given case and a sufficient number of other situations, then it is true for all such circumstances. In this technique, an issue is initially answered based on the learner's prior knowledge, thinking, reasoning, and intuition. At this point, he is unaware of any formula, theory, or approach for addressing the problem. When learners are given with a sufficient number of comparable instances, facts, or objects, they attempt to draw a conclusion for each. Consequently, they arrive at a generalization or deduce a formula by a persuasive process of reasoning and solving several situations with comparable characteristics.
Consequently, it is a way for developing a formula using a sufficient number of concrete instances. Thus, the inductive technique of instruction guides us from the known to the unknown, the specific to the general or the example to the general rule, and from the concrete to the abstract. When a number of concrete situations have been comprehended, the student can try generalization.
Example 1: Square of an odd number is odd and square of an even number is even.
Solution:
Particular concept:
12 = 1 32 = 9 52 = 25………i
22 = 4 42 = 16 62 = 36 ……..ii
General concept:
From i and ii, we get
Square of an odd number is odd and Square of an even number is even.
Example 2:
Sum of two odd numbers is even
Solution:
Particular concept:
1+1=2
1+3=4
1+5=6
3+5=8
General concept:
In the above we conclude that sum of two odd numbers is even
Example 3 :
Law of indices am x an =a m+n
Solution:
We have to start with a2 x a3 = (a x a) x (a x a x a)
= a5
= a 2+3
a3 x a4 = (a x a x a) x (a x a x a x a)
= a7
= a 3+4
Therefore, am x an = (axax….m times)x(axa …n times)
am x an = a m+n
MERITS
§ It enhances self-confident.
§ It is a psychological method.
§ It is a meaningful learning
§ It is a scientific method.
§ It develops scientific attitude.
§ It develops the habit of intelligent hard work.
§ It helps in understanding because the student knows how a particular formula has been framed.
§ Since it is a logical method so it suits teaching of mathematics.
§ It is a natural method of making discoveries, majority of discoveries have been made inductively.
§ It does not burden the mind. Formula becomes easy to remember.
§ This method is found to be suitable in the beginning stages. All teaching in mathematics is conductive in the beginning.
Demerits
§ Because some complex and difficult formulas can't be constructed, the applicability of this approach to all topics is constrained.
§ It is a tedious and time-consuming process.
§ Its length is one.
§ Its scope of applicability is somewhat narrow.
§ Because the generalization drawn from a few focused examples may not hold true in all circumstances, inductive reasoning is not completely conclusive.
2.3.2 Deductive Method
In this method, we move from the general to the specific and from the abstract to the concrete. At first the guidelines are taught and then students are requested to use these principles to answer further problems. This method is mostly utilized in Algebra, Geometry, and Trigonometry since these subdisciplines of mathematics employ distinct relations, principles, and formulas. In this method, mathematical assumptions, postulates, and axioms are utilized.
Example 1:
Find a2 X a10 =?
Solution:
General: am X an = am+n
Particular: a2 X a10 = a2+10 = a12
Example 2:
Find (102)2 =?
Solution:
General: (a+b)2 =a2+b2+2ab
Particular: (100+2) 2 = 1002 + 22 + (2 x 100 x 2)
= 10000+4+400= 10404
Merits
§ It is short and time saving method.
§ It is suitable for all topics.
§ This method is useful for revision and drill work.
§ There is use of learner’s memory.
§ It is very simple method.
§ It helps all types of learners.
§ It provides sufficient practice in the application of various mathematical formulae and rules.
§ The speed and efficiency increase by the use of this method.
§ Probability in induction gets converted into certainty by this method.
Demerits
§ It is not easy to understand.
§ It taxes the pupil’s mind.
§ It does not impart any training is scientific method.
§ It is not suitable for beginners.
§ It encourages cramming.
§ It puts more emphasis on memory.
§ Students are only passive listeners.
§ It is not found quite suitable for the development of thinking, reasoning, and discovery.
References
Ernest, P. (1986) ‘Games: A Rationale for their Use in the Teaching of Mathematics in School’, Mathematics in School 15 (1), 2-5
Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson Educational Company.
Dienes, Z. P. (1963). An experimental study of mathematics-learning. London: Hutchinson.
Dienes, Z. P. (1973). Mathematics through the senses, games, dance and art. Windsor, UK: The National Foundation for Educational Research.
Alamian, V.& Kazemi M. M (2020). Investigating the Effect of Teaching Mathematics based on Bruner Theory on Eighth-Grade Male Students' Misconceptions in Equation Solving. Arch Pharma Pract 11(1):53-60.
Bruner, J. S. (1966). Toward a theory of instruction, Cambridge, Mass.: Belkapp Press.