Mathematics, often perceived as an abstract realm of ideas, undergoes a concretization process through various mathematical structures and applications. Abstract concepts find tangible representation in the real world through the application of mathematical principles. Take, for instance, the abstract notion of a mathematical function. In real-world scenarios, functions can represent relationships between variables, such as the trajectory of a projectile or the growth of a population.
It has already indicated that mathematics is an abstract science based on logic and arguments. Through ultimately the learners are expected to appreciate this abstract nature .it could be possible only if they are mutually made concrete. This can be made possible by illustrations, common mediators in the form of familiar phenomena etc. That reduce the level of abstraction.it is in the absence of such concretization that the subject appears to be difficult and uninteresting to most learners.
Manipulatives are materials that are manipulated by the learners to be able to come up with possible understanding of the abstract concept on mathematics. Concreteness is the property which measures the degree of our relatedness to the object, how close we are to it, or if you will the quality of our relationship with object.
Geometry, another branch of mathematics, transforms abstract shapes and figures into concrete entities. The Pythagorean theorem, a classic example, elucidates the relationship between the sides of a right-angled triangle and finds practical applications in fields ranging from architecture to physics.
Moreover, algebraic equations, seemingly abstract expressions, find concrete solutions when applied to real-world problems. For instance, the quadratic formula can determine the roots of a quadratic equation, providing precise solutions to problems in physics, engineering, and finance. In calculus, abstract notions of rates of change and accumulation find expression in real-world scenarios.
Calculus is essential in understanding and predicting phenomena such as motion, population growth, and economic trends. Through the concretization of calculus, abstract mathematical ideas seamlessly integrate into fields as diverse as physics, biology, and economics.
In essence, the concretization of abstract ideas in mathematics bridges the gap between theoretical concepts and their practical manifestations, demonstrating the pervasive and essential nature of mathematical principles in diverse domains of human knowledge and experience.
In mathematics abstraction can be advantageous in the following ways
IMPROVISED AIDS
Today's schools are changing and the incorporation of digital advances into the classroom was inevitable and necessary. To prepare students for their future, education must utilize the most advanced tools and techniques available. As technology enters the classroom more and more each year, there are always new innovations being brought into the fold that can help teachers to get their points across better, and help students learn more efficiently. The benefits of interactive whiteboards in the classroom are so many that any learning institution can't afford to ignore.
It reveals deep connection between different areas of mathematics
Known results in one area can suggest conjectures in another related area
Techniques and methods from one area can be applied to prove results in other related areas
Patterns from one mathematical object can be generalized to other similar objects in the same class.
On the other hand, abstraction can also be disadvantageous in that highly abstract concepts can be difficult to learn
DIFFERENT TYPES OF LEARNING AIDS
A fact that to keep teaching interesting and make it effective we have to make use of certain material aids. The use of these materials aids makes the teaching effective, simple and interesting. The mathematics teacher we have been using text books, writing material. Geometric instruments and the black board since long as in used the equipment for mathematics classes. Mathematics is considered as a dry subject so to create interest in learning mathematics has been a constant problem for teacher. A teacher has an inherent desire that his teaching should be as effective as possible. The national policy of education 1986 and thus modified in 1992 has laid down great stress on the use of learning aids, especially improvised aids, to make teaching learning more effective and realistic.
Teaching aids or Learning aids
Teaching aids are the tools that teachers use them in the classroom such as flash cards, cassette and blackboard. A teaching aid is a tool used by teachers to help learners improve reading and other skills, illustrate or reinforce a skill, fact or idea, and relieve anxiety, fears or boredom.
Significance of learning aids
They may help in clear understanding of the learning material and for clarifying the abstract ideas by making them concrete.
They simulate pupil participation.
They are based on learning by doing.
Make teaching-learning process interesting.
Help in saving time and energy.
Help in creating a lasting impression in the mind of the learner.
Facilitate in the application of mathematics in various situations.
Different types of learning aids
There are so many abstractions in mathematics, which cannot be easily followed by the students. To make such abstract ideas rather concrete teacher takes the help of various teaching aids. These teaching aids are otherwise known as learning aids it can be classified into;
1. Graphical Display boards.
2. Graphical aids.
3. Three dimensional aids.
Graphical aids
Graphics are instructional materials that summaries significant information and ideas through some combination of drawings, words, symbols and pictures. Major graphic aids are graphs, diagrams and chart.
Graphs
A graph is a visual representation of numerical data. They are used to consolidate data, analysis, comparison and interpretation. There are many kinds of graphs
Line graphs- They are used to compare the magnitude of different measures of any given variable.
Bar graphs-The difference in their Heights could be visualized and conclusions could be drawn. Histogram is a special case of bar graph.
Sector or Pie graphs- They are used when a whole is divided into parts.
Diagrams (drawings)
Diagram is a condensed drawing consisting primarily of lines and symbols designed to represent the outline, interrelationship or key features of a process or idea.
Charts
A chart is combination of pictorial, graphical, numerical or normal materials designed to present a clear visual summary of an important process of set of relationships. For class room purpose we use two kinds of charts.
Outline and tabular charts
The organization of the content into an outline of key points and sub point, which a teacher may present systematically on the blackboard result in useful chart form. Tabular charts present information as tables, the points to be stressed being presented in rows and columns.
Flowcharts
Item such as tabulation and graphical representation of data can be subject of flow charts.
Graphical display boards
The stereotyped atmosphere of traditional class room, libraries and corridors of schools is recently giving way to attractive and educational surroundings with a variety of display materials.
1. Black boards
Black board or chalkboard still continues to be the most effective and in dispensable visual aid for the classroom teacher. Without the use of blackboard, the teaching of mathematics is impossible. Blackboard is the second tongue of the mathematics teacher. The purpose of blackboard is: To write statement of problems and their solutions graph, to write terms, symbols, formula, to draw diagrams, figures, to plot etc.
2. Roll up chalk board
Rollup chalkboards are that which can be conveniently rolled up. They are made up of rexin cloth. The charts and pictures can be drawn on it well in advance and presents as when required.
3. Bulletin board
The bulletin board, as the name implies, is a board for exhibiting bulletins and news items of absorbing interest to the students.
4. Pegboard
Pegboards can be used for compiling flat and even three-dimensional materials, the latter being supported by specially made hooks and clips.
5.Flannel board
Flannel board have fiber surface to which similar materials will adhere. This can be prepared by stretching a suitable size of flannel cloth on a heavy cardboard. Only flat, light weight teaching material can be fixed on flannel boards.
6.Magnetic board
A magnetic display will be useful to show the relative movements of element of a visual. A sheet of iron together with a piece of magnet can be used for the magnetic board.
7.Plastigraph board
Any smooth polished surface like a glass plate or rigid sheet may serve as a plastigraph board. Plastic sheet may serve as a plastigraph board. Plastic sheets of different colours cut to required shapes and patterns may serve as display material.
THREE DIMENSIONAL AIDS
Models, objects, specimen, mock ups, realia, diorama and puppets are major 3-D aids used in classroom.
Models
Models are concrete representation of objects; their size being adjusted to make if handy and conveniently observable. These are mostly three dimensional or sectional and can clearly exhibit the structure of functions of the original. Most often models prove to be more effective than the real ones.
Objects
Objects may be defined as the real things collected from their natural setting. They are the most useful and most effective means of providing direct experiences to the pupils. Examples of real objects which are generally used in mathematics teaching are beads, balls, coins, sticks, weights, balance, measuring tapes, scale etc.
Specimen
A typical object or an object, which has been remove for convenient observation.
Mock-ups
When direct first-hand experience is either impractical or impossible, mock-ups can be used. Mock-up is an imitation of the real process.
Teaching aids are important because they create a visual and interactive experience for the students. It assists students in learning. These aids consist of video, audio and hands-on tools to help involve the students and enhance the learning experience. Teaching aids are an integral component in any classroom. The many benefits of teaching aids include helping learners improve reading comprehension skills, illustrating or reinforcing a skill or concept.
Improvised learning aids refers to aid prepared with simple materials costing very little by involving teacher and student.
Improvised aids are developed by using locally available materials.
Indian schools cannot afford to have complex and sophisticated equipment and learning aids.
Improvisation is to be developed as a habit, by linking with everyday dealings in life.
Examples
MAB-place value cards and physical items such as matchsticks, buttons, ping pong balls, corks.
Food such as pizza, cake, pie etc.
Clock dials, stop watches and sand timers.
Protractors, set squares and compasses.
2D and 3D shapes- tangrams, pattern blocks and attribute blocks.
Pretend money-notes and coins Dice, cards and counters and spinners.
Trundle wheel and meter ruler.
Beakers, funnels, cylinders, scales and weights.
Advantages of Improvised Aids
It provides the joy of creation and sense of satisfaction.
It is a right direction towards inculcation of creative mathematical thinking.
Development of the dignity of labour among youngsters.
It is the way of spending leisure with pleasure by children.
Improvisation certainly leads to saving of money.
It satisfies 3 Hs-head, hand and heart.
Promotes self-confidence and generation of new ideas among students.
MODERN LEARNING GADGETS
It's back to-school time, as students change gears from the lazy days of summer and shift back to hitting the books. Today's classrooms are equipped with the latest technology to enhance instruction as technology enters the classroom more and more each year, there are always new innovations being brought into the fold that can help teachers to get their points across better, and help students learn more efficiently. Here are some favorite student and teacher gadgets for use in day-to-day learning.
INTERACTIVE WHITE BOARDS
SMART boards are popular with both teachers and students. Teachers can project presentations on to the smart boards and they can write, touch and interact with their content. Apart from bringing convenience and flexibility in education, use of interactive white board increases teacher-student engagement and boost performance in the long run. Adopting this technology can be great for any school or learning institution because it streamlines areas that were a challenge before.
Main Benefits of interactive whiteboards in the classroom Are as Follows:
Makes it Easier for the Teacher to Structure Lessons
Makes the Learning process More Fun
Increases the level of Engagement between teachers and Students
Makes Review or Revision of Lessons More Convenient
Allows Students with various Disabilities to Learn Better
Brings Convenience and Flexibility in Learning
Saves on Teaching costs
DIGITAL TEXTBOOKS
Digital textbooks are becoming more crucial as some schools are cutting back on expensive physical textbooks. Some schools are going beyond adopting a new format for education texts. Initially digital textbooks were simply the old textbooks in digital form. Nowadays, the digital versions are starting to look more like online databases or programs than their paper predecessors.
POWERPOINT
PowerPoint has become very popular because it's easy to learn and widely available. It provides the ability to equip your presentations with different types of media including images, sounds, animations and much more. This enhances the students’ abilities to retain what they are being taught, especially those who are visual learners. Teachers can focus on the class and interacting with the students instead of writing on a board, because the text and the entire presentation is already there in the form of a PowerPoint file.
AUDIO ENHANCEMENTS
Audio enhancements refer to a suite of tools from high quality speakers to wireless microphones A full system includes easy to wear microphones transmitting and instructor's voice to speakers in every corner of a classroom. The speakers can also be connected to a multimedia presentation or a person joining the class through a camera in a remote location. With quality speakers distributed around a classroom, each student is able to hear and absorb the information being shared. These systems are getting more advanced each day. Directional sound speakers enable students in different areas of a classroom to engage in activities without disturbing others. Students who have a hearing problem can tune their area speakers to provide higher amplification levels. And lectures for discussions can easily be recorded for later review or distribution. Laptops, tablets, phones, and more the presence of computers or tablets in the classroom is fairly common in developed countries. Students have used computers to take notes in class for more portable and have the same capability.
PROJECTORS
Projectors on their surface are a simple technology that has been around for decades. Historically used to project lecture notes or multimedia presentations, these classroom cornerstones can do so much more.
SMART BOARDS
A step beyond interactive projectors is the wide range of "smart boards" being utilized in classrooms. Some smart boards operate just like a whiteboard, allowing a person to write on them with real or digital markers. Smart boards then surpass their simple whiteboard ancestors by recording what is written or turning a digital marker line into a tangible visual element.
INSTRUCTIONAL SUPPORT ITEMS
TEXTBOOKS
Mathematics textbooks play a crucial role in education, providing a structured and comprehensive resource for learning mathematical concepts. Here are some reasons highlighting the importance of mathematics textbooks:
1. Systematic Learning: Textbooks are designed to present mathematical concepts in a systematic and organized manner. They typically follow a logical progression, introducing fundamental concepts before building up to more complex topics. This structure helps students understand the foundational principles before moving on to advanced material.
2. Reference Material: Textbooks serve as valuable reference materials that students can use to revisit concepts, formulas, and problem-solving strategies. Having a textbook allows students to independently study and reinforce their understanding of the material.
3. Practice Problems: Mathematics textbooks often include a variety of exercises and practice problems. These help students apply theoretical knowledge to real-world situations, reinforce their understanding, and develop problem-solving skills. Practice problems also provide opportunities for self-assessment and improvement.
4. Visual Representation: Many textbooks include diagrams, graphs, and illustrations to aid in visualizing mathematical concepts. Visual representation can enhance understanding, especially for abstract or complex ideas, by providing a concrete and graphical context.
5. Standardization: Textbooks contribute to standardization in education by offering a common resource for students following a particular curriculum. This ensures that students across different schools or regions have access to the same set of concepts and materials, promoting consistency in education.
6. Homework and Assignments: Teachers often assign homework and exercises from textbooks to reinforce classroom learning. Textbooks provide a structured source of problems for students to practice, allowing teachers to gauge understanding and provide targeted feedback.
7. Resource for Teachers: Mathematics textbooks are valuable tools for teachers. They provide a structured curriculum, lesson plans, and guidance on how to teach specific concepts. Teachers can use textbooks to plan their lessons, ensuring that they cover the necessary material in a logical sequence.
8. Preparation for Exams: Textbooks are crucial in preparing students for examinations. They typically include summaries, key concepts, and practice tests that help students review and assess their readiness for exams.
9. Self-Study: Mathematics textbooks enable students to engage in self-directed learning. Whether it's reviewing a concept, learning ahead of the class, or seeking additional challenges, textbooks empower students to take control of their learning journey.
10. Real-world Applications: Many mathematics textbooks include real-world examples and applications of mathematical concepts. This helps students understand the practical relevance of what they are learning, making the subject more engaging and applicable beyond the classroom.
HAND BOOK
Hand book or teacher text is a resource material for the teachers and it helps them to make teaching more effective. It contains list of objectives of teaching the course material, conceptual development of each topic, learning activities, teaching aids that can be used etc. Thus, a hand book provides a guide line for the teacher.
The merits of hand book are
It is a Guide for pedagogic and methodological works of teachers
Provides theoretical knowledge about course material
Locate the sources of relevant knowledge related to the content.
Make teacher more efficient and self-confident
It helps teacher for the selection of appropriate teaching methods and materials
WORK BOOK
Mathematics workbooks are supplemental resources designed to accompany textbooks or serve as standalone practice materials. They typically contain a variety of exercises, problems, and activities to reinforce and apply mathematical concepts taught in the classroom. It is an extension of text book.
MATHEMATICS LIBRARY
A mathematical library is a repository of pre-written mathematical functions and routines designed to facilitate complex computations and mathematical operations. These libraries provide a standardized set of functions, such as trigonometric, algebraic, and statistical operations, enhancing efficiency in software development. Popular mathematical libraries, like NumPy for Python or Math.NET for .NET languages, offer a diverse range of tools for numerical analysis and scientific computing. By leveraging these libraries, developers can streamline code development, reduce errors, and harness optimized algorithms, ultimately accelerating the implementation of mathematical concepts in various applications, from engineering simulations to data analysis.
MATHEMATICS CLUB
Mathematics clubs in schools serve as dynamic platforms fostering a love for numbers and problem-solving. These clubs create a collaborative space where students engage in mathematical discussions, puzzles, and activities. Through interactive sessions and competitions, members develop critical thinking skills and deepen their understanding of mathematical concepts. The clubs often organize math-related events, promoting a positive and enjoyable learning environment. By encouraging teamwork and exploration of mathematical challenges, these clubs inspire a lasting enthusiasm for the subject. Students benefit not only academically but also socially, as they build friendships centered around a shared passion for mathematics.
MATHEMATICS FAIR
A mathematics fair is an event that showcases the practical applications, creativity, and real-world relevance of mathematical concepts. Students, individually or in groups, create projects or exhibits that demonstrate various mathematical principles and problem-solving skills. These fairs provide a platform for students to explore mathematical ideas beyond the classroom, encouraging creativity and critical thinking. Exhibits often include topics like geometry, probability, algebra, and statistics, presented in interactive and visually engaging ways. Mathematics fairs not only enhance students' understanding of math but also promote communication skills as they explain their projects to peers, teachers, and visitors. Overall, these fairs celebrate the versatility and excitement of mathematics.
