Notes: Language of Mathematics | Mathematics & Pedagogy Paper 2 for CTET & TET Exams - CTET & State TET PDF Download

In this chapter, we will explore the language of Mathematics, including its symbols, vocabulary, algorithms, and methods for representing word problems as equations, and applying suitable algorithms to solve them. An analysis of previous years' CTET and state TET exams indicates that typically 1 to 2 questions are asked from this chapter each year.

Mathematics as a Language

Language of Explanation

In the classroom, teachers use ordinary language to explain mathematics—concepts, formulae, operations, procedures, and propositions. Every mathematical concept can be explained to children through ordinary language.

Language of Problem Solving

A primary aim of learning mathematics is to develop the ability to convert real-life problems into mathematical problems, solve them using known techniques, and interpret the results as meaningful solutions to real-life problems. Children develop this ability through exposure to appropriate word problems, where language plays a crucial role.

Mathematics as a Language

Mathematics is itself a language with its own symbols, words, and rules of syntax. It is based on a consistent set of assumptions and built up according to the rules of logic. The understanding and application of such logic, necessary for developing mathematical thought, depend on the level of development of ordinary language.

For example, children need to develop the ability to use conjunctions such as 'and', 'but', 'therefore', and 'or' before they are ready for mathematically logical statements such as "Every square is a rectangle, but every rectangle is not a square."

Mathematics and ordinary language also share common words, but with precise mathematical meanings, such as difference, add, multiply, and power.

Objectives of Language of Mathematics

• Explain the role of ordinary language in mathematics learning.
• Describe the difficulties children face when dealing with word problems and develop strategies to help them overcome these difficulties.
• Define what the language of mathematics is.
• Suggest ways to teach the language of mathematics effectively to help children.
• Evaluate teaching strategies in the context of teaching mathematics language.

While discussing the different aspects of the interface between mathematics and language, we will mainly focus on examples involving numbers, but this relationship applies to all areas of mathematics. As you proceed through the subsequent sections, observe how often this occurs. Such observations will help you internalize the objectives of this unit.

Learning the Language of Mathematics

Mathematics is a language composed of concepts, terminology, symbols, algorithms, and syntax unique to it. Children can only acquire this language by using it—speaking, writing, and listening to it. Throughout earlier units, we emphasized the importance of engaging children in mathematical conversations, encouraging them to discuss the activities they are doing, which helps build their mathematical language and thought.

Unfortunately, many teachers do not have the type of interactive engagement with their students that we have suggested. For instance, in Class 1, about 30 hours are allocated in the curriculum for teaching the concept of numbers and numerals from 1 to 9. How is this typically taught? The teacher picks up various objects like a bag, an umbrella, or a pen, and calls each one 'one'. She then writes its numeral, '1', on the blackboard, which the students copy. This is done in 2 or 3 minutes, and the teacher believes the concept of 'one' has been taught. Similarly, she introduces other number names and symbols, expecting the children to absorb this new spoken and written language. She is often surprised that children may be confused by this sudden introduction of terminology.

Children somehow manage to cope with such teaching methods with the help of adults around them, although many children, especially those from rural areas, lack this support. Consequently, children end up with misconceptions about mathematical concepts, processes, and skills. An example is their misunderstanding of algorithms for arithmetic operations.

Mathematical Symbols
To Understand the means of mathematics we should know about mathematical symbols and make connections between text and symbols. There are several signs and symbols used while teaching and learning mathematical concepts. Some of the notable terms are mentioned below:

Equal Sign (=)
Ohm Sign (Ω)
Not Equal to (≠)
Addition (+)

Example:

when we write 3+2 which means to add two to three. the pattern of speaking the number statement is different from its symbolic representation.

The inability to understand the basis of an algorithm often results from an inadequate understanding of why numerals are written as they are. This leads to errors like the following:

• When Jamuna was asked to add 20 and 1 using the algorithm, she wrote:
  20
1
30
+
But when she did the sum orally, she got 21.
• When 14-year-old Ajay was asked to subtract 47 from 312, he wrote:
  3 1 2
0 4 7
2 6 5
1 5
-
When asked why he changed 4 to 5, he said, "Because 2 is less than 7. So, I take one away. Then, I have to put one back below." When questioned further about why he added it to the bottom, he defensively replied, "But this isn’t wrong! If I add 265 and 47, I get 312."

These examples illustrate that while many children know what hundreds, tens, and ones are and what adding or subtracting a number means, they struggle to relate this understanding to written algorithms. They do not comprehend the steps and procedures suggested in the algorithms.

Children also encounter difficulties with various symbols, such as:

• Distinguishing between expressions like 32 and 3/2, or 2x and x^2.
• Understanding multiple ways of expressing '=' in words, such as "8 - 2 = 6" and "6 + 2 = 8."
• Recognizing that a single mathematical fact can be presented in various ways, for example, "8 - 2 = 6" can also be interpreted as "6 + 2 = 8." Children find this commutativity confusing.

As children progress, they need to move from simple sentences like "5 + 3 = 8" to more complex expressions like "5 + 3 = 3 + 5." This requires them to view equations not just as sentences but as relationships between numbers.

These difficulties have been gathered from interactions with children, indicating that the language in which mathematical questions are stated plays a crucial role in how children interpret and solve problems. Therefore, it is essential to present problems and solutions in different ways to help children develop a deeper understanding of mathematical language.

Merits of Mathematical Language

• Mathematical language is well defined, useful, and clear.
• It draws numerical inferences based on given information and data.
• It helps in developing a scientific attitude among children.
• Mathematical language is not only useful for different branches of science but also contributes to their progress and organization.
• It is highly compact and focused.
• Mathematical language plays a vital role in students' success in numeracy.

Use of Mathematical Language in Acquiring Knowledge

• Mathematics deals with abstract concepts and has created a unique world of its own with a language consisting of symbols.
• Mathematics and language help us understand the universe through interactions.
• Mathematical language is more accurate than any other language when dealing with facts.
• Proficiency in this language can be acquired through long and carefully supervised experience in using it, especially in situations involving arguments and proofs.

The linguistic theory can be used to analyze the mathematical texts in order to promote the interest of learners and comprehend the concepts of mathematics. This article provides valuable insights into the language with specific reference to mathematics as an area of study.

Relationship Between Language & Mathematics

• The language has a significant relationship with various academic genres. Language is a systematic and scientific process to exchange thoughts, emotions and symbols. There are various components to describe the linguistic forms of mathematics like Syntax, Grammar, Vocabulary, Discourse and Meaning. There is a difference between Mathematical Vocabulary and everyday vocabulary. Children often get confused between these meanings. Usage of the same word but with different meanings within mathematics and outside mathematics. Example:- Face, shape, place, limits, root etc. such examples indicate that the Mathematics vocabulary is different from Everyday vocabulary.
• The grammar of mathematics is stated in the forms of independent clauses such as greater than, lesser than or equal to. The tradition of sentence construction is followed to write from left to right. Being a part of the mathematical discourse, formal and informal styles are used. They are preferred to describe in the ‘first-person plural form.
• It is observed that the students tend to use the active voice more than the passive voice. Here, one should note that poor listening leads to poor understanding. Hence, the native speakers learn mathematics in their native language more effectively than that in any other target language. For example, if your mother tongue is English and you learn mathematics in English then it will be a comparatively easy task. However, many times students’ native language is Hindi or Spanish, when they learn mathematics in English, it becomes difficult for them to understand the subject matter and explain it. These implications work as an important tool in the mathematical structure.
• The logical conjunctions like AND, and IF-Then precede quantifiers like FOR ALL. It is observed that both existential and universal quantifiers are used in the place of each other which leads to confusion. The use of ‘any’ is worth mentioning in this regard. For example, Can anyone solve this mathematical problem? (existential) or anyone can solve it (universal quantifier). Hence, it is better to use these quantifiers accurately.

Reading Mathematics

• Some experts assert that reading skills matter while teaching and learning mathematics. It is so because it has a lot to do with the interpretation of the subject matter. ‘The transactional theory of reading’ can work well to connect the dots. It is effective for skill-gap analysis as well. The students should be trained on using the existing knowledge to link with new concepts based on strong logic and reasoning abilities. They can develop three types of connections: Text-to-self, Text-to-text and Text-to-World. All of these are possible through model exercises and appropriate guidance.
• Mathematics is learned best when the learners are proactive and regular interactions take place between the knowledge provider and knowledge gainer. The term ‘Literacy Club’ is developed to provide an experiential learning platform and simplify the content which can meet the predefined requirements from time to time. The role of language is more visible when it comes to motivating the target learners and engaging them for better performances.
• The timely interactions and discussions will be appreciated by them and create an effective support system strengthening rapport in the long term. It will also be a powerful tool to provide concrete feedback coined with identifying the areas of improvement without hurting them. After all, the teacher can create multiple learning opportunities by making effective use of language to develop interest and increase the success ratio with teaching skills. Knowing the target learners better is always helpful in the effective planning and execution of lessons.

Community of Mathematics

• Mathematics can be comprehended easily through communication in the community of teachers and pupils by this process they share their ideas and knowledge among their peers. Mathematical Communication is a collection of resources for engaging students in writing and speaking about mathematics. It provides the student with an opportunity to justify their reasoning or formulate a question, leading to gained insights into their thinking and how to solve the problem in Mathematics.
• When you become a student of mathematics, you go through four phases imagination, engagement, alignment and nature. Here, the journey starts from filling the blanks to working on small short questions and finally solving complex problems with strong logical and reasoning skills. These four phases are the identical faces of the learner of mathematics. The phase of imagination makes your learning experience wider meaning ‘mathematics for day-to-day life.’ The learner moves to the next phase when he/she thinks of mathematics as a career. The last phase is reached when you develop expertise and contribute to the area of study as a researcher, teacher, academician, curriculum developer and so on

Conclusion

All of the above discussion shows that there is a significant relationship between language and mathematics. Though both areas of study are different from each other. However, they are the two sides of the same coin. Both complement each other. The design and development of technology (software, database management systems) are notable examples of how they work hand-in-hand to perform various functions in less time and with better results. Computation has transformed the lives of people by making a balance between the two.