Why is it important to study mathematical language and symbols? In this article, we will uncover the significance and benefits for academic and real-world applications.
A. Characteristics of Mathematical Language
B. Mathematical Expressions and Sentences
C. Conventions in the Mathematical Language
Let’s start!
Language is a system of communication that involves the use of words, signs, and symbols to express ideas, thoughts, and feelings. It is a way for people to convey meaning to one another within a community or society. It encompasses not only the words themselves and their pronunciation, but also the methods by which they are combined and organized to be understood by others.
In the context of mathematical language, it refers to the system of communication used to express mathematical ideas. Mathematical language is known for its precision and clarity, as it aims to convey complex mathematical concepts accurately. Similar to other languages, mathematical language has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, idioms, abbreviations, sentence structure, and paragraph structure.
One unique aspect of mathematical language is its use of representation. Mathematical concepts and relationships are often represented through symbols, diagrams, graphs, or mathematical notation. These representations aid in understanding and communicating mathematical ideas more effectively.
Furthermore, mathematical language heavily incorporates logic. Mathematical reasoning often relies on logical deductions and proofs, which are expressed and communicated using the language of mathematics. By using logical language and reasoning, mathematicians and learners are able to make precise and rigorous arguments about mathematical concepts and their applications.
The use of appropriate language is crucial in making mathematics understandable and comprehensible. Mathematical language consists of a combination of ordinary language, technical terms, and specific grammatical conventions that are unique to mathematical discussions. Additionally, mathematical language incorporates a specialized symbolic notation for expressing mathematical formulas and equations.
As mathematical concepts become more advanced in courses such as geometry, discrete mathematics, and abstract algebra, the focus shifts from basic equation solving to understanding the intricate relationships between complex concepts. The language of mathematics serves as a powerful tool to explain and express these interrelationships.
Just like any human language, mathematical language has grammatical structures that distinguish between nouns and verbs. In mathematics, these structures are used to differentiate between objects and the actions performed on or by those objects. Mathematical nouns encompass numbers, measurements, shapes, spaces, functions, patterns, data, and arrangements, while mathematical verbs represent the main actions involved in problem-solving and reasoning.
Based on the research by Kenney, Hancewicz, Heuer, Metsisto, and Tuttle (2005), there are four main actions attributed to problem-solving and reasoning in mathematics. These actions represent the processes one goes through to solve a problem.
1. Modeling and formulating. This action involves creating appropriate representations and relationships that turn the original problem into a mathematical form. It requires identifying relevant variables, parameters, and constraints and determining how they interact mathematically. This step helps to “mathematize” the problem and set the foundation for further analysis.
Creating appropriate representations and relationships to mathematize the original problem refers to the process of converting a real-world problem or situation into a mathematical model or equation that can be analyzed and solved using mathematical techniques.
Mathematizing a problem involves identifying the relevant variables, parameters, and relationships between them, and expressing them mathematically. This allows for a systematic and quantitative analysis of the problem, enabling the use of mathematical tools and methods to find solutions or make predictions.
For example, let’s say you have a business problem where you want to optimize the production of a certain product given certain constraints like limited resources and maximum capacity. To mathematize this problem, you would define variables such as the quantity of the product produced, the variables representing the resources (such as labor, raw materials, and machines), and the constraints (such as the maximum capacity of production).
Lets have another illustration>>>
Problem: A car rental agency charges a base fee of ₱2,000 per day plus an additional ₱15 per mile driven. How can we mathematize this problem to determine the total cost of renting a car?
Representation: Let’s define the variables:
- Base fee (B): ₱2,000
- Miles driven (M)
- Cost per mile (C): ₱15
Mathematical Relationship: The total cost (T) can be represented as: T = B + (C * M)
By using this mathematical relationship, we can easily calculate the total cost based on the number of miles driven.
You would then create mathematical relationships between these variables, such as an equation representing the production process and constraints on the resources. By mathematizing the problem, you can use optimization techniques or linear programming methods to find the optimal production plan that maximizes profit or minimizes costs.
2. Transforming and manipulating. Once the problem has been mathematized, this action involves changing the mathematical form of the problem to equivalent forms that represent potential solutions. It may include simplifying expressions, applying algebraic or geometric transformations, or using mathematical operations to rearrange equations. These transformations help uncover relationships and properties that can lead to solutions.
Example Problem: Solve the equation 2x + 3 = 7.
- To transform and manipulate this problem, we want to isolate the variable x on one side of the equation: 2x + 3 – 3 = 7 – 3
- 2x = 4
- Divide both sides of the equation by 2:
- x = 2
In this example, we used algebraic operations to rearrange the equation and simplify it to find the value of x.
3. Inferring. After manipulating the mathematical form of the problem, this action involves applying derived results to the original problem situation. It requires interpreting the mathematical findings in the context of the problem and making logical inferences based on them. This step allows for deeper understanding and insight into the problem and often leads to generalizations and broader applications.
Example Problem: A car travels at a constant speed of 60 miles per hour. How far will it travel in 3 hours?
To infer from the mathematical manipulation, we can use the equation d = rt, where d represents the distance, r represents the rate (speed), and t represents the time. From the problem, we know that the rate is 60 miles per hour and the time is 3 hours. Plugging these values into the formula, we get:
d = 60 * 3 d = 180 miles
Based on the inferred result, we can conclude that the car will travel 180 miles in 3 hours at a constant speed of 60 miles per hour.
4. Communicating. The final action involves reporting what has been learned about the problem to a specific audience. It requires effectively conveying mathematical ideas, reasoning, and results in a clear and coherent manner. Communication may involve written or oral explanations, visual representations, or the use of mathematical language and notation. Sharing knowledge and findings allows others to benefit from the problem-solving process.
Problem: Determine the area of a rectangle with a length of 10 units and a width of 5 units.
Solution: Using the formula for the area of a rectangle A =l × w, where l is the length and w is the width, you multiply the length (10 units) by the width (5 units) to get an area of 50 square units.
Communicating:
- Written/Oral Explanation: “To find the area of the rectangle, I used the formula which multiplies the length by the width. So, I multiplied the given length of 10 units with the width of 5 units. This gave me an area of 50 square units for the rectangle.”
- Visual Representation: Draw the rectangle, label the length and width, and perhaps shade it in to show its area. Next to or inside the rectangle, you can write 10 × 5 = 5,010 × 5= 50.
- Mathematical Language/Notation: “Given a rectangle with l = 10units and w = 5units, the area A can be found using A = l × w. Plugging in the given values, we find A = 10 × 5 = 50 square units.”
- Sharing: Presenting your method and solution during a math class presentation, discussing it in a group study session, or illustrating it on an educational platform or tutorial.
By detailing and communicating the method in this manner, you ensure that the audience (whether it be classmates, teachers, or others) can understand not just the final answer, but also the logical steps and reasoning that led to that conclusion.
Together, these four actions provide a systematic approach to problem-solving, from mathematizing the problem to manipulating mathematical forms, inferring meaningful conclusions, and effectively communicating the results.
Undoubtedly, mathematics can be considered a language. Proficiency in this language is acquired through extensive and carefully guided experiences in its usage and application.
Objectives:
Once learners have completed Chapter II on Mathematical Language and Symbols, they will be capable of:
- Engaging in discussions about the language, symbols, and conventions utilized in mathematics.
- Describing the essence of mathematics as a language.
- Executing operations on mathematical expressions accurately.
- Recognizing the practical value of mathematics as a language.
A. Characteristics of Mathematical Language
Mathematics revolves around ideas such as relationships, quantities, processes, measurements, and reasoning. However, the language used in mathematics differs from ordinary speech in three significant ways, as detailed by Jamison (2000).
- Precision: Mathematical language is precise and specific. It uses accurate terminology and symbols to ensure clear communication of mathematical concepts. This precision helps avoid misunderstandings and allows for consistent and accurate mathematical reasoning. Example: When calculating the circumference of a circle, we use the precise value of π (pi) as 3.14159…, rather than using an approximation like 3.14. By using a more precise value, we can obtain more accurate results in our calculations.
- Conciseness: Mathematical language aims to convey ideas and concepts in a concise manner. It often uses symbols, formulas, and abbreviations to represent mathematical relationships and operations efficiently. This brevity helps convey complex concepts succinctly. In algebra, simplifying expressions is a technique that aims to make them more concise. For instance, if we have the expression 3x + 2x + 7x, we can simplify it by combining the like terms with the same variable: 3x + 2x + 7x = (3 + 2 + 7)x = 12x The simplified expression, 12x, is more concise and easier to work with compared to the original expression.
- Powerful: The ability of mathematical concepts, techniques, or tools to solve complex problems effectively. It involves efficiency, versatility, and the potential to handle challenging mathematical situations. Example: Matrices are a powerful mathematical tool that allows the manipulation of complex systems of linear equations. They can be used to efficiently represent and solve simultaneous equations in various fields, such as computer graphics and quantum mechanics.
Ordinary speech can be confusing because words can have different meanings or hidden intentions. This makes it harder for students to practice expressing themselves clearly and precisely. On the other hand, mathematical notation, the way math is written, is helpful because it can convey a lot of information in a small space and focus on what’s important. However, this compactness can also make it harder for learners because one symbol can represent many different ideas.
When students learn math, it’s not just about knowing how to solve problems. They also need to learn how to explain their thinking and understand why they’re doing certain steps. This helps them become better problem solvers. Understanding the concepts and using symbols as a way to explain ideas is more important than just memorizing formulas.
Vocabulary understanding is also important in math and other subjects. Knowing the right words and terms helps students understand the concepts better and communicate their ideas effectively. Teaching and learning the language of math is crucial for developing mathematical skills.
Once students understand how things are said in math, they can better understand what is being communicated. This understanding allows them to grasp the reasons behind mathematical concepts and solve problems more effectively.
Many people unfortunately view mathematics as a subject filled with confusing rules and strange symbols that seem unrelated to everyday speech and writing. This perception may stem from the fact that in basic math courses such as arithmetic, algebra, and calculus, the focus is often on learning specific techniques for working with numbers, symbols, and equations. While these techniques are important, they are not the main purpose of mathematics.
In more advanced math courses like geometry, discrete mathematics, and abstract algebra, the emphasis shifts towards understanding the relationships between different mathematical ideas, rather than just manipulating symbols and solving equations. These courses explore deeper concepts and connections, helping us see that math is about more than just following rules.
As children grow and develop their understanding, language and vocabulary play a crucial role in connecting their natural sense of numbers and order to learning more complex mathematical concepts. Mathematics itself is seen as a language, with its own precise, concise, and powerful way of expressing ideas. It is a clear and objective means of communication, helping us convey complex thoughts and concepts in a precise manner.
B. Mathematical Expressions and Sentences
One of the goals of learning math is for students to be able to confidently talk about it. It’s important for students to be able to explain how they solve math problems and understand the consequences if they use the wrong process. Math words, expressions, and sentences help students communicate their thoughts and ideas clearly. Using precise math terms helps students have a better understanding and appreciation of math.
It’s important to recognize the different parts that make up a math expression, as well as the basic vocabulary used when discussing math expressions. In math, there are four main operations: addition, subtraction, multiplication, and division. These operations are used to combine or manipulate numbers. Here is a table to show some words or phrases associated with each operation:
Table 2. Operational Terms and Symbols
| ADDITION | SUBTRACTION | MULTIPLICATION | DIVISION | |
| Operational Term | Plus The sum of, Increased by Total Added to | Minus, Less, The difference of, Decreased by Subtracted from | Multiply, times, product of | Divided by, The quotient of, Over Per |
| Symbol | + | − | ×, *, ( ) | ÷, / |
Given that x is commonly used as a variable in algebra, it is infrequently employed to represent multiplication. Instead, alternative symbols such as a dot, parentheses, or an asterisk can be utilized to indicate multiplication. For instance, the expression 2x + 8 could be alternatively written as 2(x) + 8 or 2*x + 8.
Multivariate mathematical expressions have more than one variables
Examples of a multivariate expression are:
- 10xy + 7x-11
- 56abc
- 7y/4x
Mathematical Expressions
Mathematical Expressions consist of terms, which are separated from each other by either plus (+) or minus (-) signs.
A term is a component or part of an expression that is separated by either addition (+) or subtraction (-) signs. It can be a combination of numbers, variables (represented by letters), and coefficients. For example, in the expression 2x + 3y – 5, there are three terms: 2x, 3y, and -5. Each term represents a specific part of the expression and can be simplified or evaluated separately.
In algebra, variables are used to represent unknown numbers. These variables can be represented by letters. An algebraic expression is a combination of numbers and variables. The variable, also known as the literal coefficient, represents the unknown quantity in the expression.
For example, in the expression 10x + 11, “x” is the variable and represents an unknown number. The numerical coefficient of “x” is the number 10, which means 10 multiplied by the value of “x”. The number 11 in the expression is a constant, which means it’s a fixed number and doesn’t change.
So, the expression 10x + 11 can be interpreted as “10 times the value of x, plus 11”.
Mathematical Sentence
A mathematical sentence combines two mathematical expressions using a comparison operator. These expressions can include numbers, variables, or a combination of both. The comparison operators include equal, not equal, greater than, greater than or equal to, less than, and less than or equal to. The symbols used to convey equality or inequality are also known as relation symbols, as they indicate how two expressions are related to each other.
Equation
An equation is a mathematical expression that uses the equal sign (=) to indicate that two quantities or expressions are equal. The equation is divided into two parts, which are referred to as its members. The members of an equation are typically separated by the equal sign.
The left-hand side (LHS) and the right-hand side (RHS) are the two members of an equation. The LHS is the expression on the left side of the equal sign, and the RHS is the expression on the right side of the equal sign.
For example, the equation “2x + 3 = 7” has two members: “2x + 3” on the left side and “7” on the right side. The equal sign shows that the two members are equal to each other.
On the other hand, a mathematical expression that uses inequality signs, such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤), is called an inequality. An inequality compares the values of two expressions, indicating that one expression is larger, smaller, or equal to the other.
For instance, the inequality “3x + 2 < 10” compares the expression “3x + 2” to “10” and states that “3x + 2” is less than “10”.
Inequalities are used to represent a range of possible values or conditions. They allow us to express relationships and comparisons between quantities or expressions that are not necessarily equal.
Both equations and inequalities are fundamental concepts in mathematics and are used in various areas, such as algebra, calculus, and real-world problem-solving. They provide a powerful framework for representing mathematical relationships and making statements about quantities.
Examples of equation:
- 4x+3 = 19
- 7y-4= 45
- 8+1 = m
- (x y z) 5 = 1
- 55 – q = 25
Examples of inequality:
- 12x-4 < 3y
- 20 > 15.5
- 88 < X
- 15 > 10xyz
- a + b + c < 888
Open Sentence
Open Sentence is a sentence or statement that contains one or more variables. These variables represent unknown values or quantities. The truth value of an open sentence cannot be determined without assigning specific values to the variables. Essentially, an open sentence is not “closed” or finalized because its truthfulness depends on what values are assigned to the variables.
For example, the statement “x + 5 = 10” is an open sentence because it contains the variable “x”. The truth of this statement depends on the specific value assigned to “x”. If we assign “x” the value of 5, then the statement becomes true. However, if we assign “x” the value of 6, the statement is false.
Closed Sentence
A Closed Sentence is a mathematical statement in which all variables have been assigned specific values, and the truth value can be determined as either true or false. In a closed sentence, all the variables have been replaced with actual numbers.
For example, the statement “2 + 3 = 5” is a closed sentence because it contains no variables. We can immediately determine that the statement is true.
More examples of open sentence:
- 5xy < 6y
- 28b > 19.5
- 7 (m + n) = 100
- 5ab- c = 1
- x + y = 10
- 8 – 7 = v
- The obtuse angle is N degrees.
- 25 m = n
- a b c = 4
- 3x + 3y – 4z= 11
Examples of true closed sentence:
- 5 (x + y) = 5x +5y
- 18 (2) > 16.5
- 2 (m + n) = (m + n) + (m + n) + (m + n)
- 8c – c= 7c
- 7 is an odd number.
- 10 1/2 = 5
- 10 – 1 = 9
- 6 – 6 = 0
- The square root of 4 is 2.
- g + g + 200= 2g +200
Examples of false closed sentence:
- 9 is an even number.
- 3 + 3 = 8
- 11 – 2 = 7
- 7 – 7 = 1
- The square root of 4 is 1.
- 100 1/3 = 10
- X + 2x + 3x = 10x
- y0 = 4
- d +6d = 6d2
- (x y z)2 = 2xyz
C. Conventions in the Mathematical Language
Symbols play a crucial role in mathematical work, highlighting the need for precision when it comes to their usage. Variables serve as a specific type of mathematical symbol. To grasp the meaning of mathematical symbols, two factors should be considered: context and convention.
Context pertains to the specific subject matter being studied, and understanding the context is essential for comprehending mathematical symbols.
Convention, on the other hand, is a practice followed by mathematicians, engineers, and scientists, where each symbol carries a specific meaning.
The arrangement of numbers and symbols relative to one another influences their interpretations. The use of subscripts and superscripts is also a significant convention. Greek and Latin letters are employed as symbols for physical quantities and special functions in mathematics, engineering, science, and various other fields. Greek letters, for instance, are commonly used in calculations. For instance, the Greek letter “π” (pi) represents the number 3.14159. Letters such as “α” (alpha), “β” (beta), and “θ” (theta) are often used to denote angles. The Greek capital letter “Σ” (sigma) is frequently used to indicate summation of multiple numbers.
Next >>> The Four Basic Concepts of Mathematics
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