Symbolic mathematics

SUMMARY: Symbols, equations, and images are all used to teach mathematical concepts and to convey mathematical information in society.

Representations are at the forefront of the focus standards of the National Council of Teachers of Mathematics to improve mathematics teaching and learning. Representations allow students to see and experience mathematics from different perspectives. The role of multiple representations in promoting students’ conceptual understanding of mathematics has long been emphasized by researchers. Thus, representations are among the essential parts of mathematics lessons. Further, in the twenty-first century, even people who had very little exposure to mathematics in school will encounter various mathematical representations in their daily lives. Familiarity with mathematical representations or representational literacy has become an essential skill. Many mathematical concepts are defined in terms of representations. A function may be represented by a Taylor series of infinite terms, which is named after Brook Taylor. There is also an entire branch of mathematics called “representation theory” that expresses algebraic structures using linear transformations.

Representations

Mathematics has its own native beauty and inspirational aesthetic to represent the physical world and the world of intellect. One of the strengths of mathematics is its resources to seek for new solutions and explore frameworks to answer problems related to the real world. To achieve this goal, mathematical representations in society should be explored and important ideas of modern mathematics should be communicated properly. Representations in mathematics can be described as constructs that symbolize or correspond to real-world mathematical entities, features, or connections. Gerald Goldin broadly defined representations as any configuration of characters, images, or concrete objects that can symbolize or represent something else. Representations take various forms, such as informal representations used in preschool settings or more formal representations used in mathematics classrooms or by mathematicians. For example, children represent groups of five with their hand or, even further, they develop proportional thinking as they relate five fingers to one hand and 10 fingers to two hands. More formally, mathematics students or mathematicians use mathematical equations, for example, to represent curves or relationships among financial variables.

Internal and External Representations

Representations can be both internal and external in nature and can be created by forming individual representations, such as letters, numbers, words, real-life objects, images, or mental configurations. Internal representations are mental images or cognitive constructs of individuals that relate to external representations or to experiences in the external world. James Kaput referred to internal representations as mental structures and defined them as instruments that are used to organize and manage the flow of an individual’s experience. Internal representation systems exist within the mind of an individual and consist of constructs to assist in describing the processes of human learning and problem solving in mathematics. Internal representations of mathematical concepts can take various forms, such as individual visualization of mathematics concepts, idiosyncratic notation systems, or attitudes toward mathematics.

External representations, on the other hand, include all external entities or symbols. External representations provide a medium to communicate mathematical ideas, concepts, or constructs. Richard Lesh defined external representations as the embodiment of internal systems of thought. Lesh also referred to external representations as mathematical representations that are simplifications of external systems. Learners use external representations, such as marks on paper, sounds, or graphics on a computer screen, to organize the creation and elaboration of their own mental structures. Unlike internal representation systems, external representation systems can be easily shared with and seen by others.

Multiple Representations in Mathematics Education

In mathematics education, there has been a shift from classic to nontraditional teaching and learning practices with multiple representations, where educators use various representations to effectively present information. Multiple representations refer to different kinds of representations that present the same mathematical ideas from different perspectives or representations that present different aspects of the same mathematical concept. For example, teaching fractions concepts using multiple representations may involve presenting fractions in real-life contexts such as partitioning a pizza or a pie, allowing students to explore equivalent fractions using kinesthetic or virtual manipulatives, or providing students with pictorial representations of fraction operations in addition to formal mathematical representations. Teaching and learning with various kinds of representations provide students with hands-on and minds-on experiences and support a better understanding of mathematical concepts. Also, using multiple representations in mathematics education can help to alter the focus from a computational or procedural understanding to a more comprehensive understanding of mathematics using logical reasoning, generalization, abstraction, and formal proof. A substantial amount of research has demonstrated the effectiveness of multiple representations in enhancing students’ conceptual understanding of mathematical concepts.

The notion of multiple representations in mathematics education commonly refers to external representations. However, one of the essential goals of mathematics education is to develop internal representation systems that interact well with external representation systems. James Kaput identified five interacting types of internal and external representations: (1) mental representations—internal representation—that learners construct by reflecting on their experiences; (2) computer representations that model mental representations through computer programs, which allow for arrangement and manipulation of information; (3) explanatory representations consisting of models or analogies that create the interaction between mental and computer representations; (4) mathematical representations, where one mathematical structure is represented by another mathematical structure; and (5) symbolic representations, such as formal mathematical notations.

To understand James Kaput’s taxonomy of internal and external representations, consider the different types of representations related to the concept of “slope.” When learning about positive slopes, a student might internally imagine a hill, which constitutes an internal (or mental) representation. This mental representation can be replicated on a computer screen. The student can create a unique model that incorporates the mental representation through a computer representation. If the model is viable, then it can be an explanatory representation for the concept of “slope.” The student, then, can sketch a similar mathematical graph of the hill and can name the steepness of the hill with the mathematical notation, “slope.” This graphical representation of slope can, then, provide support to represent the slope in a symbolic form as a rate of change (y=mx+b, where slope is represented with m and indicates the ratio of change on the y-axis to the change on the x-axis). As portrayed in this example, internal and external representations are not separate. Rather, they are intrinsically connected, and they interact continuously. Furthermore, a concept like slope is itself a type of alternative representation. In calculus, a curve is represented by the changing nature of its tangent vector, where the solution to the first derivative at a particular point is the slope of the tangent vector.

Translational Skills Among Different Modes of Representations

In addition to the importance of the effective interactions between internal and external representations in the acquisition and use of mathematical knowledge, it is essential that students develop fluency among different external representations. Richard Lesh enumerated multiple modes through which representations can be constructed: manipulatives, pictures, real-life context, verbal symbols, and written symbols. To demonstrate deep understanding of mathematics, students need to represent their mathematical ideas with different modes of representations and smoothly translate within and between those modes. For example, in algebra, students should be able to make the connection between graphical and algebraic or symbolic representations of equations. Similarly, students need to link what they learn using concrete or virtual manipulatives to both pictorial representations and abstract symbols. For instance, students who initially learn fraction operations using concrete or virtual manipulatives should be able to relate this knowledge when they later on learn fraction operations using symbolic and more abstract mathematical representations. Connecting different modes of representation simultaneously has been demonstrated to improve conceptual understanding as well as positive attitudes toward mathematics.

In mathematics education research, there is strong evidence that students can grasp the meaning of mathematical concepts by experiencing different mathematical representations and making connections and translations between these modes of representations. Using translational skills among different representational modes encourages students not to merely memorize theorems and facts but also to think analytically to reproduce and use them in real life problems or even in pure mathematical problems.

To deepen students’ understandings, teachers should provide students with multiple representations of a single mathematical concept and focus on students’ transition ability from one representation to another. Teachers need to be able to present one concept in multiple modes without relying on a single mode and provide students with appropriate transitions among these representations. Teachers should provide also students with ample opportunities to represent mathematical concepts in multiple ways and to connect these representations, thereby developing representational fluency. For example, asking a student to restate a problem in unique words, to draw diagrams to illustrate the concept, or to act out the problem are some ways to provide students with opportunities to translate among representations. If teachers fail to implement the transitioning among different representations, students will be less likely to see how different representations are related and will be more likely to develop misconceptions.

Multiple modes of representation can be used by teachers and students to enhance understanding of mathematics. Most research has shown that providing students with accurate representations improves student learning. However, different representational modes might have different impacts on student understanding. One mode might be more relevant or effective than another for teaching a specific concept. Or, some representational modes can be more appropriate at different developmental stages of the same concept. For example, research on teaching and learning of fractions has shown that students should be given the opportunity to develop mental representations of fractions using manipulatives before they are presented with symbolic representations. Thus, in addition to using multiple representations, choosing effective and appropriate presentations of information is crucial in teaching and learning. Representations that allow students to actively interact with the subject matter are more effective in student learning than representations that do not support students’ active involvement.

Despite the research support for development of higher order thinking skills afforded by different representational forms, little is understood about how students interact with multiple representations in various learning environments. Even though each representation provides similar information, the strain that each representation puts on students’ cognitive resources may differ. Not only do individual representations have different impacts on students’ conceptual understanding but integrating multiple representations may also result in interaction effects among different modes presented. Therefore, integration of multiple representations becomes an important consideration in the design of instructions. Educators should employ caution as they integrate different modes into instruction, because delivering redundant information with different modes might interfere with learning.

Mathematical Thinking and Representations in the Twenty-First Century

An increasing number of daily activities in the twenty-first century require familiarity with mathematical representations and mathematical thinking. Mathematical thinking, which is a crucial tool for every member of society, includes skills such as pattern recognition, generalization, abstraction, problem solving, proof, and analytical thinking. Most companies prefer employees who are equipped with mathematical literacy or general mathematical skills. However, many students either do not necessarily understand these qualifications or do not value them enough. It is important to emphasize that all humans use mathematical thinking tools in their every day lives and workplaces, with or without noticing they are doing so.

It is not very hard to realize the extent to which mathematical representations are integrated into mundane objects and activities. Consider the number of newspaper columns that provide their readers with different kinds of mathematical representations to explain current issues. Topics in such columns include sports, economics, advertisements, and weather reports. For example, the growth of players, the statistics and ranking of teams, and teams’ transfer budgets are represented in several representational modes, such as tabular data, textual information, visual representations, or graphical interpretation. Not only do sports fans need to understand the mathematical information provided readily to them but they also may need to use the mathematical information in problem solving situations, such as estimating the chances of their team’s victory. More surprisingly, when a rivalry game is present, the provided data get even more complicated to analyze the chances of each team.

Even though the use of mathematical representations and information in economic and weather columns in various modes is apparent, the ones used within advertisements or political columns may be overlooked. Understanding the mathematical information included in advertisements and deciding which product to buy requires effective use of mathematical thinking tools. In most advertisements, companies present several payment options with different price ranges instead of giving just one price for a product. In particular, mortgage plans to buy houses and installment plans to buy cars require serious analyses of options to choose the best for a given budget. In political columns, on the other hand, one would not be surprised to see percentages representing the proportion of the population that supports various political parties in a country or the votes of a poll. Such information is not only presented as tabular data, visual charts, or graphs, but also as textual information, which is another mode of mathematical representation.

Representations in Problem Solving

Problem solving is one of the essential tools for mathematical thinking. A person equipped with problem solving skills does not necessarily need to have the knowledge base for the solution to each problem encountered but needs to know how to approach problems, locate and access information from different resources, and process information to solve the problem. For example, when one faces a novel problem, an approach to solving that problem can be forming an analogy between the new problem and another, previously solved problem. In other words, known information from an earlier problem can be mapped onto the novel problem. Brainstorming may be another valuable approach to gather different ideas on solution paths to unfamiliar problems. If a problem is too complex, problem solvers can try to break it down into more manageable parts (more solvable problems). One approach to problem solving is solving the problem step-by-step and taking an action at each step to get closer to the goal. Another solving approach can be conducting extensive research to analyze existing ideas and then adjusting possible solutions to the problem in hand. Finally, trial-and-error may be an approach to find a solution to an existing problem. It is emphasized in problem solving that there are many solution paths to a problem and a willingness to try multiple approaches is encouraged. Multiple approaches and strategies may be available and some of these approaches may be more efficient than the others.

Problem solving in mathematics, and in other fields as well, requires both knowledge of different representational systems and representational fluency that enables flexible use of various representational systems. For example, when solving a mathematical problem that asks how many quarters there are in 2 1/2, various strategies that involve different representations exist to approach the problem. A student may choose to translate this problem, which is represented in words, into a real-life context, such as how many quarter slices of pizza there are in 2 1/2 pizzas. Another student may opt to draw a picture that represents the given problem and solve the problem using the pictorial representation. Or, some students may represent the problem using symbolic representations and solve the problem accordingly. There may be other approaches where students start with a real-life context and then translate it to a pictorial representation, or where students come up with various relevant representations and choose the most efficient one for them. In more complex problems, different parts of the problems may require different representations. Thus, representational fluency is an essential part of problem solving.

Problem solving is such an important skill that is not only required to help students solve mathematical problems but also provides them with necessary tools to approach and solve problems in the real world. Because the real word does not have recipes to solve a problem, and problem solving requires structured, thoughtful, and careful analysis of problems (especially ill-defined problems) in various situations, people equipped with problem-solving skills are highly valued by employers.

Mathematics as a Language

Mathematics is, to some extent, a language that is universal and can be understood in any part of the world without much difficulty. The mathematics language, which consists of both symbolic and verbal languages, has evolved as the most efficient medium to communicate mathematical ideas and information. Mathematics language also includes graphical images to effectively communicate mathematical concepts and ideas. Thus, different representational modes are used in communicating mathematical ideas and concepts. For example, when a mathematics teacher writes an equation and explains the equation in spoken language to a class, both verbal and written representational forms are in play. Communication in mathematics often involves a constant representational translation between symbolic and verbal representations. Symbolic and verbal languages of mathematics help to express ideas in a meaningful and efficient way. The evolution of mathematics language has been in progress for thousands of years. The goal of this progress is to improve the efficiency of communication, which is central to learning and using mathematics.

Before the emergence of mathematical notations and symbols, mathematicians found it difficult to share their knowledge with the community, even with other mathematicians. Even if a mathematician were able to prove a theorem, for example, geometrically without using mathematical notations and symbols, the mathematician might not have easily written down the proof to share it with others. Difficulties in representing mathematical ideas (writing in a concise and meaningful way using various mathematical notations and symbols) forced mathematicians to seek alternative (especially short and easy) forms to present their knowledge. The need for an effective and efficient mode of communication to convey mathematics ideas resulted in the development of the symbolic mathematical language.

Although the symbolic mathematical language is universal, the verbal mathematical language differs across societies or cultures. For example, although the American and the Japanese use the same symbolic notations to convey mathematical ideas, the verbal language each of these nations uses to communicate about mathematics is different. Differences in verbal languages to communicate mathematics have implications for teaching and learning mathematics. Verbal languages that are clearer about mathematical terms or that relate better to mathematical entities or ideas can support mathematical understanding. For example, counting in the verbal Chinese language is based on the concept of base-10 system. In Chinese, the number 11 is not an arbitrary word in the verbal language. Rather, in Chinese, 11 is “ten-one,” 12 is “ten-two,” 21 is “two-ten-one,” 22 is “two-ten-two,” and so on. In other words, the Chinese verbal language clearly conveys that there is one 10 and one 1 in 11 or there are two 10s and one 1 in 21. Such a clear relation between mathematical ideas and verbal language can be an important cognitive tool that supports mathematical understanding.

Digital Instruction of Math

In the twenty-first century, the digital educational tools used to teach math at all levels are of a technological sophistication unknown to earlier generations. Thus, paradigms for both teaching and learning math have been fundamentally altered. For example, previously, practices such as rote memorization may have proven useful in solving textbook exercises. Because of technology, new approaches to learning are possible. In an article issued by the University of Stanford Graduate School of Education, mathematician Keith Devlin suggested that technology has made it such that the emphasis should be on understanding math content rather than simply knowing how to execute a math problem.  

An additional advantage of digital mobile technology is that many accessibility barriers have been overcome. The COVID-19 pandemic accelerated the migration to instructional software and online courses, including those focused on mathematics. These innovations have themselves seen great improvement and are of a superior quality that existed before the pandemic. Improved teaching technologies have encouraged more active student learning with greater engagement and provide more flexible access to learning. Educational innovation has been moved from the periphery and is now at the center of approaches to instruction.

Bibliography

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