Recent conversations with teachers relate to the standard of the process of teaching representations and, in particular, the ability for students to show what they know in various ways, and what would be the benefit to their mathematical understanding? What does it mean to present what you know in various ways? The Carnegie Learning program is an example of the emphasis on multiple representations and the use of computational tools. [17] Specifically, Carnegie learning not only focuses the student on solving the real-world scenarios presented in the text, but also promotes literacy through the writing of sentences and the explanation of the student`s thinking. In conjunction with scenario-based text, Carnegie Learning offers an online tutoring program called „Cognitive Tutor” that uses data collected from each question a student answers to guide the student to areas where they need more help. As students begin to develop a sense of numbers and combine equivalent forms of ways in which the number can be represented, several models of that number can be introduced. In this way, some operations can be associated with rational and justified numbers, rather than approaching the operation from an algorithmic attitude. Tasks that involve the construction, use, and interpretation of multiple representations may be suitable for topic scoring[7] and other types of assessments suitable for perpetual activities. For example, if you use visualization for solving mathematical problems, several representations are manifested. [8] These multiple representations occur when each student uses their knowledge base and experience to create a visualization of the problem domain on the way to a solution. Since visualization can be divided into two main areas, schematic or pictorial,[9] most students will use one method, or sometimes both, to represent the area of the problem. The National Council of Mathematics Teachers has a standard that deals with multiple substitutions.

In part, he states [11] „Curricula should allow all students to do the following: Through these representations, students can connect different numerical models, solve problems in different ways, and move from the concrete to the abstract. Although there are many representations in mathematics, high school curricula strongly prefer numbers (often in tables), formulas, graphs, and words. [12] Students with special needs may be weaker in the use of some of the performances. For these students, it may be especially important to use multiple representations for two purposes. First, the inclusion of representations that currently work well for the student ensures an understanding of the current mathematical subject. Second, the links between multiple representations within the same topic strengthen general skills in the use of all representations, including those that are currently problematic. [2] Comparing the different visualization tools created by each student is an excellent example of multiple representations. In addition, the instructor can take elements of these examples, which he integrates into his grading section. In this way, it is the students who provide the examples and standards by which the assessment is conducted. This crucial factor puts each student on the same level and connects them directly to their performance in the classroom. [Citation required] Using multiple representations can help differentiate teaching by addressing different learning styles.

[4] [10] It is also useful for English as a Second Language/English as a Second Language (ESL/ELL) learners to use multiple representations. The more visually you can „bring a concept to life,” the more likely students are to understand what the teacher is talking about. This is also important for young students who may not have much experience or prior knowledge on the topics being taught. What are the strategies for integrating multiple representations into teaching? Learners differ in the way they perceive and understand the information presented to them. For example, people with sensory impairments (p.B. blindness or deafness); learning disabilities (p.B dyslexia); Linguistic or cultural differences, etc., may require different approaches to content. Others can simply capture information faster or more efficiently through visual or auditory means rather than through printed text. In addition, learning and the transmission of learning occurs when multiple representations are used, as they allow students to make connections within and between concepts. In short, there is no single means of representation that is optimal for all learners; Providing display options is essential. (UDL National Centre, Principle I, 2011) Visual representations, manipulations, gestures and, to some extent, grids can support qualitative thinking on mathematics.

Rather than simply emphasizing computer skills, multiple representations can help students make and develop the conceptual transition to meaning and use of algebraic thinking. By focusing more on conceptual representations of algebraic problems, students have a better chance of improving their problem-solving skills. [3] Multiple representations can also eliminate some of the gender biases that exist in math classes. For example, explaining probability using baseball statistics alone can potentially alienate students who have no interest in the sport. When teachers show a connection to real-world applications, they must choose diverse representations of interest to all genders and cultures. [Neutrality is controversial] In mathematics classes, a representation is a way to encode an idea or relationship, and can be both internal (e.B. mental construction) and external (e.B. graph). Therefore, multiple representations are ways to symbolize, describe, and refer to the same mathematical entity.

They are used to understand, develop and communicate different mathematical characteristics of the same object or operation, as well as connections between different properties. Multiple representations include graphs and tables, tables and grids, formulas, symbols, words, gestures, software code, videos, concrete models, physical and virtual manipulators, images and sounds. [1] Representations are tools for reflection for mathematics. The use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. [2] [3] [4] The choice of representation to be used, the task of making representations in other representations, and the understanding of how changes in one representation affect others are examples of these mathematically demanding activities. [Citation needed] Estimation, another complex task, can greatly benefit from multiple representations [5] Using computer tools to create and share mathematical representations opens up several possibilities. It allows the dynamic linking of several representations. For example, editing a formula can immediately change the chart, value table, and text that can be changed for the represented function in all of these ways. The use of technology can increase the accuracy and speed of data collection and enable real-time visualization and experimentation. [15] It also supports cooperation. [16] Supporting students` use of multiple representations can lead to more open problems, or at least to the acceptance of multiple methods of solution and forms of response. Project-based learning units such as WebQuests typically require multiple representations.

[Citation needed] Connected and equivalent rational number forms are fractions, decimal fractions, percentages, number line placement, circle model, and set model. Using a graphic organizer, where one of the representations is filled in and students have to fill in the other templates, gives students the ability to get the flexibility to switch between the different rational number forms. In addition, it is also criticised that caution should be exercised so that informal representations do not prevent pupils from moving towards formal and symbolic mathematics. [Citation needed] 3) The groups then provide several representations of the solution(s) of the problem posed with NAGS: number (tabular values), algebraic solution, graphical solution, sentence – explanation of what the solution offers. Curricula that support by starting with conceptual understanding and then developing procedural fluidity, e.B. AIMS FOUNDATION ACTIVITIES,[6] often use multiple representations. The Principles and Standards for School Mathematics, National Council of Mathematics Teachers (NCTM)[1] have six principles that guide teachers and schools in providing sound research-based mathematics education. These principles are: justice, curriculum, teaching, learning, assessment and technology. However, the ten standards are specific to what math lessons should allow a student to do: content – number and operations, geometry, measurement and analysis of data, and probability and process – problem solving, reasoning and proof, communication and representations. The Interactivate project [20] includes many activities that combine visual, verbal and digital representations. Currently, there are 159 different activities available in many areas of mathematics, including numbers and operations, probability, geometry, algebra, statistics, and modeling.

Several programs use widely developed manipulation systems and corresponding representations. For example, kitchen stems[13], Montessori beads[citation needed], algebra tiles[14], base blocks 10, counters.. .