representation definition in math

Representation Theory

Representation theory is fundamental in the study of objects with symmetry. 

It arises in contexts as diverse as card shuffling and quantum mechanics. An early success was the work of Schur and Weyl, who computed the representation theory of the symmetric and unitary groups; the answer is closely related to the classical theory of symmetric functions and deeper study leads to intricate questions in combinatorics.

More recently, methods from geometry and topology have greatly enhanced our understanding of these questions (“geometric representation theory”). The study of affine Lie algebras and quantum groups has brought many new ideas and viewpoints, and representation theory now furnishes a basic language for other fields, including the modern theory of automorphic forms.

All of these aspects are studied by Stanford faculty.  Topics of recent seminars include combinatorial representation theory as well as quantum groups.

Daniel Bump

Persi Diaconis

Browse Course Material

Course info.

• Prof. Pavel Etingof

Departments

• Mathematics

As Taught In

• Algebra and Number Theory
• Linear Algebra

Learning Resource Types

Introduction to representation theory, lecture notes.

Students are assigned readings in these lecture notes each week. The present lecture notes arose from a representation theory course given by Prof. Etingof in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students. The students in that course — Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Elena Yudovina, and Dmitry Vaintrob — co-authored the lecture notes which are published here with their permission.

The book Introduction to Representation Theory based on these notes was published by the American Mathematical Society in 2016. A complete file of the book (PDF - 1.1MB) is on Prof. Etingof’s webpage. [Please note: This file cannot be posted on any website not belonging to the authors.]

Complete Lecture Notes ( PDF - 1.3MB )

Introduction ( PDF )

Chapter 1: basic notions of representation theory ( pdf ).

1.1 What is representation theory? 1.2 Algebras 1.3 Representations 1.4 Ideals 1.5 Quotients 1.6 Algebras defined by generators and relations 1.7 Examples of algebras 1.8 Quivers 1.9 Lie algebras 1.10 Tensor products 1.11 The tensor algebra 1.12 Hilbert’s third problem 1.13 Tensor products and duals of representations of Lie algebras 1.14 Representations of sl(2) 1.15 Problems on Lie algebras

Chapter 2: General Results of Representation Theory ( PDF )

2.1 Subrepresentations in semisimple representations 2.2 The density theorem 2.3 Representations of direct sums of matrix algebras 2.4 Filtrations 2.5 Finite dimensional algebras 2.6 Characters of representations 2.7 The Jordan-Hölder theorem 2.8 The Krull-Schmidt theorem 2.9 Problems 2.10 Representations of tensor products

Chapter 3: Representations of Finite Groups: Basic Results ( PDF )

3.1 Maschke’s Theorem 3.2 Characters 3.3 Examples 3.4 Duals and tensor products of representations 3.5 Orthogonality of characters 3.6 Unitary representations. Another proof of Maschke’s theorem for complex representations 3.7 Orthogonality of matrix elements 3.8 Character tables, examples 3.9 Computing tensor product multiplicities using character tables 3.10 Problems

Chapter 4: Representations of Finite Groups: Further Results ( PDF )

4.1 Frobenius-Schur indicator 4.2 Frobenius determinant 4.3 Algebraic numbers and algebraic integers 4.4 Frobenius divisibility 4.5 Burnside’s theorem 4.6 Representations of products 4.7 Virtual representations 4.8 Induced representations 4.9 The Mackey formula 4.10 Frobenius reciprocity 4.11 Examples 4.12 Representations of _S_ n 4.13 Proof of theorem 4.36 4.14 Induced representations for _S_ n 4.15 The Frobenius character formula 4.16 Problems 4.17 The hook length formula 4.18 Schur-Weyl duality 4.19 Schur-Weyl duality for GL (V) 4.20 Schur polynomials 4.21 The characters of L λ 4.22 Polynomial representations of GL (V) 4.23 Problems 4.24 Representations of _GL_ 2 (F q )

4.24.1 Conjugacy classes in _GL_ 2 (F q ) 4.24.2 1-dimensional representations 4.24.3 Principal series representations 4.24.4 Complementary series representations

4.25 Artin’s theorem 4.26 Representations of semidirect products

Chapter 5: Quiver Representations ( PDF )

5.1 Problems 5.2 Indecomposable representations of the quivers A1,A2,A3 5.3 Indecomposable representations of the quiver D 4 5.4 Roots 5.5 Gabriel’s theorem 5.6 Reflection functors 5.7 Coxeter elements 5.8 Proof of Gabriel’s theorem 5.9 Problems

Chapter 6: Introduction to Categories ( PDF )

6.1 The definition of a category 6.2 Functors 6.3 Morphisms of functors 6.4 Equivalence of categories 6.5 Representable functors 6.6 Adjoint functors 6.7 Abelian categories 6.8 Exact functors

Chapter 7: Structure of Finite Dimensional Algebras ( PDF )

7.1 Projective modules 7.2 Lifting of idempotents 7.3 Projective covers

You are leaving MIT OpenCourseWare

Developing children’s mathematical understanding with a variety of representations

“We didn’t do it like that in my day.” Discover the importance of learning different methods and strategies and why teaching a variety of mathematical representations is essential to learning.

In mathematical terms, we use the word representations as one way of labelling the many methods and strategies we may encounter. This definition of representations may be useful in understanding this further:

“The act of capturing a mathematical concept or relationship in some form and to the form itself.” – National Council of Teachers of Mathematics

We also need to challenge the notion that there’s only one right way to approach a problem. It’s a reminder that mathematics is wholly creative and we should embrace the differences in the representations we see.

Can you imagine if there was only one representation of a cake, chocolate or ice-cream?

Understanding representations within the CPA approach

You may be familiar with the Concrete, Pictorial, Abstract (CPA) approach developed by Jerome Bruner.

Put simply, a mathematical problem can be represented using concrete or physical materials, the problem can then be represented using a diagram or picture and the same problem can be represented in the abstract, using symbolic notation:

Within these approaches there are variations and different representations, but they continue to represent the same problem.

How to make sure children have access to multiple representations

It’s important that children have access to different representations of the same mathematical idea or concept. Once they’ve grasped how different representations work, they’ll be on the road to understanding how and when to use them.

1. Teach representations in the correct order

To ensure different representations don’t lead to confusion, the order in which they are presented has to be well thought out. Then, any variation in the representations acts as a support and challenge in a learner’s development of new mathematical ideas.

The following example from Maths — No Problem! Textbook 5A shows a carefully planned progression through a range of representations. The lesson, as with all of the lessons in the series, starts with a single problem:

How many seats are there in this theatre?

There are 28 rows of chairs and there are 26 chairs in each row. This pictorial representation of the problem suggests three separate arrays (28 x 8, 28 x 10 and 28 x 8) which should prompt learners to realise that this is a multiplication problem. They may also realise that the three arrays can be brought together to represent the equation 28 x 26 = [ ]

At this point, children can explore the range of mathematical strategies they know to solve the problem. When these are written down or shown, they become representations of the problem. These representations help develop the ideas and concepts that can be used to solve the problem.

The learners are then given an opportunity to read the textbook and in doing so, have access to a range of carefully structured representations demonstrating the initial problem.

2. Link initial concept exploration with the representations in the textbooks

The link between the first stage of the lesson (discussing and exploring an initial problem), with the representations in the book is crucial to children’s learning. This encourages learners to compare their strategies and approaches to those shown in the book — giving them an opportunity to make connections and relate mathematical ideas.

Here’s an example:

There are 28 rows. Each row consists of 26 seats. There are 728 seats.

The first representation is pictorial. It shows the array of seats in the theatre and connects this to a multiplication strategy using number bonds to break 28 into 10, 10 and 8, and 26 into 10, 10 and 6.

Maths — No Problem! character Ravi gives support by reminding learners that the groups of seats will need to be added together to find the total number. This pictorial representation is particularly useful as it provides links to earlier ideas relating to ten that could be used to support struggling learners. The Base 10 blocks feature on the Visualiser app has been used to demonstrate this.

3. Ask learners to use their prior knowledge to solve more abstract problems

The next representation you see doesn’t have pictorial support. But because the previous example did, the link between the pictorial representation and the abstract representation has been provided. To further develop this link you can ask your learners:

“Where do we look when we read 10 x 26 = 260?” “Why are there two equations that are the same?” “How do the equations relate to the seating plan?”

These questions relate the mathematics to the problem, and children can start to see the relationship between the different representations.

4. Support learners’ understanding of mathematical concepts

The use of language, written words and reading maths is essential to a learner’s understanding. This idea is continually being developed and built upon. Revisiting these ideas again and again within a spiral curriculum is a key component of a maths mastery approach.

A lesson from Maths — No Problem! Textbook 3A , two years earlier, provided the basis for Sam to be able to relate 26 x 2 to 26 x 20.

As teachers, we can also use this earlier lesson to support children’s understanding if they are struggling to make that connection for themselves.

26 x 20 = 26 x 2 tens 26 x 2 tens = 52 tens 52 tens = 520

Sam also understands that if he doubles the number of groups, or the number of items in the group, he can systematically work through the problem. Again, you can relate this to the pictorial representation in the second example.

The final example in the book is another abstract method — often referred to as a formal written method. The orientation and directionality of the calculation has changed, however, at each step of the problem, the numbers relate to the previous examples and the original problem.

The goal of maths mastery is for learners to get to a point where they no longer have to attend to certain functions as they solve a problem. In this example, we would want the children to just know that 28 can be broken into smaller parts or just know that 6 x 8 is 48. That way, they can concentrate on new ideas that are associated (in this case, with multiplying a 2-digit number by another 2-digit number).

In the examples above, a carefully structured approach provides representations that allow children to develop their understanding of multiplication.

Teaching multiple representations is a core part of learning mathematics. It’s exciting to make new connections in the representations we have access to.

In a time where blended learning is becoming increasingly widespread, so too is the opportunity to see and interpret different representations. Whether it be representations family members, friends, online resources or books, these different representations should excite, not dull our sense of curiosity. We should use it as a challenge to see the mathematics that connects the various representations.

Adam Gifford

Browse by Topic

Your teaching practice.

Boost your teaching confidence with the latest musings on pedagogy, classroom management, and teacher mental health.

Maths Mastery Stories

You’re part of a growing community. Get smart implementation advice and hear inspiring maths mastery stories from teachers just like you.

Teaching Tips

Learn practical maths teaching tips and strategies you can use in your classroom right away — from teachers who’ve been there.

Classroom Assessment

Identify where your learners are at and where to take them next with expert assessment advice from seasoned educators.

Your Learners

Help every learner succeed with strategies for managing behaviour, supporting mental health, and differentiating instruction for all attainment levels.

Teaching Maths for Mastery

Interested in Singapore maths, the CPA approach, bar modelling, or number bonds? Learn essential maths mastery theory and techniques here.

Deepen your mastery knowledge with our biweekly newsletter

By clicking “Accept All” , you agree to the storing of cookies on your device to enhance site navigation, analyze site usage and assist in our marketing efforts.

Stack Exchange Network

Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

What is Representation Theory?

I'm beginning a course that uses representation theory, but I do not really understand what that is about. In the text I am following, I have the following definition:

A representation of the Lie group $G$ on the vector space $V$ is a continuous mapping $\cdot \colon G \times V \to V$ such that for each $g \in G$, the translation $T_{g} \colon V \to V$ given by $T_g(v) = g \cdot v$, $v \in V$, is a linear map; $T_{e} = \mathrm{Id}$ where $e$ is the identity element of $G$; $T_{gh} = T_{g} T_{h}$ for $g, h \in G$. We call the pair $(V,\cdot)$ a real (resp. complex) representation and $V$ the representation space.

What is the motivation behind this sort of definition? From my google searches I have seen different definitions, but I still don't really know why what I am reading is defined that way. Why a Lie group and not a regular group? etc.

• group-theory
• representation-theory

• 2 $\begingroup$ To answer your question "why a Lie group", the answer is that the definition works for any group. In fact, there are some groups which we only really know how to study via their representations. A representation allows us to take a group we may know very little about and study it using the familiar tools of linear algebra. $\endgroup$ –  Mathmo123 Commented Jan 27, 2016 at 0:45
• 2 $\begingroup$ This is worth reading. $\endgroup$ –  Mathmo123 Commented Jan 27, 2016 at 22:38
• 1 $\begingroup$ @MarianoSuárez-Alvarez A cursory look suggests that there are several questions on the applications of rep theory, but none that I can find asking why we would bother to define representations in the first place. This question is the closest, but seems to be asking more from a perspective of looking for applications from a position of knowledge. Of course I may have missed something. $\endgroup$ –  Mathmo123 Commented Jan 27, 2016 at 23:20
• 1 $\begingroup$ I tried searching 'motivation representation theory', but what I found didn't really satisfy what I was looking for. The answer given by @Mathmo123 has helped immensely with clearing up my confusion and hopefully will clear up the confusion of my classmates and anyone else. $\endgroup$ –  diffGeoLost Commented Jan 27, 2016 at 23:48
• 1 $\begingroup$ @Mathmo123, the asnwers here and IMO the question are subsets of those at the second question you linked. $\endgroup$ –  Mariano Suárez-Álvarez Commented Jan 28, 2016 at 0:13

2 Answers 2

One of the most common ways that groups arise "in the wild" is as sets of symmetries of an object . For example

• The symmetry group $S_n$ is the group of all permutations of $\{1,\ldots n\}$
• The dihedral group $D_{2n}$ is the group of symmetries of a regular $n$ -gon
• The Lie group $\mathrm{GL}_n(\mathbb R)$ is the group of invertible linear maps on $\mathbb R^n$

More generally, given a general abstract group $G$ , we regularly consider the case of $G$ acting on a set $X$ , and we might ask the question: given some set $X$ , what is its "group of symmetries".

Representation theory asks the converse to this question:

Given a group $G$ , what sets does it act on?

Whilst it is possible to attempt to answer this general, a useful starting point is to restrict the sets in question to sets we know an awful lot about: vector spaces.

Definition: Let $G$ be a group, and $V/k$ be a vector space. A representation of $G$ is a group action of $G$ on $V$ that is linear (so preserves the vector space structure of $V$ ) - i.e. for every $g\in G$ , $u,v\in V$ , $\mu,\lambda\in k$ $$g(\lambda u+\mu v) = \lambda g(u) + \mu g(v).$$

This is the definition that you have been given. With $V$ as before, an equivalent definition is this:

A representation of $G$ is a group homormophism $$\rho: G\to\mathrm{GL}(V)$$

Indeed, a group action of $G$ on $V$ assigns to each $g$ an invertible linear map. And given a homomorphism $\rho$ , $G$ acts on $V$ via $g\cdot v = \rho(g)v$ .

In the case that $G$ is a Lie group (or more generally a topological group), then we require this action/representation to be continuous.

Representation theory allows us to translate our viewpoint by viewing (a quotient of) our group as a group of linear maps on a vector space. This allows us to tackle problems in group theory using the familiar and powerful tools of linear algebra. For example, we can take the trace of a linear map, and the identity $\mathrm{tr}(ABA^{-1}) = \mathrm{tr}(B)$ tells us that the trace of a representation (called the character of the representation) is constant on the conjugacy classes of a group. We can also consider determinants, characteristic polynomials, dual vector spaces (or the dual representation), dimension and many more of our favourite concepts from linear algebra.

Representations are certainly powerful:

• There are theorems (for example, concerning Frobenius groups ) whose only known proofs use representation theory.
• There are groups (such as $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$ ) which we only really know how to study via their representations.

Rough answer. Before there was an abstract definition of "Lie group" mathematicians studied groups of matrices. A Lie group is a generalization of a group of matrices. It turns out that one way to try to understand a Lie group is to look at all the ways to "represent" it as a group of matrices. Each mapping in the definition in your question does just that.

You must log in to answer this question.

Not the answer you're looking for browse other questions tagged group-theory representation-theory motivation ..

• Featured on Meta
• Announcing a change to the data-dump process
• Bringing clarity to status tag usage on meta sites
• 2024 Election Results: Congratulations to our new moderator!

Hot Network Questions

• Is the 2024 Ukrainian invasion of the Kursk region the first time since WW2 Russia was invaded?
• How to Interpret Statistically Non-Significant Estimates and Rule Out Large Effects?
• Are old cardano node versions invalidated
• Conjugate elements in an extension
• How do you tip cash when you don't have proper denomination or no cash at all?
• Why would autopilot be prohibited below 1000 AGL?
• Is my magic enough to keep a person without skin alive for a month?
• Representing permutation groups as equivalence relations
• Deleting all files but some on Mac in Terminal
• What is the term for the belief that significant events cannot have trivial causes?
• Gravitational potential energy of a water column
• Is a stable quantifier-free language really possible?
• What does "dare not" mean in a literary context?
• What would be a good weapon to use with size changing spell
• How to connect 20 plus external hard drives to a computer?
• An instructor is being added to co-teach a course for questionable reasons, against the course author's wishes—what can be done?
• What`s this? (Found among circulating tumor cells)
• Are all citizens of Saudi Arabia "considered Muslims by the state"?
• Is it helpful to use a thicker gage wire for part of a long circuit run that could have higher loads?
• How rich is the richest person in a society satisfying the Pareto principle?
• Is there a way to read lawyers arguments in various trials?
• How to realize the index shift operation in quantum circuit?
• do-release-upgrade from 22.04 LTS to 24.04 LTS still no update available
• Riemannian metric for which arbitary path becomes a geodesic

Mathematical Representations

• Reference work entry
• First Online: 01 January 2020
• Cite this reference work entry

• Gerald A. Goldin 2  

1041 Accesses

7 Citations

Definitions

As most commonly interpreted in education, mathematical representations are visible or tangible productions – such as diagrams, number lines, graphs, arrangements of concrete objects or manipulatives, physical models, written words, mathematical expressions, formulas and equations, or depictions on the screen of a computer or calculator – that encode, stand for, or embody mathematical ideas or relationships. Such a production is sometimes called an inscription when the intent is to focus on a specific instance without referring, even tacitly, to any interpretation of it. To call something a representation thus includes reference to some meaning or signification it is taken to have. Such representations are called external – i.e., they are external to the individual who produced them and accessible to others for observation, discussion, interpretation, and/or manipulation. Spoken language, interjections, gestures, facial expressions, movements, and postures may sometimes...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save.

• Get 10 units per month
• Download Article/Chapter or eBook
• 1 Unit = 1 Article or 1 Chapter
• Cancel anytime
• Available as PDF
• Read on any device
• Instant download
• Own it forever
• Available as EPUB and PDF
• Durable hardcover edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Anderson C, Scheuer N, Pérez Echeverría MP, Teubal EV (eds) (2009) Representational systems and practices as learning tools. Sense, Rotterdam

Google Scholar  

Bruner JS (1966) Toward a theory of instruction. The Belknap Press – Harvard University Press, Cambridge, MA

Common Core State Standards Initiative (2018) Preparing America’s students for success. Retrieved June 2018 from http://www.corestandards.org/

Cuoco AA, Curcio FR (2001) The roles of representation in school mathematics: NCTM 2001 yearbook. National Council of Teachers of Mathematics, Reston

Davis RB (1984) Learning mathematics: the cognitive science approach to mathematics education. Ablex, Norwood

Duval R (2006) A cognitive analysis of problems of comprehension in a learning of mathematics. Educ Stud Math 61:103–131

Article   Google Scholar  

Goldin GA (1998) Representational systems, learning, and problem solving in mathematics. J Math Behav 17:137–165

Goldin GA (2008) Perspectives on representation in mathematical learning and problem solving. In: English LD (ed) Handbook of international research in mathematics education, 2nd edn. Routledge – Taylor and Francis, London, pp 176–201

Goldin GA, Janvier, C (eds) (1998) Representations and the psychology of mathematics education: parts I and II (special issues). J Math Behav 17(1 & 2)

Goldin GA, Kaput JJ (1996) A joint perspective on the idea of representation in learning and doing mathematics. In: Steffe L, Nesher P, Cobb P, Goldin GA, Greer B (eds) Theories of mathematical learning. Erlbaum, Hillsdale, pp 397–430

Gravemeijer K, Doorman M, Drijvers P (2010) Symbolizing and the development of meaning in computer-supported algebra education. In: Verschaffel L, De Corte E, de Jong T, Elen J (eds) Use of representations in reasoning and problem solving: analysis and improvement. Routledge – Taylor and Francis, London, pp 191–208

Heinze A, Star JR, Verschaffel L (2009) Flexible and adaptive use of strategies and representations in mathematics education. ZDM 41:535–540

Hitt F (ed) (2002) Representations and mathematics visualization. Departamento de Matemática Educativa del Cinvestav – IPN, México

Janvier C (ed) (1987) Problems of representation in the teaching and learning of mathematics. Erlbaum, Hillsdale

Kaput J, Noss R, Hoyles C (2002) Developing new notations for a learnable mathematics in the computational era. In: English LD (ed) Handbook of international research in mathematics education. Erlbaum, Mahwah, pp 51–75

Lesh RA, Doerr HM (eds) (2003) Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning, and teaching. Erlbaum, Mahwah

McClelland JL, Mickey K, Hansen S, Yuan A, Lu Q (2016) A parallel-distributed processing approach to mathematical cognition. Manuscript, Stanford University. Retrieved June 2018 from https://stanford.edu/~jlmcc/papers/

Moreno-Armella L, Sriraman B (2010) Symbols and mediation in mathematics education. In: Sriraman B, English L (eds) Advances in mathematics education: seeking new frontiers. Springer, Berlin, pp 213–232

Moreno-Armella L, Hegedus SJ, Kaput JJ (2008) From static to dynamic mathematics: historical and representational perspectives. Educ Stud Math 68:99–111

National Council of Teachers of Mathematics (2000) Principles and standards for school mathematics. NCTM, Reston

Newell A, Simon HA (1972) Human problem solving. Prentice-Hall, Englewood Cliffs

Novack MA, Congdon EL, Hermani-Lopez N, Goldin-Meadow S (2014) From action to abstraction: using the hands to learn math. Psychol Sci 25:903–910

Palmer SE (1978) Fundamental aspects of cognitive representation. In: Rosch E, Lloyd B (eds) Cognition and categorization. Erlbaum, Hillsdale, pp 259–303

Roth W-M (ed) (2009) Mathematical representation at the interface of body and culture. Information Age, Charlotte

Skemp RR (ed) (1982) Understanding the symbolism of mathematics (special issue). Visible Language 26(3)

van Garderen D, Scheuermann A, Poch A, Murray MM (2018) Visual representation in mathematics: special education teachers’ knowledge and emphasis for instruction. Teach Educ Spec Educ 41:7–23

Download references

Author information

Authors and affiliations.

Graduate School of Education, Rutgers University, New Brunswick, NJ, USA

Gerald A. Goldin

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Gerald A. Goldin .

Editor information

Editors and affiliations.

Department of Education, Centre for Mathematics Education, London South Bank University, London, UK

Stephen Lerman

Section Editor information

Department of Science Teaching, The Weizmann Institute of Science, Rehovot, Israel

Ruhama Even

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this entry

Cite this entry.

Goldin, G.A. (2020). Mathematical Representations. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_103

Download citation

DOI : https://doi.org/10.1007/978-3-030-15789-0_103

Published : 23 February 2020

Publisher Name : Springer, Cham

Print ISBN : 978-3-030-15788-3

Online ISBN : 978-3-030-15789-0

eBook Packages : Education Reference Module Humanities and Social Sciences Reference Module Education

Share this entry

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Publish with us

Policies and ethics

• Find a journal
• Track your research

• Math Article
• Representation Of Functions

Representation Of A Function

A function is a relation between two sets of variables such that one variable depends on another variable. We can represent different types of functions in different ways. Usually, functions are represented using formulas or graphs. There are four ways for the representation of a function as given below:

• Algebraically
• Numerically

Each one of them has some advantages and disadvantages. Let us look at them once and try to understand them.

Representation of a Function – Algebraic

The algebraic representation of a function refers to the expression of a function using an equation or mathematical model. As we know, the process of interpreting a real-world problem in a convenient function is called mathematical modelling. The resulting function and a representation of all input and output variables, including proper units of measure, are referred to as a mathematical model. Input/output diagrams often represent mathematical models. 

It is one of the usual representations of functions. Here, functions are explicitly represented using formulas, and the functions are generally denoted by lower case alphabet letters. Let us take the cube function.

Figure 1: Block diagram depicting a cube function

The standard letter to represent function is f. However, it can be represented by any variable. To denote the function f algebraically, i.e., using the formula, we write:

\(\begin{array}{l} f:x \end{array} \) → \(\begin{array}{l} x^3\end{array} \)

where x is the variable indicating the input. It can be represented by any variable.

\(\begin{array}{l} x^3 \end{array} \) is the formula of function

f is the name of the function

Though one of the easy and understandable ways of representing a function, it is not always easy to get the function’s formula. For those cases, we use other methods of representation.

Representation of a Function- Visual

This is basically the graphical representation of functions. This way of representation is very easy to understand. The input values are marked along the x-axis. For any input value, the corresponding output value is the vertical displacement from the x-axis. For e.g. at x = a, the output is equal to f(a).

Figure 2: Graph of a function

The graph shows the properties of the functions. For e.g. from figure 2, we can directly tell:

• where the graph is increasing or decreasing
• where the rate of change is more and where it is less
• where are the extreme values

Thus, graphs are very beneficial for studying the behaviour of the function. One drawback is that we can’t always get the exact values of all the outputs from the graph.

Learn more about functions and their graphs here.

Representation of a Function- Numerical

This is basically the tabular way of representing a function. The table contains two columns; one with the dependent variable and the other with the independent variable. To show an example, let us take the function f and independent variable as x. The table is given as:

Table 1: Table representing a function f(x) = 2x

Though we have the exact value of the outputs, we can only have a finite number of such outputs. The analysis of the function and study of its behaviour hence becomes difficult.

Representation of a Function- Verbal

In this way of representing functions, we use words. For e.g.

• For the input x, the function gives the largest integer smaller than or equal to x, i.e. floor function (see fig. 3).
• For the input x, the function gives the value equal to x, i.e. identity function (see fig. 4).

Figure 3: Floor function

Frequently Asked Questions – FAQs

A function can be represented in how many ways, verbal function represented by, algebraic function is represented by what, which function is represented chart, which function is represented by a continuous graph.

Each of the representations has its pros and cons. According to the information required, appropriate representation should be adopted. To learn more about functions, visit BYJU’S and fall in love with learning!

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

Visit BYJU’S for all Maths related queries and study materials

Your result is as below

Request OTP on Voice Call

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Post My Comment

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

• Skip to main content

Welcome, Fellow Math Enthusiast! I’m so happy you’re here!

• Counting & Cardinality
• Addition & Subtraction
• Multiplication & Division
• Place Value & Base Ten
• Measurement & Data
• Geometry & Fractions
• Vocabulary & Discourse
• Math Manipulatives
• Classroom Management
• Classroom Organization
• Holidays & Seasonal
• Social-Emotional Learning
• Privacy Policy
• Terms of Use
• SHOP RESOURCES
• BECOME A MEMBER
• Search this website

Teaching with Jillian Starr

teaching little stars to shine brightly

Grab Your FREE Gift

Addition Centers Freebie

Looking for fun and engaging centers to help your students practice their addition fact fluency? Be sure to grab these freebies!

Mathematical Representations Series Part 3: Symbolic Representation

Welcome back to our deep dive into mathematical representations! Today, we are taking a look at symbolic representations and how we can translate between symbolic, concrete, and visual representations. First, let’s do a two-sentence recap of this series so far:

We have already focused on concrete representations and the immense value of manipulatives, as well as the range of visual representations we want to encourage with our students. We are centering our conversation around Lesh’s Translation Model, which encompasses the range of ways we represent our thinking, and stresses the importance of making connections between representations.

Symbolic Representation

Now, that we’re all caught up, let’s take a peek at Symbolic Representations. You may have heard it called Abstract Representation, especially if you’re familiar with the Concrete-Pictorial-Abstract model (also called CRA model).

So what are symbolic representations? Well, symbolic representation is when mathematical symbols (like numerals and operation signs) are used to show a mathematical concept. They are using symbolic, mathematical language to express their understanding of a math concept.

Let’s go back to the example from concrete representation . In that example, the student used 5 yellow counters, and 4 red counters to concretely show their understanding of 5 + 4 = 9.

Then in their visual representation , they were able to create a sketch of their concrete representation. This was our first example of translating between two different representations (connecting visual to concrete).

Now, we can support students even further by helping them represent their understanding with symbols. Here the student counts the collection of five counters and writes the numeral “5” below it. They do the same for the four counters.

Finally, they count all of the counters together and write down “9”. We can help them with the operations symbols if needed, showing that “+” means that we are putting the 5 and the 4 together and that the “=” means “4+5” has the same value as “9”.

When we guide students to using numerals and operations, they are connecting their manipulatives, their sketches, AND the symbolic representation. This is such a powerful move to help deepen student understanding!

Connecting Two Symbolic Representations

It’s a fairly straightforward representation, but how do abstract representations connect to themselves in Lesh’s translation model? 

Well, this student actually wanted to write 4 + 4 +1 because that is how they solved the problem. They knew 4 + 4 = 8, so 4 + 5 = 9 because it’s 4 + 4 + 1 more. Understanding that both expressions can describe the same concept is just one example of how to translate within the symbolic representation.

What’s Up Next?

This series is going to dive deep into each of the representations discussed in Lesh’s Translation Model, and then we are going to put it all together so we can make a big impact on your math teaching this year.

If you missed Part One about Concrete Representations or Part Two about Visual Representations , check them out so you have all of the info you need before we move on!

Addition & Subtraction Resources

You May Also Enjoy These Posts:

Reader Interactions

Great post! I completely agree that symbolic representations can help students understand complex mathematical concepts more easily. I’ve seen firsthand how visual aids and manipulatives can make a difference in a student’s learning journey. Thanks for sharing your insights and experiences!

Leave a Comment

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed .

Ready to go deeper?

JOIN MEANINGFUL MATH

hello I'm Jillian

I’m so happy you’re here. I want every child to feel confident in their math abilities, and that happens when every teacher feels confident in their ability to teach math.

In my fifteen years of teaching, I sought every opportunity to learn more about teaching math. I wanted to know HOW students develop math concepts, just like I had been taught how students learn to read. I want every teacher to experience the same math transformation I did, and have the confidence to teach any student that steps foot in their classroom. I’m excited to be alongside you in your math journey!

Follow Me on Instagram!

• More from M-W
• To save this word, you'll need to log in. Log In

representation

Definition of representation

Examples of representation in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'representation.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

15th century, in the meaning defined at sense 1

Phrases Containing representation

• proportional representation
• self - representation

Dictionary Entries Near representation

representant

representationalism

Cite this Entry

“Representation.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/representation. Accessed 7 Sep. 2024.

Kids Definition

Kids definition of representation, legal definition, legal definition of representation, more from merriam-webster on representation.

Thesaurus: All synonyms and antonyms for representation

Nglish: Translation of representation for Spanish Speakers

Britannica English: Translation of representation for Arabic Speakers

Britannica.com: Encyclopedia article about representation

Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free!

Can you solve 4 words at once?

Word of the day.

See Definitions and Examples »

Get Word of the Day daily email!

Popular in Grammar & Usage

Plural and possessive names: a guide, 31 useful rhetorical devices, more commonly misspelled words, why does english have so many silent letters, your vs. you're: how to use them correctly, popular in wordplay, 8 words for lesser-known musical instruments, it's a scorcher words for the summer heat, 7 shakespearean insults to make life more interesting, birds say the darndest things, 10 words from taylor swift songs (merriam's version), games & quizzes.