Effective Teaching of Mathematics

The framers of CCSS-M have identified Standards of Mathematics Practice (SMP) based on practices and research on teaching and learning that consistently produce mathematics learners with high achievements. To be an effective teacher, one needs to know the content, pedagogy and models to deliver that content, and understand how students learn.

CCSS-M gives us what is important to teach and learn at what level. It describes the content and levels and nature of content mastery. However, teachers decide how to teach and assess. The goal of the first four to five years of students’ mathematics experience is to become proficient and comfortable in number concept, numbersense, and numeracy. After that, students use their understanding and fluency in numeracy skills to learn mathematics—algebraic and geometric models, and their integration.

The Standards for Mathematical Practice (SMP) describe ways in which students should be engaged in increasingly demanding subject matter as they grow in mathematics expertise and content throughout the elementary, middle and high school years.

To support students’ growth in mathematical maturity, designers of curricula, assessments, and professional development should connect mathematical instructional practices to the mathematics content espoused in the CCSS-M. Without connecting content standards with instruction practice standards, we cannot achieve the goals of CCSS-M. Assessment, on the other hand, is to see whether that connection has been made or not.

The Mathematical objects (e.g., numbers of various kinds—natural to complex; geometric entities—shapes, figures, diagrams, functions; operations and procedures of different types—decomposition/ recomposition, manipulations of numbers of different kinds and forms, e.g., long-division; transformations and functions—static and dynamic, congruence and similarity, matrices and determinants, etc. are examples of components and manifestation of content standards.

Mathematics practice standards, on the other hand, describe what actions teachers are to take so that students make the mathematics content—language, concepts, procedures, and skills their own. Thus while we usually pay attention to nouns in content standards, for practice standards we must pay attention to verbs. The Standards of Mathematics Practice are action steps to make the content of the CCSS-M possible to take hold in the classroom and make students learners of mathematics.

Why SMP?
The National Mathematics Advisory Panel concluded that our students have reasonable, though incomplete, factual and procedural knowledge, but poor conceptual knowledge (e.g., many do not fully understand the base-10 number system, concepts of fractions, how decimals and fractions are related, the differences in arithmetic and algebraic reasoning). This is because of our over emphasis on procedural knowledge throughout children’s mathematics experience.

At the same time, the last 20 years of mathematics education reform indicate that American students, even many college students, neither have automatized fact retrieval nor have achieved fluency with procedures. For example, many students can perform routine procedures (e.g., procedures on fractions) but cannot justify the reasons for the steps involved or provide estimated answers before they execute the procedures.

Our students’ lack of conceptual understanding is a major cause of concern and requires investment after they leave school. For example, many corporations spend large sums of money in training high school graduates in their use of simple arithmetic. The problem has also reached college and university levels. Half of the students at community colleges, and 1 in 5 students at four-year institutions, require remedial courses in writing and mathematics, with community colleges spending more than 2 billion dollars on remediation and four-year colleges $500-million. As one university leader pointed out,

Many of us in higher education have observed an increasing number of students arriving at our doorstep not fully prepared to pursue a college degree. This is our collective problem as a nation. (Chronicle of Higher Education, 2014, June 14)

Framers and supporters of CCSS-M and SMP believe that this latest educational reform, if properly implemented, can alleviate some of these problems and narrow the college and career preparation gap. Their aim is to improve instruction so that students acquire mathematics ideas with conceptual understanding and procedural fluency so they can apply mathematical tools effectively and provide reasons for what they do.

Let us consider an analogy to illustrate this point: You can watch two people swim a length of a pool. They take nearly the same time to swim the same distance, but one of them churns the water more and takes more strokes. When this swimmer gets out s/he is breathing a bit heavier but is in great shape, so it is not too noticeable. The other swimmer took fewer strokes and seemed to glide through the water.

If we assess the performance only by a stopwatch, we will conclude that they are swimmers of the same competence. In actuality, they are not.  As the lengths pile up and the task gets harder, the second swimmer will do much better. No matter what the stopwatch said, s/he is a much better swimmer than the other. And if we ask the first swimmer to swim a long distance over deep water, s/he may very well drown.

Students who only memorize facts, formulas and procedures, without understanding, are like the first swimmer. They churn and work hard, and if they are gifted with an outstanding memory, they can pull it off for a while. These students can – and often do – get by in the early grades when they can rely on their strong counting skills and contextual clues (e.g., concrete materials and pictures) to “find” a fact but they typically hit a wall sometime around 4-5th grade when they have to deal with fractions and decimal numbers with problems involving unfamiliar content and complex mathematics vocabulary words. On the other hand, children who have the rich language, robust conceptual schemas, and clear understanding of and fluency in executing procedures develop into graceful, effortless swimmers in the waters of mathematics.

In cultivating greater conceptual knowledge, effective teachers do not sacrifice procedural or factual knowledge. Procedural or factual knowledge without conceptual knowledge is shallow and unlikely to transfer to new contexts. At the same time, conceptual knowledge without procedural or factual knowledge is ineffectual and inefficient in execution. It needs to be connected to procedures so that students learn that the “how” has a meaningful “why” associated with it. It is more effective to move from conceptual knowledge to procedural knowledge. Increased conceptual knowledge helps students move from competence with facts and procedures to the automaticity needed to be good problem solvers.

When teaching conceptual, procedural and factual knowledge, effective teachers ensure that students gain automaticity. Their students know that automaticity and understanding of procedures and facts is important because it frees their minds to think about concepts and making connections. This requires some memorization and ample practice and the ability to communicate with the support of reason.

What the Research Says
The data from the 13 million students who took the Program for International Student Assessment (PISA) tests shows that the lowest achieving students worldwide are those who use memorization strategies – those who think of mathematics as methods to remember and who approach mathematics by trying to memorize facts and procedural steps. The implication is that we should not encourage rote memorization without understanding (i.e., in dividing fractions, teachers should not continue to use statements such as: “just invert and multiply”) but, instead, we should present appropriate models to create conceptual schemas and arrive at and master the procedure with proper understanding and mastery that is rooted in reasoning.

Because of the emphasis on procedural teaching, the U.S. has more memorizers than most other comparable countries. Perhaps not surprisingly mathematics teachers, driven by narrow and subjective tests (e.g., end of section tests in textbooks that value only the methods considered in the textbook), have valued those students over all others, communicating to other students that they do not belong in the mathematics class. Current research about how we learn and how our brains receive and process information shows that the students who are better memorizers may not have more ability or potential, but, unfortunately, we continue to value the faster memorizers over those who think slowly, deeply and creatively. We need students with mastery or conceptual, procedural, and factual knowledge for our scientific and technological future. Certain things should be memorized to relieve our work memory from mundane fact work to focus on creativity and applications.

Poor conceptual understanding comes at a cost. For example, if a student thinks that an equal sign means “put the answer here in the box,” she will be confused the first time she sees an equation with terms involving variables and multiple operations on both sides of the equal sign. Similarly, when a student first encounters factoring (whether in the case of whole numbers or polynomials), she ought to see its relationship to division and multiplication. But she may not be able to do so unless she has a deep conceptual understanding of multiplicative reasoning—that division and multiplication are inverse operations. She also will be slowed in factoring if she hasn’t memorized the multiplication tables, divisibility rules, prime factorization, and short division.

Factual and procedural knowledge are acquired by practice and stick-to-ness to tasks and practice. But, to be engaged in the task deeply, the task should be meaningful and the student should have understanding of the concepts and skills involved in it.

Of the three varieties of knowledge that students need, conceptual knowledge is difficult to acquire. It is difficult because knowledge is never transferred from one person to another directly nor is it developed without explorations, concrete manipulations, effective reasoning, and questioning. Rather, new concepts must build upon something that students already know when they explore the new concepts and ideas. Examples that are familiar to students and analogous to the current concept are useful to understand the concept.

SMP Principles
The writers of the CCSS-M were careful to balance the development of conceptual understanding, procedural skill and fluency, and application at each grade level. The standards are based on the idea that procedural skill and fluency expectations hinge on conceptual understanding. Fact fluency and procedural fluency help students to develop concepts, make connections, observe patterns, and form relationships between ideas, concepts, skills, and procedures thereby facilitating mathematical thinking. With mathematical thinking, students take interest in mathematics and develop mathematics stamina.

To achieve the different kinds of knowledge, we need to adopt pedagogical principles in every lesson that are informed by the Standards of Mathematics Practice:

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

Each of these standards adds to a teacher’s ability to develop the different components of knowledge, helping children acquire mathematical ways of thinking, creating interest for mathematics, and recognizing the power of mathematics. We need to incorporate these practices in our lessons if we want to have students who enjoy doing mathematics and achieve higher.

Examples of Standards
In future posts, I want to comprehensively develop how to implement each of these standards. At present, I want to consider a few illustrative examples.

Standard number one, for example, deals with developing understanding and engagement with a problem and creating mathematics stamina in solving it.

Solving a problem almost always depends on what tools (linguistic, conceptual, and procedural) one knows and how to connect those tools with the current problem. As students advance and encounter new problems, new concepts will increasingly depend on old conceptual knowledge. For example, understanding and solving algebraic equations depend on the understanding of the concepts of equality, variable, arithmetic operations, operations on fractions and integers, ability to generalize, etc. In solving problems factual, procedural, and conceptual knowledge all go together. And to stay engaged with the problem requires a student to have mastery of these prerequisite tools.

Familiarity is not the only ingredient necessary for successful problem solving. Students are more likely to understand abstract ideas when they see many diverse examples in the classroom that depict the conceptual components, schemas, and constraints of the ideas. In such instances, they can learn the essential properties in the concept of the problem (e.g., in fractions, the division of the object into equal parts and what is equal there) and which properties are incidental (e.g., in fractions, that the resulting parts need to be whole numbers).

Standard number seven: to look for structure in mathematics concepts and procedures is at the heart of understanding mathematics. Students need to realize that mathematics is the study of patterns—the underlying structures. Students frequently fail to understand the concept if they are not helped to discern patterns—to look for the structure—commonalities among examples and what is different in these examples.

Indeed, when the teacher introduces a concept through an abstract definition alone, e.g., the standard deviation is a measure of the dispersion of a distribution, students miss the conceptual understanding of the concept of standard deviation. The standard deviation is dependent on the spread; therefore, our examples should show the impact of that spread on standard deviation: Two groups of people have the same average height, but one group has many tall and many short people, and thus has a larger distribution and standard deviation, whereas the other group mostly has people’s heights right around the average, and thus has a small standard deviation.

If we introduce students to the formal procedure of finding the standard deviation too quickly, they won’t realize the relationship of spread on standard deviation. To realize the importance of such a component in the concept, we should also consider special cases and non-examples: e.g., to find the standard deviation when every piece of the data is the same, the spread has a particular meaning.

The third standard is the key to developing conceptual understanding in students. No conceptual understanding can be developed without emphasizing reasoning. To emphasize the importance of reasoning— concrete and abstract, examples, non examples, and counter examples play a crucial role.

Let us consider the definition of prime number: A number is called prime if it has exactly two distinct factors, namely, 1 and itself. The definition of prime number has two key features: it has factors and there are exactly two of them. 2, 3, 5, etc. are prime numbers as they have two factors only. It is important to give examples of prime numbers and examples of numbers that are not prime and the reasoning behind the choice. Examples emphasize the components of the concept, and “non examples” help students see the subtleties and nuances of the concept. For example, why is 1 not a prime number? (It has only one factor.) Why is 0 not a prime number? (0 is divisible by any non-zero number; therefore, it has more than two factors.)

The discussion to discern similarities and differences—comparing and contrasting examples, helps students to acquire the language and conceptual knowledge with rigor. With the help of appropriate language and transparent models, conceptual knowledge is converted into procedural knowledge. For this to happen, the conceptual (representational models—concrete and pictorial) should be congruent with abstract, procedural models. And with practice, procedural knowledge is then converted into factual knowledge. This practice should take place in a variety of problems and problem situations that are related to the procedure to arrive at the appropriate level of fluency.

If students fail to gain conceptual understanding, it will become harder to catch up, as new conceptual knowledge depends on the old. Students will also become more likely to simply memorize algorithms and apply them without understanding.

Helping Students Learn Concepts and Procedures
In our schools, much is made of the use of manipulatives to help children understand abstract concepts in mathematics, but many manipulatives and models themselves are abstract (students treat them as a symbol for something else), and not all manipulatives help learning—they sometimes impede it. This is most likely when manipulatives are so visually interesting that they distract from their purpose, when their relationship to the concept to be represented is obscure, or when they are used for rote counting. Manipulatives seem helpful because they are concrete; to be helpful, they should satisfy certain properties.

To illustrate the idea of a fraction, one might divide a cookie in two for the purpose of sharing it with a student. The concreteness of this example is likely less important than its familiarity. In contrast, suppose I cut a hexagon into two pieces and said, “See? Now there are two equal pieces. Each one is half a hexagon.” That example is concrete but less effective because it is unfamiliar; the student has no experience with divided hexagons, and the purpose of sharing is also missing.

Concreteness, in itself, is not a magical property that allows teachers to pour content into students’ minds. It is the familiarity that helps because it allows the teacher to prompt students to think in new ways about things they already know. However, familiarity also may create some misconceptions, half a pizza, half a cookie, half a glass are not precise as key characteristics of fractions may be missed. Students know a fraction when they focus on: (a) What is my whole here? (b) How many parts are there in this whole? (c) Are the parts equal? (d) Do all the parts together make the whole? (e) What is the name of each part? (f) How many of these parts will make the whole? And (g) What is the new name of the whole in the light of these parts? The teacher’s language, questions, and sequence of activities with materials transform the concrete models into representations—pictorial and abstract.

A teacher must move from familiar materials and models to the form that shows all the attributes of the concept and then can lead to abstract representations that are congruent to the abstract procedure. As concepts become more complex, it becomes harder to generate familiar examples from students’ lives to generate mathematics conceptual schemas, and teachers may have to use analogies more often. In such cases, a familiar situation is offered as analogous to the concept under discussion, not as an example of the concept.

An Example of Implementing SMP
In order to focus instruction responding to CCSS-M and applying SMP, teachers need to identify the essential language, concepts, procedure and skills in each major standard. SMP does not advocate a “one size fits all” model or a boxed curriculum. Essential Elements of each lesson informed by SMP include: Integration of language, concepts, and procedures, Multisensory (appropriate and efficient models), decomposition/ recomposition of problem components, Synthetic-Analytic (seeing patterns and analyzing the problem), Structure (logical language categories), Sequential (simple to complex), Cumulative (continually making connections), Repetitive, cognitive (meta), Diagnostic and Prescriptive (design lesson to assure progress and plan next lesson around noted errors that need additional reinforcement).

Let us illustrate this process in one specific mathematics standard. Mathematics standard 4.OA.A.3 (CCSS-M, 4th grade) says:

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

This standard includes only a few sentences, but involves several different terms, concepts, procedures, and skills. To make sense of and understand them, students need to know the meaning of these terms and concepts and execute the procedures.

There is a need to delineate these elements, focus on them, teach them, help students master them, connect them with other concepts and procedures, and then assess all of these elements.

There should be a clear understanding of what and how to represent each concept, procedure and the skill involved in this standard. Every concept and procedure involved in this standard should be transformed into a set of concepts and skills to be learned, mastered, and applied by the students. In the context of CCSS-M, teaching should be to acquire understanding; students should arrive at fluency and should be able to apply concepts and skills contextually.

Step # 1
Language and Concepts

  • Know the meaning of each word and term in order to translate from English to mathematical equations
  • Identify the unknowns and understand the role of these unknowns; know the relationship(s) between knowns and unknowns

Step # 2
Language and Concepts

  • Represent terms and words into appropriate mathematics symbols; translate multi-step word problems into/by equation(s)

Skills and Facts

  • Identify the units and the domain and the range of the variable(s) involved in the problem

Step # 3

  • Solve multistep word problems by establishing the sequence of arithmetic operations

Skills and Facts

  • Know and apply the properties of equality; mastery of arithmetic facts; execute procedures for whole numbers efficiently; know the order of operations

Step # 4
Concepts and Procedure

  • Assess the reasonableness of the answer

Skills and Facts

  • Numbersense: Use mental computations such as rounding to estimate the outcome of an operation

Step # 5
Language and Concepts

  • Interpret the answer including the remainder if involved; express the division problems in multiple ways

Skills and Facts

  • Add, subtract, multiply and divide whole numbers fluently with understanding; know the role of numbers in each operation, e.g., know the role of remainder in practical situations

Learning with rigor using SMP means that the students not only understand the concept and procedures but also see that a particular method(s) may have limitations and that the context of the problem defines the applicability and efficiency of the method.


Effective Teaching of Mathematics

Language and Number

My experience in introducing young children, with or without learning disabilities, to number concepts and number relationships shows that language alone (without any or limited concrete manipulative experiences) cannot effectively develop the concept of number. Some initial understanding of number is independent of language. For example, young children acquire the concept of number earlier than they do pluralization rules, showing that they can learn number one and generate other numbers from one. This is achieved at a concrete level. However, concrete experiences alone are not enough for the abstraction of number from specific concrete activities to a general notion of numberness and number sense. For that they need language, rich language.

For every concrete experience to reach a conceptual level and to last as a concept that can be communicated, an appropriate and stable label (language container) is needed for the concept being abstracted and acquired from the concrete experience.

Concrete experiences accompanied with language (questions being asked, commentary being made, descriptions being furnished) convert the concrete experiences into abstractions with labels. These interactions between concrete experience and language provide such labeling (language containers) and abstractions inherent in them. Persons in the child’s environment (parents, teachers, and siblings) mediate this interaction in the form of posing questions, affirming responses, and creating practical problem solving situations.

Conceptualization of number begins with concrete experiences, but its acquisition is facilitated and accelerated by language and appropriate concrete materials and actions. Scaffolding questions accelerate the symbolizing and abstracting from concrete experiences.

In other words, concrete experience is the starting point for number conceptualization, and language furthers the development of the abstraction process. In most cases, language experiences alone are not enough for number conceptualization. In the initial states, language plays a role, but it is a limited one. Initial number conceptualization is not an artifact of language because it is dependent on concrete models. The interaction of rich language and quantification with efficient concrete materials facilitates the development of number concept. Once the number is conceptualized and there are adequate language containers, in most cases, further concepts can be developed with language alone. However, it is not advisable as the purpose of concrete materials is to help to

(a) generate the language (language containers—vocabulary),
(b) create conceptual schemas, and then, finally,
(c) arrive at symbolic/procedural representation of the concept.

The role of concrete materials is not to solve every problem with them or to become dependent on them as we see in many special education settings. To gain efficiency and generalization, children should transcend the concrete materials and language containers and should be able to model practical problems using mathematical ideas and schemas.  Ultimately, children should express their understanding and problem solving ability through mathematical symbols and their manipulations such as formulas, expressions, and equations.

Mathematics Language and Native Language
Once number is conceptualized, its applications and related concepts are much more dependent on language. Number conceptualization is the beginning of the development of the language of mathematics—its terms, words, order of usage, and translations from mathematical expressions to English language and vice versa. Mathematics is a unique language. Numbers are its alphabet and number symbols are its smallest thinking units. Mathematical symbols, concepts, and language modify the natural language, and they in turn shape mathematical thinking.

In the child’s native language system and in the context of concrete experiences, numbers and quantification function as predicators and qualifiers—modifying the nouns in their scope and meaning quantitatively, such as five dishes, two toys, many books, fewer children, some books, a pen, etc. They function, in some respects, like adjectives in a sentence in the native language. Children are quite fluent in this before they begin school as pre-Kindergartners or Kindergartners.

Later, in the language of mathematics, numbers are real and even concrete and do not generally function as predicators; number characteristics are described by other predicators. For example, six hundred is a big number, an even number, or a much smaller than the number six hundred thousand. Thus, in the formal language of mathematics, numbers are abstract, singular objects, but at the same time, in mathematical operations, numbers are ‘real,’ concrete entities—for example, when a child says: “I will show you number four” and then shows four fingers. This thinking of numbers as concrete and abstract at the same time is at the core of true number conceptualization.

Holding in the mind the idea that a number is concrete and abstract at the same time is difficult for many children. At the same time, this need to understand numbers as concrete and abstract creates hurdles for many children to become fluent in its usage in other mathematics concepts and its applications outside of mathematics. Thinking of number as concrete and abstract at the same time is therefore a real challenge for many children. It is perhaps the most paradoxical thing about numbers—their concreteness and their abstract representation, and this is the feature that makes them so useful.

Transitions: Concrete to Abstract and Native to Formal Language
Acquisition of number concepts, then, requires that the child perform two types of abstractions: one abstraction when a child translates sensory, concrete representations of quantity into symbolic entities (five fingers is translated into the number 5). The second transition is when the child transforms the conception of number as a predicate in his native language to its conception as an object in the language of mathematics (I am showing you five fingers to look this is 5.). For some children, particularly learning disabled (LD) children, these transitions are not easy and need to be facilitated carefully by a knowledgeable person using appropriate language, an enabling questioning process, and efficient instructional models.

These two transitions are significant and involve complex processes. They evolve through a prolonged constructive process, which begins with native language and concrete experiences such as counting on fingers and ends in the child’s conception of number as part of the formal mathematics language and symbolic logical system. In the elementary grades, the child is constantly struggling with these roles of number and the interrelationships between the predicative function of numbers in the native language and their status as objects in the language of mathematics.

The mediation between the native language and the language of mathematics depends on the quality of concrete models and linguistic experiences provided to the child. Effective teachers provide a variety of concrete learning experiences, and their expert questioning and deft use of language facilitate these transitions.

Language and Number

Building a Mathematics Vocabulary

We cannot receive, hold and manipulate a concept without having either an image/schema or words for the concept. The image could be a picture, figure, drawing, or symbols. In the case of language it can be a word, expression, or an equation.

For effective communication of mathematics ideas, children need robust and rich images and vocabularies (language containers). Without appropriate language containers, children cannot retain and communicate mathematics ideas. Vocabulary—words, expressions, phrases—are the language containers for mathematics concepts.

Learning mathematics, then, is using, creating, extending, and modifying language containers—the vocabulary of mathematics. Students’ proficiency in mathematics is directly related to the size of the set of their vocabulary. Rote memorization of a collection of words is not enough to master the language of mathematics. Instead, one has to acquire the related schemas with understanding. Language proficiency refers to the degree to which learners exhibit control over their language.

The introduction of mathematics vocabulary and terminology should be contextual, but even direct study of quantitative and spatial vocabulary contributes significantly to improved mathematics conceptualization—learning new concepts, creating deeper and robust conceptual schemas, and more effective communication.

When children create and encounter a language container for a mathematics concept, they also create and invoke the related conceptual model in their minds. Each word and expression such as sum, product, rational number, least common multiple, denominator, rectangular solid, conic section, and asymptotic represents a concept with its related schema. For example, if a person understands the definition of multiplication as ‘repeated addition’ or ‘groups of’, then these expressions invoke the conceptual schema. The expression 43 ´ 3, will invoke: 43 repeated 3 times (43 + 43 + 43) or 3 groups of 43 (43 + 43 + 43). If multiplication is learned as the ‘area of a rectangle’, then 3 ´ 43 will invoke an image of a rectangle with dimensions 3 (vertical side) and 43 (horizontal side).

The development and mastery of mathematical vocabulary are the result of a long and continuous interactive process between native language, mathematics language and symbols, and their quantitative and spatial experiences. This begins with play and concrete experiences in children’s environment. Experiences are represented through pictorial and visual forms and means, which then may result in abstract mathematics formulations and problems that students solve. This mathematics formulation—devising of abstract symbols, formulas, and equations, is then applied to more problems, and the result of this process is communicated. Successful communications demonstrate that the child has mastered a concept. The process can be summarized as:

  • Understanding the environment (concrete experiences and use of native language).
  • Translation (native language to pictorial and linguistic forms).
  • Representation (in the native language).
  • Description and verbalization (in the native language).
  • Discussion (in the native language).
  • Mathematical formulation of the problem (in the mathematical language).
  • Manipulation of mathematical language.
  • Communication of the outcome of mathematics operations (in mathematics and native languages).

This communication furthers not only children’s mathematics achievement but also their language development.

Building the Vocabulary of Mathematics
Many of children’s mathematics difficulties are due to their limited vocabulary—its size, level, and quality. A child’s size and level of vocabulary is the intersection of three language sets:

  • The level and mastery of the native language and background the child brings to the mathematics task.
  • The level and sophistication of language that the teacher uses and the questions she asks to teach mathematics.
  • The language set of the mathematics textbook being used.

The intersection of these three language sets is the available language the child has to learn mathematics. A small intersection means the child has a limited vocabulary. The objective, then, is to increase the size of this intersection. A child’s limited mathematics vocabulary may be for many reasons.

  • The mathematics problems of the child with English as a second language in a classroom where the medium of instruction is other than the child’s native language.
  • The child’s and teacher’s economic, cultural, and geographical backgrounds differ. For example, the linguistic problems that many urban black children and immigrant children face are an example of a linguistic/cultural mismatch and the assumptions teachers make in instructing children.
  • Textbook language sets differ from the language sets of the children and the teacher.

Whatever the reasons for limited language sets, we need to help children acquire a robust mathematics vocabulary. Properly acquired and used in context, a mathematics vocabulary has a profound effect on children’s mathematics achievement and their thinking. Planned activities for developing, expanding, and using vocabulary contribute significantly to better mathematical word problem-solving ability and support learning new concepts, deeper conceptual understanding, and more effective communication.

Although more textbooks are emphasizing the language of mathematics, there is still little attempt to develop a coherent and comprehensive mathematics vocabulary in school mathematics teaching. In one textbook, the expression “find the sum” is introduced quite early. In another series, the expression is introduced much later, and then the words “find the sum” and “add” are used interchangeably. In another text, the word “sum” is used sparingly. Consequently, a child may face different language sets from grade to grade and from school to school. Although the textbooks have a large number of common language terms and vocabulary, many words are not in common. Further, some textbooks use so much language without properly introducing the terms that many children find textbooks frustrating. Exercises do not provide enough practice in basic skills, which prevents children from automatizing the language or the conceptual skills associated with them.

Strategies for Enhancing the Mathematics Vocabulary
Ways in which children’s failure to develop mathematical vocabulary may manifest as: (1) children have difficulty conceptualizing a mathematics idea; (2) they do not respond to questions in lessons; (3) they cannot perform a task; and/or (4) they do poorly on tests, particularly on word problems.

  • Their lack of conceptualization of a mathematical idea may be because they do not have the language for the concept to receive it, comprehend it or express it, such as ‘find the sum of’, ‘union of two rays…,’ ‘evaluate…’
  • Their lack of response may be because they do not understand spoken or written instructions such as ‘draw a line between…’, ‘touch the base of the triangle’, ‘place a positive sign next to the numeral,…’ or ‘find two different ways to…’
  • They are not familiar with the mathematics vocabulary words such as ‘difference’, ‘subtract’, ‘quotient’, or ‘product.’
  • They may be confused about mathematical terms such as ‘odd’ or ‘table’, which have different meanings in everyday English and have more precise meanings in mathematics.
  • They may be confused about other words and symbols like ‘area’ and ‘perimeter’, ‘factor and multiply’, ‘and’.

To enhance children’s vocabulary, every school system should have a minimal mathematics vocabulary list at each grade level. Mastery of words from such lists will prepare children to communicate mathematics. This list can also be used to assess students’ grade level language of mathematics. This list should indicate the grade of introduction of words, terms, and definitions and the level where they are mastered. It should be developmentally and linguistically appropriate. The teacher should constantly identify, introduce, develop, and display the words and phrases that children need to understand and use.

The teacher should use the same techniques to introduce mathematics words as she teaches native language. She should have a Math Word Wall for every mathematics concept she teaches. When a new word related to the concept emerges in discussion, it is added to the Word Wall. With the introduction of each word, students are exposed to several words and concepts that contain it. Then students use it in their own words, with as many examples as they can. The teacher selects a word and then asks children to use it in mathematics context. The following exchange illustrates this process.

  • Give me a sentence that uses the word ‘add.’”
  • You have $5 and I have $14. Let us add both amounts.”
  • That is great! Now use the word ‘sum’ in a sentence.”
  • That is easy. If we add our monies, what is the sum of our monies?”
  • That is great! Now I am going to write some words on the board. I want you to first to tell me and then write a sentence or two using each word. If you want, you can use more than one word in a sentence.”

The concepts are then reviewed in circular fashion, built upon, and tied into new ideas. This helps children construct a working vocabulary that is constantly augmented, and they are also learning skills to build it.

Once the key root words have been introduced to children, the teacher can begin to extend the mathematics vocabulary words. Among the easiest sets are the words formed with prefixes, suffixes and derivative words. The process is to introduce the math prefixes and roots casually and then formally. In a casual manner, parents and teachers can remark, “You know a tricycle has 3 wheels. Tri- means 3 and cycle means wheels.”

Teacher: What will be the name of the object that has three angles?
Student:  A triangle.
Teacher: Why?
Student:  A triangle has 3 angles and tri- means 3.
Teacher: Now draw a triangle on your paper.
Children draw triangles on their papers.

Teacher: The word ‘lateral’ means a side. What will you call an object that has three sides?
Student:  A trilateral.
Teacher: Now draw a trilateral on a paper.
Children draw a trilateral on their papers.

Teacher: If the word ‘gon’ means a corner, what will you call an object that has three corners?
Student:  A trigon.

Teacher:  If ‘octo’ means eight, what does ‘octagon’ mean?
Student: A figure with eight corners.

As with all language development, there is a sequence in moving from speech ability to writing ability: the input is auditory in its foundation (the child is immersed in oral linguistic experiences), then followed by speech ability (the child produces language) and later by reading and writing ability. When young children have this kind of foundation, they avoid the anxiety of making sense of key foreign words later on in a formal setting. They will be able to generalize and relate math concepts to their daily experiences.

Instructional Suggestions for Language Proficiency
There are practical reasons children need to acquire rich and appropriate vocabulary for them to participate in classroom life—the learning activities and tests. There is, however, an even more important reason: vocabulary, as part of mathematical language, is crucial to children’s development of thinking not only in mathematics problem solving but in general problem solving. Once children have control over their language usage, they begin to have control over the meta-cognitive skills that produce insights into their learning and their interactions with learning tasks. Language and thinking are interwoven in reasoning, problem solving, and applications of mathematics in multiple forms—intra-mathematical, interdisciplinary, and extracurricular. If children do not have the vocabulary to talk about a concept, they cannot make progress in understanding its applications—therefore solving word problems.

Teachers often use informal, everyday language in mathematics lessons before or alongside technical mathematical vocabulary. This may help children’s initial grasp of the meaning of words; however, a structural approach to the teaching and learning of vocabulary is essential to move to higher mathematics using the correct mathematical terminology. This also applies to proficiency. The teacher needs to determine the extent of children’s informal mathematical vocabulary and the depth of their understanding and then build the formal vocabulary on it.

It is not just younger children who need regular, planned opportunity to develop their mathematical vocabulary. All students and adults returning to education need to experience a cycle of concrete work, oral work, reading, writing, and applications.

The teacher needs to introduce new words through a suitable context, for example, with relevant, real objects, mathematical apparatus, pictures, and/or diagrams. Referring to new words only once will do little to promote the learning of mathematics vocabulary. The teacher should use every opportunity to draw attention to new words or symbols with the whole class, in small groups or with individual students. Finally, the teacher should create opportunities for children to read and write new mathematics vocabulary in diverse circumstances and to use the word in sentences.

  • Concrete work: Concrete materials/models develop images and the language for mathematics ideas. The concrete materials/models help children (a) generate the language, (b) understand the concept, and (c) arrive at an efficient procedure. Students should be encouraged to explore and solve problems using manipulative materials and asked to discuss and record the activity using pictures and symbols. The teacher or a student can also act the word out.
  • Writing work: The teacher should explain the meanings of words carefully. The teacher should refer to a similar word; give the history and the derivation of the word and write it on the board. Children should copy it in their Math Notebook. The teacher should ask the children to say the word clearly and slowly. They should rehearse the pronunciation of the word. The teacher should ask them to spell the word and ask a child to say the word and spell it with eyes closed.
  • Oral work: Students describe the work done at the concrete level, using mathematics words and expressions based on the visual and tactile experience of the meaning of mathematical words in a variety of contexts. This oral work may be facilitated in different contexts by
    • listening to the teacher or other students using words correctly
    • acquiring confidence and fluency in speaking, using complete sentences that include the new words and phrases, in chorus with others or individually
    • discussing ways of solving a problem, collecting data, organizing data and discussing the properties of the data for a variety of reasons: to generate hypotheses, develop conjectures or make predictions about possible results or relationships between different elements and variables involved in the problem
    • presenting, explaining, communicating, and justifying methods, results, solutions, or reasoning, to the whole class, a group, or partner
    • generalizing or describing examples that match a general statement
    • encouraging the use of the word in context and helping sort out any ambiguities or misconceptions students may have through a range of open and closed questions.

Because students cannot learn the meanings of words in isolation, I believe in the centrality of reading and conversation in mathematics lessons. Shared reading is a valuable context for learning and teaching not only mathematics language but also mathematics content. Strategies such as using children’s books, stories, DVDs, and videos as a vehicle for communicating mathematical ideas develops mathematical language. Reading word problems aloud and silently, as a whole class and individually, is equally important. During these readings, the teacher should ask questions involving mathematics concepts. This develops strong mathematics language and understanding. Students can be asked to read and explain:

  • numbers, signs and symbols, expressions and equations in blackboard presentations
  • instructions and explanations in workbooks, textbooks, and other multi-media presentations
  • texts with mathematical references in fiction and non-fiction books, books of rhymes, children’s books during the literacy hour as well as mathematics lessons
  • labels and captions on classroom displays, in diagrams, graphs, charts, and tables
  • definitions in illustrated dictionaries, including dictionaries that the children have made themselves, in order to discover synonyms, origins of words, words that start with the same group of letters (e.g. triangle, tricycle, triplet, trisect…), words made by coding pre-fixes or suffixes, words derived from other words.

All students from K through 12 and adults returning to education need to work on developing their mathematics vocabulary.

Building a Mathematics Vocabulary

Mathematics as a Second Language

For many students, achieving proficiency in reading and writing in their native language is a difficult task. The use of language in content specific disciplines such as mathematics makes that task even harder. In fact, students need considerable proficiency in the native and mathematical language because in a mathematics classroom, they need to constantly translate between mathematical and everyday language. For this reason, the framers of the Common Core State Standards (CCSS-M, 2010) recommend that schools and publishers of mathematics textbooks provide a high degree of familiarity with words, syntax, and grammar, as well as styles of presentation and arguments that are not part of informal talk.

Many people perceive mathematics learning as difficult, and increased linguistic demands heighten this perception. Effective teachers have always recognized language as an essential tool for the conceptualization of mathematics. The language used to convey mathematical information in teaching and learning is important because:

  • The number of students in our schools with limited English language experiences or who speak English as a second language has increased dramatically.
  • Students confront specialized and rigorous language in new mathematics textbooks.
  • The concepts in reform curricula make new demands on students’ linguistic abilities and emphasize an enlarged vocabulary.
  • The development of the conceptual schemas is not only facilitated by a strong linguistic component but also depends on it.
  • The increased realization that mathematics achievement is directly related to a student’s mathematics language development, particularly the size of the mathematics vocabulary. Early and better performance in mathematics predicts achievement in higher mathematics as well as in reading.
  • The modern technological society is built on strong numeracy as much as it is on literacy.

The Role of Language in Mathematics Learning
The vital role of language in mathematics performance is widely recognized. Most people see mathematics language only in the context of word problems, but it is much more than that. Every mathematical idea involves three components: linguistic, conceptual, and procedural. To learn an idea means to create linguistic and conceptual models for it. We need to have a language container to receive, comprehend and explain a concept. Without an internalized language container, we need to relearn the concept every time we encounter it.

The term language container means a word or a phrase to express an idea with related conceptual schema: sum, even number, least common multiple, denominator, rectangular solid, and conic section. Students’ proficiency in mathematics is directly related to the size of the set of their language containers. However, rote memorization of a collection of words is not enough to master the language of mathematics. One has to acquire the related schema with understanding.

Mathematics uses special words and phrases and many everyday words in particular ways and with special meanings to describe phenomena and concepts. The difficulty for children lies in the gap that exists between their native language and the language of mathematics. For many, this gap is a barrier to learning and using mathematics.

Poorly written textbooks cause some of the linguistic difficulties children have in mathematics. New technical terms crowd textbook pages. Explanations are unintelligible. New words and terms are introduced only as recipes, without adequate explanations and examples.

Children’s mastery of most mathematical concepts is dependent on the interplay between language, concepts, and models. Mathematical conceptualization is independent of language in early childhood. However, once a child has acquired language fluency, mathematics and language interact. Progress in one enhances development in the other (except perhaps when there is a learning disability).

Ability in mathematics is a manifestation of two different aspects: mathematical insight and knowledge and facility with language. Students who have difficulty with literacy often find themselves having similar problems with numeracy. For instance, almost 40 percent of dyslexics also show symptoms of dyscalculia.

Children from cultural and social environments with little or no emphasis on numeracy are not well prepared to learn formal mathematics as they lack language containers for these concepts. They have a backlog of numeracy learning to catch up. In contrast, children exposed to quantitative and spatial representations bring prior knowledge about quantity and space to school. Children with facility in their native language, even if it is different from the language of instruction, are better prepared for numeracy. If they possess the language containers for concepts in their native language, they can develop language and conceptual schemas in the second language.

Acquiring the Mathematics Language
Mathematics is a second language; it has its own alphabet, symbols, vocabulary, syntax, and grammar. Numeric and operational symbols are its alphabet; number and symbol combinations are its words. Equations and mathematical expressions are the sentences of this language.

Mastery of a mathematical concept is the result of an interactive process between language and quantitative and spatial experiences. Initially, concrete experiences with quantity and space form concepts and are communicated through visual representations and artifacts. Later, children learn to represent them symbolically/abstractly. Abstract symbols, formulas and equations are then applied to solving problems. This iterative and cyclic process is called mathematization.

The various linguistic activities serve different purposes in developing conceptual schemas and the acquisition of mathematical procedures and skills. To develop a mathematics language we need: Vocabulary and symbols, Syntax, and two-way translation.

Vocabulary and symbols: words, terms and symbols can represent a complex concept. Comprehending the statement of the problem (the terms and words involved) and understanding the intent of the problem (what concept and procedure is involved in the problem) requires a student to have a strong vocabulary and associated conceptual schemas. For example, on a recent state examination, some students did not answer the problems (Find the sum of 8.7 and 5.2. Find the product of 1.2 and 1.3.) because they did not know the meanings of sum and product.

Every word has at least five meanings: (a) epistemological—the origin of the word, (b) historical—the meaning acquired over time, (c) intended meaning, (d) current meaning, and (e) meaning received and understood by the reader (as Marshall McLuhan said: “message received is message sent.”).

To truly understand the meaning of a word, one needs to understand as many meanings of the word as possible.  Words are language containers for ideas and concepts. We cannot have a concept, if we do not have a language container for the concept. Similarly, a word has no value for a person, if he does not have a concept behind the word.  Understanding a word means that person has an associated schema with the word and can also use it.

Comprehension in reading and understanding in mathematics come when the child possesses the conceptual schemas behind words. This is particularly so if one wants to help children learn a second language and the meanings of words and expressions in the language. The challenge becomes more complicated when children are learning a second language like mathematics where every word is packed with complex concepts and schema. 

Syntax: organization of words and structure of mathematical expressions. Some children’s mathematics difficulties are due to not understanding the order of words in a sentence. For example: the difference between ‘subtract 5 from 3’ or ‘subtract 3 from 5.’ .75 divided by .89 or .89 divided by .75.

Translation: translating from mathematics sentences into English and from English into mathematical expressions. When students encounter a word problem, many ask the teacher to supply the operation involved in the problem. Once the teacher provides the operation, the students perform the appropriate operation and give the answer. Often, students solve a problem but do not know what the answer means. Both of these examples are problems of translation. To be proficient in mathematics, students have to navigate between mathematics and native languages.

Mathematics as a Second Language

Mathematics Words and Expressions

Transmission of knowledge requires language or symbolic representations. Most fields of knowledge require language for its communication. Each domain uses words, expressions, phrases and collections of words that have specific and contextual meanings. Mathematics, too, has special symbols, words and phrases, as well as everyday words, which are used in particular ways. It has its own vocabulary, syntax, and its structure.

At times, major gaps exist between a child’s language outside the class and the abstract symbolism and language of school mathematics. Because of the complexity of the mathematics language and because mathematics is a second language for most children, many experience difficulty in learning it.

The size of a child’s mathematics vocabulary is highly correlated with mathematics achievement. Mathematics language and concrete models help construct the corresponding conceptual schemas in the child’s mind. Through the linguistic component (lexical entries—language containers) the child understands, uses, and retains the mathematics concepts. Most words used for learning mathematics are derived from the native language of the learner (e.g., add) and many are carried over into mathematics from other languages (e.g., trigonometry). Mathematics vocabulary consists of five kinds of words or expressions:

  • Words that have the same meaning in mathematics and outside of mathematics. For example, circle, add, cylinder, etc. For these words, students already have conceptual schemas and experiences. In other words, they bring some ideas related to these words. When these words are used in mathematics, students can expand the existing schemas and meanings to accommodate the use of these words in mathematics. For this reason, these words and related mathematics are easily accessible to students.
  • Words that have different meanings in mathematics and outside. For example, distribute in common language means division, but distributive property over multiplication a(b + c) = ab + ac means multiplication by a to the sum of b and c. There are many words of this type: and, or, table, fraction, tangent, sum, product, slope, mean, mode, median, etc. Because these words have specific meanings in mathematics, the teacher needs to create new conceptual schemas either by setting patterns, concrete modeling, analogies, or reason. Relying on the meaning and usage of the word from the native language may create difficulties and misconceptions.
  • Words that are unique to mathematics and do not exist outside of mathematics. For example, exponent, quotient, numerator, denominator, trigonometric ratios (such as sine, cosine), conic sections (such as hyperbola), differentiation, integration, quadratic equation, scale, figure, etc. These words are new to most students, so a teacher must take great care in their introduction to students. Concrete modeling and practice are the best ways to introduce them. For example, terms such as conic sections (circle, ellipse, parabola, hyperbola) are best understood when students actually make sections from the cone.
  • Compound words that are formed using competing and even opposite meaning. Words and expressions in this category include: least common multiple, greatest common factor, simultaneous linear equations, etc. For example, the term Least Common Multiple of numbers of 4 and 6 is best understood when we compare this expression with the statement: John is an intelligent, handsome, tall boy. The words intelligent, handsome, tall are qualifiers of the word boy and are to be seen in a certain order. Similarly, Least Common Multiple of 4 and 6 must be seen in a certain order. The order of the activity is: first we find the multiples of 4 and 6. Then we mark (the numbers in bold) the common multiples of 4 and 6. Lastly, we select the least of these common multiples (12). Therefore, the Least Common Multiple of 4 and 6 is 12. It is important to emphasize the interaction of the words in the expression Least Common Multiple. The same process applies in the case of Greatest Common Factor of 20 and 24.

4     8   12   16   20   24   28   32   36    40   44   48, …

     6     12      18       24      30        36        42       48, …

  • Compound words formed by combining different concepts. These include: miles per hour, scale factor, density, etc. Before understanding these types of words (compound secondary concepts), students need to have a clear understanding of the corresponding component primary concepts involved in the secondary concept.

Just like in the native language, every word may have different meanings according to the context. For example, a word in the English language may be understood and its meaning found by examining its (a) entomological roots, (b) historical usage, (c) popular meaning, (d) contextual, and (e) meaning received by the listener or reader. Mathematics words, most of the time, are more consistent in meaning than they are in the native language. However, the same approach to examining the roots and history of the word may help us with the meaning of the word or expression in the context of mathematics.

Mathematics Words and Expressions