Common Core State Standards in Mathematics: Introduction

It is the significance of detail wherein the truth lies. Nadine Gordimer

Mathematics is a broad, multi-faceted, and multidimensional subject; it is about big ideas, deep concepts, elegant procedures, numerous and diverse applications. To think mathematically affords a powerful means to understand and control one’s social and physical reality. An important aspect of learning mathematics is to become aware of the fact that mathematics is indeed a useful tool for action and understanding. Yet, despite some 12 or so years of compulsory mathematical education, most children, even in the developed world, leave school with only a limited access to mathematical ideas and their vast applications. The Common Core State Standards in Mathematics (CCSSM) are an attempt to bring the big ideas and their elegance to school mathematics curriculum.

Developed in 2009 through a collective effort by educators and others across the country, the Common Core State Standards are also meant to provide some answers to our intractable school problems: 33% of all students drop out of school in the United States. Only 50% of Latino, African American, and Native American students in the United States complete high school. According to a recent Gates Foundation–funded study, 81% of those who drop out of school claim that “opportunities for real world learning” would have improved their chances of staying in school. 69% were “not inspired to work hard” and 47% said “classes were not interesting.” Significant to these findings is also the fact that only 35% of those interviewed claimed that they left because they were “failing in school.”

To solve these problems, we need to engage students in learning: set up activities and classroom where they are encouraged to ask good questions, map out solution pathways to problems, reason about complex solutions, interpret solutions, set up models and communicate in different forms. We need students who can do real mathematics, not just calculate quickly in math. Doing real mathematics is about inquiring, communicating, making connections, and representing ideas in multiple forms. Doing real mathematics is problem solving, modeling, thinking, and reasoning, as these are the mathematical abilities for the workplace and a technology rich world. This broad, multidimensional mathematics is the math that engages many learners and puts them on a pathway to lifelong success. All of these ways are encouraged by the CCSSM.

CCSSM Goals
The CCSSM’s goal is to achieve a higher level of mathematics competence for American students. For each grade level, K-8 and High School, the CCSSM list the most important mathematical concepts and “what students should understand and be able to do” with them. Unfortunately, there is little guidance for implementing the CCSSM although they are accompanied by suggestions for instructional practices—the Standards of Mathematics Practices (SMP).

Informed by best practices in national and international classrooms, the SMP describe expertise and mindsets that mathematics educators should practice and seek to develop in their students in order to implement the CCSSM. SMP suggestions for implementation of CCSSM are based on National Council of Teachers of Mathematics process standards of problem solving, reasoning and proof, communication, representation, and connections and the mathematical proficiency expectations in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out appropriate procedures flexibly, accurately, and efficiently), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

The main difference between CCSSM and previous state standards and frameworks for mathematics teaching and children’s achievement in mathematics is that the CCSSM are mathematically sound—they have focus, rigor, and coherence. They seek mastery of the material taught, not just exposure. They are aimed at helping students develop not just content mastery but also mathematical ways of thinking and the ability to apply what they learn. The CCSSM are focused and emphasize depth over breath. Moreover, as standards that set learning targets for children, they are not a curriculum. Hence, they do not mandate how teachers assist children in meeting the targets. Instead, they serve as the foundation for school curricula, mathematics textbooks, teacher preparation, lessons and assignments, student engagement and attitude, and assessment of learning.

The CCSSM provide a “staircase” of content with increasing complexity with the goal that all children become college and career ready. As such, they offer a clear design, common central goals, common language, and common high standards. Cross-curricular teaching that emphasizes problem solving, persistence, abstract reasoning, and the ability to construct arguments and critique reasoning is at the core of these standards. And this is without sacrificing the beauty and rigor of mathematics that we value.

Using the SMP, states and school systems must translate the CCSSM into well-defined curricula and executable lesson plans. When teachers understand the intent and content of these standards, align their curriculum with CCSSM expectations, and convert them into daily lesson plans coupled with appropriate progress monitoring, then children’s mathematics achievement will rise.

The CCSSM call for examining how teachers teach mathematics and how and what students at each grade level learn, know and communicate, not just what is covered. CCSSM content expected to be mastered by students is demanding. There is, therefore, an important distinction between what we as educators need to know vs. what our students need to be taught. To deliver CCSSM effectively, we need to be more than just a page, a lesson, or a concept ahead of students. We need to know much more than what we want our students to know. In addition, we need to select instructional materials wisely because they can have as great an effect on student test scores as teacher knowledge.

Content Mastery
The focus of mathematics content in the CCSSM in the elementary grades is on mastering key arithmetic concepts and procedures with deeper understanding so that students are prepared for more demanding and meaningful and comprehensive mathematics during the middle and high school years and beyond.

Such a focus means mastering certain non-negotiable skills at each grade level with conceptual understanding and fluency. Much of what we teach children, at present, during their first decade of math education relies on students’ memorization of rules, tricks, and facts. We reward correct answers, but we do not encourage students to think independently about what these rules, tricks, and facts might mean in the bigger mathematical picture.

Tricks such as telling children to think of a greater-than sign as Pac-Man or to cross-multiply when dividing fractions, or invert and multiply for dividing fractions to help children get the right answers to difficult problems have long been a staple of math education. In contrast, the intent of CCSSM and SMP is for children to know strategies, not tricks. Methods based on shortcuts and tricks are not real mathematics. If we cannot give a child the reasoning for the use of a method, procedure, or trick, then we are not teaching mathematics and the child is not being prepared for problem solving in higher grades or the work place. It is therefore never sensible to have students memorize first and understand later; this approach leaves students unprepared when they move from elementary mathematics to complex problem solving.

At any grade level, the non-negotiable skills are the basis of all other concepts, procedures and skills at that grade level. If the non-negotiable skills are mastered, then other skills and concepts are easily learned. For example, in CCSSM, the concepts of fractions are introduced in earlier grades (e.g., adding and subtracting fractions in the third and fourth grades with same denominators), but by fifth grade, children have mastered the operations on fractions with deeper understanding. Thus, fractions—understanding and mastering of the concept and procedures, is a non-negotiable skill in the fifth grade. However, the mastery of fractions is dependent on (a) mastery of multiplication facts, (b) divisibility tests, (c) prime factorization, and (d) short-division that are learned by the end of fourth grade.

Multiplication is introduced in the second grade as repeated addition, equal groups of objects, and arrays. By the end of third grade, however, children should have mastered the concept of multiplication as repeated addition, groups of, arrays, and the area of a rectangle so that they understand the distributive property of multiplication over addition and subtraction, have automatized multiplication facts, and are prepared to multiply fractions in fifth grade using the area model of multiplication and then apply multiplication of fractions to master the operations of addition and subtraction on fractions with efficiency and understanding. Using the same principles, they should have automatized division facts by the end of fourth grade and division of fractions by sixth grade.

With the current focus on “covering” and “spiraling” many children lack true mastery in concepts, procedures, and skills. They move with “shallow” mastery from grade to grade and with “holes” in their conceptual and skill sets. Because of this lack of mastery teachers presently devote a great deal of time on preview, pre-testing, and review of previous material and do not teach the grade level content—language, concepts, and procedures, with a level of mastery needed to progress in higher mathematics. In such situations, they hurriedly cover the material, and some students never see meaningful mathematics. With a focus on mastery of non-negotiable skills, teachers can pay attention to children’s errors or lack of mastery and provide appropriate remedial instruction in proper time.

Ability Grouping
Our classrooms have become so diverse in the levels of mastery of content and preparation for mathematics that every teacher has to make several preparations to present the material to the whole class. She ends up teaching the class in several smaller groups, giving limited instructional time to each group while expecting to achieve a year’s growth in return for a fraction of a year in instruction for each child (and therefore the whole class). Each student gets limited attention in the allotted time for instruction; as a result, the teacher finds meeting individual academic needs to be very difficult.

These multiple preparations for the same class with only “shallow” coverage as a focus rather than mastery send many students to the next grade with holes in their preparation and limited achievement and mastery. Even the best strategies of differentiation, in such situations, are inadequate to make up the gaps. On the other hand, breaking each grade into sections according to ability deprives some children the opportunity of novel thinking strategies and others of communication and collaboration. Most ability groupings in American schools are based on students’ computational facility. As a result, ability grouping before sixth grade sends a message that mathematics is just a collection of computational tasks. Recognizing the need for some differentiated instruction, CCSSM does make provision for accelerated mathematics programs at and after sixth grades for students who have conceptual understanding, computational fluency, and adequate cognitive and mathematical thinking that they can handle higher order of thinking and problem solving. These students are the future of the country. They are our future mathematicians, natural and social scientists.

Proper implementation of CCSSM will solve many instructional and achievement problems in our classrooms—particularly the diverse levels of students’ mathematics mastery. SMP and CCSSM are steps in the right direction.

This is the first in a series of blogs examining SMP and CCSSM in American schools.

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Common Core State Standards in Mathematics: Introduction

Mastering the Concept of Number: Numbersense

Numbersense
Just like language, the awareness of quantity and space is socially mediated. Culturally expected achievement related to quantity and spatial skills initiates and supports the awareness and mastery of number related skills. The cultural tools – language, games, toys, social interaction, and related goals and expectations – maximize the formation of the idea of number and its usage. Acquiring number concept and the usage of numbers is a cultural tool in most advanced cultures and the first of a series of tools for being productive as a citizen, solving survival problems, and demonstrating cognitive potential—mathematical ways of thinking.

Just like learning the use of any tool, the outcomes can be enhanced if we learn how to use these tools efficiently, appropriately, and effectively. For this reason, the introduction to and use of these tools by children should begin as early as possible.

Apart from language, calculation is perhaps the only culturally determined system in the modern world that the majority of the population is expected to master. For success in today’s technological world, it is important to have good number and spatial sense. In our world, one needs to be literate and numerate.

Being numerate means a person has flexibility with the use of numbers. It means having a good sense of number and number relationships. And it means an ability to make use of number skills, which enables an individual to cope with the practical quantitative demands made by everyday life, for instance the numerical trends in graphs, charts, or tables, or in reference to percentage increase or decrease. Numerates—practitioners of numeracy have advantage in the modern world.

Numeracy, like literacy, is a complex phenomenon. Numeracy is the demonstration of proficiency in various number related skills. Numeracy is the ability to execute four whole numbers operations, in the standard form, correctly, consistently, efficiently, and with understanding. To be fluent in numeracy includes having a good numbersense. Flexibility in handling quantity is called numbersense. Numbersense refers to a person’s ability to look at the world quantitatively and make quantitative and spatial comparisons and decisions using mental calculations. Technically, it is the integration of (a) number concept, (b) number relationships (arithmetic facts), and (c) place value. With practice and experience, this proficiency and fluency in numbersense translates into numeracy.

 Numbersense describes a cluster of ideas such as the meaning of a number, ways of representing numbers, relationships among numbers, the relative magnitude of numbers, and proficiency in working with them (ultimately leading to mastering arithmetic facts and their usage).

Numbersense is not a set of discrete skills but a set of integrative skills. Students with good number sense can move effortlessly between the real world of numbers and formal numerical expressions. They can represent the same number in multiple ways depending on the context and purpose. In operations with numbers, individuals with a good sense of number can decompose and recompose numbers with ease.

Understanding the concept of numbersense provides a window into children’s arithmetic difficulties, particularly dyscalculia. Dyscalculia is a child’s difficulty in conceptualizing number, mastering number relationships, and producing outcomes of number operations.

The difficulty may be the result of a child’s assets—neurological, neuropsychological, and cognitive reasons and/or environmental factors—poor teaching, poor curriculum, or lower expectations. When these difficulties exist in spite of a child having intact neurological, neuropsychological, or other cognitive assets, then they are purely because of environmental factors – the term for such difficulties is acquired dyscalculia.

Dyscalculia or acquired dyscalculia results in the manifestation of difficulties in the integration of number concept, numbersense, and numeracy. But just as most dyslexics can learn to read with efficient teaching methods, in most cases, those with acquired dyscalculia and even dyscalculia can learn mathematics with effective and efficient strategies. Thus, one can have dyscalculia or acquired dyscalculia, but effective and efficient teaching can give skills so that the effect of dyscalculia or dyscalculia is mitigated. A person would still have dyscalculia, but he/she will not be disabled.

More current definitions of dyscalculia are critical. Children, especially gifted children, may be able to compensate for even considerable deficits using one or more of their equally substantial strengths. For a while, children with tremendous memory and oral comprehension might be able to cope with lack of arithmetic fact fluency to produce adequate arithmetic results when mathematics is still fairly simple. But if they have deficits in the understanding, fluency and applicability of number concept and numbersense and procedures, then they are disabled for future mathematics. Unfortunately, school officials and parents often have an insufficient understanding of the connection between dyscalculia and arithmetic. As a result, they may assume that all conditions related to dyscalculia equate to disability.

True number concept is at the basis of the development of fluent numbersense. It is difficult to master (understanding, fluency and the ability to apply) arithmetic facts without proper number concept. And, numbersense—number relationships, arithmetic facts, and place value, is necessary for number work, procedures and meaningful problem solving.

Every child should acquire a sense of what numbers represent and be fluent with arithmetic facts such as addition and subtraction, number relationships, multiplication tables, and division facts. Every child should be able to use what he knows to calculate accurately and efficiently, both mentally and on paper, by taking advantage of a range of calculation strategies. These skills are necessary for building a solid foundation for numeracy. The acquisition of these skills is dependent upon appropriate teaching and learning experiences.      

Number Conceptualization: Numberness/Numerosity
To most people, knowledge and use of the first nine natural numbers (one, two, three… up to nine) appear to be a simple and straightforward process. To them, learning to count is merely a matter of reciting a string of words like a nursery rhyme, a feat that most young children can master surprisingly early. Most children, in the natural course of living, are able to progress from working with objects to representing these experiences in pictures and icons, to representing them in abstract symbols, and then to manipulating those numerical symbols.

Numbers can be represented in three main formats: Hindu-Arabic (numbers in numerical format), verbal (graphemic or phonological word format), and magnitude-related. Learning number is based on a functional relationship between these different representations and their processing characteristics. Of particular interest is understanding magnitude information because magnitude information is the semantic aspect of numerical processing. This is so because each number, whatever its format, is a symbolic representation of a magnitude or quantity. Just counting objects is not a number concept. For his reason, we want to introduce another representation of quantity in the form of visual clusters. Visual cluster representation subsumes and extends the subitizing, on the one hand, and the magnitude of number, on the other. Thus, understanding of number is the integration of (a) Hindu-Arabic representation (grapheme, (b) verbal (phonemic) and (c) visual cluster of the number.

An average child takes around five years, from about age two to six, to learn to handle numbers and to apply them to everyday situations to solve simple quantitative problems accurately and consistently. Yet many children have difficulty in mastering and applying this skill according to a socially acceptable timetable or an acceptable level of mastery. For a variety of reasons, this process may be longer and difficult for those who are not secure in their ability to read and write numerals and to visualize sets of objects or who are not secure in their sense of number and their applications.

If environmental factors are ruled out, difficulty in acquiring numbersense appears to be related to particular deficits in the learning mechanism and specific learning disabilities. Fortunately, early instruction involving activities that develop numbersense can limit and even prevent failure in numeracy and in later mathematics. It is therefore important to teach the integration of numbersense activities with a focus on “sight number fact” automaticity. As the acquisition of sight vocabulary plays a big role in early reading, similarly, the acquisition of sight facts plays a significant role in early numbersense mastery.

Mastering the Concept of Number: Numbersense

Mastering the Concept of Number

Lexical Entries: Number Names
Most children come to school able to recite the string of alphabets as a song—a rhyme. Similarly, many children begin counting by rote—the recitation of an ordered sequence of responses using concrete objects, long before school, particularly in homes where there is emphasis on learning letters and numbers.

In many cases, this rote counting does not mean real understanding of number or its effective usage. Children acquire this skill by observing others using a series of number words in their day-to-day living. Exposure to other people counting or a child’s counting adds to the development of a larger vocabulary for number names—a prerequisite for number concept. It is somewhat similar to a child acquiring sight vocabulary words. Just as a child should know the letters of the alphabet by rote first and then the meaning, similarly, children should acquire the string of number names, particularly the numbers up to about 30.

Counting by rote is an indication of the presence of lexical entries relating to number (number names). Though knowing number names (these lexical entries) is essential, the recitation of numbers is not sufficient for fluent number conceptualization and number usage. Counting as a rote activity can be taught easily, but the child may or may not know what he is doing except saying the words in a sequence—a string of words. Let us consider the following example of a five-year child working with number.

Teacher: Can you tell me how many cubes there are on the table? (Points to a collection of cubes on the table.)

❒   ❒   ❒   ❒     ❒     ❒   ❒

Child: (Child sequentially counts by touching each cube once) Seven.
Teacher: When you were counting, what number came just before seven?
Child: I don’t know. Is it one? Three? Five? I don’t know.
The child thinks and says: Lemme see.
Teacher: Please count the cubes again. (Points to the same collection.)
Child: (The child counts the cubes again using the same strategy of touching each one as he counts them.) Seven.
Teacher: Can you give me six cubes?
Child: I don’t know. I don’t think I have enough here. Do I have enough to give you? Maybe I do. (He counts six cubes one by one and gives them to the teacher.)
Teacher: That is right. (She gives them back to the child.) Please put them back the way they were.
The child rearranges them.
Teacher: How many cubes are there on the table now?
The child counts them by touching each one once and answers.
Child: Seven.
Pauses and says: Oh! Seven again. The same seven, I guess.
Teacher: Yes!

This example shows that the child knows the number words but does not know what those number words mean. He has learnt these number words by rote. Thus, rote counting may not help a student in problem solving such as the comparison of two sets or operations on numbers. Still, this rote counting process is essential as it develops the lexical entries for numbers. With rote counting as the only skill, children may not use the simple one-to-one counting as a strategy to compare two sets even when they are given perceptual reminders of this correspondence. This rote counting is a starting point for number conceptualization.

To extend the lexical entries for numbers beyond the first few, it is important that teachers devote a few minutes each day to sequential counting. This helps children to see the structure and patterns of numbers. Regular rote counting is like reading to young children (even as young as two years old). When we read to very young children, they are acquiring the structure of the language—the intonation, pronunciation, and sound—orthographic symbol correspondence. They are also becoming familiar with the affect associated with words/language. Similarly, when we count with children, they become familiar with the sequence, the structure, and the ability to differentiate the number words and non-number words. This process of regular and frequent counting activates the imaginary number line in children’s minds, just like the exposure to rich language activates the grammar and structure of the language of that culture. At the end of Kindergarten, children should be able to count, forward and backward, by 1, 2, and by 10 from any number at least up to 100.

From Egocentric Counting to the Cardinality of the Set
To be meaningful, counting behavior must have the underlying cognitive structures and processes and the support of language. Many children of ages 4 through 6 can count objects in a rote manner where, for them, the last number uttered represents the “cardinality of the set”; however, they may not know what that means. For them, the cardinality of the set is the outcome of their counting process, not the property of the collection. They think of this number as the outcome of their action (“These are six blocks because I just counted them.”) rather than the affirmation of the quantity representing that collection—a property of the collection (“These are six cubes.” “There are six cubes here.”). The transition from simple counting to an understanding of the number as the property of the collection is a key transition in number conceptualization. The following illustrates how to achieve this transition. Let us consider the example of an interaction with another child during number work in a Kindergarten class.

Teacher: Can you show me your right hand?
Child: With a little hesitation he raises his right hand.
Teacher: Good!
Teacher: Can you tell me how many cubes there are on the table? (Points to a collection on the table.)

❒   ❒   ❒   ❒     ❒     ❒   ❒

Child: (Child sequentially counts by touching each cube once) Seven.
Teacher: Very good! How many did you say?
Child: I think seven. Let me see. (He counts again in the same manner.) Seven. I told you there are seven. I counted them again. There are seven. Right?
Teacher: Yes! You are right. There are seven cubes. Let me ask you something. When you were counting, I noticed that you started counting them from left to right (points to the direction) what if you counted them from right to left? (Points to the direction.)
Child: I don’t know. Let me try.
The child thinks and counts.
Child: It’s seven.
Teacher: That’s right.
Child: (As the teacher acknowledges child’s answer, the child is thinking about something and then touches the cubes. The child counts the cubes again using the same strategy of touching each one as he counts them, first from left to right and then right to left.) Seven. You know what. It is the same thing. It’s seven. It doesn’t matter how you count; it is the same number. It’s seven. See! (Counts again once from left to right and then right to left.) I guess it is always seven.

In the first example, the child is only reciting a string of words. He is not connecting the different number words with each other and with the property of the collection. The number (cardinality of the set) arrived at by counting only shows that the child thinks the number is the property of the counting process not of the collection. The child thinks he has produced that number by counting. Assigning a number to the collection always by counting only and even change the number based on recounting is called egocentric counting. The number produced is the product of that counting. Many young children when asked to count the same collection again may produce a different answer and may not even question themselves. Most children with exposure to counting and usage of this count transcend this type of egocentric outcome of the counting process.

In the second case, the child acquired an important concept: the number (cardinality of the set) is not the property/function of the counting process but the property of the collection. Therefore, children’s use of number requires a true understanding of number, not just rote counting. Associating a number to a collection and considering that number to be the property of the collection is the first step in conceptualizing number properly.

The types of questions asked in these two activities are the means to converting a child’s concrete experiences (egocentric counting) into abstract number conceptualization (number represents the cardinality of the set—it is the property of the set, not of the counting process). The above examples illustrate how the structure of a problem and nature of our questions influence the strategies and concepts children develop to solve problems.

The distinction between the two ways of learning mathematical concepts (drills vs. understanding relations) has a long history in the pedagogy of mathematics learning and teaching. The debate is whether individuals can learn the same mathematical skill either as a set of discrete, non-related rote activities and simple rules or whether they learn concepts and higher order rules with understanding—knowing the interrelationships of component concepts. This artificial distinction also exists in the learning and teaching of number conceptualization. Although the two approaches may result in similar outcomes on tests of isolated skills and the same level of performance on an immediate mastery test, learning with understanding generates broader and deeper learning outcomes with superior transfer to related concepts and longer retention of learned material.

Mastery, in the second example, is not temporary and is not problem and format specific. For example, when a child can derive the sum: 7 + 3 = 10 by counting on fingers, he may not be able to extend it and even remember it from one setting to the other. Counting strategies do not develop “sight number facts” as the results are derived by counting the number each time children want the outcome. When the results are derived by rote counting, there is no visual representation of the number relationships. In contrast, when the child knows 7 + 3 by visualizing it (for example, the display of 10 hearts as a pattern on a visual cluster card and decomposing it as two clusters—one of seven and the other of three hearts), with understanding and automatized mastery – fluency with understanding), then she can extend that 7 of something plus 3 of the same thing is 10 of the same thing.

We have known for a long time that skilled readers are able to read almost every word without activating the same phonological processor they used when first starting to read. They reached that point because they practiced that phonological processor, which allowed them to make words automatic. Reading a word as a sight word is not a strategy, it is a goal. So, how do we get there? Sight words are acquired by continuous and multiple exposures. Likewise, looking at a collection of objects (a cluster of objects up to ten), recognizing it, and instantly giving it a numerical name is the goal, but to get there we need to focus on smaller clusters and even some counting. In reading a word, we do not focus on each letter, nor do we begin with the whole word memorization. Similarly, in numberness, we neither focus on one object at a time nor on the whole cluster to start with.

Mastering the Concept of Number

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