Mathematics Learning

What does it mean to suggest that an individual possesses good number sense? The ability to see patterns and relationships between numbers, to work flexibly with operations and procedures, to recognize order and relative quantities, and to utilize estimation and mental computation are all components of what is termed number sense. Individuals who quickly calculate a 15 percent gratuity at a restaurant, know that the seven-digit display 0.498732 is approximately 1/2, or recognize that calculating 48 × 12 will be less problematic than calculating 48 × 13 are said to manifest qualities associated with good number sense.

Most mathematics educators agree that developing number sense is important, yet there is no single definition that is unanimously accepted. Number sense is highly personalized and thought to develop gradually. It includes self-regulation, an ability to make connections in number patterns, and an intuition regarding numbers. Number sense "refers to a person's general understanding of number and operations along with the ability and inclination to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for handling numbers and operations" (McIntosh et al., p. 3).

Historical Background

Before the term number sense came into use, the word numeracy was coined in 1959 to denote those within the realm of mathematics who had a propensity to comprehend higher-level mathematical concepts. Yet the general public took numeracy to be the mathematical analogue of literacy, and therefore reduced its meaning to connote the propensity to comprehend basic arithmetic. A book by John Allen Paulos, Innumeracy: Mathematical Illiteracy and Its Consequences (1988), demonstrated the dangers of a population that lacks a basic understanding of mathematics and that views the subject as enigmatic due to poor education or psychological anxiety. Many of those involved with mathematics education felt that math pedagogy was in need of serious reform due to a superficial acquisition of knowledge based merely on a procedural understanding of mathematics (e.g., "just follow this algorithm").

In the late 1980s and the 1990s researchers and educators saw a heightened need to examine the role of computation as it related to elementary mathematics, reflecting on both the process and result of employing algorithmic strategies. It was during this period that the term number sense gained wide acceptance, epitomizing the desired outcome for the teaching and learning of mathematics. Yet due to its implicit nature, succinctly describing how number sense is revealed can be problematic. The mathematician Stanislas Dehaene, in his 1997 book The Number Sense: How the Mind Creates Mathematics, states, "Our number sense cannot be reduced to the formal definition provided by rules or axioms" (p. 240). In addition, James Greeno relates, "We recognize examples of number sense, even though we have no satisfactory definition that distinguishes its features"(p. 171).

Similar to the ambiguous implications of common sense, number sense is open to a variety of interpretations. The National Council of Teachers of Mathematics, in Curriculum and Evaluation of Standards for School Mathematics (1989), defines number sense as "an intuition about numbers that is drawn from all the varied meanings of number. It has five components: (1) having well-understood number meanings, (2) developing multiple relationships among numbers, (3) understanding the relative magnitudes of numbers, (4) developing intuitions about the relative effect of operating on numbers,(5) developing referents for measures of common objects" (pp. 39–40). However, others would argue that such descriptors and boundaries for the nature of number sense do not characterize it in forms that guide instruction. Lauren Resnick and Judith Sowder categorize number sense as an open-ended form of reasoning that is nonalgorithmic, complex, and involves uncertainty. These multiple views are highlighted merely to show the somewhat amorphous nature of number sense and qualities ascribed to it.

Examples of Number Sense

Most often, number sense is recognized through example. One ascribed attribute is the ability to use numbers flexibly when mentally computing an abstract numerical operation. This flexibility evolves through infixing connections and relationships between numbers and their representations. By augmenting the number of connections to analogous situations, more flexibility and utility ensues. For example, a simple computation involving subtraction is the problem 7-4. The ability to place this abstraction of symbols into multiple situations signifies a certain number sense, such as: (1) a set or group–seven cookies take away four cookies; (2) a distance–in order to move from space 4 to space 7 in a board game, 3 moves are required; (3) a temperature reading–to change from 7° C to 4° C, the temperature must drop 3° C. These mental models seem natural to most adults and children who have been guided to think with such models. With the simple transition of this problem, reversing the minuend and subtrahend mandates an ability to move into negative numbers: 4 - 7 equals what? For a child who has only the group mental model (4 cookies take away 7 cookies), this operation seems problematic or impossible. A child who has multiple models can utilize the one that gives a more intuitive representation of the abstract operation–if the temperature is 4° C and then falls 7°C, then the new temperature would be negative (or minus) 3° C.

In addition, being able to compare the relative size of numbers would be a sign of number sense. Students should recognize that 4,562 is large compared to 400 but small compared to 400,000. There should also be emphasis placed on providing context to compare large numbers. For example, a million and a billion are ubiquitous quantities in many economies. Therefore, to recognize that it takes roughly eleven-and-a-half days for a million seconds to pass and nearly thirty-two years for a billion seconds to pass connotes a deeper appreciation for the relative magnitude of quantities.

Number sense extends beyond the set of whole numbers and integers. Consider a more frequent area of concern for many school children, fractions. Consider the following example: ² §3 + ¹/4. For conceptual understanding, fractions and ratios necessitate the skill of proportional reasoning in order to make sense of this abstract representation. Considering a part-to-a-whole relationship, the adroit student can recognize the necessity to compare equal size parts (and therefore find a common denominator) before total parts can be computed:

In contrast, a child who has no intuitive grasp for fractions will most likely commit the error of adding the numerators and adding the denominators. This algorithmic error might also be attributed to those who rely on a strictly procedural understanding, because this procedure is correct when it relates to multiplying fractions,

and students often confuse these two rules. Furthermore, this nonconventional result for addition can be justified with concrete examples. If Barry Bonds plays in both games of a double-header, and he bats 2 for 3 in the first game and 1 for 4 in the second game, then his correct batting average for the day is 3 for 7, which, in terms of the traditional procedure for adding fractions, is not conventionally correct:

Therefore, number sense involves knowing when a specific model is applicable.

Sometimes, number sense can be grasped intuitively through visual clues as well. Some people have an affinity for understanding visual models, which they might then internalize and incorporate into their personal number sense. Figure 1 contains no symbolic representation of numerals; rather, actual quantities are depicted as the objects themselves. The question at hand is to compare the available cake for girls and for boys and determine in which of the two groups does an individual receive more cake. Students who are versed in strictly procedural understandings might set up ratios that symbolize the situation, then try to rely on memorized algorithms to simplify the symbols:

Someone with a more flexible understanding might simply notice that for the boys there is one cake for a group of three; therefore, an equal ratio based on three cakes would be a group of nine girls. From this equivalency, they would deduce that since there are less than nine girls, then each girl must receive more cake than each boy.

Developing Number Sense

The acquisition of number sense is often considered to develop as stages along a continuum, rather than as a static object that is either possessed or not. Dehaene reports that most children enter preschool with a well-developed understanding of approximation and counting. Dehaene presents research from cognitive psychologists, such as Jean Piaget, Prentice Starkey, and Karen Wynn, suggesting contradictory results about what skills are innate, when skills are developed, and how they are acquired. Part of the complexity to succinctly describe a development of number sense stems both from the subtlety of multiple factors it encompasses and the lack of explicit demonstrability. For example, with the problem 18 × 5, someone demonstrating number sense might recognize the relationship of the quantity 5 comparedto 10 is simply half, and knowing that, taking half of this result would give the desired result, 90. This sophisticated innovation may be entirely internal, with only the final solution given and no account of the process. Although we can recognize number sense when we see it, the question as to how one's cognitive process completes individual tasks is less certain. It is similar to mathematicians' demands for valid proofs to be rigorous, though they are unable to adequately describe what is meant by rigor.

There are several factors regarding the development of number sense that mathematics educators have come to agree upon from empirical research during the 1990s. Results from Paul Cobb et al., Judith Sowder, Sharon Griffin and Robbie Case, and

FIGURE 1

Eddie Gray and David Tall have provided a more clearly agreed upon framework regarding advantageous skills for building number sense. Sowder notes that computational estimation and mental computation are important links to building number sense. Both Cobb et al. and Greeno state that both the use of mental models and creating a conceptual environment are necessary facilitators to make these links. Educators see a necessity to incorporate rich examples that guide students toward conceptual understandings, instead of superficial procedures that are not considered malleable. Developing mental models and utilizing mental computation are increasingly considered vital skills in mathematics; however research about reasoning with mental models is in a preliminary state.

Current Trends and Their Effects on Mathematics Education

The twentieth century saw its share of reforms in mathematics pedagogy–from the algorithmic framework of connectionist theory attributed to Edward L. Thorndike to the axiomatic formalization of modern mathematics pursued by Bourbaki (a pseudonym taken by a group of French mathematicians) and Piaget's constructivist theory, which dominated the second half of the century and emphasized individuals as constructing their own knowledge through a process of abstraction, generalization, and concept formation. The concern in the 1990s surrounding a superficial (or merely procedural) understanding of mathematics with a lack of conceptual understanding was the catalyst that galvanized a push toward interpreting mathematics not as rote and memorization, but as problem solving, intuitive reasoning, and pattern recognition. The concept of number sense sprang forth from these shifts in philosophy regarding mathematics education. With this shift, a question arises: What significance does number sense have on mathematics education and pedagogy?

Another difficulty in encapsulating pedagogy that develops number sense stems from the fact that most mathematicians fail to recognize their own number sense and how they employ it. Their ability to move beyond procedures and definitions into the realm of concepts is rarely a conscious process. To a mathematician, the act is incorporated into their thinking process such that its nature becomes an involuntary action, like blinking or breathing. The mathematical paradox of striving for efficiency, both in notation and procedures, can oftentimes add to a lack of understanding for the student. To communicate efficiently, all those involved must be fluent in the language of mathematics.

Obviously, some students are successful in mathematics regardless of the pedagogical approach used. If this were not the case, the explosion within the new fields of mathematics that occurred after 1950 would not have occurred. Philip Davis and Reuben Hersh attest that more than half of all of mathematics was discovered after World War II. The question, then, is what percentage of those who completed traditional education were finding this success. Gray and Tall speculate that only 30 percent of students were able to develop an intuitive grasp of mathematics and higher-order thinking with previous pedagogies. So what about the other 70 percent? The research by Paul Cobb et al., Sharon Griffin and Robbie Case, and others consider this a central focus of current pedagogical issues.

This search for conceptual understanding seems to be the focus of research and pedagogy in the beginning of the twenty-first century. Empirical evidence supports a curriculum that stresses practical, intuitive, and rich real-world examples within mathematics. The Rightstart project, developed by Case and Griffin in 1997, is one such example. Their research focused on children living in urban, low-income communities who were lagging behind their peers in terms of age-level mathematics abilities. After participating in forty twenty-minute sessions that incorporated numerical games and concrete materials (using thermometers, board games, and number lines) these children were propelled to the top of their class, and they maintained this placement over a longitudinal study lasting several years. This success was achieved by focusing on two main goals: (1) to help students to develop a set of symbolic states and operations that are intimately tied to real-world quantities, and (2) to develop students' explicit knowledge of notational systems in conjunction with their implicit and intuitive knowledge, thus ensuring that these two types of knowledge act as natural companions to each other. Both of these goals coincide with the parameters of developing number sense.

The mathematician Warren McCulloch (1965) once observed, "What is a number, that a man may know it, and a man, that he may know a number?" The answer to this question, which has been posed in various forms since antiquity, changes with the understanding of mathematics. Since the 1990s, mathematics educators have been researching how number sense ameliorates students' understanding of mathematics. Mathematics educators have embraced this shift toward a pedagogy that strives to merge intuition, formal notation, and conceptual understanding. Number sense helps students eschew the notion that mathematics is merely a collection of rules to memorize. Number sense fosters students' ability to make judgments about the reasonableness of solutions and to build on their intuitions and insights. Number sense helps convince students that mathematics makes sense.