Rational Number Project Home Page Post, T., & Cramer, K. (1989, March). Knowledge, Representation and Quantitative Thinking. In M. Reynolds (Ed.) Knowledge Base for the Beginning Teacher - Special publication of the AACTE (pp. 221-231). Oxford: Pergamon Press.

CHAPTER 19

Knowledge, Representation, and Quantitative Thinking1

THOMAS R. POST
University of Minnesota, Minneapolis

KATHLEEN A. CRAMER
University of Wisconsin, River Falls

Mathematics is pervasive! Everyone, regardless of background or interests, uses numerical, spatial, and analytic ideas in thinking about and describing the real world. Therefore it is important for teachers - and students - to understand how these ideas are (or should be) developed and maintained and how to distinguish between bona fide numerical concepts and the rote manipulation of symbols.

Unfortunately, many new teachers have grown up with mathematics curricula that lacked development of deep mathematical structures, concentrating instead on drill and practice and providing standard answers to familiar questions. Fortunately, major advances are underway in theory development and enlarging the scope of school mathematics. Mathematics programs of the early 1990’s will not resemble those of even five years earlier. Geometry, estimation, graphical interpretation, computer literacy, calculator usage and probability and statistics are likely to play a much more prominent role in the school mathematics program of the future. Teachers, therefore, will not be able to teach the way they have been taught, nor will they have a priori knowledge of all appropriate content areas. Virtually all beginning teachers will need to acquire new pedagogical skills as well as new mathematical understandings. This is a formidable yet challenging task! This chapter outlines the nature of those impending changes and suggests ways they can be implemented.

What is important for all beginning teachers to know about the acquisition, use, and maintenance of quantitative thinking skills? To answer this question it is important first to consider (a) how children learn mathematics, and (b) what mathematics should be taught.

In the following sections we will examine both issues. We will explain the importance of conceptual and procedural knowledge in teaching quantitative understandings to children, and the role of representation in the acquisition of mathematical concepts. We also will discuss current thinking related to the question of what kinds of skills should be taught and the present and future influences of technology on the scope, sequencing, and presentation of mathematics curricula.

How Children Learn Mathematics

THE IMPORTANCE OF CONCEPTUAL AND PROCEDURAL KNOWLEDGE

Issues relating to skill learning (Gagne, 1985; Thorndike, 1922) as opposed to principle learning or understanding (Brownell, 1935; Bruner, 1960), concepts as opposed to procedures (Piaget, 1978), and "knowing that" and "knowing how to" (Scheffler, 1965), have received a good deal of attention and debate during the last century.

In earlier years the debate characterized these basically different types of knowledge-concepts and procedures-as having diametrically opposed and seemingly unrelated natures. Current discussions, however, have emphasized the relationships between concepts and procedures. There is growing recognition in the mathematics education research community of the importance of the interaction between the two forms of knowledge and the role each can play in the development and maintenance of the other (Davis 1984; Davis & McKnight, 1980). Mathematics educators generally agree that knowledge of concepts is the foundation for intuitions and procedures and that a teacher should be concerned with the development of both conceptual and procedural knowledge.

Conceptual knowledge is knowledge that is rich in relationships. It can be thought of as a connected web of knowledge, a network in which the linking relationships are as important as the discrete pieces of information. By definition, a piece of information is part of conceptual knowledge only if the holder recognizes its relationship to other pieces of information (Hiebert & Lefevre, 1986). When previously independent pieces of information are organized and related to one another there is a dramatic and significant cognitive reorganization (Lawler, 1981). Piaget referred to this process as accommodation. It is through such reflective mental activity that intellectual growth occurs.

Relationships can be established on two levels (Lawler, 1981). On one level, understanding emanates from the ideas embedded within the context in which they are presented, that is, context-specific understanding. On the second level, relationships are understood in a "context free" environment where appropriate abstractions have been made. This latter level is the sine qua non of the professional mathematician, but too often ignored or unrecognized at the school level.

Procedural knowledge in mathematics is composed of two parts: knowledge of the formal language of mathematics, that is, symbols and syntax; and the rules, algorithms, or procedures used to solve mathematical tasks (Hiebert & Lefevre, 1986). The former implies only an awareness of superficial features, not a knowledge of meaning or underlying structure; the latter consists of step-by-step instructions that define precisely how to complete mathematical tasks or exercises in a predetermined linear sequence. The rules lend themselves to mechanical reproduction and can be implemented without conceptual understanding, although this is not necessarily the case. In contrast, conceptual knowledge must always be learned meaningfully.

Unfortunately, most school mathematics curricula are overly concerned with developing procedural knowledge in the form of speed and accuracy in using computational algorithms rather than the development of higher order thought processes, such as those used in problem solving, deductive reasoning, and logical inference. This is true throughout the elementary, junior high, and senior high school levels. For example, at the elementary level, students are found practicing the long division algorithm, while at higher levels they spend considerable time on the rote manipulation of algebraic sentences or on solving quadratic equations. In short, procedural knowledge dominates school mathematics curricula at virtually all levels.

Shirley Hill, a past president of the National Council of Teachers of Mathematics, commented on phenomenon. "It struck me as supremely ironic that at the very time we are on the threshold of teaching machines to reason, we are spending an inordinate amount of our educational energies teaching our children mechanistic skills" (Hill, 1979, p. 2). Unfortunately, things have not changed very much. A 1987 survey (Post & Orton, 1987) of elementary and junior high school teachers in a Minneapolis school district indicated that slightly over 50% of mathematics instructional time is spent developing speed and accuracy in paper and pencil calculations. Another 16% is spent on textbook word problems. Thus 2/3 of students' time is spent dealing with text-related activities. Yet it is generally agreed that concepts initially evolve through interaction with the environment, not with the printed page. Since Minneapolis is considered one of the nation’s more progressive urban educational systems, the percentage of time spent on procedural knowledge might be even higher in the nation's schools as a whole.

Here are a few examples showing how concept knowledge and procedural knowledge differ:

1. Procedural knowledge: A child can simply memorize addition or multiplication facts, but has difficulty reconstructing a fact when memory fails.
2. Conceptual knowledge: A child can understand that 9 + 4 can be rewritten as 9 + (1 + 3), or (9 + 1) + 3, or 10 + 3 = 13. This type of procedure can be used to generate any addition fact, even multidigit ones, such as 27 + 36 = 27 + (3 + 33) = (27 + 3) + 33 = 63. Likewise: 7 x 6 can be interpreted (6 x 6) + 6 = 42. These are instances of concept knowledge since they focus on the relationships between the numbers and the arithmetical operations.

3. Procedural knowledge: A child can divide two fractions using the invert and multiply procedure: 1/2 + 2/3 = 1/2 x 3/2 = 3/4. This is not usually a meaningful activity because most individuals (adults included) use the procedure without understanding it.
4. Conceptual knowledge: From a conceptual perspective this complex procedure requires basic understanding of the relationships between division, multiplication, the notion of inverse, a thorough understanding of fractions, and the ability to interpret the magnitude of the result in terms of the original situation (i.e., since 2/3 is greater than 1/2, the quotient should less than 1). The depth of understanding needed suggests that this skill should be delayed until greater knowledge has been acquired. The fact that the invert and multiply procedure is so easily accomplished does not justify its inclusion in the curriculum.

5. Procedural knowledge: If 7/8 were added to 12/13 would the result be closer to 1,2, 19, or 21? This problem appeared in a recent version of the National Assessment for Educational Progress (Post, 1981). Only 23% of the nation's 13-year-olds selected the correct answer. Over half (55%) thought the answer was 19 or 21, the result of adding numerators or denominators. Such answers indicate complete misunderstanding of the concept.

Conceptual knowledge: A correct, conceptually based approach would be, "7/8 is about 1, and 12/13 is about 1, therefore their sum is about 2." Instead of using this commonsense approach based on conceptual knowledge, the majority of students tried a procedure (probably from memory) to find a solution.

The issues surrounding procedural- and conceptual- based knowledge no doubt can be found in other disciplines as well. Sometimes the interplay between them is subtle. At other times it is quite overt. For our purposes it is enough to be aware of the issue and the educational implications of requiring students to process and retain significant amounts of information which is not conceptually based. Procedural knowledge can be learned and applied without the appropriate conceptual underpinnings. Under such conditions, higher order thought processes (which may deviate from learned procedures or require the application or extension of learned ideas) are highly improbable.

Both procedural and conceptual quantitative knowledge are vital. Mathematics requires the execution of a wide variety of procedures. Ideally these procedures are derived from fundamental conceptual understandings, which extends the process beyond rote manipulation and increases the potential for intellectual growth.

One way to gauge individual growth in mathematics is to observe the degree and extent to which conceptual knowledge is reinterpreted as procedural knowledge. That is, the degree to which the individual has enlarged the nature and scope of the mathematical situations which can be reacted to in a comfortable and routine fashion. The danger, of course, lies in premature abstraction and the mindless manipulation of symbols without the appropriate conceptual underpinnings.

THE ROLE OF REPRESENTATION

Mathematics is often used to represent the world in which we live. As such there must be a correspondence between some aspects of the represented world and some aspects of the representing world. The abstract relationships or mapping between these two can be thought of as applied mathematical structures. They are useful in the discipline at all levels and also form the essence of mathematical conceptual development in school-aged children. This relationship is illustrated in Figure 19.1.

Figure 19.1 Manipulative materials and learning

Representations can be viewed as the facilitators which enable linkages between the real world and the mathematical world. Formulae, tables, graphs, numerals, equations, and manipulative materials all are mathematical "objects" used to represent various real world ideas and relationships. At a more advanced stage these "objects" themselves can be represented by formulae, tables, and so forth. That is, mathematical concepts can be viewed as tools to help with understanding new situations and problems and also as objects that can be investigated in their own right. Translations within and between these "objects" constitute the essence of mathematical activity.

Questions about the nature of knowledge, the nature of one who knows (or the one who learns), the "transmitability" (or lack thereof) of knowledge and conditions under which learning most effectively occurs have persisted since 500 B.C. A problem results from a view of knowledge which requires a match between the cognitive structures themselves and what those structures are supposed to represent. The difficulty here is that it is never possible to determine how well our (or others') mental structures represent what they are intended to represent, for such assessment "lies forever on the other side of our experimental interface" (Von Glaserfeld, 1987). Since we can never step outside of ourselves and achieve a truly objective perspective, a different view of what it is "to know" is required, one which is not based on a correspondence with reality.

Such a view was stated by Osiander in 1627 in his retort to the critics of Copernicus' revolutionary idea that the Earth was not the center of the universe. He said, "There is no need for these hypotheses to be true, or even to be at all like the truth; rather one thing is sufficient for them - that they yield calculations which agree with the observations" (Popper, 1968).

This second conception of knowledge, one that fits our observations, has profound implications for education and instruction and for the organization of experience. Piaget (1952) characterized this situation as follows: "Intelligence organizes the world by organizing itself." Thus, an individual's experiences and their subsequent reorganization become the beginning, middle, and end points of conceptual development and the concomitant evolution of intelligent action.

But how can one directly experience a number which is, by its very nature, an abstraction? The answer is that one cannot, but it is possible to experience representations of it. Representation is, therefore, a crucial component in the development of mathematical understanding and quantitative thinking. Without it, mathematics would be totally abstract, largely philosophical, and probably inaccessible to the majority of the populace. With it, mathematical ideas can be modeled, important relationships explicated, and understandings fostered through a careful construction and sequencing of appropriate experiences and observations. It is currently held that it is the translation between different representations of mathematical ideas, and the translations between common experience and the abstract symbolic representation of those experiences, that make mathematical ideas meaningful for children.

Bruner (1966) suggested three modes of representation-enactive, iconic, and symbolic-for modeling mathematical ideas for children. Intuitively these modes suggest a linear temporal ordering: first, enactive; second, iconic (pictures); and last, symbols. This logic has been so persuasive that several decades of mathematics textbooks have sequenced their content with conscious or unconscious adherence to this model. Unfortunately, this issue is more complex than Bruner implies, and almost surely involves nonlinearity and requires additional modes. Lesh (1979) suggested adding two additional modes, spoken language and real world problem situations. Most importantly, he stressed the interactive nature of these various types of representations, that is, often several modes will exist concurrently in a problem solving setting and individuals will routinely re-employ a variety of representations and sequences of representation as they reorganize problem components and interrelationships between them. Figure 19.2 illustrates Lesh's model.

Figure 19.2 Lesh's model for translations between modes of representation.

The Rational Number Project (Post, Behr, Lesh, & Wachsmuth, 1985), in a series of research projects supported by the National Science Foundation, has, over the past decade, utilized this framework in the development of its own theory-based instructional units. This project corroborated the interactive nature of student approaches and has found the use of predetermined mode translation tasks to be a powerful stimulus to the acquisition of various rational number concepts (fraction, part-whole ideas, ratio, decimal, operator, a measurement interpretations).

For example, the concept of adding fractions can taught using several translations. A translation from written symbol to manipulative mode can be shown by asking a child to model the sum 1/2 + 1/4 with manipulative materials, such as fraction circles. Similarly, using the illustration shown in Figure 19.3, translation from pictorial mode to written symbols can be achieved by asking the child to write a number sentence for each step in the addition problem (1/2 + 1/4; 1/2 = 2/4; 2/4 + 1/4 = 3/4).

Figure 19.3 Translation from pictorial mode to written symbols.

A within-mode translation (manipulative to manipulative) occurs when a student is given a manipulative display showing, for example, fraction circles, and ask to model the same idea with fraction bars, as shown Figure 19.4. Such translations cannot be made unless the person understands the concept under consideration as presented in the original mode. Further, such translations require the reinterpretation (or reorganization to use Piaget's term) of the concept in order to display it in another mode or with another material in the same mode. Such understandings and reinterpretations are important cognitive processes and are to be encouraged in the teaching/learning process. It is for this reason that the Lesh model is such a powerful tool for the classroom teacher. While good teachers probably have been inadvertently using aspects of these ideas in their interactions with children, the model makes explicit what has been implicit up to this point. In general, it is the translations within and between modes of representation that make ideas meaningful for children. Such a view promotes a dynamic rather than a static view of mathematical conceptual development. In addition, it promotes recognition of appropriate patterns, identification of similarities, recognizing and sorting crucial and noncrucial variables, and ultimately encourages applying new ideas to traditional methods.

Figure 19.4 Manipulative to manipulative translation

Such translations are useful not only in mathematics learning and quantitative thinking, but can be applied as well to instruction in other content domains, notably the natural and social sciences. For example, the ability to understand and interpret ratios and rates is essential. In both natural and social science domains. Concepts such as speed, density, mixture, voltage, exchange rates, percentages, force, solubility, acceleration, concentration, unit costs, pricing, productivity, supply and demand, income distribution, GNP, unemployment, and inflation are all expressed as rates; students, when encountering these concepts in other disciplines, should focus on the similarities and differences of their meaning and interpretation, for it is the realization of their similarities which will foster the evolution of more encompassing cognitive schemas (Kaput, 1987). Such reorganizations are at the core of intellectual development (Piaget, 1952, 1960).

What Mathematics Should be Taught

REDEFINING THE BASIC SKILLS

Several national and international reports recently have bemoaned the relatively low levels of student achievement in mathematics along with concern about the overall quality of mathematics instruction. The Cockcroft Report published in 1982 revolutionized mathematics instruction in the United Kingdom. Its major recommendations are relevant to mathematics instruction everywhere. The report suggested initially that mathematics instruction at all levels should include exposition by the teacher. The study revealed that in classes where instruction is strongly guided by the textbook, most of the students' time is spent in independent seatwork. Too often students do not benefit from the teacher's insights relating to the content. More of the teacher's time needs to be spent teaching mathematics, during which time students and teacher should be involved in verbal exchanges about the subject.

The report also stresses the need for practical, "hands on" activities to develop mathematical concepts before emphasizing the practice of skills and procedures. Instruction that includes problem solving, applications, and an interdisciplinary approach to mathematics instruction was also recommended.

Organizations in this country also are looking at the structure of our schools' mathematics programs. There is a great deal of similarity between the Cockcroft Report and the mathematics reform movement in this country.

Revising the mathematics curriculum to better reflect the needs of an expanding technological society has been the major concern of mathematics and supervisors since the mid-1970s. Those concerned have organized to counter the influence of the "back to basics" movement of the 1970s. The back-to-basics group suggested that the skills needed by children in the 20th century are in fact the same skills that were needed by children in the 19th century. Textbook companies responded by producing books devoted primarily to developing computational skill with paper and pencil and emphasizing speed and accuracy in addition, subtraction, multiplication, and division with whole numbers, decimals, and fractions.

For the schools to emphasize facility with paper and pencil computation at the expense of higher level cognitive processes at a time when calculators and computers are commonplace tools in the adult world is reactionary. To emphasize computation when international studies show that American children rank in the lower half of nearly every mathematics category compared to children in other leading industrial nations is unacceptable. For example, in the Second International Mathematics Study (International Association for the Evaluation of Educational Achievement [IEA], 1984), eighth grade children in the United States ranked below students in Japan and Canada in every category assessed. In measurement and geometry, U.S. children scored in the bottom quarter of 20 developed countries. In a more recent international study comparing performance of children in comparable cities in Minnesota, Japan, and China, the highest average score of an American fifth grade class assessed was below the average score of the lowest Japanese fifth grade class assessed (Stevenson, Shin-Ling, & Stigler, 1986).

The National Council of Teachers of Mathematics (NCTM), in an attempt to upgrade the quality of mathematics instruction, clarify goals, and promote change, issued a report in 1987 entitled The Standards for School Mathematics. The report specifies five basic goals for all students: (a) to become a mathematical problem solver, (b) to learn to communicate mathematically, (c) to learn to reason mathematically, (d) to learn to value mathematics, and (e) to acquire confidence in his or her ability to do mathematics. A new teacher should be aware of the changes that educators hope will occur as a result of the Standards (Commission on Standards for School Mathematics of the National Council of Teachers of Mathematics, 1987).

In 1977 the National Council of Supervisors of Mathematics (NCSM) published a position paper presenting a new definition of basic skills. The position paper emphasizes strongly that all students who hope to participate successfully in adult society must be knowledgeable in these domains (NCSM, 1977). This paper was supported by all national and state mathematics teachers' organizations. Current textbooks are only beginning to reflect this expanded view of what it means to be mathematically literate. It is important that all teachers understand the thrust of viable mathematics programs and that computational ability is no longer the sole indicator of the mathematically literate individual. The skills considered essential by the NCSM to further educational opportunities are: (a) problem solving; (b) applying mathematics to everyday situations; (c) alertness to reasonableness of result; (d) estimation and approximation; (e) appropriate computation skills; (f) geometry; (g) measurement; (h) reading, interpreting and constructing tables, charts, and graphs; (i) using mathematics to predict, and (j) computer literacy (NCSM, 1977).

As stated previously, computation is seen as only one facet of basic mathematical skills.

The role of computational skills in mathematics must be seen in the light of the contributions they make to one's ability to use mathematics in everyday living. In isolation, computational skills contribute little to one's ability to participate in mainstream society. (NCSM, 1977, p.1)

The NCSM (1988) has recently updated the list of basic skills. The new position paper lists 12 critical areas of mathematical competence for all students. Problem solving, applying mathematics to everyday situations, alertness to reasonableness of results, estimation, geometry, and measurement will be reaffirmed as basic skills. Five new categories are: (a) mathematical reasoning (b) communicating mathematical ideas, (c) algebraic thinking, (d) statistics, and (e) probability. The spirit the earlier report remains intact. Additional positions relating to instruction include the following:

Curriculum should focus on meaningful understanding developed from concrete experiences. Learning environment should incorporate high expectations for all students regardless of sex, race or socioeconomic status.

Calculators should be used by all students at all grade levels during instruction and testing.

Computers should be used to support instruction.

There should be a moratorium on standardized tests until they reflect updated program objectives. (NCSM, 1988, pp. 3-5)

QUANTITATIVE THINKING

Within this larger conceptualization of mathematic understanding, thinking quantitatively about numbers also assumes an expanded definition. Quantitative thinking involves the following:

1. Mental Arithmetic: facility with single digit computation; ability to work with powers of 10; ability to use number relationships and properties to facilitate mental computations.

2. Number Sense: knowing when to use a particular operation; when to use mathematics relationships; ability to monitor one's performance when computing, as example, judging reasonableness of answer with respect to an applied problem or by what one knows about numbers.

3. Computational Estimation: process of obtaining approximate answer without recording devices.

4. Thinking about Number Relationships without Numbers.

What is meant by mental arithmetic skills? Here are two examples. The first is taken from Hope's article in the 1986 Yearbook of the NCTM. Consider the problem 99 x 8. An unskilled mental calculator will be tied into the tedious paper and pencil algorithm to find the answer: Nine times eight is 72. Record the two and carry the seven. Nine times eight is 72 plus seven is 79. The answer is 792. A skilled mental calculator sees 99 x 8 as one group of eight less than 100 x 8. Eight hundred minus eight is 792. A skilled mental calculator could compute 25 x 480 simply by using the number relationship: 25 = 100/4. Instead of calculating with paper and pencil, the problem computes mentally as 100 x 480 = 48000/4 = 12000. There are other types of calculations, and with practice, children can learn and develop their own techniques. Flexibility is key. Note in the examples above that the focus is on the relationships between the numbers, described previously as conceptual knowledge.

Research has shown that children who are quick to learn the basic arithmetic facts create efficient mental strategies for obtaining answers. For example, a child's thought process behind 8 + 7 might be: "I know 8 + 8 is 16 so 8 + 7 must be one less, 15." A quick way to find the product of 8 and 7 involves this type of thinking: "4 x 7 = 28; I have twice as many groups of 7 so 8 x 7 is 28 + 28."

Number sense is an important component of quantitative thinking. Here are the results from two examples from the Third National Assessment (Lindquist, Carpenter, Silver, & Mathews, 1983). The first deals with a decimal estimation problem given to 13-year-olds. Students were not given enough time to calculate the answer with paper and pencil.

 ESTIMATE the answer to 3.04 x 5.3 Percentage choosing response 1) 1.6 28% *2) 16 21% 3) 160 18% 4) 1600 23% 5) I don't know 9%

Performance was dismal, indicating an obvious lack of number sense. Students could not see that 3.04 is about 3, 5.3 is about 5, and their product is about 15. The only reasonable answer was 16. It is important to note that on a similar computation item when students were given time to compute, 57% arrived at the correct answer. Procedural knowledge does not add to students' number sense. Only conceptual knowledge can.

The following open-ended word problem, part of the same assessment, was given to 13-year-olds. Students could use a calculator.

An army bus holds 36 soldiers. If 1,128 soldiers are being bussed to their training site, how many buses are needed?

 Percentage responding *1) 32 7% 2)   31.33; 311/3 etc. 16% 3) 25% 4) wrong operation 20% 5) no answer 32%

Again, performance revealed a lack of number sense and also perhaps a lack of common sense. A calculator in the hands of students without number sense is no help. Most students could not determine what operation to choose. Forty-one percent of those who were able to choose the correct operation did not know how to use the results with respect to the constraints posed by the problem.

Estimation is a critical part of quantitative thinking. It permeates our daily lives. If one listed the types of computation done in the course of an average day, the times when an estimate is sufficient will greatly exceed the number of times when an exact answer is required. This is true in all subject areas and in and out of school. When an exact answer is needed, a calculator is used for its speed and accuracy, not a paper and pencil calculation. Research has shown that good estimators have several common characteristics (Reys, Bestgen, Rybolt, & Wyatt, 1982). They are able to quickly produce "ballpark" estimates in solving a problem. They rely on mental computations, not recording devices. They have a tolerance for error and are comfortable with estimates. They have developed strategies for estimating not normally taught in school and their use of a particular strategy depends on the numbers in the problem. (In other words, they are flexible.) Other common characteristics include: quick and accurate fact recall; understanding of place value; skill with multiples of 10; insisting on compensating their original estimate.

Here are examples of processes used by expert estimators (Reys et al., 1982):

(347 x 6)/43-"I look for nice numbers or multiples to round to 347 to 350, 43 to 42. I cancel 6 and 42 which gives 350/7 or 50." or

"347/43 is about 9; so 9 x 6 is 54, but it must be less because 347/43 is below 9. So I'll take off some. That leaves… I'll say 50."

The 1979 Super Bowl netted \$21,319,908 to be equally divided among 26 NFL teams. About how much does each team get? - "I rounded to 26 million divided by 26 which is 1 million each, It has to be less because of my rounding procedure, say 850,000 each."

An important part of quantitative thinking is the ability of the student to reason about number relationships without numbers. For example, in the last example the student understood that by increasing the dividend in a division problem (keeping the divisor constant), the size of the answer increases (given a/b = c; if a is increased, then c increases). Other examples of this type of knowledge are:

Adding two positive numbers yields a quantity larger than either added. (1/2 + 1/3 cannot equal 2/5 because 2/5 is less than 1/2, one of my original addends.) Dividing by a whole number yields a quantity less than the dividend, but division by an amount less than 1 yields a larger quantity.

Given a fraction or ratio, a/b, if a is increased, the size of the fraction increases; if b is increased the size of the fraction decreases; if both are increased, change in the fraction value cannot be determined.

INFLUENCES OF TECHNOLOGY

Computers and calculators influence mathematics instruction and mathematics usage in two major ways. First, technology raises the question, "How should the mathematics curriculum be changed given the availability of calculators and computers?" Our answer to this question has already been addressed. The curriculum should de-emphasize speed and accuracy in paper and pencil algorithms and emphasize development of a wider range of mathematics competencies such as those listed in the NCSM report (NCSM, 1988). Since the elementary program emphasizes the development of computational proficiency, the use of calculators should have a greater impact on change in the elementary schools than in the secondary schools. (In fact, calculators are accepted more by secondary teachers than elementary teachers, probably because they have less effect on what they teach.) It follows that calculators should be used in other disciplines as well.

The rapidly expanded power of computers raise analogous questions about the algorithmic aspect found in the secondary curriculum (Fey, 1984). Computer software programs are currently available to do much of the symbolic manipulation which continues to constitute a major portion of the mathematical time in classes from algebra to first-year calculus. MuMath, TK!Solver, and the Geometric Supposer are three software programs that have an interactive rather than a tutorial format (Sunburst, 1987). At the time of this writing it is possible to purchase a calculator for about \$60 that will graph all of the types of equations encountered in secondary school. Likewise, a geometry proof checker will evaluate the adequacy of a student's logic when doing two-column geometric proofs. How can these advances fail to have an impact on the curricula?

The second question raised by this technology is, "How should the way mathematics is taught be changed, given the availability of calculators and computers?" The NCTM recommends that mathematics programs take full advantage of the power of calculators and computers at all grade levels (NCTM, 1980). Using calculators in problem-solving situations allows all students to participate in vastly expanded types of activities. Lack of computational ability need not separate students into those who work on low level skills and those who have the opportunity to work on higher level activities. Tedious calculations need not limit the type of problems students are given. Applied problems with real data need not be avoided since the calculator can do the messy work. Using calculators in problem-solving situations is not limited to the mathematics classroom. Data collected in science experiments, and making predictions or understanding trends in the social sciences can all be processed with calculators.

Calculators are not just for number crunching; calculators also can be used to develop new mathematics concepts and to reinforce previously learned concepts. Below are three activities which use a calculator to reinforce and extend a basic concept. The first deals with place value, the second productivity, and the third relates to conservation of natural resources.

1. Enter 6,425 on your calculator using only the numbers 1000, 100, 10, 1 and + , = keys.

2. A social studies class might want to explore the impact on productivity (productivity = output/input) when the output increases and the input remains stable or vice versa; or even more interesting, when both increase or both decrease. Students can quickly generate large numbers of examples. These can form the basis important generalizations.

3. A social studies class dealing with the conservation of natural resources might choose to determine the number or amount of paper products used by the school or community in one year. While data collected would be too tedious to work with by hand, calculators would allow processing to take place with greater ease. Discussion could then focus on ways to conserve and extend these resources rather than on the more mundane calculations.

Computers can deliver instruction in a variety of ways. These include drill and practice for skill learning, simulations for problem solving, and student programming (Shumway, 1988). Teachers should avoid using computers or calculators to train children to do tasks better done by the computer or calculator in the first place. Computers should be used for doing what they do uniquely well and not for duplicating what can be done just as well in other ways. For example, using a drill and practice computer program to practice multiplication problems is inappropriate. First, a worksheet would do just as well; second, increasing speed and accuracy in that skill is a questionable goal. Similarly, calculators should not be used to check the answer to a long division problem solved with paper and pencil. The student should do the problem on the calculator originally.

Computers do have a place in the school curriculum. Drill and practice of basic facts and reinforcement of other mental arithmetic skills continue to be appropriate in the mathematics classroom and can be handled with the use of existing software. Other drill and practice programs are appropriate for spelling, typing, and certain areas where rote memorization is necessary and appropriate, such as learning states, capitals, or presidents. Software manufacturers are beginning to produce materials which emphasize conceptual rather than procedural domains. This is a welcome addition. As more of these higher quality programs find their way into the school curricula, computers will become increasingly important.

Software programs have been created that present problems and monitor student performance. These allow both teacher and students to keep track of progress. Drill and practice programs do not teach new skills; they simply provide opportunity for the practice of learned skills.

Experiences that are too time-consuming or inconvenient to conduct can be simulated by a computer. Teachers can write their own simulation programs or use a wide range of commercial programs. For example, a teacher, without too much difficulty, can write a program to simulate tossing a coin or die a large number of times, enabling students easily to collect data for a lesson on probability and statistics. The computer graphic capabilities provide a unique opportunity to explore problems in geometry and algebra. The Geometric Supposer and Presupposer allow students to explore informally many ideas in geometry. This can lead to conjectured theorems that eventually will be formally proved. Other software packages are available for learning strategies for solving problems. How many diagonals in a 100-sided figure? A software program helps students solve this problem by drawing 3-, 4-, 5-, 6-, or up to 10-sided figures and their corresponding diagonals one at a time. Students use this program to collect data to build a table and look for number patterns.

Simulations also are available in other subject domains. A social studies teacher planning a unit on settling the West in the early and middle 19th century might use the "Oregon Trail," software which counterbalances the many decisions faced by the early pioneers and the consequences of those decisions in terms of conserving limited resources, protecting personal health and safety, and eliminating or limiting other costly errors in judgment or data-based decision making.

Programming can be an effective use of the computer in the elementary and secondary schools. Teachers can learn to program the computer to help children explore a variety of mathematical ideas, such as counting, multiples, and prime numbers (Shumway, 1987). Students also can learn to do the programming. Two commonly used languages are LOGO and BASIC. Advocates suggest that computer programming by students teaches concept learning and problem solving.

Another use of computers is for instructional management. Software packages are available to evaluate tests and to give teachers detailed printouts of student achievement. Such information can also provide longitudinally oriented feedback to students and teachers. This can be useful in grouping for instruction.

Calculators and computers have not yet greatly influenced what mathematics is taught or how it is delivered. However, mathematics educators will continue to push for change in schools in these directions.

Conclusion

All is not well in the mathematics classrooms of the nation's schools. Phrased more positively, there is ample room for improvement in the nation's mathematics classrooms. The latter version, however, fails to communicate the seriousness of the problem. International comparisons of the U.S. and leading industrial nations consistently rank this country in the lower half in nearly every major mathematical category. Major differences in achievement patterns have caused much concern and have stimulated a series of major reports (National Commission on Excellence, 1983; Second International Study, IEA 1984; Stevenson et al., 1986). At the time of this writing, talk continues, and though followed by relatively little action, the horizon looks brighter.

During the past two decades enormous changes have occurred in the scope and depth of the research in mathematics education. New organizations, steady (although insufficient) federal funding, and the evolution of the concept of cooperative research has appreciably improved the extent to which we understand how students learn mathematical and quantitative ideas. Large-scale and long-term projects have emerged dealing with how children learn early number concepts, geometry, and rational number concepts (fraction, ratio, decimal); how children develop concepts of multiplication and division and learn estimation strategies and processes; the influence of sex-related variables on mathematics performance; the impact of calculators and computers; and other aspects of thinking and concept development. There is still much that is not fully understood, but real progress has been made. Schools have not kept abreast of such progress. In fact, mathematics curricula and methods have not changed very much at all in the last 20 years. Paper and pencil calculations and text-related worksheets still dominate in the mathematics classroom.

However, new attempts at upgrading are underway. More money is available for research and the federal government is once again becoming involved in large-scale curriculum development. This is true for science as well as for mathematics. As these changes slowly infiltrate school curricula, new and higher achievement levels ideally will become commonplace among our students. Ideas similar to those expressed here will provide a conceptual framework for those efforts.

Clearly the development of quantitative thinking is an important goal, not only for the mathematics teacher and classroom but for all teachers of all school subjects. An understanding of the world in which we live cannot progress very far unless quantitative issues are addressed. How many? How far? How much? How has it changed? Can you predict? What would happen if? These are questions which are found in all subject domains. The answers to these and numerous other inquiries require serious consideration of the quantitative aspects of the phenomena.

1 The draft version of this chapter was reviewed by : Gary Griffin, University of Illinois-Chicago; Perry Lanier, Michigan State University; and Dick Lesh, WYCAT Systems. Each made valuable suggestions, but the final version is totally the product and responsibility of the authors. AACTE expresses appreciation to all the individuals who contributed to this chapter.

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Annotated Bibliography

Charles, R., & Lester, F. (1982). Teaching problem solving: What, why and how. Palo Alto, CA: Dale Seymour Publications. This very readable booklet addresses issues associated with mathematical problem solving, specifica1ly, why problem solving is important and how to integrate problem solving into the mathematics program. Ideas come directly from experiences with teachers and children at several grade levels.

Hiebert, J. (Ed.). (1986). Conceptual and procedural knowledge: The case of mathematics. Hi1lsdale, NJ: Erlbaum. This reference for researchers contains 10 chapters dealing with current thinking on this very important topic. The authors represent a cross-section of influential mathematics evaluators who discuss the basic procedural/conceptual issue from a number of perspectives and from the vantage point of a number of mathematical topics.

Janvier, C. (Ed.). (1987). Problems of representation in the teaching and learning of mathematics. Hi1lsdale, NJ: Erlbaum. This reference for researchers is part of a series of edited publications designed to update the research community on developments in important areas relating to mathematics education and in areas where important research has been and is being conducted, including studies in representation.

Lesh, R., & Landau, M. (Eds.). (1983). Acquisition of mathematics concepts and processes. New York: Academic Press. This edited book presents some of the more promising and productive research in mathematics learning and problem solving.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books. This book presents the author's vision of computers in education. He describes how education can be changed by incorporating his computer language LOGO into the curriculum. Papert's view of the future has created important debates on the nature and importance of mathematics in school curricula.

Post, T. R. (Ed.). (1988). Teaching mathematics in grades K-8: Research-based methods. Boston, MA: Allyn & Bacon. Each chapter in this mathematics methods book is written by nationally or internationally recognized leaders in the field discussed. Ideas and activities presented are based on the most current research developments. Topics include the nature of mathematics learning; problem solving; measurement; operations with whole numbers; rational numbers; geometry; estimation; ratio and proportion; calculators and computers; gender issues; and evaluation.

Reys, R., Suydam, M., & Lindquist, M. (1984). Helping children learn mathematics. Englewood Cliffs, NJ: Prentice-Hall. A methods book for teachers of elementary school mathematics. The first section provides information on the changing mathematics curriculum and how children learn mathematics. The second section discusses strategies and teaching activities related to early number concepts, place value, whole number operations, measurement, fractions, decimals, ratio, proportion and percent, estimation, and geometry.

Schoen, H. (Ed.). (1986). Estimation and mental computation: 1986 yearbook. Reston, V A: National Council of Teachers of Mathematics. Each year the National Council of Teachers of Mathematics publishes a yearbook addressing a timely topic related to the contemporary mathematics curriculum. The topic in 1986 was estimation. Articles provide specific classroom activities on such topics as computational estimation, estimation in measurement, estimating fractions, estimating decimal products, and using money to develop estimation skills with decimals.

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