# Does whole-class teaching improve mathematical instruction

# Does whole-class teaching improve mathematical instruction?

**Viewpoint:** Yes, whole-class teaching improves mathematical instruction by cutting down on preparation time for teachers and providing teachers with a consistent method to assess student development.

**Viewpoint:** No, whole-class teaching does not improve mathematical instruction; it is inefficient and ineffective except in situations where class size is very small and student ability falls in a narrow range.

At the heart of this debate is the age-old question, "How much is too much?" Should class time center around "whole-class teaching" methods, or should it be tailored toward individual needs? Or should it, ideally, be a mixture of methods?

Most of us have suffered from a twinge of math anxiety at least once, especially when faced with the challenge of solving a seemingly impossible-to-solve equation. And yet, by all accounts, basic mathematical skills are considered by many to be essential for success in the business world.

Success, of course, is measured in a variety of ways. Test scores commonly measure one form of success, which is academic achievement. Korean students, for example, frequently outrank other students with regard to test scores. Proponents of whole class learning are quick to point out that the average Korean class size is made up of 40 or more students. This, they say, allows teachers to utilize a wider variety of nontraditional teaching methods more effectively. But are higher test scores a result of larger classes, or a societal attitude toward learning in general? Certainly study habits play a part in academic achievement.

Success can also be measured in other, less tangible, ways such as a student's confidence level and ability to apply knowledge to real-life situations. After all, not all students are gregarious and outgoing. Some are less comfortable with class participation than others. For the shy, perhaps math-phobic student, individual attention can be quite beneficial. Working in small groups might allow the less confident student to ask questions he or she might otherwise feel uncomfortable to ask in front of the entire group. Many students dread asking what could considered to be a "stupid question" and would rather remain in the dark than look foolish in front of their peers. So, when considering whether whole class learning is the best approach to teaching mathematics, one must also think about the feasibility of class participation as a key element in the learning process. The issue is clearly complicated, and is not likely to be decided any time soon.

It would seem that we must first agree on the goals of mathematical instruction and on what constitutes success, and that alone is a monumental task. Some people value personal achievement, while others say that measurable success (a high test score) is paramount. Making math seem less intimidating is certainly one goal; making it more interesting is another.

Ideally, a teacher's job is to create an environment in which learning can take place, but how does one do that when students enter a classroom at different academic levels? Does the teacher team them together, utilizing technological aids and modern resources, in the hope that demonstrating mathematical concepts on a broader basis will drive the finer points home? Or should the teacher focus on the detail work, breaking mathematical analysis up into smaller bits (and in smaller study groups) in the hope that individual needs will be met? Whether it is best to focus on the collective or on the individual depends on the educator you talk to.

Some say the focus should be situational. In other words, some aspects of math can be taught on a whole-class basis, while others can be explored on an individual basis. The nontraditional approach embraces active rather than passive learning, which is said by some to bring an element of excitement to the classroom. Others would argue that there is something comforting about the old-fashioned approach—the teacher puts the problem on the board and explains in a step-by-step manner how to solve it. This method, some would argue, serves as an equalizer; regardless of their skill level, students can grasp the general concepts of mathematics if they simply go to class and pay attention. Still others say that is too optimistic an approach; keeping students' attention in this highly visual, technological world is a constant challenge. Perhaps the solution can be found in a combination of methods; as in mathematics itself, there is often more than one way to solve a problem.

—LEE ANN PARADISE

## Viewpoint: Yes, whole-class teaching improves mathematical instruction by cutting down on preparation time for teachers and providing teachers with a consistent method to assess student development.

In the real world, people must use mathematical skills in their daily lives. In the ideal world of mathematical instruction, students are provided with a good overall sense and understanding of mathematics, as well as of its importance to solving real-world problems. Unfortunately, things are often far from the ideal. Some people actually have a phobia of mathematical concepts, and a lower-grade anxiety about math is present in even larger numbers of people. In any case, a majority of the population is apathetic toward math, and ignorant concerning its true nature.

At present, educators put great effort into trying to develop mathematical skills in students using traditional methods. For a majority of students, however, these traditional methods fail. Instead of focusing upon the utility of mathematics for its application to a wide variety of life experiences, traditional methods rely far too much upon rote learning and a mechanical attitude towards math. This traditional, mechanical approach all too often leads to student disinterest in, and even distaste for, mathematics.

### A Big Difference

Whole-class teaching (WCT) and its associated technology, make it easier to effectively teach mathematics. WCT makes a big difference by cutting down on preparation time for teachers, and helps to keep the class focused on the math concept at hand, thus freeing the teacher from classroom management issues (so more time can be spent on effectively teaching math), and providing a consistent way to assess student development and learning. The whole-class methodology requires a fundamental change in teaching; with a change in the emphasis on how to teach, changes in the assessment of student performance, use of an integrated curriculum, and the use of manipulative and activity-based instruction.

In spite of the many favorable attributes, including those referred to above, that its supporters contend are genuine, WCT does have its detractors. One of the main objections relates to class size. Opponents of WCT make the contention that whole-class teaching is only possible in small classes; that it breaks down in large classroom settings. However, the Third International Mathematics and Science Study (TIMSS), a project of the International Study Center at Boston College, stated in a report that "69% of the students in Korea were in mathematics classes with more than 40 students and 93% were in classes with more than 30 students. Similarly, 98% of the students in Singapore, 87% in Hong Kong, and 68% in Japan were in classes with more than 30 students." The report continues, "Dramatic reductions in class size can be related to gains in achievement, but the chief effects of smaller classes often are in relation to teacher attitudes and instructional strategies. The TIMSS data support the complexity of this issue. Across countries, the four highest-performing countries at the fourth grade—Singapore, Korea, Japan, and Hong Kong—are among those with the largest math classes … the students with higher achievement appear to be in larger classes." It is reported that these teachers relied more on whole-class instruction and independent work than the United States, two practices which are currently opposed by many school reformers. The TIMSS study shows that WCT is effective not only in small classes, but also in large classroom environments.

In additional support of the TIMSS studies, psychology professors Jim Stiegler of the University of California at Los Angeles, and Harold Stevenson of the University of Michigan recommended in *The Learning Gap* (1992) larger class sizes (in the context of using WCT in those classrooms) in order to free teachers to have more time to collaborate and prepare.

### Oral, Interactive, and Lively

High-quality WCT has been described as "oral, interactive, and lively." WCT is not the traditional method of teaching math that uses the all-too-simplistic formula of lecture, read the book, and "drill and practice." It is a two-way process in which pupils are expected (and encouraged) to play an active role by answering questions, contributing ideas to discussions, and explaining and demonstrating their methods to the whole class. For example, the June 1998 report entitled *Implementation of the National Numeracy Strategy* written by David Reynolds, a professor of education at the University of Newcastle upon Tyne, Great Britain, talks about why WCT is more active than passive in its practices throughout the classroom. Reynolds says, "Direct teaching of the whole class together does not mean a return to the formal chalk and talk approach, with the teacher talking and pupils mainly just listening. Good direct teaching is lively and stimulating. It means that teachers provide clear instruction, use effective questioning techniques and make good use of pupils' responses."

### Achievements

The National Numeracy Strategy is an important educational project involved in raising math standards in Great Britain. According to the project's *Framework for Teaching Mathematics*, WCT succeeds in a variety of important math-related teaching areas:

- Teachers are effectively able to share teaching objectives. This ability guides students, allowing them to know what to do, when to do it, and why it is being done.
- Teachers are able to provide to the student the necessary information and structuring that is essential to proper instruction.
- Teachers are able to effectively demonstrate, describe, and model the various concepts with the use of appropriate resources and visual displays, especially by using upto-date electronic devices for conveying information.
- Teachers are able to provide accurate and well-paced explanations and illustrations, and consistently refer to previous work to reinforce learning.
- Teachers are able to question students in ways that match the direction and pace of the lesson to ensure that all pupils take part. This process, when skillfully done, assures that pupils (of all abilities) are involved in the discussions and are given the proper amount of time to learn and understand. Only WCT allows this to happen because all students are part of the group, allowing for more effective responses from the teacher.
- Teachers are able to maximize the opportunities to reinforce what has been taught through various activities in the class, as well as homework tasks. In parallel, pupils are encouraged to think about a math concept, and when ready to talk through a process are further encouraged either individually or as part of the group. This process expands their comprehension and reasoning, and also helps to refine the methods used in class to solve problems. It also allows students to think of different ways to approach a mathematical (or for that matter, any other) problem.
- Teachers are able to evaluate pupils' responses more easily, identify mistakes, and turn those mistakes into positive teaching points by talking about them and clarifying any misconceptions that led to them. Along these lines, teachers discuss the justifications for the methods and resources that students have chosen, constructively evaluating pupils' presentations with oral and written feedback. This is done with all of the students, so all learn from their fellow students' mistakes.
- Teachers are able to review the students near the end of each lesson with respect to what had been taught and what pupils have learned. Teachers identify and correct any misunderstandings, invite pupils to present their work, and identify key points and ideas. Insight into the next stage of their learning is introduced at this time. Uniform review is important, and is most effectively performed with WCT as the predominate style of teaching.

As implied above, teachers are an essential component in the success of WCT. Debs Ayerst, the education officer at the British Education, Communications, and Technology Agency states that in relation to WCT, "The role of the teacher or assistant is seen as paramount in order to demonstrate, explain and question, stimulate discussion, invite predictions and interpretations of what is displayed, and to ask individual children to give an instruction or a response."

### Key Technological Aids and Techniques

WCT allows for the successful deployment of technology in the classroom. A number of key techniques utilizing electronic and other technologies supplement WCT with respect to mathematics. These techniques are instituted primarily to increase the effectiveness of math instruction through the overall WCT process. Some of the more vital technologies include television screens, large-screen monitors, data projects, graphing calculators used with an overhead projector, LCD panels/tablets that sit atop an overhead projector unit, plasma screens, and interactive whiteboard technologies. These devices all aim to allow access to and use of digital resources for the benefit of the whole class while preserving the role of the teacher in guiding and monitoring learning.

Computers are often either underused or incorrectly used—or both—in the classroom. But with today's technology (and that technology will only get better over time) each student can be equipped with a computer and follow the class lesson while the teacher demonstrates the lessons with results projected on a large overhead projection screen. Again, WCT will advance as teachers are better trained with computers and gain a more informed understanding of technology as applied to the classroom.

### The Benefits of Technology

The speed and automatic functions of the technologies discussed above enable teachers to more easily demonstrate, explore, and explain aspects of their instruction in several ways. First, the capacity and range of technology can enable teachers and pupils to gain access to immediate, recent, or historical information by, for example, accessing information on CD-ROMs or the Internet. Second, the nature of information stored, processed, and presented using such technology allows work to be changed easily, such as using a word processor to edit writing. Third, pupils learn more effectively when they are using, for example, a spreadsheet to perform calculations rather than performing them by hand. In this way they can concentrate on patterns that help to enhance the learning process. In summary, such technologies are geared toward WCT because all students can easily follow along and actively participate.

### Examples of Successful WCT Programs

One good example of a WCT mathematics resource is Easiteach®, an annual subscription service developed in Great Britain which is designed to aid teachers in delivering lessons, and provides an online collection of ready-made downloadable teaching activities. Easiteach Maths® combines a teaching tool full of familiar math resources—such as number lines, number grids, and place-value cards—together with a collection of ready made online teaching activities to help teachers deliver math lessons. The advantages of Easiteach Maths® are that it: (1) saves the teacher's planning time by providing familiar resources plus ready-made WCT activities to use with them; (2) gives access to "best-practice" methods through expertly written, flexible activities that integrate with existing classroom resources; (3) reduces classroom management issues by providing a wide range of familiar resources that are managed centrally; and (4) enhances learning through its WTC-based interactive and dynamic nature.

WCT suits the study of mathematics because there are so many real-life examples that can be used to teach and understand math. As an example, a former middle school teacher near Detroit, Michigan, designed a practical supplement for middle school curriculum that follows the thirteen weeks of the National Association of Stock Car Auto Racing (NASCAR) Winston Cup season. Students use newspaper results of the automobile races to focus on topics such as rounding, estimating, integers, proportions, graphing, percents, probability, and statistics. In this setting the whole class is introduced to the NASCAR curriculum; assignments are given that can be done individually, in small groups, and with the whole class; and discussion is used to analyze facets previously learned. This whole-class setting is much more conducive to learning than a small-group setting, where results are not commonly compared and discussed, and different viewpoints not often shared.

Other examples of a successful WCT program is that created by Creative Teaching Associates, a company started in 1971 by three educators, Larry Ecklund, Harold Silvani, and Arthur Wiebe. Ecklund, Silvani, and Wiebe wanted to assist teachers to help students become more efficient learners and to help them enjoy learning, and created a variety of activities and games from real-life experiences to reinforce math skills. The company developed a "Money and Life" game that represents a real-life bank account to teach students from grades 5 to 12 about the variety of processes involved in having a checking account. The game board allows students to write checks, make deposits, and keep records of such transactions as buying groceries, paying taxes, making car payments, and paying medical bills.

### Conclusion

What *does* constitute success in mathematics? In order to count the learning experience as successful, students need to enjoy math, and this in turn means mathematics should become less intimidating, more relevant to students, and involve more active learning. The position taken in 1989 by the NCTM Standards and the National Research Council's *Everybody Counts* report lent support to the idea that math needs to be thought about in new ways; more specifically, in the way it is taught and the way it is evaluated.

In the traditional classroom setting, mathematics is taught separately from other courses. It is force-fed to students as a set of necessary skills, and this-force feeding, for the most part, turns off the majority of students. Later in life, when those same people need math skills, they are typically too intimidated and unmotivated to acquire that knowledge. The whole-class technique incorporates mathematics within all coursework in order to teach basic skills that are usable in everyday life.

The main goal of WCT as it relates to the teaching of math is to make students aware of the nature of mathematics and the role math plays in contemporary society. To do this, mathematics must be taught in much the same way as history, geography, or English literature, as part of learning about society. This is best accomplished in the whole-class environment, in which mathematics is taught as a part of the culture in order to provide a motivating situation for more students. With this aim in mind, a mathematics education will be more likely to produce an educated citizen who can use mathematics in everyday life, and not someone who shudders at the term "math." Finally, in a WCT atmosphere, teachers are better able to identify those students that are not keeping up with the rest of the class, as well as those who are gifted students.

—PHILIP KOTH

## Viewpoint: No, whole-class teaching does not improve mathematical instruction; it is inefficient and ineffective except in situations where class size is very small and student ability falls in a narrow range.

The reform of mathematics education seems to be a high priority in the United States and the rest of the world. The National Council of Teachers of Mathematics (NCTM) established a Commission on Standards for School Mathematics in 1986 for just this reason. The commission's conclusions led to the 1989 publication of *Curriculum and Evaluation Standards for School Mathematics* (known simply as *Standards* ), which includes numerous standards for grades kindergarten through 4, 5 through 8, and 9 through 12, as well as procedures for evaluating the mathematical proficiency of students within these three groups based upon the standards. The NCTM standards established five goals for improving mathematical literacy. These goals are geared so that students: (1) become mathematical problem solvers; (2) become confident in their ability to perform mathematics;
(3) learn to value mathematics; (4) learn to communicate mathematically; and (5) learn to reason mathematically. When properly used in the classroom, whole-class teaching (WCT) can be a useful "component" of instruction in the achievement of these five important goals. However, whole-class teaching does not, in and of itself, improve mathematical instruction.

### The Difficulties

By and large, whole-class instruction does not significantly improve the performance of mathematics teaching when used alone. It is often very difficult to use whole-class approaches, for instance, when there are multiple languages spoken in the classroom, when students with little or no mathematical background (with respect to fellow students) enter into a new classroom, or when a wide diversity of students (with respect to math abilities) is taught in the classroom. It is also increasingly difficult to effectively teach whole-class with respect to mathematics when the classroom size is very large. Sonia Hernandez, deputy superintendent for the Curriculum and Instructional Leadership Branch of California State's Department of Education, noted that in some countries where math students are among the highest achievers in the traditional classroom environment, the diversity of the student population is very small. Hernandez asserts that in some major urban school districts "the instructional strategies have to be much different from what can be put into place when one has a pretty uniform student population." Hernandez continues, "By and large, we do have whole-class instruction to the extent that it is reasonable, but it is very difficult to use only whole-class approaches when there are multiple languages in the classroom, or when students with no educational background whatsoever are coming into California." Contentions such as these show why WCT is inadequate for the general teaching of students, and specifically for the teaching of mathematics.

Implementing WCT often results in unequal opportunities for learning. Most classrooms do not contain students with a narrow range of math abilities, but usually those with skills ranging from low to high. In addition, unmotivated students can easily disrupt the time the entire class needs to learn math. Highly motivated students are easily bored with a slow process, or with the constant repetition of concepts that they have already learned. Disciplinary problems, late-comers, and the need to repeat instructions are several ways in which time can be wasted with WCT. In contrast to whole-class methods, multiple sizes for groups, along with learning in pairs and as individuals, all help to reduce unequal opportunities that exist to a greater or lesser extent in most schools.

Perhaps the most recurring criticism of mathematical whole-class instruction is that it is extremely difficult to meet the needs of both high and low achievers. Students have diverse learning styles that make WCT very inefficient to use, or at least to use solely, in the classroom setting. It is essential to have a mix of classroom techniques, from group time to individual time, so that students can demonstrate different aptitudes in different settings. It is generally accepted that all students of all abilities welcome nontraditional math activities, but each student has their own likes and dislikes among these activities. However, when only one avenue is available (as with WCT), students who dislike this way of learning will suffer academically.

### What Does Work?

Students usually enjoy a change from the traditional class work involving lecturing, memorization, and tests, and especially enjoy the variety inherent in different instructional techniques used in the classroom throughout the day, week, and semester. In order to most effectively teach mathematics to students, the teacher must successfully guide them through diverse avenues involving individual, small group, and whole-class activities. The introduction of a new math concept with a new set of vocabulary terms is easy within the whole-class setting. Such a discussion, for example, can build listening and response skills, and the teacher can talk with the entire class about this new topic. However, the effectiveness of WCT normally ends there. It is much more beneficial to then break the class down into smaller groups, pairs, or (occasionally) individuals in order to achieve better comprehension of mathematical concepts. The same challenging math curriculum is followed throughout each group, but the depth of work can vary, and the topics explored in different ways, depending on the needs and skills of each group. The teacher is critical to this process, as students remain in certain groups or are moved depending on changing needs and/or requirements. The fact remains that one teaching style (WCT, for instance) is typically not effective if used all the time.

A whole-class environment is often useful while working with math that applies to everyday life. However, assignments are more effectively carried out when they are assigned to smaller groups that perform the actual work, then report their results back to the main group with a summary and related math questions. Typical topics might be: "If five gallons of water flows from the faucet into the bathtub in one minute, how much water would be in the bathtub after 12 minutes?" and "The Mississippi River delivers *x* gallons of water each hour to the Gulf of Mexico. How much water is this in one day? … one week? … etc.?" This type of math encourages interesting discussions, but, for the most part, is more effective when students first interact in small groups, then relate their experiences back to the main group. Small-group work allows students to talk about the math tasks at hand while they solve nonroutine problems. Smaller groups, such as pairs and individuals, are practical for computer lab work. Moreover, individual work settings ensure that all students process lessons at their own rate of learning.

### Traditional Teaching

Dr. Keith Devlin, dean of science at Saint Mary's College of California, and a senior researcher at Stanford University's Center for the Study of Language and Information, stated in an article of the Mathematical Association of America that "any university mathematics instructor will tell you that the present high school mathematics curriculum does not prepare students well for university level mathematics." In response to such criticisms, the NCTM *Standards* call for the implementation of a curriculum in which, *through the varied use* of materials and instruction, students are able to see the interrelatedness of math. The emphasis here is on a mix of traditional (with lecturing, memorization, and tests and homework designed to reinforce the lecture), and whole-class instructional approaches. A pronounced shift away from traditional teaching methods to the newer whole-class approach can run into several inherent difficulties of WCT. For instance, in WCT, students who are not interested in math quite often hamper those students that are interested. When there is a mix of hard-working students and low-achievers in a whole-class environment, the quality learning time of the entire class suffers. Students too often view the whole-class environment as a time for social interaction rather than for academic effort.

### Effective Math Teaching Methods

Hands-on and interactive materials, computer lab assignments for individuals and pairs, and participative whole-class discussion and problem-solving are all ways to contribute to an effective math class. A flexible classroom setting is necessary for these activities in order to maximize the learning for each and every student. Small groups (as opposed to whole-class) that are composed of a wide mix of abilities are especially appropriate when creative brainstorming is sought. Regardless of ability, students generally enjoy this level of grouping when working on a math problem. At other times, it is more suitable to use small groups that are more or less homogeneous. Even in smaller, homogeneous groups, students who show the most aptitude do not necessarily show the highest aptitude in all assignments and areas of math. In fact, it is often obvious that students who may be perceived to be of low ability are very proficient in a particular skill, and such students are able to effectively interact in groups when the need to use that special skill arises.

Even though nontraditional teaching methodologies, including whole-class learning, are important new tools in the teaching of mathematics, older, traditional techniques still have an important place in the classroom. Homework, quizzes, and examinations, although often very unpopular, often act as motivators in a highly diverse classroom. Homework can be given in several levels of difficulty because of the diversity of group size. Gifted students normally work the more difficult assignments, while others choose a level they feel more comfortable with, given their expertise in a particular topic. All students, regardless of their level of expertise, benefit when the class discusses problems from each assignment in detail. Average students gain confidence more easily by working in various groups, and can strive toward their capabilities when they perceive advanced work as more attainable. They became motivated by the challenge of more difficult assignments, while always learning from the opportunity of resubmitting corrected assignments. Because all students take the same quizzes and tests—but from a diverse (i.e., as regarding group size) classroom environment—the entire process becomes less intimidating.

So-called "at-risk" (those perceived to have low potential) students certainly benefit from having good role models. Discipline problems are greatly reduced when these students are placed in a class where the majority of students want to learn. Slower students gain confidence because they can easily receive individual and smaller group help from the teacher or fellow students. Students who had previously been unable to take advanced, enhanced, or high-ability math classes, now have the opportunity to develop their true potential. No longer relegated to a mediocre curriculum with no diversity, these students have a much better chance to become both assisted and challenged by the higher achievers.

Diverse grouping also offers benefits for advanced students. They now have the opportunity to examine certain math topics in depth and to explore some unusual areas of the subject, and they often benefit from helping others. Some may be under a great deal of parental pressure to continue as high achievers, but this pressure is often reduced when average students are intermingled within the classroom setting. Within such intermingled classes, advanced students become less threatened than they would be by learning in a class with an overwhelming majority of advanced students.

### The Future

Many mathematics teachers appreciate the advantages of flexible teaching, which gives longer time for classes, additional laboratory time in math, more time for presenting projects and extracurricular activities, and more opportunity to involve students in in-depth and active-learning scenarios. These teachers believe that it is essential to have plenty of quality time to learn math, and that this quality time will encourage the students' commitment to learning. As opposed to the strictly whole-class approach, where the entire class "brainstorms" and discusses mathematical concepts, flexible grouping arrangements normally produce fewer discipline problems, and provide more time for learning. By the end of the semester, teachers may note advantages associated with their students remaining with the same group for all academic subjects. Students often request that groups remain intact, and their self-confidence when giving oral reports and answering questions has often increased because of the camaraderie within the groups.

Organizing into diverse groups gives all students in math classes the opportunity to be exposed to a challenging curriculum. A combination of traditional and nontraditional arrangements of students—individual, pair, small group, and whole-class teaching environments—provides the best possible avenue to teach students to become well versed in mathematics, and to feel comfortable with their ability to perform necessary math skills.

—WILLIAM ARTHUR ATKINS

## Further Reading

Carnegie Council on Adolescent Development. *Turning Points: Preparing American Youth for the 21st Century.* New York: Carnegie Corporation, 1989.

Davidson, Neil, ed. *Cooperative Learning in Mathematics: A Handbook for Teachers.* Menlo Park, CA: Addison-Wesley Publishing, 1990.

Department for Education and Skills, London, England. "The Final Report of the Numeracy Task Force." <http://www.dfes.gov.uk/numeracy/>.

Glasser, W. *The Quality School: Managing Student Without Coercion.* New York: Harper and Row, 1990.

Kennedy, Mary M., ed. *Teaching Academic Subjects to Diverse Learners.* New York: Teachers College Press, 1991.

Kitchens, Anita Narvarte. *Defeating Math Anxiety.* Chicago: Irwin Career Education Division, 1995.

Langstaff, Nancy. *Teaching in an Open Classroom: Informal Checks, Diagnoses, and Learning Strategies for Beginning Reading and Math.* Boston: National Association of Independent Schools, 1975.

Lerner, Marcia. *Math Smart.* New York: Random House, 2001.

Mertzlufft, Bonnie. *Learning Links for Math.* Palo Alto, CA: Monday Morning Books, 1997.

National Assessment of Educational Progress. *The Mathematics Report Card: Are We Measuring Up?* Princeton, NJ: Educational Testing Service, 1988.

National Council of Teachers of Mathematics. *Curriculum and Evaluation Standards for School Mathematics.* Reston, VA: National Council of Teachers of Mathematics, 1989.

National Research Council. *Everybody Counts: A Report to the Nation on the Future of Mathematics Education.* Washington, DC: National Academy Press, 1989.

Oakes, J. *Keeping Track: How Schools Structure Inequality.* New Haven, CT: Yale University Press, 1985.

——. *Multiplying Inequalities: The Effects of Race, Social Class, and Tracking on Opportunities to Learn Mathematics and Science.* Santa Monica, CA: The Rand Corporation, 1990.

Schlechty, P. *Schools for the Twenty-First Century: Leadership Imperatives for Educational Reform.* San Francisco: Jossey-Bass, 1990.

Valentino, Catherine. "Flexible Grouping."<http://www.eduplace.com/science/profdev/articles/valentino.html>.

## KEY TERMS

### CD-ROM:

Acronym for Compact Disc-Read-Only Memory. A CD-ROM is a rigid plastic disk that stores a large amount of data through the use of laser optics technology.

### HOMOGENEOUS:

Having a similar nature or kind.

### INTEGERS:

Any positive or negative counting numbers, or zero.

### LCD:

Acronym for Liquid Crystal Display. An LCD panel is a translucent glass panel that shows a computer or video image using a matrix of tiny liquid crystal displays, each creating one pixel ("picture element" or dot) that makes up the image.

### PROBABILITY:

Ratio of the number of times that an event occurs to the larger number of trials that take place.

### STATISTICS:

Discipline dealing with methods of obtaining a set of data, analyzing and summarizing it, and drawing inferences from data samples by the use of probability theory.

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