It is intended to be read by school personnel who are responsible for developing and improving the district's curriculum, as well as by teachers who wish to understand their own instructional activities within a K-12 context. It is also intended to be read by parents who have reviewed the Parents' Guide and are interested in getting a more detailed overall picture of what mathematics educators in the state perceive to be a comprehensive mathematics education.
The ten content standards address the following areas:
Following this chapter are ten chapters in each of which one of these content standards is addressed individually. Each chapter is intended to be a self-contained discussion of that content area. It begins with a K-12 overview (taken from this chapter) and continues with appropriate information about that standard for each of five grade level clusters.
The materials for each of the grade level clusters, K-2, 3-4, 5-6, 7-8, and 9-12, are also intended to be self-contained discussions of that content area for that grade level. For each of these clusters, the chapter contains a grade level overview (consistent with the K-12 overview of that standard), expectations for that grade level, and activities which illustrate how the expectation can be addressed in the classroom.
Altogether over 1500 activities are described here that support the implementation of the ten content standards of the New Jersey Mathematics Standards.
The materials have thus been arranged so that, for example, 5th grade teachers can focus on the grade level 5-6 sections of Chapters 9-18. Note, however, that the 5th grade teacher should also review the grade level 3-4 material ~ to find out what the student brings to the 5th grade ~ and the grade level 7-8 material ~ to find out what the student will be expected to do at the next grade level; the goal however is that every teacher should be familiar with the mathematics standards and expectations at all grade levels.
The next version of the New Jersey Mathematics Curriculum Framework will be accompanied by a disk so that districts can print the material in various ways for various audiences. Treating the material as a database will also permit updated versions, with extended lists of activities, to be issued periodically on disk.
Although each content standard is discussed individually both in this chapter and those that follow, it is not the intention that each be treated separately in the classroom. Indeed, as noted in Chapter 1, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences. Many of the activities provided in Chapters 9 through 18 are intended to convey this message; you may well be using other activities which would be appropriate for this document. Please submit your suggestions of additional integrative activities for inclusion in subsequent versions of this curriculum framework; address them to Framework, P.O.Box 10867, New Brunswick, NJ 08906.
| All students will develop their number sense through experiences which enable them to investigate the characteristics and relationships of numbers, represent numbers in a variety of forms, and use numbers in diverse situations. |
Our students often see too little relationship between what is happening in their mathematics classrooms and their daily lives. It is essential that the mathematics curriculum build on the sense of number that students bring to school. Problem solving offers the major avenue to that link and to the development of number sense. Problems and numbers which arise in the context of the students world are more meaningful than traditional textbook exercises and help to promote the relationship between the operations and the numbers involved. Frequent use of estimation and mental computation are also important ingredients in the development of number sense as are regular opportunities for student communication. The discussion of their own invented strategies for problem solutions helps to strengthen the students intuitive understanding of numbers and number relations.
Number sense is not inherent in a person's ability to perform numerical computations. A "sense-building mode" is best established when students are provided with opportunities to explore number relationships, are encouraged to challenge and question, and are allowed to experiment to discover strategies and techniques of their own that ease the path to the solution of mathematical problems.
The development of personal meaning for numbers should be reinforced in the middle grades with an extension to other numbers and notations such as percents and roots. Student experiences should include exploration of the properties of sets of numbers. High school students should extend their meaning of number to the real number system and a recognition that still other number systems exist. They should have the opportunity to develop intuitive proofs of properties of operations and sets of numbers such as closure, commutativity, and associativity.
Students must also develop a facility for working with different forms of numbers. To intelligently select the right form of numbers to use for a particular task, a student must be very comfortable with order and comparison relations and with various approaches to establishing equivalence among the variety of types of numbers we use regularly in our society. Statements about one particular quantity might best be expressed with a fraction, a percent, a decimal, a ratio, an approximated whole number, or some other number form. The correct choice depends on the context of the use of the number, but must be built upon an adequate understanding of each form and the interrelationships among them.
One empowering way to achieve these understandings in the classroom is through the identification and description of number patterns and the use of pattern-based thinking. For example, the examination and modeling of many pairs of fractions with equal numerators help students develop the generalization that the fractions with larger denominators represent smaller quantities. Activities promoting pattern-based thinking can assist students in making similar generalizations about other number forms and their relationships as well as to build initial notions of yet other types of important number concepts such as odd and even, prime numbers, factors and multiples.
Graduates of our schools also must be able to wisely use numbers wherever they are encountered in real life. They must develop an awareness of numbers and their uses. Numbers are used as counts, as measures, as labels, and as locations and each use has unique restrictions on appropriate forms and operations. The opportunity to develop the needed familiarity with all of these uses comes through the regular presentation of problem situations which emphasize them. Some activities should focus on the explicit uses of the numbers themselves, however. An elementary grades discussion of why it makes no sense to add the numbers on a town's roadway welcome sign which lists the population, elevation, and year of founding would help challenge thoughtless computational manipulation of numbers as well as establish the standard number uses.
In summary, the commitment to develop number sense requires a paradigm shift in the way students learn mathematics. Our students will only develop strong number sense to the extent that their teachers use pedagogy which encourages the understanding of mathematics as opposed to the memorization of rules and mechanical application of algorithms. Every child has the capability to succeed as a user of mathematics, but the degree of success is directly related to the strength of their number sense. The way to assure that all students acquire a good sense of number is to have them consistently engage in activities which require them to think about numbers and number relationships and to make the connection with quantitative information encountered in their daily lives.
| All students will develop their spatial sense through experiences which enable them to recognize, visualize, represent, and transform geometric shapes and to apply their knowledge of geometric properties, relationships, and models to other areas of mathematics and to the physical world. |
Traditionally, geometry in schools has been taught as the prime example of a formal deductive system. While this view of the content is important, its domination has led to the exclusion of other, less readily formalized topics and applications. Geometry instruction should not be limited to formal deductive proof and simple measurement activities, but should include the study of geometric transformations, analytic geometry, topology, and the connection of geometry with algebra and other areas of mathematics. Posing and solving problems in this more richly defined geometry allows students to use geometric intuition to develop more generic mathematical problem-solving skills.
Although the levels are not completely separate and the transitions are complex, the model is very useful for characterizing levels of children's thinking. One particularly pertinent finding that shows up consistently in the research is that appropriately targeted instruction is critical to children's movement through these levels. Stagnation at early levels is the frequent result of a geometry curriculum that dwells on identification of shapes and their properties.
It is not difficult to conceive of a curriculum that adheres to the van Hiele model. By virtue of living in a three-dimensional world, having dealt with space for five years, children enter school with a remarkable amount of intuitive geometric knowledge. The geometry curriculum should take advantage of this intuition in expanding and formalizing the students' knowledge. In early elementary school, a rich, qualitative, hands-on study of geometric objects helps young children develop spatial sense and a strong intuitive grasp of geometric properties and relationships. Eventually they develop a comfortable vocabulary of appropriate geometric terminology. In the middle school years, students should begin to use their knowledge in a more analytical manner to solve problems, make conjectures, and look for patterns and generalizations. Gradually they develop the ability to make inferences and logical deductions based on geometric relationships. In high school, the study of geometry expands to include coordinate geometry, trigonometry, and both inductive and deductive reasoning.
The study of geometry should make abundant use of experiences that require active student involvement. Constructing models, folding paper cutouts, using mirrors, pattern blocks, geoboards, and tangrams, and creating geometric computer graphics all allow opportunities for students to learn by doing, to reflect upon their actions, and to communicate their observations and conclusions. These activities and others of the same type should be used to achieve the goals in the seven specific areas of study that comprise this standard and which are described below.
In their study of spatial relationships, young students should make regular use of concrete materials in hands-on activities designed to develop their understanding of objects in space. The early focus should be the description of the location and orientation of objects in relation to other objects. Additionally, students can begin an exploration of symmetry, congruence, and similarity. Older students should study the two-dimensional representations of three-dimensional objects by sketching shadows, projections, and perspectives.
In the study of the properties of geometric figures, students deal explicitly with the identification and classification of standard geometric objects by the number of edges and vertices, the number and shapes of the faces, the acuteness of the angles, and so on. Cut-and-paste constructions of paper models, combining shapes to form new shapes and decomposing complex shapes into simpler ones are useful exercises to aid in exploring shapes and their properties. As their studies continue, older students should be able to perform classic constructions with straight edges and compasses as well as with appropriate computer software and to formulate good mathematical definitions for the common shapes, eventually being able to make deductions and solve problems using their properties.
The standard geometric transformations include translation, rotation, reflection, and scaling. They are central to the study of geometry and its applications in that these manipulations of figures offer the most natural approach to understanding congruence, similarity, symmetry, and other geometric relationships. Younger children should have a great deal of experience with flips, slides, and turns of concrete objects, figures made on geoboards, and paper-and-pencil figures. Older students should be able to use more formal terminology and procedures for determining the results of the standard transformations. An added benefit of experience gained with simple and composite transformations is the mathematical connection that older students can make to functions and function composition.
Coordinate geometry provides another strong connection between geometry and algebra. Students can be informally introduced to coordinates as early as kindergarten by locating entries in tables and finding points on maps and grids. In later elementary grades, they can learn to plot figures on a coordinate plane, and still later, study the effects of various transformations on the coordinates of the points of two- and three-dimensional figures. High-school students should be able to represent geometric transformations algebraically and interpret algebraic equations geometrically.
Measurement and geometry are interrelated, and understanding the geometry of measurement is necessary for the understanding of measurement. In elementary school, students should learn the meaning of such geometric measures as length, area, volume and angle measure and should be actively involved in the measurement of those attributes for all kinds of two- and three-dimensional objects, not simply the most standard, uniform ones. Throughout school, they should use measurement activities to reinforce their understanding of geometric properties. Eventually all students should understand such principles as the quadratic change in area and cubic change in volume that occurs with a linear change of scale. Another of the interdependencies between geometry and measurement is seen in high school when students learn to use trigonometry to make indirect measurements.
Geometric modeling is a powerful problem-solving skill and should be modeled for and frequently used by students. A simple diagram, such as a pie-shaped graph, a force diagram in physics, or a dot-and-line illustration of a network, can illuminate the essence of a problem and allow geometric intuition to aid in the approach to a solution. Visualization skills and understanding will both improve as students are encouraged to make such models.
The relationship between geometry and deductive reasoning originated with the ancient Greek philosophers, and remains an important part of the study of geometry. A key ingredient of deductive reasoning is being able to recognize which statements have been justified and which have been assumed without proof. This is an ability which all students should develop, in all areas, not just geometry, or even just mathematics! At first, deductive reasoning is informal, with students inferring new properties or relationships from those already established, without detailed explanations at every step. Later, deduction becomes more formal as students learn to state all permissible assumptions at the beginning of a proof and all subsequent statements are systematically justified from what has been assumed or proved before. The idea of deductive proof should not be confused with the specific two-column format of proof found in most geometry textbooks. The object of studying deductive proof is to develop reasoning skills, not to write out arguments in a particular arrangement. Note that proof by mathematical induction is another deductive method which should not be neglected.
IN SUMMARY, students of all ages should recognize and be aware of the presence of geometry in nature, in art, and in woman- and man-made structures. They should believe that geometry and geometric applications are all around them and, through study of those applications, come to better understand and appreciate the role of geometry in life. Carpenters use triangles for structural support, scientists use geometric models of molecules to provide clues to understanding their chemical and physical properties, and merchants use traffic-flow diagrams to plan the placement of their stock and special displays. These and many, many more examples should leave no doubt in students' minds as to the importance of the study of geometry.
| All students will develop their understanding of numerical operations through experiences which enable them to construct, explain, select, and apply various methods of computation including mental math, estimation, and the use of calculators, with a reduced role for pencil-and-paper techniques. |
The major shift in this area of the curriculum, then, is one away from drill and practice of pencil-and-paper procedures and toward real-world applications of operations, wise choices of appropriate computational strategies, and integration of the numerical operations with other components of the mathematics curriculum. So what is the role of pencil-and-paper computation in a mathematics program for the year 2000? Should children be able to perform any calculations by hand? Are those procedures worth any time in the school day? Of course they should and of course they are.
Most simple pencil-and-paper procedures should still be taught and one-digit basic facts should still be committed to memory. We want students to be proficient with two- and three-digit addition and subtraction and with multiplication and division involving two-digit factors or divisors. But there should be changes both in the way we teach those processes and in where we go from there. The focus on the learning of those procedures should be on understanding the procedures themselves and on the development of accuracy. There is no longer any need to concentrate on the development of speed. To serve the needs of understanding and accuracy, non-traditional pencil-and-paper algorithms, or algorithms devised by the children themselves, may well be better choices than the standard algorithms, which were built mostly for speed. The excessive use of drill, necessary in the past to develop reflexive automaticity, is no longer necessary and should play a much smaller role in todays curriculum.
For procedures involving even larger numbers, or numbers with a greater number of digits, the intent ought to be to bring students to the point where they understand a pencil-and-paper procedure well enough to be able to extend it to as many places as needed, but certainly not to develop an old-fashioned kind of proficiency with such problems. In almost every instance where the student is confronted with such numbers in school, technology should be available to aid in the computation, and students should understand how to use it effectively. Calculators are the tools that real people in the real world use when they have to deal with similar situations and they should not be withheld from students in an effort to further an unreasonable and antiquated educational goal.
In summary, numerical operations continue to be a critical piece of the school mathematics curriculum and, indeed, a very important part of mathematics. But, there is perhaps a greater need for us to rethink our approach here than there is to do so for any other component. An enlightened mathematics program for todays children will empower them to use all of todays tools rather than require them to meet yesterdays expectations.
| All students will develop their understanding of measurement and systems of measurement through experiences which enable them to use a variety of techniques, tools, and units of measurement to describe and analyze quantifiable phenomena. |
This standard is also, in many ways, the prototypical "integrated" standard because of its strong and essential ties to almost every one of the other content standards. Measurement is an ideal context for dealing with numbers and numerical operations of all sorts and at all levels. Fractions and decimals especially appear very naturally in real-world measurement settings. In fact, metric measures provide perhaps the most useful real-world model of a base-ten numeration system we can offer to children. Geometry and measurement are similarly almost impossible to think about separately. Very similar treatments of area and perimeter, for instance, are called "measurement" topics in some curricula and "geometry" topics in others because they are, quite simply, measurements of geometric figures. Yet another of the content standards which is inextricably linked to measurement is estimation. Estimation of measures should be one of the focuses of any work that children do with measurement. Indeed, the very concept that any measurement is inexact -- is at best an "estimate" -- is a concept that must be developed throughout the grades.
Think about how many different content standards are incorporated into one simple measurement experience for middle grades students: the measurement of a variety of circular objects in an attempt to explore the relationship between the diameter and circumference of a circle. Clearly involved are the measurement and geometry of the situation itself, but also evident are opportunities to deal with patterns in the search for regularity of the relationship, estimation in the context of error in the measurements, and number sense and operations in the meaning of the ratio that ultimately presents itself.
Much research has been done into the development of children's understanding of measurement concepts and the general agreement in the findings leads to a coherent sequence in curriculum. Young children start by learning to identify the attributes of objects that are measurable and then progress to direct comparisons of those attributes among a collection of objects. They would suggest, for instance, that this stick is longer than that one or that the apple is heavier than the orange. Once direct comparisons can consistently be made, informal, non-standard units like pennies or "my foot" can be used to quantify how heavy or how long an object is. Following some experiences pointing out the necessity of being able to replicate the measurements, regardless of the measurer or the size of the measurer's foot, these non- standard units quickly give way to standard, well-defined units like grams and inches.
Older students should continue to develop their notions of measurement by delving more deeply into the process itself and by measuring more complex things. Dealing with various measurement instruments, they should be asked to confront questions concerning the inexact nature of their measures, and to adjust for, or account for, the inherent measurement error in their answers. Issues of the degree of precision should become more important in their activities and discussion. They need to appreciate that no matter how accurately they measure, more precision is always possible with smaller units and better instrumentation. Decisions about what level of precision is necessary for a given task should be discussed and made before the task is begun.
Students should also begin to develop procedures and formulas for determining the measures of attributes like area and volume that are not easily directly measured, and also to develop indirect measurement techniques such as the use of similar triangles to determine the height of a flagpole. Their universe of measurable attributes expands to include measures of a whole variety of physical phenomena (sound, light, pressure) and a consideration of rates as measures (pulse, speed, radioactivity).
The growth of technology in the classroom also opens up a wide range of new possibilities for students of all ages. Inexpensive instruments that attach to graphing calculators and computers are capable of making and recording measurements of temperature, distance, sound and light intensity, and many other physical phenomena. The calculators and computers, when programmed with simple software, are then capable of graphing those measurements over time, presenting them in tabular form, or manipulating them in other ways. These opportunities for scientific data collection and analysis are unlike any that have been available to math and science teachers in the past and hold great promise for some true integration of the two disciplines.
IN SUMMARY, measurement offers us the challenge to actively and physically involve children in their learning as well as the opportunity to tie together seemingly diverse components of their mathematics curriculum like fractions and geometry. It also serves as one of the major vehicles by which we can bring the real worlds of the natural and social sciences, health, and physical education into the mathematics classroom.
| All students will develop their understanding of estimation through experiences which enable them to recognize many different situations in which estimation is appropriate and to use a variety of effective strategies. |
People who use mathematics in their lives and careers find estimation to be preferable to the use of exact numbers in many circumstances. Frequently, it is either impossible to obtain exact answers or too expensive to do so. An air conditioning salesperson preparing a bid would be wasting time and money by measuring rooms exactly. Astronomers attempting to determine movements of celestial objects cannot obtain precise measurements. Many people use approximations because it is easier than using exact numbers. Shoppers, for example, use approximations to determine whether they have sufficient funds to purchase items. Travelers use rough estimates of time, distance, and cost when planning trips. Commonly reported data often use levels of precision which have been accepted as appropriate, even though they may not be considered "exact". Astronomers always report information to two significant digits, and baseball players always report their batting averages as three place decimals.
Instruction in estimation has traditionally focused on the use of rounding. There are times when rounding is an appropriate process for finding an estimate, but this standard emphasizes that it is only one of a variety of processes. Computational estimation strategies are a new and important component in the curriculum. Clustering, front-end digits, compatible numbers, and other strategies are all helpful to the skillful user of mathematics, and can all be mastered by young students. Selection of the appropriate strategy to use depends on the setting and the numbers and operations involved.
The foregoing discussion describes a new emphasis on the use of estimation in computational settings, but students should also be thoroughly comfortable with the use of estimation in measurement. Students should develop the ability to estimate measures such as length, area, volume, and angle size visually as well as through the use of personal referents such as the width of a finger being about one centimeter. Measurement is also rich with opportunities to develop an understanding that estimates are often used to determine approximate values which are then used in computations and that results so obtained are not exact but fall within a range of tolerance.
Estimation should be an emphasis in many other areas of the mathematics curriculum in addition to the obvious uses in numerical operations and measurement. Within statistics, for example, it is often useful to estimate measures of central tendency for a set of data; estimating probabilities can help a student determine when a particular course of action would be advisable; problem situations related to algebraic concepts provide opportunities to estimate rates such as slopes of lines and average speed; and working with sequences in algebra and increasing the number of sides of a regular polygon in geometry yield opportunities to estimate limits.
In summary, estimation is a combination of content and process. Students ability to use estimation appropriately in their daily lives develops as they have regular opportunities to explore and construct estimation strategies and as they acquire an appreciation of its usefulness through using estimation in the solution of problems.
| All students will develop their understandings of patterns, relationships, and functions through experiences which enable them to discover, analyze, extend, and create a wide variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena. |
When solving difficult problems, we frequently suggest to students that they try to solve a simpler problem, observe what happens in a few specific cases (that is, look for a pattern) and proceed from there. This pattern-based thinking, in which patterns are used to analyze and solve problems, is an extremely powerful tool for doing mathematics. Students who are comfortable looking for patterns and then analyzing those patterns to solve problems can also develop understanding of new concepts in the same way. Most of the major principles of algebra and geometry emerge as generalizations of patterns in number and shape. For example, one important idea in geometry is that: For a given perimeter, the figure with the largest possible area that can be constructed is a circle. This idea can easily be discovered by students in the middle grades by examining the pattern that comes from a series of constructions and measurements. Students can be given a length to use as the perimeter of all figures to be created, say 24 centimeters. Then they can construct and measure or compute the areas of a series of regular polygons: an equilateral triangle, a square, and a regular hexagon, octagon, and dodecagon (12 sides). The pattern that clearly emerges is that as the number of sides of the polygon increases (that is, as the polygon becomes more "circular"), the area increases.
All of the content standards have close interconnections, but this is one that is very closely tied to all of the others since an understanding of patterns can be either content or process. When the patterns themselves and their rules for generation are the objects of study, they represent the content being learned. However, when pattern-based thinking, or the search for patterns, is the approach taken to the discovery of some other mathematical principles, then patterning is a process, and the approach can easily be applied to content in numeration, geometry, operations, or the fundamentals of calculus. There is a very special relationship, though, between patterns and algebra. Algebra provides the language in which we communicate the patterns in mathematics. Early on in their mathematical careers, children must begin to make generalizations about patterns that they find and try to express those generalizations in mathematical terms. Examples of the different kinds of expressions they might use are given in the discussion below.
Young students should frequently play games which ask them to follow a sequence of rules (arithmetic or otherwise) or to discover a rule for a given pattern. Sequences which begin as counting patterns soon develop into rules involving arithmetic operations. Kindergartners, for example, will make the transition from 2, 4, 6, 8 ... as a counting by twos pattern to the rule "Add 2" or "+2." The calculator is a very useful tool for making this connection. Through the use of the calculators constant function feature, any calculator can represent counting up or counting down by any constant amount. Students can be challenged to guess the number that will come up next in the calculators display and then to explain to the class what the pattern, or rule, is.
At a slightly higher level, input-output activities which require recognition of relationships between one set of numbers (the "IN" values) and a second set (the "OUT" values) provide an early introduction to functions. One of these kinds of activities, the Function Machine games are a favorite among first through fifth graders. In these, one student has a rule in mind to transforms any number which is suggested by another student. The first number inserted into the imaginary Function Machine and another number comes out the other side. The rule might be plus 7, or, times 4 then minus 3, or even the number times itself. The classs job is to discover the rule by an examination of the input-output pairs.
In the middle grades, students begin to work with patterns that can be used to solve problems within the domain of mathematics as well as from the real world. There should also be a more obvious focus on relationships involving two variables. An exploration of the relationship between the number of teams in a round robin tournament and the number of games that must be played, or between a number of coins to be flipped and the number of possible resulting configurations, provides a real-world context for pattern- based thinking and informal work with functions. Graphing software is extremely valuable at this level to help students visualize the relationships they discover.
At the secondary level, students are able to bring more of the tools of algebra to the problem of analyzing and representing patterns and relationships. Thus we expect all students to be able to construct as well as to recognize symbolic representations such as y = f(x) = 4x+1. They should also develop an understanding of the many other representations and applications of functions as well as of a greater variety of functional relationships. Their work should extend to quadratic, polynomial, trigonometric, and exponential functions in addition to the primarily linear functions they worked with in earlier grades. They should be comfortable with the symbol f(x), both as the application of a rule of correspondence and as a "value" corresponding to x, in much the same way that elementary students have to view 3+2 both as a quantity and as a process.
The use of functions in modeling real-life observations also plays a central role in the high school mathematics experience. Line- and curve-fitting as approaches to the explanation of a set of experimental data go a long way toward making mathematics come alive for students. Technology must also play an important role in this process, as students are now able to graphically explore relationships more easily than ever before. Graphing calculators and computers should be made available to all students for use in this type of investigation.
In summary, an important task for every teacher of mathematics is to help students recognize, generalize, and use patterns that exist in number, shape, and the world around them. Students who have such skills are better problem solvers, have a better sense of the uses of mathematics, and are better prepared for work with algebraic functions than those who do not.
| All students will develop their understanding of probability and statistics through experiences which enable them to systematically collect, organize, and describe sets of data, to use probability to model situations involving random events, and to make inferences and arguments based on analysis of data and mathematical probabilities. |
Understanding probability and statistics is essential in the modern world with the print and electronic media full of statistical information and its interpretation. The goal of mathematical instruction in this area should be to make students sensible, critical, users of probability and statistics, able to apply their processes and principles to real-world problems. We do not want students to think that those people who did not win the lottery yesterday have a greater chance of winning today! We also do not want them to believe an argument merely because various statistics are offered. We would like them to understand the issues underlying the reliability of election polls. In short, they should be able to judge whether the statistics are meaningful and are used appropriately.
In the area of probability, young children start out simply learning to use probabilistic terms correctly. Words like possibly, probably, and certainly have definite meanings, implying increasing likelihood of an event happening, and it takes children some time to begin to use them correctly. Beyond that, though, elementary age children are certainly able to understand the probability of an event. Starting with phrases like once in six tosses, children progress to more sophisticated probabilistic language like chances are one out of six, and finally to standard fractional notation for the expression of a probability. To motivate and encourage that maturation, students should be regularly engaged in predicting and determining probabilities.
Experiments leading to discussions about the difference between experimental and theoretical probability can also be done by older elementary and middle grades students. Theoretical probability is the probability based on an analysis of the physical properties and behavior of the objects involved in the event. For instance, we know enough about the properties of a fair die to know that each face is equally likely to wind up on top. Experimental probabilities are those determined by data gathered experimentally. For example, students may be able to determine the experimental probabilities of rolling a sum of seven or a sum of four with two dice long before they can explain why the two probabilities are different from each other from a theoretical point of view.
Older students should understand the difference between simple and compound events (like rolling one die versus rolling two dice), and the difference between independent and dependent events (like flipping one coin repeatedly five times versus picking five marbles out of a bag of ten). Again, the best way to approach this content is with open-ended investigations that allow the students to arrive at their own conclusions through experimentation and discussion. Eventually, students should feel comfortable representing real-life events using probability models.
In statistics, young children can start out as early as kindergarten with data collection, organization, and graphing and a focus on those skills, with obviously increasing sophistication, should last throughout their schooling. Students must be able to understand the tables, charts, and graphs used to present data, and they must be able to organize their own data into formats which make it easier to understand. While young students can do exhaustive surveys about some interesting question for all of the members of the class, older students should focus some time and energy on the questions involved with sampling. Identifying and obtaining data from a well-defined sample of the population is the most important job of a professional pollster.
As students progress through the elementary grades, an increased focus on central tendency and later on variance and correlation are appropriate. Students should be able to use mean, median, and mode and should know the differences in their uses. Measures of the variance from the center of a set of data, or dispersion, also provide useful insights into sets of numbers and can start early with box-and-whisker plots for upper elementary students and progress to measures like standard deviation for older students.
The reason statistics grew as a branch of mathematics, however, is to provide tools that are helpful in analysis and inference and that focus should permeate everything we do with students in the area. Whenever they look at data, there should be some question they are trying to answer, some position they are trying to support. K-12 students should have many opportunities to look for patterns, draw conclusions, and make predictions about the outcomes of future experiments, polls, surveys, and so on. They should examine data to see whether it is consistent with some hypotheses that a classmate may already have made, and learn to judge whether the data is reliable or whether the hypotheses might need revision.
In summary, probability and statistics hold the key for enabling our students to understand, process, and interpret the vast amounts of quantitative data that exist all around them. To be able to judge the truth of a data-supported argument presented to them, to discern the believability of a persuasive advertisement that talks about the results of a survey of all of the users of a particular product, or to be knowledgeable consumers of the data-intensive government and electoral statistics that are ever-present, students need the skills that they can learn in a well-conceived probability and statistics curriculum strand.
| All students will develop their understanding of algebraic concepts and processes through experiences which enable them to describe, represent, and analyze relationships among variable quantities and to apply algebraic methods to solve meaningful problems. |
The algebra which is appropriate for all students in the twenty-first century moves away from a tight focus on manipulating symbols to include a greater emphasis on conceptual understanding, on algebra as a means of representation, and on algebraic methods as problem-solving tools. These changes in emphasis are a result of changes in technology and the resulting changes in the needs of society.
The vision proposed by this Framework stresses the need to prepare students for a world that is rapidly changing in response to technological advances. Throughout history, the use of mathematics has changed with the growing demands of society as human interaction extended to larger groups of people. In the same way that increased trade in the fifteenth century required businessmen to replace the use of Roman numerals with the Hindu system and teachers changed what they taught, today's education must reflect the changes in content required by today's society. More and more, the ability to use algebra in describing and analyzing real-world situations is a basic skill. Thus, this standard calls for algebra for all students.
What will students gain by studying algebra? In a 1993 conference on Algebra for All, the following points were made in response to the commonly asked question, "Why study algebra?"
As students develop their understanding of different arithmetic operations, they also need to focus on the properties of these operations as examples of patterns. Students should describe the patterns that they find in looking at these operations both in words and in symbols.
In the middle grades, problem situations should provide opportunities to generalize patterns and use additional symbols such as names and literal variables (letters). This development should continue throughout the remainder of their program, ensuring that the relationship between the variables (unknowns) and the quantities they represent is consistently stressed. Middle school students should extend their ability to use algebra to generalize patterns by exploring different types of relationships. They explore and generalize patterns which arise from nature, including non-linear relationships. As students move into the secondary grades, the graphing calculator and graphing software provide tools for examining relationships between intercepts and roots, between turning points and maximum or minimum values, and between the slope of a curve and its rate of change. As the student continues through high school, similar experiences should be provided for other functions, such as exponential and polynomial functions, developing these functions from situations to which students can relate.
The use of algebra as a tool to model real world situations requires the ability to represent data in tables, pictures, graphs, equations or inequalities, and rules. Through exploration of problems and number patterns, elementary students are provided with opportunities to develop the ability to use concrete materials as well as the representations mentioned above. Having students use multiple representations for the same situation helps them develop an understanding of the connections among them. The opportunity to verbally explain these different representations and their connections provides the foundation for more formal expressions.
A fundamental skill in algebra is the evaluation of expressions and the solution of equations and inequalities. This ability can be difficult to understand unless it is related to situations which give them meaning. Expressions, equations, and inequalities should arise from students exploration in a variety of areas such as statistics, probability, and geometry. Elementary students begin constructing and solving open sentences such as those encountered in missing addend problems. The use of concrete materials and calculators allow them to explore solutions to real-life situations. Gradually, students are led to expand these informal methods to include graphical solutions and formal methods. The relationship between the solutions of equations and the graphs must be stressed regularly.
In summary, there are algebraic concepts and skills which all students must know and apply confidently regardless of their ultimate career. To assure that all children have access to such learning, algebraic thinking must be woven throughout the entire fabric of the mathematics curriculum.
| All students will develop their understanding of the concepts and applications of discrete mathematics through experiences which enable them to use a variety of tools of contemporary mathematics to explore and model a variety of real-world situations. |
However, during the past 30 years, discrete mathematics has grown rapidly and has become a significant area of mathematics. Increasingly, discrete mathematics is the mathematics that is being used by decision-makers in business and government; by workers in fields such as telecommunications and computing that depend upon information transmission; and by those in many rapidly changing professions involving health care, biology, chemistry, automated manufacturing, transportation, etc. Discrete mathematics is the language of a large body of science and underlies decisions that individuals will have to make in their own lives, in their professions, and as citizens.
Discrete mathematics has many practical applications that are useful for solving some of the problems of our society and that are meaningful to our students. Its problems make mathematics come alive for students, and helps them see the relevance of mathematics to the real world. Discrete mathematics does not have extensive prerequisites, yet poses challenges to all students. It is fun to do, is often geometrically based, and stimulates an interest in mathematics on the part of students at all levels and of all abilities.
Students should use a variety of strategies to systematically list and count the number of ways there are to complete a particular task. For example, elementary school students should be able to make a list of all possible outcomes of a simple situation such as the number of outfits that can be worn using two coats and three hats. Middle school students should be able to list and count the number of towers that can be built using four blue and red blocks, or the number of possible routes from one location on a map to another, or the number of different "words" that can be made using five letters. High school students should be able to determine the expected payback of a $1 investment in the state lottery.
Students should use discrete mathematical models such as graphs and trees to represent and solve a variety of problems based on real-world situations. For example, elementary school students should recognize that a street map can be represented by a graph and that routes can be represented by paths in graphs; middle school students should be able to find cost-effective ways of linking sites into a network using spanning trees; and high school students should be able to use determine efficient methods of ordering the tasks in a larger project using directed graphs.
Students should recognize and apply iterative patterns and processes in a variety of settings in nature and art, as well as in mathematics. For example, elementary school students should recognize and investigate sequences on pine cones and patterns on floor tilings; middle school students should generate fractal curves and construct tessellations; and high school students should understand the long-term behavior of infinite sequences.
Students should explore different methods of arranging, organizing, analyzing, transforming, and communicating information, and understand how these methods are used in a variety of settings. Elementary school students should investigate ways to represent and classify data according to attributes like color or shape, using relationships like family trees, and into structures like tables. Middle school students should be able to read, construct, and analyze tables, matrices, maps and other data structures. High school students should understand the application of discrete methods to problems of information processing and computing such as sorting, codes, and error correction.
Students should explore different methods of solving real-world problems, and determine what is the best solution using a variety of algorithms -- where best may be defined, for example, as most cost-effective or as most equitable. For instance, elementary school students should discuss different ways of dividing a pile of snacks, and should determine the shortest path from one site to another on a map laid out on the classroom floor; middle school students should be able to plan an optimal route for a class trip (see the vignette in Chapter 1 entitled "Short-circuiting Trenton") and, pretending to be the manager of a fast-food restaurant, devise work schedules for employees which meet specified conditions yet minimize the cost; high school students should be conversant with fundamental strategies of optimization and recognize both the power and limitations of computers in solving algorithmic problems.
In summary, discrete mathematics is an exciting and appropriate vehicle for working toward and achieving the goal of educating informed citizens who are better able to function in our increasingly technological society; have better reasoning power and problem-solving skills; are aware of the importance of mathematics in our society; and are prepared for future careers which will require new and more sophisticated analytical and technical tools. It is an excellent tool for improving reasoning and problem-solving skills.
| All students will develop their understanding of the conceptual underpinnings of calculus through experiences which enable them to describe and analyze how various quantities change, to build informal concepts of infinity and limits, and to use these concepts to model, describe, and analyze natural phenomena. |
Middle school students should be moving beyond the concrete and pictorial representations used in the elementary grades to more symbolic ones, involving functions and equations. They should use graphing calculators and computers to develop and analyze graphical representations of the changes represented in the tables, and to produce linear and quadratic regression models of the data. They should apply their knowledge of decimals to solving problems involving interest, making use of a calculator to determine for example the yield of a given investment or the length of time it would take for an investment to double. In high school, students can apply their knowledge of exponents, algebra, and functions to solve these and other more difficult interest problems algebraically and graphically.
Throughout their school years, students should be examining a variety of situations where populations and other quantities change over time, and use the mathematical tools at their disposal to describe and analyze this change. As they progress, the situations considered should become more complex; students who experiment with constant motion in their early years will be able to understand the motion of projectiles (a ball thrown into the air, for example) by the time they complete high school.
Similarly, students should be aware of the effect of change on measurement, such as the effect that changes in the linear dimensions of an object have on its area and volume. In the early years children should learn through appropriate hands-on experiments; for example, they might find that doubling the diameter of a circular can increases the volume four-fold by filling the smaller can with water and emptying it into the larger one. By the time students are familiar with variables, this intuition will provide them with the information they need to understand formulas such as those involving volume.
In many settings, the kind of change that takes place over time is repetitive and an important question that should be discussed is what happens in the long run. The principal tool for understanding and discussing such questions is the concept of infinite sequences and the types of patterns that emerge from them. Thus a second central theme is that of "infinity".
There is nothing more fascinating to all students than the mysteries of the infinitely large and, later, the infinitely small. Children are excited about large numbers and "infinity," and that excitement should be nourished and be used, as with other "teachable moments," to motivate the learning of more mathematics. Primary students enjoy naming their "largest" number or proudly declaring that there is no largest! In the early years, large numbers and their significance should be discussed, as should the idea that you can extend simple processes forever (e.g., keep adding 2, keep multiplying by 3).
Once students have familiarity with fractions and decimals, these notions can be extended. What happens when you keep dividing by 2? By 10? Can you find a fraction between 0.499 and 1? What decimal comes just before 1? Such applications can be related to compound interest and population explosion. Students should explore and experiment with infinite repeating decimals and other infinite series, where they can make tables and look for patterns. They should learn that by repeated iteration of simple processes you can get better and better answers in both arithmetic (with increased decimal accuracy) and in geometry (with more accurate estimates of the area and volume of irregular objects).
Although the concept of a limiting value (or a limit) may appear inaccessible to K-8 students, this basic notion of calculus can be explored through the process of measuring the area of a region. Students can be provided with diagrams of a large circular (or irregular) region, say a foot in diameter, and a large supply of tiles of different square sizes. By covering the space inside the region (with no protrusions!) with 4" tiles, then with 2" tiles, then with 1" tiles, then with .5" tiles, students can get an appreciation that the smaller the unit, the larger the area. They will recognize that the space cannot be filled completely with small tiles, yet, at the same time, the sum of the areas of the small tiles gets closer and closer to that of the region.
In summary, these kinds of experiences will provide a good foundation for the notions of limits, infinity, and changes in quantities over time. Such concepts find many applications in both science and mathematics, and students will feel much more comfortable with them if we begin their development in the early grades.