Mathematical Content
Dimensions of Mathematical Power
Mathematical Thinking
Mathematical Communication
Mathematical Ideas
Mathematical Tools and Techniques
Strands
Functions
Algebra
Geometry
Statistics and Probability
Discrete Mathematics
Measurement
Number
Logic and Language
Ideas That Do Not Fit Within Strands
Unifying Ideas for the Elementary Grades
How many? How much?
Finding, making, and describing patterns.
Representing quantities and shapes.
Unifying Ideas for the Middle Grades
Proportional relationships.
Multiple representations.
Patterns and generalizations.
Unifying Ideas for High School
Mathematical modeling.
Variation.
Algorithmic thinking.
Mathematical argumentation.
Multiple representation.
Strands play a pivotal role in the design and implementation of complete mathematical programs. They represent continuous threads running throughout the curriculum, each being developed in appropriate ways at all grade levels, kinde
rgarten through grade twelve. Strands help evaluate whether the mathematical content of a curriculum is broad enough and well balanced at all grade levels.
However, by themselves the strands are not sufficient to help identify the most important ideas for the curriculum. This Framework provides guidance here through another way to look at mathematical content--through unifying ideas, which focus understanding on a few very important and deep mathematical ideas and are central goals for student learning in each grade span.
For example, proportional relationships is a unifying idea for the middle grades. Elementary students work with proportions, as do high schoolers; but in the middle grades students focus most intently on proportional relationships as a mathematical
principle that unifies a broad range of concepts and applications. Another unifying idea is patterns, which begins to play a role in the elementary grades and whose role extends in middle school to include generalizations of many kinds.
Unifying ideas allow a focus on mathematical themes that bridge many strands. The example of proportional relationships draws from virtually every strand: scale drawings from measurement; sampling from statistics; percent and ratios from number; linear relationships from functions; solution of proportions from algebra; and similarity from geometry. Unifying ideas bind the curriculum together through the year and give a focus to understanding.
In 1989 the National Council of Teacher of Mathematics (NCTM) adopted the Curriculum and Evaluation Standards for School Mathematics. These standards include a classification sim
ilar to the 1985 strands. In keeping with the national consensus, this Framework incorporates the standards as desired outcomes of the curriculum for kindergarten through grade twelve. The strands used in this Framework correspond closely to
many of the infividual NCTM standards, and the unifying ideas used in this Framework represent themes that appear in several different strands (and standards). Unifying ideas add a different dimension to content description and help to integrate s
ubjects too often kept separate. These issues of content are discussed fully in Chapter 3.
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Lauren Resnick, a psychologist who has examined in detail how children learn to think mathematically, points out that although defining higher-order thinking skills is difficult, such thinking can be recognized as follows:
Thinking skills resist the precise forms of definition associated with the setting of specified objectives for schooling. Nevertheless, it is relatively easy to list some key features of higher order thinking . . . .
Consider the following:
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Besides helping to achieve the purpose for the task, communication provides an opportunity for teachers (and peers) to assess students' thinking and depth of understanding. This communication can be the most concrete product of the student's performance.
(This matter will be discussed further in Chapter 2 under "Assessing Student Work for Mathematical Power.")
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Strands are familiar subject-matter categories widely used to broaden school mathematics since the 1985 Framework. The NCTM Standards adopted a similar classification. Usi
ng strands helps ensure that programs are sufficiently broad. The use of appropriate ideas from every strand at every grade level helps to maintain a healthy balance.
Unifying ideas are few, ubiquitous, and deep and bridge most or all of the strands. These very important themes of mathematics, such as patterns and proportional relationships, have a simplicity th
at makes sense to the youngest thinkers and a depth and power that students can appreciate only after a time. They knit the curriculum together and provide continuity throughout the years. Even so, they are by themselves too abstract and all-encompassing
to use as a basis for manageable instructional units. Instead, the mathematics underlying an instructional unit will cut across several strands and unifying ideas. Units will present complex but manageable chunks of mathematics. The collection of units th
at make up a year's work will develop all of the unifying ideas at that grade level and contain substantial mathematics from all of the strands. (Units are described in greater detail in Chapter 3.)
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Students can use tools and techniques to:
The ability to select appropriate tools and techniques and use them effectively is an essential part of mathematical power. (The use of technology in the classroom will be discussed in greater detail in Chapter 2, pages 56-60.)
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Strands provide a referent for assessing the balance of content; they help broaden the scope of school mathematics. It is not acceptable for elementary school mathematics to be concentrated solely on arithmetic or for high school mathematics to be concent
rated solely on algebra and geometry. Appropriate material from each strand should appear at each grade level. Together, the strands describe the range of mathematics important for today's students.
Yet it is not enough simply to represent all the different strands somewhere in the curriculum. The strands are meant to interweave or integrate with one another. Some content in fact involves particular mathematical principles that belong e
qually to several different strands. These principles will be discussed under unifying ideas.
Real mathematical problems rarely involve just one strand. Rather, they demand that the problem solver integrate ideas from several strands to arrive at a meaningful result. Therefore, every unit of mathematical work must involve an appropriate mixture of
ideas from more than one strand. Each of the strands is described as follows:
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Functions remains as a strand in the Framework because it represents important and useful mathematics. By labeling the strand functions, however, the formal mathematical definition of function (a mapping of all members of one set to m
embers of another) is not being emphasized. Instead, the functions strand explores quite broadly the many kinds of relationships among quantitites and the manner in which those relationships can be made explicit - but not necessarily symbolic.
In the early grades functions almost always appear in conjunction with another
strand. For example, when primary students explore the number of cookies they need for different numbers of friends, they are exploring
functions as well as number. And when elementary students learn to find the area of a rectangle, they are studying functions as well as geometry. Conversion formulas such as F = 32 + 9/5 C and disbributions of die rolls are functions as well. Students making polygons with LogoTM
discover how the angle at which the turtle turns depends on the number of sides.
Functions are a backbone of mathematics but need not be the formal, abstract, imposing, and opaque endeavors that traditionally terrorize Algebra II students. Thoughtful experiences with a variety of functions, formal and informal,
in early grades will help all students make functions part of their intellectual repertoire when the functions are met later on.
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The relationship between the algebra and functions strands deserves particular attention. Algebra helps in the manipulation of expressions and the finding of solutions; functions describe
relationships among quantities. The point emphasized previously - that strands must be integrated with one another - applies here especially. These two subjects remain far too separate in most treatments of school mathmatics.
Specifically, teaching formal algebra without appropriate ideas about functions makes the algebraic ideas more mysterious, harder to grasp, and harder to link with future work. (Some high school students miss the fact,
for example, that polynomials are functions and that the essence of solving equations involves finding points where two functions have common values.) At the same time an understanding of functions (when expressed symbolically) requires algebra from the very start to provide the variable
expressions that describe functions.
For similar reasons the curriculum needs much more of an integration of algebra and geometry to replace the typical separation of these subjects at the high school level into year-long, single-strand segments.
Specifically, students need the tools of algebra (such as formulas, functions, and equations) to describe and clarify geometric relationships; and they need the vehicle of geometry to provide graphic illustrations of algebraic relationships.
In the early grades student should not be expected to use symbols to represent variables and solve equations on paper. Instead, they can be expected to develop the concept of a variable and of algebraic operations informally. In solving some problems with
manipulatives, for example ("Take a block and let that be the number of beans Jack has in his bag"), students use concrete objects as symbols they can manipulate. The block becomes a variable years before x appears.
Missing addend problems help students develop intuition not just about number but about what must occur around an equals sign in order to arrive at a solution. As students' work in all the strands of mathematics becomes more sophisticated, their knowledge of and
ability to use algebra as a language should develop correspondingly. Older students use algebra to express symbolic relationships, model interactions between quantities, and extract critical information about these relationships.
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Older students gradually build on their foundation of hands-on experiences. They become more familiar with the properties of geometrical figures and get better at using them to solve problems. They explore symmetry and proportion and begin to relate geome
try to other areas of mathematics - to the benefit of both. For example, graphical representations of functions can help explain and generalize geometric relationships while geometrical insights inform the study of functions.
As students become more familiar with geometrical figures, they are better equipped for mathematical argumentation in that field. They focus on making convincing arguments with a rigor appropriate to the situation rather than on being forced into two-colu
mn proofs. The goal is to develop fluency with basic geometrical objects and relationships and to connect that fluency with spatial reasoning and visualization skills.
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Discrete in this context means focusing on discrete and separate entities rather than on measures of continuous quantities; it does not mean that everything not continuous is to be considered discrete mathematics.
Arithmetic with integers, for example, is treated under number, not under discrete mathematics.
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Measurement is used in all occupations and in everyday life to compare. Numbers are assigned to quantitative aspects of the world by being compared to a scale of standard or non-standard units, such as inches, paper clips, kilograms, heartbeats, paces, or
degrees Celsius. The measurement strand, by focusing on obtaining numbers through direct interaction with the universe, makes a physical connection between numbers and the world through the action of the student.
Work in measurement begins with comparison: bigger-smaller, heavier-lighter, warmer-colder. Students then create nonstandard units to help with the comparison. Later, they learn about standard systems of units, including the international metric system, e
specially units of time, distance, angle, weight, and temperature. Students also learn to combine units to find measures of other properties, such as area, volume, speed, acceleration, density, and pressure, and learn to apply other mathematics (e.g., trigonometry) to indirect measurement tasks.
During their careers students learn to choose suitable units, estimate, and allow for measurement error. They learn to judge the degree of precision appropriate for a given situation and the importance of accurate measurement and calculation; devise ways
to assign numerical values to quantities they are interested in (a chocolate quality index or Consumer Reports-style ratings, for example); and even assess whether that process is appropriate.
Although this strand is closely allied with geometry through linear and angular measurement (angle, length, area, and volume), measurement involves more than using a ruler and a protractor. Measuring diverse quantities makes connections within mathematics
, especially to statistics, and outside, to the natural and social sciences.
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Students learn about numbers from experience: they count how many, measure
how much, and label objects in a collection. Over time they develop number
sense, a sense for quantity in increasingly complex situations. Early on,
students experience the power of mathematics to go beyond direct measurement
and counting to answer how many? and how much? questions by using the basic operations of addition, subtraction, multiplication, and division.
The number system has power that is deeper than counting and collecting. Through exploration, usage, and reflective thought, students learn to make the system work for them. They learn how to use different kinds of
numbers (integer, rational, real, complex, and vector) and what the basic operations mean. They learn about special numbers and properties (properties of 1, 0,
This strand permeates all of mathematics. Numbers appear with all of the other strands--on coordinte axes and in labels on graphs and as the products of measurement, together with associated errors. Numbers describe scaling and proportionality and are the
coordinates of data points and the representations of probabilities.
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The logic and language strand focuses on the power of careful reasoning carried out in natural language to show things that are important but not obvious. However, formal deductive systems should not be introduced prematurely. (Very young students,
especially when working together, can solve quite complex informal deductive problems.) Nor should explicit proofs of results that are intuitively obvious be emphasized. Proofs should make difficult things clear, not make clear things difficult.
Nor should geometry be allowed to dominate what is done here because the ability to reason well and understand carefully worded explanations is too important to be associated with a single strand.
Mathematical Content
The 1985 Framework identified seven strands of mathematical content: number, measurement, geometry, patterns and functions, statist
ics and probability, logic, and algebra. The 1992 document endorses those strands and adds another, discrete mathematics. In addition, this document expands logic to logic and language, acknowledging the importance of language in presenting mathematical ideas unambiguously; and changes patterns and functions to functions, recognizing that patterns are part of every
strand. (Patterns are promoted to an even more important role in the curriculum and are discussed later under unifying ideas.)
Dimensions of Mathematical Power
Mathematical Thinking
Mathematical thinking at its most powerful grows out of the kinds of thinking that are naturally part of everyone's repertoire. Many of the words used to describe mathematical thinking--such as classify, plan, analyze, conjecture, design, evaluate, for
mulate, investigate, model, and verify--have a natural meaning more general than their mathematical meaning. There are other activities we label less frequently in everyday life, such as deducing, inferring, hypothesizing, and synthes
izing. As a group these activities are often referred to as higher-order thinking skills and are characterized in three of the first four standards contained in the NCTM Standards; that is: reasoning, problem solving and making connections
3L.B. Resnick, Education and Learning to Think Washington, D.C.; National Academy press, 1987, pp. 2-3.
Mathematical Communication
As students do mathematics, they communicate their thinking and understanding to themselves, their peers, their parents, their teachers, and other adults. Students can communicate in many ways: informal conversations, verbal presentations, written text, d
iagrams, symbols, numbers, graphs, tables, models, and algebraic expressions. Communication helps to clarify a student's thinking, and feedback (formal or informal) can provide useful information for revision. In turn, all students are enabled to improve
the quality of their work.
Mathematical Ideas
In this Framework, mathematical ideas refers to content, the specific subject matter of mathematics as distinct from the other dimensions of mathematical power. Communication, reasoning, tools, and problem solving do not make a person mathematicall
y powerful unless they can be used in conjunction with particular mathematical content. (Content is described in terms of strands and unifying ideas, which are discussed in greater detail in Chapter 3.)
Mathematical Tools and Techniques
To do complete work, students must put thinking and ideas to use. Involved here are the classic intellectual tools and techniques of mathematics as well as problem-solving tools, including technology. Mathematical tools and techniques enable students to u
se their hands and eyes to experiment and explore relationships. In effect these tools and techniques extend the students' thinking power and translate ideas into action.
The computer is a flexible tool and is ubiquitous in the workplace. For example, students can use word processors to express themselves coherently and legibly, and they can use spreadsheets, graphing programs, and data bases to display results and see pat
terns. Students can also use Microworlds and modeling programs to ask and answer what if? questions.
Strands
The strands for mathematics in kindergarten through grade twelve are the traditional, widely used subject categories of mathematics. Strands have appeared in California mathematics frameworks for about 30 years.
Functions
The patterns and functions strand has been replaced simply by a functions strand because the role of patterns in mathematics is too important and too broad to link with one particular strand such as functions. In fact, patterns play a promin
ent role in all strands of mathematics; functions (functions often represent a way of generalizing a numerical pattern); geometry (e.g. the role of patterns in tessellations); number (e.g.
, the Fibonacci sequence); algebra (e.g., the binomial theorem); and statistics (where a main goal is to find patterns in the data). Because it crosses many strands, the theme of patterns is
an example of what this Framework calls a unifying idea. Unifying ideas are another way to analyze curriculum content; they focus attention on ideas that apply to many strands.
Unifying ideas are discussed later.
Algebra
Algebra is generalized arithmetic. It helps make the specific universal, describe situations, and derive relationships with elegance and power. By itself it is the language of variables, operations, and symbol manipulation. In addition, every other strand
uses algebra to symbolize, clarify and communicate. It is important, therefore, to use algebra in the context of problems and situations arising in other strands.
Geometry
Through the study of geometry, students link mathematics to space and form in the world around them and in the abstract. In this strand students are exposed to and investigate two-dimensional and three-dimensional space
by exploring shape, area, and volume; studying lines, angles, points, and surfaces; and engaging in other visual and concrete experiences. In the early grades this process is informal and highly experiential; students explore many objects and
discover and discuss the attributes of different shapes and figures.
Statistics and Probability
In an age of rapid communication and immediate access to information, data abound. Descriptive statistics help students learn to collect and organize information in a variety of graphs, charts, and tables to make
those data easier for the students and others to comprehend. Students will also learn to interpret data and to make decisions based on their interpretations. Probability is a part of this strand because statistical data are often used to predict the likelihood of future events and outcomes.
Students learn probability, the study of chance, so that numerical data can be used to predict future events as well as record the past. A command of statistics and probability is essential in all aspects of adult life.
Discrete Mathematics
The discrete mathematics strand did not appear in the 1985 Framework, although some of its ideas appeared under statistics and probability. Discrete mathematics includes a cluster of related ideas, such as principles for counting
arrangements of discrete objects (permutations, combinations, selections); other counting principles (the inclusion/exclusion principle, the pigeonhole principle); some basic and useful ideas from set theory (unions and intersections); the study of discrete
structures (networks, graphs, and tree structures); recurrence relations ( such as the Fibonacci relation, Fn = Fn-1 + Fn-2); and the analysis of algorithms.
Measurement
Student activity for this strand centers on the physical activity of measurement. Students use real tools to measure real objects and events.
Number
Where do numbers come from and how do people use them? How does the system of numbers and operations work? The number strand weaves these two questions through the kindergarten through grade twelve curriculum.
, reciprocals, and the square root of 2, for example). They learn how numbers relate to each other (order, inequality
, betweenness, and density; factors, multiples, sums, and ratios). They learn to think with and communicate in the language of numbers (using various notations, verbal expressions with numbers, and number sentences). They become facile with techniques for
computing (in their mind only or with the aid of pencil and paper or a
calculator; through estimation or the use of algorithms). In the end students have an understanding of the system itself--how its simple elements give rise to a structure capable of
representing relations among quantities in the real world and in the imagination.
Logic and Language
The logic strand has been renamed logic and language to emphasize the importance of (1) language, in clarifying mathematical thinking and making valid arguments; and (2) mathematics, in giving natural language powerful tools for communicatin
g complicated ideas clearly and precisely. In this way language connects mathematics to all other disciplines at all grade levels.