Some Organizing Principles for K-4 Mathematics



    This document is an evolving working paper about school mathematics initiated by groups of mathematicians who met at the Park City Mathematics Institute (PCMI) in the summers of 2004 and 2005. It is a companion to a collection of essays on key ideas in school mathematics being written by mathematicians from these groups. This essay project, organized by Roger Howe and Alan Tucker, begins to distill some of the thinking about school mathematics that has evolved in the past decade as mathematicians have become involved in the national debate about strategies to improve school mathematics education. The 2004 and 2005 PCMI meetings, supported by the National Science Foundation, benefited from discussions with groups of school teachers, mathematics educators and state mathematics supervisors who were meeting concurrently at Park City. Continuing input from these groups is an integral part of this project.

   This working paper presents a set of organizing principles for K-4 mathematics curriculum and standards. It is written in broad terms, with details left to the accompanying essays on key ideas. Companion essays on middle school and high school mathematics will follow. A major goal of this project is to develop a collection of essays that can provide concrete guidance to committees that are revising state mathematics standards and curricula. The essays are also meant to be a resource for pre- and in-service professional development. The process of developing these essays and getting broad input to them is as important as their content. This project is sponsoring discussions to find common ground about school mathematics among representative groups of mathematicians and between mathematicians and other concerned constituencies.  Seeking such common ground is the other major goal of this project.

    Since their publication in 1989, the National Council of Teachers of Mathematics (NCTM) Standards, and the revision published in 2000, have served as the basis for most school mathematics standards, curricula, and textbooks as well as the accreditation of teacher preparation programs. In initial efforts to reform school mathematics, very few mathematicians were active participants.  In the past few years, a number of mathematicians have been invited to play a major role in creating standards and curricula in some states and in national efforts, such as Achieve. This essay project seeks to draw upon the experiences of these mathematicians to assist similar efforts of other mathematicians in the future. To be effective, mathematicians need to do more than express their own individual views about school mathematics; they need to present consensus views.  This project aims to produce documents that present these views.

   The original stimulus for this paper was a reaction to state mathematics standards, which contain long lists of terms and skills broken down by Œstrands¹ and further subcategorized.  The mathematicians at Park City rebelled against these lists of standards that seemed like collections of isolated leaves on the tree of mathematics. (These lists of standards are also a growing concern in the mathematics education community.) The trunk and branches of this tree — the intellectual framework of mathematics — was lost. This essay started as an attempt to re-think state standards in terms of this intellectual framework. 


1. Introduction

    The value of a mathematics education and the power of mathematics in the modern world arise from the cumulative nature of mathematical knowledge and reasoning. A small collection of simple facts combined with appropriate theory is used to build layer upon layer of ever more sophisticated mathematics.

    The essence of mathematical learning is a two-way process of moving forward by using previous layers to understand the next layer and looking backward by using the next layer of knowledge to gain better mastery of previous layers. A critical aspect of this mathematical learning is Œunpacking¹ concisely formulated mathematical structures and ideas in order to understand them well, and then to reason with them in compact form as building blocks for more complex mathematical structures and ideas. These learning processes are hard to test but critical to retention and cumulative learning in mathematics. Students¹ first experience with this unpacking—and perhaps the most powerful example of it— is the use of place value and the distributive law to build multi-digit arithmetic algorithms.

   A similar type of unpacking occurs in solving mathematical problems, whether abstract or applied, e.g., show that the product of two even numbers is even; or if 6 school buses each with 24 seats are used to transport 135 students, how many empty seats will there be on these buses.

   A major challenge facing U.S. K-16 mathematics education is poor cumulative learning, sometimes characterized as superficial learning.  While most knowledge slips away over time if infrequently used, basic mathematical knowledge, e.g., solving a linear equation, should not have to be re-taught in every year of secondary school, and again in precalculus classes in college.  What students do retain reflects this superficial learning—they remember formulas but not the concepts that underlie them and thus are unable to use the formulas properly.  The challenges to achieving better long-term learning involve every stage of K-16 mathematics instruction.  Mathematics faculty often see engineering majors in Calculus II who have forgotten basic properties of logarithms that were taught in Calculus I, as well as earlier in high school.

    The Park City mathematics group decided to focus first on the elementary grades, believing that the more soundly mathematics in the earliest grades is learned the better the learning in later grades will be.  In recent years, Liping Ma, Deborah Ball and other mathematics educators have reawakened mathematicians to the depth and substantial nature of the mathematical ideas involved in learning and teaching elementary mathematics. (Elementary grade mathematics was studied by some eminent mathematicians in the past, such as Hassler Whitney.) The 2001 National Research Council report, Adding It Up [3], a joint effort of mathematicians and mathematics educators, summarizes research about learning mathematics in grades K-8 in a thoughtful framework that illustrates the current intellectual respect for mathematical instruction before high school. 

    Another reason to focus on early grades comes from U.S. social traditions.  The United States was founded as a commercially oriented country with success thought to come from more savvy and hard work, that from education.  While the first widespread public education in any country started in New England in the early 1700s, the original goal of teaching basic Œreckoning skills¹ for running a business (or farm) or working in a business has never really changed. See R. Hofstadter¹s Anti-Intellectualism in American Life [1] for details. For this reason, compound interest calculations in dollars and cents were the ultimate goal of most students¹ mathematical education, even into the middle of the 20th century. Algebra was thought to have limited commercial value.  Hofstadter asserts that mandatory high school attendance was instituted 100 years ago not to teach students algebra and other high school level knowledge but largely to keep young people out of (possibly dangerous) factories, where they undercut wages of adults.

    Today, these traditional Œreckoning skills¹ are no longer required by business.  Even the simplest commercial arithmetic computations, e.g., making change at a counter, are done with computers. Instead, an increasing array of jobs requiring only a high school degree involve creating spreadsheets, making multi-step quality-control calculations, following instructions involving formulas (expressed in symbols), and more.   Students graduating from high school now need lasting knowledge of simple algebraic reasoning and problem-solving beyond basic arithmetic.  Of course, for the majority of high school graduates who eventually go to college, more mathematics is needed.  Too many college freshmen today retain little knowledge of the skills that all high school graduates now need.

   Given the recognition of the intellectual substance of elementary mathematics and that teaching it properly lays a sound foundation for later mathematics, we offer the following principles for a more coherent organization of mathematics instruction in the early grades.  Effective yearly learning outcomes should be a dividend of this program for long-range mathematical learning. However, our recommendations have much in common with the goals of NCTM¹s Curriculum Focal Points project.


II. Principles for K-4 Mathematics

1. The place value number system and whole number arithmetic are the central mathematical themes in K-4 grades.  Measurement is an important, closely connected theme. Preparatory work for functions and algebra is naturally integrated with these themes.  These connections provide Œhorizontal¹ reinforcement of learning across topics as well as multiple models for understanding place value and arithmetic.  Instruction of this content also involves Œvertical¹ (level-by-level) reinforcement of learning, in which work at each level solidifies lower levels of knowledge and skills and anticipates the next levels.

2. In every grade the mathematics curriculum needs to focus on a small number of core topics with the objective of deepening mastery of these topics. Along with horizontal and vertical coherence in content, this focused curriculum should also develop a similar coherence among the mathematical skills of computation, problem-solving and reasoning.

3. The role of geometry throughout K-12 instruction needs to be re-thought. Geometry was developed by the Greeks as a qualitative type of mathematics, while the growing role of mathematics in the modern world focuses on the quantitative.  However, geometry is the field that links mathematics to the physical world.  As such, geometric models enrich the learning of almost all mathematics and provide the framework for interesting applications of mathematics.  In early grades geometry should be studied primarily in connection with numbers, arithmetic and measurement.  As more complex geometric models are introduced, e. g., right angle geometry and trigonometry it is necessary to study core geometric concepts and reasoning carefully in their own right.  The challenge is to reduce the wide current divide between Euclidean geometry and simple analytic geometry.  We suggest using similar triangles as a theme to bridge this divide.

4. Data analysis, statistics and probability have emerged as a major area of mathematics used in daily life. Most instruction in this area in elementary grades involves data analysis— collecting, displaying and interpreting information.  Data analysis requires a context for collection and interpretation. For this reason, we recommend that most K-4 work with data analysis should involve using it as a tool for quantitative discussions in social science and natural science instruction, i.e., as mathematics across the curriculum.


III. Discussion of Principles

1. The place value number system and whole number arithmetic are the central mathematical themes in elementary grades.  Measurement is an important, closely connected theme.  Preparatory work for functions and algebra is easily integrated with these themes.  These connections provide Œhorizontal¹ reinforcement of learning across topics as well as multiple models for understanding place value and arithmetic.  Instruction of this content also involves Œvertical¹ (level-by-level) reinforcement of learning, in which work at each level solidifies lower levels of knowledge and skills and anticipates the next levels.

   Some grade-by-grade standards documents seem open to the misinterpretation that mathematics is a forest of distinct, albeit inter-connected, strands, and the strands in turn are composed of lists of concepts, procedural skills and types of reasoning.  There are far too many items in these lists to be learned and retained as isolated pieces of knowledge. Further, these items each appear to be of equal value. This is misleading. A mathematical education involves much more than a collection of such items.  School mathematics should have a more unified organization. The place value number system and integer arithmetic provide a natural core for developing most mathematics in elementary grades.

   Mastering addition, subtraction, multiplication and division needs to be an incremental, evolving process, which carefully extends previous knowledge and constantly lays a solid foundation for future knowledge.  This process can begin by using addition facts of integers ¾ 5 to add larger single-digit numbers: a depiction of 8 + 9 can show the 8 items displayed as a group of 5 and a group of 3, and similarly for the 9 items; then the answer is found by combining 5+5 = 10 with 3 + 4 = 7. Informal multiplication can begin very early with counting by 2¹s, 3¹s, 4¹s or 5¹s.  As in addition, more difficult multiplication facts can be built on easier ones.  For example, multiplication by 7, say, 6 x 7 can be introduced with a depiction of six groups of 7 items in which each group of 7 in broken into a subgroup of 5 and subgroup of 2, so that multiplication by 7 is reduced to multiplication by 5 and 2. With this approach, as students work to memorize single-digit arithmetic facts, they do not need a book to remind them what 6x7 is; they can derive it by computing 6x5 + 6x2.

     Connecting multiplication with division is critical to developing a sound understanding of division. Division is possibly the most important of the basic arithmetic processes since it leads to fractions and proportions, topics which too many U.S. students have great trouble learning.

    From an early age, students need to be developing an understanding of the algebraic structure underlying arithmetic. For example, starting from missing number problems, such as such as 7 + __ = 12 and ³four times what is 12,² students can build a sound understanding that subtraction is the inverse of addition and later that division is the inverse of multiplication.  They need to become proficient in using the commutative and distributive laws for arithmetic in explicit calculations and applied interpretations; for example, when simplifying the calculation 37*42 + 63*42 to 100*42. Such proficiency sets the stage for successful learning of fractions and algebra.

    Mastery of the place value system evolves similarly, possibly starting with sums of pennies and dimes.  In time, students need to become proficient at decomposing numbers by powers of 10, e.g., 435 = 4x100 + 3x10 + 5x1, and applying the associative, distributive and commutative laws to perform the multi-digit algorithms in terms of such decompositions for addition and subtraction and later for multiplication. An early example of an important indirect form of commutativity is the fact that 37 + 92 = 32 + 99 (the latter being an easier problem).  An extension of this reasoning is used when one reorganizes the digits in each column of a multi-number addition in order to find pairs of digits that sum to 10.  Howe¹s essay on place value Œunpacks¹ the many facets of the place value system and leads to a careful discussion of multi-digit arithmetic algorithms.  

   These algorithms are a quintessential example of how a powerful mathematical theory is constructed.  From single-digit addition facts, one derives the facts for subtraction and multiplication, and from multiplication comes division. Then multi-digit algorithms use place value to extend single-digit arithmetic. These algorithms extend further to the arithmetic of fractions and decimals.  Many complicated calculations in algebra and later in college mathematics are done using incremental extensions of the reasoning behind these basic algorithms.  These future developments make a thorough understanding of the multi-digit algorithms of arithmetic absolutely essential for all students.

   Howe¹s essay discusses how a careful study of numerical estimation in middle school can be used to rethink decimal arithmetic in a fashion that develops valuable insights into the structure of place value arithmetic.

   Most other mathematical themes in early grades are easily connected to numbers and arithmetic. Measurement-- for time, money, weight, and the physical dimensions of length, area, and volume—offers constructive contexts in which to visualize and work with numbers and arithmetic. Of course, measurement is also one of the most common ways that numbers and arithmetic arise in daily life.    Clock time offers a rich mathematical context, including counting by 5¹s and 10¹s and representing the multiples of 5¹s of 10¹s with angles. Determining areas of figures decomposed into unit squares is a model of multiplication. Such unit square models also provide a concrete verification of the commutative law of multiplication.  Problems involving money illustrate place value, e.g., counting in pennies, dimes and dollar bills use powers of 10; dollars and cents presage decimals.  Measuring simple fractional units, such as quarters of sticks of butter and similar ingredients in a recipe, provides early experience with fractions.

    Problems with number patterns that anticipate rules defining sequences and functions, such as 3, 6, 9, _, _, 18, _, _ , have similar connections. [Aside to mathematicians: while theoretically there are many number patterns that could start with 3, 6, 9, and have 18 as the sixth term, this pattern is explicit enough for second graders to discover.] Histograms and other data displays can be used starting in first grade to help students visualize and organize basic counting and comparisons of quantities. These few examples hint at a tapestry of elementary mathematics based on numbers and arithmetic. 

   Connections need to be emphasized between different situations that model the same mathematical problem. E.g., one can ask students to create a problem in terms of nickels that is mathematically equivalent to measuring a period of time by counting by five-minute marks on a clock.  The level-by-level development of mathematics has obvious lines of development, such as how repeated addition leads to multiplication, but the other important vertical connections.  One example is working with units, which in many countries are seen as a critical prerequisite to working with fractions.  Counting by 1¹s can be generalized to counting by 5¹s and then by unit fractions, such as fourths (for instance, using a picture of pies, each divided into 4 equal parts).  This can be further generalized to counting with multiple units, e.g., pennies, nickels, dimes, quarters; here pennies are the unit in which all the other units can be commensurately expressed.

      Awareness of, and practice with, these horizontal and vertical connections need to be an integral part of virtually every day of mathematical instruction.  Accustoming students to these connections is part of the development of what Adding It Up terms a productive disposition to mathematics thinking. At a more general level, instruction with numbers and whole number arithmetic normally needs to involve three viewpoints: analytical (computation), algebraic, and geometric (measurement).

    We are giving only a few selective hints of how to develop an integrated school mathematics curriculum built on a foundation of number and arithmetic.  We recognize that the pedagogical expertise for the detailed implementation of such a curriculum lies with school mathematics teachers and mathematical education faculty.  Our goal here is to help frame an overall structure for early school mathematics instruction that enhances mathematical reasoning and long-term learning for success in later grades and college. From this viewpoint, we emphasize the importance of a firm foundation in arithmetic and the place value system, both as preparation for mastery of later school mathematics and as a model for the power of mathematical methods. 


2. In every grade the mathematics curriculum needs to focus on a small number of core topics with the objective of deepening mastery of these topics. Along with horizontal and vertical coherency and interconnections in content, this focused mastery should also be used to develop similar coherency and interconnections among the mathematical skills of computation, problem-solving and reasoning.

   As listed in Principle 1, the core topics in K-4 mathematics are number and arithmetic along with measurement. ŒPre-algebra¹ and Œpre-function¹ material can readily be integrated with these topics.  Geometry is traditionally considered a core topic throughout the K-12 curriculum.  As discussed in Principle 3, we envision a major re-thinking of geometry so that at the K-4 level it arises not as its own strand but primarily through geometric models, e.g., measurement in one-, two- and three-dimensions; the number line, and physical world applications. Probability and data analysis have emerged in recent years as another major topic in the K-12 mathematics curriculum.  However, we recommend that at the K-4 level this topic should be a relatively minor part of the mathematics curriculum. We believe that the principal role of data analysis at this level should be as a quantitative tool in social and natural science instruction; see Principle 4 for details.

   Reasoning starts with finding and articulating patterns in simple counting and arithmetic and with constructing new arithmetic facts and procedures from current knowledge. Early examples are learning the place value representation of numbers between 10 and 99, and adding two large digits (> 5) by writing each digit as 5 plus a smaller digit.

   Breaking problems into pieces and assembling final answers from solutions to these pieces is a critical habit of mind that should underlie as much instruction as possible.  For example, students need to have the mental image of all the ways to break numbers ¾ 18, especially 10, into a sum of two single digits.  This knowledge makes single-digit subtraction easy.  It begins to set the stage for abstracting statements about numbers to statements about symbols, e.g., a+b = 10.  Pairs of digits summing of 10 are used in the strategy for adding a column of numbers by looking for pairs (or triples) of digits that sum to 10.  This Œshortcut¹ for multi-digit addition using the laws of addition is a good example of the valuable experience that comes from working problems involving a column of (say, at least 6) multi-digit numbers.  Calculators may eliminate the need for facility with such computations but it does not diminish the pedagogical value of working some such problems.

      Likewise, more complicated mental arithmetic is needed in later grades to develop familiarity with the other algebraic properties of whole number arithmetic.  The more Œtheoretical¹ side of reasoning in early grades involves, among other topics, an understanding of the standard multi-digit algorithms of arithmetic.  These algorithms need to be presented with (grade-appropriate) precision.

   There is wide agreement today that all students need to be able to add, subtract, multiply and divide integers with accuracy and confidence.  With calculators, there is less need for drill of arithmetic computations with large numbers than there was 50 years ago.  However, computational practice is needed for more than mastering single-fact arithmetic facts and simple multi-digit arithmetic. The role of computational practice in building understanding and reasoning associated with arithmetic dictates that more complicated computations are needed than most state standards currently recommend.  To demonstrate true understanding of multi-digit arithmetic procedures, students need experience multiplying a 3-digit number times 3-digit number, and dividing a 5-digit number by a 2-digit number. While division, even by a 2-digit number, is admittedly tedious, it lays an essential foundation for polynomial division.  Few calculus students today know how to divide a quadratic polynomial by x - 2—we suspect a major of this difficulty is inadequate practice with multi-digit division.

    We believe that students will have an easier time mastering the steps for simplifying and evaluating algebraic expressions if they have first had extensive practice simplifying and evaluating numerical expressions, such as (7 x 18) - (144/12) x 9 + 27 (where each term has 9 as a factor).  There is fairly wide agreement that algebra should initially be taught as generalized arithmetic (with symbols replacing numbers).  It follows that work with algebraic expressions should be introduced as a generalization of previous work with arithmetic expressions.  See the problems chapters in this project for more thoughtful examples of complex arithmetic calculations.

   As standard algorithms are mastered, students can also learn various shortcuts, such as writing 69 as 70-1 in the problem 8 x 69.  Knowing a number of such shortcuts and selecting an appropriate one for a particular problem is a valuable exercise combining computation and reasoning; sometimes no shortcut will exist and the standard algorithm will be needed.

    Children¹s experience with mathematical problem-solving starts with visual depictions of objects to be counted by enumeration or by performing an arithmetic procedure.  Single-step arithmetic problems may also be stated in words.  Problem-solving at this level is a source of concrete examples of computation and reasoning.  In the later elementary grades (it is not within our expertise to suggest exactly when), problem-solving becomes a major component of mathematical learning as multi-step problems are introduced.  Students can be helped by carefully planned increments in difficulty in the transition from one-step to two-step and from two-step to three-step problems.  For example, the one-step problem, how many boxes of 6 apples can be made using 24 apples, moves to the two-step problem (with easy numbers) of how much money does it cost to buy 8 apples if apples are sold in boxes of 2 and a box $1 costs. In the next grade, the two-step problem might become, how much does it cost to buy 104 apples if apples are sold in boxes of 8 for $1.60 a box.  Restatements of the problem would be studied such as, if 91 apples are placed in boxes holding 7 apples and sold for $.90 a box, how much money will be received.  Other types of two-steps problems would also be learned in similar increments.  These set the stage in middle school for a three-step problem such as, if a merchant purchases 40 apples from a farmer for $10 and packages them in boxes of 5 that sell for $2 a box, how much money will the merchant make from buying and reselling the apples? 

   Such multi-step problems are standard in fifth grade in many countries.  We believe it is possible to achieve comparable performance in U.S. schools by developing the proper foundation in reasoning while learning arithmetic in early grades, as outlined above. Here we are focusing on building up skill at multi-step problem-solving from single steps.  This is just the start.  The harder part of problem-solving involves decomposing a new type of problem into simple steps.  This process also starts with gentle variations of types of problems that have already been analyzed step by step.  Only fairly simple problem decomposition is within the scope of K-4 school mathematics.

   The reasoning used to solve multi-step problems, even when the steps are carefully organized as in the above example, is an excellent foundation for learning to solve proportion problems in middle school and word problems in algebra.  Moreover, the problem-solving skills that allow a student to decompose and solve multi-step problems are of universal value. These skills are related to the reasoning involved in Œunpacking¹ concise mathematical structures.

   Problem-solving can also be used to build understanding and reasoning through exercises that ask students to make up stories associated with arithmetic calculations. (Liping Ma¹s book [2] publicized the weakness of U.S. teachers in formulating such problems.)

   There is one aspect of problem-solving currently being emphasized in many curricula that especially concerns us, namely,  Œreal world¹ applications, which in our view can be inappropriately complicated.  While it sounds appealing to connect mathematics to real world situations, the result too often is artificial, wordy problem statements that have limited mathematical value.  The reading level of the problems can easily be higher than the mathematical level. Using Œrealistic¹ numbers, with many significant digits, in problems also seems to be mathematically counterproductive, since the only result of such numbers is to allow students to punch buttons on a calculator.  On the other hand, finding problems that genuinely engage students¹ interests and use age-appropriate language is worthwhile.  More complex applications, and with them calculators, have their place in school mathematics, but not in the K-4 grades, we believe.  For example, a natural middle-school use of a real-world problem requiring calculators is determining the height of a building from the length of its shadow, using similar triangles.

   We have concentrated on arithmetic so far in the discussion of Principle 2.  Measurement can be developed in a comparable incremental fashion, e.g., moving from lengths to areas to volumes.  It has many natural connections to computation, reasoning and problem-solving.  In particular, we mention the widely appreciated role of measurement as a practical introduction to the number line and fractions.  Likewise, topics that prepare students for algebra and functions are readily integrated into such a curriculum; e.g., computing with formulas, devising a formula for a (simple) function based on a table of function values, and solving a simple linear equation.

    The focused program outlined here for K-4 mathematics seeks to develop, in a carefully measured fashion, ever deepening skills and understanding in computation, problem-solving and reasoning.  The goal of such K-4 instruction is to have students fully prepared for the greater challenges of working with fractions and their applications in middle school and with algebra and related topics in high school.  This preparation has two components: a deeper understanding of K-4 mathematics; and an anticipation of mathematics in future grades.  The introduction of material that does not promote such preparation should be minimized. While new topics are constantly entering the curriculum, we want them to connect with, and build on, the core knowledge and skills discussed here so that they can be introduced more efficiently and acquire their own intellectual depth.


3. The role of geometry throughout K-12 instruction needs to be re-thought. Geometry was developed by the Greeks as a qualitative type of mathematics, while the growing role of mathematics in the modern world focuses on the quantitative. Geometry is the field that links mathematics to the physical world.  Geometric models enrich the learning of almost all mathematics and provide the framework for interesting applications of mathematics.  In early grades geometry should be studied primarily in connection with numbers, arithmetic and measurement.

    In the first half of elementary school, there is essentially no explicit role for geometry.  Geometry arises implicitly in measurement.  There is physical measurement of lengths and later rectangular areas.  There is circular geometry underlying measuring time (on analog clocks). A little qualitative geometry in early grades, such as recognizing different n-gons or counting the number of corners on a figure, is reasonable.

   There are plenty of opportunities in the second half of elementary school to use geometry in a fashion that connects with measurement and arithmetic and that involves study in depth.  As one extended example, students can learn the term Œangle,¹ practice identifying angles in pictures, count angles on figures and then restrict these activities to right angles.  They can learn that a figure with four right angles will be a rectangle whose area (assuming, at this stage, integral sides) is easy to calculate.  They can progress to determining that the area of a triangle generated by a right angle with adjacent sides of length one equals one half the area of a unit square; that is, its area is ½.  Then by counting whole and half unit squares, they can determine the area of any figure drawn on a unit grid when the figure¹s corners are at grid points and the figure¹s sides are formed by vertical, horizontal, and diagonal (slope +1 or -1) lines. To speed up the counting of unit squares within a figure, students can decompose as much of the figure as possible into rectangles so that rectangular collections of unit squares can be counted by multiplication.  This geometric decomposition exercise illustrates the type of quantitative geometric reasoning we want to promote that connects geometry with arithmetic and helps prepare students for later work in secondary school with algebra and analytic geometry. In contrast, we want to minimize the type of qualitative geometric decomposition problem in which students, say, use a set of triangles and rectangles of different sizes to construct a given figure.

     Assuming students have previously learned to plot coordinate points, students can be asked to draw such a figure themselves given the coordinates of its corners, and then find its area.  The next year, a geometry chapter that extends the preceding work could start by proving that the area of a triangle with integer base b and integer height h, is bh/2. This proof employs the Euclidean geometry construction of using the attitude line segment to split a triangle into two right triangles.  Now students can draw a broader set of figures whose area they can calculate, namely, those figures that can be decomposed into rectangles and triangles and whose corners have integral coordinates.

     These examples illustrate the focus on triangles, especially right triangles that we advocate.  Elementary school work with triangles can serve the dual objectives of learning geometric problem-solving, which is of value in its own right, and laying a foundation for the more complicated uses of triangles in middle school and high school.  In middle school, along with traditional similar triangle problems, right triangles can be used to visualize rates and proportions.  Triangles are pervasive in analytic geometry, because a point (x,y) in the Euclidean plane is naturally associated with the triangle whose other corners are (0,0) and (x,0) (or (0,y).  Trigonometry is the study of ratios in right triangles.  Slopes of lines, and, later, derivatives of functions involve triangles.  At the same time, a large amount of Euclidean geometry involves triangles.  


4. Data analysis, statistics and probability have emerged as a major area of mathematics used in daily life. Most instruction in this area in elementary grades involves data analysis— collecting, displaying and interpreting information.  Data analysis requires a context for collection and interpretation. For this reason, we recommend that most K-4 work with data analysis should involve using it as a tool for quantitative discussions in social science and natural science instruction.

   Data analysis starts with the first work with numbers.  A number can be Œgraphed¹ by darkening the proper number of squares (or other shapes) in a row. When this process is done for several numbers, the representation produces a histogram.  Such elementary data analysis anticipates the number line, measurement, and other geometric representations of numbers.  It also allows a visual/geometric interpretation of the < relation.

     The ability to organize information in simple histograms can be used immediately for simple experiments in science instruction. Reasoning with quantitative information should come to permeate science and social studies instruction in elementary grades, reflecting the heavy use of quantitative information in modern daily life.  More extensive activities involving the collection and analysis of data in histogram form can be part of science and social studies instruction in later elementary grades. We do not recommend reasoning about histograms that is not tied to a specific context, such as the problem: given a particular histogram, in which of the following contexts might it have arisen.

     Data analysis can enrich instruction about numbers in several ways.  For example, in second grade, stem-and-leaf plots can help teach place value with two-digit numbers. And after students have been asked to place a set of numbers in order, they can count halfway through the list to find the median.  Even better, if the students formed a stem-and-leaf plot to order, say, 25 numbers, the students can find the median by adding successive stem sums until they exceed 13 and then counting along the stem containing the median.

    In later elementary grades, students have the arithmetic skills to compute averages of data.  There are many interesting variations on the calculation of the mean, such as simplifying the calculation by a change of scale.  For example, if the numbers being averaged are near 80, one can subtract 80 from each number, determine the new average, and add 80 to it.  Now the goal is to cancel positive partial sums against negative partial sums.  This same strategy can also be modified to sum a collection of numbers grouped near 80.

     Another valuable tool is pie charts.  They provide visual representations of rational numbers between 0 and 1.  Simple pie charts are now introduced in first grade for counting in unit fractions, e.g., counting fourths of a pie.  In middle school, they are used to break a population into subcategories by percentages. 

     Probability is a more complicated subject, but because of its rich mix of problem-solving, logical reasoning, and computation, along with its wide applicability in daily life, the subject should be an important part of school mathematics.  As soon as children learn to count with unit fractions, simple probability situations can be discussed.  For example, suppose a pizza is cut into fourths, with sausage on one piece (one fourth), pepperoni on one piece, and the other two pieces are plain (nothing but cheese). What are the chances of your getting a plain piece, if someone hands a piece to you and each of your three friends (without asking you what you want).  Some of the set theory that underlies probability can be introduced in early grades.  For example, given songs A and B that all students know, find out from the students the number who like song A, the number who like song B, and the number who like both songs; now ask for the size of other Boolean combinations of the sets of students who like each song (these sets would be described in words).  First the answers can be obtained by asking for students in each of sets that are components of the Boolean expression to raise their hands to be counted, and then ask students in the composite expression to raise their hands; e.g., everybody who likes song A raise your hand, now everyone who likes song B, and now everyone who likes neither song A nor B raise your hand. Then class can try to solve the problem with set-theoretic arithmetic.

    This discussion is meant to suggest the spirit of the use of data analysis, statistics, and probability in elementary grades.  Statistics education experts are needed refine and improve on this brief overview.