NUMBER AND OPERATIONS: K-2 Benchmarks, page 211

NCTM Principles & Standards, page 79

Introduction

The most fundamental concept in elementary school mathematics is that of number.

Number sense gives students the confidence to solve problems and communicate ideas. Young children need opportunities to develop efficient strategies to compute fluently and to solve problems. In addition, students should have a variety of experiences investigating numbers in order to become numerically powerful. This power goes beyond the ability to compute; it encompasses an understanding of various meanings, relationships, properties, and procedures associated with numbers and operations.

The Research Base Benchmarks, page 334 and page 358

NCTM Principles & Standards, page 35

Adding it Up, page 160

“Counting in the absence of perceivable objects is the culmination of a rather intricate development of an ability to create unit items to be counted, first on the basis of conscious perception of external objects and then on the basis of internal representations” (Steff, 1994). “Typical student beliefs about mathematical inquiry include the following: There is only one correct way to solve any mathematics problem; mathematics problems have only one correct answer; mathematics is done by individuals in isolation; mathematical problems can be solved quickly or not at all; mathematical problems and their solutions do not have to make sense; and that formal proof is irrelevant to processes of discovery and invention.” (Schoenfeld, 1985, 1989a, 1989b). “Developing fluency requires a balance and connection between conceptual understanding and computational proficiency. On the one hand, computational methods that are over-practiced without understanding are often forgotten or remembered incorrectly.” (Hiebert 1999; Kamii, Lewis, and Livingston, 1993; Hiebert and Lindquist 1990). “On the other hand, understanding without fluency can inhibit the problem-solving process.” (Thornton, 1990). “Researchers and experienced teachers alike have found that when children in the elementary grades are encouraged to develop, record, explain, and critique one another’s strategies for solving computational problems a number of important kinds of learning can occur.” (see e.g., Hiebert 1999; Kamii, Lewis and Livingston, 1993; Hiebert and Lindquist, 1990.) “Research suggests using word problems as a basis for teaching addition and subtraction concepts, rather than teaching computational skills first and then applying them to solve problems.” (Carpenter & Moser, 1983).

MEASUREMENT: K-2

Introduction

Measurement is an integral part of each of the strands of mathematics. It bridges two fundamental areas of school mathematics – geometry and number. A measure is the numerical value given to an attribute of an object. It answers questions such as how big, how long, how far, how much. In primary grades, students need many experiences using nonstandard and standard forms of measurement.

The Research Base Adding it Up, pages 282-283

“Children’s first understanding of length measure involves the direct comparison of objects. (Lindquist, 1989) Younger children often employ resemblance as the prime criteria for selecting a unit of area measure. Teaching experiments with area measure have revealed that second graders could develop a comprehensive understanding of area measure when they began by solving problems involving portioning and redistributing areas without measuring.”

GEOMETRY: K-2

Introduction

Proficiency in geometric reasoning develops in stages. These sequential stages are associated with age. Children can be assisted by a progression of experiences that take them from recognizing shapes as wholes to recognizing explicit properties.

Students need concrete experiences to develop spatial awareness and geometric knowledge. Students need to identify shapes in the world around them and to compare and sort these shapes according to their properties. These experiences lay the groundwork for further exploration in analyzing the characteristics and properties of two and three-dimensional shapes.

The Research Base NCTM Principles and Standards, page 41

Benchmarks, page 352

“Geometry is more than definitions; it is about describing relationships and reasoning. The notion of building understanding in geometry across the grades, from informal to more formal thinking, is consistent with the thinking of theorists and researchers.” (Burger and Shaughnessy, 1986; Fuys, Geddes, and Tischler, 1988; Senk, 1989; Van Hiele, 1986)

“Students advance through levels of thought in geometry. Van Hiele has characterized them as visual, descriptive, abstract/relational, and formal deduction. At the first level, students identify shapes and figures according to their concrete examples.” At the next level, students identify shapes according to their properties. (Van Hiele, 1986; Clements & Battista, 1992).

ALGEBRA: K-2 Benchmarks, page 217

Introduction

Algebra is the fundamental language of mathematics. From the earliest grades of elementary school, students can begin to use simple algebraic thinking in their mathematical activities. They can observe that over time and across certain circumstances mathematical patterns occur. The can learn about functions by identifying and observing how changes in one variable may cause changes in other situations. A teacher’s ability to help all students learn algebra depends in part on his or her awareness of the most important concepts and ideas: symbols, variables, structure, representation, patterns, graphing, expressions, equations, rules, and functions.

The Research Base NCTM Principles and Standards, page 93

Math Matters, page 123

“It is essential for students to learn algebra as a style of thinking involving the formalization of patterns, functions, and generalizations , and as a set of competencies involving the representations of quantitative relationships.” (Silver, 1997)

“Two central themes of algebraic thinking are appropriate for young students. The first involves making generalizations and using symbols to represent mathematical ideas, and the second is representing and solving problems.” (Carpenter and Levi, 1999)

DATA ANALYSIS AND PROBABILITY: K-2 SMART Mathematics

Course of Study, page 52

Introduction

Informal comparing, classifying, and counting activities can provide the mathematical beginnings of developing learners’ understanding of analysis of data, and statistics. Throughout the K-2 years, students should pose questions to investigate, organize responses, and create representations of their data. Students should be encouraged to think clearly and check new ideas against what they already know. This will allow them to develop concepts of making informed decisions.

Ideas about probability in grades K-2 should be informal and focus on judgments that children make through their experience. Activities that underlie experimental probability, such as tossing number cubes, should occur at this level, but the primary purpose of these activities is focused on other strands, such as number.

The Research Base Benchmarks, page 353

“The process of organizing and reducing data incorporates mental actions such as ordering, grouping, and summarizing. The process of analyzing and interpreting data incorporates recognizing patterns and trends in data and making inferences and predictions from the data.”

NUMBER AND OPERATIONS: 3-5

Introduction

Number sense gives students confidence to use mathematics in everyday life. In grades

3 – 5, students’ understanding of the base-ten number system is extended to larger numbers and decimals. Using benchmark values, common fractions are compared to each other and to whole numbers.

Computational fluency is essential and may be accomplished using various methods. The focus at this level is multiplication and division. This fluency should be developed with an understanding of arithmetic operations and problem solving.

Estimation is encouraged to judge the reasonableness of an answer. A range of strategies should be employed and students should be able to explain their thinking both orally and in writing.

When appropriate, calculators and computers can enhance and extend mathematical understanding at this level.

The Research Base Benchmarks, pages 350, 358-360

Whole numbers. “Elementary and middle school students may have limited ability with place value (Sowder, 1992a). Sowder reports that middle school students are able to identify the place values of the digits that appear in a number, but they cannot use the knowledge confidently in context (for example, students have trouble determining how many boxes of 100 candy bars could be packed from 48,638 candy bars).”

Operations with whole numbers. “Students make a variety of errors in multi-digit addition and subtraction calculations (Brown & Van Lehn, 1982). Given traditional instruction, a substantial number of 4^th and 5^th graders are not able to subtract some whole numbers successfully. (Fuson, 1992). Student errors suggest students interpret and treat multi-digit numbers as single-digit numbers placed adjacent to each other, rather than using place-value meanings for the digits in different positions (Fuson, 1992). With specially designed instruction, 2^nd graders are able to understand place value and to add and subtract four-digit numbers more accurately and meaningfully than 3^rd graders receiving traditional instruction (Fuson, 1992). Research also suggests students interpret multiplication of whole numbers mainly as repeated addition. This interpretation is inadequate for many multiplication problems and can lead to restrictive intuitive notions such as ‘multiplication always makes larger’ (Greer, 1992).”

Operations with fractions and decimals. “Elementary and middle school students make several errors when they operate on decimals and fractions (Benander & Clement, 1985; Kouba et al., 1988; Peck & Jencks, 1981; Wearne & Hiebert, 1988). For example, many middle school students cannot add 4 + 0.3 correctly or 7 1/6 + 3 ½ (Kouba et al., 1988; Wearne & Hiebert, 1988). These errors are due, in part, to the fact that students lack essential concepts about decimals and fractions, and they have memorized procedures that they apply incorrectly. Interventions to improve concept knowledge can lead to increased ability by 5^th graders to add and subtract decimals correctly (Wearne & Hiebert, 1988).”

“Students of all ages misunderstand multiplication and division (Bell et al., 1984; Graeber & Tirosh, 1988; Greer, 1992). Commonly held misconceptions include ‘multiplication always makes larger,’ ‘division always makes smaller,’ ‘the divisor must always be smaller than the dividend.” Students may correctly select multiplication as the operation needed to calculate the cost of gasoline when the amount and unit cost are integers, then select division for the same problem when the amount and unit cost are decimal numbers (Bell et al., 1981).”

Rational numbers: “Upper elementary and middle school students often do not understand that decimals and fractions represent concrete objects that can be measured by units, tenths of units, hundredths of units, and so on (Hiebert, 1992). For example, students have trouble writing decimals for shaded parts of rectangular regions divided into 10 or 100 equal parts (Hiebert & Wearne, 1986). Other students have little understanding of the value represented by each of the digits of a decimal number or know the value of the number is the sum of the value of its digits. Students of all ages have problems choosing the largest or the smallest in a set of decimals with different numbers of digits to the right of the decimal points (Carpenter et al., 1981; Hiebert & Wearne, 1986; Resnick et al., 1989). Upper elementary school students can establish rich meanings for decimal symbols and do a variety of decimal tasks well after specially designed instruction using base-10 blocks (Wearne & Heiberts, 1988, 1989).”

Converting between fractions and decimals. “Instruction that focuses on the meaning of fractions and decimals forms a basis on which to build a good understanding of the relationship between fractions and decimals. Instruction that merely shows how to translate between the two forms does not provide a conceptual base for understanding the relationship (Markowits & Sowder, 1991).”

Number comparison. “Lower elementary students do not have procedures to compare the size of whole numbers. By 4^th grade, students generally have no difficulty comparing the sizes of whole numbers up to four digits (Sowder, 1992). Students are less successful when the number of digits is much larger or when more than two numbers are to be compared. This might be due to increased memory requirements of working with more or larger numbers (Sowder, 1988). Upper elementary and middle school students taught traditionally cannot successfully compare decimal numbers (Sowder, 1988, 1992). Rather, they overgeneralize the features of the whole number system to the decimal numbers (Resnick et al., 1989). They apply a ‘more digits make bigger’ rule (according to which .1814 > .385). After specially-designed instruction which develops good meanings for decimal symbols, many students are able to compare decimal numbers with understanding by 5^th grade (Wearne & Hiebert, 1988). Upper elementary and middle school students taught traditionally, cannot compare fractions successfully (Sowder, 1988). Students’ difficulties here indicate they treat the numerator and the denominator separately. Specially-designed instruction to teach meanings for fractions can help to improve ordering fractions by as early as the end of the 5^th grade (Behr et al., 1984).”

Calculators. “The use of calculators in K-12 mathematics does not hinder the development of basic computation skills and frequently improves concept development and paper-and-pencil skills, both in basic operations and in problem solving (Hembree & Dessart, 1986; Kaput, 1992). The use of calculators in testing produces higher scores than paper-and pencil efforts in problem solving as well as in basic operations (Hembree & Dessart, 1986).”

Estimation. “Middle school and even high school students may have limited understanding about the nature and purpose of estimation. They often think it is inferior to exact computation and equate it with guessing (Sowder, 1992b), so that they do not believe estimation is useful (Sowder & Wheeler, 1989). Students who see estimation as a valuable tactic for obtaining information use estimation more frequently and successfully (Threadgill-Sowder, 1984).”

“Good estimators use a variety of estimating tactics and switch easily between them. They have a good understanding of place value and the meaning of operations, and they are skilled in mental computation. Poor estimators rely on algorithms that are more likely to yield the exact answer. They lack an understanding of the notion and value of estimation and often describe it as ‘guessing’ (Sowder, 1992b). Before 6^th grade, students develop very few estimation skills from their natural experiences (Case & Sowder, 1990; Sowder, 1992b). As a result, some researchers caution that teaching estimation to young children may have, as its single effect, that they master specific procedures in a superficial manner (Sowder, 1992b).

MEASUREMENT: 3-5

Introduction

Measurement is a link that connects ideas within areas of mathematics and bridges mathematics to other disciplines. Using measurement, students in grades 3-5 explore questions related to their environment. They investigate real-world situations involving measurement of temperature, perimeter, angles, area, and volume.

Students should select appropriate tools and units of measurement and recognize factors that affect precision. In addition, it is important that students realize that all measurements are approximations.

The Research Base Benchmarks,

Teaching and Learning Mathematics: p.36-40

Math Matters: page 177, 195

Adding it Up, pages 88, 282

In order to realize that arbitrary measures are not reliable, a child must reconcile the varying lengths and numbers of arbitrary units and reason transitively. On the other hand, to use a standard device such as a 30cm or meter ruler to make direct comparisons of lengths of objects is a less demanding task. It also has the advantage that it appears to be a real-world meaningful activity. (Boulton-Lewis et al., 1994, p.130).

It was concluded…the use of drawing in the development of area concepts helps children to develop abstractions and to recognize the units that go to make up a shape….(Wheatley and Reynold, 1996).

“Most researchers agree that there are three components of measuring: conservation, transitivity, and units and unit iteration.” (Chapin & Johnson, page 177). “Students in the United States must become proficient in using both the English system and the metric system of measurement.” (Chapin & Johnson, page 195).

“Tools…help children reason about the mathematically important components of an activity so that invariants like unit are represented physically and then mentally.” (Lehrer & Schauble, page 282). “Students find it very difficult to decompose and then recompose shapes or even to see one shape as a composition of others, an idea that is fundamental to conservation.” (Lehrer, Jenkis, and Osana, page 88).

GEOMETRY: 3-5

Introduction

As students investigate the attributes and properties of geometric shapes, they develop more precise descriptions of the relationships they discover. They are learning to reason and to make, test, and justify conjectures about these relationships. Students need to extend geometric knowledge and develop spatial reasoning ability by visualizing geometric relationships. Spatial understanding is necessary for interpreting, understanding, and appreciating our inherently geometric world.

The Research Base Benchmarks, page 352

NCTM Principles and Standards, page 41

Students advance through levels of thought in geometry. Van Hiele has characterized them as visual, descriptive, abstract/relational, and formal deduction (Van Hiele, 1986; Clements & Battista, 1992). At the first level, students identify shapes and figures according to their concrete examples. For example, a student may say that a figure is a rectangle because it looks likes a door. At the second level, students identify shapes according to their properties, and here a student might think of a rhombus as a figure with four equal sides. At the third level, students can identify relationships between classes of figures (e.g., a square is a rectangle) and can discover properties of classes of figures by simple logical deduction.

Progress from one of Van Hiele’s levels to the next is more dependent upon instruction than age. Given traditional instruction, middle school students perform at levels one or two (Clements & Battista, 1992). Some evidence suggests it is possible for students to understand the abstract properties of geometric figures by 5^th grade (Clements & Battista, 1989, 1990, 1992; Wirszup, 1976).

With well-designed activities, appropriate tools, and teachers’ support, students can make and explore conjectures about geometry and can learn to reason carefully about geometric ideas from the earliest years of schooling. Geometry is more than definitions; it is about describing relationships and reasoning. The notion of building understanding in geometry across the grades, from informal to more formal thinking, is consistent with the thinking of theorists and researchers (Burger and Shaughnessy 1986; Fuys, Geddes, and Tischler, 1988; Senk 1989; Van Heile, 1986).

ALGEBRA: 3-5

Introduction

Algebra is a style of thinking where students study patterns and relationships and learn to use them in daily life. Patterns are the basis for reasoning about regularity and consistency. As students move into upper elementary, they need to generalize these patterns and express the relationships using language symbols, tables, and graphs.

Change is an important mathematical idea that can be studied using the tools of algebra. Research indicates that this is not an area that students typically understand with much depth. Using graphs and tables, student in grades 3-5 start to notice and describe change. As they look at sequences, they can begin to distinguish between arithmetic growth and geometric growth.

The Research Base Benchmarks, pages 334, 351-352

NCTM Principles and Standards, pages 40, 163

Preliminary research hints that students have difficulty making connections between mathematical expressions, sentences, and sequences that share common structural patterns. They focus instead upon incidental similarities or differences (Ericksen, 1991).

Students of all ages often do not view the equality sign of equations as a symbol of the equivalence between the left and the right side of the equation, but rather interpret it as a sign to begin calculation (Kieran, 1992). Students who are encouraged initially to use trial-and-error substitution develop a better notion of the equivalence of the two sides of the equation and are more successful in applying more formal methods later on (Kieran, 1988, 1989).

DATA ANALYSIS AND PROBABILITY: 3-5

Introduction

The analysis of data helps students begin to understand the world around them. Books, newspapers, the Internet, and other media are filled with graphical displays. With such widespread use, data analysis becomes very critical. Hence it is important that students in grades 3-5 progress from reading data to interpreting tables and graphs.

Moreover, students should formulate questions to investigate relevant issues in their lives. Furthermore, they must develop the skills of collecting valid data, organizing it, describing its central tendency and variability, and creating meaningful representations that can be used to make predictions and inferences.

Students at this level will also begin to investigate the concepts of probability. Through experiments, students will explore the frequency of various outcomes and use the results to make predictions.

The Research Base Benchmarks, pages 353-354

Research suggests that a good notion of representativeness may be a prerequisite to grasping the definitions for measure of location like mean, median, or mode. Students can acquire notions of representativeness after they start seeing data sets as entities to be described and summarized rather than as “unconnected” individual values. This occurs typically around 4^th grade (Mokros & Russell, 1992).

Research suggests students should be introduced first to location measures that connect with their emerging concept of the “middle,” such as the median, and later in the middle school grades, to the mean. Premature introduction of the algorithm for computing the mean divorced from a meaningful context may block students from understanding what averages are for (Mokros & Russell, 1992; Pollatsek et al., 1981).

The concept of the mean is quite difficult for students of all ages to understand even after several years of formal instruction. Several difficulties have been documented in the literature: students of all ages can talk about the algorithm for computing the mean and relate it to limited contexts, but cannot use it meaningfully in problems (Mokros & Russell, 1992; Pollatsek, Lima, & Well, 1981); upper elementary and middle school students believe that the mean of a particular data set is not one precise numerical value but an approximation that can have one of several values (Mokros & Russell, 1992).

Research presents somewhat contradictory results on elementary children’s understanding of probability. Piagetian research says lower elementary children have no conception of probability (Piaget & Inhelder, 1975; Shayer & Adey, 1981), but other studies indicate that even lower elementary school children have probabilistic intuitions upon which probability instruction can build. Falk et al. (1980) presented elementary school students with two sets, each containing blue and yellow elements. Each time, one color was pointed out as the payoff color. The students had to choose the set from which they would draw at random a “payoff element” to be rewarded. From the age of six, children began to select the more probable set systematically. The ability to choose correctly precedes the ability to explain these choices.

Upper elementary students can give correct examples for certain, possible, and impossible events, but cannot calculate the probability of independent and dependent events even after instruction on the procedure (Fischbein & Bazit, 1984). That is partly because students at this age tend to create “part to part” rather than “part to whole” comparisons (e.g., 9 men and 11 women rather than 15% of men and 10% of women).

Extensive research points to several misconceptions about probabilistic reasoning that are similar at all age levels and are found even among experienced researchers (Kahneman, Slovic, & Tversky, 1982; Shaughnessy, 1992). One common misconception is the idea of representativeness, according to which an event is believed to be probable to the extent that is “typical.” For example, many people believe that after a run of heads in coin tossing, tails should be more likely to come up. Another common error is estimating the likelihood of event based on how easily instances of it can be brought to mind.

NUMBER AND OPERATIONS: 6-8

Introduction

Students in grades 6-8 must develop number sense, computational estimation, mental computation, and number size in order to thoroughly understand the real number system. The primary focus in the middle grades should be on fractions, decimals, percents, integers, and rational numbers. Students should apply their understanding of factors, multiples, and prime factorization to problems involving fractions. Students need to develop an understanding of decimals as fractions whose denominators are powers of 10. The concept of fractions should be extended to include rates, ratios, and proportionality. Percents can be thought about in ways that combine aspects of both fractions and decimals, paying particular attention to percents less than 1 or greater than 100. Applications with integers will develop the notation that they represent relative changes in values. As a result of the studies in numbers and operations, students will be able to judge the advantages and disadvantages of various representations of numbers.

Students in middle grades must also understand the meaning of operations and how they relate to one another. In addition to developing proficiency with fraction, decimal, percent, integer, and rational number computations, students should be able to determine the reasonableness of their answers. Technologies such as calculators and computers can aid in connecting basic skills and calculation procedures to a deeper mathematical understanding. Students should also have experiences solving problems in context, choosing the appropriate computational method, and deciding whether to use approximate or exact values.

The Research Base Benchmarks, page 350

NCTM Principles and Standards, pages 216,218

Science for All Americans, page 131

Middle school students are able to identify the place values of the digits that appear in a number, but they cannot use the knowledge confidently in context (Sowder, 1992a). Upper elementary- and middle-school students often do not understand that decimal fractions represent concrete objects that can be measured by units, tenths of units, hundredths of units, and so on (Hiebert, 1992). Other students have little understanding of the value represented by each of the digits of a decimal number or know the value of the number is the sum of the value of its digits. Students of all ages have problems choosing the largest or the smallest in a set of decimals with different numbers of digits to the right of the decimal points (Carpenter et al., 1981; Hiebert & Wearne, 1986; Resnick et al., 1989). Upper elementary- and middle-school students may exhibit limited understanding of the meaning of fractional numbers (Kieren, 1992).

From their experience with whole numbers, many students appear to develop a belief that “multiplication makes bigger and division makes smaller.” When students solve problems in which they need to decide whether to multiply or divide fractions or decimals, this belief has negative consequences that have been well researched (Greer, 1992). Also, a mistaken expectation about the magnitude of a computational result is likely to interfere with students’ making sense of multiplication and division of fractions or decimals (Gaeber & Tannenhaus, 1993). For example, fewer than one-third of the thirteen-year-old U.S. Students tested in the National Assessment of Education Progress (NAEP) in 1988 correctly chose the largest number from 3/4, 9/16, 5/8, and 2/3 (Kouba, Carpenter, and Swafford, 1989). Students’ difficulties with comparison of fractions have also been documented in more recent NAEP administrations (Kouba, Zawojewski, and Strutchens, 1997).

Students are allowed much more flexibility in mathematics with the use of integers, which can be thought of in terms of a number line (AAAS, p. 131). They can now analyze numbers in terms of below sea level, debt, and left of zero on the real number line.

Middle-school and even high-school students may have limited understanding about the nature and purpose of estimation. They often think it is inferior to exact computation and equate it with guessing (Sowder, 1992b), so they do not believe estimation is useful (Sowder & Wheeler, 1989). Students who see estimation as a valuable tactic for obtaining information use estimation more frequently and successfully (Threadgill-Sowder, 1984).

MEASUREMENT: 6-8

Introduction

It should be recognized that students bring to the middle grades many diverse experiences from prior classroom instruction and life experiences. Important aspects of measurement at this level should include choosing and using appropriate units for attributes being measured, estimating measurements, solving problems involving perimeter, area, surface area, and volume. In addition, students should become proficient in using measurement tools while working within both metric and customary measurement systems.

Students should become proficient in composing and decomposing two- and three-dimensional shapes in order to find lengths, areas, and volumes of complex objects. Through these investigations, students can discover formulas and use them to solve problems involving perimeter, area, and volume. Student should explore the effect on perimeter and area when dimensions are proportionately changed.

Measurement concepts should be used throughout the school year by providing connections to other mathematics strands. Many measurement topics are closely related to what students learn in geometry.

The Research Base Research Ideas for the Classroom,

Middle Grades Mathematics, Chapter 5

Most students’ estimation skills are not well developed, especially for metric units; only 30% of 13-year-olds could estimate the length of a segment to the nearest centimeter (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981). About 70% of seventh graders could choose the best estimate of the height of a tall man in feet, but only about 35% could do so in meters. More experience with estimation in both systems of measurement appears to be needed.

Development of measurement formulas is an important part of middle grade mathematics. Formulas should be a product of exploration and discovery. This is appropriate at middle grades for concepts like area and perimeter, if students have spent time measuring these in their own ways. Among seventh graders, 56% could compute the area of a rectangle with length and width given, but only 46% could compute the area of the same rectangle drawn on grid paper without the dimensions written in. Only 33% of students could compute the volume of a rectangular solid with dimensions written in (Lindquist & Kouba, 1989).

GEOMETRY: 6-8

Introduction

Students come to the middle grades with an informal knowledge about geometric concepts. They have had experience in visualizing and drawing lines, angles, triangles, and other polygons. Moreover, they have developed intuitive notions about geometry from interacting with objects in their daily lives.

Middle school geometry programs should allow students to investigate relationships by drawing, measuring, visualizing, comparing, transforming and classifying two-dimensional and three-dimensional geometric objects. Geometry provides opportunities for developing mathematical reasoning (inductive and deductive), making and validating conjectures, and investigating properties that lead to the classification of geometric shapes. In the middle grades, students begin to use the coordinate plane to investigate transformations, congruence, similarity and symmetry.

Many topics included in the measurement strand are closely connected to the study of geometry.

The Research Base Research Ideas for the Classroom, Chpt. 11

Benchmarks, p. 352

Students advance through levels of thought in geometry. Van Hiele has characterized them as visual, descriptive, abstract/relational, and formal deduction (Van Hiele, 1986; Clements & Battista, 1992). At the first level, students identify shapes and figures according to their concrete examples. For example, a student may say that a figure is a rectangle because it looks like a door. At the second level, students identify shapes according to their properties, and here a student might think of a rhombus as a figure with four equal sides. At the third level, students can identify relationships between classes of figures (e.g., a square is a rectangle) and can discover properties of classes of figures by simple logical deduction.

Progress from one of Van Hiele’s levels to the next is more dependent upon instruction than age. Given traditional instruction, middle-school students perform at levels one or two (Clements & Battista, 1992). Some evidence suggests it is possible for students to understand the abstract properties of geometric figures by 5^th grade (Clements & Battista, 1989, 1990, 1992; Wirszup, 1976).

Since learning geometry requires student to recognize figures, their properties, and their relationships, spatial visualization skills are essential and contribute in an important way to the learning process. Professional educators point out the need to equip students with mathematical methods that support the full range of problem solving. These methods include the use of imagery, visualization, and spatial concepts and emphasize activities that use concrete representations to improve the perception of spatial relationships (Lappan & Schram, 1989; Young, 1982). Many researchers have hypothesized the differences in students’ spatial visualization skills are one cause of their problem solving difficulties. Major findings indicated a strong correlation between spatial ability and problem-solving performance, suggesting that spatial visualization skill is a good predictor of general problem solving (Tillotson, 1985).

Transformations bring a spatial-visual aspect to geometry that is as important as logical-deductive aspects. Also, transformation geometry has important real-world applications such as fabric patterns, mirrors, symmetry in nature, photos and enlargements (Sanok, 1987).

ALGEBRA: 6-8

Introduction

Algebra and algebraic thinking are fundamental to the basic education of all middle school students. Algebraic thinking is a natural extension of arithmetical thinking, but while arithmetic is effective in describing static pictures of the world, algebra is dynamic and a necessary vehicle for describing a changing world. Students in middle grades should investigate patterns expressed in tables, graphs, words, or symbols, with the emphasis on patterns that exhibit linear relationships (constant rate of change). They should explore notions of dependence and independence as the values of variables change in relation to one another. Students should connect this rate of change to the slope of a line and be able to interpret its meaning. In addition, they should develop facility in recognizing the equivalence of mathematical representations they can use to transform expressions, to solve problems, and to relate graphical, tabular, and symbolic representations. Students should be given the opportunity to solve linear equations as well as inequalities. Whenever possible, the teaching and learning of algebra in the middle grades should be integrated with other topics in the curriculum.

The Research Base Benchmarks, pp. 351-352

Research Ideas for the Classroom –

Middle Grades Mathematics, p. 94

Students have difficulty translating between graphical and algebraic representations, especially moving from a graph into an equation (Leinhardt et al., 1990). Results from the second study of the National Assessment for Educational Progress showed, for instance, that given a line with indicated intercepts, only 5% of 17-year-olds could generate the equation (Carpenter, et al., 1981). Students sometimes resist dealing with multiple representations because they do not find them helpful in solving problems (Dufour-Janvier et al., 1987). Rather, they see generating any representation as an end in itself, demanded by the requirements of the teacher of the text rather than by the needs of the problem. In addition, students confound the slope of a graph with the maximum or the minimum value and do not know that the slope of a graph is a measure of the rate (McDermott et al., 1987; Clement, 1989).

Beginning algebra students use various intuitive methods for solving algebraic equations (Kieran, 1992). Some of these methods may help their understanding of equation and equation solving. Students who are encouraged initially to use trail-and-error substitution develop a better notion of the equivalence of the two sides of the equation and are more successful in applying more formal methods later on (Kieran, 1988). By contrast, students who are taught to solve equations only by formal methods may not understand what they are doing. Students who are taught to use the method of “transposing” are found to only mechanically apply the change side/change sign rule (Kieran, 1988, 1989).

Additional research on classroom practices can be found in Research Ideas for the Classroom – Middle Grades Mathematics, Chapter 10.

DATA ANALYSIS AND PROBABILITY: 6-8

Introduction

Students in grades 6-8 should build on past experiences with data analysis to answer questions about populations and samples. To do this, students should begin to develop and conduct more complex studies. Data collection can be extended to other resources such as websites, and spreadsheets can facilitate data collection and organization. Acquiring new techniques to express the distribution of data will aid in the analysis and interpretation. Interpretation of results should contain appropriate uses of measures of central tendency and spread and construction of lines of best fit. Furthermore, the use of box plots, scatter plots, as well as histograms, circle graphs, and stem-and-leaf plots, will facilitate the representation of relationships between two populations.

In the middle grades, students should have numerous opportunities to engage in activities that promote probabilistic thinking. The resulting observations and inferences should be discussed using appropriate terminology. Comparison of theoretical and experimental probability should be undertaken. In addition, students can use and further develop their emerging understanding of proportionality to make predictions about future experiments.

The Research Base Benchmarks, pp. 353-354

Science for All Americans, p. 19

The concept of the mean is quite difficult for students of all ages to understand even after several years of formal instruction. It is important for students to be able to express ideas about the results of their research in order to see whether it says something useful about the original data (AAAS, p. 19). Several difficulties have been documented in the literature: students of all ages can talk about the algorithm for computing the mean and relate it to limited contexts, but cannot use it meaningfully in problems (Mokros & Russell, 1992; Pollatsek, Lima & Well, 1981); upper elementary and middle school students believe that the mean of a particular data set is not one precise numerical value but an approximation that can have one of several values (Mokros & Russell, 1992); some middle school students cannot use the mean to compare two different-sized sets of data (Gal et al., 1990).

Students of all ages often interpret graphs of situations as literal pictures rather than as symbolic representations of the situations (Leinhardt, Zaslavsky & Stein, 1990). “In addition, students confound the slope of a graph with the maximum or the minimum value and do not know that the slope of a graph is a measure of rate” (McDermott, et al., 1987; Clement, 1989). When constructing graphs, middle school students have difficulties with the notions of interval scale and coordinates even after traditional instruction (Kerslake, 1981). “Finally, students read graphs point-by-point and ignore their global feature. This has been attributed to the fact that they are rarely asked questions about maximum and minimum values; intervals over which a function increases, decreases, or levels off; or rates of change” (Herscovics, 1989).

Extensive research points to several misconceptions about probabilistic reasoning that are similar at all age levels and are found among experienced researchers (Kahneman, Slovic, & Tversky, 1982; Shaughnessy, 1992). One common misconception is the idea of representativeness, according to which an event is believed to be probable to the extent that it is “typical.” For example, many people believe that after a run of heads in coin tossing, tails should be more likely to come up.

NUMBER AND OPERATIONS: 9-12

Introduction

Students in grades 9-12 should see number and operations from a more global perspective. Their understanding of numbers is the foundation for their understanding of algebra, the core of all mathematics. High school students should understand more fully the concept of a number system, how different number systems are related, and whether the properties of one system hold in another. Students will develop an increased ability to estimate the results of an arithmetic computation and judge the reasonableness of results obtained through technology.

Students in high school will understand the meaning of exponents and how to apply their properties in computations. The real number system will be explored for its use in matrices and vectors. High school students will relate complex numbers to problems for which there are no real solutions. Students will apply these concepts in a variety of problem-solving situations.

The Research Base NCTM Principles and Standards, page 32

Regardless of the particular method used, students should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general. Students also need to be able to estimate and judge the reasonableness of results. Computational fluency should develop in tandem with understanding of the role and meaning of arithmetic operations in number systems (Hiebert, et al., 1997; Thornton, 1990).

MEASUREMENT: 9-12

Introduction

Opportunities to use and understand measurement arise naturally during high school in various disciplines. By ninth grade, students will have a good understanding of measurement concepts and well-developed measurement skills. Electronic measurement instruments aid the students in collecting, storing, and analyzing real-time measurement data. High school students should be able to make reasonable estimates and sensible judgments about the precision and accuracy of the values they obtain.

Students in high school will distinguish between precision and accuracy of measurements. With the widespread use of calculator and computer technologies for gathering and displaying data, students will understand that selections of scale and viewing window become important choices. Through logarithmic scaling, students will graphically represent some naturally occurring phenomena. Students will understand how unit analysis can be used to make decisions about which units are most appropriate.

The Research Base Benchmarks, pages 350, 351, 360

Students who can use measuring instruments and procedures when asked to do so often do not use this ability while performing an investigation. Typically, a student asked to undertake an investigation and given a set of equipment that includes measuring instruments, will make a qualitative comparison even though she might be competent to use the instruments in a different context (Black, 1990). It appears students often know how to take measurements, but not what or when.

Middle-school and even high-school students may have limited understanding about the nature and purpose of estimation. They often think it is inferior to exact computation and equate it with guessing (Sowder, 1992b), so that they do not believe estimation is useful (Sowder & Wheeler, 1989). Students who see estimation as a valuable tactic for obtaining information use estimation more frequently and successfully (Threadgill-Sowder, 1984).

Students of all ages often interpret graphs as literal pictures rather than as symbolic representations of the situations (Leinhardt, Zzaslavsky, & Stein, 1990; McDermott, Rosenquist, & Van Zee, 1987). Many students interpret distance/time graphs as the paths of actual journeys (Kerslake, 1981).

GEOMETRY: 9-12

Introduction

High school students should develop capacity with several ways of representing geometric ideas. These representations will allow multiple approaches to solve geometric problems. Geometry offers a means of describing, analyzing, and understanding the world; its ideas can be useful both in other areas of mathematics and in applied settings. By the ninth grade, students will have explored and discovered relationships among two- and three-dimensional geometric shapes. The students’ high school experiences in geometry will enhance their ability to discover patterns and formulate conjectures. Technology will be a useful tool for accomplishing this task.

The Research Base Research Ideas for the Classroom,

High School, page 151

Research Ideas for the Classroom,

Middle School, page 219

Probably the most comprehensive study of an alternative to traditional Euclidean geometry instruction is Usiskin’s investigation of the feasibility of a transformation approach (Cosford & Usiskin, 1971; Usiskin, 1969). Neither approach was clearly superior overall. On some measures, particularly attitudinal, the transformational approach seemed more successful on some measures of achievement.

“It is not enough….to learn about properties of shapes and the vocabulary of geometry; they [students] must understand what geometry is and how it relates to the real world and other topics in mathematics. Research has shown that our students must be active learners engaged in the process of discovering, conjecturing, and thinking at higher levels” (Fortunato, 1993).

ALGEBRA: 9-12

Introduction

Algebra is the core of mathematics. High school students’ experiences in mathematics should provide insights into algebraic abstractions and structures. These insights can help students develop a deeper understanding of real-world phenomena. By the ninth grade, students will have explored various ways of representing linear and non-linear qualities. At the same, time, working in real-world contexts may help students make sense of the underlying mathematical concepts and may foster an appreciation of those concepts. Using technology, students can model and analyze a wide range of phenomena.

Students in high school will become competent with their use of algebra. They will create models that satisfy applications of exponential and other non-linear functions. The development of function notation will assist students to better understand the effects of translations on graphs. In addition, students will recognize the effects of parameter changes. Functions notation will also help students identify how a relation might be represented through the use of parametric equations. Having gained deeper insight into the applications of mathematics, students will be able to use technology to solve a variety of problems and to identify the reasonableness of the answers that they obtain.

The Research Base Benchmarks, page 351

Research Ideas for the Classroom –

High School, pp. 202, 204-205

Students have difficulty understanding how symbols are used in algebra (Kieran, 1992). They are often unaware of the arbitrariness of the letters chosen to represent variables in equations (Wagner, 1981). Middle-school and high-school students may regard the letters as shorthand for single objects, or as specific but unknown numbers, or as generalized numbers before they understand them as representations of variables (Kieran, 1992).

Another study focused on the mathematical performance of upper secondary students who had regular and prolonged access to graphing calculators (Ruthven, 1990). These students developed specific calculator techniques for finding symbolic rules for graphically represented functions. Interestingly, the graphing-calculator group outperformed students who did not have such access on tasks that required symbolization.

A study involving beginning high school students in learning mathematical modeling while using a computer for symbolic manipulation also suggested conceptual gains without noticeable skill loss (Heid, Sheets, et al., 1988). In the Heid study, distinctive patterns of classroom interaction were noted in the experimental course. The activities that characterized the experimental course included: making, defending, and debating mathematical conjectures; interpreting and reasoning about mathematical representations; and suggesting and justifying mathematical models (Heid, 1988).

DATA ANALYSIS AND PROBABILITY: 9-12

Introduction

Upon entering high school, students should be familiar with designing simple surveys and experiments; gather data through the use of tables, charts, and graphs; and summarizing that data in various ways. Students will have computed probabilities of simple and some compound events, and will have performed simulations, comparing the results of the simulations to predicted probabilities.

In grades 9-12, students should gain a deep understanding of the issues entailed in drawing conclusions in light of variability. They should learn to ask questions that will help them evaluate the quality of survey, observational studies, and controlled experiments. Students can use their skills in algebra to model and analyze data, with increasing understanding of what it means to fit data well.

High school students should link probability to other topics in mathematics, especially counting techniques, area concepts, and relationships between functions and the area under their graphs. Students should learn to determine the probability of a sample statistic for a known population and draw simple inferences about a population from randomly generated samples.

The Research Base Benchmarks, page 361

NCTM Principles and Standards, page 50

Research Ideas for the Classroom,

High School, page 188

Research has shown that students in grades 5-8 expect their own judgment to be more reliable than information obtained from data (Hancock, Kaput, and Goldsmith, 1992). In the later middle grades and high school, students should address the ideas of sample selection and statistical inference and begin to understand that there are ways to quantifying how certain one can be about statistical results.

Even researchers trained in the use of statistics entertain statistical misconceptions. For example, they may erroneously believe that when conducting a replication studys’ [sic] even smaller sample sizes than the first study’s are sufficient, since sample should be “representative” of the population regardless of its size (Tversky & Kahneman, 1971). If trained researchers have trouble with statistical concepts, it should not surprise us that students have misconceptions of some of the most elementary concepts, such as mean and variance.

A basic problem appears to be understanding the distinction between a variable making no difference and a variable that is correlated with the outcome in the opposite way than the students initially conceived (Kuhn, et al., 1988).

References:

American Association for the Advancement of Science. (1993). Benchmarks For

Science Literacy. New York, New York: Oxford University Press.

American Association for the Advancement of Science. (1990). Science For All

Americans. New York, New York: Oxford University Press.

Chapin, S. H., and A. Johnson. (2000). Math Matters: Understanding The Math You

Teach, Grades K-6. Sausalito, CA: Math Solutions Publications.

Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, N.J.:

Lawrence Erlbaum Associates.

National Council of Teachers of Mathematics. (2000). Principles and Standards for

School Mathematics. Preston, VA: National Council of Teachers of Mathematics.

National Research Council. (2001). Adding It Up: Helping Children Learn

Mathematics. Washington, D.C.: National Academy Press.

Van de Walle, John. (2001). Elementary and Middle School Mathematics: Teaching

Developmentally, Addison Wesley Longman, Inc.

Wagner, Sigrid, ed. (1993). Research Ideas for the Classroom. National Council of

Teachers of Mathematics.

Mathematical Processes

Mathematical Processes Standard

Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas

A mathematics curriculum is more than a set of isolated content strands. The framework that connects the concepts consists of five equally important process strands that are interwoven and interdependent. The NCTM recognizes problem solving, representations, communication, reasoning and proof, and connections as the process strands that are vital to a comprehensive understanding of mathematics.

Problem solving, an essential tool for learning and applying mathematics, should be embedded in all aspects of the curriculum. Moreover, integration of content to other disciplines through problem solving gives meaning and purpose to the acquisition of mathematics skills. Flexibility in application of problem solving skills enables various strategies to be applied to a single situation. Reflecting upon strategies used and reasonableness of solutions develops habits of self-assessment.

Representation is the key to understanding mathematics at all levels. As students progress through school, learning and the ability to represent ideas develop over time by using physical models, informal representations, symbols, equations, charts, and graphs. As students communicate their thinking about mathematics to others, these representations serve as tools for thinking about and solving problems. “If mathematics is the ‘science of patterns’ representations are the means by which those patterns are recorded and analyzed.” (NCTM, 2000)

Communication of mathematical thought is vital in a society saturated with advanced technology. In order to share thinking, students have to organize and clarify their thought processes and learn to listen carefully and critically to others. It is important that mathematics language and vocabulary be emphasized at the developmentally appropriate time, leading to precise and formal explanations. Participation in discussions, analyzing multiple strategies and solutions, and providing written arguments will facilitate language development. The use of technology in the forms of calculators, computers, and the Internet will become increasingly important in the lives of today’s students.

Reasoning and proof are indicative of logical thought and reflection among concepts and situations. It guides the learner on a journey through all the facts, procedures, and concepts necessary to make sense of problems and their solutions. Questioning, hypothesizing, testing and analyzing conjectures contribute to the justification and communication of conclusions. Reasoning and proof lead to conclusions about general properties and relationships and encourage self-expression and self-assessment.

Connections among mathematics concepts occur when students link prior knowledge to new concepts across the curriculum. Although mathematics instruction is often portioned, a curriculum that emphasizes the interrelatedness of the content strands empowers the learner with the utility of mathematics. The learning of mathematics should build upon previous experiences rather than repeat what has already been learned. School mathematics experiences at all grade levels should include learning opportunities for students to apply concepts to problems arising in contexts outside mathematics.

A mathematics curriculum is the sum of all its parts. The process strands are the glue that cement the content strands together. The process strands and the content strands converge to strengthen the underlying unity of the mathematics curriculum.

The benchmarks for mathematical processes articulate what students should demonstrate in problem solving, representation, communication, reasoning and connections at key points in their mathematics program. Specific grade-level indicators have not been included for the mathematical processes standard because content and processes should be interconnected at the indicator level. Therefore, mathematical processes have been embedded within the grade-level indicators for the five content standards.

Mathematical Processes Benchmarks

By the end of the A. Use a variety of strategies to understand

K-2 program: problem situations; e.g., discussing with peers,

Stating problems in own words, modeling problems with diagrams or physical materials, identifying a pattern.

B. Identify and restate in own words the question or problem and the information needed to solve the problem.

C. Generate alternative strategies to solve problems.

D. Evaluate the reasonableness of predictions, estimations and solutions.

E. Explain to others how a problem was solved.

F. Draw pictures and use physical models to represent problem situations and solutions.

G. Use invented and conventional symbols and common language to describe a problem situation and solution.

H. Recognize the mathematical meaning of common words and phrases, and relate everyday language to mathematical language and symbols.

I. Communicate mathematical thinking by using everyday language and appropriate mathematical language.

By the end of the A. Apply and justify the use of a variety of

3-4 program: problem-solving strategies; e.g., make an organized list, guess and check.

B. Use an organized approach and appropriate strategies to solve multi-step problems.

C. Interpret results in the context of the problem being solved; e.g., the solution must be a whole number of buses when determining the number of buses necessary to transport students.

D. Use mathematical strategies to solve problems that relate to other curriculum areas and the real world; e.g., use a timeline to sequence events; use symmetry in artwork.

E. Link concepts to procedures and to symbolic notation; e.g., model 3 x 4 with a geometric array, represent one-third by dividing an object into three equal parts.

F. Recognize relationships among different topics within mathematics; e.g., the length of an object can be represented by a number.

G. Use reasoning skills to determine and explain the reasonableness of a solution with respect to the problem situation.

H. Recognize basic valid and invalid arguments, and use examples and counter examples, models, number relationships, and logic to support or refute.

I. Represent problem situations in a variety of forms (physical model, diagram, in words or symbols), and recognize when some ways of representing a problem may be more helpful than others.

J. Use mathematical language to explain and justify mathematical ideas, strategies and solutions.

By the end of the A. Clarify problem-solving situations and identify

5-7 program: potential solution processes; e.g., consider

different strategies and approaches to a problem, restate problem from various perspectives.

B. Apply and adapt problem-solving strategies to solve a variety of problems, including unfamiliar and non-routine problem situations.

C. Use more than one strategy to solve a problem, and recognize there are advantages associated with various methods.

D. Recognize whether an estimate or an exact solution is appropriate for a given problem situation.

E. Use deductive thinking to construct informal arguments to support reasoning and to justify solutions to problems.

F. Use inductive thinking to generalize a pattern of observations for particular cases, make conjectures, and provide supporting arguments for conjectures.

G. Relate mathematical ideas to one another and to other content areas; e.g., use area models for adding fractions, interpret graphs in reading, science and social studies.

H. Use representations to organize and communicate mathematical thinking and problem solutions.

I. Select, apply, and translate among mathematical representations to solve problems; e.g., representing a number as a fraction, decimal or percent as appropriate for a problem.

J. Communicate mathematical thinking to others and analyze the mathematical thinking and strategies of others.

K. Recognize and use mathematical language and symbols when reading, writing and conversing with others.

By the end of the A. Formulate a problem or mathematical model in

8-10 program: response to a specific need or situation, determine information required to solve the problem choose method for obtaining this information, and set limits for acceptable solution.

B. Apply mathematical knowledge and skills routinely in other content areas and practical situations.

C. Recognize and use connections between equivalent representations and related procedures for a mathematical concept; e.g., zero of a function and the x-intercept of the graph of the function, apply proportional thinking when measuring, describing functions, and comparing probabilities.

D. Apply reasoning processes and skills to construct logical verifications or counter-examples to test conjectures and to justify and defend algorithms and solutions.

E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas.

F. Use precise mathematical language and notations to represent problem situations and mathematical ideas.

G. Write clearly and coherently about mathematical thinking and ideas.

H. Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner.

By the end of the A. Construct algorithms for multi-step and non-

11-12 program: routine problems.

B. Construct logical verifications or counter-examples to test conjectures and to justify or refute algorithms and solutions to problems.

C. Assess the adequacy and reliability of information available to solve a problem.

D. Select and use various types of reasoning and methods of proof.

E. Evaluate a mathematical argument and use reasoning and logic to judge its validity.

F. Present complete and convincing arguments and justifications, using inductive and deductive reasoning, adapted to be effective for various audiences.

G. Understand the difference between a statement that is verified by mathematical proof, such as a theorem, and one that is verified empirically using examples or data.

H. Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations.

I. Communicate mathematical ideas orally and in writing with a clear purpose and appropriate for a specific audience.

J. Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation.

MATHEMATICS COURSE OF STUDY

Committee Members

DISTRICT BELIEFS, VISION AND MISSION

BELIEFS

MISSION STATEMENT

WE EDUCATE FOR EXCELLENCE…

Mathematics Program Philosophy

NCTM Principles & Standards, page 79

NCTM Principles & Standards, page 35

MEASUREMENT: K-2

Introduction

The Research Base Adding it Up, pages 282-283

GEOMETRY: K-2

ALGEBRA: K-2 Benchmarks, page 217

DATA ANALYSIS AND PROBABILITY: K-2 SMART Mathematics

Course of Study, page 52

The Research Base Benchmarks, page 353

MEASUREMENT: 6-8

GEOMETRY: 6-8

ALGEBRA: 6-8

DATA ANALYSIS AND PROBABILITY: 6-8

MEASUREMENT: 9-12

GEOMETRY: 9-12

ALGEBRA: 9-12

DATA ANALYSIS AND PROBABILITY: 9-12