Adding It Up Helping Children Learn Mathematics (2001) / Chapter Skim
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8 Developing Mathematical Proficiency Beyond Number
8 Developing Mathematical Proficiency Beyond Number
Pages 255-312
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From page 255... ...
Algebra Beginning Algebra For most students, school algebra with its symbolism, equation solving, and emphasis on relationships among quantities seems in many ways to signal a break with number and arithmetic. In fact, algebra builds on the proficiency that students have been developing in arithmetic and develops it builds on the ,.
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From page 256... ...
The difficulties associated with the transition from the activities typically associated with school arithmetic to those typically associated with school algebra have been extensively studied. In this chapter, we review in some detail the research that examines these difficulties and describe new lines of research and development on ways that concepts and symbol use in elementary school mathematics can be made to support the development of algebraic reasoning.
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From page 257... ...
. Proficiency with representational activities involves conceptual understanding of the mathematical concepts, operations, and relations expressed in the verbal information, and it involves strategic competence to formulate and represent that information with algebraic equations and expressions.
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From page 258... ...
lust as in arithmetic, aspects of conceptual understanding and strategic competence interact with each other and with procedural fluency in transformational activities in algebra. Lastly, there are the generalizing andjustifying activities.
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Data from the National Assessment of Educational Progress (NAEP) further reveal the shortcomings of traditional school algebra.
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From page 260... ...
Since much of this research has been carried out with students making the transition from arithmetic to algebra, it casts light on the kinds of thinking that students bring with them to algebra from the traditional arithmetic curriculum centered on algorithmic computation that has been predominant in U.S. schools.~7 Indeed, many studies have been oriented toward either developing approaches to teaching algebra that take this arithmetic thinking into account or, more recently, developing approaches to elementary school mathematics that build foundations of algebraic reasoning earlier.
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From page 261... ...
The Representational Activities of Algebra \/Vhat the Number-Proficient Student Brings rat , ... .~ 1 .~ r 1 1 .1 r traditional representational activities ot algebra center on the formation of algebraic expressions and equations.
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From page 262... ...
A number of recent intervention studies have shown how selected modifications of elementary school mathematics might support the development of algebraic reasoning. One approach infuses elementary mathematics with a systematic use of problems requiring students to generalize, to determine values of a literal term that satisfy quantitative constraints (with or without equations)
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From page 263... ...
Further, these third graders outperformed fourth graders on items testing number sense from a mandated statewide assessment.26 A third approach to modifying elementary school mathematics focuses on helping teachers understand their students' thinking when the students are asked to generalize operations and properties from arithmetic. In one combination first-and-second-grade class, the teacher focused on number sentences twice a week during the school year.
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From page 264... ...
and in improving their attitude toward algebra (at the beginning of the study, they "hated algebra, didn't understand it" and complained that "letters are stupid; they don't mean anything"~.3~ Later research in which students used actual computers confirmed these results, both with respect to increasing the students' motivation and developing their understanding of algebraic expressions as general computational procedures.32 Representational activities of algebra can interact with well-established natural-language-based habits. These interactions are particularly clear in the well-studied class of tasks exemplified by the so-called students-andprofessors problem:33 At a certain university, there are six times as many students as professors.
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From page 265... ...
A series of teaching experiments conducted over three years during the late 1980s in Mexico and the United Kingdom demonstrated the potential of computer spreadsheets to help students grasp the meaning of variables and algebraic expressions, including students who had been having difficulty with traditional approaches to algebraic symbolism.38 Further, spreadsheets can provide a vehicle for introducing students to formal symbolism.39 For an example of how a student can profit from the use of a spreadsheet, see Box 8-3. This student was a tenth grader in a low mathematics track of a school in England who had little previous experience with algebra.
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From page 266... ...
" She wrote the following, which shows that she was now able to represent the problem using the literal symbols of algebra (note that the syntax of many spreadsheets requires the entry of an equal sign before the algebraic expression) : - x = xx 4 = xx 4 + 10 SOURCE: Sutherland, 1993, p.
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From page 267... ...
Interviews and tests of one cohort of students at the end of their first year of algebra showed that the experimental group did significantly better than their counterparts from conventional classes in improving their problem-solving abilities and in comprehending the notion of variable. For example, in constructing mathematical representations, the success rates were 48% versus 21%; in interpreting mathematical representations, 78% versus 28%; and in planning solutions and solving problems, 77% versus 66%, respectively.49 A similar approach to teaching algebra that involves graphing calculators has been implemented in a three-year high school mathematics curriculum used in several states.50 When students from three schools at the end of their third year in this curriculum were compared with students nearing the end of their high school algebra experience in advanced algebra classes in three other schools, the students in the new curriculum did better than the comparison group on algebraic tasks that were embedded in applied problem contexts when graphing calculators were available (43% correct for the project group
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From page 268... ...
Used by permission of the National Council of Teachers of Mathematics.
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From page 269... ...
In fact, when the equation-solving tasks were presented in a contextualized form, such as the example shown in Box 8-5, the students in the new curriculum were more successful than the comparison students (61% correct vs. 45%~.5~ The ways that graphing calculator use can produce improved student performance were examined more deeply in a recent study.52 The study used a three-condition pretest-posttest design to study the impact of prolonged use of the graphing calculator throughout the entire school year for all topics of the mathematics curriculum (i.e., functions and graphs, change, exponential and periodic functions)
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From page 270... ...
The Transformational Activities of Algebra What the Number-Proficient Child Brings In the previous section, we discussed some of the perspectives brought to the study of algebra by students emerging from traditional elementary school arithmetic. These perspectives included the following: An orientation to execute operations rather than to use them to represent relationships; which leads to Use of the equal sign to announce a result rather than signify an equality; · Use of inverse or undoing operations to solve a problem and the corresponding absence of a notion of describing a situation with the stated operations of a problem; and A perception of letters as representing unknowns but not variables.
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From page 271... ...
These notions, however, are not always as well cultivated in elementary school mathematics as they should be if they are to serve as a basis for algebraic reasoning. Students emerging from six or seven years of elementary school mathematics are ordinarily aware of the close relationship between addition and subtraction.
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From page 272... ...
These findings suggest that students learning formal methods of equation solving may benefit from well-timed prior instruction in the informal technique of "cover up." Another study found that students who were entering their first algebra course showed one of two preferences when solving simple linear equations in which there was only one operation: Some used trial-and-error substitution; the others used undoing.57 For two-step equations involving two operations such as 2x - 5 = 11, the latter group of students spontaneously extended their right-to-left undoing technique: Take 11, add 5 to it, then divide by 2.
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From page 273... ...
(They had to ignore the last operation of multiplication because they had run out of operands.) A preference for the undoing method of equation solving seemed to work against the students when they were later taught the procedure of performing the same operation on both sides of an equation.
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From page 274... ...
This result has been found repeatedly, even in recent studies: "Few students tcan] do the kinds of basic symbolic calculation that are common fare on college-admission and placement tests."60 The Role of Technology Transformational activities of algebra have benefited substantially less than representational activities from the use of computer technology to help develop meaning and skill.
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From page 275... ...
In 1989 one mathematics educator noted that "the unanswered question standing in the way of reducing the manipulative skills agenda of secondary school algebra is whether students can learn to plan and interpret manipulations of symbolic forms without being themselves proficient in the execution of those transformations."63 Very little research has been conducted since then to help resolve the question; however, the research that has been done is quite telling. A recent study investigated the impact on algebra achievement of a three-year integrated mathematics curriculum in which technology was used to perform symbolic manipulations as well as to link various representations of problem situations.64 In this study, which involved over 300 high school students in 12 schools, some support was found for the notion that learning how to interpret results of algebraic calculations is not highly dependent on the ability to perform the calculations themselves.
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From page 276... ...
Several of these activities require a certain level of skill in representing and transforming algebraic expressions, as well as in adaptive reasoning. Two problems from the research literature help illustrate the issues (see Box 8-8~.
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From page 277... ...
translating from a verbal representation to a symbolic representation through the use of a letter as a variable to represent "any number," (b) manipulating the algebraic expression to yield simpler equivalent expressions with the underlying aim of arriving at an expression indicating "3 more than the number she started with," and (c)
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From page 278... ...
The evolving sequence of simplified algebraic expressions can permit a perception of "x + 3-ness" in a way that is not so readily available from simply reading the problem. Thus, the algebraic representation can induce an awareness of structure that is much more difficult, if not impossible, to achieve using everyday language.
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From page 279... ...
Algebra students have more difficulty deriving the latter rule, y = Ax + 1~/2, than the former.68 The use of computer technology can enable students to engage in activities like those above without having to generate or transform algebraic equations on their own.69 But students have to learn how to use the equations produced by the technology to make predictions, even if they do not actually generate them by hand. Through an emphasis on generalization, justification, and prediction, students can learn to use and appreciate algebraic expressions as general statements.
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From page 280... ...
By focusing on ways to use the elementary and middle school curriculum to support the development of algebraic reasoning, these efforts attempt to avoid the difficulties many students now experience and to lay a better foundation for secondary school mathematics.74 From the earliest grades of elementary school, students can be acquiring the rudiments of algebra, particularly its representational aspects. They can observe that over time and across different circumstances, numerical quantities may vary in principled ways the essence of the concept of variable.
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From page 281... ...
Studies conducted in the last two decades, however, have generally failed to support the contention that there is a tight coupling between understanding a spatial measure and knowing when it is conserved.76 Length Measure Length needs to be understood from several perspectives: for example, as magnitude, as a span, as the distance traveled, or as motion.77 Proficiency in the measurement of length requires the learner to restructure space so that There is much pedagogical value in returning geometry to its roots in spatial measu ret
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From page 282... ...
In a recent teaching experiment on measuring length, children used computer tools that provided them experience with a unit and the repetition of units to get a measurement. The tools helped the children mentally restructure lengths into units.83 In other studies, researchers have placed a premium on transitions from active forms of length measure, like pacing, to recording and symbolizing these forms as "foot strips" and other kinds of measurement tools.84 Tools like foot strips help children reason about the mathematically important components of activity (e.g., pacing)
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From page 283... ...
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. For example, students in grades 1 to 3 often cannot think of a rectangle as an array of units.89 By the end of the elementary grades, students typically understand core concepts like using identical units and covering the object for length measure but not for area measure.
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From page 284... ...
Developing Geometric Reasoning Early work on geometric reasoning suggested that proficiency in geometry develops in a sequence of stages associated with aged and that children can be assisted, through appropriate activities, to move to more advanced levels of reasoning.94 Recent work has confirmed the effectiveness of appropriate activities even as it has called into question the notion of a stage-like sequence.95 Reasoning About Shape and Form Children enter school with a great deal of knowledge about shapes. They can identify circles quite accurately and squares fairly well as early as age four.96 They are less accurate at recognizing triangles (about 60% correct)
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From page 285... ...
On a set of geometry items from NAEP,99 the performance of both groups well exceeded the performance by the high school students in NAEP. Moreover, on measures of abstracting and applying geometric properties for reasoning, the fourth graders who had used Logo as a construction tool significantly outperformed their contemporaries.l°° Although previous work had suggested that children's reasoning about geometric figures is based on global appearances, primary school children in one studying routinely used a variety of attributes of shape and form to describe how two shapes, in either two or three dimensions, were alike yet different from a third shape.
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From page 286... ...
angle as a measure, a perspective that encompasses the other two.~°8 Although as preschoolers, they encounter and use angles intuitively in their play, children have many misconceptions about angles. They typically believe that angle measures are influenced by the lengths of the intersecting lines or by the angle's orientation in space.~09 The latter conception decreases with age, but the former is robust at every age.~° Some researchers have suggested that students in the elementary grades should develop separate mental models of angle as movement and angle as shape.
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From page 287... ...
Several researchers have looked at the effects of introducing children to ideas about modeling space. In these studies, middle school students made significant progress in developing their conceptions of proportion and scale when they used a computer-assisted-drawing (CAD)
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From page 288... ...
in a classroom game of tag.~3~ Some research has focused on relationships between spatial models and learning about science. For example, middle school students' understanding of area and volume measure was found to make a significant contribution to their understanding of concepts like buoyancy,~32 and the idea of similarity in substance helped in developing their understanding of similarity of shapes.~33 Engineering problems involving stability have also been employed to help middle school students understand the relationship between geometry and the success or failure of architectural structures.~34 Collectively, research on geometry points the way to a significant expansion of what is meant by the study of shape and form in school mathematics.
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From page 289... ...
The process is essentially what has been called reading the data,~37 and researchers have found that the majority of students in the elementary and middle school grades can read data displays accurately.~38 Although children in the primary grades often give idiosyncratic descriptions of data, explorations with categorical and numerical data in instruction that incorporates technology produce more focused and less idiosyncratic descriptions.~39 Organizing Data The process of organizing, and reducing, data incorporates mental actions such as ordering, grouping, and summarizing.~40 Data reduction also includes the use of representative measures of center (often termed measures of central tendency) such as mean, mode, or median, and measures of spread Research on organizing data at grades such as range or standard deviation.
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From page 290... ...
Representing Data Representing data in visual displays requires the generation of different organizations of data according to certain conventions. Many elementary students have difficulty creating visual displays of data.~47 graders' knowledge of how to represent data appears to be constrained by difficulties in sorting and organizing data, and technology has been found to be helpful in overcoming those difficulties.~48 Studies of middle school students have revealed substantial gaps in their abilities to construct graphs from given data.~49 Processes like organizing data and conventions like labeling and scaling are crucial to data representation and are strongly connected to the concepts and processes of measurement.
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From page 291... ...
Although some children as young as seven years can use efficient procedures for listing all outcomes,~58 other children in grades 4 through 6 are reluctant or unable to list them all.~59 Probability of an Event Although probability tasks used in research with elementary and middle school students have typically involved equally likely outcomes, a number of
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From page 292... ...
of instances favoring the target event.~64 Conditional Probability A number of studies have addressed elementary and middle school students' thinking in conditional probability situations their ability to recognize when the probability of an event is or is not changed by the occurrence of another event.~65 For example, the conditional probability of drawing a white ball, given that you have already drawn and not replaced a white ball from a bag containing three white balls and three red balls, is 0.4, not 0.5. When fifth, sixth, and seventh graders were asked to determine conditional probabilities, the performance of the sixth and seventh graders was dramatically lower when the tasks involved selection without replacement compared with selection with replacement.
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From page 293... ...
The school mathematics curriculum, although separated into domains for the purposes of this report, needs to be experienced by the learner as a unified whole. In general, the arithmetic thinking of number-proficient students emerg~ng from the typical elementary school mathematics program is different from the thinking that is central to algebra.
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From page 294... ...
The National Council of Teachers of Mathematics, 1997, offers four organizing themes for school algebra: functions and relations, modeling, structure, and language and representation. Kaput, 1995, identified five aspects of algebra: generalization and formalization; syntactically guided manipulations; study of structure; study of
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From page 295... ...
18. See Swafford and Langrall, 2000, for research using exponential and inverse variation functions with sixth graders; Rojano, 1996, for research involving systems of linear equations; and Bednarz, Radford, and tanvier, 1995, and Radford, 1994, for research using situations with more than one unknown.
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From page 296... ...
Z96 ADDING IT UP symbols. Other students who gave the same incorrect answer drew a diagram showing six students and one professor and seemed to think of S and P as units of measure rather than variables.
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From page 297... ...
93. Van Hiele, 1957/1984b, 1959/1984a; Van Hiele-Geldof, 1957/1984.
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From page 298... ...
118. The Geometer's Sketchpad, Cabri, and other "dynamic geometry" software allow students to construct geometric figures on the computer screen just as students and mathematicians for centuries have used a ruler (or straightedge)
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From page 299... ...
DEVELOP/NO MATHEMAT/CAL PROF/C/ENCY BEYOND NUMBER Z99 139. tones, Thornton, Langrall, Mooney, Wares, Perry, and Putt, 1999; tones, Thornton, Langrall, Mooney, Perry, and Putt, 2000.
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From page 300... ...
(1996~. Mathematics achievement in the middle school years: IEA's Third International Mathematics and Science Study (TIMSSJ.
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From page 301... ...
Paper presented at the meeting of the American Educational Research Association, Montreal. Carpenter, T
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Elementary School Journal, 98, 171-186. Clements, D
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(Doctoral dissertation, University of Toronto, 1986~. Dissertation Abstracts International, 47~10)
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From page 304... ...
Paper presented at the meeting of the American Educational Research Association, Boston. Gutierrez, A., Jaime, A., & Fortuny, J
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From page 305... ...
, Developing mathematical reasoning in grades K-12 (1999 Yearbook of the National Council of Teachers of Mathematics, pp.
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From page 306... ...
Paper presented at the meeting of the American Educational Research Association, New York. (ERIC Document Reproduction Service No.
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(1990~. Embedded figures and the structures of algebraic expressions.
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From page 308... ...
(1992~. Development of a middle school statistical thinking framework (Doctoral dissertation, Illinois State University, 1992~.
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From page 309... ...
(1994~. Patterns and functions (Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8, F
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From page 310... ...
(1988~. Elementary school children's use of strategy in playing microcomputer probability games.
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From page 311... ...
, The teaching and learning of algorithms in school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics, pp.
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From page 312... ...
Tischler, English translation of selected writings of Dina Van Hiele-Geldaf and P M Van Hiele (pp.
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