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A MULTIDIMENSIONAL APPROACH TO UNDERSTANDING IN MATHEMATICS TEXTBOOKS DEVELOPED BY UCSMP
Denisse R. Thompson Sharon L. Senk
University of South Florida Michigan State University
Tampa, FL, USA East Lansing, MI, USA
HYPERLINK "mailto:thompson@tempest.coedu.usf.edu" thompson@tempest.coedu.usf.edu HYPERLINK "mailto:senk@math.msu.edu" senk@math.msu.edu
ABSTRACT
This paper describes two features of the textbooks written by the University of Chicago School Mathematics Project. The first, called the SPUR perspective, is a multidimensional approach to understanding that guides the development of lessons and chapters. The second, called the CARE perspective, guides the development of the problem sets to ensure that students engage with the essential concepts, have an opportunity to apply concepts to more challenging problems, and review previously learned concepts to maintain and strengthen understanding. Data about the relation between SPUR objectives and students' achievement, and teachers' use of the CARE problems are also provided.
The Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000) and the earlier Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) have guided the development of curriculum materials and state curriculum frameworks in the United States since their release. These documents outline a vision for mathematics for students in grades preK12. In particular, they describe the content that students should be expected to learn as well as mathematical processes with which they should engage; the standards documents emphasize the importance of a balanced perspective relative to procedural fluency and conceptual understanding. Such concerns about broadening the mathematics curriculum are also evident in curriculum materials developed in other countries (e.g., Realistic Mathematics from the Netherlands (Gravemeijer 1994), curriculum textbook materials from Singapore (Seah & Bishop 2000) or Japan (Watanabe 2001)).
Many mathematics educators have argued for considering multiple perspectives in the learning of mathematics content. For instance, Freudenthal (1983) considers the different ways in which a topic might be used and how those different perspectives lead to different understandings. The synthesis of research about children's understanding of mathematics compiled by Kilpatrick, Swafford, and Findell defines mathematical proficiency as consisting of five intertwined strands: procedural fluency, adaptive reasoning, conceptual understanding, productive disposition, and strategic competence (National Research Council 2001). These strands are interconnected and interdependent. Kilpatrick et al. argue that students need to develop competence in all five strands concurrently to develop a robust understanding of mathematics. Krutetskii (1976) showed that, at least among gifted students of mathematics, one can identify students who regularly use algebraic or analytic approaches to solve problems and others who use geometric or spatial approaches. Hence, curriculum materials that use a multidimensional perspective present a balanced view of mathematics that accommodates classrooms with a range of students having different mathematical strengths and learning styles.
The purpose of this paper is to illustrate how a multidimensional perspective on mathematics is embodied in the set of textbooks developed in the United States by the University of Chicago School Mathematics Project (UCSMP). UCSMP is a K12 curriculum research and development project that began in 1983 (Usiskin 2003). The project has three main goals: (1) to upgrade student achievement; (2) to update the curriculum to include appropriate technology (e.g., graphing calculators, dynamic geometry software, computer algebra systems) and content important for the 21st century (e.g., statistics, probability, modeling, discrete mathematics); and (3) to increase enrollment in mathematics beyond algebra and geometry with mathematics appropriate to students' needs. In particular, the secondary curriculum integrates arithmetic, algebra, geometry, and statistics so that students who complete the entire curriculum would be prepared for postsecondary courses in calculus as well as in areas of discrete mathematics.
Beginning in 2005, UCSMP began revising its materials for grades 712, roughly ages 1318, to develop third editions of those materials and to write a new course for grade 6 to bridge the curriculum from the upper primary grades to lower secondary. This work has resulted in the following seven secondary textbooks:
PreTransition Mathematics (primarily for grade 6) is designed to help students solidify their arithmetic proficiency (particularly fractions and percents) while studying geometry, measurement, and basics of data analysis. Algebra is used for generalization and to solve simple equations.
Transition Mathematics (primarily for grade 7) connects applied arithmetic, algebraic concepts, and geometry. Variables are used to generalize patterns, in formulas, and as unknowns; variables are graphed on a number line and in the coordinate plane. Properties of geometric figures and two and threedimensional measurement provide opportunities to make connections with algebra.
Algebra (primarily for grade 8) introduces equations and functions (e.g., linear, quadratic, and exponential) via tables, graphs, and symbols. Statistics, probability, and geometry are used to make connections with algebraic concepts.
Geometry (primarily for grade 9) connects coordinates and transformations and includes early work with measurement and threedimensional figures. Proof is carefully sequenced from singlestep proofs to more complex proofs.
Advanced Algebra (primarily for grade 10) provides opportunities for students to develop facility with a range of functions, including the use of models to describe realworld situations. Geometry is assumed as a prerequisite so that geometric connections can be made.
Functions, Statistics, and Trigonometry (primarily for grade 11) has a strong modeling theme within the three content areas of functions, data analysis and probability, and trigonometry. Enough trigonometry is included to constitute a typical precalculus course with trigonometric functions.
Precalculus and Discrete Mathematics (primarily for grade 12) is designed to provide sufficient background in precalculus for a subsequent calculus course, including work with limits and the conceptual underpinnings of the derivative and integral, while also presenting discrete mathematics topics such as graph theory and logic. Formal proof is integrated throughout.
FEATURES OF THE UCSMP SECONDARY TEXTBOOKS
All of the secondary textbooks have several common features. For instance, lessons are generally written to be completed in a single day. Activities are embedded throughout the lessons to provide opportunities for students to engage actively with the content and to work collaboratively with peers. Appropriate technology is used in the curriculum, with graphing calculators assumed for all texts, dynamic drawing software assumed in Geometry, and computer algebra systems assumed beginning with Advanced Algebra.
The SPUR perspective guides the manner in which content is introduced and studied within lessons, chapters, and the text as a whole. The CARE perspective guides the manner in which the problem sets are constructed so that students have opportunities to interact with the content and practice important skills. We discuss both of these perspectives in detail from a conceptual basis in the next two sections, using examples from Transition Mathematics. We then follow this discussion with data from Transition Mathematics on how these approaches relate to students' achievement and to teachers' use of the textbooks.
The SPUR Perspective
The UCSMP textbooks embody a multidimensional approach to understanding mathematics known by the acronym SPUR for Skills, Properties, Uses, and Representations. Skills represent those procedures that students should master with fluency; they range from applications of standard algorithms to the selection and comparison of algorithms to the discovery or invention of algorithms, including procedures with technology. Properties are the principles underlying the mathematics, ranging from the naming of properties used to justify conclusions to derivations and proofs. Uses are the applications of the concepts to the real world or to other concepts in mathematics and range from routine "word problems" to the development and use of mathematical models. Representations are graphs, pictures, and other visual depictions of the concepts, including standard representations of concepts and relations to the discovery of new ways to represent concepts. At times, there is a fifth dimension, History, which includes cultural understanding of mathematics, such as names and origins of ideas and relationships between cultures and mathematics.
As authors develop chapters and write individual lessons, SPUR provides a focus and framework for their thinking. Authors ask themselves, "Which dimensions of understanding are addressed by this concept?" or "What skills are embodied in this concept? To what extent can properties or uses or representations provide another perspective to understand this concept?" Then, as authors develop the lesson, they attempt to address as many of these dimensions of understanding as possible; with rare exception, all dimensions are addressed within a chapter. In addition to guiding the development of individual lessons, each chapter review is structured around SPUR, with the objectives of the chapter classified as Skills, Properties, Uses, and Representations, as well as History when appropriate.
Consider, for instance, the case of the text Transition Mathematics, designed primarily for students in 7th grade who are at grade level, roughly 13 years old. Chapter 8 addresses multiplication in algebra. Table 1 contains the SPUR objectives for the chapter, including a reference for the lesson in which that objective can be found. Notice that several lessons contain objectives from more than one dimension of understanding. For instance, Lesson 83 has objectives that are skills, properties, and representations; Lesson 89 has objectives that are skills, properties, and uses. Figure 1 illustrates sample SPUR items from this review. (Note: Examples of SPUR questions from other UCSMP textbooks can be found in Hirschhorn, Thompson, Usiskin, & Senk (1995) and Senk (2003).)
Table 1. SPUR Objectives from Transition Mathematics Chapter 8: Multiplication in Algebra
Dimension of UnderstandingLabelLesson(s)ObjectivesSkillsA83Multiply positive and negative numbers.B86, 88Solve and check equations of the form ax = b and ax + b = c.C89Solve and check inequalities of the form ax + b < c.PropertiesD81, 83Recognize and use the RepeatedAddition Property of Multiplication and the Multiplication Properties of 1, 0, 1, and positive and negative numbers.E86, 89Recognize and use the Multiplication Properties of Equality and Inequality.UsesF82Apply the RateFactor Model for Multiplication.G86Find unknowns in real situations involving multiplication.H89Solve inequalities arising from real situations.I85Answer questions involving percents and combined percents.J84Calculate probabilities of independent events.RepresentationsK83Perform expansions or contractions with negative magnitudes on a coordinate graph.L87Graph equations of the form y = ax + b. From Viktora et al. (2008), pp. 549551.
Skills: Solve and check by substitution.
EMBED Equation.DSMT4
Properties: Multiple choice. Suppose x is positive and y is negative. Then xy is
A. always positive
B. zero
C. always negative
D. sometimes positive, sometimes negative.
Uses: Harry ate 30% of a pie. Ted ate 60% of what was left. How much of the whole pie did Ted eat?
Representations: Let K = (0, 6), I = (2, 0), T = (0, 4), and E = (2, 0). Graph KITE and its image under a size change of magnitude EMBED Equation.DSMT4 . ______________________________
Figure 1. Sample items illustrating the four dimensions of understanding from Transition Mathematics Chapter 8 Review: Multiplication in Algebra (Viktora et al. (2008), pp. 549551)
The use of SPUR also plays a role in assessments. For instance, the chapter tests in the textbook are intended as a practice for students prior to sitting for a chapter test administered by the teacher. Each chapter test appears in the text prior to the Review and is correlated to the SPUR objectives. This correlation ensures that every objective is assessed at least once. Further, it enables students and teachers to individualize endofchapter reviews. Students can focus on those objectives (and related dimensions of understanding) where they still need work, thus using their strengths in one area to enhance their understanding in other areas.
Additional supplementary practice materials available to teachers are also designed with this SPUR perspective in mind. By using SPUR as a framework for lesson development, assessments, reviews, and supplementary practice, the developers provide teachers with a balanced mathematics curriculum across the four dimensions of understanding and make it less likely that mathematics study becomes narrowly focused on skills.
The CARE Perspective
In many countries, textbooks consist of narrative prose which discusses the mathematics to be learned, and problems which provide an opportunity for students to practice and demonstrate their understanding of the mathematics. The problem sets developed as part of each UCSMP lesson are generally designed so that students should complete almost all of the problems in one assignment. This perspective contrasts with the perspective in most mainline commercial mathematics textbooks in the United States which contain lengthy exercise sets with the expectation that teachers will pick and choose problems (e.g., evens, odds, multiples of 3).
Each UCSMP problem set consists of four types of problems, known by the acronym CARE: Covering the Ideas, Applying the Mathematics, Review, and Exploration. Covering the Ideas problems focus on the essential ideas of the lesson; students who are successful with these items have grasped the core of the mathematics in the lesson. Many of the problems are similar to those in the lesson narrative. Applying the Mathematics problems provide an opportunity for students to apply the core ideas to new settings or in new ways that have not necessarily been discussed in the lesson; these problems are more challenging than those in the Covering the Ideas section. Review problems serve to give students a chance to continue working on concepts studied earlier in the chapter or earlier in the text. UCSMP uses a modified mastery learning approach in which students are not necessarily expected to master a concept on the first day it is discussed; rather, students continue to work on these concepts through the review problems that are part of every lesson. Finally, Exploration problems provide an opportunity for students to extend the mathematics of the lesson in an exploratory manner, sometimes with a historical connection or with exploration via the Internet; these problems are not intended to be done by every student every day. Many teachers choose some of the Exploration problems that they find particularly interesting to do with the class or they permit students to complete them for extra credit.
Within each problem set, the authors ensure that there are problems emphasizing skills, properties, uses, and representations according to the objectives of the lesson. Thus, the SPUR and CARE perspectives guide authors as they write the curriculum materials. SPUR ensures a balanced view of the mathematics. CARE ensures that problem sets cover a range of difficulty levels, from the core concepts to extensions of those concepts and opportunities to maintain and extend understandings from lesson to lesson. This is the format that has been used throughout all the textbooks developed by the secondary component for 25 years.
USE OF THE UCSMP TEXTBOOKS
As UCSMP develops its curriculum, whether a first edition or a third edition, it conducts a formative evaluation of the textbooks prior to their commercial publication. Materials are fieldtested from as demographically varied a sample of schools as possible. In addition to collecting data on students' achievement, teachers evaluate each chapter and report the lessons taught and the problems assigned. In the next sections, we describe how SPUR and CARE provide insights into students' achievement with and teachers' use of the curriculum.
Achievement in Terms of SPUR
Throughout the evaluation of the curriculum materials, SPUR provides a lens through which to consider the results. When UCSMP constructs tests, whether directly tied to chapters in the curriculum materials or as part of its evaluation studies, there is always a conscious decision to ensure that the assessment includes problems from all four dimensions. For instance, on a 40item endofyear multiple choice assessment of Transition Mathematics (Third Edition) covering variables and uses, equations and inequalities, measurement, transformations and symmetry, geometric figures and properties, and arithmetic, there were 10 (25%) skills problems, 5 (12.5%) properties, 10 (25%) uses, and 15 (37.5%) representations. For 6th and 7th grade students using Transition Mathematics, Table 2 shows their overall performance according to SPUR. The results show that performance is not the same across all four dimensions. In this instance, students at both grades did better on items focusing on properties, such as writing a rule to represent a table of values, identifying an arithmetic property of numbers, or identifying a property of rectangles. For many of the 7th grade students, achievement with representations of concepts was higher than achievement with skills or uses. Nevertheless, the results suggest that students can study a curriculum emphasizing multiple dimensions of understanding and maintain skill proficiency.
Table 2. Mean Percent and Standard Deviation on Items by SPUR for Students Using UCSMP Transition Mathematics (Third Edition)
Grade nSkills
(10 items)Properties(5 items)Uses
(10 items)Representations
(15 items)mean %sdmean %sdmean %sdmean %sd695791585167315781171426024702554216520Based on Thompson and Senk (in preparation).
SPUR also provides insights into how different curricula may influence the extent to which a balanced view of mathematics is pursued. In its evaluation studies, UCSMP administers pretests and posttests not only to students using the UCSMP curriculum but also to students using the comparison curriculum already in place at the school. When the posttests are administered, both UCSMP and comparison teachers are asked to indicate if they taught or reviewed the content their students needed to answer each of the posttest items. Their responses are used to control for differences in opportunities to learn by considering items common to all the teachers in the study, regardless of the curriculum being used.
Over studies for multiple courses since the 1990s, we have found that skills dominate these common items. For instance, among 8 teachers in a study of UCSMP Geometry, four of whom were teaching the UCSMP curriculum, content for only 19 of 40 items on a standardized test was reported as taught by all the teachers; 12 (63%) of these were skills and 6 (32%) were properties (Thompson et al., 2003). Likewise, among 8 teachers in a study of UCSMP Advanced Algebra, four of whom were teaching the UCSMP curriculum, content for only 15 of 36 items on a projectconstructed test was taught by all; 8 (53%) of these were skills items, 5 (33%) were representations, and 2 (13%) were properties (Thompson et al. 2001). So, if a textbook does not provide a balanced view of mathematics across dimensions of understanding, it is less likely that such balance will occur in instruction, and subsequently, in assessment.
Teachers' Use of Textbooks and CARE Problems
It is natural to wonder if teachers use the UCSMP materials as intended. As part of the formative evaluation of the Third Edition materials, teachers rated each lesson within a chapter, and indicated the problems from that lesson that were assigned to students. Table 3 reports the percent of lessons covered as well as the percent of Covering the Ideas, Applying the Mathematics, and Review problems that seven UCSMP Transition Mathematics teachers assigned from the lessons they taught; assignment of the Exploration questions is not considered as these problems are not intended to be assigned to all students.
Table 3. Percent of Covering, Applying, and Review Problems Assigned From the Lessons Taught by Transition Mathematics Teachers
TeacherPercent of Lessons Taught in Text
(n = 141)Covering the IdeasApplying the MathematicsReviewA79978991B73907649C165928748C255948682D67915650E81936527F89978277Note: Teachers C1 and C2 are in the same school. Based on Thompson and Senk (in preparation).
First, note the range in the percent of lessons within the text that teachers taught. Teacher C2 taught slightly more than half of the lessons and Teachers C1 and D taught about twothirds of the lessons. These teachers did not address any of the content in the final third of the 12 chapters. In contrast, Teacher F taught almost 90% of the lessons.
Second, the results provide insight into the ways in which teachers used the problem sets as intended. All teachers assigned more than 90% of the Covering the Ideas problems, ensuring that students had an opportunity to engage with the core mathematics concepts of the lessons taught. However, there is more variability in assigning the Applying the Mathematics and Review problems. In particular, Teachers D and E assigned considerably fewer of the Applying the Mathematics problems than the other teachers. Hence, their students may have had less opportunity to engage with novel or challenging problems where they were expected to determine how to use the mathematics concepts they studied. Teachers B, C1, and D assigned only about half of the review problems and Teacher E assigned only about a quarter of the reviews. The textbook authors consider the review problems to be a critical component of the modified mastery learning approach because these problems provide an opportunity for students to develop fluency with the skills and concepts within the lessons in a chapter and to maintain proficiency throughout the school year. Hence, it is questionable whether these students had sufficient opportunity to develop and maintain the desired skills.
CONCLUSION
This paper describes two characteristics of UCSMP textbooks: SPUR and CARE. The SPUR perspective enables authors to present a balanced view of mathematics, that is, a view of mathematics that is more than just fluency with skills. It also provides opportunities for teachers to structure lessons that help students understand these different aspects of mathematics. SPUR provides opportunities for students who have different strengths as learners to make sense of mathematics in ways that are meaningful to them and to build competence in other aspects of understanding.
The CARE perspective scaffolds the mathematics from core essentials to more complex aspects. In particular, the review questions help students deepen their understanding over the course of a year and beyond.
Finally, the data in this paper indicate that teachers do not always use materials as intended by curriculum developers. More research into how and why teachers make their decisions about the use of textbooks is needed.
REFERENCES
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel.
Gravemeijer, K. (1994). Developing realistic mathematics education. Utretcht: Freudenthal Institute.
Hirschhorn, D. B., Thompson, D. R., Usiskin, Z., & Senk, S. L. (1995). Rethinking the first two years of high school mathematics with the UCSMP. Mathematics Teacher, 88, 640647.
Krutetskii, V. (1976). The psychology of mathematical abilities in school children. (translated by J. Teller. Edited by J. Kilpatrick and I. Wirszup). Chicago, IL: University of Chicago Press.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Seah, W. T., & Bishop, A. J. (2000). Values in mathematics textbooks: A view through two Australasian regions. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA. (ERIC Reproduction Services ED 440 870)
Senk, S. L. (2003). Effects of the UCSMP secondary school curriculum on students' achievement. In Senk, S. L. & Thompson, D. R. (Eds.), Standardsbased school mathematics curricula: What are they? What do students learn? (pp. 425456). Mahwah, NJ: Lawrence Erlbaum.
Thompson, D. R., & Senk, S. L. (in preparation). An evaluation of the Third Edition of UCSMP Transition Mathematics. Chicago, IL: University of Chicago School Mathematics Project.
Thompson, D. R., Senk, S. L., Witonksy, D., Usiskin, Z., & Kealey, G. (2001). An evaluation of the Second Edition of UCSMP Advanced Algebra. Chicago, IL: University of Chicago School Mathematics Project.
Thompson, D. R., Witonsky, D., Senk, S. L., Usiskin, Z., & Kealey, G. (2003). An evaluation of the Second Edition of UCSMP Geometry. Chicago, IL: University of Chicago School Mathematics Project.
Usiskin, Z. (2003). A personal history of the UCSMP secondary school curriculum: 19601999. In Stanic, G. M. A., & Kilpatrick, J. (Eds.), A history of school mathematics, Volume 1 (pp. 673736). Reston, VA: National Council of Teachers of Mathematics.
Viktora, S. S., Cheung, E., Highstone, V., Capuzzi, C. R., Heeres, D., Metcalf, N. A., Sabrio, S., Jakucyn, N., & Usiskin, Z. (2008). The University of Chicago School Mathematics Project: Transition Mathematics. Chicago, IL: Wright Group/McGraw Hill.
Watanabe, T. (2001). Content and organization of teachers' manuals: An analysis of Japanese elementary mathematics teachers' manuals. School Science and Mathematics, 101(4), 194205.
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