Mathematics Curriculum Implementation and Linear Functions in Secondary Mathematics: Results from the Comparing Options in Secondary Mathematics Project Óscar Chávez Douglas A. Grouws James E. Tarr Dan J. Ross Melissa D. McNaught University of Missouri — Columbia Please address all correspondence to: Óscar Chávez University of Missouri College of Education 121E Townsend Hall Columbia, MO 65211-2400 578-882-4521 [email protected]Paper presented at the Annual Meeting of the American Educational Research Association San Diego, April 13–17, 2009
Transcript
Mathematics Curriculum Implementation and Linear Functions in Secondary Mathematics: Results from the Comparing Options in Secondary Mathematics Project
Óscar Chávez Douglas A. Grouws
James E. Tarr Dan J. Ross
Melissa D. McNaught
University of Missouri — Columbia Please address all correspondence to: Óscar Chávez University of Missouri College of Education 121E Townsend Hall Columbia, MO 65211-2400 578-882-4521 [email protected]
Paper presented at the Annual Meeting of the American Educational Research Association
San Diego, April 13–17, 2009
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Mathematics Curriculum implementation and Linear Functions in Secondary Mathematics: Results from the Comparing Options in Secondary Mathematics
Project
Óscar Chávez Douglas A. Grouws
James E. Tarr Dan J. Ross
Melissa D. McNaught
University of Missouri The purpose of this paper is to report findings related to secondary mathematics teachers’ use of curriculum materials. In particular, we report results of a content analysis of these textbooks where we examined differences and similarities in pedagogical approaches and content selection regarding the topic of linear functions in order to document how these teachers use their district-adopted textbooks to provide students opportunities to learn the topic of linear functions. This study is part of an NSF-funded research project: The Comparing Options in Secondary Mathematics: Investigating Curriculum (COSMIC). This project involves a three-year longitudinal comparative study of integrated mathematics curricula and subject-specific mathematics curricula on student learning at the high school level (McNaught, Tarr, & Grouws, 2008).
Schmidt, McKnight, and Raizen (1997) have described the role of textbooks in the U.S., as “bridges between the worlds of plans and intentions, and of classroom activities shaped in part by those plans and intentions” (p. 53). Even if students are the intended audience for curriculum materials, teachers mediate between their students and their textbook. As teachers use curriculum materials to teach mathematics, they are expected to clarify, expand upon, or smooth out difficulties in the interaction with the text (Love and Pimm, 1996). Teachers’ interactions with curriculum materials involve a decision-making process shaped by their knowledge, skills, and views of mathematics and mathematics teaching (Chávez, 2006; Stein, Remillard, & Smith, 2007) but it is also the last part of a process of “narrowing down from the universe of possible activities to those considered desirable for use in the classroom” (Bishop, 1988). As Bishop points out, the first stages of this process are established by the government, the state, or the school, well before the teacher is able to make any decision. But as teachers interpret, understand, and use the ideas contained in curriculum materials, they make decisions regarding pacing, sequence, supplementation and about the tasks that they choose to present to their students relying heavily on their mathematics textbooks to make those decisions (Grouws & Smith, 2000; Tarr et al., 2006). As a result, students’ opportunity to learn (OTL) is affected by such decisions. In this paper we consider two decisions: (1) the decision made by the developers about what topics to include in a textbook and how to present them, and (2) the teacher’s selection of topics and tasks from the textbook to use with his or her students and how to use these tasks. In the findings reported here, we narrow the focus to a particular topic —linear functions— because of its importance in subsequent learning
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of secondary mathematics, and because it was common to the two types of curriculum materials.
The data we collected during the study served to document the extent to which a causal link could be established between the textbooks used by the teachers and student outcomes. Rigorous criteria for evaluating curricular effectiveness (NRC, 2004) have elevated the importance of documenting the use of mathematics curriculum materials in order to examine their resulting influence on student achievement. Therefore, we do not equate a “faithful” implementation with good or exemplary teaching practices.
Methodology
Participants The teachers participating in the study reported here teach in schools in five states where two parallel curriculum options exist: a sequence of courses that reflect an integrated approach to the content, and the more conventional sequence of Algebra I, Geometry, and Algebra II courses. In these schools, students have a choice between courses with the subject-specific approach or courses with the integrated approach. The sample is comprised of 113 classes (60 integrated and 53 single-subject) and approximately 2600 students. In this paper we report only our analysis of teacher data.
Data sources We conducted a content analysis of the textbooks used by the teachers in the study. In particular, we examined how the topic of linear functions developed in both the curriculum materials with an integrated approach (Core-Plus Mathematics Course 1) and in those developed with a subject-specific organization (Algebra I textbooks by different publishers). We documented the extent to which the topic was present in each textbook and the pedagogical approaches.
Table 1. Number of teachers using different textbook series
Textbook No. of teachers
Glencoe Algebra I 10
McDougal Littell Algebra I 6
Interactions 4
Holt, Reinhart, Winston Algebra I 1
Prentice Hall Algebra I 2
Core-Plus Mathematics Course 1 21
Teachers in our study who used subject-specific curriculum materials used Algebra I textbooks produced by different publishers. While these books are not identical, all of them treat the topic of linear functions in a similar way. Glencoe Algebra I is usually considered as one of the high school algebra textbooks with a larger market share, and it was the most commonly used subject-specific textbook in our sample. For these reasons,
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most of the content analysis for the subject-specific curriculum materials was conducted on Glencoe Algebra I. Table 1 shows that a majority of the teachers in our study who were using subject-specific textbooks used either Glencoe Algebra I or McDougal Littell Algebra I (see Table 1). All teachers using integrated textbooks used Core-Plus Mathematics Course 1. Our discussion of implementation data will focus on the data from teachers using these three textbooks.
Teachers’ use of textbooks
Three instruments were developed to facilitate teachers’ self-reporting data related to their use of the textbook: (1) a written survey (2) a Table of Contents Record and (3) a Textbook-Use Diary. The written survey served to gather information regarding teacher demographic data and some information about the use of their textbook during instruction. In the Table of Contents Record, teachers provided information regarding textbook coverage and the extent to which the content was altered for each lesson. The Textbook-Use Diary focused on a particular chapter associated with linear equations. Teachers used this instrument to provide information about supplementary materials used, specific examples and textbook pages utilized during instruction, etc. McNaught, Tarr, & Grouws (2008) provided a detailed description of these instruments.
In addition to the collection of self-report data, COSMIC Project personnel observed each teacher three times during the school year. Upon completion of each observation, the observer completed a summary form where he or she recorded whether specific features of the curriculum materials were used in the classroom and whether the teacher’s teaching practices were consistent with the developers’ recommendations, as expressed in the notes and suggestions to teachers included in the Teacher’s Edition of each textbook. These data provided a way to triangulate the teachers’ self-reported data. Not all classroom observations took place when teachers were teaching content related to linear functions.
The findings we report in this paper correspond to data gathered from the Table of Contents Records and the classroom observations.
Results
We compared the treatment of linear functions in Core-Plus Mathematics Course 1 and Glencoe Algebra I. The following is a description of our content analysis.
Linear functions in an integrated mathematics curriculum The first unit in Core-Plus Mathematics Course 1 is titled Patterns in Data. The content of this unit introduces the basis for a consistent approach that is followed throughout the program (Courses 1-3), namely the use of problems set in realistic contexts to help students develop mathematical concepts and methods. In Patterns in Data, students use actual data and different graphical representations to develop models. These models are examined to determine similarities and differences in order to classify them as linear, quadratic, etc. The introduction of these models is informal, the defining characteristics of these models and the corresponding terminology is formalized later on. The emphasis on connecting different representations is deliberate and occurs throughout the algebra strand in the different courses. As Huntley et al. (2000) observed, “[A] sense of connection among representations (such as tables, graphs, and expressions) for the
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concept of a linear function, for example, is enabled through a variety of experiences with representational activities and applied-problem settings” (p. 332). Linear functions and recursive formulas used to describe linear functions are introduced in Unit 2 within the context of studying change. Linear (and nonlinear) patterns are described in terms of tables, (recursive) rules, and graphs.
Unit 3 focuses on linear models. Functions are not formally defined, and therefore the authors simply refer to “linear models” throughout the unit. In this unit, the authors introduced the concept of slope, intercepts, closed and recursive forms for linear equations, solving linear equations, and determining lines of best fit. The teachers guide is explicit about the goals for the unit: “The emphasis in this unit is on linear models of real-world relations, not on solving equations or simplifying linear expressions. The approach in the unit assumes that an understanding of linearity in realistic situations will make the manipulating of symbols more meaningful” (p. T153). There is an explicit goal for students to be able to explain relationships between the graph, the equation, and a table of values for a linear model. The concepts of slope and intercepts are tied to their meaning within the realistic contextual problems. The unit comprises four lessons, each of which spans several days: Predicting from Data; Linear Graphs, Tables, and Rules; Linear Equations and Inequalities; and Looking Back. Each of these lessons includes up to four investigations.
Linear functions in subject-specific mathematics curricula
In Glencoe Algebra I, linear equations are introduced in chapter 3. The emphasis is on the acquisition of skills related to proficient symbol manipulation. The focus of the chapter is on solving linear equations, and detailed methods are introduced for doing so: using addition and subtraction, using multiplication and division, and multi-step equations. Each of these methods is treated separately in different sections. Each section includes worked out examples followed by a number of similar exercises that students are expected to complete. Chapter 4 introduces graphs of relations and functions. There is no distinct emphasis on linear functions, but rather the emphasis is placed on the connection between equations in two variables and their representations on the coordinate plane. One section deals with Graphing Linear Equations; one with Arithmetic Sequences, where linear patterns are identified; and one is about Writing Equations from Patterns, where some of these patterns lead to linear functions. In the next chapter, the concept of slope is introduced, and different forms of the equation of a line are studied in detail. The last two sections of Chapter 5 extend linear functions by examining connections to other areas of mathematics. For example, Section 5-6 is titled Geometry: Parallel and Perpendicular lines and section 5-7 is titled Statistics: Scatter Plots and Lines of Fit. The concept of slope in integrated curricula In Core-Plus Mathematics Course 1, mathematical concepts and methods are introduced through problems based on realistic contexts. Core-Plus introduces linear functions (and other functions) as models for studying data. The emphasis is on the use of different representations of linear relationships. In this sense, applications precede the formalization of definitions and procedures. In addition, change is inherent to the context in which these data are presented. For this reason, Core-Plus Mathematics Course 1 uses recursive expressions as a means to introduce the concept of function.
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The concept of slope in subject-specific curricula The authors of the subject-specific curriculum materials treat the topic of linear functions formally and they place emphasis on fluency with symbols and different forms of the equation of a line. The use of linear functions as models for contexts in which change takes place is usually left to the end of a section or chapter as an example of applications. In each section, definitions and examples are included and students are expected to solve exercises and problems where the concepts introduced and the methods exemplified can be applied. This approach was consistent across the subject-specific curriculum materials included in our study.
Summary of the differences in instructional approaches These different approaches have implications on how other important topics are introduced. Most notably, slope is presented as an attribute of the geometric characteristics of a line on the coordinate plane in the subject-specific books, while in the integrated materials it is a measure of change. Thus, the concept of slope provides a compelling illustration of the consequences of following these different approaches. For example, in the teacher’s guide of Core-Plus Mathematics Course 1, an explicit distinction is made for teachers who may be familiar with other Algebra textbooks and are now using Core-Plus Mathematics Course 1:
There are many different possible forms of equations for linear relations, each carrying information about the relation in its own way. While it is not uncommon for algebra texts to march through a sequence of exercises developing and practicing each form (“standard,” point-slope, slope-intercept, two-point, etc.), we have chosen primarily one form that we believe is the most natural in modeling linear relations. The focus on change is central. In the form y = a + bx, the value of y starts at a (when x = 0) and changes in steps of b as x increases in unit increments. (p. T181)
In Core-Plus Mathematics Course 1 the study of change takes precedence over geometric attributes, as is exemplified in the following excerpt from the teacher’s guide to Course 1:
By now you’ve probably noticed the following key features of linear models and their graphs.
• Linear models always have a constant rate of change. That is,
€
ΔyΔx
is a
constant.
• The constant rate of change can be seen in the slope of the linear graph. The slope is the direction and steepness of a walk along the graph from left to right.
• The y-intercept of the linear graph is the point where the graph intersects the y-axis. (p. 186)
On the other hand, Glencoe Algebra I discusses slope in terms of the geometry of the line:
The slope of a line is a number determined by any two points on the line. This number describes how steep the line is. The greater the absolute value of the
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slope, the steeper the line. Slope is the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move from one point to the other. (p. 256)
Figure 1 shows a diagram that maps the conceptual development of slope in Core-Plus Mathematics Course 1 and Glencoe Algebra I.
Figure 1. The concept of slope as presented in the two types of curriculum materials.
Although not every topic in Core-Plus Mathematics Course 1 can be matched with a
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topic in Glencoe Algebra I, it is apparent that the order of the conceptual progression develops in opposite directions. On the one hand we have a progression from problems and representations of data to linear functions, and on the other a progression from definition to applications.
Implementation data
In the Table of Contents Record (TOC), teachers indicated whether they used their textbook or other resources to teach each section in their book. In Table 2 we summarize the corresponding data from the teachers using Glencoe Algebra I. It is clear that most teachers using Glencoe Algebra I taught linear functions using their textbooks as their main resource, or in conjunction with other resources. It is important to note that the only sections that were skipped were either investigations or graphing calculator activities.
Table 2. Table of Contents Record data for Glencoe Algebra I Section Primarily
from textbook
Some supplementation
Primarily from alternative to
textbook
Did not teach content
4-5 Graphing Linear Equations
6 4
4-7 Arithmetic Sequences 7 2 1
4-8 Writing Equations from Patterns
6 3 1
5-1 Slope 7 3
5-2 Slope and Direct Variation
7 3
Investigating Slope-Intercept Form
2 3 5
5-3 Slope-Intercept Form 3 7
Graphing Calculator Investigation: Families of Linear Graphs
3 2 5
5-4 Writing Equations in Slope-Intercept Form
3 7
5-5 Writing Equations in Point-Slope Form
4 5 1
5-6 Geometry: Parallel and Perpendicular Lines
7 3
5-7 Statistics: Scatter Plots and Lines of Fit
4 5 1
Graphing Calculator Investigation: Regression and Median-Fit Lines
2 2 2 4
The content in McDougal-Littell Algebra I is organized in a similar way, except that the content is spread over more lessons. Teachers using McDougal-Littell Algebra I also reported using it as the main resource for most of the lessons.
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Table 3. Table of Contents Record data for McDougal-Littell Algebra I Section Primarily
from textbook
Some supplementation
Primarily from alternative to
textbook
Did not teach content
4-1 Coordinates and scatter plots
5 1
Graphing calculator: Graphing a scatter plot
2 4
4-2 Graphing linear equations 3 3 4-3 Quick graphs using
intercepts 3 3
4-4 Slope 3 3 Concept activity:
Investigating slope 1 3 1 1
4-5 Direct variation 2 1 2 1 4-6 Quick graphs using slope-
intercept form 4 1 1
Concept activity: Graphing families of linear equations
1 2 2 1
Graphing calculator: Graphing a linear equation
2 4
4-7 Solving linear equations using graphs
4 1 1
5-1 Writing equations in slope-intercept form
4 2
5-2 Writing equations given the slope and a point
2 4
5-3 Writing equations given two points
3 3
5-4 Fitting a line to data 2 3 1 Graphing calculator:
Best-fitting lines 3 2 1
5-5 Point-slope form of a linear equation
3 1 1 1
5-6 The standard form of a linear equation
3 2 1
Concept activity: Investigating the standard form of a linear equation
2 1 3
5-7 Predicting with linear models
2 2 1 1
Concept activity: Investigating linear modeling
1 3 2
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A different perspective can be appreciated when looking at what individual teachers decided to teach from the textbook, teach using supplementation, or omit. The following figures illustrate the data reported by teachers using Algebra I textbooks.
In Figures 2 and 3, each bar represents the lessons on linear functions taught by a teacher, who is identified by a number and an initial for the textbook used. Each shaded rectangle represents a lesson on linear functions, as listed in Tables 2 and 3, respectively, and the shade indicates whether the content was taught primarily from the book, with any supplementation, or omitted (as indicated in the key). Teachers grouped within a bracket teach in the same school.
G 52
G 53
content taught primarily from textbookcontent taught from the textbook with some supplementationcontent taught primarily from an alternative sourcecontent not taught
G 51
G 47
G 46
G 56
G 58
G 41
G 11
G 12
Figure 2. Lessons taught and resources used by teachers using Glencoe Algebra I to teach linear functions.
M 72
M 73
content taught primarily from textbookcontent taught from the textbook with some supplementationcontent taught primarily from an alternative sourcecontent not taught
M 21
M 22
M 27
M 28
Figure 3. Lessons taught and resources used by teachers using McDougal-Littell Algebra I to teach linear functions.
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These data suggest that decisions about what to teach, what to supplement, and what to omit are not taken in an entirely individual way, for teachers in the same school exhibit a similar pattern, identical in some cases. Some of the resources used to supplement, as reported by the teachers, included teacher-made worksheets or test preparation materials, which may be related to needs created by state tests, as perceived by school administrators or curriculum coordinators.
In Core-Plus Mathematics Course 1, lessons usually span several days, and each lesson includes several investigations. In Table 4, we have enumerated the investigations within each lesson, to provide a finer grained analysis of the topics taught.
Table 4. Core-Plus Mathematics Course 1
Unit-Lesson-Investigation Primarily from
textbook
Some supplementatio
n
Primarily from alternative to
textbook
Did not teach
content
3-1 Predicting from Data
Investigation 1 (Examining data from an experiment)
11 6 2 2
Investigation 2 (Examining data from a survey)
11 5 2 3
Investigation 3 (Fitting lines to scatter plots)
10 6 3 2
3-2 Linear Graphs, Tables, and Rules
Investigation 1 (Equations of the form y = a + bx)
14 5 2
Investigation 2 (Finding linear equations from a graph, using the y-intercept and slope, from two points, and using a graphing calculator)
11 6 4
Investigation 3 (Graphs of lines and their equations)
9 5 4 3
3-3 Linear Equations and Inequalities
Investigation 1 (Using tables and graphs to solve equations)
10 6 4 1
Investigation 2 (Solving linear equations)
9 5 4 3
Investigation 3 (Using graphing calculators to solve systems of linear equations)
11 6 4
Investigation 4 (Equivalent equations)
10 5 5 1
3-4 Looking Back (Chapter summary)
2 3 1 15
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Teachers using Core-Plus Mathematics Course 1 are more likely than their Algebra I counterparts to use some degree of supplementation. As many as half of the teachers using Core-Plus Mathematics Course 1 used supplements to teach linear functions. Data collected in the Textbook-Use-Diaries suggest that these teachers used worksheets and resources that emphasized practice, rather than investigations. These teachers also reported using test preparation materials from other curriculum series.
School policy seems to influence what gets taught and what is omitted by teachers using Core-Plus Mathematics Course 1. Figure 4 displays implementation data for the unit on linear functions. Each shaded rectangle represents an investigation from the unit on linear functions, as listed in Table 4, and the shade indicates whether the content was taught primarily from the book, with any supplementation, or omitted (as indicated in the key). Teachers grouped within a bracket teach in the same school.
content taught primarily from textbookcontent taught from the textbook with some supplementationcontent taught primarily from an alternative sourcecontent not taught
C 84
C 82
C 85
C 83
C 11
C 73
C 71
C 72
C 74
C 93
C 21
C 22
C 23
C 26
C 29
C 27
C 52
C 47
C 56
C 57
C 59
Figure 4. Investigations and resources used by teachers using Core-Plus Mathematics Course I to teach linear functions.
Although there is some variability by school, it is clear that in some schools the Core-Plus materials are supplemented often, to different degrees.
McNaught et al. (2008) introduced several measures of implementation by comparing what is taught, what is omitted, and what is taught with supplementation. The Opportunity to Learn (OTL) index indicates what content was taught, and it is the ratio between lessons whose content was taught primarily from the textbook or from other
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sources and the total number of lessons in the textbook. Therefore, the OTL index represents the percent of lessons taught. The Textbook Content Taught (TCT) index is a calculated by weighting each of the first three options provided to the teachers in the Table of Contents Record. A weight of 1 was given to a lesson taught primarily from the textbook. A weight of two-thirds and one-third, respectively, was given to lessons taught with some supplementation or primarily from an alternative source. Lessons not taught were not considered. Reported on a scale from 0 to 100, the TCT index indicates the degree to which the content actually taught was taught from the book.
Table 5. Implementation indices (OTL and TCT), by teacher. Textbook Teacher
1Core Plus Mathematics Course 1 2 Glencoe Algebra I 3 McDougal-Littell Algebra I
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While McNaught et al. used these indices to document the degree of implementation throughout a whole course, we used them here to compare the emphasis given to the lessons on linear functions by the teachers in this study. In Table 5, we summarize the OTL index and the TCT index relative to the lessons on linear functions (LF). In order to compare these measures with the implementation indices for the whole course, we included the corresponding indices. The last two columns include emphasis indices (Tarr et al., 2006), with which we compare the emphasis given to linear functions, relative to the implementation indices for the whole course.
The OTL Emphasis index is the ratio between the OTL index for the lessons on linear functions and the OTL index for the whole course. An index of 1 would indicate that the emphasis given to linear functions was the same emphasis the topic had originally in the curriculum materials. An OTL Emphasis index greater than 1 would mean that a teacher gave greater attention to the lessons on linear functions than would be expected given the composition of the textbook. The TCT Emphasis index is the ratio between the TCT relative to linear functions and the TCT index for the whole course. This index provides a measure of the extent to which the teacher relied on the textbook for the content she or he taught to teach linear functions compared with her or his use of the textbook for the rest of the content taught. Thus, a TCT Emphasis index greater than 1 would indicate that a teacher used his or her textbook to teach the lessons on linear functions more often than she or he used it to teach other lessons.
Every one of the teachers in this study gave an equal or greater emphasis to linear functions than would be expected, given the content of the curriculum materials. On the other hand, teachers were about as likely to use their textbook to teach linear functions as other topics in the course, with the exception of some teachers using Core-Plus Mathematics Course 1.
Not many observations were conducted when teachers were teaching linear functions. Tables 6–8 summarize the data collected during these observations regarding textbook use. In most cases, teachers used their textbooks to teach their lessons. The most common supplementary material was teacher-made worksheets. Only two teachers, both of them using Core-Plus Mathematics Course 1, used other textbooks as resources, specifically Algebra I textbooks.
Table 6. Summary of curriculum materials used by teachers using Glencoe Algebra I during observations Teacher School Lesson Materials used Content
Table 7. Summary of curriculum materials used by teachers using McDougal-Littell Algebra I during observations. Teacher School Lesson Materials used Content
Table 6. Summary of curriculum materials used by teachers using Core-Plus Mathematics Course 1 during observations Teacher School Lesson Materials used Content
For each of these observations, observers made a determination of Content Fidelity and Presentation Fidelity (see Tables 6–8). The former refers to what mathematics content in the intended curriculum was taught during the observed lesson. The latter refers to how the content was presented in relation to the structure of the curriculum materials and the recommendations of the developers. Our content analysis informed the determination of this measure. These measures are intended to be descriptive, rather than measures of quality. The rubric that observers used describes in detail what these measures mean, as indicated in Figure 5.
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Figure 5. Rubric for assigning Content and Presentation Fidelity ratings.
Tables 6–8 include the Content and Presentation Fidelity measures for each observation. All teachers had a moderate to high Content Fidelity measure. The average Presentation Fidelity was low to moderate. In other words, while teachers used the textbook to determine the content of their lessons, they made decisions that departed from the recommendations for teachers set forth by the developers of their textbooks. In some cases they used other resources to teach the same content, in others they decided not to pursue activities that were recommended by the textbook authors.
Discussion
The topic of linear functions is appropriately considered an important component of the preparation in algebra that high-school students are expected to receive. Whether teachers use an integrated curriculum or a subject-specific course in algebra, linear functions occupy a prominent place in their curriculum materials. Unlike other topics, such as probability and statistics, lessons on linear functions are unlikely to be omitted. Indeed, some teachers give linear functions a greater emphasis than would be expected given the composition of the book. At the same time, it cannot be assumed that the sequence of concepts and activities, as outlined in Figure 1, will reflect what students will experience during instruction. With few exceptions, all teachers in this study did some supplementation, which in some cases shifted the priorities that curriculum designers have established in developing curriculum materials. As pointed out by McNaught et al. (2008), lower Presentation Fidelity ratings are an indication that a change in curriculum materials does not imply a resulting change in teaching practices.
As long as the purpose of the materials used to supplement is to provide students with more opportunities for practice or to prepare for state tests, teachers using subject-specific curricula will depart little from the pedagogical approach and the learning sequence in their curriculum materials. In contrast, teachers using integrated curricula may alter in a more significant way the pedagogical goals implicit in their textbooks whenever they substitute other resources for an investigation, or a whole lesson, in their
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materials. This is not good or bad in itself, but it is an important consideration when comparing the effectiveness of curriculum materials. An understanding of what Tarr et al. (2006) called a discordant implementation of mathematics curricula is an important component of studies on curriculum.
Our study has examined how an important topic, linear functions, is developed conceptually in different types of curriculum materials and also how teachers have used these materials to teach this particular topic. Mathematics textbooks have been used as an intervention strategy to improve student learning. It is therefore crucial that policy makers, school and districts personnel make decisions based on sound research rather than on surface characteristics of curriculum materials. Curriculum developers need also to be informed about how their decisions on curriculum design impact mathematics teachers’ practices. According to Remillard (2005), much of the research on teachers’ use of curriculum materials is conducted within an underdeveloped theoretical and conceptual terrain. We believe that this study provides empirical results that may contribute to the development of theoretically grounded analytical and methodological frameworks to guide future research on mathematics curriculum.
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