# Components of probability lessons in textbooks.

Abstract

A study designed to examine the structure of probability lessons within mathematics textbooks written for students in the middle grades is described. Lesson components and sequences from eight textbook series, spanning four decades, are identified and compared. The four popular textbook series had fairly similar lesson sequences. Lesson sequences within the alternative textbook series varied considerably. One component - reflecting on the mathematics of the lesson--was unique to the Connected Mathematics series. Recommendations for curriculum development are offered.

Introduction

Over a decade ago, Shaughnessy (1992) stated, "No comprehensive surveys of how much probability and statistics is taught in grades K-4, or 5-8 is currently available, but until very recently one could confidently say practically none" (p. 466). Examining the content of mathematics textbooks provides a lens to view what is potentially taught in school mathematics classrooms. Due to the growing emphasis on the topic of probability in recommendations from professional organizations over the past half-century (e.g., College Entrance Examination Board, 1959; National Council of Teachers of Mathematics [NCTM], 1980, 1989, 2000), one might reasonably expect to observe changes in textbooks. Unfortunately, there is little research to support this expectation. To date, there have not been any systematic examinations of the composition of textbooks as they have evolved over time, particularly in relation to the mathematical branch of probability.

In this paper, I report on the findings of a study designed to examine the structure of probability lessons within mathematics textbooks written for students in the middle grades. I analyzed lessons that focused on probability topics and identified the components of those lessons and how those components were sequenced for lessons in a particular series. In particular, I sought to answer the research question: How are probability lessons structured, and how have the structures of probability lessons changed over the past 48 years?

Theoretical Framework

The research design of this study is informed by numerous theoretical considerations that collectively serve to influence the selection of curriculum analysis procedures. The U.S. mathematics curriculum is influenced by many factors, including the recommendations of prominent national organizations (e.g., NCTM, 1989) and high-profile criticisms (McKnight et al., 1987; Schmidt, McKnight, & Raizen, 1997). Therefore, textbooks should be analyzed from a historical perspective in an attempt to gauge how textbooks may have changed in response to these factors.

Given that textbooks are ubiquitous elements of U.S. mathematics classrooms (Tyson-Bernstein & Woodward, 1991) and are frequently used by both teachers and students (Grouws & Smith, 2000; Weiss, Banilower, McMahon, & Smith, 2001), it is important that the most "popular" textbook series are included within the sample of textbooks to be analyzed - we should examine the books that were most likely to be used by students. Textbook series that were written as alternatives to the "popular" textbook series should also be examined, in an effort to gauge trends within and across time periods.

Finally, not only have recent mathematics reform efforts advocated broadening the scope of the curriculum to include probability, but a robust body of research (Garfield & Ahlgren, 1988; Konold, Pollatsek, Well, Lohmeier, & Lipson, 1993; LeCoutre, 1992; Shaughnessy, 1992) has revealed that probability misconceptions are both prevalent and persistent among people of all ages. For these reasons, it is important to study the components and structure of probability lessons in middle grades mathematics textbooks with an eye for how these factors may support or hinder the process of addressing and correcting misconceptions.

Sample Selection and Methodology

The sample for this study was comprised by textbooks written for students in grades 6-8 from two series, one Popular and the other Alternative, within each of the four most recent eras of mathematics education: New Math, Back to Basics, Problem Solving, and Standards. It is difficult to determine the precise beginning and end of these eras, and a significant event that marks the start of a new era (e.g., the publication of the Curriculum and Evaluation Standards for School Mathematics in 1989) does not necessarily immediately impact the textbooks that are published that year or the next. Furthermore, change from one era to the next is not instantaneous, and there exists a period of transition between eras. For the purposes of my study, however, it is necessary to define some time frame for each era. Therefore, I refer to the years 1957-1972 as the New Math era, 1973-1983 as the Back to Basics era, 1984-1993 as the Problem Solving era, and 1994-2004 as the Standards era.

Market share data were used to identify the popular series from the three most recent eras. The popular series from the New Math era and the four alternative series were identified through a "professional consensus" of mathematics educators affiliated with the Center for the Study of Mathematics Curriculum and familiar with middle grades mathematics curricula during these eras. The textbook series included in the sample for this study are listed below[I], along with the number of probability lessons within each series.

* New Math-Popular (11 lessons): Modern School Mathematics: Structure and Use 6 and Modem School Mathematics: Structure and Method 7 & 8

* New Math-Alternative (5 lessons): Mathematics for the Elementary School, Grade 6 and Mathematics for Junior High School, Vols. I & II

* Back to Basics-Popular (11 lessons): Holt School Mathematics: Grades 6, 7, & 8

* Back to Basics-Alternative (24 lessons): Real Math: Levels 6, 7, & 8

* Problem Solving-Popular (11 lessons): Mathematics Today: Levels 6, 7, & 8

* Problem Solving-Alternative (5 lessons): Math 65/76/87: An Incremental Development

* Standards-Popular (20 lessons): Mathematics: Applications and Connections: Courses I, 2, & 3

* Standards-Alternative (29 lessons): Connected Mathematics

I examined all of the probability lessons within each series to determine the components of lessons used in a series, such as definitions, worked examples, and oral or written exercises. Lesson components were determined through the analysis of probability lessons using open coding within the development and assignment portions of the probability lessons. I then noted how those components were sequenced within a given series.

The intent of this methodology was to identify the components in their existence and placement within a lesson. For that reason, these components identified segments of instructional material, and were not indicative of the amount of material within each segment. For example, two components coded as "description of probabilistic situation" may have contained different amounts of narrative measured in terms of page length, and two components coded as "written exercises" may have contained different numbers of questions and tasks.

I compared the lesson components and lesson sequences for the eight series. I devoted particular attention to the percentage of lessons with particular components, as well as any variation of the typical lesson sequences within each series. Finally, I described a typical lesson sequence for each series, when possible.

Analysis and Results

Lesson Components Across the eight series, there were eleven types of components represented: seven within the development of a lesson and four within the assignment portion. These components are listed below.

Development portion:

(1) Descriptions of probabilistic situations

(2) Definitions of vocabulary or statements of formulas related to probability

(3) Questions related to description of a probabilistic situation

(4) Activities related to probability requiring physical materials or working in groups

(5) Worked examples related to probability

(6) Oral exercises related to probability

(7) Written exercises designed for guided practice

Assignment portion:

(1) Written exercises or problems related to probability

(2) Exercises or problems unrelated to probability

(3) Activities unrelated to probability

(4) Questions designed for reflection on the mathematics of the lesson

The majority of lessons in each series (and all lessons within six series) contained descriptions of probabilistic situations in the development portion of probability lessons. Almost all lessons contained written exercises as part of the assignment portion. Only four of the 24 probability lessons in the Back to Basics-Alternative series did not contain any written exercises; these same four lessons did not contain an assignment portion.

The development portions of probability lessons generally began with descriptions of probabilistic situations. Definitions were commonplace in all series with the exception of the Back to Basics-Alternative (33%) and Standards-Alternative (14%) series. Worked examples were present in all lessons from the Problem Solving-Popular, Problem Solving-Alternative, and Standards-Popular series and about half of the lessons from the New Math era, but virtually nonexistent in the two series from the Back to Basics era and the Standards-Alternative series. Guided practice became a part of each lesson beginning in the Problem Solving era and continued into the Standards-Popular series.

The components within the assignment portions of probability lessons were relatively the same through the New Math and Back to Basics eras, consisting solely of written exercises related to probability. However, all probability lessons in the Problem Solving-Alternative and Standards-Popular series offered exercises not related to probability within each lesson, while the Standards-Alternative series concluded each investigation with a reflection on the important mathematical ideas from the lesson.

The absence of definitions in most probability lessons of the Back to Basics-Alternative and Standards-Alternative series represents a significant departure from the typical lesson structures of the six other series. Tasks within these series required students to develop and test conjectures and develop and refine algorithms. The lesson structures within these series may reflect the authors' desire to create true alternatives to contemporary popular series. Furthermore, it is surprising that the fewest worked examples appeared in the era titled "Back to Basics," while all lessons within the era titled "Problem Solving" contained worked examples with suggested solution strategies.

The inclusion of exercises not related to probability within probability lessons, particularly in the Problem Solving-Alternative and Standards-Popular series, may reflect the spiral curriculum philosophy that constant, systematic review will help students maintain their skills. One may conjecture that the publishers of the Standards-Popular series may have decided to incorporate a type of spiral review after observing the commercial success of the Problem Solving-Alternative series; on the other hand, it is possible that these similarities were purely coincidental. The emphasis on activities and mathematical reflections in lessons within the Standards-Alternative series is a departure from the typical lesson structures of the other seven series. This is possibly in response to the call of NCTM (1989, 2000) and the authors' views of how middle grades students learn mathematics.

Lesson Sequences The sequences of lesson components for the four popular series were remarkably similar. Lessons tended to begin with a description of a probabilistic situation and definitions, followed by worked examples or questions related to the probabilistic situation. The lesson concluded with oral exercises or guided practice items, and written exercises related to probability.

The sequences of lesson components for the four alternative series were not as homogeneous. Qualitative analyses revealed that there was no typical lesson sequence for the New Math-Alternative series. In particular, this series contained five probability lessons, organized into four different sequences, within the 7th and 8th grade textbooks. Each of these lessons began with descriptions of probabilistic situations and concluded with written exercises, but the sequences of components in the middle of lessons varied considerably.

Similarly, no lesson sequence was typical for the Back to Basics-Alternative series. In particular, there were 24 probability lessons in this series, and 15 different lesson sequences. One-third of lessons consisted of only an assignment portion, and one-sixth of lessons consisted of only a development portion. The most frequent lesson sequence was comprised of written exercises related to probability; seven lessons (29%) followed this sequence.

By way of contrast, lessons in the Problem Solving-Alternative and Standards-Alternative series each followed a particular sequence. All five of the probability lessons in the Problem Solving-Alternative series had a development portion similar to the popular textbook series. The lesson sequence in this series is different than the others in that the assignment portion of each lesson begins and ends with written exercises not related to probability. In a set of 25 to 30 exercises, one or two related to probability.

The Standards-Alternative series contained 29 probability lessons, all within the 6th and 7th grade textbooks. Most of the lessons (86%) followed the exact sequence of description of a probabilistic situation, activity, questions related to the activity, written exercises related to probability, and questions designed for reflection. All lessons began with a description of a probabilistic situation and concluded with the same assignment sequence.

Conclusion

Some series had the same structure for all or almost all lessons, while other series contained lessons with a diversity of sequences and components. This raises the question, "Is it appropriate that all textbook lessons are structured in the same way?" It is possible that different lesson structures may best address particular topics. For example, listing the complete sample space and understanding the relationship between theoretical and experimental probability may necessitate different instructional approaches; the former topic may be approached quite procedurally, while the latter requires more attention to mathematical concepts.

Specifically, in an effort to address and correct misconceptions, it may be more beneficial to present students with an activity and series of questions designed to bring attention to a common misconception and assist students in understanding the concept, as opposed to providing worked examples and sets of similar written exercises. Finally, nearly every lesson contained written exercises designed to help students develop fluency with procedures. By also providing a reflection component within a lesson, textbook authors may help students refocus their attention on the main ideas of each lesson, thus assisting in the development of conceptual understanding and procedural fluency.

Finally, as stated in the Equity Principle of Principles and Standards for School Mathematics (NCTM, 2000), "reasonable and appropriate accommodations [should] be made as needed to promote access and attainment for all students" (p. 12). Therefore, curriculum developers and textbook selection committees should seriously question whether a predetermined lesson structure is necessary or desirable, and if so, determine how that structure addresses the needs of all students across and within the various branches of mathematics.

References

College Entrance Examination Board. (1959). Program for college preparatory mathematics. New York: Author.

Garfield, J., & Ahlgren, A. (1988). Difficulties in learning basic concepts in probability and statistics: Implications for research. Journal for Research in Mathematics Education, 19(1), 44-63.

Grouws, D. A., & Smith, M. S. (2000). Findings from NAEP on the preparation and practices of mathematics teachers. In E. A. Silver & P. A. Kenney (Eds.), Results from the Seventh Mathematics Assessment of the National Assessment of Education Progress (pp. 107-141). Reston, VA: National Council of Teachers of Mathematics.

Konold, C., Pollatsek, A., Well, A., Lohmeier, J., & Lipson, A. (1993). Inconsistencies in students' reasoning about probability. Journal for Research in Mathematics Education, 24(5), 392-414.

LeCoutre, V. P. (1992). Cognitive models and problem spaces in "purely random" situations. Educational Studies in Mathematics, 23, 557-568.

McKnight, C. C., Crosswhite, F. J., Dossey, J. A., Kifer, E., Swafford, J. O., Travers, K. J., et al. (1987). The underachieving curriculum: Assessing U.S. school mathematics from an international perspective. Champaign, IL: Stipes Publishing Company.

National Council of Teachers of Mathematics. (1980). An agenda for action: Recommendations for school mathematics of the 1980s. Reston, VA: Author.

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.

Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education. Boston: Kluwer Academic Publishers.

Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 465-494). Reston, VA: National Council of Teachers of Mathematics.

Tyson-Bernstein, H., & Woodward, A. (1991). Ninteenth century policies for twenty-first century practice: The textbook reform dilemma. In P. G. Altbach, G. P. Kelly, H. G. Petrie & L. Weis (Eds.), Textbooks in American society: Politics, policy, and pedagogy. Albany, NY: State University of New York Press.

Weiss, I. R., Banilower, E. R., McMahon, K. C., & Smith, P. S. (2001). Report of the 2000 national survey of science and mathematics education. Chapel Hill, NC: Horizon Research, Inc.

Endnote

[1] For complete bibliographic information for the textbooks included in the sample for this study, please contact the author.

Dustin L. Jones, Sam Houston State University

Jones, Ph.D., is an Assistant Professor of Mathematics Education in the Department of Mathematics & Statistics
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