Introduction
This is part of a series of second, fifth, and seventh grade Mathematics Program Reviews. This review includes a summary of the structure of the program, evaluations of a selected set of content areas, and evaluations of program quality. Ratings in these areas were made on a scale from 1 (poor) to 5 (outstanding). The overall evaluation was made using the traditional system of letter grades. For details of the methods used in this evaluation see Methods for Seventh Grade Program Reviews.
Student Text Structure
The student text includes:The twelve chapters are arranged by mathematical topic.
Each chapter is composed of about 7-9 lessons. Lessons are clearly laid out, and generally focus on clearly defined goals, with direct teaching of the skills or concepts to be learned. There are a 1-4 reasonable examples with good exposition, followed by a guided practice section reviewing the concepts and manipulations of the lesson. The exercises contain straightforward practice of the lesson plus word problems, as well as a few standardized test practice questions. About every second to fourth lesson, there is a "spiral review" incorporating problems from earlier in the book. The exercises are of moderate difficulty. Each exercise set has a section of a continuing "project" that goes through the chapter. In every chapter there are about 3-4 "labs" involving a variety of topics. In each chapter there is also an unnumbered technology lesson, most of which describe either how to punch numbers into a particular calculator or how to read the display. Each chapter also comes with a mid-chapter assessment, a summary, a review and a final assessment. The summary, review, and assessments are clear and free of nonsense.
The book is moderately illustrated with 23 pages with extraneous pictures between pages 101 and 150. The pictures are generally small and not particularly distracting.
Content Area Evaluations
Properties, Order of Operations [3.0]
The coverage of this topic provides a midlevel introduction to order of operations, properties and powers as they apply generally to positive whole numbers and relatively simple variable expressions (i.e. 54 = 5 x 5 x 5 x 5). Late in the course (chapter 10) when integer multiplication is presented, powers of negative numbers come in. The commutative and associative properties are clearly defined. The distributive property is introduced in the fractions chapter, as a mental math tool as well as a tool for simplifying algebraic expressions.
A note on presentation, the initial introduction to commutative and associative properties is an exercise in algebra tiles that is more appropriate to a much earlier grade and the introduction of multiplication and addition facts. This is followed by a decent explanation, although it has the students define commutative and associative properties of multiplication without leaving them with the appropriate definition in the text where they can go back to it if necessary.
Exponents, squares, roots [2.5]
This is a less than adequate treatment of this topic. The book introduces the concepts of powers and roots without getting into much algebraic simplification. Negative integers are covered, as are powers of negative numbers. Scientific notation is introduced but is not used for calculations, indeed, there appear to be no problems in multiplication of division with exponents of any kind. Finally, the meat of the coverage of this topic does not come until nearly 5/6 of the way through the course, making it difficult to build the skills and concepts into other lessons.
Fractions [2.7]
All basic computations are explained. Generally an "investigation" with paper, pencils, markers, etc. precedes each lesson. The manipulative presentations are often less than enlightening and perhaps more confusing to the average student. The approach is certainly too simplistic for pre-algebra students and likely to be time wasting for seventh grade students who need to learn the steps and move on. There is relatively little use or computation with negative fractions. There are relatively few problems involving converting fractions to decimals. Just about the first example on this topic deals with converting 2/9 and 7/8 to decimals. This is the only teaching of repeating decimals. The correct answer given in the book is "use a calculator to divide the numerator by the denominator." This is inappropriate and indicates, along with the use of so many manipulative investigations, a somewhat less than rigorous approach.
Decimals [2.5]
The calculator appears to be the tool of choice for manipulating decimals and converting fractions to decimals. The problems are of reasonably good quality otherwise, but the students need the experience of manipulating numbers! Negative rational numbers are not considered in this text. There is no conversion between terminating or repeating decimals and fractions. In the absence of the relatively good problems, the lack of key topics and the emphasis on calculators would lead to an even lower rating on this topic.
Percents [3.0]
The proportion model of introducing percents ( x is what percent of y translates into x/ y = %/100), sometimes referred to as the "is over of" method, is used in this text. This builds directly from the definition of percent, fits well with converting from word problems, and works well for many students at this level. It also leads naturally, via cross multiplication, into the "decimal x 100 = %" formulation. Percent increase/decrease, simple interest, and large and small percents are all introduced. Unfortunately, there are problems with presentation. The 10 x 10 squares modeling examples are confusing. Calculators are used at totally inappropriate times. For example, in a simple interest problem, the students are instructed to use a calculator to compute 300 x 0.06 x 1/12 . So much for number sense. This is exactly the kind of computation seventh graders should be learning to do with mental math or pencil and paper!
Proportions [4.0]
The text does a good job with ratio, rates and proportions. There is an excellent section on converting between rates of speed by canceling units, although there is not conversion between systems. Of course, even the good problems tend to use calculators needlessly. Cross multiplication is clearly presented as the method for solving proportions. The word problems are varied and of appropriate depth.
Expressions and Equations - Simplifying and Solving [2.0]
This is a less than satisfactory coverage of this topic for students in the year before algebra. It may be adequate for pre-pre-algebra. One step equations are introduced early and include negatives. Two step equations are given a very brief introduction at the end of the text. This is too late, two step equations should follow right after one step equations. The unit on two step equations is brief, but both decimals and fractions are covered and the step-by-step examples show the properties of equality in use, even if they are not named. There is no simplifying using the distributive property nor are there problems with variables on both sides of the equation. No inequalities appear. It is not clear that the algebra tiles at the start of essentially every section contribute to student learning.
Expressions and Equations - Writing [2.0]
This is a less than satisfactory coverage of this topic for the course before algebra, although it may be sufficient for a pre-pre-algebra course. The coverage is generally thin on writing expressions and equations. The skills are presented correctly, but at a basic level. There is not a lot of "translation" from English to math and there are only sprinklings of the kinds of word problems one might expect in pre-Algebra. On the other hand, as noted above, the proportion writing exercises are at a pre-algebra level. There are no inequalities.
Graphing [1.5]
This book does not cover linear functions in an appropriate manner. Graphing functions only occurs at the level of plotting "data points." Two coordinate graphs are referred to as "scatter plots." The relationship between a linear function and its graph is not pursued. Tables for functions are presented and the "data" are graphed on a scatter plot. Students do not learn the definition of slope or how to use it to plot a line given a point and the slope. In short, far below satisfactory.
Shapes, Objects, Angles, Similarity, Congruence [2.0]
This book has a limited approach to plane geometry. Angle measures and circle vocabulary are presented. Parallel lines, adjacent, vertical, complementary and supplementary angles are defined. Triangles are only classified according to their angle measures. The treatment of polygons is limited to the classification of squares, rectangles, parallelograms, pentagons and hexagons via the use of tangrams, not by logical deduction. While similar polygons are defined, regular polygons are not. There is little or no construction and no treatment of central or interior angles of polygons.
Area, Volume, Perimeter, Distance [4.0]
This is an above average treatment of this topic with only a few gaps. Formulas for areas of parallelograms, triangles, trapezoids (with an interesting derivation, p 396) and circles are given and used in problem sets. Irregular shapes are included. Some "shaded area" problems are also provided. The Pythagorean theorem is stated and used in a brief problem set. It is also proven, via the classic geometric (essentially non-algebraic) proof (p 410-411) in a "lab exercise". This is commendable. The surface area unit emphasizes nets and "views" of solids, but the formulas are also presented. Cones are left out of the volume section but some extras appear. For example, there is a good introduction to polyhedrons in which they are classified according to their vertices, faces and so on. At the end of the chapter, there is a discussion of similar prisms giving the students some extra work with proportions.
Program Quality Evaluations
Mathematical Depth [2.7]
This book appears to be deliberately written at a level that places it below a pre-algebra book, essentially at the level of a pre-pre-algebra book. There is a need for such books and good ones should be noted, but this target level will generally cause a book to fall short of the particular content criteria we have used. Even given this constraint, the level of the topics varies widely. Some are well covered and others are covered rather poorly.
Quality of Presentation [2.4]
Many of the presentations are overly dependent on manipulatives and calculators, distracting from or outright displacing important skills and concepts.
Although not as long as some of the true pre-algebra books, this book is long. Substantial decisions will have to be made by teachers, or their supervisors, so that the course can focus on essential mathematical content and not on less important topics or activities which are not efficient at mediating learning.
The "labs" are of mixed value, some are very good (e.g. lab 8.8 Discovering the Pythagorean theorem) and some have less to recommend them (e.g. lab 4.6 Investigating algebra tiles).
Quality of Student Work [3.1]
Within the constraints of the content level and some of the presentation difficulties, the work presented in this book is satisfactory to bring a significant fraction of students up to the level of the material
Overall Program Evaluation
The book earns a relatively low rating based largely on a content level below that necessary to proceed on to algebra in the next year.
It is not possible to recommend this book for general use as a pre-algebra book. The question arises as to whether it will support high levels of preparation for pre-algebra in the next year. Although a student could well go on to succeed in pre-algebra from this book, some of the content and teaching method deficiencies are large and might have a negative impact on preparation for later years. In addition, there are other books covered as part of this review that are likely to work much better as pre-pre-Algebra books.
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