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 contains:This book is organized differently from nearly every math book on the market (with the exception of other books from the same publisher) and is different in presentation and homework from so called traditional math programs. Given the difference between the style and philosophy of this book and the rather uniform style and philosophy of other books for use at the same level, it would be informative for anyone interested in this series to read the preface to the teacher's edition. This book is specifically designed to be used in the year before a student takes algebra. The 137 lessons follow one to the next with no chapter divisions. This number of lessons can clearly be completed in a 180 day school year, with plenty of time for tests, enrichment activities and days lost to assemblies and so on. Each lesson has a clearly defined and limited topic, with a clear exposition, examples very closely linked to the exposition and a few practice problems to be worked in class. The homework, about 30 problems, contains a very small number of problems related to the lesson of the day, but problems related to the lesson will appear in at least the next 5 lessons and in many lessons after that. Thus, students do as many problems related to any particular lesson as in any other program, but the problems are spread out over an extended period of time. Those topics chosen for particular emphasis are closely related to skills needed for success in algebra and are practiced essentially continually through the book. Every problem set begins with a series of word problems based upon material already covered. There is a substantial treatment of geometry as the book starts a series of three algebra-related books into which the material of a geometry course is integrated.
The lack of a chapter format sometimes results in seemingly unusual juxtapositions of topics. In the middle of a set of obviously related lessons will be one, two or several lessons that are completely unrelated to the topic. It is unclear if this is a positive for student learning.
There are no extraneous illustrations or color pictures. Everything is directed at mathematics. Neither word problems nor examples use currently popular and soon to be outdated people, movies or other issues as subjects. Indeed, the book seems to take pride in stressing the generality of problem solving skills rather than the particularity of any specific context. There is no use of manipulatives, calculators or computers.
Content Area Evaluations
Properties, Order of Operations [4.0]
This book gives an above average treatment of this topic. Expressions with as many as three terms and three variables are given for evaluation. Powers of positive and negative numbers are included in evaluation and simplification problems. The text pays particular attention to "like terms" and drills the combination of like terms throughout. Students might have an easier time in future courses if the traditional names of the properties were explicitly presented. The distributive property is well defined but commutative, associated, identity and inverse are not named as such, although they are certainly applied often and appropriately.
Exponents, squares, roots [5.0]
The treatment of this topic is excellent. Attention is given to both positive and negative powers of whole numbers and of fractions, as well as of variables and variable expressions. The product and quotient rules for powers are presented and drilled in multistep problems. Squares and square roots of both whole numbers and fractions are discussed with minimal reliance on calculators. There are problems with higher order roots (third and fourth roots) of both whole numbers and fractions. Scientific notation is not only defined, it is used in multiplication problems.
Fractions [5.0]
The coverage of this topic is excellent in this book. All sub-topics are covered and covered well and to an appropriate depth. The objectives of every lesson are clear as are the lessons themselves. There are no distractions within or between lessons. As with other topics, there are sometimes intervening lessons on different topics that may somewhat interrupt the flow of lessons for some students. All things considered, this book supports top level student learning in this topic.
Decimals [5.0]
This book provides an excellent, thorough treatment of decimals, although it provides relatively few examples at each stage. It appears that the author is assuming, correctly, that skills in decimal arithmetic should have been mastered before pre-algebra. The only subtopic that is missing is the conversion of repeating decimals to fractions.
The author has an interesting method to help students in moving between fractions, decimals and percents. He chooses a set of reference numbers (51/100, .51 and 51%) for students to memorize and use for guidance when performing similar conversions. The key is that 51/100 cannot be reduced, thus making obvious the relationship between a terminating decimal and its fractional equivalent.
Percents [5.0]
The book presents clear explanations and challenging problems. Diagrams are used frequently to help students organize their thinking. Complicated decimal numbers and fractional percents are simplified in a step-by-step manner without reverting to calculators. The word problems are clever and provide marvelous vocabulary lessons as well as math drill. This book might work better for average students if it provided more drill problems of each step.
Proportions [4.5]
This topic is well treated. The computational practice includes ratios with fractional components {e.g. (3 1/4) / p = (2 1/6) / (3 2/10) }, the word problems are clever and amusing. They cover rates, changes of scale, and similarity relationships. Conversions within the English system and within the metric system are explained carefully, although no conversion between systems is discussed. The use of unit multipliers is stressed throughout the book as length, area and volume are presented.
Expressions and Equations - Simplifying and Solving [4.0]
This book provides an excellent treatment of equalities. Students are taught to solve equations by transformation in a step-by-step, systematic way. As the book progresses, equations involving integers, fractions and decimals are presented, as are equations with variables on both sides. The symbols are introduced and one variable number line graphing is covered.
The process of solving inequalities by transformation is not covered, although graphing simple ( e.g. x >3 ) inequalities on a number line does appear. If this topic had been covered in anything resembling the level that equalities are covered, this topic would be rated a 5.
Expressions and Equations - Writing [4.0]
The concept of a "let" statement (e.g. let A = the number of apples) is not used in this text, although at many places examples have implicit let statements, as there is a direct translation from a verbal sentence to a number sentence where the first letter of the missing item is used as the unknown (i.e. the equation is written such that "A" clearly stands for apples). Surprisingly, the idea of translating expressions and sentences into mathematical statements is not presented as a unit in the early part of the course.
In spite of this reservation about order of topics, students do learn to write equations with variables and get lots of practice with word problems. The students write many proportions and solve them by cross multiplication, generating linear equations for solution. They identify parts of a percent relationship and write equations from these. The solve some "number" problems using N for the unknown number.
This topic would be improved by a more explicit and larger emphasis on instruction in translating verbal descriptions to mathematical statements. Addition of inequalities, as noted in the previous section, would also help.
Graphing [3.0]
This book presents data graphing (e.g. histograms, circle graphs) adequately. It introduces linear equations in two variables and has students fill in tables and graph the ordered pairs. The book does not introduce the concept of slope or y-intercept. It does not graph 2 variable inequalities or systems of equalities. While the book does not waste time or space on irrelevant graphing projects, more attention should be paid to the linear equation and its properties, slope in particular.
Shapes, Objects, Angles, Similarity, Congruence [4.0]
The treatment of these topics is above average, although there are some gaps. The construction units are very well done. Line segments, angles, angle bisectors, perpendicular bisectors and various angles are presented as construction exercises. Angles and triangles are classified, as are quadrilaterals. Circle vocabulary is presented and the term "regular" is defined. Similarity proportions are well taught. The rating for this topic is reduced because there is no discussion of diagonals, interior angle sums for polygons with more than four sides and neither central angles nor exterior angles are defined.
Area, Volume, Perimeter, Distance [5.0]
The book is very strong in the areas of area, volume, perimeter and distance. All essential formulas are derived and their use demonstrated. The problems involving computation of perimeter, area and volume of irregular shapes are particularly well done. These add challenge and variety to what could be dull, computational procedures. The Pythagorean theorem and the use of pi are presented clearly without a lot of emphasis on the use of a calculator. There is no proof of the Pythagorean theorem.
Program Quality Evaluations
Mathematical Depth [4.1]
The high topic by topic ratings indicate a generally high rating in mathematical depth
The coverage of the key topics at this grade level is very good. Students mastering the material of this book will be extremely well prepared to succeed in algebra and geometry. There are a small number of deficiencies, noted below, but these will in no way keep students from doing well at the next level. Indeed, students who master the material in this book will be a significant way through meeting the Algebra standards of some states, especially those states in which first year algebra material officially or practically ends prior to in depth coverage of quadratics.
The content of some sections (as noted) is less than we might hope, but still not enough to keep students from being prepared for algebra. There may be some deficiency in top end content in this book relative to some of the other highly rated texts. On the other hand, as noted in the presentation section, a very high fraction of the students in the course are likely to master the material at a level leading to success in algebra.
Quality of Presentation [4.0]
In addition to covering the material at high level, the instructional design of the program is such that one might expect a high proportion of the prepared students entering the course to master the material. Within individual sections, the book received high ratings for clarity of objectives, explanations, examples and efficiency of learning, range of depth and scope and proof/mathematical logic. This is not a trivial consideration. A book of a high content level that hides or dilutes the key elements with less important topics or confusing, jumbled expositions will not allow the high level success for many that we wish to see.
The book is strikingly free of any lessons that distract from the major topics of importance. There is no dilution of the key topics with less important lessons. Unlike most of the other books covered in this review, this book has little tendency toward the "inch deep and mile wide" presentation for which some have criticized American textbooks. A teacher starting at page 1 and progressing to the end is very likely to finish in a school year and give her or his students a high chance of learning the material. There is relatively little need for teachers to pick and choose among lessons and therefore much less possible variance in coverage between teachers. What variance there is will be relatively unimportant and will reflect whatever outside enrichment activities individual teachers choose to add. Some appropriate enrichment may be important for the top end students to be challenged at the level they could be challenged in some of the other highly rated programs.
The one area in which there is a perceived relative weakness is the sequence of presentation. As noted above, coverage of some major topics seems to be broken up by inclusion of pieces of other topics.
In contrast to most other books currently in use at this level, this book stresses the mental over the mechanical and eschews calculators, thus leading to both greater calculational competence and to development of a deeper "feel" for the concepts involved.
Quality of Student Work [4.5]
The design of the problem sets is such that important material has a very high probability of being learned and over learned, such that it will always be available to students. This is excellent. The spiral learning in this book is substantially more advanced than in most other programs and is clearly focused on important topics.
Overall Program Evaluation
This high overall rating reflects a high level of content and particularly effective presentation and student work ratings.
This book is among the top three rated books in this survey. The level of content is more than adequate to prepare students for algebra, although it is not as high as in the other two books in the top tier. On the other hand, the presentation and student work make it more likely that more students will actually master the material at an appropriate level in the average classroom.
This creates a dilemma for those choosing textbooks in that they need to balance between mathematical depth and presentation AS IT WILL OCCUR IN THEIR SCHOOLS. This book is may be the safest choice as it is fully satisfactory for content and least subject to teacher to teacher variance.
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