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 eleven chapters are arranged by mathematical topic.
Each chapter consists of 7-10 lessons including two "problem solving" exercises, one usually an application exercise and the other teaching some "critical" problem solving skill such as "guess and test", as well as two unnumbered "math toolbox" technology or exploration exercises. In the problem solving lessons, a whole array of problem solving strategies is listed, with last strategy being "Write an equation." The problem solving strategy lesson on this comes in chapter 11. Except for the last problem solving practice, this lesson on writing an equation to solve problems is the very last such lesson in the book. About four lessons per chapter contain a "work together" exercise, most of which do not require more than one person to do. Chapters end with a review and assessment section including such things as a journal, a portfolio, a chapter wrap up, Chapter assessment and a cumulative review. Each chapter also includes a chapter project which goes throughout the chapter.
Individual lessons contain an exposition on the topic, a set of about 2-3 examples that work through a problem with explanations, and then exercises for students to do. There does not appear to be any guided practice or scaffolding of initial student work. Each problem set contains about 25 -30 new problems and up to 15 mixed review exercises. The exercises allow heavy use of calculators, even for relatively simple problems like 1.52 or 26. Algebra tiles are used moderately often. The teacher's edition contains a number of hints for different "learning styles." Interestingly, there is no indication that these should be limited to students with the particular learning style. As examples of the kind of learning style presentations, one tip deals with auditory learners and suggests that "students work in pairs. One partner reads the question aloud, the other listens and writes the equation." Another suggestion for kinesthetic learners is that the class form human factor trees with each student representing either a prime factor or a node.
The book is relatively lightly illustrated, with 14 pages in the interval from page 101 to 150 having extraneous illustrations.
Content Area Evaluations
Properties, Order of Operations [4.0]
This treatment is above average. The properties are accurately defined with number and variable examples. After introducing the properties in the decimal chapter, they are reviewed when integers are explained in the following chapter. Positive powers of positive and negative numbers, numerical substitution, and order of operations are all well presented. Negative exponents do not appear until the end of the book, and then only in the context of scientific notation. Over use of calculators lowers the rating on this topic.
Exponents, squares, roots [3.0]
This topic is covered in an average manner. This is very much a pre-algebra approach to exponents, roots, and scientific notation. Exponents are well defined and expressions involving them are simplified, but multiplying and dividing monomial expressions with exponents is not covered. Negative exponents are only shown briefly in a one page scientific notation "extension".
Fractions [2.7]
This is a good solid unit on positive fractions, but it lacks negative fractions. In addition to missing content, there is one major complaint which negatively effects the rating: A significant number of the problems sets for manipulation of fractions give the students the choice of pencil and paper, mental math or CALCULATORS to find the answer. What is the point of teaching the mental skills and understanding of fractions if the skills are drilled with a calculator that may neither require nor ingrain either skills or understanding!
The proportion method for changing fractions to percents ( x is what percent of y converts to x/y = %/100) is not presented, only the fraction to decimal then decimal x 100 method is taught. This may well be because this method is faster on the calculator, even if it does not foster understanding as well.
Decimals [3.5]
All decimal skills, excluding converting repeating decimals to fractions, are adequately covered with commendable emphasis on the properties. Negative decimals are not presented in a lesson since integers come after decimals in the book. The word problems are appropriate and varied. A metric system "skills review" is offered, fits in nicely, and has excellent problems. As noted for "Fractions", the over permissiveness with calculators could significantly decrease learning. A wise teacher will remove the calculator option in most cases.
Percents [4.0]
This chapter is well done. All essential computational skills are taught and a couple of nice extras, such as "mental computation of tips" are added. The word problems concentrate on discounts, markups and percent increase/decrease. Simple interest is covered later in the book than most percent-related topics. This material could use a few more word problems. As with "Fractions" and "Decimals", students are too often given the calculator option.
Proportions [4.0]
The chapter covering most of this topic progresses smoothly and clearly from ratios, to unit rates, to proportions with variables and on to cross multiplying. The word problems focus on similarity and scale drawings and are preceded by plenty of examples with step-by-step solutions. In the chapter that largely deals with fractions, there is a nice section on changing units within the customary system for lengths and volumes.
Expressions and Equations - Simplifying and Solving [4.0 (no calculators)]
[2.0 (calculators)]
The properties of equality are clearly stated with arithmetical and variable examples, although inequalities are not covered. One and two step equalities are solved with step-by-step justifications. This is all good.
Unfortunately, the examples in which a calculator is used for solving such equations as n/3 + 2 = 6 are frightening. They send the absolutely wrong message about how algebra should be done.
Expressions and Equations - Writing [3.0]
This texts treatment of expressions and equation writing is about average. The students are asked to "translate" from English to mathematical sentences and use "let" statements (let A = the number of apples) to identify variables and write equations for word problems. The practice is thin - - there are only a handful of actual word problems. At the end of the text, a few more word problems for which one must write an equation are integrated into a generalized "problem solving" section. This tendency to treat writing and solving equations as just one of many possible problem solving strategies, along with "guess and check", "draw a picture" and "make a table", is worrisome, since equations are the heart of algebra and of much of mathematical problem solving.
Graphing [3.0]
Reading graphs, making tables of functions, graphing ordered pairs and recognizing linear functions are all well treated. Slope is not presented, nor are coordinate graphs of inequalities or systems of graphs. A good introduction, but incomplete.
Shapes, Objects, Angles, Similarity, Congruence [4.0]
This is a thorough treatment of the essential topics. The sequence of lessons makes sense (with the exception of a section on "draw a diagram" which interrupts the continuity of the topic). For instance, circles are defined in one section and circle graphs are introduced right afterwards. Some procedures, such as using all the diagonals from one vertex of a polygon to help in determining the sum of the measures of all interior angles, are buried in the technology pages. This is too bad and unnecessary. Most of the use of computers or calculators is overkill and probably a time waster.
Area, Volume, Perimeter, Distance [3.5]
This book contains a good treatment of area and perimeter. Derivations for the formulas are demonstrated and irregular shapes are presented in the problem sets. The Pythagorean theorem and its converse are clearly explained with sufficient practice. Prisms, cylinders, pyramids, spheres and cones are all described. Volume and surface area are only computed for prisms and cylinders since these can most easily be derived from "nets" and diagrams. Included in the chapter are some interesting word problems such as using the Pythagorean theorem twice to find the length of a diagonal of a box, and determining how many boxes can be stacked in a storage unit given all the dimensions.
There is a major problem with this presentation as well. There is, within the coverage of this topic, a section dedicated to "Guess and Test". All the problems in this section could be approached with the algebraic and geometric skills students should have learned from the course, but instead they are asked to us Guess and Test, first cousin to trial and error and definitely not a powerful analytical tool we need to be teaching to our children.
Program Quality Evaluations
Mathematical Depth [3.7]
The mathematical content of this book is good, but at a level just below that of the true pre-algebra books. On the other hand, the content level could support reasonably good preparation for algebra. This book certainly has very good pre-pre-algebra content. It is critical to note that the content as it reaches the student significantly depends on the exact presentation used in the particular classroom.
Quality of Presentation [2.5]
After chapter 1, the order of topics within the book is quite good. Things flow well from one important topic to the next with relatively little interruption by less important topics. Appropriate order from topic to topic applies to the ordering of many lessons as well.
Unfortunately, the book has many flaws of presentation. There is far too much tendency to allow students to use calculators to do things that should be well within the ability of the students to do without electronic assistance. Similarly, there is far too much emphasis on algebra tiles as a teaching method.
As noted in the course description, the problem solving lessons down grade the most powerful of algebraic problem solving strategy : Write and solve an equation. Instead they make it clear to students that almost any approach other than writing and solving an equation should be tried first. This is completely wrong and reflects a major flaw in the book. Not all strategies are created equal and to stress the less efficient over the more efficient and more general is not good teaching.
Finally, the book, although not as long as others, is still relatively long and has important content in the last chapter. Thus, teachers must make every effort to reach the last chapter. It seems unlikely that this will happen if a class does all of the working together exercises or uses all of the different tips for teaching students with various different learning styles. After all, how much time can we spend making human factor trees or having one student read the problems to another, to say nothing of each chapter project and technology lesson.
Quality of Student Work [2.7]
Student work is very dependent on the choices a teacher makes with regard to use of calculators. If a student uses a calculator every time he is given a choice of the best method (mental math, paper and pencil, calculator), then there will be little value to most of the work.
Overall Program Evaluation
This book has moderate potential to be used as a successful pre-algebra book and good potential for use as a pre-pre-algebra book. For real pre-algebra applications it is difficult to recommend this book over books written specifically for use as pre-algebra books, but it might meet certain needs. The overuse of technology and manipulatives generally lowers the rating of this book and the ability to recommend it. In the absence of knowing that the right choices will be made in the implementation of this book, there is the real possibility of a not very good course coming out of this potentially good book.
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