HRW Algebra: Explore - Communicate - Apply

Section I - Organization and Features

The student text for HRW Algebra: Explore - Communicate - Apply contains 758 pages organized into 14 chapters. The chapters are arranged and identified by math topics, not by context topics.

The student text contains an index with a moderate number of entries. Index entries are not based on context references, they are references to math topics.

The student text also contains a glossary with a small number of entries. The entries in the glossary include page number references. The breadth of coverage of mathematics terms in the glossary is moderate.

There are many answers to problems for students to check their own work.

There are many pictures within the text beyond those that clearly illustrate the material being presented.

The student text includes self-testing sections.

Section II - Major Topic Summaries

A) Linear equations in one variable

This topic is covered at a fair level allowing a less-than-comprehensive level of achievement. The overall presentation is at a moderate level. Worked out examples contain justification for each step and strategies for word problems are well presented. Unfortunately, far too much time is spent on teaching how to use manipulatives to solve equations. There are also exercises teaching how to key things into a calculator. This is unnecessary at this level of building the foundations of algebra.

The level of problems seldom exceeds "easy" and when it does there are far too few problems. For example, there are few, if any, problems in which there are multiple terms with the same variable on the same side of the equal sign, there is little need to apply the rules for order of operations, and nothing but easy word problems.

This topic is treated poorly. Compound inequalities are missing and there are far too few problems - at far too low a level - for each subtopic. The pages are "fussy," so key items, such as properties, do not stand out. Few or no problems require more than one step for solution.

This topic is treated poorly. The right topics are covered, at least at an easy/medium level, but the text does not systematically teach the necessary skills. There is too much graphing with calculators and "exploring" without the conclusion being clearly stated in the text and strengthened with mathematical logic. Extra topics such as statistics further dilute the content. There is a decent discussion of the different forms of a linear equation (slope intercept, standard, point-slope) and an in depth discussion of horizontal and vertical lines and their equations.

Except for some uses of algebra tiles, the presentation of factoring is excellent (chapter 11). The exposition is clear, there is good mathematical logic, and intelligent problems building to a mix of skills. On the other hand, chapter 12 is far weaker. It directs much effort to non-analytical or formula solutions, or solutions by factoring only for simple equations. Chapter 12 also fails to extend factoring to special higher order polynomials factorable by techniques covered in chapters 11 and 12. Completing the square is introduced before students learn to solve factorable polynomials by setting them equal to zero. The presence of the factoring and solving equations by factoring toward the end of the book may keep some classes from covering them in depth.

This topic suffers from numerous drawbacks. When material is explained, the explanations are generally clear. However, more often then not, the details are left unexplained in the effort

to have students infer them. The graphic approach to systems has generally adequate coverage, although the systems with no solutions and infinite solutions are deferred for several sections. The number of clearly worked out examples is lower than desired. There is reference to the properties of equality for addition, but not for multiplication. Otherwise, justification of steps is not generally adequate.

The number of sample problems is sometimes adequate and sometimes not. The problems are almost all at the easiest level. The exception is the section that covers word problems, which does

get to some more difficult cases, but the exercise count is still low. Linear programming is mentioned but not covered. Solutions of 3 equations in 3 unknowns are not addressed. Although this text develops work with matrix algebra through inversion, the application to systems is barely touched upon. The drawbacks combine to prohibit a positive rating for this major topic.

The presentation generally leads students through a few simple cases and expects the student to infer the underlying property. Often, the relevant information is not presented until several pages later, and there is no real attempt to show why these properties are true.

The quotient of powers is given, but the power of a quotient is not. Fractional exponents are not covered.

There are few sufficient examples and the problem set is both limited and at a low level. There is a very small number of problems of a moderate difficulty level. In general, however, the treatment of this major topic is insufficient to promote effective student learning even at modest levels.

The arrangement and method of introduction is a bit unusual, and follows what might be called a functional approach. Thus, the topic begins by addressing the solutions to problems that can be solved by taking the square root of both sides. The basic introduction covers most of definitions and introductory subtopics, including approximating roots. Irrationals are introduced but their relation to roots is not entirely clear.

Properties are given short treatment in exposition and examples, which also fits the functional theme. There is a better presentation for finding solutions of radicals, also fitting this theme.

The presentation of the Pythagorean theorem is clear but the argument/proof is not. Examples are few here, as are problems. The distance formula is given, but very briefly without much exposition or examples. The problem count is again low and only to an easy difficulty level.

In general, the presentation is clear and there usually are enough examples. There are modest problem counts in general that may be sufficient to cover low difficulty cases.

The functional approach encourages the transition between numeric representations and those with variables, which is what gives this text at least some strength.

The general form of introduction of a subtopic is through an example with tabled values. Calculators or graphing are used quite a bit in elaborating function properties, thus resulting in a decreased emphasis on analytic methods. However, to the book's credit, it frequently and clearly draws the distinction between graphic methods and analytical methods.

Section III - Overall Evaluation

Mathematical Depth and Breadth

The coverage of algebra content is mixed from one topic to the next. Sometimes the needed contents are addressed, but other times there are omissions.

The depth of coverage is generally poor. The support and experience that students need to achieve at basic levels may be provided, but extension beyond this is rare.

There is a surprising variability in coverage from one topic area to the next, so that some topics receive good treatment while others are poor.

Overall the quality of presentation is often fair in terms of clarity and understandability. Terms, concepts, and procedures are often clearly stated and explained. Other times, however, students are left guessing.

Examples are often fair in quality, and may reference properties in justifying steps. However, this presentation is quite variable as sometimes explorations are substituted for examples. Examples do not go beyond the simplest cases in most instances.

While the emphasis on properties is sometimes good, derivations and proof receive little attention.

The use of technology is sometimes excessive in application without underlying understanding.

The number of student exercises is often inadequate. Worse yet, the student work rarely extends beyond basic difficulty level. Thus, the presentation does not consistently provide fair support for even moderate levels of achievement.

The book provides only a modest opportunity for student learning, and seems to target low levels of student achievement.

The greatest strength of the program is in a good treatment of factoring, but coverage of other topics is mixed.

The greatest weakness of this text is that it does not consistently provide sufficient support for even moderate achievement levels. Difficult problems, including application problems, are rare and proof and derivation are not well addressed.