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 text is divided into 8 "modules." Each module is divided into 4-6 sections, each of which contains 1-3 "explorations." At the end of each section are one to two pages giving a very brief statement of "key concepts," practice and application exercises covering the entire section, a spiral review containing questions from earlier topics, and extra skills practice exercises. The module concludes with a two page review and assessment problem set.
Each module is, artificially, built around a "real world" topic rather than a connected series of mathematical themes. The modules are
Within a module the themes jump about without a continued logical development. For example, within the module "The art of motion" the section topics and explorations are
Within each exploration the topic is developed through a series of "discovery" problems. From these the student is to infer various aspects of the topic, but there is little closure on each discovery section, and the discovery conclusions are not clearly linked, in writing, in the text, to the exploration at hand. A great burden is placed on the teacher to make sure that students discover the right thing, and the value of the text itself as a resource the student can refer back to when confused is lessened. By stressing inference and discovery there is a general downgrading of deductive logical development or logical testing.
The book is moderately illustrated, with 19 irrelevant pictures in the 50 pages from page 101 to 150.
The Toolbox (15 pages) contains a brief description of particular procedures, a few annotated examples and a few practice problems. In general, the topics, such as adding and subtracting decimals, finding equivalent fractions, perimeter and using a ruler, and finding the mean, should all have been mastered well before the course and are not explained sufficiently to allow a student who did not master them previously to succeed.
The Glossary has a moderate number of entries which contain pointers to relevant pages in the text. Although of reasonable size, there are a few items that are missing or incomplete. A number of three dimensional shapes such as prisms and pyramids are described as "space figures" yet "space figure" is never defined. Tessellations are defined as tilings of the plane using "congruent polygons." This implies that only a single kind of polygon is used, yet among the most famous of tessellations is the tiling achieved with Penrose tiles, which are of two different shapes.
Content Area Evaluations
Properties, Order of Operations [2.5]
Material in this section falls far short of high level expectations of students at this grade level. Order of operations is described for use with positive and negative numbers, and simple roots and powers. The properties of the real number system are only briefly mentioned (commutative, associative, distributive) or not mentioned at all (identity, inverse). They are not explicitly used in the evaluation of numerical or algebraic expressions, nor are they used in justifying steps in simplification of expressions or in solving equations. Thus, an opportunity to build mathematical logic and the groundwork for proof is lost.
Exponents, squares, roots [2.0]
There is at least superficial coverage of most but not all topics. Notably weak are multiplying, dividing or simplifying fractions using exponent rules, evaluation of monomials (other than single integers) with powers, and any treatment of operations with expressions involving variables raised to a power. Scientific notation is presented, but there appear to be no problems involving any calculations with scientific notation, i.e. scientific notation is presented in a way that totally disconnects it from much of its practical value. The instruction in both integers raised to integral powers and scientific notation are entirely based on calculator manipulation. For example, scientific notation is introduced with a manipulation in which students are directed to multiply 10 times itself at least 20 times and see what the display looks like. This teaching method, plus the lack of actual use of scientific notation in calculations, makes scientific notation seem like something invented because calculator displays are too small.
Fractions [2.0]
Although nearly every important topic (e.g. operations with fractions, conversion of fractions to decimals or percents) is covered to some extent, the particular presentation and level of discussion and problems substantially limits the quality of this section. The examples and problems seldom go beyond the easy level, and there are too few problems, thus neither mastery nor higher skill levels are well supported. The critical topics are jumbled with other largely unrelated topics. The explanations are too short and are buried in other unimportant material. Far too much time is spent on estimating approximately what percent value corresponds to a particular fraction and far too little time is spent doing the actual calculations. This error carries over into the word problems, where too many problems deal with approximate, estimated answers rather than solving exactly. Finally, this section relies entirely too much on fraction calculators for both teaching and homework. Not only does this place a significant financial burden on the school or pupils, it robs the students of the mental skills of fraction manipulation that are critical for the abstract manipulations and generalized arithmetic of algebra.
Decimals [2.0]
Although many subtopics within this topic are covered in at least a superficial manner, the general depth and quality of coverage here is below what is needed to meet even minimal levels of mastery and achievement. The material is notably lacking in the use of negative decimals, either for ordering and comparing, for addition/subtraction, or for multiplication/division. There is coverage of conversion of decimals to percents and terminating decimals to fractions, but there is no coverage of conversion of repeating decimals to fractions. The objectives of each section seemed poorly defined. The exposition occurs entirely in discovery problems with no closure in the lesson itself other than as the teacher may summarize and there are too few examples. Students are asked to infer important elements of the method from a few examples but there is no real development of the logic behind the final procedure other than inference. This is particularly evident in the discussion of division by zero, which is largely dealt with in a single exploration in which one first predicts what will happen if a number is divided by zero on a calculator and then the student tries the problem on the calculator to discover that he/she gets and error message. Thus, the decision of a calculator becomes the determining factor in exploring number theory. In this, as in other sections, the use of computer/calculator games or "labs" distracts from the mathematics.
Percents [3.0]
This topic is covered moderately well, certainly better than fractions or decimals. All appropriate subtopics to prepare a student for algebra are represented at a depth that allows a moderate level of learning. Coverage on markups and discounts is present but thin. There is, as in other parts of the book, too much jumping between unrelated topics. For example, the material on markup, discounts, profits, commission and interest is separated by an entire module on geometry from the coverage of percent increase/decrease and the coverage of percents greater than 100.
Proportions [2.0]
This section falls below a level allowing moderate levels of achievement. There are generally too few problems at too low a level to allow mastery of the topic. There is very little conversion between measurement systems (e.g. metric vs. English) or monetary systems, nor is there sufficient emphasis or practice on following or canceling the units through the solution of various proportion problems. As this is a critical tool in later physics and chemistry courses, failure to explicitly deal with the units is notable. As with most other topics, coverage of proportions and problems involving proportions is disrupted by unrelated topics, in this case exercises on scatter and box-and-whisker plots.
Expressions and Equations - Simplifying and Solving [1.7]
The coverage of algebraic skills in this book is very thin and coverage of this topic is very poor. There are not nearly enough problems, there is no coverage of collection of like terms, there is no use of the distributive property and the level of problems never extends beyond easy. Decimals are seldom used in either equations or inequalities and the properties of the number system are not used to justify steps in equation solving or simplification of expressions. Direct variation is never described as such and there are only a handful of problems on direct variation. To top things off, the presentation of key topics is poor. For example, the introduction of solution of two step linear equations, a clear analytical algebraic skill, begins with an introduction on translating points in the plane. This is confusing to people who know how to solve equations, how must it appear to students?
Expressions and Equations - Writing [2.0]
This topic is covered at a less than moderate level. Although most essential elements are covered in at least a superficial way, there is far too little depth or difficulty and too few problems to get the skills across to the student.
Graphing [2.0]
This topic is covered at a below minimum level. There is coverage of all pre-grade 7 topics, at some level, but the coverage of linear functions and their graphs is very weak. The small amount of graphing linear functions is based on plotting a few points, not on recognizing a linear function or plotting on any basis other than plugging in few points. If direct variation and its graphs are covered as a topic, it is cryptic. The concept of slope is only presented near the end of the book. Although it is in the context of linear formulas, it does not appear to be made clear that constant slope is a characteristic of linear functions. There is no graphing of inequalities on the coordinate plane, nor are there graphs of higher order functions by plotting points. Little of the coverage of graphing is in the context building the base for analytical geometry and algebra, rather it is involved with data analysis. Even the few explorations of plotting linear functions give as much space to instruction in the use of graphing calculators as to actually plotting things or understanding the nature of the graph. As with most parts of this book there is too little work at too low a level. The format is distracting and it is difficult to discern the actual focus of any lesson.
Shapes, Objects, Angles, Similarity, Congruence [2.0]
This material is covered at a below adequate level. Most of the pre-grade 7 topics are covered, but some grade 7 topics are lacking. The presentation of many important topics is not as clear as it could be and misses the opportunity to develop mathematical logic. There is little or construction of even simple kinds. Diagonals of polygons are discussed and an extra section deals with the general formula for the sum or the interior angles of a polygon, although no direct link is made to the measures of the angles of regular polygons. The discovery method of presentation used for this topic is both cumbersome and leads to confusion about what is exactly true, and provably so, and what is close but maybe not exact. Similarity is introduced via an extended exercise involving cutting out two different sized and rather complicated drawings of a biplane and then measuring the distances apart the two cutouts must be for one image to exactly occlude the other image. Although this may make a link to proportionality in the real world, it is virtually certain to suffer from measurement errors, thus obscuring the real accuracy of the proportions. Later, the students are directed to draw triangles with given angle measures, using a protractor, for two angles and different side lengths. They then measure the missing angles and all the sides of the two triangles. Since various measurement errors are likely to occur at each step, many groups will determine that the measures of the third angles, and the proportions between corresponding sides, are close to the expected values, but not exact. This is not really the lesson we want. Similar sorts of discovery exercises, divorced from related informal proofs or formal, are present for other topics such as the relation between angles in a triangle or in parallel lines cut by a transversal. It is reasonable to show that one can experiment in mathematics to develop a hypothesis, but this should then be linked to the rigor of mathematical logic. Interestingly, there is one attempt at a clever proof-like argument. This is an "extension" exercise on the sum of the interior angles of polygons. Unfortunately, it is done as an exercise in patterns and the key logical point making it work is not developed.
Area, Volume, Perimeter, Distance [3.0]
This topic is covered at a moderate level. The basic points are covered with the exception of the Pythagorean theorem and its applications, which is missing, and a relative paucity in the coverage of finding the areas of irregular shapes by breaking them into triangles, rectangles and parts of circles, or finding areas or volumes of irregular three dimensional shapes. The sub-topics are spread out throughout the book, leading to a lack of continuity within the topic and the emphasis on rigid motions and translations is overdone. As with other topics, the geometry sections often seem to be disrupted by exercises in probability or data analysis.
Program Quality Evaluations
Mathematical Depth [2.5]
The book generally down grades essential algebraic skills and concepts by giving equal weight to both algebraic skills and less important non-algebraic material such as probability, statistics and tessellations. Even in algebraic topics there is a general de-emphasis of the key skills of writing and solving equations. When lessons are generally related to key topics, there is much material that de-emphasizes algebraic or logical geometric patterns of thought. These problems are apparent in essentially every topic rating and description.
Quality of Presentation [2.0]
The low overall rating (below) is reflected in the possible pedagogical subscores. All possible categories for consideration under presentation (clarity of objectives, clarity of explanations, clarity and usefulness of examples, freedom of the lesson from distractions and busy work, logical sequence of presentation and appropriate use of technology and manipulatives) would earn low ratings if rated separately. All of these low ratings together cannot lead to anything but a low rating on each topic and a low overall rating.
The organization of the text by non-mathematical themes, the resultant jumping about, and the emphasis on discovery learning seriously decreases the likelihood of high level student learning. If it weren't for the very short "key concepts" items at the end of each section, there would be no closure or written summary of any lesson in the book. If there is a relative positive besides the key concepts, it is that some of the brief extra skills sections in the Student Resource Toolbox are actually of some value, even if they are not long enough.
Quality of Student Work [2.0]
The relative weaknesses of Presentation carry over into the student work. As such, this topic earns a low rating as well.
Overall Program Evaluation
This low rating reflects weakness in content, weakness in presentation, and weakness in student work as discussed immediately above and in each of the content topic reviews. It is not possible to recommend this book to anyone for any purpose.
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