Publisher: McDougal-Littell Title: Concepts & Skills - Grades 6-7

Reviewer’s Name: Bishop, Friedman, Kerckhoff, Poon

__Conclusion__

With regard to mathematics content only, this program sufficiently addresses the content standards and applicable evaluation criteria to be recommended for adoption (with corrections and edits as specified below), but only for Grades 6 and 7. In regard to some of the additional criteria, however, there is substantial concern. The submission was for Grades 6-8 with Algebra 1 at Grade 8. The format for the standards is sufficiently different prior to and for algebra that the clearly negative Grade 8 report will be submitted separately.

This submission meets each of the grade level content standards explicitly, and for this reason this submission is recommended for adoption. The authors have clearly taken the standards seriously and made a serious attempt to meet each of them. The book has been written carefully and there seem to be very few actual errors. The submission is disappointing however in the approach which was adopted which ignores the spirit and the intention behind the standards.

The submission is inconsistent in the extent to which it develops concepts and justifies results. The Mathematical Background Notes in the Teachers editions are strong. The understanding of the standards-based approach that California has imposed is most evident in these lessons for teachers but since they are separate from the text pages and much more sophisticated mathematically, it is not clear whether or not the teachers who most need deeper understanding would benefit from this material.

The worked examples are a strong point of these texts. These examples usually provide clear justifications for each step.

The books themselves are not as coherently written as they could be. The pages, especially the teacher’s editions, are garish in the publisher’s efforts to make sure that teachers do not miss the significance of particular topics in reference to the standards. The presence of extraneous visual material in the student books will make it difficult for some students to focus on the mathematical content. Examples of solution techniques in the text are generally clear and well presented, but somewhat disjointed. The books appear to be more disjointed than they really are through trying to achieve an impossible goal, address the needs of both those students who are working at grade level and those who are far below who would be more appropriately placed in a mathematics environment a year or more earlier. This is not evident in the procedural comments and basic development of the books but it is quite evident in the exercise sets, especially those designed as review. For example, P. 499 of Course 1, at the end of a chapter on data analysis and statistics there is a review of how to add and multiply common fractions, 16 exercises that are far off of the Grade 6 standards. For example, 6/7 + 5/7 and 1 - 3/5. These are examples of the third grade standards (NS 3.2), not almost the end of Grade 6. The books would be better without them and the goal, admirable though it is, cannot be realized. The teacher’s editions provide pacing recommendations for each chapter for ‘’transitional’’, ‘’average’’, and ‘’advanced’’ students, but these are almost identical. A better approach for students with severe deficiencies would be to offer pacing suggestions for covering the same material over an extended period of time.

__Mathematics Content/Alignment with StandardsMathematics Content/Alignment
with Standards__

*A systematic review of determinations regarding the criteria in this
section. Citations of standards not adequately addressed (if any)
are of particular importance with regard to Content Criterion 1.*

Content Criterion 1. The content supports teaching the mathematics standards at each grade level (as detailed, discussed, and prioritized in Chapters 2 and 3 of the framework).

MEETS

Meets, although there are weaknesses in the coverage of some standards.

For example, in Course I:

MG 2.3, There are only four problems on page 369 and three problems
on page 379 which ask students to "Draw quadrilaterals and triangles from
given information about them…" The teacher’s edition gives virtually
no help to the teacher who might be confused about how to do this, merely
stating that the teacher should "check drawings".

Statistics Standards 1.2, 1.3, and 1.4 are barely covered. The teacher’s edition gives little help to teachers here. Teachers are told, "In many case, the question of what is the "best" measure of central tendency depends on how this measure will be used." But no further guidance is given as to how to determine the best measure of central tendency in any given situation. Students are expected to be able to answer such questions on the basis of a single example. Coverage of the rest of these standards is as weak.

In Course II:

NS 1.4 and 1.5 are barely covered. This is discussed further under Criterion 10.

AF 1.3. Very few of the citations given involve having the student "justify the process used."

These texts are generally weaker in their coverage of the Mathematical
Reasoning Standards. The organization of the text, the pervasiveness
of extraneous material, and the quick movement to solving problems without
development of concepts, makes the coverage of these standards appear even
weaker than it is. The text does not appreciate the special role of these
standards or their importance.

Content Criterion 2. A checklist of evidence accompanies the submission
and includes page numbers or other references and demonstrates alignment
with the mathematics content standards and, to the extent possible, the
framework.

MEETS

Generally the standards map is complete and accurate. A small problem exists in a positive direction. Some of the references are bolded and appear to be applicable only to the teacher’s edition when, in fact, they appear in both. A couple of times, helpful references were omitted; e.g., pp 221-259 for standard AF 1.0, or pg 661 for AF 1.2 in Course II. Occasionally references are given which do not actually refer to the standard or indicate a misinterpretation of the standard.

Examples:

In Course II:

AF 1.5 pgs. 105-109.

There are some other minor errors in the standards map. For example:

In Course I MR 2.1 cites pg. 557 (exs. 105-132),
but there are no exs. 105-132 on that page.

Content Criterion 3. Mathematical terms are defined and used appropriately, precisely, and accurately.

MEETS

These sometimes are problematic but not fatally so. Often definitions are implicit, just a bold or italics word in a description, perhaps, when an explicit definition would be preferable, but it is not fatal at this level of sophistication. For example, the standard descriptors of triangles on pages 366-367 of Course 1 could be formal definitions instead of two lists of classifying adjectives.

On pages 362-363 of Course I, the terms inductive reasoning, and deductive reasoning are used. These terms do not appear to be defined in the text, they do not appear in the index or the glossary.

A related problem with these texts is that numerical examples are not
clearly labeled as such. Often in the same high-lighted box a property
or concept is described or defined symbolically with the label ``In algebra'',
and also a numerical example is given with the label ``In arithmetic''
sometimes also a verbal description is given with the label ``In words''.
This would tend to encourage confusion between general statements and specific
examples. See for example the inverse property of addition on page 117
of Course II, the area of a triangle on page 372 of Course I or the cross
products property on page 269 of Course I.

Content Criterion 4. Concepts and procedures are explained and are accompanied by examples to reinforce the lessons.

MEETS

This is problematic in these texts. The texts often jump too quickly to examples without taking time to properly motivate and develop the concepts. On the other hand the examples are well-chosen and clearly explained.

In most sections only a very few sentences serve to introduce a concept before the text begins to solve problems. In some cases these few sentences may be sufficient. But in many cases they are not. Some of the many examples where there should be more explanatory material at the beginning of a section include:

Course I:

Section 1.9 pg. 43. This section is on dividing decimals. The introductory text:"To divide a decimal by a whole number, use long division as you would with whole numbers. Line up the decimal places in the quotient with the decimal places in the dividend." This is entirely procedurally focused. No motivation is given.

Section 9.2. pg. 411. This section begins by stating that you need to use the correct units for length area and volume. It states that we measure length in feet and area in square feet etc. It then merely states the volume of a rectangular prism.

Section 10.1, Goal 2 pg. 454. The fact that outliers affect the mean more than the median or mode is stated without any explanation or justification.

Course II:

Section 3.7, pg. 140. Before the examples this section on solving
equations has only the single sentence:

"You can use properties of equality to solve equations." It would
be helpful to remind students of these properties.

Section 6.6, pg. 294. This section is on solving equations with rational numbers. No text at all precedes the first examples.

Section 12.3, pg. 626. A single sentence " You can add polynomials by combining like terms"; introduces adding polynomials.

Section 12.6. pg. 642. This section is about graphing y=ax^2 and y=ax^3. The only sentences in this section outside of the examples are the introductory sentences "In Chapter 11 you learned that the graph of a linear function is a line. In this lesson you will study the graphs of two types of non-linear functions". No motivation, no context is provided.

Algebra tiles are over-used. In particular, using algebra tiles to develop
the concept of polynomial multiplication is liable to make this concept
more confusing rather than less so (Course II pgs. 636-637).

Content Criterion 5. Opportunities for both mental and written calculations are provided.

MEETS

Content Criterion 6. Many types of problems are provided: those that help develop a concept, those that provide practice in learning a skill, those that apply previously learned concepts and skills to new situations, those that are mathematically interesting and challenging, and those that require proofs.

DOES NOT MEET

The text does provide a large number of problems. It has plenty of problems which provide practice with skills. It is weaker when it comes to conceptual problems, more challenging problems and problems that require proof. Many of the problems in the texts which are labeled as mathematical reasoning problems are not.

Some problems labeled mathematical reasoning merely require the student to fill in a blank, without providing any explanation for his/her answer. For example:

Course I: pg. 129 #47, 48, 49. pg. 137 #35, 36, 37. pg. 184 #46, 47.

Course II: pg. 242 #32, 33, 34. pg. 463 #25, 26.

Some problems labeled mathematical reasoning are essentially just word problems, for example Course II pg. 332 #2.

Too few problems are given which require students to provide justifications
or reasons for steps in a calculation or derivation. More problems like
Course II pg. 304 #9 should be given.

Content Criterion 7. Ample practice is provided with both routine calculations and more involved multi-step procedures in order to foster the automatic use of these procedures and to foster the development of mathematical understanding, which is described in Chapters 1 and 4.

MEETS

Content Criterion 8. Applications of mathematics are given when appropriate, both within mathematics and to problems arising from daily life. Applications must not dictate the scope and sequence of the mathematics program and the use of brand names and logos should be avoided. When the mathematics is understood, one can teach students how to apply it.

MEETS

No problem.

Content Criterion 9. Selected solved examples and strategies for solving various classes of problems are provided.

MEETS

No problem.

The solved examples are a strength of this text.

Content Criterion 10. Materials must be written for individual study as well as for classroom instruction and for practice outside the classroom.

MEETS

However, there are concerns. In some cases there is not enough explanatory text in the student books to enable the student to understand the concepts or to do the problems. For example, pgs 453-455 in Course II. An example is given of an irrational number 0.10100100010001... and the increasing number of zeros is noted, but how this differs from an expansion that repeats is not explained. Students are told that some numbers have rational square roots and some do not, but are not told how to determine which is which although they are expected to be able to do this in the exercises on page 455.

Another example occurs in Course I, pgs 453-456. Students are expected
to be able to determine which measure of central tendency mean, median
or mode provides the most useful information. This is Statistics Standard
1.4. No information is given on how a student is to make this determination,
except for one single example where the mean is not as good as the others
due to the presence of an outlier. Admittedly, the level of sophistication
of this standard cannot be very deep at Grade 6.

Content Criterion 11. Mathematical discussions are brought to closure. Discussion of a mathematical concept, once initiated, should be completed.

MEETS

No problem.

Content Criterion 12. All formulas and theorems appropriate for the grade level should be proved, and reasons should be given when an important proof is not proved.

MEETS

There are some problems here. The key words are "appropriate for the grade level", but that is not intended to be deprecating. For example, that a/b = c/d if and only if the cross products are equal is proved and the connection with its converse and contrapositive also. That will pass over most in the class but it will plant useful seeds for some.

Some examples of places where the text should offer more justification and reasons for facts include:

Course I pg. 376. Here two properties of quadrilaterals are stated, without any kind of justification.

Course II pg. 309. In a discussion of scientific notation it is noted that "the exponent of 10 tells you how many places to move the decimal point". No justification is given for this fact.

Course II pg. 219. The sieve of Eratosthenes is presented. No explanation is given for why this method will work. The justification of the method is left as an exercise.

Course II pgs. 303-305. In these pages, the book gives explanation for the correct interpretation of a non-zero number raised to a non-positive power. Although the book correctly states the rules and notes that the base cannot be zero, no explanation of any sort is given of this restriction. Nor is it stated that zero to the zero power is undefined.

Course II pgs. 617-619. This section presents several rules of exponents.
The discussion is too brief; not enough development and explanation of
the rules is provided.

Content Criterion 13. Topics cover broad levels of difficulty. Materials must address mathematical content from the standards well beyond a minimal level of competence.

MEETS

No problem.

Content Criterion 14. Attention and emphasis differ across the standards in accordance with (1) the emphasis given to standards in Chapter 3; and (2) the inherent complexity and difficulty of a given standard.

MEETS

No problem.

Content Criterion 15. Optional activities, advanced problems, discretionary activities, enrichment activities, and supplemental activities or examples are clearly identified and are easily accessible to teachers and students alike.

MEETS

No problem.

Content Criterion 16. A substantial majority of the material relates directly to the mathematics standards for each grade level, although standards from earlier grades may be reinforced. The foundation for the mastery of later standards should be built at each grade level.

MEETS

There are a couple of problems here; one was described in the general response at the beginning in regard to occasional review that is too far off the mark. Another concern is that most of the first 200 pages of Course 1 and 250 pages of Course 2 are material from earlier standards. Instead of everybody’s pacing and assignment guides to be the same, as essentially indicated therein, there will be the need to do these in the thoroughness indicated for transitional students but not for advanced ones or they will be needlessly bored. Literally taken at random, P. 43, Exercises 8-11 has students evaluating 27 - t, t + 2, 16t, and 84/t where t = 7. This is below grade-level. Some review at the beginning of a course is appropriate but months of review is not. Taking Course 1 and Course 2 over three years instead of two, with the intent of not starting algebra until high school for transitional students, is not only more reasonable, that is how it should be done.

Content Criterion 17. An overwhelming majority of the submission is devoted directly to mathematics. Extraneous topics that are not tied to meeting or exceeding the standards, or to the goals of the framework, are kept to a minimum; and extraneous material is not in conflict with the standards. Any non-mathematical content must be clearly relevant to mathematics. Mathematical content can include applications, worked problems, problem sets, and line drawings that represent and clarify the process of abstraction.

DOES NOT MEET

There is a problem here. The books are too busy. Too many colors, photos and other visual distractions clutter these books, distracting from the mathematics.

The emphasis on state test practice is also misplaced. This is
a mathematics text not a test-taking text. Test tips should not be part
of this book. Many of these tips are very general in nature and have nothing
at all to do with mathematics such as, "If you become anxious, take a few
deep breaths to relax," on page 541 of Course I. These are pervasive
throughout the text, see Course I pgs. 53, 105, 301, or Course II
pgs. 159, 232, 557 for a few other non-mathematical examples.

Content Criterion 18. Factually accurate material is provided.

MEETS

No problem.

Content Criterion 19. Principles of instruction are reflective of current and confirmed research.

MEETS

The CRP members generally agreed that they would not comment on this criterion.

Content Criterion 20. Materials drawn from other subject-matter areas are scholarly and accurate in relation to that other subject-matter area. For example, if a mathematics program includes an example related to science, the scientific references must be scholarly and accurate.

MEETS

No problem.

Content Criterion 21. Regular opportunities are provided for students to demonstrate mathematical reasoning. Such demonstrations may take a variety of forms, but they should always focus on logical reasoning, such as showing steps in calculations or giving oral and written explanations of how to solve a particular problem.

MEETS

The text is weak in this area.

Content Criterion 22. Homework assignments are provided beyond grade three (they are optional prior to grade three).

MEETS

No problem.

Additional Comments and Citations.

Every chapter and every section of these books contain multiple-choice State Test Practice problems and test-taking tips. These send a dangerous message. In the hands of some teachers these could become the main focus of the course.

The pervasiveness of this material weakens these texts.

Corrections and Edits.

Although not mandatory for approval, it is strongly recommended that the Pacing and Assignment Guides be reworked to be more appropriate for the different audiences as suggested in the comment of Criterion 16. That is, the publisher should be offered the opportunity to recommend Course 1 and Course 2 over three years instead of two for transitional students, with the intent of not starting algebra until high school. Associated with that recommendation, there should be some abbreviation of the earlier material for regular students and almost complete elimination of it for advanced students. Of course these ideas need review but mathematics is cumulative so earlier ideas do get reviewed as new material is studied. Deliberately going over sub-grade standard material should be held to the minimum needed for comfortably going ahead or the later material of the books will be at risk of non-coverage and we will fail to stimulate young minds that are ready for more and deeper mathematics.

In all cases examples should be clearly labeled as such. In highlighted boxes throughout the text the words "In Arithmetic" should be changed to "example", "numerical example", "for example" or something of that sort. Pages on which this change needs to be made include:

Course I: pgs. 39, 139, 182, 269, 372

Course II: pgs. 32, 74, 75, 117, 365, 366, 617, 618

Although not mandatory for approval, it would be highly desirable to
remove all state test practice problems, or as many as is possible from
the student books. It should be possible at this point to remove
from the student books the full pages of test practice problems which occur
at the end of each chapter.