Publisher:  McDougal-Littell Title: Algebra 1
Reviewerís Name: Bishop, Friedman, Kerckhoff, Poon


With regard to mathematics content only, this program does not sufficiently address the content standards and applicable evaluation criteria to be recommended for adoption.

This submission nominally does cover the standards of algebra 1 so asserting that it does not requires explanation, but it is an easy explanation.  So much of the important material is in last third of the book, even the last chapter, with so much review in the first two thirds, that reasonable use would be almost a duplication of the material of Course 1 and 2 in the same submitted series, not a course that is fundamentally algebra.   A quick summary of the situation is given by a glance at the first pages of the same Section 9.1 of Course 2 (P. 449) and of Algebra 1 (P. 499).  For all intents and purposes, they are interchangeable.  If it were only that page, and the algebra book went deeper faster, there would be no grounds for complaint but that is not the case.  There is far too much duplication, redundancy, and review that it should not be considered adequate; the first quarter of the book is exclusively reviewing  the standards of Grade 7 and below.

Although the basis for understanding linear equations and functions is a Grade 7 standard, the repetition and more thorough treatment of the topic is universally understood as appropriate in algebra I.  How well does this submission do?  At this level, students should be being forced to hone their skills in writing appropriate equations from word problems, cookbook though they may be.  In 70-80 problems per section, there are at most a couple - often none at all - where the student is required to independently assign a variable and write an appropriate equation that mathematically describes the verbal situation.  For example, consider the 69 problems in Section 3.5, a section entitled "More on Linear Equations".   Of the seven word problems, none requires the student to write his or her own equation that describes the situation mathematically.  Item 40 pretends to but, in fact, it is all but given.  In 41, students choose between two given choices, and in 43, a choice among four.  Thatís it.  Sections 3.6, has exactly one, Item 43.  Sections 3.7 and 3.8 have none.   This is representative of real word problems throughout the book.  There are so few that they can be avoided entirely while students get "algebra"credit for meeting 7th, 6th, or even 4th or 3rd grade standards.  Looking again at Section 3.5, Items 64-69 are all subtracting decimals; e.g., Item 64 is 11.9 - 1.2 .  That is, the last six items of the exercise set address only the Grade 3 standard, NS 2.1.  Jumping far ahead to Section 10.4 (literally randomly chosen), P. 593, Items 75-86 are twelve problems factoring natural numbers of a trivial nature, 12, 20, 18, up to 112.  Thatís Grade 4, NS 4.2.

This submission fails in another way, the mathematical coherence implied in Criteria 11 and 12, that mathematical discussions are to be brought to closure in the sense that a discussion of a mathematical concept, once initiated, should be completed and that formulas and theorems should be proved as appropriate for the grade level, and that reasons should be given when an important proof is not proved.  This submission has a serious organizational problem in that concepts are introduced and used, but the needed development is then presented much later, so it becomes effectively moot as to whether or not the standard has, in a real sense, been met.  Consider, for example, the quadratic formula.  It is simply given in a highlighted box in Section 9.6 with a note that it will be derived later.  In fact, completing the square and a derivation of  the quadratic formula are delayed beyond the end of the school year for all but the most advanced students.  Because the explanation, through the algebraic process of completing the square, will not be seen by most students, this text does not adequately cover this standard (19.0).  Understanding the derivation of the quadratic formula is grade-appropriate and an important part of the standard. In contrast, the level of coverage here is well below that of an algebra text; it has been reduced simply to the mechanical process of plugging values into a formula which magically gives the right answers.

A similar situation arises with exponential functions in Chapter 8.  Only exponential expressions with integer exponents have any meaning so far in the book.  No mention is made as to what meaning, if any, might be assigned to 2^1.5 (Example 2b of Section 8.3) because fractional exponents are not dealt with until Chapter 12, beyond the course for most students.  Even after rational exponents are introduced, the connection with what had been done previously in Chapter 8 is never made.  Instead, exponential functions are treated simply as a button on the calculator.  Students are told that certain situations are modeled by exponential functions but no reason is given; no reasonable explanation could be given at this stage since the function itself has been given no meaning.  Students are further asked to identify the domain and range simply by looking at the picture.  Not only is this a below-standard method of dealing with this standard (17.0), but it is likely to lead to incorrect responses since the graph quickly disappears from view.

The presentation of both the quadratic formula and exponential functions simply as "black boxes" are representative of the textís inadequate attention to Mathematical Reasoning and logical discussion.  In particular, it fails to meet the Content Criterion 11, which is of critical importance.

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).


As explained in the general criticism above, the critical words here are "supports teaching the mathematics standards at each grade level [subject matter content for algebra and beyond]" not "can somehow be referenced".  Under the latter interpretation, the submission would pass, under the former, it does not.

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.


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


There are a few problems with the definition of mathematical terms. For example on page 710 it is not made clear that a^(1/n) is the principal nth root of a if more than one root exists. In fact, it is stated that ``because 8^2=64, you know that 64^(1/2)=8''. But it is also true that (-8)^2=64.

On page 588, the text defines what it means for a polynomial to be in factored form, but gives several examples of polynomial equations where a factored polynomial is equal to zero. This could lead students to confuse polynomials, with polynomial equations.

The algebra text is an improvement over the previous two texts in the series in that, when concepts or properties are defined and examples given, the examples are clearly labeled as such.  For example, the clear labeling of the examples of the distributive property on page 101 of this book compare favorably with the confusion of example and general statement of the distributive property on page 139 of Course I.

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


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



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.


As describe in the introduction, the problems are misleadingly trivial or are impossible except as concept-limited calculator exercises.

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.


The earlier described problem with algebraically based word problems is beyond rectifiable.  It is clear that the authors perceive that they are hard for students so instead of building on an acceptable base from the preceding Courses 1 and 2, they have avoided the problem entirely.  That alone would be fatal but it is exacerbated with too many other "problem" situations being delayed too long to be developing anything close to automaticity.  Factoring, and its various multi-step uses, and completing the square come quickly to mind.

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.



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


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


In some places there is not enough explanatory text. For example on page 568, a monomial in one variable is defined, and examples are given, sqrt(x) and 1/x, of expressions that are not monomials. But it is not explained why these are not monomials, this is something which a student may not understand at first reading.

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


Exponential functions and quadratic equations were discussed in the initial summary.

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.


  An example is the treatment of radical expressions in Section 5.3.  Sqrt(ab) = sqrt(a) sqrt(b) credited to the "Product Property of Radicals", with no justification nor explanation, let alone proof.  It would be easy enough to lead students through a proof, just square the right side and use properties of multiplication and the definition of square root given at the beginning of the chapter, but the idea of proof is underplayed quite consistently throughout the book.   One could assume that the authors avoided the proof since fractional exponents are not yet taken up and this is only a special case of that more general property of exponents but it is the wrong approach. It is reasonable to "do" square roots more deliberately first, in part as motivation for the more general setting.  Showing how the proof of new situations can be based on known properties, as in this case, is an important part of that developmental process.

Another example is Section 10.3 pgs 581-587 regarding special products. The proofs of these are well within the capacity of students at this level, since they only depend on basic properties of numbers. This opportunity to provide proofs is missed except for the use of an area model to justify one of the special products on page 583.

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


As commented at the beginning, the student involvement in appropriate use of the ideas developed is below adequate, much less beyond.

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.


The guidance of Mathematics Framework Chapter 3 is not followed, particularly in regard to quadratic functions.  Factoring is not taken up until Chapter 10 (and the word "factor" is not explained, although "term" was, much earlier) so its importance in solving quadratic equations of one variable is delayed and its importance in identifying the zeros of a quadratic function are postponed.  For many students, its importance will be lost entirely.  Just plug numbers into the quadratic formula or let the graphing calculator do it all.  The fact that this produces an adequate approximation is irrelevant when judged against the first paragraph of the quadratic function portion of the Algebra 1 section of Chapter 3.

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.


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.


Too much is below grade level.

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.


But as with the other books of the series, the books are too busy.  Too many colors, photos and other visual distractions clutter them, distracting from the mathematics.
 The emphasis on multiple choice 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 ``Start to work as soon as the testing time begins. Keep working and stay focused on the test'' on pg 194.  Other examples are on pages 60, 264, 463, 752.


Content Criterion 18.  Factually accurate material is provided.


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

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.

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.


It pretends to offer this but it falls far short.  The text offers problems throughout which are labeled logical reasoning. Some of these are good, but too often they do not require sufficient thought. Some of the logical reasoning problems only ask the students to make a conjecture based on a number of examples without requiring the student to explain his/her reasoning. See page 92, for example, problems #9, 10, 11.  Other ``logical reasoning'' problems only ask the student to fill in a blank, again without explaining their reasoning, see for example #50, 51, 52, -53 on page 334, or #28, 29l, 30, -31 on page 358.
Other problems do ask the student to explain his/her reasoning or justify steps in a computation for example #61 on page 137, #43 on page 148, #137 on page 161, #24 on page 176, #5, 6, 7 on page 415. More of this type of problem would strengthen the text.

The last section of the book which is about proof, has a very few problems requiring a student to "use properties of numbers to construct simple, valid arguments for ... claimed assertions", only #10, 11, 20, 21 on pages 744-745 and #43 on page 750. Given the difficulty and importance of proof in mathematics more practice should be provided. Also all the problems in this section tell the student what needs to be done, i.e., find a counter-example, construct an indirect proof etc. There should be some problems of the prove or disprove type.

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


Additional Comments and Citations.

Corrections and Edits.