November 21, 2000
California State Curriculum Commission and
California State Board of Education
California Department of Education
721 Capitol Mall Sacramento, CA 94244
Dear Commissioners and Board Members,
As former members of CRP panel # 3, we are writing to the Commission to provide further clarification of our report on the algebra text, Algebra 1, Concepts and Skills, published by McDougal Littell. While we realize that this letter has no official status, we believe that the treatment of this text has been subject to certain errors and misunderstandings that, if not corrected, would effectively circumvent the role of the CRP.
In short, the opinion of the CRP was, and still is, that this text, even including the supplementary material in the "California Standards Key Concepts Book", fails to cover adequately all the California Standards for Algebra and hence should not be recommended for adoption. The IMAP assigned to this text (IMAP # 6) chose to disagree with this recommendation. Although they provided no specific examples of how they believed the relevant standards were, in fact, covered, they seem to have made certain implicit assumptions about the CRP report in choosing to disregard it. We believe that, if these are accepted by the Commission and the State Board, the focus will be diverted from the real issue which is the inadequate coverage of the Standards by this text.
There are two issues that we believe are relevant. First of all, between the submission of the CRP's final report and the IMAP's report, the publisher, contrary to explicit rules for the meeting, handed out an extensive rebuttal to a preliminary draft of the CRP report. Because most of the CRP members had left before an analysis of the publisher's material was possible, there was no opportunity to address the statements made in that material. Although this material was then withdrawn from the IMAP members the following day, it clearly had been read by several of them and appeared to have influenced their opinions.
The second and more central issue concerns the supplementary text, "California Standards Key Concepts Book" (CSKC). The IMAP report, echoing the publisher materials referred to above, states that the "CRP did not review the California Standards Key Concepts Book....", implying that this justifies their opinion that all the Standards are adequately covered. The CSKC was, in fact, read carefully by the primary author of the preliminary report on this text. This did not alter the opinion that certain standards were not met. Furthermore, because there is no reference to this text in either the Student or Teacher Edition or any indication of how it was to be used, the entire CRP decided that it was not appropriate to use it for analysis of the standards.
We feel it is important to explain this decision. All of the CRP's were sent only the Student Text and the Teacher's Edition for their content reviews and told to restrict their attention to these texts in their content analyses. The reason for this policy, as was clearly and explicitly stated by Professor Wu at the opening of the summer CRP/IMAP training session, was that it is essential that coverage of all standards be contained in the Student Text, that is, in material available to all students and to any family members, such as parents, guardians, or siblings, who might wish to help the students. Materials available only in class or only to the teacher are not relevant to this analysis. Furthermore, coverage of standards must be done in a logical and coherent manner. This does not necessarily mean that it must be contained in a single, bound volume; however, it cannot be contained in a hodge-podge of materials whose inter-relationship is unclear. Finally, the Teacher's Editions were sent to the CRP's so that they could better understand how the Student Texts were to be used.
The CSKC text, which was sent to the CRP members as part of the original submission, contains a review of pre-grade level material (pages P1-P104), a discussion of further topics (T1-T25), and some brief sections on selected Standards (S1-S95). Only these last sections are relevant to the question of standards coverage. The sections discussing the standards that the CRP felt were not adequately covered are extremely short, mostly 4 pages long. The most important point here is that there is no connection between this text and the main textbook. Not only are they not co-ordinated in any fashion whatsoever, but the CSKC is not even mentioned in either the Student or Teacher Edition. In particular, there are no instructions at all to the teachers on how to use this text. We emphasize that this is not a question of some standards being covered in one text, others in the other text. Instead, single standards are somehow supposed to be covered by using a couple of pages from one text and a couple from the other.
The publisher has now agreed to make the CSKC part of its standard Student package, but this does not address the primary issue of use of the text. They have agreed to put page references to the CSKC in the primary text. However, this still does not give any information on how the supplementary text is to be used. Furthermore, although putting in page references could be considered an edit, claiming that this somehow provides adequate coverage of the standards is something that would have to be checked by a mathematically trained panel; this is clearly beyond the correction/edit criteria.
Given that there seems to be some confusion concerning the validity of considering the CSKC text as part of the Student text, we feel that it is important to provide the Commission with further details of our original analysis and explain how, EVEN WITH THE CSKC text, several of the Standards are not adequately met.
Standard 3.0 is NOT COVERED.
This Standard concerns absolute value, in particular, equations and inequalities involving absolute value. This standard is not discussed at all in the CSKC text so whether or not that text is considered is irrelevant in this case. In order to cover the topic of absolute value, one must understand "and" and "or" statements. The discussion of these topics is inadequate. They are never related to the concepts of intersection and union; students are simply told that "and" corresponds to one type of inequality and "or" to another. Furthermore, the transition from equations involving absolute value to inequalities involving absolute value is utterly confused. In section 6.6, students are told that a solution to | ax +b | = c is a solution to either ax +b = c OR ax + b = -c. In the very next section, 6.7, they are told that a solution to | ax + b | < c is a solution to ax + b < c AND ax + b > -c. A solution to | ax + b | > c is a solution to ax + b < c OR ax + b > -c.
NO EXPLANATION whatsoever is given for these statements. Any reasonable student would wonder where the "and" came from and why only in one of the inequalities. A student might also notice that, if "c " is not required to be positive, (and the text does not require this in the inequality case), then the statements about the inequalities are incorrect. This is a totally misleading and completely unacceptable treatment of this standard.
Standard 18.0 is NOT ADEQUATELY COVERED
This standard is concerned with the concept of a relation and how to decide whether or not a relation is a function. Students are supposed to be able to justify their answer.
The problem with the discussion of this standard is similar to that of many topics in this text. Students are repeatedly given rules to apply or formulas to use with absolutely no explanation or justification. In this case, the students are told that, for a relation to be a function, its graph must pass the "Vertical Line Test": each vertical line must intersect the graph in at most 1 point. Again, no justification or explanation is given for this fact. Nor is it stated that the input is assumed to be the x-coordinate and the output the y-coordinate, without which the statement is incorrect.
The discussion in the CSKC text is essentially identical to that in the main text. There is no further explanation or anything else additional that is not in the main text.
Standards 14.0 and 19.0 are NOT ADEQUATELY COVERED
These standards deal with the quadratic formula and completing the square. The text provides the quadratic formula in section 9.6 with no explanation. As noted in the discussion of the previous Standards, this is a problem that is pervasive in the text; procedures and formulae are given to the students with no justification; they are to be used without the students knowing where they come from.
The quadratic formula is eventually derived, but not until almost 200 pages later and in a totally unmotivated fashion in the middle of a chapter consisting of largely unrelated topics. There is no reason given for why one would want to complete the square; there is only one simple example provided before the derivation of the quadratic formula is given. There is no need to delay this central topic in algebra until the very end of the text because the intervening material is essentially unused. In particular, the form of a perfect square quadratic polynomial is barely mentioned in the 2 pages in which completing the square is presented and the quadratic formula is proved. The text simply states that, "By using FOIL to expand ..., you can show that this pattern holds for any real number b." The text itself does not show this fact nor does it even refer to the section, a hundred pages before, where FOIL is explained.
The CSKC has 4 pages on completing the square in which it provides a few, slightly more complicated, examples. However, it still doesn't provide any justification for the completing the square formula; it doesn't even mention FOIL. Even using the material in both the main text and the CSKC, which are not even vaguely co-ordinated, the connection between the process of completing the square in the specific examples and what is done in a general form in the proof of the quadratic formula is never made.
We want to emphasize that our objection to the presentation of the quadratic formula and completing the square is not that it is done in a non-standard order. It is that it is done in a manner that is antithetical to the goal teaching students logical reasoning. A formula is provided and used to compute values without the students having any understanding of what they are doing. When a explanation is finally given, it is so unmotivated and disjointed as to be useless. Also, the degree of sophistication here is considerably below grade level. This is a totally inadequate coverage of these Standards.
Steve Kerckhoff, Wayne Bishop
Jane Friedman, Yat-Sun Poon