Content Review of CPM Mathematics
Department of Mathematics and Computer Sciences
California State University, Los Angeles
| NOTE: CPM withdrew its application to California so this report is not based on its formal submission but, instead, on the document that CPM supplies as part of the Teacher’s Version entitled, “Correlation of CPM Mathematics 1, 2nd ed. (Algebra 1, v. 6.0) and the California Mathematics Standards”, hereafter, “Correlation”. Although Professor Bishop was a member of both the 1999 and 2001 state adoption cycle Content Review Panels, any official role as a CRP member ended with the conclusion of the 2001 cycle so this report is that of an experienced private citizen, not an official CRP review. Nonetheless, the criteria used herein were developed from the state criteria that Professor Bishop used for the official reviews of the 2001 adoption cycle. He is, however, more than happy to testify informally, by legal deposition, or in person, as to the quality and consistency of this report in comparison with those which he formally helped to prepare. |
Overall Summary
| With regard to mathematics content, this program does not sufficiently address the content standards and applicable evaluation criteria to be recommended for adoption. |
Evaluation of Content Criteria
| 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). | |
| 2. Mathematical terms are defined and used appropriately, precisely, and accurately. | |
| 3. Concepts and procedures are explained and are accompanied by examples to reinforce the lessons. | |
| 4. Opportunities for both mental and written calculations are provided. | |
| 5. 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. | |
| 6. 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. | |
| 7. 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. | |
| 8. Selected solved examples and strategies for solving various classes of problems are provided. | |
| 9. Materials must be written for individual study as well as for classroom instruction and for practice outside the classroom. | |
| 10. Mathematical discussions are brought to closure. Discussion of a mathematical concept, once initiated, should be completed. | |
| 11. All formulas and theorems appropriate for the grade level should be proved, and reasons should be given when an important proof is not proved. | |
| 12. Topics cover broad levels of difficulty. Materials must address mathematical content from the standards well beyond a minimal level of competence. | |
| 13. 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. | |
| 14. 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. | |
| 15. 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. | |
| 16. 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. | |
| 17. Factually accurate material is provided. | |
| 18. 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. | |
| 19. 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. | |
| 20. Homework assignments are provided beyond grade three (they are optional prior to grade three). |
Notes on Individual Criteria
1. The idea of the shortfall here is explained in more detail in the Additional Comments at the end of the review. From the Correlation, a sufficient percentage of the CA Algebra Standards are addressed in some form as to ostensibly meet Criterion 1 but there is much less present than CPM indicates. In regard to some standards, more than three-fourths of the indicated citations are stretched beyond the limit of what the standards writers clearly had in mind.
2. CPM-1 is deliberately constructivist in regard to such things. It is unfortunate as well because, if a student has not accurately built his Tool Kit, there will be severe difficulties since there is no glossary or even clearly stated terms.
4. Some might disagree with this assessment because too much written work is often required. The problem is that it is often a misdirected effort that does not provide sufficient mental and written calculation of a genuinely algebraic nature – for example, a Guess and Check table of values in word problems. More opportunity for standard “by-type” approaches is needed and more use and confirmation of standard skills such as arithmetic or rational function..
7. See #4 above. The program delights in major “daily life” problems except that the forest is lost for its trees. The entire Unit 7 “Big Race” is an example, and the introductory section of Unit 12 entitled “Problems Solving with Distance, Rate, and Time.” It is almost beyond belief that students could then never have seen d = rt but it is true. The authors hold such an anti-“by type” bias that it happened; e.g., no such items are mentioned in the Assessment Handbook for either Team or Individual tests.
8. See #7 above. Another example is the absence of I = Prt; there is not so much as a mention of the terms. Similarly with the ideas of direct and inverse proportionality that the Framework deliberately discusses in Chapter 3. The self-proclaimed goal of the program is simply not met and, ironically, somewhat by design.
9. The program has such a pedagogical bias toward group work that it is not clear what, if anything, is expected of students outside of the classroom environment and includes such little direct instruction that it would be extremely difficult for a student who had to miss class to fill in the gaps. There is an accompanying Parent Guide but it is not clear that all parents would have a copy and, beyond that, it really is not much help.
11. Properties of exponents are shortchanged. The Pythagorean Theorem is just given (9: 12) when algebraic proofs are easily available, the quadratic formula is used without proof for a couple of chapters before a proof is given that only a leap of insight would call a proof.
12. Nothing close.
13. There is almost no emphasis given to standards of any kind, let alone the ideas of Chapter 3 of the Framework. For example, the second subheading is “Basic Skills for Algebra 1” and includes Standards 4.0-7.0, 9.0, and 15.0. A glance at the list in Criterion 1 shows that these basic skills are inadequately developed in CPM-1.
14. There is a great deal of irrelevancy, especially in Volume 1, but these are not supplemental.
15. The entire Volume 1, so the first half the course, would be better left in the closet. Almost anything mathematical is Grade 7, if not below, yet the time requirements are huge. For example, Unit 3:1 is a silly “Algebra Walk”, literally, a human graphing exercise that is at the Grade 5 standard, AF 1.4 and 1.5, yet would take an entire class period to get organized, go outside, an conduct the exercise.
19. Regular opportunity is present but, reiterating Criterion 1, Standard 24.0 is not met. The entire course confuses heuristics and inductive reasoning, one form of mathematical reasoning, with logical argumentation.
20. It is not clear, even from the Teacher’s Version that says “Homework begins here,” what is to be homework and what is to be done as work in class as a team.
Standard by Standard Evaluation
| Standard 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:1.1 Students use properties of numbers to demonstrate whether assertions are true or false. | Met. |
| Standard 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. | Not met, no fractional exponents and properties of exponential expressions with the same base are not confirmed until Unit 10: 40 and 43. |
| Standard 3.0 Students solve equations and inequalities involving absolute values. | Very weakly met, only the simplest of absolute value equations. |
| Standard 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12. | Met, but “cups and tiles” all the way through Volume 1! |
| Standard 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. | Met. |
| Standard 6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). | Weakly met in 12: 99 ff but weakly assessed and not in the Two-Year Final at all. |
| Standard 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. | Met. |
| Standard 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. | Weakly met. 12: 110, 120 meet the perpendicular specification but they are not assessed or used regularly enough to be confirmed. |
| Standard 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. | Met. |
| Standard 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. | Met, but inadequate. No division of polynomials except simplification. |
| Standard 11.0 Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. | Met, but inadequate. Essentially no perfect square trinomials but 13: 79 hints at it. |
| Standard 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. | Met, but too many are already in factored form and the skills are barely assessed. |
| Standard 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. | Met, but too many are already in factored form and the skills are barely assessed. |
| Standard 14.0 Students solve a quadratic equation by factoring or completing the square. | Inadequate. There are quite a few by factoring so “or” is satisfied. (See #19). |
| Standard 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. | Not met. There is too much of a program bias against “by type.” |
| Standard 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. | Inadequate. |
| Standard 17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. | Inadequate. |
| Standard 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. | Inadequate. |
| Standard 19.0 Students know the quadratic formula and are familiar with its proof by completing the square. | Not met. The standard is to “know”, while it is always available in the student’s Tool Kit and the proof is weak since completing the square is weak and not assessed. |
| Standard 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. | Weakly met and not assessed, the last topic of the course. |
| Standard 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. | Met, but weakly, more systematic methods are needed. |
| Standard 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. | Weakly met |
| Standard 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. | Very weakly met. 13: 77 purports to address it but most won’t see the relation. |
| Standard 24.0 Students use and know simple aspects of a logical argument:24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
24.2 Students identify the hypothesis and conclusion in logical deduction. 24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. |
Not met. The entire course confuses heuristics with logical argumentation. |
| Standard 25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step. 25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never. |
Weakly met. |
Comments Regarding Assessment Philosophy
Clear indication of the fact that the CPM program is not serious about students meeting the CA Mathematics Content Standards for Algebra follows from looking critically at their own words about, and examples of, assessment. For example, from the CPM Assessment Handbook, Math 1, “A good first step is trying group tests. [P. 3]” “We recommend for this course that you don’t give the usual type of in-process quiz that evaluates students’ mastery of skills while they’re still in the early stages of learning them. [P. 9]”
Why the de-emphasis on early skill mastery? It is philosophical, “When we test the students mastery of skills too early, the students focus is diverted from understanding where and how the skill is to be used.”
As shown below, CPM has carried this philosophy, that does have some element of validity, beyond logic to a point that lack of mastery of skills – ever – still allows students to get good grades in the course without having developed skill mastery. For example, the model final included in the CPM Outline for a Two-Year Program has no mention of parallel or perpendicular lines, no items that require an equation given two points or given one point and the slope, one simple item that implies factoring (solve x2 + 6x + 8 = 0) but no simplification of rational expressions, no mention of completing the square or quadratic formula, etc.
A strong student working at the level of the CA Grade 7 standards, i.e., a good pre-algebra background, would do very well on this final with no formal course, let alone two years of “algebra”! The proposed Team Final does address some of these in a minimal way but the two addition of rational expressions items are very easy and all but one of the expressions in the two quotient items are already in factored form, etc. Again, there is no mention, or use of, slopes in regard to parallel or perpendicular lines, completing the square, the quadratic formula, etc. So to argue, as CPM does, that the CA standards are met, just a bit delayed until mastery has had time to sink in, is nonsense. They are not assessed because they have not been mastered by sufficiently many students.
To put this in perspective, the reviewer’s daughter is in a school that uses one of the California approved texts at the appropriate grade level. Three-quarters of the way through her sixth grade program she was presented the first-year final of the CPM two-year program and had no trouble setting up the percentage word problems (“43 is what percent of 125”, etc.) in mathematical equation form out-loud as she read them. When she was shown the title, “Algebra – Part 1, Final Exam”, she laughed and then excused herself, “I’m sorry, but it’s kind of funny.” Excluding the CPM specialized “Diamond Problems” that supposedly lead to eventual factoring (of which there are none, even on the Part I Team Final), she would have been able to do nearly every item correctly, and will be able to do all of them by the middle of seventh grade per the California Standards for that grade.
Indicative is the outrageous presentation of the only real word problem, “Solve the problem and write an equation. You may do this in either order. If you do not need a guess and check chart to solve the problem use it to define your variables.” That is, use algebra to “solve” it or don’t. A table of trials is perfectly OK half-way through algebra, in fact its inclusion is mandated even for those students perfectly capable of solving the problem entirely algebraically.
Looking more deeply at the assessment philosophy, “the emphasis should be on the mathematical thinking evident in the work and on what the student knows, not on what the student does not know.[P. 9]” Several pages of the Assessment Handbook are devoted to scoring holistically. “Holistic scoring means just writing the score by the problem 0, 1, 2, 3, or 4 and not making corrections on the students’ papers. [P. 9, Bold is original.]” Still, the language, including that of the portfolios and the journals is sufficiently imprecise as to allow the possibility of clear, objective, individual student evaluation, “Assessment includes testing basic knowledge and skills, but it encompasses much more.”
Comments Regarding Research Support
The reality of this program is that the standards are not met and genuine assessment would quickly confirm that fact. The evidence and testing, both globally for all CPM students and locally as a teacher tries to assess a student’s knowledge, are entirely inadequate and the conclusions of studies as described in the Teacher’s Version is not nearly as conclusive as the writers imply.
For example, the first paragraph of the page entitled “Research Summary, Comparison of CPM and Traditional Students” is in regard to use of the CSU/UC Mathematics Diagnostic Testing Program (MDTP), data from eight schools that purport to verify that students in seven of the eight learned more in CPM-1 than in their traditional counterparts. Ignoring the fact that “traditional” is not defined and the schools are not named so it is impossible to see exactly what CPM-1 was being compared against, this study did not use the MDTP. It used only 20 of the available 50 MDTP Elementary Algebra items. Two word problems specifically designed for CPM evaluation were also included in the test. Key components of a traditional Algebra I course which are largely or completely absent from CPM -1 were omitted. The following tables indicate the breakdown of the original MDTP items into its subscales as well the distribution of MDTP questions used in this study:
in MDTP |
in CPM Test |
|
Another measure of CPM “success” is the state SAT-9 scores, in which a page of data purports to prove that CPM schools are more successful than “their peers who use other curriculum materials,” but there is so much missing as to make the data almost meaningless. As a start, the SAT-9 is not algebra! Of course, there are some exercises, ratio and proportion problems for example, that lend themselves to nice algebraic representation, but it is not algebra at the level of then President Clinton’s assessment, “Algebra is algebra!”
A comparison of the California Algebra Standards Test would be useful data, but even that (which CPM chose not to publish) would be comparing CPM-1 students against a far less homogeneous group, some using an even more aggressively “reform” curriculum. Beyond that, some schools use CPM for regular classes and a more traditional program for more advanced ones. That could be taken as evidence, probably supportable, that CPM is preferable to “general math” but hardly an argument for using it in place of a traditional college preparatory curriculum as its name would imply.
Finally, CPM cannot be trusted to give us an honest picture. The “MDTP” study that they continue to use is a clear indicator of that fact with its 20 of the 50 MDTP items. Since the public lacks the names of the CPM schools in the CPM summary sheet, it is impossible to do a quick comparison of SES factors, for example, to see if most of the CPM schools in these counties might have had a head start even before any choice of mathematics curriculum.
Comments Regarding Assessments in CPM
A much clearer vision of how far short CPM falls on more traditional end-of-course assessments is contained within their own Teacher’s Version and, most explicitly of all, within their “Outline for a Two-Year Program”, the guide to teachers for setting up the same program but over a two-year, less-demanding schedule referred to above. This document supplements the Teacher’s Version guides for constructing unit tests and tells what the designers really have in mind for verified competence. It is far off the algebra standards of California or, beyond that, of any other set of standards for algebra.
Here is one way that the assessment materials are designed to appear to be sufficiently demanding of standards-level competence when they are not. After Unit 3, the Assessment Handbook itself does not have model exams (they are in the Two-Year Program guide), but it does have item banks, by unit, along with instructions for constructing the tests themselves. Indicative in these instructions is “If you give an individual test …” That is, even the act of having an individual, bottom-line assessment can not be taken for granted in a CPM environment. Going onward with the quote, “it would be best to make this a very short test.”
Most indicative of all, however, is the test bank itself. The instructions recommend, “no more than one question of any type,” which would be reasonable advice if the items in each set were, in fact, of the same type. That is obviously not the case, the sets are constructed so that it is possible to avoid confirmation of the ideas involved at the level that a cursory look could imply.
Standards Representation in CPM Test Items
The test-bank items are not numbered (so as to make it easy to omit the item entirely) but some representative examples are the following from the indicated unit with item number counting from the first item in that unit looking at every reference given for the particular standard in the Correlation for the CA Standards for Algebra I .
Standard 8 Students understand the concepts of parallel and perpendicular lines and how these slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
Unit 7: 49, and 73 These words are not mentioned in these references, nor in the assessment-bank, Team or Individual. Ref. 7:83 looks at parallel for slope of 2/3 but not in general.
Unit 11: 101 This item does have students note that the slope of one specific line is -5/3 when the original line was 3/5 and that the lines are perpendicular, but without verification other than they look like it. Students are to “Record your observation in Your tool kit.” These words are not mentioned in the Study Team Questions (Team) but in #2 of the Individual Test (Individual), parallel is to be recognized by slope, without graphing. No student constructed equations are expected except when given two points.
Unit 12: 110, 122 Both of these meet the standard for perpendicular slope. Neither parallel nor perpendicular is mentioned in either the Team or Individual test banks.
Unit 13: 28, 61, 102 Items 28 and 102 meet the standard and 61 is borderline.
Standard 11. Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
Units 0-8: diamond problems 0: 3 ff, 2: 53, 78, 4:52, 6:70, 82, 96 All of these references from Volume 1 are so far off the standard as to be open to a charge of lying. For example, the last, 6: 96 consists of 11 exercises of multiplying a monomial or binomial times a binomial, not the reverse. Granted, converting expressions in factored form into un-factored form is helpful but it is does not begin to address, let alone meet, the standard.
Unit 8: 2-4, 10, 12 These are only preliminaries to factoring; e.g., 4 is only recognizing if an expression is written in any kind of factored form, 10 and 12 are tile-pushing with the factored forms already given. 8: 18 finally is an actual factoring but only with algebra tiles, 19 and 20 tie factoring into the earlier “generic rectangle” extension thereof, 31 is factoring out monomial common factors. 8: 50-51 does treat the difference of two squares and 8: 52 is ten mixed factoring problems, including one of them. 8: 57-63 meet the standards along with 70-71 and 77-80. The Unit 8 Team #1 has six expressions to be factored, including one difference of two squares and #3 uses factoring to solve quadratic equations in one variable, none of which is a difference of two squares, exactly as in the Individual, four quadratic equations to be solved, three that still need to be factored, two that are trinomial, and none that are the difference of two squares. In the Two-Year Program, that includes models of actual tests, not “select from the following,” the Unit 8 Individual Test includes six factoring items, five with assistance and one stand alone. There is one difference of two squares item, but not “by type” but only by coincidence. The test includes no quadratic equations to be solved.
Unit 9: 90 does FOIL factoring (not in the CA Standards) and this is the only reference in the Correlation but, in fact, they missed some, 9: 21, 42, etc. However, the Team Questions only include two factoring exercises and one quadratic equation to be solved. The Individual test includes a choice of three or two factoring items, one of which includes a difference of two squares, and block of five equations to solve that includes one quadratic. By contrast, the Two-Year Unit 9 Mid-Unit Individual Test (there is no unit test because of the up-coming First Semester Test, contains no factoring items nor quadratic equations to be solved. The First Semester Individual Final has two, x2 – 7x + 12 and x2 + 5x. There are no difference of squares items and no perfect square trinomials.
Unit 10: 1-3 ff, 17 Recurring exercises do confirm the ideas. The quadratic formula is simply given in 10: 86 so it is not clear whether or not factoring will continue in solving quadratic equations. In regard to assessment, several Team and Individual items have radical expressions or decimal approximations indicating that they are not to be factored since completing the square is not introduced until three units later. Somewhat surprisingly, given the end-of-course exams, the Two-Year Unit 10 Individual Test does have several factoring problems including the advice to look for difference of two squares and a simplification of a rational expression that requires factoring both numerator and denominator (See CA Standard 11).
Unit 11-13 practice in homework 13:79 hints at perfect square trinomials but this standard is not met under almost any level of generosity. The Two-Year Part 2 end-of-course Individual Final Exam is the most indicative, here. Instead of demonstrating that students finally have mastered these ideas, there is exactly one factoring exercise, #6b) Solve: x2 + 6x + 8 = 0. There are no simplification of rational expressions, let alone multiplication or division of them.
Standard 12. Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to lowest terms.
There are some, but see the last line of the Standard 11 remarks above. Since the assessment specifications allow for picking and choosing – they’re deliberately not numbered – it is impossible to say to what extent the program expects individual student competence.
Standard 13. Students add, subtract multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
There are some, but see the last line of the Standard 11 remarks above. Since the assessment specifications allow for picking and choosing – they’re deliberately not numbered – it is impossible to say to what extent the program expects individual student competence. Indicative of how far short the program is of the intended standard, there is one example given in the 1999 Mathematics Framework to demonstrate the intent of this standard: Solve for x and give a reason for each step: 2 / (3x + 1) + 2 = 2/3. There is no equation of this level of difficulty to be solved in the two volumes.
Standard 14. Students solve a quadratic equation by factoring or completing the square.
The same remarks apply; completing the square is an afterthought at the very end of the book, Unit 13: 67, and only with the CPM insistence on an overuse of so-called “algebra tiles” belying the problem with an odd or fractional middle term, and students are simply not expected to use it. In fact, the disclaimer at the beginning of the Unit 13 Individual Test admits as much, “We really do not expect many students to begin to master the topics in Unit 13. So, we provide a little extra assistance so we can still test them on these topics,” followed by inclusion of the quadratic formula with no items that require completing the square, not even with an even middle term and a pile of algebra tile. The Part 2 Individual Final is the most indicative, of course, neither is ever needed. The one quadratic equation is already in standard form and factors easily.
Standard 15. Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
From the Correlation comment one would expect this standard to be well met, “Word problems and investigations are at the core of the CPM program. Students regularly solve word problems without categorizing them “by type.” It is the second sentence that belies the first. Nearly all of the Unit 4-9 exercises are below the level of the algebra standards, many at the 6th grade standards, just far wordier and often with a great deal of irrelevant fluff. A good example is Unit 7, the entire “Big Race” premise. It is a complicated d = rt exercise without saying so but therefore not on standard at all, but a year or two off. Even among the last exercises that the Correlation indicates, Unit 13: 1, 12-14, 52, 62, 72, 78, 101, only 62, 78, and 101 meet the intended standard and nothing of the kind appears on the Two-Year Final. The only genuine word problem leads immediately to a pair of simultaneous linear equations. That qualifies but only as a small part of the intent of the standard. More ordinary problems are described in words and that is to be commended, but that is not an acceptable excuse for avoiding the others.
Other standards are far wide of the mark as well, and by design. CA Standard 24 is claimed to be met in the Correlation by lots of “Explain your answer…” and there is much of that in CPM-1. Most of them are not close to what the standard says and means. The books can be opened almost anywhere to see examples but using one that they chose:
Unit 10: 33 The item consist of three parts, solving a quadratic equation in standard form by factoring, graphing the corresponding function with these points as zeros and finally the supposed logical argumentation, “How are (a) and (b) related?” This not an unreasonable exercise, even good. The problem is that the informal argumentation of the exercise has nothing to do with the listed standards:
Standard 24. Students use and know simple aspects of logical argument:
24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
24.2 Students identify the hypothesis and conclusion in logical deduction.
24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.
Or another example on the same theme, 10: 96. Again, the exercise is reasonable; it is asking students to expound on the connection between the discriminant term of the quadratic formula and factorability. In fact, it would meet 24.1 if were it a bit more direct, something like, “Argue that if b2 – 4ac is a perfect square, then …” instead of “Explain what the values of sqrt(b2 – 4ac) tells you about factorability of the polynomial?” Few students would recognize that this new tool constitutes proof that the polynomial can or cannot be factored; they will not distinguish it from an earlier “logical argument” 10: 74 prior to the introduction of the quadratic formula, “Is the expression x2 + 6x + 2 factorable? Explain your answer.” Even assuming knowledge of uniqueness of factorization (which should get acknowledged in a formal setting), lacking the rational roots theorem, the ability to complete the square that (unconscionably!) is not introduced until three chapters later, or the quadratic formula, it is not at all clear what a “good” explanation would be! “I tried everything I could think of!” perhaps?
Final Observations Regarding The Curriculum Materials
In spite of being too far off the algebra standards to warrant state approval but on the positive side, the books themselves have, or rather, Volume 2 has some distinct advantages over more traditional texts. The regular, mixed review of exercises is excellent. So is requiring students to present sensible explanations in support of their conclusions. If the authors of the Teacher Version introduction materials were to abandon their strong allegiance to what’s often known as “Authentic Assessment,” i.e., team tests, journals, portfolios, and individual observation and get back to lower case authentic assessment that algebra teachers everywhere have traditionally done, the program would be greatly enhanced. Yes, there should be more “by type” word problems and they should be part of the regular mixed review exercises, more formal logical argument, actual use of completion of the square in a way that students are expected to be able to use it, etc., but that is not the worst problem. The worst problem is pedagogical.
Volume 2, in the hands of a competent traditional algebra teacher, could be reasonably effective; short of the standards, but effective. If all of it, through Unit 13, is covered and if individual students do all of the exercises, they will master enough of the skills to be considered competent at the level of algebra 1, capable of going forward successfully in mathematics that depends on these concepts and skills. The problem is, and it is convicted as fatal by the included Individual Tests in the Two-Year Program guide, there is no assurance – nor any genuine effort to confirm – that this is what will happen. A little group work from time-to-time is fine, having stronger students help weaker ones master the material likewise, but that is not the described philosophy; in fact, it is completely the reverse. Similar is the work with algebra tiles. A little at the beginning? Sure, why not. Plastic toys instead of mental manipulation far into the course is very different. But, although serious, the worst parts are not the leisure introduction to factoring, that does have supportable logic, or too much graphing calculators, or being too far off the standards. They are two; one is pretending that everyone is mastering algebra while letting strong students carry weak or lazy ones, and the other is the heavy-handed, non-algebra, time-wasting in Volume 1.
The latter problem is associated with the first pedagogically and philosophically although the supporters would phrase it differently, of course. From the Introduction and Overview, “Telling has little to do with promoting learning – students must construct their own understanding.” In a sense, this is true since we do not learn by “direct download” but neither do we learn much mathematics by activity-based insight. Being told wrongly by a convincing fellow student, a situation common in un-led team settings, is far worse than being told what is correct by a competent teacher. Yet, “The daily activities in this course will require much more work in study teams and much less introduction and explanation of ideas by the teacher.” Much of Volume 1 actually detracts from developing algebraic competence. Almost all of the mathematical content is at the level of the Grade 7 standards or below, e.g., the equations to be solved all are, but the activities are still very time consuming and sometimes frustrating. The worst of all, however, is not teaching the power of algebra itself. Unit 4: 123 is TOOL KIT CHECK UP and it is mandated that it contain Guess and Check tables, and “cups and tiles to model solving equations.” This is not algebra and it is not college preparatory math, no matter what it calls itself. Eventually, Volume 2 starts teaching some algebra but it is too little and too late.