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California Review:
"Math Trailblazers"

    The following review was prepared by Wayne Bishop when he participated in the state of California's "Content Review Panel" evaluating math programs and curricula under consideration for that state's schools. "CA" in this document refers to California. The items marked are the selected descriptive phrases chosen by the reviewer from among multiple choices in the California review process.

    This review makes frequent reference to specific items in the California Mathematics Framework. You can access the PDF version of that framework here.

    Reviewer's Name: Wayne Bishop

    AB 2519 Content Review Panel

    Name of Publisher: Kendall/Hunt Publishing Co.
    Name of Program: Math Trailblazers
    Identification Number: None Indicated in Standards Map
    Intended Grade Level(s): 4-5

    "Content-Only Summary Recommendation"

    Based upon evaluation of its content only, this submission is NOT recommended for adoption as either a BASIC or a PARTIAL program.

    Notes and Citations

    In Summary, too many of the CA standards - both in number and in their relative importance are not met by this submission. See detail in the block on Addressing of Standards.

    In fact, there may be somewhat more coverage than is indicated herein. This submission is unbelievably poorly organized as to determining whether a standard is or is not met. Neither the students' books nor the teacher's manuals have indexes and the lessons in the Tables of Contents have "cutesy" names that are not very indicative of their content, e..g., "A Matter of Survival", and "Searching the Forest". In addition, the teachers' guides are boxed as individual units so considerable jumping around is needed to follow their map since the consecutive page numbers given are to these separate booklets instead of the students' books. Still, the conclusion remains accurate.


    Type of Program

    This submission is not comprehensive, and it does not contribute clearly and significantly to the course of study.

    Notes and Citations

    This submission is comprehensive but it does not meet the objectives of the standards, see the block on Addressing of Standards.

    Fundamental Skills

    This submission is not based upon the fundamental skills required by mathematics, including, but not limited to, basic computational skills.

    Notes and Citations

    This submission is well below the CA standards for fundamental skills rather consistently. Some examples are the following:

    NS 2.0 The standard requires calculations of the four functions with fractions and decimals. The standard is not met. Multiplication appears to only be in Unit 9, Lesson 5, and division is not included at all. Addition and subtraction are done with subdivisions of rectangles that represent a whole. Almost as an afterthought, the last half of Lesson 7 of Unit 5 take up the topic without actually drawing fraction rectangles although a step in the process is to "think of dot paper rectangles". The one good feature is that students are regularly reminded to think about reasonableness of answers, even before beginning the actual computation.

    NS 2.2 requires "proficiency" of long division with multiple digit divisors (single was the grade 4 standard). Although a somewhat awkward algorithm is presented in Unit 9 Lesson 2, instead of proficiency we have (Unit 9, Lesson 5, P. 315), "Estimate the size of each quotient. Use your calculator to find the answer to each division problem. Change each remainder [the decimal part] to a whole number." Amazingly, even proficiency with addition and multiplication is not clear. It appears that students have calculators at the ready and may know little about basic computation with decimals.

    See the block on Addressing of Standards for more examples.

    This submission does not provide for basic skills instruction that is systematic and explicit.

    Notes and Citations

    See the above.

    Addressing of Standards

    This submission does not enable instruction in almost all (if not all) of the individual standards for the intended grade level(s) or discipline(s), either in whole or in one or more of the subject area(s) or discipline(s) listed above, in a cohesive, clear, systematic, and significant fashion.

    Notes and Citations

    Note: I only looked at NS, AF, and MG because the submission was failing so badly on these most fundamental ones that considering other areas of the standards seemed pointless.

    Grade 4

    NS 1.6 There is such a decimal-off-of-the-calculator orientation to this submission that even this standard of connecting the most elementary ordinary fractions such as 1/2 and 3/4 to their decimal equivalents is not met.

    NS 1.9 There is no number line in Unit 6, Lesson 4, let alone one that includes "relative positions of fractions, mixed numbers, and positive decimals" as required by the standard. That section has some ordering from smallest to largest as does Unit 12, Lesson 3, for ordinary fractions (using a fraction strip chart of various denominators) but there is no number line in the book except as rulers and edges of graphs.

    NS 2.1 The standard indicates two decimal addition and subtraction. No decimals, not even dollars and cents representation is in the book until Unit 10 (not 3 and 6 as indicated!) . The submission has a most interesting split personality: It is heavily calculator dependent but there is little development of what the numbers read off a calculator might mean.

    NS 2.2 Again the submission takes liberty with the standard that calls for rounding two-place decimals. Decimals have not yet been introduced at the indicated standards map reference, Unit 5, Lesson 3. These are positive whole numbers and nothing more! The fact that they are centimeters may make them feel like decimals to someone more experienced but they are still only positive integers.

    NS 4.0 The idea of factoring is not in the book.

    AF 1.2 Expressions with parentheses do not exist. Their recommended reference, Unit 7, Lesson 1, is the simplest succession of key strokes for "computations" such as 319/2. This submission is extremely limited in regard to any algebraic or functional symbolism. AF 1.3-2.2 (the end of the AF section) are not met.

    MG 1.4 Formulas are essentially nonexistent in this submission. The standard is for perimeter and area of rectangles by formula but the reference only has the simplest numerical examples.

    MG 2.1 The standard is not met there are no functions of any kind in the book. Standards 2.2-2.3 are also not met.

    MG 3.2 Despite the unwillingness of the publishers to acknowledge non-alignment with the standards, none of the specifically identified words, radius, diameter, or even circle itself(!), appear at the given reference. Instead, Grace and Tanya make a "spinner" using a protractor. Most of MG 3.3-3.8 are either not addressed at all or else are addressed but then dropped very ineffectively.

    Grade 5

    NS 1.5 Unit 3, Lesson 3 finally has number lines but they don't include negatives, decimals, nor mixed numbers

    NS 2.1 The only division found was not at the reference but at Unit 9. Lessons 4-5. Decimals are not included. Multiplication by a decimal is in Unit 7, Lesson 6, as indicated but only by a single digit and there are no negatives. Conventional algorithms for arithmetic appear only minimally. There appears to be no systematic goal to learn them well - nor even to learn them at all.

    NS 2.3 Division by two digit numbers is considered in Unit 9, Lesson 2-5 (which the standards map does not indicate) but includes no positive decimal work and division by two-digit divisors is sufficiently confusing that the authors do not show that this works for three-digit divisors, or anything higher.

    NS 2.4-5 These are not met; there is no division of ordinary fractions!

    AF 1.0 There is insufficient experience with "variable"; this submission is very weak with respect to algebra readiness preparation. AF 1.2 is related to this formulas for area of rectangles and triangles are essentially the only use of variables.

    AF 1.3 For the publishers to call their given example, Unit 11, Lesson 3, an exemplar of the standard that students are to "know and use the distributive property in equations and expressions with variables" is completely inadequate. Not even a numerical example is given, let alone an algebraic one. Since there is no index, it would be difficult to guarantee but most likely there is not a singe case of the use of the distributive law (except intrinsically in place value calculation) in the entire book.

    AF 1.5 Plotting linear functions with integer values. Since there are no functions, there are no linear ones. The publishers' reference is to the entire Unit 10 which is closer to the game of Battleship than meeting the standard. MG 1.0-2.0 None of these standards are close to being met. For example, 1.1 requires that students derive the formulas for area of a triangle and a parallelogram, relating them to an appropriate rectangle. The triangle formula is almost just given, Unit 15, Lesson 4. Although somewhat motivated by geoboard sub-division into two right triangles, since there is no symbolic manipulation in this book, formally adding the areas of the two is not attempted, the formula just appears. Non-rectangular parallelograms don't appear at all, let alone their areas.

    Standards Alignment

    This submission is not aligned with the standards for the intended grade level(s) or discipline(s), meaning that (1) it does not enable successful instruction in the individual standards it covers, (2) it includes something fundamentally contrary to the standards, or (3) some content extraneous to instruction in the standards does detract from the ability of teachers to teach readily and students to learn thoroughly the content specified in the standards.

    Notes and Citations

    See Addressing of Standards

    Factual Accuracy

    This submission is not factually accurate and the inaccuracies cannot reasonably be corrected.

    Notes and Citations

    In a very real sense, this submission is grossly inaccurate. Not in regard to individual mathematical facts, but in a larger sense. This submission gives the impression that mathematics "works" in ways quite differently than how it does. Mathematical conclusions are not made by trying a few small values on a calculator and extrapolating ad infinitum. There are reasons for using symbolic representations, formulas and the like (not hiding them from students), and there are reasons for becoming comfortable with symbolic manipulation to an extent that new relationships can be proved, not just concluded to be true because of some geoboard observations or paper cutting and folding. It is an inherent problem in many mathematics education settings - should a useful result be put off until it can be done "right'? Not necessarily by any means, and if there's a useful formula to learn, then learn it. If you can prove it, or at least see the essence of a situation that will later become a proof, so much the better.

    Research Base

    To the extent this submission incorporates principles of instruction that are not reflective of current and confirmed experimental research, the bases of those principles are either not appropriately identified or are misleadingly presented as experimental research.

    Notes and Citations

    This submission continues to survive in spite of the existence of experimental research that it is not effective. The student books, evident from looking at the teachers' guides, are really a set of separate units that were widely distributed under the name TIMS, a curriculum project out of the University of Illinois at Chicago. More than ten years ago, this reviewer heard the lead author discuss the concept and curriculum which had already been class tested for several years. In response to a question from the audience about student performance, he admitted that student performance in mathematics "had not gone up", which many of us interpreted as had, in fact, gone down. He assured us that the understanding of mathematics did go up with his curriculum but did not tell us how that had been determined. He also reassured us that the understanding of science, another goal of this integrated curriculum, did increase. No data was provided for either mathematics or science and the science conclusion should be considered speculative since 4th and 5th graders are not often externally tested in science (California's STAR scores, for example, do not indicate a science grade until grade 9). What instruments they may have been using to make their conclusions is unknown. The mathematics component, however, is known. Objectively measured student performance did not improve.

    Adequacy of Coverage

    This submission is not adequate in its coverage.

    Notes and Citations

    See Addressing of Standards.

    Other Comments

    This submission falls far short of the standards for the grades at which it was submitted. It should not seriously be entertained as a candidate for these grades.

    AB 2519 Content Review Panel

    Other related pages:
  • Return to main MATH page of Illinois Loop
  • Colorado parent review of Math Trailblazers

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