May 1997
Prepared at the request of the Core Knowledge Foundation, this is a review of mathematics materials from several different publishers at two grade levels, second and fourth, with an emphasis on the comprehensiveness of their coverage as measured against the stated standards of the Core Knowledge (CK) Sequence of the Core Knowledge Foundation. Materials from four publishers were evaluated:
It is reasonable to extrapolate to other nearby grades from my remarks herein; there is a philosophy of learning that drives materials preparation coherence. One instance where that is less valid is in the Saxon materials from grade 4 to grade 5. The K-4 materials were separately written and tested by Nancy Larson, whose name appears on the materials, prior to being published by Saxon. The Mathematics 65 book, the grade 5 book for appropriately prepared students, has a distinctly different presentation than the materials for earlier grades.
Summary
After having looked closely at these materials, my ranking from first to last is Saxon, CMC, Sadlier, with UCSMP a distant last. In more detail, I rank Saxon and CMC both as solidly acceptable, Sadlier less so but still acceptable, and UCSMP unacceptable. Both the Saxon and the CMC have many more word problems than Sadlier and represent a continuing use of the ideas that will help develop longer, even lifelong, retention of them. The Sadlier concentrates much more heavily on computational competency; to my way of thinking, excessively so. More good applications to develop appropriate use of algorithmic competence and thinking skills with fewer drill exercises would be more appropriate.
In some situations, I might interchange the ranking of CMC and Saxon. Some elementary school teachers are ill prepared to teach mathematics, and very inexperienced teachers can seriously misjudge how to present the material appropriately to their classes. The CMC program provides specifically worded scripts for teachers to follow in presentation of the material. An experienced and mathematically competent teacher would find these unnecessary and perhaps even offensive. A wise administrator might allow such a teacher (assuming student performance indicators remain high) to depart from the text when more comfortable doing so.
Additional Detail
Sadlier-Oxford
Sadlier is clearly inferior to the Saxon and the CMC materials The greatest difference is that the Sadlier materials suffer somewhat from some of the objections mathematics education reformers level at their perception of traditional school mathematics, page after page of computations with too little emphasis on using the ideas they represent. Although automaticity of fundamental operations is important - in fact, crucial - it will develop over time with regular repetition, and varying the exercises within each set helps keep students' minds in gear as well as their pencils. These materials have insufficient variation within a given exercise set although plenty of practice overall.
There are minor embellishments to make the Sadlier books acceptable to modern mathematics education decision makers such as use of lots of color, an occasional problem-solving situation, and questions about greater or less than which would not have been present 30 years ago. Everything else is not much different. I do not mean to imply that this is bad. In comparison with the materials that California has ranked very high and which are being rapidly adopted across the state, students and teachers would be much better off with the Sadlier series.
The problem is not lack of coverage, nearly all of the specifications of the CK Sequence are met by the Sadlier books. The few discrepancies that I noticed could be worked around without difficulty. For example in grade 2, the thermometers only read to the nearest 5 degrees instead of the nearest 2. Line segments are not identified in the AB form that is specified. Clock time is not distinguished as to a.m. or p.m.
The great difference is that these books offer far less "incremental review" than do the Saxon or CMC. By incremental review I mean that, once a concept or skill is introduced, it is consciously and regularly presented in later exercises, gradually increasing in depth and difficulty. This feature in mathematics materials helps to cultivate the automatic recognition of what is to be done, essential for using mathematics effectively, as well as steadily reinforcing the mechanics themselves. It also helps minimize student frustration with a new, and not well understood, concept. Each day's student work should include some items that are "old hat" thereby removing the "I didn't understand" excuse for not turning in a homework paper. While straight arithmetic is heavily covered in the Sadlier materials, almost all of the applications of the mathematics are covered in the chapter where they are developed but then seldom revisited. In grade 2, telling time from a "face" clock is covered in Chapter 5 but the only clock items in the rest of the book are in the first Cumulative Review at the end of Chapter 6. No work with telling time is included in the later Cumulative Reviews, let alone thrown in every once in a while to stay current. Thermometer reading is similar (though inherently easier and less critical in second grade). Number families represent an easy idea that is helpful in developing mental facility Again, they are covered in Sadlier, but too much in one place and not enough everywhere else. I didn't see any past Chapter 5.
Similar comments apply to the grade 4 materials. For example, negative numbers are introduced in the middle of the book and then not seen again until late. In addition to that, they are only present in the context of one very specific situation the first time, a thermometer, and two specific situations, temperature and elevation, when revisited late in the book. The idea of the number line itself is introduced early but used very little through the book until revisited with decimals toward the end (P. 450). The negative numbers have not been included on number lines even though there is a thermometer with negatives on the opposing page. Presenting pictures of the general situation when no effort is required to do so, and no confusion will arise from doing so, is important and that idea doesn't seem to be grasped.
Pages of 20 or 30 problems of exactly the same type, for example pages 181,183, 185, 187, and 189, lend themselves to legitimate criticism from math reformers. No matter how many colors the book is printed in, they are boring. Better to have 5, or at most 10, and mix up the exercises with more word problems (there will never be too many - the Russian Grade 3 has 1091 of them and nothing else!) as well as some review exercises of a computational nature perhaps, but of various types.
As with grade 2, the content of Sadlier grade 4 is a little below that of the CK Sequence. Square roots aren't done at all and squares are "stuck in" and dropped rather than "playing with them" in some interesting exercises from time to time. Roman numerals don't include M. Decimals go to hundredths, not thousandths. A much greater problem is the lack of deliberate embedding of the ideas in occasional word problems after they are introduced. Though there may be an adequate number of word problems overall, seldom do they require any knowledge other than fairly obvious simple arithmetic operations. To be sure that units of measure are really known as specified in the CK Sequence, they should get used from time to time (maybe with a "recall from page ..." for awhile) rather than covered and then almost never seen again.
CMC (Primarily Level D)
The organization of this material is clearly superior to Sadlier. There appears to be ample practice without running it into the ground and, after a topic is introduced, something similar will show up regularly in the word problems. The materials do not provide an index nor table of contents which makes it harder to check on specific CK Sequence topics but the Objective Facts of the Teacher's Guide is helpful. There are some items that are not covered, thousandths for example. I was not able to find congruence and similarity nor Roman numerals. Overall, however, my judgment is that the mathematical retention and subsequent ability to use the mathematics in applications will be substantially better with CMC than with Sadlier.
One of the most obvious differences between the CMC materials and those of any other publishers that I have seen is the fully scripted format of the Presentation Books with additional suggestions in the Teacher's Guide. For example, from page 69 of the Teacher's Guide:
Incremental review is well developed in the CMC materials. Take, for example, the concept of the area of a rectangle. In the first few lessons, square units are pictured. In Lesson 14, students finally see a rectangle, 5x7, without the squares shown. Area problems are a regular part of exercise sets thereafter. For example, in Lesson 32, students are to sketch the diagram and find the area of a wall 23 feet wide and 8 feet tall. By Lesson 97, it is a garden 37 yards long and 45 yards wide for the most obvious - but important - kind of incremental review, harder arithmetic. Another shows a more sophisticated kind; it involves division instead. The exercise is a ranch of 54 square miles that is 6 miles wide. The problem, of course, is to find its length.
Saxon
Saxon is clearly superior to Sadlier and, in my judgment at least, slightly superior to CMC. The K-4 Saxon materials were written by Nancy Larson in the format of daily worksheets.. Students do one side at school with some guidance and the other at home as homework. That format and the materials themselves work very well. Nearly all of the CK items are covered and the level of incremental review is so thorough that students will almost certainly know the material. As an example, take one side of Lesson 131 near the end of grade 2. The first exercise is an exercise in words involving 6 cars. Students are to write a number sentence describing the number of wheels (multiplication, not any "child developed strategy") and give the answer. Exercise 2 involves use of the ideas and the vocabulary of parallel and perpendicular. Exercise 3 is rounding, including rounding numbers first and then finding the sum. Exercise four is doubling two items. Exercise 5 is regrouping 5 dimes and 18 pennies into 6 dimes and 8 pennies but with no pictures of coins to be drawn since, by this point, such problems have become strictly mental. The final exercise is four problems of two- digit addition and subtraction, two written vertically and two horizontally. Tomorrow these will be division at the level of 16÷2 and yesterday, they were multiplication problems at the level of 6×4, concepts that are being developed late in second grade.
By fourth or fifth grade, however, a transition is appropriate so the change of format to textbook form is warranted. I have not yet seen the Nancy Larson Grade 4 materials but am quite familiar with the Saxon Math 54, so my remarks are based on that book. Math 54 presents a sequence of lessons (142) rather than being broken into chapters. Once they have been introduced, the ideas are used throughout the book . Nearly everything from the Core Knowledge Sequence is present in the book and, more than that, used. In the numbers, fractions, computation, and measurement categories, I only saw use of the square root sign missing. Geometry is a little below the CK list, there are no parallelograms or trapezoids, for example, nor use of the ideas of similar and congruent figures. The latter is a little surprising since congruence, and perhaps similarity too, have already been introduced in the K-3 materials.
Saxon 54 doesn't have the formula for the volume of rectangular prisms, either, but I do not consider it missing; in fact, quite the opposite. There are exercises over several lessons that are building the concept to get ready for its formal introduction in the following book. This is first introduced on p. 424 in an example:
Example 3: This rectangular solid is made of how many small cubes? (With a picture of a 3×2×2 box of cubes.)
Solution: We see that the rectangular solid is made of two layers of cubes with 6 cubes in each layer. 2×6 = 12. The rectangular solid is made of 12 cubes. (The two layers are explicitly identified in a second picture and the 6 cubes of one layer individually identified 1-6.).
It shows up as a 4×2×3 exercise a few pages later, as a cube of cubes, etc. and for the rest of the book. It would be hard for any student not to be getting the idea of volume so deeply that V = lwh will be only a formalization of an idea already well understood rather than a formula to be memorized.
The real strength of all of the Saxon materials is their fully developed philosophy of incremental review. Topics are introduced gradually. Immediate competence is not expected since different students pick up new ideas at different rates. Eventually, however, every student is doing every topic with reasonable proficiency. This philosophy persists all the way down to the kindergarten materials and it works very well.
UCSMP
At the second grade level, the Chicago Everyday Mathematics materials include a little book entitled "Minute Math+" that is reminiscent of the CMC scripting without being quite so overbearing. The items do appear to be good classroom discussion items.
There is a book of "Home Links" with some good ideas for children's activities away from school along with some silly ones.
There are a couple of "Activity Books" that in total are not one-fourth that of the Saxon homework sheets.
There are a couple of "Journal" books that aren't. They are really two more activity books that still don't bring the total up to half of Saxon.
There is a K-3 "Assessment" book with a lot of pages of nothing worthwhile but several pages of student-by-student evaluation of each item of a list similar to the CK Sequence. It would take a lot of time but those pages would be genuine assessment if used correctly.
There are two large volumes of "Teacher Manual & Lesson Guides." A standard day would be to have some small group record some observations (what is the temperature on the thermometer, how many different rectangles could you make on a geo-board, etc.). This may or may not be tied into a "Journal" activity or a "Home Link."
In normal classrooms with normal teachers, I would characterize these materials as "dangerous." My impression is that it would be very difficult to be sure that appropriate material has been covered adequately. One can expect a very high degree of teacher variability. Knowledgeable teachers, well grounded in the materials, may be able to pull it off; at least it's clear from the assessment book that there are some things that the children are supposed to know. There is almost no routine practice, although a small amount is built into the activities.
The fourth grade format is similar, a "Student Record Book"(let), a booklet entitled "World Tour" with maps and data, a couple of Journal books that aren't (they are activities), a "Study Links" activity book, a teacher's Reference Manual, and two large "Teacher's Manual and Lesson Guides." The same concerns are appropriate here as well. An example, randomly chosen, p. 444, Lesson 98, Multiplication and Division Number Stories, half way through Volume B, so three-quarters of the way through the year, starts with a "Math Message" for the day. "There are 6 rows of chairs with 4 chairs in each row. How many chairs there in all? A bit later is the "Follow-up." "Ask several students to give their answers to the Math Message and describe their strategies." Long before the end of third grade there should be only one "strategy." Multiply, and do it correctly! Contrast this exercise with the self aggrandizing words to the student at the beginning of Journal 1. "Fourth Grade Everyday Mathematics builds on this basic training [the preceding paragraph] and begins to make the transition to mathematics concepts and ways of using mathematics that are more like what your parents and siblings may have done in high school. We believe, along with many other people, that fourth graders in the 1990's can learn more and do more than was the case ten or twenty years ago."
The fourth grade package of materials is even more dangerous than the second grade. I simply cannot tell if the CK Sequence items are being mastered or not. Everything is up to teacher competence to fill in the gaps, provide the routine drill, and confirm that it is happening. Much will be, of course. For example, page 118 of Journal 1 has the student fill in missing fractions on a number line. But the opposite page, Exercise 3, says "List the countries on page 32 of your "World Tour Book" whose flags have a circle". That is a kindergarten exercise, first grade at most.
Conclusion
Whether or not all of the "reform movement" curricula - Quest 2000, Math Their Way, etc. - are as susceptible to omission, or at least lack of certifiability, of the CK Sequence specifications as the UCSMP curriculum would have to be confirmed on a case-by-case basis. Given that the UCSMP materials are often perceived as the best of the genre, however, I would be extremely wary. One that I have looked at in some detail, though not as carefully as I did with UCSMP, is California's current largest seller, MathLand, by Creative Publications. Based on my cursory perusal and substantial California student performance data, I would go from "dangerous" to "poisonous." Any more traditional text such as Sadlier is preferable. Even more preferable is one that does not overlook the importance of automaticity, of course, but does it in the style of pervasive incremental review. Both CMC and Saxon do this well and my impression is that Saxon does it better, at least in the hands of competent teachers.
Wayne Bishop, Ph. D.
Department of Mathematics & Comp. Sci.
California State University, LA
Los Angeles, CA 90032