In 1989, the National Council of Teachers of Mathematics (NCTM) released Curriculum and Evaluation Standards for School Mathematics. These standards quickly became so widely accepted by the math establishment that they were, and still are, considered by many to be the best available guidelines for teaching mathematics. These standards have lead us past the "new math", the "back to basics movement", and on to the "new new math" (Finn, 1993). This new thinking suggests that the right answer isn’t as important as the methods students use to reach their answers. The main ingredient in this recipe is problem solving. The NCTM encourages students to do complex exercises in groups and to worry more about how to approach a difficult problem than whether they’ve misplaced a decimal point (Mathews, 1993). Some disagree with this method; among them are Frank Allen, a former president of the NCTM. According to Allen (1988), the NCTM advocates a flawed approach to math that says, in essence, "if we emphasize the applications of school math in a wide variety of problem-solving situations, two good things will happen; (1) the student will somehow learn the mathematics needed to solve these problems, and (2) seeing that mathematics is useful, he will be motivated to learn more math. According to my experience , students must know the mathematics before they can apply it. Or to say it differently, they cannot apply math they do not know. To expect them
to learn math in the process of applying it is preposterous. It is like trying to teach people to play water polo before they know how to swim." (p. 2).
Many educators have become dissatisfied with mediocre test scores and the seemingly haphazard approaches to the teaching of math. During this particular moment in education, the buzzword is "accountability", and standardized test scores are the most commonly used measurement for holding schools and teachers accountable. Because of an emphasis on test scores and a growing discontent with the instructional strategies of traditional math programs, educators across the country have sought new methods to improve student achievement.
Enter John Saxon. This man’s appearance on the educational scene with the publication of an algebra text, Algebra I: an Incremental Approach (1981) ignited a discussion, which is still simmering concerning what constitutes "best practices" in math (Hill, 1993). His success, and the acidic reaction to it, has exposed a deep split in the American educational-reform movement between classroom teachers and reformers (Mathews, 1993).
The NCTM stresses real-life problem solving, and most traditional text- books try to emphasize such strategies in their word problems. The colorful graphics in a traditional text might picture someone in a wheelchair, or have a character called "Flat Matt" when studying plane figures. The problems may describe situations in bakeries or warehouses. Saxon calls them "pseudo-real life problems" and states, "You must design the problem so that the concept is the only thing that’s real in the problem," (Hill, 1993). He maintains that such problems, like colorful graphics and photographs, only distract students from the main task at hand, learning math. Word problems in a Saxon text might resemble this: "Four-fifths of the pixies in the kingdom had sad faces. If 840 pixies had sad faces, how many pixies lived in the kingdom?" His word problems may contain words that students have never heard before, i.e. "the ratio of the erudite to the unlettered", but Saxon defends this practice by saying if students learn to work problems with unfamiliar words and realize they have no effect on the actual calculations, than they will not have trouble with chemistry problems containing words like trinitrotoluene (Mathews, 1993).
With the opportunity to choose many colorful, "tasteful" textbooks from the math menu, almost anyone would find the Saxon text bland. It is a sea of gray ink compared to newer traditional texts. Some teachers criticize the books because of the absence of traditional chapters. Others find it mechanistic, repetitive, and boring (Stengel, 1981). Many criticize Saxon’s textbooks as sterile and uninviting. An evaluator for the state of Virginia turned down one of Saxon’s algebra books with this rationale: "The text does not adhere to the NCTM Standards using manipulatives, calculators, computers, discovery methods. . .The review sections are not grouped. The text would not attract the attention of the average ninth grader because it has no color or real people."(Hill, 1993).
To state education departments and textbook advisory committees, Saxon represents the worst of the old school. Saxon’s opponents claim that his methods remain unproven. Mary Lindquist, past president of the NCTM, cites studies in Pittsburgh and a small town in the Southwest showing little or no improvement in students taught with Saxon’s textbooks. Saxon’s emphasis on drill and practice is similar in some ways to the Kumon system, which is popular in Japan. Even Japan has its share of education reformers, and though they have consistently produced students who out-perform their Western counterparts, Japanese officials now worry that their methods "yield only skillful mathematicians who are not sufficiently creative."(Mathews, 1993).
A key aspect to the Saxon books is the prevalence of the spiraling technique. Students learn different algorithms horizontally over time rather than vertically all at once. This gives students, even those with learning difficulties, time to learn. The books are heavy on traditional word problems which many teachers tend to leave out when making assignments (Hart, 1996). "I finally figured out that you learn to work problems by working them repetitively, over a long period of time," says Saxon (Hill, 1993). In one study, students who had worked one word problem every night for four months more than tripled the score of those who had studied out of a book with no repetition (Saxon, 1984). This spiraling, or incremental, technique is sometimes called continuous distributed review (CDR). Spiraling requires comprehensive coverage of all material on a regular basis (every two or three days) without increasing the length of the assignments as the year progresses. Students always encounter concepts addressed in a previous (or current) lesson, and each lesson’s concepts will appear in some form throughout the remainder of the year (True, date unknown ). Typically, most math textbooks now in use spend three to five pages developing one concept. The students are tested on the material at the end of the chapter. Then the book moves on to the next topic and, except for a few reviews questions now and then, the concept is dropped in order to move on to a new one (Saxon, 1984).
Critics of Saxon’s approach maintain that what he calls "gentle repetition" is nothing more that another term for "drill and practice" or "rote learning". According to Saxon, the reforms he suggests are not a return to boring drills or mind-numbing "back to basics", but sound practices that will make fundamentals automatic (Saxon, 1984). Even education guru Benjamin Bloom acknowledges that procedures must become automatic in order to free up the mind for higher-order thinking and reasoning.
Some studies conducted over the past two decades seem to give credence to the effectiveness of an incremental approach such as Saxon’s. In 1980-81, the incremental approach of Saxon Math was used in a pilot program in twenty Oklahoma schools involving 1,365 ninth grade algebra students. In tests of skills at the end of the year, students from the Saxon group scored, on average, more than twice as high as other students (Saxon, 1984). Richard Wheeter, of the Oklahoma Planning, Research, and Evaluation Department, directed a study of the Oklahoma City Schools for the 1992-93 academic year using the Saxon Math program. Saxon was compared to the Scott- Foresman program and scores from the Iowa Test of Basic Skills used as an indicator of achievement. The results showed that Saxon Math students scored higher in all normal curve equivalent comparisons. Teachers using this program reported improved active learning, greater attention span, and more obedience from their students. They also felt their students were more challenged and less intimidated by math, and that they retained more information from their lessons (Wheeter, 1993).
In 1985, the head of the math department of North Dallas High School began using the Saxon program. In three years the percentage of students passing the Texas State Skills Math Test jumped from ten percent to 91 percent. Scott High School in Huntsville, Tennessee began using Saxon texts is 1989. In three years the average ACT math scores increased from 13 to 22 (30 is a perfect score on the ACT) (Durham, 1995). In a 1992 study by Robert Calvery, 190 second and third grade students were assessed, with one section in each grade receiving instruction in Saxon Math and the other three sections using the Holt Mathematics series. Students using the Saxon method scored significantly higher than those in the Holt book (Calvary, 1993).
At San Fernando High School in Los Angeles, California, the Saxon method in Algebra 2 and Calculus were implemented during the 1993-93 school year. The Saxon students nearly doubled the performance of those students still using the Merrill series on the UC/CSU Pre-Calculus Readiness Test. In this same school district, Compton Elementary School was one of ten elementary schools targeted to help at-risk students. From the 1993 CTBS to the 1995 test, the math scores increased from the 35th percentile to the 55th percentile, an impressive gain (Hart, 1996).
Mathematics tends to be a textbook-oriented course, in which the textbook often determines the instructional process. When a class follows a textbook that is based on principles of sound instruction, increased achievement should result. Conversely, when a class follows a textbook that depends on the teacher to incorporate material from the text into an overall process of sound instruction, the results fluctuate with the teacher’s subject knowledge and his ability to engage student learning (Klingele and Reed, 1984). The NCTM doesn’t believe that textbooks should drive instruction: "Rather, other materials that support the standards such as manipulatives and courseware, must be developed, in addition to new textbooks." (Hill, 1993).
There is no such thing as a foolproof recipe in the kitchen or in education. In comparing the Saxon and traditional basal approaches to teaching mathematics, it is hoped that the ingredients for success in math will be highlighted.
PURPOSE:
This study will examine the math achievement of fourth graders who have been taught the Saxon Math program with an incremental approach to instruction as compared to students who have been taught with a Basal Math text using a unit-based approach. Achievement of these students will be measured by scores on the Iowa Test of Basic Skills (ITBS).
BACKGROUND:
Pepperell Elementary School is a brand new school built in 1998 located in Lindale, Georgia, a small mill town on the outskirts of Rome. It houses only fourth and fifth grades, which come from three feeder primary schools, and has a school population of approximately 550 students. Two fourth grade classes from this school will be used in this study. Both classes are heterogeneous, including a mixture of average, gifted, and special education students. One group is using the Macmillan text, Mathematics in Action, while the other group is receiving instruction from Saxon’s Math 4 curriculum.
Pine Log Elementary School is a small school of just over 400 students in a rural section of north Bartow County. It houses grades preK - fifth and has only five minority students in the entire school. Since Pine Log uses ability grouping to determine math classes, an advanced class from the 1998-99 academic year (which received basal instruction) will be compared to the advanced class from 1999-2000 which is currently using the Saxon Math 54 text. In addition, an average class from 1998-99 which was taught with a basal text will be compared to a class of similar ability from the 1999-2000 academic year which is also using the Saxon Math 54 text.
Both schools were given the opportunity to pilot the Saxon program in mathematics for the 1999-2000 school year. The teachers involved in this new program were dissatisfied with the traditional "study-learn-drop" approach of the various traditional basals being used which does not allow enough practice for true mastery of skills. The incremental approach to instruction afforded by Saxon was appealing because several teachers hoped that it would allow students to develop mastery and automaticity through continuous repetition and practice. These schools both come from counties with parallel city and county school systems (Floyd County/Rome City; Bartow County/Cartersville City) in which the city schools are already using the Saxon Math, where it has proven to be successful in achieving high scores on standardized tests. In 1988, following a spring term pilot of Saxon textbooks, Cartersville Middle School began with the Saxon series for grades 6-8. Prior to the introduction of Saxon textbooks the school board reported the eighth-graders scored in the 59th percentile on the ITBS. By the spring of 1990, the eighth graders scored in the 74th percentile and ranked first in the state of Georgia.
HYPOTHESIS:
Because we as Saxon teachers predict that students’ scores will improve, we have chosen a directional hypothesis: fourth grade students who receive incremental instruction using Saxon Math will achieve higher math scores on the Iowa Test of Basic Skills as compared to those who received traditional instruction from a Basal Text.
DEFINITIONS:
Incremental development is the introduction of topics in easily understandable pieces (increments), permitting the assimilation of one facet of a concept before the next facet is introduced. Both facets are then practiced together until another is introduced. Incrementalization combined with continuous practice and review provides the time required for concepts to become familiar and internalized.
Unit-based instruction is the format of most traditional math textbooks. Concepts are grouped into units and chapters, with support materials such as practice, reteach, and enrichments masters. These concepts are introduced, concentrated on, tested, then dropped, except for occasional review.
Saxon Textbooks
1. Teachers lecture in class for only 10-15 minutes, leaving the rest of the time
for exercises.
2. Much of the classwork reviews previous lessons, trying to reinforce skills
before they are forgotten.
3. Books have few frills, no pictures and short explanations.
4. Includes timed drills every day.
Saxon Math 4 (Pepperell Elementary)
This method is an approach that is teacher-centered, consumable, manipulative-based, and does not have a textbook. Math 4 is for teachers who do not yet wish to move students to the more theory-based, paper-and-pencil approach of the middle and secondary grade books. The concepts covered in Math 4 and Math 54 are the same, but the order of topics is different.
Math 54 (Pine Log Elementary)
This textbook method is incremental and provides a non-consumable alternative for fourth graders ready for transition to a textbook. It is the first textbook in Saxon’s series of books for grades 4-8.
Traditional Textbooks (Harcourt/Brace, Macmillan, Holt)
1. Teacher is in charge of class, may decide to apportion time for lectures or
exercises in any way he/she wishes.
2. Emphasis is on learning concepts while acquiring math skills.
3. Books are colorful, graphic-laden; feature long explanations.
METHODS AND INSTRUMENTS:
Part of the sample for this study were students grouped heterogeneously in two fourth grade math classes from Pepperell Elementary School during the 1999-2000 school year. The first class of fourth graders (experimental group) consisted of twenty-seven students who used the Saxon series for fourth grade. The control group who used the Macmillan basal contained 29 students. Different teachers taught these two math classes.
The other parts of the sample were the four homogeneous groups from Pine Log Elementary. These classes were from two consecutive years. The 1998-99 advanced class (control group) consisted of 18 students, taught from Mathematics Plus by Harcourt Brace, while the advanced 1999-2000 class of 24 students received instruction in Saxon Math 54 (experimental group). Both of these groups had the same teacher. The average group (control) from 1998-99 was taught using the Holt Mathematics Unlimited, while the average group (experimental) in 1999-2000 used the Saxon Math 54 text. The average ability classes had different instructors. In this group of samples, advanced students are being compared to advanced students, and average to average.
The pretest used in this experiment is the math scores from the Iowa Test of Basic Skill at the end of the students’ third grade year. This breaks down into three areas of mathematics: (1) math concepts/estimation, (2) problem solving/data interpretation, and (3) computation. The posttest is the ITBS, administered at the end of the students’ fourth grade year. These scores will then be compared to assess which group, if any, made the most significant gains.
CONCLUSIONS
The posttest results showed a greater increase in scores of each of the Saxon groups; however, the increases were not statistically significant. The chart below will show that in the heterogeneously grouped Pepperell Elementary classes, Group B started with a lower pretest score and ended with a higher posttest score, as compared to Pepperell Group A, traditional math text. This was also true of the average placement Pine Log Elementary group (Group D). In essence, the Saxon groups Pepperell B and Pine Log Average D, began with lower pretest scores but ended with a higher posttest mean than groups using the traditional textbook. Pine Log Advanced Placement (Group B, Saxon Posttest) had an increase of +1.49 grade equivalent. Again, the Saxon groups showed a greater increase in test scores.
Advantages and Disadvantages of Saxon
Erin Hansen 4th grade teacher, Pepperell Elementary, Saxon Math 4 (consumable, no text book) found that after using Saxon Math 4 curriculum for a year, required a greater than average amount of teacher preparation to plan for and make ready the hands-on activities for each day’s lesson. Each lesson is scripted and many can be quite lengthy, often taxing the attention span of fourth graders. Hansen found that using the same type of worksheet everyday could get boring.
However, Hansen considered some of the activities in the workbook fun, with real-life applications; such as a check registry, and merchandise catalog where the students could spend money and subtract purchases from the registry. This activity taught the importance of spending wisely. The math meeting each day recapped specific skills and problem solving techniques. Focusing upon math skills every day gave students the opportunity to ask questions and master new applications. Posttest results indicate that a majority of students in the Math 4 group mastered math skills. All of the student’s math scores were significantly higher in fourth grade as compared to third grade scores after using the Saxon method. When Mrs. Hansen saw individually how scores had improved she felt that the extra time spent planning and preparing for lessons was worth the effort.
Kay Greene and Macy Fowler, 4th grade teachers at Pine Log Elementary, users of Saxon Math 54, (textbook version) felt that lessons were easy to teach and not particularly intrusive with regards to preparation time. Daily timed fact tests were used with axon Math 54 to improve practice and mastery of skills. Saxon Math 54 students used a pre-numbered form on which to do the daily lesson. This forced each student to organize problems on the sheet. Each lesson also contained work in problem solving and finding number patterns. For some concepts, Greene and Fowler had to supplement with work sheets from traditional text books when students needed more exposure to a skill than was given in the Saxon 54 method. Through the heavy emphasis on drill and practice required with Saxon 54, students seemed to get more comfortable doing word problems over time. Both teachers enjoyed using the Math 54 text.
Student Views on Saxon
Mrs. Hansen’s students stated that the Saxon Math 4 (incremental) was boring. They did enjoy the daily timed fact practice and using the Math 4 student notebook, which contained real-life, hands-on experiences. Some favorites: Mapping time zones, charting and graphing activities, keeping the check registry, and check writing. (They thought that was really cool!). Mrs. Hansen’s students were extremely pleased when each compared third grade test scores to fourth grade test scores after using the Saxon approach. They decided that the Saxon way wasn’t so bad after all.
The Pine Log Elementary students overall, enjoyed the Saxon 54 text method. Surprisingly, students did not find the lack of graphics in the textbook to be a problem. Students particularly liked the forms used for problem sets. Because they were written in the same format as the daily lessons, tests were not particularly dreaded or anxiety inducing. Students also stated that they were better prepared for math computation on standardized tests. Each problem of the Saxon 54 text contained a small box equipped with the lesson it came from in the text so student could refer to the book if needed. The students liked that aspect especially.
Parent Views of Saxon
One parent, a high school math teacher of a Pepperell Elementary student expressed in the beginning of the year a negative viewpoint about the Saxon method, but
when her child’s math total score went from 4.7 GE (3rd grade) to 8.9 GE (4th grade), her opinion changed somewhat. Other Pepperell parents felt that the Saxon Math 4 was too difficult for the children. Some felt that the lessons seemed too advanced for fourth graders. Some parents did not like the consumable method because it was one more paper that the children had to keep track of. One parent noted that "A paper is easier to lose than a textbook." Most parents stated that they were pleased to see the growth in mathematical ability in their children regardless of the method used.
An interview with a group of Pine Log parents were interviewed about the Saxon 54 textbook revealed that they were pleased with the Saxon 54 approach. They felt the pre-formatted lesson sheets forced the student to be organized. Another strength cited by parents was the incremental lessons with constant review built in the each lesson. Saxon emphasized student practice, which parents felt was very helpful to the children.
A Final Word
Although Saxon Math has received some severe criticisms from many in education, we found some positive results among third and fourth graders using the Saxon Math 4 and Saxon 54 approaches. The biggest drawbacks to the Saxon approach are a certain blandness that evolves over time through its emphasis on drill and review. However, even the students did not find the repetitious nature of Saxon to be insufferable. In fact, many students seemed to attribute their success in math to Saxon’s incremental style. Although the classes receiving the Saxon approach to mathematics did not score significantly higher on posttest, teachers, students and parents seemed to believe that the Saxon approach has some merits. Although many in the research community have damned the Saxon approach, the results of our study would seem to indicate that Saxon certainly does not harm, either in regard to student attitudes towards mathematics or their academic performance. In essence, there is not a perfect math method. Saxon may not be the right recipe for your taste buds. We found it to be deliciously tempting.
References
Allen, Frank B. (1988, April). Language and the Learning of Mathematics. Speech delivered at the NCTM Annual Meeting.
Calvary, Robert, Bell, David, &Wheeler, George. (1993, November). A Comparison of Selected Second and Third Graders’ Math Achievement: Saxon vs. Holt. Paper presented at the Annual Meeting of the Mid-South Educational Research Association.
Durham, Helen. (1995, September 30). Multiplication of skills. World, 19.
Finn, Chester. (1993, January 20). What if those math standards are wrong? Education Week, 36-37.
Hart, Dan. (1996). A Tale of Two Schools: LAUSD and Saxon. Document from San Fernando High School, San Fernando, CA.
Hill, David. (1993, September). Math’s angry man. Teacher Magazine, 24-28.
Klingele, William E. and Reed, Beverly W. ( 1984, June). An examination of an incremental approach to mathematics. Phi Delta Kappan, 15-16.
Mathews, Jay. (1993, March 1). Psst, kid, wanna buy a used math book? Newsweek.,34-35.
Saxon, John. (1984, May). The way we teach our children math is a disgrace. American Education, 10-13.
Stengel, Richard. (1981, December 21). New angle on algebra. Time, 54-55.
True, Greg N. (Date unknown). The effect of continuous distrubuted review on mathematics achievement. Document from the Indiana University School of Education.
Weeter, Richard. (1993). Saxon Mathematics Program Evaluation Report.
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