CPMP III versus Algebra II.

Abstract

The research question was: When a non-paper-and-pencil test is used as the source of measurement, do there appear to be differences by students' mathematics curriculum, Algebra II or CPMP III, in the mastery level of problem-solving and algebra? Students from 3 classes of CPMP III and 3 classes of Algebra II (resulting in 76 CPMP III students and 71 Algebra II students), all in the same school district, were given a four-item interview test. To answer the research question only in terms of correctness, there was equal performance on two non-routine problems. On two algebraic items, the Algebra II students scored significantly higher. In terms of Polya's stages entered (did not attempt, understanding the problem, devising a plan, carrying out the plan, looking back), there were no significant differences between the two groups, nor was there a difference in terms of strategies employed. CPMP III students took significantly more time in completing the test than the Algebra II students did.

CPMP III versus Algebra II

Standards-oriented mathematics curricula are present in numerous secondary mathematics classrooms (National Council of Teachers of Mathematics, 2000). These curriculum projects were based on research and project personnel continue to do extensive research on student outcomes (Senk & Thompson, 2003). It may appear then that further research on standards-oriented curricula is not necessary. However, the vast majority of the research has been conducted by project personnel. Although no one is accusing project personnel of faulty or unethical research, it is a concern that researcher bias may occur to some extent when high stake studies are conducted by the very people with high stakes (Campbell & Russo, 1999; Crano & Brewer, 2002; Mehra, 2002).

Berg (1988) cautions that "[i]t is misleading to assume that because researchers are scientists they are not human beings. Research is an activity embedded in a profession.... As a result, all of the anxieties associated with earning a living (e.g., advancement, security, recognition) are also part of the research process." (p. 219). Qualitative research has taught us that "all research is participatory" (Brown, 1996, p. 15) and that as researchers conduct research they "naturally experience a range of emotions and thoughts, some of which threaten to bias, distract, and even disable them" (Brown, 1996, p. 15).

Researcher bias in studies of standards-oriented curricula versus traditional curricula can occur in several manners, and may even seem appropriate to the reader. For example, it is common to have researchers select teachers who most faithfully implement the standards-oriented curricula. At first glance, this might seem appropriate. However, teachers are implementing standards-oriented curricula across the nation. If it takes a perfectly committed and extensively trained teacher to implement the curriculum successfully, then it should be considered whether the curriculum is a realistic curriculum to use. It would be better to attempt to select teachers in a random fashion. If teachers are selected in a random fashion, the studies will not be so much about what is possible, as they will be about what is happening. Thus it is timely that more non-project researchers conduct studies regarding the student outcomes in standards-oriented curriculum. This paper documents one such study.

Besides having this study conducted by a researcher not connected to a standards-oriented project, this study contributes to the research in another significant manner. Some of the existing research reports on qualitative results using a single standards-oriented curriculum (without comparison to another standards-oriented curriculum, or to a traditional curriculum). Other studies offer comparisons but compare quantitative results of standards-oriented students' scores on paper-and-pencil tests to traditional students' scores (Kilpatrick, 2003). While both of these types of studies are important, comparison is important, also, and there is no reason to believe that paper-and-pencil tests are the best means of comparison. In fact, there is reason to believe that non-paper-and-pencil tests may be fairer to students in standards-oriented curriculum. One example may clarify. Students in standards-oriented curriculum spend a rather large portion of class time on mathematical communication. This communication includes written communication, but also includes verbal communication. Verbal communication is not measurable in strictly paper-and-pencil format (which many of the paper-and-pencil standardized tests are).

In sum, this study contributes to the research base in three manners. This study (1) is conducted by non-project personnel, (2) compares students from a standards-oriented curriculum to students in a traditional curriculum, and (3) the instrument used is a non-paper-and-pencil instrument, thus possibly measuring characteristics not measurable in paper-and-pencil format.

The standards-oriented curriculum used in this study was the Core-Plus Mathematics (CPMP) curriculum (Coxford et al., 1997, 1998, 1999, 2001). CPMP has developed student and teacher materials for three-year high school mathematics curriculum for all students and a fourth-year course for college bound students. The main theme of CPMP is mathematics as sense making. Students investigate problems set in real-life contexts within an integrated curriculum that includes algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics. The curriculum for each year is seven units and a capstone section which is "a thematic two-week, project-oriented activity that enables students to pull together and apply the important mathematical concepts and methods developed in the entire course" (Schoen & Ziebarth, 1998, p. 153). Mathematical modeling is emphasized throughout the curriculum. Graphing calculators are used. Additional characteristics of CPMP are that the curriculum is designed to be accessible to all students, to engage the students in active learning and to provide multidimensional assessment (Hirsch & Coxford, 1997). The assessments are embedded within the curriculum and include students' answers to questions in class, group work, students' journals, quizzes, in-class and take-home end-of-unit assessments, cumulative written assignments, and extended projects (Hirsch, Coxford, Fey, & Schoen, 1995).

Inclusion of topics in the CPMP curriculum is based on the merits of the topics themselves; that is, the topics must be important in their own right (Schoen, Fey, Hirsch, & Coxford, 1999). The instructional sequence follows a four-step process labeled as launch, explore, share and summarize, and apply. The "launch" sets the context for what is to follow and consists of a class discussion of a problem. The "explore" is usually a cooperative group or pair activity in which students investigate the problems and questions. "Share and summarize" brings the class back together to discuss key concepts and methods. "Apply" is time in which individual students practice what has been learned (Hirsch, et al., 1995).

Project personnel have conducted numerous and varied studies of the effectiveness of the CPMP curriculum. Space limitations prevent a report of the details from each study. However, see Table 1 for a listing of abilities, skills, and results for which CPMP students are significantly higher, no difference, or lower than non-CPMP students (or in some cases, national norms). In each case, a reference to a research study, journal article, or chapter is given. (In Schoen & Hirsch, 2003, the article/chapter summarizes numerous separate research studies.) In addition, a bulleted list follows the table giving the highlights of some especially important results.

* At the end of Course 1, CPMP students' average score on the Ability to Do Quantitative Thinking (a subtest of the nationally standardized Iowa Test of Educational Development, ITED) was significantly higher (p < .05) than algebra students in traditional curricula with 11 schools used (Schoen and Hirsch, 2003b). Other studies, however, show no significant differences between CPMP students' average score on the ITED and traditional students' average scores (Schoen and Hirsch, 2003a).

* At the end of Course 3, CPMP students performed significantly better (p < .05) on concept and application tasks but significantly poorer on algebraic manipulation tasks when compared with Algebra II students in traditional curricula with 6 schools used (Huntley et al., 2000).

* Using SAT 1 mathematics scores, CPMP III students versus Algebra II students from 8 schools resulted in no significant differences. When ACT Mathematics means were used, the Algebra II students had significantly higher scores (Schoen, Cebulla, & Windsor, 2001).

* Using placement tests constructed with items from a test bank from the Mathematical Association of America, CPMP Course 4 (N=164) versus Precalculus (N=177) students showed no significant differences on algebraic symbol manipulation skill but a significant difference in favor of CPMP on concepts and methods needed for the study of calculus (Schoen & Hirsch, 2003a).

In sum, in CPMP conducted research, CPMP students scored higher or there was no significant difference, except when it came to measures of algebraic manipulation skills and procedural skills. In those cases, traditional students sometimes performed better. (CPMP students never performed better than traditional students on algebraic manipulation skills.)

The non-standards-oriented curriculum used was a non-accelerated Algebra II. The course lacked an emphasis on groupwork and graphing calculators, while emphasizing symbolic manipulations. The curriculum had an emphasis on separate units of mathematical content, which includes an emphasis on procedures (although conceptual understanding is present as well). The teachers do often serve as the "tellers" of information (i.e., the students were not engaged in discovery work). Students most often work individually. Testing is usually easily accomplished through short-answer or even multiple-choice items. Computation is important. The curriculum certainly included solving algebraic equations with solution processes not dependent on technology. The curriculum lacked an emphasis on problem-solving strategies. The textbook used by all involved students was Glencoe's Algebra 2 (Glencoe, 2001).

The Research Question

Since CPMP students sometimes do not do as well on symbolic manipulation skills, one would expect that this loss is met with gain in some other area. CPMP authors state:

  On the whole, the evidence suggests that it is possible to streamline
  the traditional components of high school mathematics and incorporate
  important concepts and methods of statistics, probability, and
  discrete mathematics, while significantly improving students'
  understanding of the mathematical content and its applications. A
  trade-off in somewhat lower traditional paper-and-pencil algebraic
  skills may result, although the revisions in the Course 4 field-test
  materials appear to have reduced the deficit. (Schoen & Hirsch, 2003,
  p. 120)

This trade-off is probably acceptable if indeed there is a trade-off. Are skills in other areas superior?

The current study compares end-of-the-year CPMP III students to Algebra II students. The instrument used contains both items of a problem-solving nature (nonroutine) as well as routine algebraic items. In a small pilot study of the items used, the author found that CPMP students seemed to perform better when interviewed than when tested with multiple-choice paper-and-pencil tests. The research question is: When a non-paper-and-pencil test is used as the source of measurement, do there appear to be differences by students' mathematics curriculum, Algebra II or CPMP III, in the mastery level of problem-solving and algebra?

Students

A Midwestern school district serving an urban city was involved in the study. In the given school district, there are three high schools. In each high school, there are three strands of mathematics curricula: CPMP, non-accelerated traditional, and accelerated traditional. One class of CPMP III and one class of Algebra II (within the non-accelerated traditional strand) were randomly selected from each high school.

This resulted in 6 classes with 6 different teachers, 76 CPMP III students, and 71 Algebra II students. Note that the accelerated traditional students were not a part of this study. It seemed unduly cumbersome to have three groups to study (CPMP, Algebra II non-accelerated, Algebra II accelerated) when it would seem clear that the Algebra II accelerated would score higher than either group (those students were in the accelerated program because of superior past performance in mathematics). To lump the accelerated with the non-accelerated and call all of those students traditional seemed unfair.

The question of interest, of course, is whether students were equivalent before entering the CPMP sequence versus the traditional sequence. It does seem likely that the very best students were not a part of this sample. Here is why: Students were advised whether to take CPMP, traditional, or accelerated traditional. Before students entered eighth grade, students' scores on statewide mandated tests were examined. Those with scores above 87% were advised to take the accelerated route. The remaining students took either CPMP or traditional. This study compares those students who took either CPMP or traditional (not the accelerated). Thus, it does seem to be a fair assumption that the very best students were not in CPMP, nor were they in the traditional sample of this study.

Let us return to the students' scores on the statewide mandated tests (given before students entered the eighth grade). For the involved subjects, the end-of-eighth grade was before students were to enter Algebra I or CPMP I. Since the sample was formed randomly, the scores on these mandated standardized mathematics tests were gathered for all students in the sample by searching the students' school records. There was no statistically significant difference (p < .05) on these test scores between the students who first took CPMP I and who first took non-accelerated traditional (Algebra I).

Another question of interest is whether the CPMP III and Algebra II teachers fully implemented the intended curriculum. Since the classes were selected randomly, there was not an attempt to find the most enthused CPMP teacher or the most enthused Algebra II teacher for that matter. On the other hand, selecting classes randomly might create unevenness in the study. If one views the implementation of a curriculum along a continuum from not implemented at all to fully implemented, the CPMP classes may end up in a different spot on this continuum than the Algebra II classes.

In an attempt to understand the situation (even though the situation will then have to be accepted as is, and not manipulated to lose the randomness), the teachers were interviewed by the author. As luck would have it (at least there is no other identifiable reason), the three classes of CPMP and the three classes of Algebra II seemed to have fairly net equal implementation levels. To illustrate, quotes follow which represent the level of enthusiasm for the curriculum for each of the CPMP III and the Algebra II teachers. There is no claim that enthusiasm and implementation are identical. However, the intention of using the enthusiasm quotes is to illustrate the possibility that not every teacher may have implemented the given curriculum as faithfully as possible.

CPMP Teacher 1 (considered high level of enthusiasm): "When my child is in high school, I will make sure she takes the Integrated [CPMP] class."

CPMP Teacher 2 (considered medium level of enthusiasm): "Oh, it is like everything else. There are good and bad things about it."

CPMP Teacher 3 (considered low level of enthusiasm): "I'm sorry, but I just don't think it prepares kids for college."

Algebra II Teacher 1 (considered high level of enthusiasm): "Traditional is still the tried and true."

Algebra II Teacher 2 (considered medium level of enthusiasm): "When I teach Integrated [CPMP], I really enjoy myself. But, this is fine too. I know this stuff is important."

Algebra II Teacher 3 (considered low level of enthusiasm): "You know you teach all these little procedures that are not common sense. There is no reason students need to know these things, and kids won't remember them."

Again, the classes were selected randomly, and there is no claim that a scientific method has been employed here to justify the balance of the classrooms used in this study. The quotes are offered to give some reason for the researchers' belief that the classrooms may have ranged in degree to which the curriculum was fully implemented. In addition, a teacher may actually perfectly implement a curriculum for which he or she has no enthusiasm. The classrooms were picked randomly, and the researchers want to avoid a quantitative measure of implementation. Through the interviews, however, there is reason to believe that the CPMP III classrooms differed across the continuum, as did the Algebra II classrooms.

Method

At the end of the eleventh-grade, two researchers administered orally a test to the students. The student was told to "think out loud" as he or she solved four mathematical problems. The problems were written on paper, as well as read to the students. The students were also told that it was more important how they solved the problem than the actual answer to the problem. Students were prompted to think out loud if they were silent. See Figure 1 for the exact problems.

[FIGURE 1 OMITTED]

To select the first two problems the researchers selected a large set of problems judged to require students to make sense of a novel situation and use some sort of strategy. (The strategy needed in the case of the selected problems involved how to keep track of which rectangles or digits have been counted.) The researchers contacted experts who had published in refereed journals in the area of mathematical problem solving (approximately 25 experts were contacted and 13 responded). Each expert was asked to look at the problems (there were approximately forty) and vote for the two that the expert judged to be the best in terms of two things: nonroutine to most eleventh graders, and yet capable of measuring problem solving skills. Two problems with the most votes were selected to be items one and two on the instrument of this study.

Expert analysis is one measure of validity. In addition, a small pilot study was run. Approximately 20 students from CPMP III and 20 students from Algebra II (in the same school district, but not part of the sample for this study, as the pilot study occurred the year before this study) were asked to work the problems (this process was videotaped, and the problems were administered in the same manner as in the larger study). This pilot process revealed no concerns with the problems, and teachers reported that students who scored well in the pilot study were those who were expected to score well. Thus, some evidence toward validity has been gathered. (Reliability is more difficult to determine because of the small number of problems involved. Of course, reliability increases with quantity of problems. Rater reliability will be discussed later in this section.)

To select the last two problems (items three and four in this study), the researchers took two problems from a university placement test that were part of the section where a low score would place the student into college algebra. Validity and reliability evidence are available for the placement test (see Latterell & Regal, 2003). Of course, only two of the problems out of the twenty that are on the placement test are being used here, but there is certainly evidence (including item analysis) supporting the items in the placement test.

The test administration was videotaped. Three copies of each tape were made, as well as annotated transcripts, and given to three mathematics education researchers. To analyze the tests, the three researchers separately filled out a scoring rubric while viewing the videotape (see the appendix). Note that using the scoring rubrics had been practiced by these same researchers with a smaller set of students (ten from CPMP III and ten from Algebra II) during a pilot study.

The rubric contained three parts for each of the four problems: a score from 0 to 3 on correctness, a score of 0 to 4 identifying the last stage ever entered in the problem-solving process and a listing of any strategy used, and a part for additional observations. The listing of strategies used included, but was not limited to, the strategies listed in Polya's How to Solve It (1945/1973). Under additional observations, the amount of time the student took to complete the test was recorded, and researchers looked for additional characteristics, and if present, made a note of them. These additional characteristics were to include rich mathematical language used by the students, students' enthusiasm for working the problems, and any special characteristic or happening the researcher wanted to note.

After each of the three researchers separately filled out a rubric for each student, the three researchers together viewed the videotapes. During this second viewing process, the three sets of rubrics (one from each researcher) were compared and all differences were discussed until agreement was reached. Nearly 98% of the correctness scores matched with all three researchers before discussion, and 100% agreement was reached after discussion. Of course, there was 100% agreement on the amount of time (this was simply a function of the videotape, which had a time/date stamp).

The qualitative measures created more discussion than the quantitative scores. In these cases, the researchers looked for consensus of the researchers that the information on the scoring rubrics had captured important aspects of the students' problem solving. To facilitate this process, annotated transcripts were created and qualitative software was used to search and code for keywords in the annotated transcripts. Another meeting of all three researchers occurred, as the three researchers viewed the tapes for a third time (the second time in the company of each other). The result of the analysis was one filled out scoring rubric per student (which was the combination of the three separate scoring rubrics that each researcher filled out for a given student, as well as the discussions as the researchers viewed the videotapes together twice, and the results from the annotated transcripts and qualitative software).

Results

The results will be given in three parts: correctness, stages entered and strategies, and additional characteristics and time spent.

Correctness

The researchers scored each of the problems from 0 to 3 in terms of correctness of solution, with 3 being fully correct. On problem one, there was no significant difference between the two groups (F = 0.17, p = .68). On problem two, there was no significant difference between the two groups (F = .01, p = .89). On problem three, there was a significant difference in favor of the Algebra II students (F = 11.74, p = .00). On problem four, there was a significant difference in favor of the Algebra II students (F= 61.91, p = .00). On the total score, results do favor the Algebra II students (F = 29.59, p = .00). See Table 2 for the means and standard deviations.

Looking more closely at the first item (How many keystrokes are needed to put page numbers on a book with 124 pages?), a typical low-level answer was to oversimplify the situation. Consider this student.

    Student: Well, hmmm. It is just 124 divided 2, right?
    Interviewer: Why would you say divide by 2?
    Student: There are two pages, if the book is open? Right?

Although other low-level answers did not necessarily divide by 2, they contained other simple operations that made little mathematical sense. Of course, some students simply said that they did not know and, thus, did not do anything.

Other students had more of the general idea, but made mistakes in reasoning or calculation. Under correctness, the researchers could not distinguish between a typical CPMP III answer and a typical Algebra II answer. Those students who were at least partially correct attempted to sum the number of single-digit, double-digit, and triple-digit numbers.

Several examples follow.

    Student: There are 1 through 10, and that is 11. Then, 11 through
    99, which is 2, and there are 90, so it is 263.
    Interviewer: Wait. What was that?
    Student: There are 11 numbers with one number, and 90 with two, so
    11 plus 180, plus the numbers with three, which is 263 when it is
    all put together.

    Student: It is 9 plus 89, plus 25, so 113.
    Interviewer: Where do you get the 9, 89, and 25?
    Student: There are 9 with one number, 89 with two numbers, and 25
    with three numbers.

    Student: I can write an equation. Where can I write it? [Interviewer
    hands student a piece of paper, and student writes this: 1 +
    (100x2) + (124x3) = 1 + 200 + 372 573]

    Student: I'm not ... [profanity] ... you would have to write out all
    the page numbers and count each number of digits. No.

On that note, let us examine a few examples from the second item, in which students were to count the number of rectangles. Many students, of course, just said 9. Others tried to keep track through some method (for example, by starting with the smallest rectangles and working up to larger rectangles). Again, there was no typical distinction between a CPMP III answer and an Algebra II answer.

    Student: I'll group them.
    Interviewer: Group them? How?
    Student: Count all that are the same size, and then another size.

    Student: I'll mark the top of each rectangle. To keep track. [pause]
    No. That won't work. I'll count them from left to right.

    Student: Draw lines. Some lines are one unit. Some lines are two
    units. Lines this way [drew a horizontal line] and lines this way
    [drew a vertical line].

    Interviewer: So, what are you doing with the numbers?
    Student: I'm putting a little number by each rectangle.

It is difficult to offer excerpts for the two remaining problems (the algebra ones). The CPMP III students were not particularly capable of solving them, and were then inarticulate on a solution process. The Algebra II students who did solve the problems were unwilling or unable to give a verbal description of their thought process. On the graph of the equation, some of the Algebra II students plotted points, and others stated that it was a parabola (or a quadratic or had a square) and then either plotted one point, usually (0, -3), or said that it opened up. On the algebraic equation, the Algebra II students wrote out the mathematical steps for solving the equation. A large number of the CPMP III students simply wrote a question mark next to item four.

Stages and Strategies

The researchers scored each problem by the stage in the problem-solving process (using Polya's stages, 1945/1973) that the student ever entered:

0 = Did not attempt

1 = Understanding the problem

2 = Devising a plan

3 = Carrying out the plan

4 = Looking back.

It was not always easy to determine which stage a student entered. Nevertheless, three researchers independently assigned a stage based on students' talking about the problem (from the videotape). All differences in ratings were discussed and consensus was reached. The particular interest with stages entered is whether CPMP III students entered the looking back stage more often than Algebra II students did. Since pre-standards research has shown that few students ever enter the looking back stage (Schoen & Oehmke, 1980), it is interesting if students in standards curricula do enter the looking back stage. However, there were no significant differences on stages entered. One attempt to see differences in stages entered was to total for each student the number of times the student looked back. An F-test was used, then, to compare CPMP III students with Algebra II students on this result. However, this too did not result in a significant difference (F = 1.41, p = .24).

The researchers also made lists of strategies that students used. Before the interviews, researchers hypothesized that students might employ any of the following strategies: use a simpler case, work backward from the problem, look for a pattern, draw a picture or series of pictures, and guess and check. Researchers further hypothesized that CPMP III students and Algebra II students might differ in the quantity, variety, and quality of strategies employed.

An interesting result is that students (both CPMP III and Algebra II) were not inclined to employ a variety of strategies, at all. Those students who correctly solved the first problem did so by summing up the number of each type of number (single, double, or triple-digit). Those students who correctly solved the rectangle problem had some method for keeping track of the rectangles. No nameable strategy was used on the remaining two problems, unless one calls it "algebraic strategy." There was no distinction between CPMP III students and Algebra II students on strategies used. Neither group used strategies more often or used a larger variety of strategies. It is possible, of course, that the nature of the items did not bring out a difference that did exist between the CPMP III students and the Algebra II students.

Additional Characteristics and Time

Researchers also looked for additional problem-solving characteristics (besides correctness, and stages and strategies) but were not successful at obtaining good measurements of the additional characteristics. For example, when language was used that researchers noted as particularly mathematical (terms specific to mathematics, beyond simply multiply, add, etc.), researchers noted such language and, later, coded it in the annotated transcript. However, this type of language was nearly absent, and no differences were observed between CPMP III and Algebra II students in this area.

Researchers also attempted to make a judgment whether the student entered the problem-solving process willingly and with enthusiasm. However, the researchers soon realized that the students were placed in too artificial of a situation to make a good judgment of mathematical enthusiasm. Rather, the researchers seemed to be obtaining a judgement of the students' self-confidence in novel situations. (Recall that students were interviewed individually and videotaped.) Perhaps had an entire classroom been observed (even with researchers walking around to individuals as they remain in the classroom situation), the researchers might have obtained a better measure of problem-solving willingness. Or perhaps in the interview setting, researchers could ask directed questions that may prove to be a measure of the students' enthusiasm. As it stands in this study, there is not enough evidence to make conclusions about students' mathematical enthusiasm, and certainly not enough evidence to make a comparison between CPMP III and Algebra II students on this characteristic.

Turning to the time involved in the test, there is no claim made that it is better to take less or more time when working mathematical problems. Taking more time may signify willingness to work at a set of problems. Taking less time (and being correct) may signify deep understanding. However, this relationship is unclear. The reader may wonder why even record time, then. Depending on the results, it may very well be an intresting factor. For example, if one group of students does not score as well as the other and takes more time, this is at least not a good thing. For situations such as that one, time was recorded. There was a significant difference, with CPMP III students taking more time (F = 38.15, p = .00). The mean time taken for Algebra II students to work all four problems was 277 seconds (DS = 11.46 seconds) and for CPMP III students the mean time taken to work all four problems was 400 seconds (D.S. = 16.08 seconds).

Discussion and Implications

Based on the items given, there are two content/process issues: problem solving and routine algebraic problems without a context. There was no significant difference between the two tracks in problem solving.

However, in the area of routine algebraic problems without a context, the CPMP III students were considerably below the traditional. Both of the items were part of the college placement test. In fact, both of these items were in the part of the college placement test that decided between college algebra and precalculus. (Obviously, those students who placed into precalculus should have been able to solve these items.) Students who pass three years of high school mathematics that is intended to be a college preparation sequence should not be placing into college algebra, regardless of their career goals or mathematics curriculum. Even if there is no comparison to be made in this study, all students should be able to answer these types of items. All students should absolutely exit any three-year college preparatory high school mathematics program (and CPMP is billed as a college preparatory curriculum by the school district as well as by CPMP authors) with the ability to test at least into college precalculus. It is understood that not all students enter college, but should not all students have the opportunity to enter college without being severely behind in their mathematics requirements? If this is not the case, then too many shortcomings are accepted from the curriculum even if students do succeed at problem solving. Recall that the traditional students in this study were also succeeding at problem solving.

Of course, this is not an attempt to return us to traditional curricula (which certainly has its own concerns). Rather, it is a call to find research support to determine what adjustments need to be made to reform curricula. In general, this author agrees with Schoenfeld (2002) when he states that reform curricula "can be made to work as hoped" (p. 19). This study does not try to claim that traditional or CPMP is a better curriculum. Rather, it attempts to make comparisons so as to point out where CPMP is possibly succeeding and where it is not. In the given study, CPMP students appear to be succeeding in problem solving, but not in algebraic problems.

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Schoen, H. L., & Oehmke, T. (1980). A new approach to the measurement of problem-solving skills. In S. Krulik and R. E. Reys (Eds.), Problem solving in school mathematics: 1980 yearbook (pp. 216-227). Reston, VA: National Council of Teachers of Mathematics.

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Senk, S. L., & Thompson, D. R. (Eds.). (2003). Standards-based school mathematics curricula. Mahwah, NJ: Lawrence Erlbaum.

Appendix

Scoring the Test

Part I: Score from 0 to 3 points.

Problem 1:

Pages 1-9, one digit, 9 x 1 = 9 keystrokes

Pages 10-99, two digits, 90 x 2 = 180 keystrokes

Pages 100-124, three digits, 25 x 3 = 75 keystrokes

9 + 180 + 75 = 264 keystrokes

0: not attempted

1: any 2 of these:

- Separated like # digit numbers into 3 groups

- Tried to determine how many pages in each group

- Recognized the need to multiply the three groups by the number of digits

- Added the three numbers together

2: put the above together in a plan, but made minor mistakes

3: correct answer

Problem 2:

There are 36 rectangles.

0: not attempted

1: between 1 and 10

2: between 11 and 27

3: 28 or greater

Problem 3:

0: not attempted or e

1: a or b

2: d

3: c

Problem 4:

0: not attempted

1: attempted problem

2: 1 simple arithmetic mistake

3: x = -l

Part II: What stages, if any, were entered in the problem-solving process? Score from 0 to 4.

1. Getting to know the problem

2. Choosing what to do

3. Doing it

4. Looking back

Problem 1 ____

Problem 2 ____

Problem 3 ____

Problem 4 ____

Part III: Additional notes including total time spent, mathematical language used, and high or low enthusiasm/interest in working the problems.

Carmen M. Latterell

University of Minnesota Duluth

Table 1. Summary of CPMP Studies

CPMP
Significantly     Higher            No differences    Lower

Conceptual        Schoen & Hirsch,
Understanding     2003a
                  Schoen & Hirsch,
                  2003b
                  Huntley, et al.,
                  2000
Interpretation    Schoen & Hirsch,
of Mathematical   2003b
Representations
and Calculations
Problem Solving   Schoen & Hirsch,
in Applied        2003b
Contexts          Huntley, et al.,
                  2000
Algebraic                                             Schoen & Hirsch,
Manipulative                                          2003b
Skills                                                Huntley, et al,
                                                      2000
Calculus          Schoen & Hirsch,
Readiness         2003b
Placement                           Schoen & Hirsch,
Testing of an                       2003b
Algebraic
Nature
Procedural Skill                                      Huntley, et al.,
                                                      2000
                                                      Schoen & Hirsch,
                                                      2003a
ITED Results      Schoen & Hirsch,  Schoen & Hirsch,
                  2003b             2003a
NAEP Results      Schoen & Hirsch,
                  2003a
ACT Results                                           Schoen, et al.,
                                                      2001
SAT Results                         Schoen, et al.,
                                    2001

Table 2. Means (Standard Deviations)

Problems  CPMP III     Algebra II

Number 1  1.36 (0.98)  1.42 (0.96)
Number 2  1.83 (0.62)  1.85 (0.42)
Number 3  1.64 (1.11)  2.24 (1.10)
Number 4  1.30 (1.33)  2.55 (0.48)
Total     6.13 (5.66)  8.06 (3.45)

(Problems are scored 0-3)
* p < .01.