The effect of a problem centered approach to mathematics on low-achieving sixth graders.

ABSTRACT:

Twenty-six sixth grade low achievers in the rural south experienced a problem centered mathematics curriculum for nine weeks. Potentially meaningful tasks were utilized in the class, which was divided into small collaborative groups of two or three like-ability students. The groups then presented and defended their solutions and strategies before their peers. The students showed marked increases in achievement and positive attitude towards mathematics when compared to a control group at the same school that was experiencing a traditional approach.

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The Effect of a Problem Centered Approach to Mathematics on Low-Achieving Sixth Graders

The results from the Third International Mathematics and Science Study (TIMSS) and TIMSS-R (TIMSS-Repeat) have become a source of concern for mathematics educators in the United States (USDE, 1996). Our nation is consistently out-ranked by the other industrialized countries that participated in these tests. Considering the money spent on education in this country and the economic and technological advantages we enjoy, educators are not satisfied with such low performance in a globally competitive market.

Japanese students consistently scored among the top three in these tests. Prior investigations by Stigler (1991) and Stigler and Perry (1988) at UCLA infer possible reasons for this superiority. In Japan, students spend a great deal of classroom time analyzing meaningful situations and working for long periods to solve non-routine problems. In contrast, U.S. students spend the majority of their time independently practicing a specific procedure demonstrated by the teacher. This educational strategy is contrary to research that shows that knowledge of rote procedures often interferes with students' attempt to build on their informal knowledge (Mack, 1990). In fact, traditional drill-and-practice teaching can even inhibit understanding, reify the divide between school and the "real world," and suppress the transfer of knowledge (Boaler, 1996).

Low achievers present special difficulties when considering any type of teaching approach. Apparently, the act of grouping students by ability level can of and by itself have an influence on motivations, perceptions, and eventual achievement of students (Boaler, 1997). A child who is labeled as a low achiever may experience detrimental consequences that last throughout their entire school career. In support of this effect, research has indicated that low achieving students tend to score lower and lower each subsequent year in comparison to others on standardized tests (Denvir, Stoltz, & Brown, 1984). Given the current level of their performance, it is unlikely that remedial work will be sufficient to close the gap between these children and their higher achieving peers, especially when that remediation is focused on skill deficits (Hankes, 1996).

Recent publications have suggested that a problem-centered approach might improve the mathematics competency of low achieving students (Hankes, 1996, Nicholls et al., 1991). Silver and Lane (1995) were able to demonstrate that middle school students from low-income disadvantaged backgrounds were able to outperform their peers in a demographically similar school when they participated in the Quasar Project, a program that emphasized reasoning, problem solving, and understanding. Research by Ginsburg-Block and Fantuzzo (1998) also showed that instruction that emphasized problem solving and peer collaboration enhanced the mathematics achievement, motivation and self-concept of low-achieving third and fourth graders. In fact, problem centered learning has been shown to foster high mathematics achievement and meaningful communication for all students in the second grade (Cobb, Wood, & Yackel, 1991a; Cobb et al., 1991; Thompson, 1985; Wood & Sellars, 1996).

Sfard (2000) asserts, "thinking is subordinate to, and informed by, the demands of communication" (p. 297). Thus organizing students in small groups to complete mathematics tasks and then present their solutions to the class has the potential of promoting thinking. These opportunities to communicate play a decisive role in mathematics learning. Further, Sfard argues that it is through this process that individuals construct the mathematical objects that constitute knowledge.

In a problem-centered learning strategy, activities are designed to emphasize communication and meaning making. One possible explanation of low scores on standardized and classroom tests is that students have an inability to "make sense" of topics studied. Teachers of low achievers tend to focus on memorization of facts and practicing procedures, thinking that it is the procedure itself that students may not understand (Hankes, 1996). Unfortunately, many of these students barely "remember" methods long enough to pass classroom unit exams. Repeated demonstrations on the board do not create any understanding. Some low achievers retain procedures for a limited time and are able to "pass" timely tests, but they do so without constructing essential meaning for the mathematics. Hence, they are unable to connect their shallow understandings in any significant way beyond an isolated task at hand. As long as they are practicing procedures just illustrated by the teacher, they might look competent. But these students do not own the mathematics; it does not mean anything to them beyond a set of unrelated procedures. Out of context on a standardized test with a variety of problem types, they are bewildered and achieve low scores (Pesek & Kirshner, 2000).

Studies suggest that students benefit from using their own insights to make meaning of mathematics (Cobb, Wood, & Yackel, 1991a; Cobb et al., 1991; Nicholls et al., 1991; Thompson, 1985; Wood & Sellars, 1996). They have to become empowered. They need to trust their own experiences and realize that there are many acceptable ways to do mathematics. They must develop confidence that they can understand mathematics (Nicholls et al., 1991). Von Glasersfeld has characterized this type of empowered learning during the past two decades. He has argued that children "construct their individual mathematical realities by reorganizing their personal experiences in an attempt to resolve what they find problematic" (1991, p. 209). Critics have suggested that such an interpretation of reality in fact denies that any "true" reality exists. Constructivists have continued to expand their theory into a coherent framework that is useful for dialog about the teaching and learning of mathematics. Thus, Steffe and Cobb (1988) suggested that the constructivist view of learning be described as follows:

  Mathematical learning is viewed as consisting in the adaptations
  children make in their functioning schemes as a result of their
  experiences to neutralize perturbations that can arise in one of
  several ways ... Problem solving conceived of as goal directed
  activity is a crucial aspect of learning mathematical knowledge.
  (p. 5)

The implication of this viewpoint is that mathematics instruction should be problem centered. The constructivist model asserts that the teacher's role is to continually present students with problematic situations that are designed to meet defined classroom goals. By creating goal-appropriate tasks, the teacher creates the opportunities children need to construct an experiential body of knowledge in the most personal, significant manner. As G. Polya so aptly stated in early thoughts on problem centered research in 1971:

  A teacher of mathematics has a great opportunity. If he fills his
  allotted time with drilling his students in routine operations he
  kills their interest, hampers their intellectual development, and
  misuses his opportunity. But if he challenges the curiosity of his
  students by setting them problems proportionate to their knowledge,
  and helps them to solve their problems with stimulating questions, he
  may give them a taste for, and some means of, independent thinking.

When considering the development of mathematical thinking in low achieving children, the age of the student becomes significant. An abundance of research has emphasized the need to reach students during their middle school years--the critical age when children make many permanent decisions about themselves, their abilities, and their future (Manning, 1997; Rothenberg, 1997; Seidman, 1994; Thorndike-Christ, 1991). "By about age 12, students who feel threatened by mathematics start to avoid math courses, do poorly in the few math classes they do take, and earn low scores on math achievement tests" (Bower, 2001). Therefore, the purpose of this study was to compare the mathematics learning of low-achieving middle school children by comparing two instructional strategies: one group having the experience of problem centered learning and the other group using a conventional approach.

METHOD

Design

The experiment was designed to study sixth graders who had poor scores on the Iowa Test of Basic Skills (ITBS) in mathematics by examining their achievement and their attitudes before and after the project. A public middle school in the rural south agreed to participate in the investigation. The students were divided into two groups, the control and the experimental.

The control group was randomly distributed throughout the sixth grade with four certified mathematics teachers. These teachers had experience ranging from eight to twenty-four years and all had taught at this middle school for at least five years. The control group was required to study mathematics by following the sequential order in a popular conventional textbook. The four teachers claimed to use traditional explain-practice teaching methods. In this instructional strategy, the focus is on procedures and the teacher demonstrates how algorithms are to be performed. Working individually, the students are then assigned exercises to practice the procedure that was demonstrated. Teachers grade students' papers, marking the exercises right or wrong.

The experimental group attended a special class for the first nine weeks of school with the first author, an experienced teacher, as the teacher-as-researcher. They learned mathematics through problem-centered instruction--a strategy that went beyond typical problem solving. Problem-centered learning is an instructional strategy that involves the selection of problematic tasks where students work in collaborative groups and present their solutions to the class for validation. The teacher neither explains procedures nor serves as arbiter of correct solutions to the problems. The model for Problem-Centered Learning (PCL) was described by Wheatley (1999) as:

    The class begins with a problem posed by the teacher, or perhaps by
  a student. The class is then organized into small groups (two or three
  students of similar capabilities) and the students work collectively
  in groups on the tasks posed. After about 25 minutes, the students are
  assembled for class discussion in which students present to the class
  their solutions for consideration by the group which then serves as a
  community of validators. During the class discussion the teacher is
  nonjudgmental and the viability of solution methods is determined by
  the class, not the teacher. In problem centered learning the teacher
  has three main roles: selecting appropriate tasks based on her
  knowledge of the students, organizing the groups and listening
  carefully as they work and finally, facilitating the class discussion.

Wood, Cobb, Yackel, and Dillon provide a detailed look at the problem-centered classroom in the Journal for Research in Mathematics Education Monograph Number 6 (1993). Working with second graders in the Midwest, they emphasize the interactive nature of three levels of mathematics activity: "activity by the individual class member, activity within the classroom community, and activity within the larger sociopolitical context" (p. 114). Full explanations of each type of interaction are cited to illustrate the issues involved and the mathematics reform required to implement a problem-centered instructional strategy. The instructional strategy referred to as "Problem-Centered Learning" is based on constructivist epistemology (von Glasersfeld, 1991).

Participants

The school district in this study used the ITBS as their standardized measure, and they identified low achievers as having scores below 40% national percentile on this test. Before the researchers began the experiment, the 650 sixth-graders in the school had already been ranked in descending order based on their mathematics score on the ITBS. The school agreed to designate 26 sixth grade students to be in the experimental class, while assigning 26 others to a control group from a potential population of 153 low achievers.

All of the 153 potential choices took a pre-test. Then 52 students were randomly assigned by their computer number to be in either the control or the experimental group. No consideration was given to their ITBS rank or pre test score in this selection. Six students were chosen as alternates in case some of the original 52 did not return to the school as planned in the fall when the experiment would begin. Four alternates replaced the original choices due to this reason. Also, two students (both African American females) in the experimental group exchanged places with those in the control group due to their reconsidered desire to participate in the experiment once school began. Data on the two groups (see Table 1) showed that together they fairly represented the demographics of the entire population of low achievers. Of the 153 students, the school related the following data: 60% African-American, 30% White, 8% Hispanic, 2% Other, and 58% female and 42% male. The demographics of the study were 54% African-American, 33% White, 10% Hispanic, 4% Other, and 62% female and 38% male.

Instruments

The entire population of 153 available low-achieving students took a qualifying pre-test during the final week of fifth grade. Teachers at the school were consulted and agreed that this free-response test characterized age-appropriate mathematics' content. They made this determination based on their experience with their curriculum and the ITBS test. Topics assessed included patterns, number sequencing, fraction concepts, and arithmetic word problems. The pre-test served two purposes: 1) it confirmed a student's identification as a low achiever, in addition to their ITBS score (none of the potential 153 students scored above 54% correct on the pre-test), and 2) it gave an independent measure of mathematics achievement for comparison at the end of the experiment, since the ITBS would not be administered until the following spring.

A similar post-test was administered to the 52 participants after nine weeks, along with parent and student attitude questionnaires, to identify any changes that occurred. This post-test was exactly like free-response pre-test taken except that the order of the questions was rearranged and the digits were changed. The same concepts were tested: pattern matching, number sequencing, fraction concepts, and arithmetic word problems. A few new free response questions were added to the post-test, but these were not scored. The tests were graded by another person not involved in the study using a rubric developed by the researchers.

The validity of the test instruments was determined by closely matching test items to the mathematical content to be taught to both groups during the nine weeks of the experiment (see Table 2). A nationally known curriculum was used by the school system, and this textbook was chosen because it closely matched the subject matter that would be tested later that year on the ITBS. So, as the test instrument was based on the specified school curriculum, it contained questions very similar in format and content to the practice ITBS-test the school provided to teachers. The only difference was that there were no multiple-choice answers; students had to write their own response.

Description of Instructional Materials

During the nine weeks, both classes would be exposed to the same mathematical content (Table 2). It was critical that the experimental group learn the same topics as the control group, because at the end of the study they would return for the remainder of the school year to the regular classroom and continue on with the traditional curriculum. Thus, while the two groups studied the same mathematical topics, the experimental design was to present this content using innovative activities.

However, although the mathematical content of the two groups in the study was identical, the instructional strategies had little in common. The activities in the experimental PCL classroom were predominately problem centered, with some teacher-led activities and discussion. An "explain-practice" strategy was never used in the PCL classroom, while explain-practice was the predominant instructional strategy for the rest of the sixth grade teachers at the school. Nevertheless, all teachers, including the PCL teacher, gave individual assessments (tests) of content every three weeks. Also, the school mandated that all classrooms spend some class time "practicing facts" such as multiplication tables, so both groups participated in this activity.

The nine-week curriculum for PCL was sensitive to the issues discussed in previous sections, particularly those recommended in 1989 by NCTM's Standards and 1998's Discussion Draft. Important considerations included student empowerment, communication of ideas through student-teacher, student-student, and teacher-student interaction (NCTM, 1991), collaborative group work, potentially meaningful tasks, supportive atmosphere of acceptance and respect, and journal writing--all couched in problem-centered instruction that emphasized making sense of mathematics. The Equity Principle was underscored, with the teacher having "high expectation and strong support for all students" in an effort to enhance their individual success (NCTM, 2000, p. 12). Progressively difficult goals were identified and tasks were designed to meet both social and academic objectives (Table 3).

Sixth grade meant new experiences for most children. At their new middle school, they now rotated classes and had a variety of teachers. Students were open to the prospect of having no math book (since the PCL materials were not in a book), even if this meant taking home worksheets. They seemed unaware of any special qualifications to get into the class. The teacher told them "the computer had picked them" to explain their random assignment. Their low-achieving status was not mentioned to them, and this label was disregarded once school began.

During the first week of school, the class negotiated the social norms of respect, sense making, collaboration, and self-validation. As NCTM has stated, "creating an environment that fosters a (positive) intellectual environment is essential" to an effective mathematics classroom (1991, p. 56). Teachers must develop each "student's mathematical power by respecting and valuing their ideas, ways of thinking, and mathematical dispositions" (1991, p. 57). Quick Draw figures (Wheatley, 1996) were used extensively on the overhead projector to persuade students to create a classroom culture where they actively participated by verbalizing their thinking and explaining their ideas to their peers.

The execution of the Quick Draw activity was straightforward. A single picture (see Figure 1) was shown for a brief period (i.e. 3 seconds) on the overhead projector. The figure was then covered and students were asked to "draw what you saw." The past tense was used because students had to make their drawings based on the mental images they had constructed since the figure was not observable while they were drawing. Finally the figure was uncovered and students volunteered the mental tactic they had used to draw their illustration. A discussion of their interpretations and method of drawing was an integral part of the Quick Draw activity. Quick Draw figures are designed to develop powerful mental imagery which will come into play in both numerical and geometric settings and to encourage students to recognize there is more than one way to solve a problem.

[FIGURE 1 OMITTED]

When a figure is shown, a wide variety of interpretations are possible and learning that other persons see it differently can be liberating. That is, students come to believe that there are no wrong explanations for any ilustration; in fact, the teacher encourages multiple interpretations rather than enforcing the idea that there is just "one way" to do a task. This latter belief can be debilitating. As students are engaged in doing mathematics, they are likely to fear forgetting THE way to do the task and develop a level of anxiety that destroys their confidence and capacity to construct mathematical ideas for themselves. Once students come to believe that mathematics is memorizing and applying rules, it is very difficult for them to respond meaningfully to tasks that require decision-making. Thus, for the group of children in this study where past failure and current anxiety were high, Quick Draw activities were chosen for the initial week of class because of their potential to empower individuals to think creatively without fear of being wrong. Quite simply, for many of them it was a revelation to realize that they could be successful at solving a problem. Discussion of interpretations of a figure is an essential part of the Quick Draw activity because it is this exchange that so effectively facilitates the development of the non-judgmental classroom atmosphere. A transcript of the dialogue that resulted after showing the first drawing (Figure 1) follows:

    Teacher: What did you see?
    Student 1: I see a V and an upside down V.
    Student 2: I see a box with two X's in it.
    Student 3: It looks like a drum.
    Student 4: There is a diamond in the middle.
    Student 5: There are triangles and a diamond.
    Student 6: I see teeth.

All of these answers were equally correct and accepted with enthusiasm by the rest of the class. Students often came forward to explain their interpretations and point at different parts of the figure so that others could understand their vision. During their explanations, the teacher was careful not to make comments that showed she valued one interpretation above another. She monitored her own behavior vigilantly to make sure she sent a clear expectation to the children: student's opinions were valued and desirable. "Teachers must think through what they really expect from their students and then ensure that their own behavior is consistent with those expectations" (Good and Brophy, 2000, p. 127).

The class also wrote in journals several times per week in the beginning of the study. This activity fostered the belief that expressing their ideas in writing was an important component in learning mathematics. None of them had ever been required to keep a journal in their mathematics class and several expressed surprise that "you did anything but numbers in math." They were always given a standard prompt, such as, "What I learned in math today was ..." or "The thing I still don't understand about today's lesson was ..." Sometimes they were given a choice of two prompts or a problem to solve. Journal writing cultivated the belief that students were responsible for their own learning. They were encouraged to recognize their strengths and weaknesses, to self-assess their progress, and to congratulate themselves on their successes. As the study progressed beyond the third week, journal writing became less frequent, occurring once or twice a week. Journals were only "graded" for participation points. The teacher picked them up on a rotating schedule and wrote non-judgmental comments in response to the entries to show that she valued student explanations and opinions. Journals were very helpful in informing classroom practice.

  Christi, like many others in the class, did not volunteer answers
  readily-but waited to be called on. She acted according to her
  previous experience in mathematics class, writing in her journal that
  in math "you just have to listen carefully and do what you are told."

For instance, such a journal entry during the first week coupled with teacher observation showed the necessity of developing classroom strategies that would encourage self-confidence and participation. So, during the early part of the study, the teacher used a seating chart to call on every student by name almost every day, thus ensuring that all of the children had many opportunities to come forward and publicly justify their responses. Children learned to listen respectfully to each other and then to respond to what they heard. Verbal participation in mathematics was a new experience for many students.

An algebraic reasoning activity called "What's My Rule?" also encouraged students to participate. The teacher began by writing any whole number on the board (for example, a "4.") Then the teacher mentally identified an arithmetic transformation (such as "multiply by 2.") Drawing an arrow, the teacher would say, "My rule makes this number a ______," while writing the transformation for this first case (for our example, 4 multiplied by 2, so the teacher would write 4 [right arrow] 8.) Next, the teacher called on a student to provide a new number (i.e. the student volunteers a "6.") The teacher wrote this number on the board, saying, "My rule makes this number a ______." (For our example 6 multiplied by 2 is 12, so she would write 6 [right arrow] 12.) After two or three examples, the students guessed the transformation by raising their hand and calling out, "Rule!" To prove their guess was correct, the teacher gave the student a new number and the child was responsible for stating the number transformation and the arithmetic rule. The teacher always let two students give responses in case the first rule-guesser was incorrect. The students' names were written alongside their responses in recognition for all to see.

Each problem had to be considered individually, since the pattern could involve any computation in any combination. The class liked this game so much that by the fourth day of class, enthusiastic children took over the lead of the activity for the teacher. For instance, Timmy wrote: 10 [right arrow] 50, and underneath that 30 [right arrow] 150, and the students excitedly tried to guess what rule had been applied. He then wrote 12 [right arrow] ______, and waited until Ricky said, "60!" He wrote 6 [right arrow] ______, and Tyler quickly replied, "30!" Since Ricky was the first to guess the correct rule, he would then take over the lead for the activity and Timmy would sit down. (If Ricky had been wrong, Tyler would have taken the lead.)

One of the advantages of the students leading this activity on their own is that it freed the teacher to walk around the room and interact quietly with individual students. The majority of the class was always actively engaged in the game, so the teacher could easily do on-the-spot interviews with particular children to get an idea of what they were thinking. This procedure allowed the teacher to further inform and refine her classroom practice. Also, the student leading the activity learned to communicate effectively with his peers and develop appropriate leadership skills.

  Christi was confused. I quietly asked her what Timmy was doing--how
  did 10 get to be 50? She whispered, "You add 40."
    "Then how does 30 get to be 150? Do you add the 40 again? It has to
  be the same pattern," I asked. She knew that adding 40 wouldn't be
  right, and struggled with the answer. Christi was unable to see the
  multiple of 5 and the class was moving on, so I quickly showed her how
  it worked on her paper and encouraged her to be patient.
    Ricky came up and wrote: 5 [right arrow] 35. Christi knitted her
  eyebrows in puzzlement. "That's not multiply by 5," she said. Then
  underneath that example Ricky wrote 2 [right arrow] 14. Her eyes lit
  up as she recognized that in the second step he multiplied by 7. When
  Ricky put 3 [right arrow] ______ she worked slowly but got the answer.
  A triumphant "21!" she squealed a minute after he had already written
  it. She was smiling, proud to be getting it right. I told her I knew
  she could do it. Obviously, Christi had memorized her "times tables,"
  even if she didn't know what to do with them on an ITBS test!

By the end of Week 2, the teacher noticed significant changes in student behavior that had developed coincident with the change in instructional content. The class was now engaged in "Mental Math," where numbers were being mentally partitioned into convenient addends. The teacher wrote horizontal addition problems on the overhead projector and solicited student input for strategies to solve them. As students responded, their suggestions were written on the overhead along with their name. The teacher took as many solutions as possible to develop an inclusive classroom culture that maximized student participation. The following transcript illustrated typical classroom discourse during this activity. Characteristic of PCL, the students' solutions are validated by peers.

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[ILLUSTRATION OMITTED]

It was during these Mental Math activities that changes began to occur in student behavior. When class was over, half of the children left for their next class while others voluntarily stayed behind to continue work. Perhaps the smaller stay-behind group had a less competitive feel and gave students confidence, so they were hesitant to depart. Perhaps they simply enjoyed the activity of "splitting numbers" as they called it. Another transformation was observed during whole class discussion: the number of children who spontaneously raised their hands to volunteer answers instead of waiting to be called upon was growing steadily each day. Students apparently enjoyed the opportunity to reason mathematically in this setting.

  We were doing mental arithmetic, and Christi raised her hand to answer
  a problem. This was the first time she had volunteered, and her body
  language spoke volumes. She was wiggling in her seat, grinning from
  ear to ear as she called out her suggestion in a clear, loud voice.
  "25 + 15. You can break the 15 down into three 5s, and just count up
  by 5s from 25. That's 25, 30, 35, 40!" I wrote her name and suggestion
  on the overhead as she talked. She slapped her pencil down with
  satisfaction when she was through.

By the third week, these low-achieving reticent students regularly engaged in lively discussion. The teacher introduced several kinds of tasks each day so various students had a chance to excel. For addition and subtraction, "Math Squares" became the class favorite (see Figure 2). They called it "a game because we have to think." "It's never the same," was their perception for this departure from a traditional row of computation problems. Math Squares provided rich opportunities for students to think in tens and refine their mental arithmetic in a meaningful setting (Wheatley and Reynolds, 1999). They encouraged students to consider alternatives in determining what operations to perform next. Students had to think about Math Squares and then do what made sense rather than following fixed procedures.

[FIGURE 2 OMITTED]

All the children appeared to be actively engaged. Students rarely put their heads down on their desk. Many hands eagerly went up with each problem, and someone was always disappointed when another answered first. When not explaining to the class, students shared drawings and solutions with close neighbors. A low rumble of animated conversation became the norm after each task was presented. Comments like, "See how I did it?" and "Look at mine!" were often heard.

The classroom culture continued to evolve as inclusive and communicative. So, in Week 3 students were organized into small groups and began to solve more difficult non-routine mathematics problems. First, the students defined their own set of "Rules for Group Work" by consensus as the teacher wrote on the overhead. Most suggestions were expected (one person talks at a time, respect your group members, everyone has a job, we all get the same grade), but interestingly they added this rule: "There are many ways to look at things." That idea received an almost unanimous vote.

Group assignments were based on the teacher's judgment of student competence evidenced by class activity and homework. Students were placed in a small group with others of similar mathematics ability so they would more likely feel free to challenge each other and not become dominant or dependent/passive. We did not want one individual to become the mathematics "authority" in their group and inadvertently usurp the autonomy of the other group members. No one needed to slip back into the habit of copying what he or she perceived as someone else's "superior" mathematics without making meaning of it for themselves. Personality traits were also a factor. Two like-ability, introverted, and quiet children might be placed together to necessitate their participation in class discussion. Two extroverted, out-spoken students might be in the same group to cultivate their respect for each other. (Later reflection showed that groups did not necessarily match children with similar ITBS scores.)

  Christi was designated to work with teammates Donald and Jennifer.
  These three seemed to be the most hesitant, unsure, and slowest
  workers in the class. By the end of the third week, this trio was the
  most focused group in the class. They were well-engaged and worked
  steadily. On average, they completed fewer problems than the rest of
  the class, but their solutions were elaborated and correct. Whenever
  I came over to check their progress during an activity, they were
  Deeply engrossed. Sometimes they were not even aware of my presence as
  they collaborated or wrote in their journals. They effectively shared
  responsibility and negotiated meaning for the mathematics tasks they
  were assigned.

By Week 4, groups successfully solved and justified solutions to tasks such as the one shown in Figure 3. Use of non-routine problems was a major component of the curriculum. These activities required students to make sense of their mathematics by using it in meaningful situations. Hence, instead of memorizing a procedure to find an average, students collected real data and devised their own strategies.

[FIGURE 3 OMITTED]

Placing students together in a group of similar abilities may have been successful because it allowed them to participate in mathematics meaning-making in a way they had not been previously afforded. Instead of trying to keep up with students who moved faster than they, they finally had time to construct their own mathematics. They could discuss their methods with less threat of ridicule. There was evidence that students responded well to this arrangement because they often asked, "Do we get to work in groups today?"

Students developed confidence in using their own strategies to enhance standard algorithms as the nine weeks progressed. For example, at one point the class was practicing multi-digit multiplication. Eight students had trouble getting a correct solution when the multiplier on the bottom had more than one digit and found the customary method confusing even when using grid paper to keep the digits "lined up." But then several children worked together using their knowledge of multiplication procedures to create a system of covering up the numbers on the bottom row not currently being used as the multiplier with their fingers. After multiplying one digit by everything on top, they moved their fingers to uncover the next number on the bottom row to be used. They justified their procedures and helped all the others learn to use it successfully.

About a month into the study, students were introduced to "Balances" (Figure 4). These activities were adapted from similar tasks that had proven successful for number development in younger children (Cobb et al., 1991). Children elaborated and enriched their constructions of number patterns and relationships by finding ways to "balance" the numbers on the scales so that they were equal on both sides, and then writing a corresponding number sentence for their solution. They used a variety of strategies and computations to solve the problems, such as trial-and-error, partitioning, addition, subtraction, multiplication, division, doubles, or thinking strategies. Students often commented on the challenge and enjoyment they found in discovering diverse ways to solve Balances. In fact, many routinely solved a Balance problem in more than one way and then shared their different solution methods and respective number sentences with the whole class. Balances were found to be effective and meaningful in sixth grade mathematics when used with progressively larger whole numbers, decimals, fractions, and percents.

[FIGURE 4 OMITTED]

By week six, the class was doing "Two-Ways" (Figure 5). This task discouraged using rote procedures. Students had to make sense of each problem since the omitted information appeared in a variety of unpredictable combinations. Two-Ways helped students elaborate on the mathematical patterns and relationships they were developing and further enhanced mental arithmetic methods. The self-checking nature of the Two-Ways fostered interest, self-confidence, and made it easier for students to identify and correct their errors.

[FIGURE 5 OMITTED]

During the seventh and eighth week, students were using fraction bars, diagrams, and a variety of schemas for dividing whole numbers into "fair shares." The teacher purchased some of these instructional materials, like Fair Shares workbooks (Tierney & Berle-Carman, 1995). For making sense of fraction concepts, additional problems like those in Figure 6 were very helpful in developing students' analytical reasoning about partitions. This format was particularly beneficial because students had seen simpler versions of "pattern" questions like these in their elementary school years. The idea of looking for a pattern in a sequence was faimiliar--and it was easy to make connections between past experience and new fraction concepts.

[FIGURE 6 OMITTED]

Homework assignments during Week 8 and 9 involved a series of Math Squares and Balances that contained increasingly difficult fractional computations. We began with fractions totaling less than one whole and with the same denominators, and progressed to fractions with related factors in the denominators, like 2, 4, and 8. Next we worked with combinations of unrelated denominators, such as 3, 7, and 10. We introduced mixed numbers in the same manner, beginning with similar denominators and ending with worksheets that contained mixed numbers of any size or denominator. These were the final instructional materials of the study. At the end of nine weeks, the class returned to the traditional textbook and curriculum along with their peers throughout the school.

RESULTS

Quantitative Methodology

A number of statistical tests were used to examine the quantitative data obtained in this study. First, the pre-test and post-test were subjected to an item analysis to determine the validity of the test instrument. Validity assures that the test results are useful; i.e. the test results measure what they were intended to measure. Item analysis involved a statistical correlation between the score students obtained on each test question and their total score on the test. Results are shown in Table 4.

This discriminate analysis yielded only positive correlations. In other words, if children scored well on a given question, they also obtained relatively high scores on the entire test. If students scored poorly on a test item, they also tended to have low scores overall. The mean and the median correlation for test items in the control group was 0.41 and 0.42, respectively. For the problem-centered group, the mean and median correlations were 0.44 and 0.46. These values are important indicators of the quality of the test and give supporting evidence that the test instrument was valid.

Next, Hoyt's formula was used to obtain a reliability coefficient for the test instrument (Hoyt, 1941). Reliability indicates that the test can be counted on to give consistent results on subsequent administrations. The Hoyt reliability is designed to determine the degree to which test responses reflect relative student performance. If student achievement and test question variability can account for all the variance in individual item scores, then the reliability will approach a perfect score of 1.

Hoyt's formula is r(ii) = (a-c)/a, where a = mean squares for individuals and c = residual mean squares (error). Standard analysis of variance with [alpha] = 0.05 was computed on the test item scores in both groups to obtain mean squares. The result was a reliability of r = 0.88, suggesting the content of the test is reliable.

Descriptive statistics and ANOVA were used to further examine the results of the pre-test and post-test scores of both groups of children (see Table 1 for scores). The data for "Pre-test" and the "Increase in Score" both had normal distributions and met the assumptions for ANOVA (see results shown in Tables 5 and 6). The gain in test score was judged to be the most appropriate method for analyzing the data because it accounted for natural ability. A single factor regression was chosen because in each case a single set of data for each sample was compared (e.g. the control group's pre-test scores vs. the experimental group's pre-test scores--with one score and one trial for each student). An alpha equal to 0.05 was selected for a confidence level of 95%.

The analysis shows there was no significant difference between the experimental and the control group at the onset of the study on the pre-test at the 5% level. The mean of the experimental group was 27.5 and the control group was 27.3.

The results of examining the data obtained by subtracting each student's pre-test score from their post-test score (or the Increase in Score due to both approaches) does, however, show a significant difference. The p-value for the ANOVA is less than .01. We can state with 95% confidence that the gain in the experimental group's post-test score was significant and attributed to the experimental treatment.

Qualitative Methodology

Qualitative results from the surveys, journal writings, and interviews were also analyzed. All children in the control and experimental group took a survey at the end of the nine weeks (see Table 7). These questionnaires were completed anonymously to increase the likelihood of honest response. There were 20 questions that were analyzed on a Lickert scale, followed by three open-ended questions. Their parents took the same survey expanded with 5 more Lickert-type questions regarding their perception of their child's achievement at the class, school, and national level.

There were twenty-six students in both groups, all of who responded because the survey was given at school. Parent surveys were sent home to be returned in sealed unmarked envelopes to ensure confidentiality. Fourteen parents in the control group and eighteen in the problem-centered group returned their surveys. These survey results were summarized in simple uncoded form Table 8.

An outside individual used an inductive approach to encode the data from the surveys. Data was categorized into meta-codes and sub-codes. The number of responses in each cluster was counted in order to determine significant clusters. Certain patterns and themes emerged and these were triangulated with the results obtained in interviews and observations. Notes were taken on surprise responses and negative responses. Positive themes were generated: students in PCL were more interested and eager to do their work, their grades improved and they enjoyed mathematics. Negative themes were that student work was improved but could still be better, group work was slower than doing it alone for some individuals, and not all students shared equally in the group.

All in all, student survey results showed sharp contrasts between the two groups. The control group had clusters in the middle-to-negative range for attitude. Only those students who reported liking mathematics prior to this year continued to give positive responses. The majority reported mathematics was the same as it had always been: consistently boring, uninteresting, and they did not enjoy homework or talking about mathematics outside of class. The problem-centered group, on the other hand, had clusters in the positive range on the Lickert scale in 17 of 20 questions. Conversely, they described mathematics class as "fun," enjoyed doing homework, and said they often talked about mathematics at home. Fifteen of the twenty-six students reported changing their feelings about mathematics during the study from negative to positive.

Significantly, the control group wrote in few if any comments in the free-response area of the survey instrument, while 22 of the 26 children in the problem-centered group wrote some comment about small group work, problem solving, or their feelings about their competency, self-confidence, and self-esteem. One might attribute this difference to the fact that students were regularly encouraged to write and explain their solutions in PCL, whereas in the traditional curriculum students seldom wrote explanations or stated opinions. This behavior may have extended to the survey forms. Another explanation might be that students in PCL felt that their opinion "counted" because this belief was supported by the classroom practice of encouraging individual responses. In the traditional curriculum 4 students usually only made contributions by giving solutions to mathematical questions posed by the teacher or the textbook. As the survey questions were not mathematical, they may have been viewed as insignificant.

Parent's responses on surveys paralleled their respective children with regard to nearly every question. Eleven parents in the PCL group wrote additional comments regarding their child's increased motivation and improved attitude towards mathematics, while parents in the control group mentioned having too much homework or complained of their child's lack of motivation and achievement. Three parents in the problem-centered group wrote that they were dissatisfied with the child's mathematical achievement to date in their schooling, but they also had positive comments about the program.

Three students (one performing well, one middle, and one low) and their parents were interviewed from the experimental group by the outside observer or researcher. None of the parents gave negative feedback and all expressed enthusiastic support for the PCL. The three students also gave predominately positive responses, with the only derogatory comments referring to occasional incompatibilities with other group members or desire to spend more time on computers.

Students in the experimental group wrote in their journals at least once a week for ten minutes. Usually they responded to lead-in questions like, "This week in math I learned ..." or "What I need to know more about is ..." or they explained some mathematical concept or problem in their own words. These journal writings supported the anonymous answers from the surveys and the interview responses. The most common themes were "math is fun now," because "we get to figure out things for ourselves and work in groups." The children reported they felt "important" (i.e. valued) and "able to say my ideas" (i.e. empowered) by the learning environment they negotiated.

DISCUSSION

Unfortunately, remediation efforts with low achieving children have proven a difficult task for teachers (Hankes, 1996). By sixth grade, such children are accustomed to consistently obtaining the "wrong" answer in mathematics and they are reluctant to trust their own reasoning. (Indeed, it was their previous lack of correct responses that was the basis for labeling them low-achievers in the first place.) We believe this lack of a meaning-making orientation in children is related to the kind of response they got from authority figures. The more often the teacher or perceived "smart kid" said they were wrong, the more often low-achievers regressed into just trying to remember what they saw without trying to make any sense of it. They assumed they didn't know how to think anyway because they were often incorrect, so why not just mimic an authority's method? These students had little confidence in their own reasoning (Mack, 1990; Wearne & Mebert, 1988).

Furthermore, to maintain some self-respect, low achievers may have "learned" to avoid humiliation by shying away from offering in-depth solutions in public because they lacked confidence (Hankes, 1996). Their hesitancy to talk in class then actually inhibited their learning even more. Therefore, the first challenge in the experiment was to get students to express themselves and recognize the reward of asserting themselves mathematically.

The observations of an outside evaluator are particularly helpful in a discussion of how children appeared to have overcome this self-defeating behavior by the seventh week of the study. During the last two-and-a-half weeks, the university engaged an individual who was by profession an education/curriculum evaluator sent from abroad to earn an advanced degree by her government. She had no connection to the researchers, the teachers, or the school. She wrote in her reports that the "children worked with enthusiasm ... the groups were working on different worksheets--some are further ahead than others." She observed that groups had developed their own rules for interaction. Some were slow to get to work, needing to chat before tackling the task, while others started to work promptly. Some students drew pictures; others used a more traditional numerical technique, others an abstract schema. There were groups that shared mathematics through language, and groups whose members tended to work independently. In spite of children having freedom to move around the room, she observed that their "behavior was not disruptive."

The observer particularly noted the students' active participation during whole group discussion after small-group problem solving. As a program evaluator for her government, she had recently visited many middle school classrooms and felt she had a basis for comparison. She found it difficult to believe that this was a class of low achievers. She asked the teacher: "How did you get them to be so excited and involved--I've never seen anything like it! You would think this was an advanced class from the way they wave their hands in the air, each trying to volunteer answers." The kids appeared excited about mathematics--they readily shared their ideas before the whole class and challenged each other's solutions. Their behavior was in sharp contrast to the traditional middle school classroom she had observed in the past.

Late in October, the program evaluator witnessed a review lesson where overhead transparencies and marking pens were distributed to the groups. The students were instructed to choose two problems from their worksheets to display and explain to the class. Her observation of this activity was:

  The group stands round the projector, heads together over the
  transparency. [Eager to explain. They are very excited about using the
  overhead projector, that's usually the teacher's role, not theirs. It
  makes them feel important]. The teacher points out to them anything
  they've forgotten to do in solving the problem. Other students watch
  and listen [some attentive, some are not]. The teacher corrects them
  kindly. There is no expression of embarrassment on their faces when
  they make an error. [It is all right to be wrong, they feel safe]. The
  problems reflect real life situations, e.g. "You have so much money
  and you want to buy ..." The students' eyes are focused on the teacher
  when she explains [attentive]. Some hang around at the end of class.
  [It seems they would rather stay than go]

The teacher had also noted this student behavior--staying voluntarily and without invitation after class. After the first week together, up to ten students remained after the dismissal bell each day after everyone else departed. Though they knew they had to get to their next class, they would tarry for as long as possible--often up to 15 minutes. Attempts to walk them up the hall after 5 minutes just to keep them from accumulating tardies were unsuccessful. So, when the 15-minute mark arrived, the teacher literally resorted to a good-natured physical push to force them out the door.

At first we believed the children stayed late because they had a bathroom break at the beginning of the next period and were willing to skip that. Leisure time in one place was the same as leisure time in another, and they were already here. But this was not a completely satisfactory explanation given the circumstances. While they waited, the laggards usually continued to do mathematics. Some stayed huddled in groups of two or three, puzzling over a worksheet solution. Others clustered at the blackboard or projector, writing what they called "hard problems" for each other. It soon became apparent the reason they stayed was to continue their engagement with mathematics. These "low achievers" seemed to have developed the sense that mathematics was fun, for they enjoyed the experience sufficiently to want to prolong it.

  During an interview with the program evaluator, Christi
  enthusiastically confirmed that she "understood stuff now. It's fun in
  our class--math is fun." She further elaborated to the interviewer
  that fun meant a variety of activities and that "every problem is
  different." Apparently she did not feel threatened by the continual
  variation of tasks placed before her. Unlike her prior mathematics
  experience, there was no teacher procedure to copy and then follow
  with rote practice on a set of similar examples. She had to make
  meaning of every problem because each was unique. Christi was
  liberated into a world of her own understanding, unmarked by the
  forced pressure of an imposed method that relegated her thinking to
  inferior status. Empowerment was fun! That's why she stayed behind to
  "play more games on the board." She even demonstrated the strength of
  her meaning-making by getting good grades on classroom tests that
  included non-routine word problems.

Did a non-judgmental classroom culture influence students' attitude towards mathematics? Evidence suggests that the PCL model may have had a positive input. On the last day of class, two decorated envelopes were received. One note said:

  We will never forget how you helped us with our math and how you made
  math fun. I just want to say 'we thank you' for letting me take a test
  and then letting me come into your class and then letting us know that
  you did care. We will miss you.

Seven students had signed the note. The other envelope contained a personal note echoing the same theme, supplemented by a smiling school photo. Further evidence of an attitude shift was found in final journal comments--three of the low achievers from the class now stated that they wanted to be a math teacher when they grew up.

Christi's mother confirmed these attitude changes two weeks after the study:

    My daughter has enjoyed the class and understood math a whole lot
  more. Christi is the kind of kid who has to understand things or she
  won't do it. For the first time, she's been able to do her math and is
  excited and talking about it every day. She always has her math work
  out, showing it to me and saying, 'Look at this! Look what I did,
  Mama!' She shows me worksheets with squares, and word problems. She
  has to explain them to me sometimes--I can't even figure them out. She
  has a good grasp ... a good understanding ... she likes doing homework
  and thinks it's fun.
    She didn't seem like my child at all. She really hated math before
  this. It's always been her worst subject. Now she's really
  interested--all of a sudden in this class. It has made a big
  difference in her life. It was the best thing that ever happened to
  her. I have a new child!

One should note that the results of this study could have been influenced by the characteristics of the researcher-as-teacher. As an experienced mother and teacher who has worked extensively with youth both professionally and personally, the researcher concedes she enjoys children and is often perceived as a nurturing personality. However, the other four teachers in the study might likewise claim the same credentials.

The role of homogeneous grouping may also have had an effect on the experience of these low-achieving students. Whereas the low achievers in the control group were mixed with the rest of sixth grade, the experimental low achievers were grouped in a homogeneous classroom. They were obligated to assume leadership roles both in the class and in their peer groups. This situation may have increased motivation in the PCL group and influenced achievement. Further investigation in heterogeneous classrooms is needed to determine the effect of PCL on low achievers who are mainstreamed as well as the effect on students of diverse ability.

Nonetheless, evidence suggests that the combination of approaches in the classroom may have helped the students make sense of mathematics because this understanding was reflected in measurably higher test scores. Statistical analysis provided earlier in this paper showed a significant gain in scores for the children in the experimental group compared with those in the control group. Although both groups studied the same content for only nine weeks, the experimental treatment increased scores 34%, while the control group gained 19%. Additional study involving a longer period of time and collection of annual ITBS scores is needed to determine the longitudinal effect of PCL.

Qualitative results were also supportive of PCL. From the experimental group, none of the three parents interviewed by the outside observer or researcher gave negative feedback. The three students interviewed also gave predominately positive responses. Student surveys of the two groups showed sharp contrasts. Students in the control group reported mathematics was the same: consistently boring, uninteresting, and they did not enjoy homework or talking about mathematics outside of class. On the other hand, the PCL group said that mathematics was fun now, they enjoyed homework, and they often discussed mathematics at home.

Parents responded in basically the same way to surveys as their children in the respective group. Eleven parents in the PCL group wrote additional comments regarding their child's increased motivation and improved attitude towards mathematics. However, a limitation of the study is that these results were post-experimental. No student or parent attitude surveys were given at the beginning of the experiment, so the responses represent reflections on the past combined with current beliefs.

Student attitude towards mathematics may well have had an impact on how students achieved and thus may have contributed to the difference in post test scores when one considers that the same content was taught to all students. A positive attitude increases motivation, and motivation has been positively correlated with achievement in many studies over the years (Karsenti & Thibert, 1995; Passe, 1996). Because the children said they felt valued and empowered by the learning environment they negotiated in problem-centered mathematics, they may have been motivated to participate more actively in whole class discussions and small group work. They appeared to take responsibility for their own learning and appreciate the development of their intellectual autonomy. The effect of a non-judgmental, respectful, and communicative classroom such as that fostered by the PCL strategy thus deserves strong consideration for further study.

TABLE 1: Student Data

                 Control Group
    Race  Sex  Pre-  Post-         Increase
               Test  Tes[t.sup.+]  in Score

 1   W     F    22       44           22
 2   A     M    16       56           40
 3   O     F    40       73           33
 4   W     F    48       62           14
 5   A     M    28       44           16
 6   W     F    20       24            4
 7   A     F    12       22           10
 8   W     M    42       74           32
 9   A     F    16       26           10
10   A     F    22       40           18
11   A     M    34       46           12
12   A     F    22       37           15
13   A     F    22       47           25
14   A     F    52       60            8
15   A     F     8       20           12
16   W     F    46       64           18
17   A     F    47       48            1
18   A     F    10       20           10
19   A     F    18       46           28
20   W     M    36       79           43
21   A     F    26       60           34
22   A     M    12       40           28
23   W     M    28       43           15
24   A     M    14       47           33
25   H     F    20       27            7
26   A     F    54       57            3

                  Experimental Group
    Race  Sex  Pre-          Post-         Increase
               Tes[t.sup.+]  Tes[t.sup.+]  in Score

 1   W     F       46             87           41
 2   W     F       30             85           55
 3   O     F       22             64           42
 4   A     F       30             59           29
 5   W     M       12             59           47
 6   H     M       22             45           23
 7   W     F       14             49           35
 8   H     F       14             50           36
 9   A     M       40             69           29
10   A     M       14             77           63
11   A     F       36             80           44
12   W     M       30             49           19
13   W     M       12             52           40
14   A     M       28             71           43
15   A     M       48             79           31
16   W     M       28             47           19
17   H     F       42             54           12
18   A     F       10             30           20
19   W     F       22             74           52
20   W     M       38             26          -12
21   A     F       36             62           26
22   A     M       18             35           17
23   W     F       10             52           42
24   A     M       38             77           39
25   A     F       34             82           48
26   H     M       36             70           34

Key to Demographics: W=White, A=African American, H=Hispanic, O=Other,
M=Male, F=Female

TABLE 2. Mathematical Content Studied by Both Groups

Main topics addressed during the nine-week study were:
      1. Addition and subtraction of whole numbers, decimals, and
           currency
        2. Multiplication and division of whole numbers, decimals, and
           currency
        3. Place value to the billions
        4. Order of operations
        5. Sequences and writing patterns
        6. Factors of whole numbers
        7. Estimation of whole numbers
        8. Fractions: drawing partitions and shading parts, relative
           size, addition and subtraction
        9. Ratio and proportion
       10. Finding averages
       11. Reading, interpreting, and making tables, bar graphs, line
           graphs

TABLE 3. Curriculum Focus in the Experimental Class

Week 1 and 2 Goal: Negotiate non-judgmental social climate, encourage
             student verbal participation

    Tasks:  *  Partner interviews and oral introductions
            *  *Quick Draw figures with multiple interpretations (see
                 Figure 1)
            *  *Pattern and sequencing games (like "What's My Rule?")
            *  Mental Math (variety of mental strategies)
            *  *Student journals (self-evaluation and attitude checks)
Week 3 Goal: Develop small group interactions potentially meaningful
             tasks
    Tasks:  *  Students define group rules (articulation with respect)
            *  *Math Squares (approaching each problem as unique--see
                 Figure 2)
            *  *Challenge cards (meaningful word problems--see Figure 3)
            *  *Balance Activities (using a variety of strategies--see
                 Figure 4)
Week 4 and 5 Goal: Reach more sophisticated levels of problem solving
    Tasks:  *  *Data collection and graphing (meaningful situations)
            *  *Decimals and money (real-life applications)
Week 6 and 7 Goal: Focus on non-routine addition/subtraction
             strategies, multiplication
    Tasks:  *  *Two-Ways (see Figure 5)
            *  Multiplication developed using two-dimensional arrays
            *  Sharing problems (text format encourages diagrams &
                 variety of procedures)
            *  *Speed tests (relate to data collection, graphing)
            *  Advance from 1-digit to 3-digit multipliers
Week 8 and 9 Goal: Understand fraction concepts
    Tasks:  *  Fraction bars (make sense of division, partition schemas)
            *  *Sequences (develop number sense--see Figure 6)

*Once introduced, these activities were used with increasingly difficult
content (e.g. single-digit whole numbers [right arrow] three-digit whole
numbers [right arrow] decimals [right arrow] fractions.)

TABLE 4. Test Item Analysis

Question #  Correlation for  Correlation for
               Control       Experimental

    1          0.2142           0.6570
    2          0.4661           0.6570
    3          0.2517           0.2262
    4          0.1734           0.1694
    5          0.2542           0.4879
    6          0.1530           0.5347
    7          0.4818           0.4152
    8          0.1255           0.3115
    9          0.6325           0.6099
   10          0.4289           0.6600
   11          0.5917           0.7423
   12          0.3992           0.6708
   13          0.4972           0.4621
   14          0.4149           0.4619
   15          0.1228           0.5745
   16          0.2272           0.3807
   17          0.4233           0.5977
   18          0.3813           0.4313
   19          0.5269           0.5836
   20          0.5663           0.2975
   21          0.5378           0.5886
   22          0.3931           0.5650
   23          0.2764           0.5506
   24          0.4930           0.4106
   25          0.6086           0.5379
   26          0.6747           0.5019
   27          0.2253           0.3439
   28          0.3745           0.1943
   29          0.6304           0.2267
   30          0.4968           0.4024
   31          0.5704           0.3032
   32          0.4047           0.4035
   33          0.3917           0.3757
   34          0.7101           0.2579
   35          0.6768           0.2974

TABLE 5. ANOVA Single Factor on Pre-Test Scores

SUMMARY
Groups          Count     Sum      Average   Variance

Column 1
  (Control)       26      715.000   27.500   194.660
Column 2
  (Experimental)  26      710.000   27.308   137.902

Source of
  Variation     SS        df        MS       F         P-value  F crit

Between Groups     0.481    1.000    0.481     0.003   0.957    4.034
Within Groups   8314.038   50.000  166.281
Total           8314.519   51.000

TABLE 6: ANOVA Single Factor on Increase in Score

SUMMARY
Groups            Count      Sum      Average   Variance

Column 1
  (Experimental)     26      874.000    33.615   246.806
Column 2
  (Control)          26      491.000    18.885   139.546

Source of
  Variation        SS        df       MS        F         P-value

Between Groups     2820.942    1.000  2820.942    14.603   0.001
Within Groups      9658.808   50.000   193.176
Total             12479.750   51.000

SUMMARY
Groups

Column 1
  (Experimental)
Column 2
  (Control)

Source of
  Variation       F crit

Between Groups    4.034
Within Groups
Total

TABLE 7: Sample Survey Questions

1. How often did you do math homework in 5th grade?
   Every day  Most of the time  Sometimes  Once in a while  Never

2. How often do you do math homework now?
   Every day  Most of the time  Sometimes  Once in a while  Never

3. How many times did you say good things about math in 5th grade?
   Every day  Most of the time  Sometimes  Once in a while  Never

4. How many times do you say good things about math this year?
   Every day  Most of the time  Sometimes  Once in a while  Never

5. When you talk about math this year, do you feel excited?
   Usually    Most of the time  Sometimes  Once in a while  Never

TABLE 8: Survey Responses

           Cluster Category  Response   # of      % of   # of     % of
                             Level      Students  Total  Parents  Total

CONTROL    Likes sixth       Better     10        38%    3        21%
GROUP      grade math,       Same       13        50%    8        58%
(26        motivated         Less        3        12%    3        21%
students,  Understands,      Better      9        35%    6        43%
14         can do            Same       14        54%    6        43%
parents)   problems          Less        3        11%    2        14%
           Says good         Usually     6        23%    3        21%
           things about      Sometimes  15        58%    8        58%
           math, excited,    Never       5        19%    3        21%
           fun
           Bored, doesn't    Usually     7        27%    5        36%
           want to learn     Sometimes  11        42%    7        50%
           more math         Never       8        31%    2        14%
PCL GROUP  Likes sixth       Better     16        62%   13        72%
(26        grade math,       Same        8        31%    3        17%
students,  motivated         Less        2         7%    2        11%
18         Understands,      Better     18        69%   12        67%
parents)   can do            Same        8        31%    5        28%
           problems          Less        0         0%    1         5%
           Says good         Usually    13        50%   14        78%
           things about      Sometimes  11        43%    2        11%
           math, excited,    Never       2         7%    2        11%
           fun
           Bored, doesn't    Usually     2         7%    3        17%
           want to learn     Sometimes   9        35%    4        22%
           more math         Never      15        58%   11        61%

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Candice L. Ridlon

Brigham Young University