Conference MaterialStudents' perceptions and attitudes in a standardsbased high school mathematics curriculum
STUDENTS' PERCEPTIONS AND ATTITUDES IN A STANDARDSBASED HIGH SCHOOL MATHEMATICS CURRICULUMHarold L. Schoen, University of IowaJohnette Pritchett, University of Iowa Paper presented at the 1998 Annual Meeting of the American Educational Research Association, San Diego, California, April 16, 1998 The research reported in this paper was supported by the National Science Foundation (Grant #MDR9255257). The views herein are those of the authors and do not necessarily reflect those of the Foundation.
In the mid1980s, the Commission on Standards for School Mathematics of the National Council of Teachers of Mathematics, the National Research Council's Mathematical Sciences Education Board, and the Mathematical Association of America began to work together in a coordinated effort to influence the direction of change in the mathematics curriculum from kindergarten through undergraduate mathematics. One document from this effort, the NCTM's Curriculum and Evaluation Standards for School Mathematics (referred to herein as the NCTM Standards), recommends that students go beyond memorizing formulas and applying formulas and applying procedures (NCTM, 1989). Students, it is recommended, should not only understand mathematics, but also do mathematics; for example, pose and solve problems, make conjectures, look for patterns, and justify and explain their mathematical thinking. The CorePlus Mathematics Project (CPMP or Core Plus) is a comprehensive curriculum development project funded initially by a fiveyear grant from the National Science Foundation. It is developing student and teacher materials for a threeyear high school mathematics curriculum for all students, plus a fourthyear course continuing the preparation of students for college mathematics. The curriculum builds upon the theme of mathematics as sensemaking. Throughout it acknowledges, values, and extends the informal knowledge of data, shape, change, and chance that students bring to situations and problems. Each year the curriculum features multiple strands of algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics, which are connected by fundamental themes, by common topics, and by habits of mind. The curriculum also emphasizes mathematical modeling, especially the modeling concepts of data collection, representation, interpretation, prediction, and simulation. Numerical, graphing, and programming/link capabilities of graphing calculators are assumed and capitalized on throughout the curriculum. This technology helps to facilitate the emphasis in the curriculum and instruction on multiple representations (numeric, graphic, and symbolic) and on goals in which mathematical thinking is central. Instructional practices promote mathematical thinking through the use of rich applied problem situations that involve students, both in collaborative groups and individually, in investigating, conjecturing, verifying, applying, evaluating, and communicating mathematical ideas (Hirsch, Coxford, Fey & Schoen, 1995). The CPMP curriculum and instructional model are described in more detail elsewhere (Hirsch, Coxford, Fey & Schoen, 1995; Schoen, Bean & Ziebarth, 1996; Hirsch & Coxford, 1997), and the textbooks, Contemporary Mathematics in Context, Course 1 and Course 2 are now available in published form (Coxford, Fey, Hirsch, Schoen, Burrill, Hart, Watkins, Messenger, & Ritsema, 1998). Carefully developed with teacher input over a threeyear period, each CPMP course is field tested in 36 high schools in Alaska, California, Colorado, Georgia, Idaho, Iowa, Kentucky, Michigan, Ohio, South Carolina, and Texas. A broad crosssection of students from urban, suburban, and rural communities with ethnic and cultural diversity is represented. Course 1 was field tested in ninthgrade classrooms in 199495, Course 2 was field tested in tenthgrade classrooms in 199596, and Course 3 was field tested in eleventhgrade classrooms in 199697. A great deal of quantitative and qualitative data were collected during the CPMP field test. The data includes information about various student outcome variables as measured by standardized tests and by constructed response or performance assessments, teachers' and students' attitudes and beliefs, level of implementation of the curriculum and instructional model, and specific site characteristics and experiences. Achievement outcomes are reported elsewhere (Schoen, Hirsch, & Ziebarth, 1998), and reports of several focused research studies conducted in Core Plus classrooms are cited in the appendix to this paper. This report focuses on students' attitudes and perceptions of their experiences in CPMP.
According to McLeod (1992), "...there appear to be at least three major facets of the affective experience of mathematics students that are worthy of further study. First, students hold certain beliefs about mathematics and about themselves that play an important role in development of their affective responses to mathematical situations. Second, since interruptions and blockages are an inevitable part of the learning of mathematics, students will experience both positive and negative emotions as they learn mathematics; these emotions are likely to be more noticeable when the tasks are novel. Third, students will develop positive or negative attitudes toward mathematics (or parts of the mathematics curriculum) as they encounter the same or similar mathematical situations repeatedly." (p. 578) Some of the main features of the Core Plus curriculum, as briefly sketched above, require many fairly abrupt changes in content, emphases and teaching methods. Changes of this magnitude and frequency require adjustments by students and teachers. While most CPMP teachers have had the benefit of professional development experiences to prepare them for the changes required of them, there is no similar advance preparation for students. They are thrust into the new classroom environment with a few initial days of orientation to this unfamiliar view of mathematics, how it is taught, and what is expected of them. Such discrepancies between students' expectations and classroom reality can elicit strong emotions, which through repeated experience may become a more stable (positive or negative) attitude or belief about mathematics and how it is taught and learned (Mandler, 1989; McLeod, 1992). Further insight into the nature of likely discrepancies between what students encounter in Core Plus and what they expect of a mathematics class can be seen by examining beliefs students have typically brought to mathematics, especially to problem solving. A compilation of these student beliefs from the work of several researchers in the 1970s and 1980s is provided by Schoenfeld (1992).
Virtually all the beliefs in the above list are challenged in the Core Plus curriculum, suggesting rich possibilities for research on affect. That there appear to be no previous studies of students' attitudes in a Standardsbased high school mathematics curriculum makes the motivation for the present study even more compelling.
CPMP Classes In most CPMP classrooms, students are actively engaged in investigating mathematical ideas and monitoring their emerging understanding of those ideas. For example, Course 1 teachers at midyear reported from 10% to 80% (Mean = 47%; SD = 17%) of class time was spent on small group work with the remainder being a mix of teacherled discussion, student presentation, individual work, and student assessments. Teachers also report using a broader range of assessment techniques than is typical in traditional classes, including written and oral reports, group observations, and takehome tests to supplement the usual inclass quizzes and endofunit examinations. The field test schools were encouraged to group students heterogeneously or at least in classes that included students with a wide range of achievement and interest in mathematics. Limitations at local sites did not always make this possible. Course 1 teachers' descriptions of their entering CPMP students are summarized in Table 1.
Table 1.
About onefifth of the teachers reported that their classes included the full range of ninthgrade students. The most common CPMP class (as reported by 43.0% of the teachers) was comprised of students with a wide range of prior mathematics achievement and interest. Often, however, honors students were not included because they completed the grade nine course in eighth grade and moved on to a tenthgrade mathematics course in grade nine. Thus, the CPMP field test sample, as reported by the teachers, included students with a wide range of prior achievement and interest in mathematics, but accelerated honors students are probably underrepresented. Traditional Comparison Classes At the beginning of Course 1, eleven field test schools volunteered to pretest and posttest students in traditional, comparison classes using standardized and projectdeveloped achievement measures. The comparison classes were comprised of 20 algebra 1, five prealgebra, three general mathematics, and two honors geometry ninthgrade classes. Ten of these schools also administered the CPMP Student Belief Survey to their comparison students. The nature of the instruction in the comparison classes was not specified in advance, but at the end of the year comparison teachers described what transpired. For example, a variety of traditional textbooks were used. Small group work was reported to be used either not at all or less than once a week by about 80% of the comparison teachers. About 74% of the comparison teachers reported that their students used calculators more than once per week, although there is little information about how it was used. Solving linear equations in one variable was the main instructional goal for an average of 23% of the class time for the year, with up to 50% of the time spent on this topic in some algebra I classes. The Course 2 comparison group consisted of all students who were in the Course 1 comparison group, completed a traditional sophomore mathematics class, and completed the Course 2 posttests. Only five of the 11 schools who administered achievement tests to comparison groups in Course 1 were able to do so at the end of Course 2, and four of these five also administered the Student Belief Survey. The main reason for this drop in number is that the Course 1 comparison students enrolled in a variety of mathematics classes in their sophomore year and were difficult to locate and posttest at the end of the year. By the end of Course 3, the number of comparison students from the original pretested group that were available for posttesting was so small that a Course 3 comparison group was not feasible. CPMP Student Belief Survey The CPMP Student Belief Survey (SBS) is a 50item likert scale with ten subscales, followed by an openended writing prompt. For Courses 1 and 2, the writing prompt was, "Think of a friend or relative in another school who is about your age. Write a letter to this person describing your experience in mathematics class this year." For Course 3, several items in the survey were changed to gather some information that was of particular interest at this point in the field test and to account for the fact that the same students had completed this survey three times in the previous two years. One change was in the writing prompt, which in the Course 3 SBS read, "Describe your experience in mathematics class this year." The SBS was developed by the CPMP evaluation team in 199293 and is based partially on a mathematics belief survey used by Deborah Ball (personal communication) in some of her research in the 1980s with elementary school children. Nine of the subscales are generalized belief scales, each consisting of three to six likert items that by logical and correlational analysis appear to measure related aspects of a belief about mathematics. The scales are named to reflect what they measure as follows: Self Assessment, Challenge versus Ease, Creativity versus Curiosity, Cooperation versus Competition, Creativity versus Rote, Genetic versus Effort, Utility, Affect, and Cooperation with Others. The tenth scale contains 15 items and is entitled, Attitude Toward Your Mathematics Course. This last scale along with the openended writing prompt are the focus of this paper. The CPMP Student Belief Survey was administered as a pretest in September 1994 at the beginning of Course 1 to both CPMP and comparison students. The Attitude Toward Your Mathematics Course scale was not part of the pretest instrument, since students had not yet completed a high school mathematics class. At the end of Course 1 (May 1995) and again at the end of Course 2 (May 1996), the complete Student Belief Survey was administered to both CPMP and comparison students. Finally, at the end of Course 3 (May 1997) the slightly revised Course 3 Student Belief Survey was administered to CPMP students only. The data analytic approach taken here is meant to be consistent with McLeod's (1992; p. 591) position, "The debate over qualitative versus quantitative research methods appears to be almost over, and the time for intelligent use of multiple research methods that fit the research problems is here." In the present study, the results are a combination of quantitative survey data from the Attitude Toward Your Mathematics Course subscale of the CPMP Student Belief Survey and qualitative data from students' responses to the writing prompts on the Student Belief Surveys each year. The approach is to present a quantitative summary of the survey data in each of several logical categories with a discussion that draws on CPMP student' written responses to help explain or elaborate findings in each category. This is followed by an attempt to synthesis of the findings across categories and in the context of previous research on affect in mathematics education.
For Courses 1 and 2, respectively, ten and four of the field test schools had comparison groups who completed the Student Belief Survey. Comparative results for six logical groupings of the 15 items on the Attitude Toward Your Mathematics Course scale are presented and briefly discussed below. The groupings are: Course Difficulty; Problem Solving, Reasoning and Sense Making; Learning in Groups; Communicating Mathematics; Graphing Calculators; and Realism and General Interest. Finally, a single item that was unique to the Course 3 survey is discussed. This item concerned the role of Core Plus in keeping the students in mathematics courses for three years. The Course 1 results are based on 834 CPMP students and 634 comparison students in eight schools, and the Course 2 results are based on 221 CPMP students and 134 comparison students in four schools. In the tables, item means are computed by assigning 1 for strongly disagree, 2 for disagree, 3 for not sure, 4 for agree, and 5 for strongly agree. The "% agree" column is the combined percent of students who either agreed or strongly agreed with that item. To test the statistical significance of the item means, a school by treatment group (CPMP versus comparison) MANOVA with the 15 items as dependent variables was run for Course 1 and then for Course 2. In each case, the multivariate school by treatment interactions as well as the school and treatment group main effects were significant (p < .01). Followup univariate school by treatment ANOVAs were then run for each item for each course, and the treatment main effect for each item and course was used to identify significant differences (p < .05) in treatment group means. Course Difficulty CPMP and comparison students did not differ significantly after either course in their average perceptions of their mathematics grades, how well they understood the mathematical ideas, and the readability of their text materials. See Table 2. As one Course 1 student wrote, "It [CPMP] challenged me a lot more than regular math, but I got a way better grade in regular math than Core Plus. But I will tell you something, you learn a whole lot more [in CPMP]." Another Course 1 student wrote, "My experience in math this year can be summed up in one sentence. It was challenging and difficult." However, some students found Course 1 to be relatively easy. "This year math was easier and more understandable & logical."
Table 2.
Items
CPMP and comparison means do not differ significantly (p = 0.05) on any of the items. By the end of Course 2, some CPMP students were more articulate about their learning and their grades. For example, one wrote, "I passed with at least a C average. I figured that all you need to do is homework and be able to understand what it is you are doing. And I did it. The books and problems are easy." Another Course 2 student wrote, "It's really easy but the book is hard to understand sometimes, and everybody gets different varieties of answers, but once you get it it's basic. It's a lot more related to real life than other math classes." Finally, this student provided some further insights about CPMP. "Not all of it was easy or fun, but if you worked with it you could understand it. I think if we moved slower I would of got the class more." Overall, these data describe CPMP as a curriculum in which the content and grading standards are at least as challenging as the traditional curriculum. Yet, many students also recognized that with work they could understand the mathematical ideas of the course. There is also some indication that CPMP students were more satisfied with their level of understanding after two CPMP courses than after one, while the trend for the comparison students was in the opposite direction. Problem Solving, Reasoning and Sense Making Relative to the comparison students, CPMP students were more positive about their ability to solve problems and to reason mathematically, and they also thought CPMP helped them see that mathematical ideas make sense. See Table 3. Differences in means were statistically significant on all three items at the end of Course 2. The increased feeling of confidence seems to be closely connected for CPMP students to a recognition that the content made sense to them. As one student wrote at the end of Course 1, "I began to understand difficult ideas and methods and they didn't seem as hard." Another wrote, "I understand it now and I'm not afraid to apply myself to mathematical situations." In a similar vein, a third Course 1 student wrote, "I really think I learned more because I understood the work."
Table 3. Items
* This mean is significantly greater (p < 0.05) than the mean of the other group on this item. This theme of understanding leading to improved confidence and ability to solve problems and reason mathematically can be seen at the end of Course 2 as well. One Course 2 student wrote, "...it [CPMP] made it much easier to learn and figure out the problems and how to apply [the mathematics]." Another echoed this sentiment, "...the most important thing was that it [CPMP] made me think and understand what I was doing." One Course 3 student was probably the most articulate about understanding and its related payoffs, "I learned to comprehend at a technical level, with that I was able to view mathematical meanings and reasoning with an open mind, therefore allowing me to further my growth in a field I struggled to get by in." Learning in Groups CPMP developers recommend that the investigations be completed by students working in small groups or pairs. Teacher survey and classroom observation data reported earlier in this paper indicates that, on average, just under half (but in some classes as much as 80%) of class time in Course 1 field test classes was spent in small group or pair arrangements, so small cooperative group instruction is an important component of the CPMP instructional model. Students were directed to skip the items given in Table 4 if they did not use group work in their classes, and about 19% of the comparison students did so. Both CPMP and comparison students, on average, enjoyed group work and believed it helped them learn mathematics. The only statistically significant difference between CPMP and comparison means occurred at the end of Course 1 when the CPMP students agreed more strongly than comparison students that group work helped them learn mathematics.
Table 4. Items
* This mean is significantly greater (p < 0.05) than the mean of the other group on this item. The enjoyment of working in groups and its value for social learning is expressed well by this Course 2 student, "I don't think I have ever had so much fun doing problems and solving them in a group. I learned a lot of how to work with people." The experience of seeing how other people attack problems and the mutual supporting of efforts by group members seems to be the key to learning in groups. As this Course 1 student wrote, "Math was pretty cool this year because we got to be in groups...the groups were helpful because if I didn't get a problem there was a chance someone else knew what it was." Another student (in Course 3) enjoyed the role of helper for others in the group, "I also had fun working with groups and helping them get the answers." On the other hand, a few CPMP students (about 15%) did not like working in groups, mainly because some students did not contribute enough or they perceived the learning to be inefficient. As this Course 3 student wrote, "I would have learned better in this course if I would have learned it from the teacher the first time instead of learning it wrong in groups." Communicating Mathematics In CPMP, students in groups read through the mathematical material and investigation questions. They also discuss the mathematics orally in their groups and often present what their group learned to the whole class. The CPMP assessments require more written description of the student's mathematical thinking than most traditional assessments. In addition, many CPMP teachers make occasional assignments that involve oral or written reports. Given all this opportunity to communicate mathematics, perhaps it is not surprising that CPMP students at the end of each course agreed more strongly than comparison students that their mathematics course made them better at both talking and writing about mathematical ideas. See Table 5.
Table 5. Items
* This mean is significantly greater (p < 0.05) than the mean of the other group on this item. In spite of the consistently positive perceptions of communicating mathematics on the attitude survey, students did not often mention this aspect of CPMP in openended written descriptions of their experience in mathematics class suggesting that most did not see it as a particularly important aspect of CPMP. One Course 1 student saw writing as generally important. "The writing helps a lot considering there's a lot of it in life..." Another said, "I have learned to express my mathematical ideas in words which before this class I was not able to do." A third Course 1 student did not like writing, "The class is fine. But I can't stand writing in complete sentences. All the other math classes just have to write the answer. We have to, and in complete sentences." Graphing Calculators The pattern of student attitudes about the graphing calculator is similar to the pattern for group work. Both CPMP and comparison students consistently indicated that they enjoyed using graphing calculators and, a little less consistently, that they learned more mathematics by using the calculator. Students were instructed to skip these items if they had not used calculators in their mathematics class, and about 11% of comparison students did so. On the other hand, all CPMP students used graphing calculators which were intended to be a natural and always available tool. Results on the two graphing calculator items are given in Table 6. The only statistically significant difference was at the end of Course 1 when CPMP students agreed more strongly than comparison students that they learned more mathematics by using the graphing calculator.
Table 6. Items
* This mean is significantly greater (p < 0.05) than the mean of the other group on this item. Most of the students who wrote about the calculator mentioned that they had learned more. Some specific content references were made by several students. "The calculators taught me a lot about graphs and tables" and "I learned by using a calculator to graph and find the yintercepts." Other comments like the following were more general. "...we use an advanced calculator which helps us expand our knowledge of math," "The calculator was helpful, especially when you're making graphs or dealing with a lot of data and statistics," and "I learned so much on the calculator." The few students who expressed negative opinions about the calculator appeared to be concerned that they would become too dependent on it. As one student said, "In real life, I might not have a calculator around wherever I go." Realism and General Interest Courses 1 and 2 The three items in Table 7 were grouped primarily because realism of the problem contexts in CPMP is the variable that students most frequently connect to their interest in the course. Realistic problems clearly make a mathematics course more interesting for many students, as a Course 1 student wrote, "...we do more realistic math problems and that makes it most interesting." Finally, to connect the third item to the first two, students are likely to want to take another math course taught in the same way if they have found that this one was interesting. As a Course 3 student wrote, "I do plan to take this type of course next year, because it's more realistic than the regular math class."
Table 7. Items
* This mean is significantly greater (p < 0.05) than the mean of the other group on this item. First, to elaborate on the realism of the problems, a Course 1 student wrote, "I've really enjoyed learning problems that I might actually need to use in real life." Another wrote, "The problems are realistic and make sense. I never found myself asking questions, like `When will I ever use this in life?'" A third Course 1 student related the realism to ease of understanding, "It had a lot of real life situations which made the math easier to understand." A Course 2 student related the integration of topics and realism, "...an integrated kind of math that takes math ideas and teaches them to us in a realistic way." A second Course 2 student saw the realism as good preparation for the future, "...the problems are realistic...definitely helps prepare students for their future." A Course 3 student described the value of the course in some detail, "This was my first year in [CPMP]...before this I took algebra, geometry and advanced algebra...for the first time I could relate math to everyday life. Learning geometry and things like tan, sin, cos, actually made more sense than learning it in traditional math." Second, to elaborate on the general interest level of the course, a Course 1 student wrote, "This math class is the first class that I have took [sic] that I like in a long time." Other students expressed a general pleasure with Courses 1, 2, and 3, such as, "...how much fun I had this year in math class," "It was a funner way of doing math, I actually looked forward to coming to class," and "It has been a delightful experience throughout." Besides the realism of the problems, some students mentioned other interesting aspects of the course. Some examples are, "there's more hands on activities and that makes it fun to learn," "this course makes you think for yourself and solve problems for yourself," and "...this course makes us think. It is very interesting." Third, to elaborate on the decision to take a math course taught in the same way next year, several students simply said they would do so, "I'm going to take it next year," "I really hope that they teach math this way next year," and "I will probably take this course next year. I really like the teaching method." A quote given earlier attributed the decision to take next year's CPMP course to the realism of this year's course. Another student, this one in Course 3, attributed the decision to having learned a lot, "I learned a lot and I feel knowledgeable. I enjoy math and plan to take a fourth year of it." A few CPMP students answered negatively concerning their interest in the Core Plus course and in continuing in the next course. Written comments suggest that negative responses on this item may have been motivated by one of three issues: (1) a worry that there was not enough work on algebraic skills (e.g., "I didn't take algebra so now I'm stuck in this stupid course."; (2) finding the Core Plus course to be too difficult (e.g., "I only understood maybe 2 of the books and we had at least 10 different books. My grade improved from last year, but I think this class isn't a good class for me. It's mostly hard stuff and it gets frustrating and confusing."); or (3) finding the course uninteresting or boring (e.g., "I wish it wasn't so rythmatic [sic]. We do the same thing every day, boring....but I did learn more than last year."). Course 3 Related to the issue of wanting to continue in more Core Plus courses is the following item, administered only on the Course 3 Student Belief Survey: It is mainly because of Core Plus that I took a third year of high school math. Of the 1,944 Course 3 students completing this survey, 297 (15.3%) strongly agreed and 223 (11.5%) agreed with this statement. Thus, over one in four of the Course 3 students attributed their being in a third year of mathematics mainly to the Core Plus curriculum.
Students perceive the Core Plus curriculum as quite difficult, at least as challenging as traditional collegeprep mathematics courses. Nevertheless, Core Plus students were significantly more positive about various aspects of the curriculum and of their classroom experience than students in traditional classes in the same schools. A common perception of students is that Core Plus is challenging and makes them think, but with effort it is possible for them to understand the ideas and their applications. Indeed, achievement results available to date for the wide range of field test students in Core Plus classes are strong, especially in areas of understanding, reasoning and problem solving (Schoen, Hirsch, & Ziebarth, 1998). It seems reasonable to conclude that affective and cognitive factors are working together to lead to the positive findings in each domain. The following hypotheses draw on the CPMP field test data presented here and in Schoen, Hirsch, & Ziebarth (1998), as well as on classroom observations and interviews with teachers and students. Each hypothesis is meant to suggest an area in which more research is needed.
The findings of this study suggest that Core Plus students see frequent counterexamples of the commonlyheld beliefs compiled by Schoenfeld (1992) and described earlier in this paper. Particular instances are the following.
Several elements of the curriculum and instructional model working together appear to explain the positive cognitive and affective results that are emerging from the Core Plus field test. More research is needed to better understand these elements and their interactions in classrooms that use Core Plus or other Standardsbased curricula.
Coxford, A. F., Fey, J. T., Hirsch, C. R., Schoen, H. L., Burrill, G., Hart, E. W., Watkins, A. E. with Messenger, M. J. and Ritsema, B. (1997). Contemporary Mathematics in Context: A Unified Approach, Course 1 and Course 2. Chicago: Everyday Learning Corporation. Hirsch, C. R. & Coxford, A. F. (1997). Mathematics for all: Perspectives and promising practices. School Science and Mathematics, 97(5), 232241. Hirsch, C. R., Coxford, A. F., Fey, J. T., & Schoen, H. L. (1995). Teaching sensible mathematics in sensemaking ways with the CPMP. Mathematics Teacher, 88(8), 694700. Mandler, G. (1989). Affect and learning: Causes and consequences of emotional interactions. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving (pp. 319). New York: SpringerVerlag. McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualization. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575596). Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics, Commission on Standards for School Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: The Council. Schoen, H. L., Bean, D. L., & Ziebarth, S. W. (1996). Embedding communication throughout the curriculum. In P. C. Elliott and M. J. Kenney (eds.), Communication in Mathematics: K12 and Beyond, 1996 Yearbook of the National Council of Teachers of Mathematics. (pp. 170179). Reston, VA: The Council. Schoen, H. L., Hirsch, C. R., & Ziebarth, S. W. (1998). An emerging profile of the mathematical achievement of students in the CorePlus Mathematics Project. Paper presented at the 1998 Annual Meeting of the American Educational Research Association. San Diego, CA. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127146). Reston, VA: National Council of Teachers of Mathematics. Sowder, L. (1989). Searching for affect in the solution of story problems in mathematics. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving (pp. 104113). New York: SpringerVerlag.
Coxford, A. F., & Hirsch, C. R. (1996). A common core of math for all. Educational Leadership, 53 (8), 2225. Coxford, A. F., Fey, J. T., Hirsch, C. R., Schoen, H. L., Burrill, G., Hart, E. W., Watkins, A. E. with Messenger, M. J. and Ritsema, B. (1997). Contemporary Mathematics in Context: A Unified Approach, Course 1 and Course 2. Chicago: Everyday Learning Corporation. Flowers, J. (1995). A study of teachers' indirect influence in CorePlus Mathematics Project classes. Unpublished paper, University of Michigan. Hart, E. W. (1997). Discrete mathematical modeling in the secondary curriculum: Rationale and examples from the CorePlus Mathematics Project. In J. Rosenstein and F. Roberts (eds.), Discrete Mathematics in the Schools. Providence, RI: DIMACS Series in Theoretical Computer Science and Discrete Mathematics, American Mathematical Society. Hart, E. W. (in press). Algorithmic problem solving in discrete mathematics. In L. Morrow and M. J. Kenney (eds.), Teaching and Learning of Algorithms in School Mathematics, 1998 Yearbook of the National Council of Teachers of Mathematics. Reston, VA: The Council. Hart, E. W., & Stewart, J. (In press). Composing a curriculum: Reflections on high school reform and implications for middle schools. In L. Leutzinger (Ed.), Mathematics in the Middle Grades. Reston, VA: National Council of Teachers of Mathematics. Hirsch, C. R., Coxford, A. F., Fey, J. T., & Schoen, H. L. (1995). Teaching sensible mathematics in sensemaking ways with the CPMP. Mathematics Teacher, 88(8), 694700. Hirsch, C. R. & Coxford, A. F. (1997). Mathematics for all: Perspectives and promising practices. School Science and Mathematics, 97(5), 232241. Hirsch, C. R. (in press). The CorePlus Mathematics Project (CPMP). In L. S. Grinstein and S. I. Lipsey (eds.), Mathematics Education: An Encyclopedia. Washington, D. C.: Taylor and Francis. Hirsch, C. R. & Weinhold, M. L. W. (in press). Everybody countsIncluding the mathematically promising. In L. Sheffield (ed.), Developing Mathematically Promising Students. Reston, VA: National Council of Teachers of Mathematics. Kett, J. R. (1997). A portrait of assessment in mathematics reform classrooms. Unpublished doctoral dissertation, Western Michigan University. Lloyd, G. M., & Wilson, M. R. (1997). Secondary mathematics teachers' experiences using a reformoriented curriculum to encourage student cooperation and exploration. Paper presented at the Nineteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Lloyd, G. M., & Wilson, M. R. (in press). Supporting innovation: The impact of a teacher's conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education. Lloyd, G. M., & Wilson, M. R. (1997). The role of high school mathematics teachers' beliefs about student cooperation and exploration in their interpretations of a reformoriented curriculum. Paper presented at the Annual Meeting of the Eastern Educational Research Association. Lloyd, G. M., & Wilson, M. R. (in press). The impact of teachers' beliefs about student cooperation and exploration on their interpretations of a secondary mathematics curriculum. In J. Dossey (ed.), Proceedings of the Nineteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Columbus, OH: The ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Schoen, H. L., Bean, D. L., & Ziebarth, S. W. (1996). Embedding communication throughout the curriculum. In P. C. Elliott and M. J. Kenney (eds.), Communication in Mathematics: K12 and Beyond, 1996 Yearbook of the National Council of Teachers of Mathematics. (pp. 170179). Reston, VA: The Council. Schoen, H. L., & Ziebarth, S. W. (1997). A progress report on student achievement in the CorePlus Mathematics Project field test. Unpublished manuscript, University of Iowa. Schoen, H. L., & Ziebarth, S W. (1997). A progress report on student achievement in the CorePlus Mathematics Project field test. NCSM Journal of Mathematics Education Leadership, 1(3), 1523. Schoen, H. L., & Ziebarth, S. W. (1998). Assessment of students' mathematical performance: A CorePlus Mathematics Project field test progress report. Unpublished manuscript, University of Iowa. Schoen, H. L., & Ziebarth, S. W. (1998). High school mathematics curriculum reform: Rationale, research, and recent developments. In P. S. Hlebowitsh & W. G. Wraga (eds.), Annual Review of Research for School Leaders. Pp. 141191. New York: Macmillan Publishing Company. Schoen, H. L., & Ziebarth, S. W. (1998). Mathematical achievement on standardized tests: A CorePlus Mathematics Project field test progress report. Unpublished manuscript, University of Iowa. Truitt, B. A. (1998). How teachers implement the instructional model in a reformed high school mathematics classroom. Unpublished doctoral dissertation, University of Iowa. Tyson, V. (1995). An analysis of the differential performance of girls on standardized multiplechoice mathematics achievement tests compared to constructed response tests of reasoning and problem solving. Unpublished doctoral dissertation, University of Iowa. Van Zoest, L. R., & Ritsema, B. E. (1998). Fulfilling the call for mathematics education reform. NCSM Journal of Mathematics Education Leadership, 1(4), 515. Wilson, M. R., & Lloyd, G. (1995). Sharing mathematical authority. Paper presented at the Seventeenth Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Wilson, M. R., & Lloyd, G. M. High school teachers' experiences in a studentcentered mathematics curriculum. Paper presented at the Seventeenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Wilson, M. R., & Lloyd, G. M. (1997). Ways of experiencing curriculum: A phenomenographic analysis of two high school teachers' attempts to promote cooperation and exploration in their mathematics classrooms. Manuscript submitted for publication. (Journal for Research in Mathematics Education)
