A response to the NCTM Standards: confidence and competence project ([C.sup.2]).

Abstract

The purpose of this paper is to share the development and the design of a mathematics reform project, Confidence and Competence ([C.sup.2]). [C.sup.2] was a series of workshops that developed the content and pedagogical knowledge of teachers that is necessary to support reform-oriented mathematics instruction and assessment. [C.sup.2] emerged in a manner that is very different from most educational reform projects. This article describes the conceptualization of [C.sup.2] and the results of the evaluation in the fall of 1998.

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Project Conceptualization

A few years ago, three friends were discussing the changes that were taking place in the Jefferson County School District. Jefferson County is a suburban school district that is near Denver, CO and serves approximately 88,000 students. The district was in the process of implementing Standards Based Education. One of the friends was an experienced mathematics teacher within the district and another was the Jefferson County School District Mathematics Project Coordinator. Both women had witnessed the district's teachers struggling to reform their classrooms based on the districts (Jefferson County Public Schools, 1996), state (Colorado State Department of Education, 1995) and the NCTM Standards Documents (1989; 1991; 1995).

The third friend was a faculty member in the Mathematical and Computer Sciences Department at the Colorado School of Mines. She was well aware of the research in mathematics education that suggests that elementary, middle school and high school teachers mathematical knowledge is often incomplete and/or fragmented (e.g., Post, Harel, Beher & Lesh, 1991; Ball, 1990; Simon, 1993). Additionally, research suggests that teachers from other countries often have a deeper conceptual understanding of mathematics than do their counter parts in the United States (Yang & Cobb, 1995). Adding to this problem is the fact that a large number of middle school and high school teachers are providing instruction in a field other than the field in which they are certified (Ingersoll, 1999).

Through the course of their discussion, these women determined that a lack of sufficient content knowledge was unlikely to be the sole contributor to the difficulties that the districts' teachers were experiencing. Recent efforts to reform mathematics education (NCTM, 1989; 1991) had resulted in changes to the recommended approaches to mathematics instruction. In the past, teachers were expected to transfer their content knowledge to students and students were expected to passively absorb this information. Many researchers currently believe that learning occurs when students actively integrate new concepts into their knowledge structure (Baxter & Glaser, 1998). In this view of the learning process, the teachers' role has changed from presenting information to providing activities that stimulate students' interests and active participate in the learning process.

Also as a result of the reform effort, perceptions of classroom assessment have changed (NCTM, 1995). In the past, assessment was conceived as a method for measuring students' knowledge after instruction had taken place. Assessment is currently considered to be a method for examining students' knowledge throughout the learning process. Assessment evidence may be used to examine the effectiveness of previous instruction and to inform how future instruction should proceed (Van den Heuvel-Panhuizen, 1994). The three friends hypothesized that the districts' teachers whose pedagogical training had preceded the current reform effort were unlikely to have the necessary skills to implement the previously described visions of instruction (Kober, 1993) and assessment (Stiggins, 1990; 1991; 1999). They decided to create a professional development program for teachers that would address not only mathematical content, but also the knowledge of mathematics that is essential to effective teaching. In addition, they recognized the importance of the pedagogy of instruction and assessment. They wanted the teachers who participated in the program to learn to use assessment both to evaluate previous instruction and to motivate future instruction. To complete the program, they also wanted to provide the teachers with some form of classroom assistance. The result of their efforts was the Confidence and Competence Project ([C.sup.2]).

Project Design

The [C.sup.2] project was funded for six semesters through grants from the National Science Foundation and the Colorado Commission of Higher Education Eisenhower funds. Each semester, the workshops addressed different grade levels and a specific district mathematics standard. The focus of this article is on the Fall of 1998. At that time, 29 fourth, fifth and sixth grade teachers were participating in the workshops. The focus of the workshops was on the district's Algebra Standard, "Students use algebraic methods to explore, model, and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate with appropriate mathematical language and reasoning used in solving these problems" (Jefferson County Public Schools, 1996, p. 10).

The structure of the workshops, which remained the same throughout the period of funding, is shown in Figure 1. Each semester consisted of four cycles and each cycle consisted of a content workshop, a pedagogy workshop and a classroom practice component. Further detail concerning each component is provided in the sections that follow. See issue's website <http://rapidintellect.com/AEQweb/win01.htm>.

Content Workshops

The participating teachers attended four workshops designed to improve their algebraic content knowledge. These workshops were seven hours long and were led by a high school mathematics teacher who had received the Presidential Teaching Award in 1991. Throughout these workshops, an emphasis was placed on problem solving and the use of multiple representations. The instructor often illustrated how problems could be represented and solved numerically, visually and algebraically. Each representation, she explained, provides a slightly different insight into the nature of the problem. This emphasis on multiple approaches and multiple representations is consistent with current recommendations (NCTM, 1989; 1991; 1995; 2000). Additionally, both problem solving and the use of multiple representations are common elements found in many mathematics reform projects (e.g., Huntley, Rasmussen, Villarubi, Sangtong & Fey, 2000; Usiskin, 1993).

In one activity, the teachers examined lines drawn on graph paper and visually determined the slope of the line. This was used to develop the well-known algebraic formula (i.e., y2-y1 over x2-x1) for the slope of a line between two points. In another activity, dried spaghetti noodles were distributed. The teachers were instructed to break the noodles into three uneven parts and form triangles out of the pieces. Several of the teachers were unable to form a triangle from their pieces. This led to determination that the sum of any two sides of a triangle must be greater than the third side of that same triangle.

The primary purpose of the content workshops was to develop the teachers' algebraic content knowledge. However, the activities that were used during these workshops were consistent with the recommendations for teaching reform-oriented mathematics (NCTM, 1989; 1991; 1995). This provided the teachers with a model of instruction for their own classrooms. Some of the activities completed during the workshops could be easily implemented in a fourth, fifth or sixth grade classroom - such as the noodle activity described above. Other activities were designed to challenge the participating teachers' current level of algebraic knowledge. As will be described in the next section, during the pedagogical workshops the teachers discussed which activities would be appropriate or which activities could be changed to be appropriate to the grade level that they taught.

Pedagogy Workshops

On the first Saturday that followed the content workshop, the pedagogy workshop was held. A total of four pedagogy workshops were held over the course of the semester. The pedagogy workshop was designed to reinforce the concepts studied earlier in the week and to assist the teachers in extending these ideas to classroom practices. The pedagogy workshop was seven hours long and was led by the district's Mathematics Project Coordinator.

During the content workshop, the participating teachers had been assigned homework that reinforced the algebraic concepts that they were studying. Each pedagogy workshop began with the opportunity to ask questions of the content workshop instructor. By providing a question/answer session, the teachers were able to clarify questions that they had before developing a lesson for their students. The question/answer session was followed by a group discussion concerning how the mathematical content that they had learned could be restructured into a lesson that was appropriate to the grade level that they taught. At least two hours of each pedagogy workshop were dedicated to the discussion of recent developments in mathematics education. These discussions focused on a broad range of topics, such as standards based education, alignment of instruction and assessment, scoring rubrics, and performance assessments. The instruction that was provided during the content workshops was used as an example of the type of instruction that is consistent with the reform process.

During the afternoon portion of the pedagogy workshops, the teachers divided into groups of three or four. Each group consisted of teachers who taught the same grade level. The teachers worked to develop a set of objectives, a lesson plan and an assessment activity for their own classrooms. Often, the lesson plan that the teachers developed mimicked the activities that had been completed during the content workshops. A selection of reference books that concerned standards based education and assessment were available (e.g, Danielson, 1997a; 1997b; Jacobs, 1997; Lewin & Shoemaker, 1998; NCTM, 1989; 1991; 1995).

Each of the pedagogical workshops concluded with the teachers sharing the lesson plans and assessment activities with the class. Classroom discussions focused upon the extent to which the proposed activity was likely to result in the proposed learning outcomes and the appropriateness of the proposed activity to the respective grade level. The teachers made suggestions as to how to improve the proposed lesson plans and assessment activities.

During the next pedagogy session, the teachers discussed the effectiveness of their lesson with the other members of their team. Since each of the team's teachers had taught the same lesson, they were able to compare the components of instruction they found to be effective. This supported the teachers in recognizing how to further improve instruction. This type of reflective process has been found to be an effective method for improving instruction in countries whose students are outperforming their U.S. counterparts in mathematics (Stigler & Hiebert, 1999; Ma, 1999).

Instructional Practices

Three weeks separated each pedagogical workshop and the next content workshop. During this time, the teachers implemented the lesson plan and assessment activity in their classrooms. After completing the lesson plan, the teachers discussed their lesson either in person or over the phone with a cognitive coach. A cognitive coach assists teachers as they reflect upon instruction and how to improve instruction (Costa & Garston, 1994). The role of the cognitive coach is not to judge the effectiveness of a lesson or to provide advice on how to improve a lesson, but rather to ask questions that stimulate the teacher to evaluate their own efforts. Discussions with the cognitive coach usually last about an hour.

In the typical model of cognitive coaching, the coach observes instruction (Costa & Garston, 1994). The cognitive coaches in the current project did not observe instruction, but rather each coaching session began with the teacher describing their lesson. This modification to the model of cognitive coaching was the result of the financial constraints of the project.

Evaluation Instruments

This section describes the evaluation instruments that were used. These instruments were selected or developed with the purposes of determining whether the teachers' confidence and competence in the algebraic content and in the pedagogy of teaching and assessing algebraic concepts improved as a result of project participation.

Attitude Survey

Teachers' confidence in their knowledge of algebraic concepts, and how to teach and assess algebraic concepts was assessed using an attitude survey. An earlier version of this survey was used in previous semesters of [C.sup.2] (see Juraschek & Wooley, 1997; Shaw, 1998). The attitude survey addressed the teachers' attitudes with regard to their knowledge of algebraic content, instructional pedagogy, assessment pedagogy and the broader category of teaching. The categorization of each survey question is displayed in Table 1. The teachers were asked to indicate on a five-point scale the extent to which they agreed with each statement. This survey was administered during the first content workshop and on the last pedagogy workshop. Further details concerning this scale and the teachers' responses will be discussed in the "Results" section of this article. See issue's website <http://rapidintellect.com/AEQweb/win01.htm>.

Reflective Survey

Fifteen questions comprised the structured response portion of the Reflective Survey. In the first three questions, the teachers were asked to evaluate the effect that the workshops had upon their content knowledge. This was followed by 7 questions that concerned instructional pedagogy and 5 questions that concerned assessment pedagogy. For each of these questions, the teachers circled one of the following: "No Increase in Understanding", "Slight Increase in Understanding", "Large Increase in Understanding", and "Substantial Increase in Understanding". The specific questions that comprised this instrument are provided in the "Results" section.

Algebraic Pre and Post Assessments

The same problem was used to examine the teachers' knowledge before and after participating in the [C.sup.2] workshops. This problem was developed through the combined efforts of the content knowledge instructor and the evaluator. An effort was made to create a problem that could be completed using a variety of different solution strategies. Since the participating teachers were expected to have a limited algebraic background at the start of the workshop, the problem was developed to allow for both algebraic and non-algebraic solutions (e.g., guess and check and graphical solutions). The selected problem is shown in Figure 2. This problem was administered in standardized manner on the first and last day of the [C.sup.2] workshop. See issue's website <http://rapidintellect.com/AEQweb/win01.htm>.

Results

The results are discussed in the sections that follow. One teacher did not attend the final pedagogical workshop and therefore, did not complete the post assessment activities. The analysis that follows is restricted to the twenty-eight teachers that completed both the pre and post assessment activities.

Attitudes Survey

Each of the questions on the attitude survey can be categorized as either positive or negative statements. Agreement with positive statements, such as "I feel prepared to make algebraic connections to other mathematical standards", suggests a positive attitude with regard to the individual's ability. Agreement with negative statements, such as "Algebraic concepts are difficult for me", suggests a negative attitude with regard to the individual's ability. Five questions on the attitude survey were negative statements. For analysis purposes, a response of "Strongly disagree", "Disagree", "Neutral", "Agree", and "Strongly agree" in reaction to a positive statement was coded as 1, 2, 3, 4 and 5, respectively. A response of "Strongly disagree", "Disagree", "Neutral", "Agree", and "Strongly agree" in reaction to a negative statement was coded as 5, 4, 3, 2, and 1, respectively.

For each of the categories of "Content Knowledge", "Instructional Knowledge", "Assessment Knowledge" and "General Teaching Knowledge", a total score was acquired by summing the numerical value assigned to each of the teachers' responses to the questions within each category. The maximum total score for "Content Knowledge", "Instructional Knowledge", "Assessment Knowledge" and "General Teaching Knowledge", was 25, 30, 35, and 35, respectively. A paired t-test was used to determine whether there was a statistically significant change in attitudes in each category of knowledge from the beginning to end of the [C.sup.2] workshops. As suggested by Table 2, a statistically significant change occurred in all categories of knowledge.

Reflective Survey

For analysis purposes, the teachers responses, "No Increase in Understanding", "Slight Increase in Understanding", "Large Increase in Understanding", and "Substantial Increase in Understanding" were mapped into the values of 1, 2, 3, and 4, respectively. Table 3 displays the median, mode and range of the teachers' responses. Examination of this table suggests that for the majority of questions the teachers indicated a "Large" increase in understanding. In response to questions 2 and 6, the majority of teachers indicated only a "Slight Increase in Understanding." A commonality between questions 2 and 6 are the references to technology. Although the teachers indicated that their knowledge had increased with regard to using technology as a result of the workshops, they did not judge their knowledge gains within these areas to be as substantial as their knowledge gains in the other areas. See issue's website <http://rapidintellect.com/AEQweb/win01.htm>.

Algebraic Pre and Post Assessments

On both the algebraic pre and post assessments, the total number of correct solutions that each teacher provided was counted. The average number of correct solutions across teachers on the pre assessment was 1.54 and on the post assessment was 2.64. A paired t-test (t = 5.89, p = .00) indicated a statistically significant change in the average number of correct solutions provided on the algebraic pre and post assessments.

On both the algebraic pre and post assessments, the total number of correct algebraic solutions that the teachers provided was counted. Algebraic solutions were defined to be equations that involved the calculation of a missing variable. The average number of correct algebraic solutions provided on the pre assessment was .5 and on the post assessment was .71. A paired t-test indicated a statistically significant change in the average number of correct solutions provided on the algebraic pre and post assessments.

On both the algebraic pre and post assessments, the total number of correct graphical solutions that the teachers provided were counted. Graphical solutions were defined to be a drawing of two intersecting lines that appeared to be used to calculate the correct answer. The average number of correct graphical solutions provided on the pre assessment was. 14 and on the post assessment was .71. A paired t-test (t = 5.28, p = .00) indicated a statistically significant change in the average number of correct graphical solutions provided on the algebraic pre and post assessments.

Conclusions

The primary goals of [C.sup.2] were to improve teachers' confidence and competence in both content and pedagogy. The results presented in this paper support the success of these efforts. The outcomes of the attitude survey suggest that the teachers' attitudes with respect their own knowledge of algebraic content, instructional pedagogy, assessment pedagogy and general teaching improved over the course of [C.sup.2] participation. Also, the pre and post algebraic assessment provided evidence that the number of correct answers that the teachers provided and that the teachers use of algebraic and graphical approaches increased over the course of the semester. On the reflective survey, the majority of participating teachers reported that they believed that their content and pedagogy knowledge had improved as a result of project participation. In other words, the participating teachers evaluated the project as having been effective in reaching its goals.

A word of caution is necessary in the interpretation of these findings. The purpose of [C.sup.2] was to develop teachers' confidence and competence with respect algebraic content and pedagogy. It is hoped that the participating teachers' improved confidence and competence would have a positive impact upon their classroom instruction and assessment. This, however, was not directly examined in the current project. Determining whether participation in [C.sup.2] directly impacted classroom instruction and student learning are both left for future research.

The findings that have been reported here are consistent with the results of the five other semesters in which the [C.sup.2] project was implemented (see Juraschek & Wooley, 1997; Shaw, 1998; Moskal, 1999). Each of these reports supports the assertion that participation in the [C.sup.2] project resulted in increased teacher confidence and competence with respect to the mathematical content that was under investigation and how to teach and assess that content. In order for future programs to use [C.sup.2] as an example for teacher development, a better understanding is needed as to why [C.sup.2] was effective in reaching its goals.

Hiebert (1999) has argued that there are four core features necessary to support the learning of teachers. Each of these features was present in the [C.sup.2] design. One feature is having specific learning goals to guide student attainment. Each semester that [C.sup.2] was offered, a standard was selected from the Jefferson County Mathematics Standards (Jefferson County Public Schools, 1996). Based on this standard, the teachers defined specific learning objectives before they constructed instruction and assessment plans. Instruction was then planned with these learning objectives in mind; Assessment was designed to examine the extent to which the students reached the given objectives. The existence of the learning objectives provided the teachers and their teams with a focus for developing instruction and a framework for guiding assessment.

A second feature that was discussed by Hiebert was ongoing collaboration among teachers for the purpose of improving instruction. In [C.sup.2], the teachers worked in teams to develop instruction and assessment plans. Although this type of interaction is most effective when it occurs over a period of years, [C.sup.2] introduced the participating teachers to the collaborative process. Additionally, many of the [C.sup.2] teachers selected to complete multiple semesters of the workshops, each focusing on a different Jefferson County mathematics standard. This provided these teachers with extended exposure to the collaborative process.

The third feature of an effective teacher development program (Hiebert, 1999) is paying close attention to the following: students' thinking, the curriculum and pedagogy. The importance of each of these was recognized and emphasized throughout [C.sup.2]. Teachers learned to develop and use open-ended assessment tasks with the purpose of eliciting evidence of complex learning outcomes, such as student reasoning. The teachers also learned to develop scoring rubrics or scoring schemes that emphasized the reasoning process. They then administered these tasks to their students and used the scoring rubrics to evaluate their students' responses. Since the responses were drawn from the teachers' classrooms, the teachers had a reason to scrutinize and reflect on student performances. The teachers also had the opportunity to discuss their findings with the other members of their teams. This led to new insights and these insights were used to suggest changes to future curriculum, instruction and assessment.

The fourth and final feature that Hiebert (1999) discussed is providing exposure to alternative ideas and teaching methods and then providing teachers with the opportunity to reflect on the effectiveness of these approaches in their own classrooms. Throughout the content and the pedagogical workshops, the teachers were exposed to standards based instructional approaches and were encouraged to use these approaches in their classroom. Since [C.sup.2] occurred over the course of the semester, the teachers had the immediate opportunity to implement new ideas in their classrooms. The cognitive coaching component of this project and the team discussions concerning the effectiveness of their lessons encouraged the teachers to reflect on their instruction and how to improve future instruction.

In addition to the features identified by Hiebert, [C.sup.2] assisted participating teachers in developing a deeper understanding of algebraic content and instructional pedagogy. Pedagogy in the current project was conceptualized as consisting of two components: instructional pedagogy and assessment pedagogy. Fennema and Franke (1992) have argued that teachers' knowledge of both content and pedagogy contributes to the appropriateness of the instructional decisions that they make in a classroom. This suggests that an effective teacher development program should account for the development of teachers' knowledge in both of these areas. The content workshops in the current project focused upon the development of the teachers' algebraic content knowledge. The pedagogy workshops addressed the pedagogy of instruction and assessment.

The authors of the current article believe that the success of the [C.sup.2] program is attributable to a design that had the features of an effective teacher development program that were identified by Hiebert (1999) and that addressed the concerns that were raised by Fennema and Franke (1992). The authors hope that by presenting this design similar projects will surface in other school districts. A summary of this design is as follows:

   1. Learning objects were used to guide the development of lesson plans and
   assessment activities,

   2. The teachers worked in teams with the purpose of improving curriculum,
   instruction and assessment,

   3. An emphasis was placed upon making sense of the students' reasoning
   process throughout instruction and assessment,

   4. The teachers were continually exposed to alternative instructional and
   assessment techniques, and

   5. A dual emphasis was placed on the development of teachers' content and
   pedagogical knowledge.

Acknowledgements

The authors would like to thank Ken Berry (pedagogical instructor for the first two years), Jill Fellman (pedagogical instructor for the last year), Judy Metz (head of cognitive coaching component), Marilyn Stor (a member of the project development team and the content knowledge instructor) and Glenda Sadler (a member of the project development team and the project facilitator) for their contributions to the project. We would also like to thank the participating teachers. Barbara Bath, an author on the current paper, was also a member of the original project development team.

References

Ball, D. (1990). "Prospective elementary and secondary teachers' understanding of division." Journal for Research in Mathematics Education, 21 (2), 132-144.

Baxter, G.P. & Glaser, R. (1998). "Investigating the cognitive complexity of science assessments." Educational Measurement: Issues and Practice, 17 (3), 37-45.

Colorado State Department of Education (1995). Colorado Model Content Standards: Mathematics. [Available: http://www.cde.state.co.us/math.htm]

Costa, A. L. & Gramston, R. (1994). Cognitive Coaching: A Foundation for Renaissance Schools. Norwood, MA: Christopher-Gordon.

Danielson, C. (1997a). A Collection of Performance Tasks and Rubrics: Upper Elementary School Mathematics. Larchmont, NY: Eye on Education Inc.

Danielson, C. (1997b). A Collection of Performance Tasks and Rubrics: Middle School. Larchmont, NY: Eye on Education Inc.

Fennema, E. & Franke, M. (1992). "Teachers' knowledge and its impact." In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 147-164). New York: MacMillan Publishing Company.

Hiebert, J. (1999). "Relationships between research and the NCTM Standards." Journal for Research in Mathematics Education, 30 (1), 3-19.

Huntley, M. A., Rasmussen, C. L., Villarubi, R. S., Sangtong, J., & Fey, J. T. (2000). "Effects of standards-based mathematics education: A study of core-plus mathematics project algebra and functions strand." Journal for Research in Mathematics Education, 31 (3), 328-361.

Ingersoll, R.M. (1999). "The problem of underqualified teachers in American secondary schools." Educational Researcher, 28 (2), 26-37.

Jacobs, H. H. (1997). Mapping the Big Picture: Integrating Curriculum & Assessment k-12. Alexander, VA: Association for Supervision and Curriculum Development.

Jefferson County Public Schools (1996). Mathematics Content Standards. [Available: http://204.98.1.2/isu/math/math.html#standards].

Juraschek, B & Wooley, C. (1997). Evaluation [C.sup.2] for k-4 Mathematics Teachers. Internal document: Colorado School of Mines.

Kober, N. (1993). What we know about science teaching and learning? Washington, DC: Council for Educational Development and Research, US Department of Education.

Lewin, L. & Shoemaker, B.J. (1998). Great Performances: Creating Classroom-Based Assessment Tools. Alexandria, VA: Association for Supervision and Curriculum Development.

Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teacher's Understanding of Fundamental Mathematics in China and the United States. Mahway, NJ: Lawrence Erlbaum Associates.

Moskal, B. (1999). Formative and Summative Evaluation of the C2 Project. Internal Document: Colorado School of Mines.

National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, Virginia: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (1995). Assessment Standards for School Mathematics. Reston, Virginia: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (1991). Professional Standards for Teaching Mathematics. Reston, Virginia: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, Virginia: National Council of Teachers of Mathematics.

Post, T.R., Harel, G., Behr, M. J. & Lesh, R. (1991). "Intermediate teachers' knowledge of rational number concepts." In E. Fennema, T.P. Carpenter & S.J. Lamon (Eds), Integrating research on teaching and learning mathematics (pp. 177-198). Albany, NY: SUNY Press.

Shaw, R. (1998). Results of C2 Evaluation. Internal document: Colorado School of Mines.

Simon, M. (1993). Prospective elementary teachers' knowledge of division. Journal for Research in Mathematics Education, 24 (3), 233-254.

Stiggins, R. J. (1999). "Evaluating classroom assessment training in teacher education programs." Educational Measurement: Issues and Practice, 18 (1), 23-27.

Stiggins, R.J. (1991). "Relevant classroom assessment training for teachers." Educational Measurement: Issues and Practice, 10 (1), 7-12.

Stiggins, R.J. (1990). "Toward a relevant classroom assessment research agenda." The Alberta Journal of Educational Research, 36 (1), 92-97.

Stigler, J. & Hiebert, J. (1999). The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom. New York: The Free Press.

Usiskin, Z. (1993). "Lessons from the Chicago mathematics project." Educational Leadership, 50 (8), 14-18.

Van den Heuvel-Panhuizen, M. (1994). "Improvement of (didactical) assessment by improvement of problems: An attempt with respect to percentage." Educational Studies in Mathematics, 27 (4), 341-72.

Yang, M.T. & Cobb, P. (1995). "A cross-cultural investigation into the development of place-value concepts of children in Taiwan and the United States." Educational Studies in Mathematics, 28 (1), 1-33.

Dr. Moskal is the Associate Director of the Center for Engineering Education and an Assistant Professor in the Mathematical and Computer Sciences Department. Dr. Bath is the Director of Undergraduate Studies and Associate Professor in the Mathematical and Computer Sciences Department. Currently Dr. Bath is on sabbatical as a Program Director in the Education and Human Resources Directorate, Elementary, Secondary, and Informal Education Division, at the National Science Foundation.