Report on NSF 0087892: Math for Future Secondary Teachers

DUE-CCLI Proof-of-concept project
January 1, 2001—June 30, 2004

James J. Madden, Department of Mathematics
David Kirshner, Department of Curriculum and Instruction
Louisiana State University, Baton Rouge LA 70808

Summary

Rationale: The typical undergraduate mathematics program in the USA equips students with a subset of the knowledge, skills and dispositions possessed by research mathematicians. In such a program, math majors who plan to become high-school teachers receive very little training that is directly relevant to their chosen careers. These students should have opportunities to develop the kind of mathematical understandings that a career in secondary teaching routinely calls upon.

Goals: We proposed a strategy to cultivate “knowledge of mathematics for teaching” during the major. This strategy had two distinct but interdependent components:

1. Develop instructional modules that could be used to enhance existing upper-level university mathematics courses.
2. Develop a small community of scholars dedicated to designing and testing such materials and integrating them with existing programs.

Activities: Four collaborative teams in four Louisiana locations were recruited. Each team included a university faculty member in mathematics, a faculty member mathematics education, practicing teachers and students intending to become high-school teachers. Each team worked under central support and direction to author and test curriculum materials in algebra, geometry, analysis, or statistics that addressed the problem identified in the rationale.

Products: Three of the teams successfully completed course-supplements that included in-depth design rationale, useable curriculum materials and extensive reports on field-tests. Incidental to this work, a number of additional items were developed.

Findings: We arrived at two sets of findings, corresponding respectively to the two goals listed above.

1. The high-school curriculum is rich in traditions relating to content, delivery and expectations regarding student performance. These traditions are evidently shaped by the demands of school-life, which range from the basic need to maintain discipline to the need to promote specific intellectual values. Intending teachers seem to be aware of the prevailing traditions and they show great interest in university-level courses that seek to analyze, explain or elaborate the mathematical meanings that are in them. (For an example of a curriculum module that seems to have been successful in this regard, see http://www.math.lsu.edu/~madden/Sec_Math_Site/TriangleArea/index.htm.) On the other hand, university mathematicians (with some notable exceptions) tend to be insulated from high-school traditions. What they may propose by way of curriculum can easily fail to be viable in the “cognitive ecology” of high-school. This constraint having been having been observed, however, we found instances where significant, deep mathematics can fit with the customs and traditions of high-school learning.
2. Based on experiences in this project, as well as experiences in several other projects with similar design, we conclude that:
   • Curriculum development is a skill that—like the ability to compose music or write for the theater—is difficult and time-consuming to learn and may be dependent upon unusual talents and dispositions.
   • Simple monetary incentives are not effective in promoting curriculum innovation, but opportunities for professional recognition and career advancement are powerful and effective motivators.
   These finding help to explain why we found it more challenging to keep the teams on-track and moving toward meaningful objectives than we had envisioned in the original proposal.

Conclusions and recommendations: The rationale for the project remains persuasive. We need to consider very carefully the pragmatic concerns and the traditions that shape high school mathematics education in the United States. While some practices may appear to be in conflict with what university mathematicians view as optimal for mathematics learning, they may be difficult to change because they have important functions in high school life. There are opportunities within existing frameworks where the right attention at the right time can lead to significant enhancement, and mathematics-teacher-educators should equip future teachers with the understanding they need in order to recognize and use these opportunities. Promising strategies include:

• Refer to state standards and to a variety of high-school textbooks.
• Dig deep into the mathematical content. What it means, precisely, to do this is hard to describe. Creating multiple representations or realizations of an idea and carefully and explicitly drawing out and describing the analogies between the different representations is part of it.
• Investigate the historical origins of mathematical ideas appearing in the high school curriculum.
• Listen carefully to undergraduate-reactions to curriculum innovation.
• Develop long-term working relationships with high-school teachers and seek to understand high-school students.

There are areas where deep and drastic changes in schooling might improve student learning. Mathematicians have an important role to play because they have a unique view of the intellectual possibilities. By teaming with other key players—teachers, school administrators, business and political leaders—they can be part of the process.

Rationale Elaborated

Major national surveys of the mathematical preparation of elementary teachers were made in the 1970s and again in the late 1980s. During the 1990s many colleges and universities reformed the mathematical training of elementary-school teachers. In Louisiana for example, the NSF-funded Collaborative for Excellence in the Preparation of Teachers (LaCEPT) powerfully influenced mathematics courses for elementary teachers at virtually every public institution of higher education.

The training of secondary teachers is a far more complex and may be a more critical problem, for the most serious problems in U.S. mathematics education occur at the secondary level. (TIMSS showed that the mathematical performance of U.S. students as compared to their international peers declines as grade levels go up, bottoming in grade 12.) Yet the role of university mathematics departments in preparing middle- and secondary-school mathematics teachers has not been carefully studied and we know little of the range of options that are out there. Scattered reform efforts are just beginning to get underway.

In general, mathematics departments still tend to offer to future secondary school teachers very little outside the traditional vertical sequence for math majors. This sequence emphasizes access to higher, advanced, modern mathematics. Students are encouraged to climb rapidly and to acquire some subset of the knowledge, skills and dispositions of researchers. There is mounting evidence that such experiences omit much that is critical in developing excellent secondary teachers. We now understand that effective teachers possess richly interconnected understandings of the “horizontal” relationships among foundational concepts. This enables them to respond more effectively to student needs, to interpret and compare numerous versions and formulations of critical basic ideas in response to student thinking, to create interesting intellectual opportunities for students with a range of conceptual repertoires and generally to provide students with meaningful, globally coherent mathematical experiences.

Goals Elaborated

The project set out to develop materials for use in those advanced college mathematics courses that are taken by intending middle- and high-school teachers. The materials would be based on proven theoretical models of mathematics learning, addressing specific middle- and high-school topics and goals as delineated in recognized standards. They were to include:

• tested and proven lesson plans and classroom materials, " meta-discussions" that would a) describe the mathematical intent of the lessons and appropriate ways of using them, b) outline relationships to curriculum standards for grades 7-12 and c) provide a perspective on the mathematical content by developing mathematical themes, exploring historical connections, or by other appropriate means,
• a record of actual classroom experience (capable of being updated each time the lesson is used), including well-selected examples of real student work accompanied, where appropriate, by teacher commentary, and
• evaluation materials for determining the readiness of a college class for the lesson and the extent to which the lesson's objectives are realized in a class in which it has been used.

The project was intended to support and develop a “community of practice” with participants at several colleges and universities. A basic goal was to create a system of cooperating scholars who would share professional pedagogical knowledge with one another. To this end, four production teams were selected at four different Louisiana universities. Each included a mathematics faculty member, a mathematics education faculty member, a practicing teacher recruited from a local school and one or more students. Each team had the responsibility of producing one completed lesson package meeting the specifications listed above. The project provided a range of support.

Timeline of Activities

• In spring 2001, four curriculum development teams, representing LSU-Baton Rouge, LSU-Shreveport, U. L. Monroe, and Southern Univ. Baton Rouge, were formed based on applications received following a solicitation.
• Workshop was held July 12--15, 2001, at LSU. Teams were provided with information on project goals and given guidance and advice on curriculum development. Teams planned their curriculum projects.
• Teams carried out initial phases of planned curriculum writing and testing between August 2001 and April 2002.
• April 2002. PI visited and interviewed three of the teams.
• In July 2002, LSU-BR team completed a report on its findings. This was reviewed by project advisor Wilkerson.
• In September 2002, LSU-Shreveport team completed a preliminary report on its findings and submitted a report.
• In December 2002, the UL-Monroe team submitted a preliminary report to PIs.
• Summer 2003. The team leaders did not view their modules as complete, and therefore did not request full payment of the stipends due team members. The PI requested and received a no-cost extension, moving the end of the project to June 30, 2004. (Total project duration: 3.5 years.)
• Monroe team submitted a final report in January 2004.
• Madden taught a geometry course in spring 2004 incorporating materials developed in this project. See: www.math.lsu.edu/~madden/M4005s2004.
• May 2004. Two-day workshop. Undergraduate math majors with concentration in secondary-education reviewed and evaluated products.
• Summer 2004, Madden developed additional materials, especially materials related to ratio and proportion.
• A web site with links to the products developed in this project was created.
• Other activities. This project influenced the master's theses of 3 LSU mathematics graduate students: Belinda Brand (M.S. 2003), Mabrouck Faradj (M.S. 2004) and Summer Armstrong (M.S. 2004). Each chose a topic with connections to the high-school curriculum, and followed design principles developed within this project. Brand has now become site coordinator for an outreach project that is providing professional development to local high-school teachers, and Faradj has become a high-school teacher.