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Mathematics Education

Teaching & Learning Problems in Mathematics

Curated by Cheryl Richardson with Introduction by Jacqueline Dewar

Jackie Dewar Photo

Lee Shulman (Annual Report 2004)

“Indeed, one of the most important missions of the Carnegie Foundation is to demonstrate through our work the ways in which all levels of education are intimately connected and interdependent. We can witness this commitment by taking a close look at our work in just one field, the learning and teaching of mathematics…”
This exhibit documents the work, courses or programs of five mathematics instructors spanning 5th grade through graduate school mathematics programs…work prompted by “teaching (and learning) problems” that these instructors encountered. Although referred to as snapshots, some of these websites are so comprehensive they could be called photo albums. No matter what they are called, these vignettes enlighten us about the situations and challenges today’s teachers of mathematics encounter and tell us how reflective practitioners have responded to these challenges. 

Communication clearly emerges as a common theme across all five snapshots. In USC’s new research apprentice approach, a graduate student in applied mathematics now has opportunities to present work orally and in writing much earlier than in traditional program. In Calculus Conversations Salem and Michael use technology to encourage student-to-student communication and then study the results to find out about student learning in first-year calculus. Da Luz explores how community and communication enable previously unsuccessful ninth grade students to overcome “academic language” barriers. Boerst investigates the role of multiple representations in making sense of and communicating mathematics in a fifth grade classroom.  Bennett has the “goal of producing conversation and student motivated thoughts” about mathematics in a course for future secondary mathematics teachers.

The motivating force of interdisciplinary work appears several times at different levels (for example, art in the service of algebra in the ninth grade and graduate research modeled as a cooperative interdisciplinary enterprise).  We also see how engaging the use of technology can be, both for instruction (computer spreadsheets and programmable calculators in Bennett’s course and a web-based threaded discussion conducted by Salem’s students) and for documentation (da Luz’ video clips).

The 5th and 9th grade teachers (Boerst and da Luz) clearly acknowledge the influence of state standards on their curricula.  These K-12 instructors very purposely reflect on their students’ previous learning and its effect on their practice as well as on their role in preparing students for future learning.  Bennett, as a teacher of future teachers, also keeps the demands of his students’ future careers at the forefront of his thinking when designing his instruction.

As models of the scholarship of teaching and learning (SoTL) in the discipline of mathematics, these snapshots demonstrate the value of situating a project within the literature and the importance of presenting and interpreting student work as evidence.  In the case of the newly designed program for a doctorate in applied mathematics at USC, the lack of evidence beyond student and faculty testimonials is readily conceded as a weakness.

These projects illustrate the potential of SoTL to transform a teacher’s thinking about teaching and learning.  Boerst’s case provides a powerful example of this.  He comments that initially he viewed multiple representations as a way to address the needs of his advanced students, but he came to see it as integral to a “Math for All” curriculum. Bennett too observes that some of the desired outcomes for his course were never explicitly stated, but that, in reflection, they can be recognized.

Teachers will likely find viewing this collection triggers new ideas and reflections about teaching, assessing student learning, and documenting teaching at all levels K-16 and beyond.  Much, much more is available here for anyone with the time and inclination to explore like an archeologist mining a site.

Department of Mathematics

University of Southern California

The Carnegie Initiative on the Doctorate engages with several doctoral programs that are committed to restructuring their programs to better prepare students to be stewards of their disciplines. Directors of this initiative have asked the departments to share, among other aspects, an "exemplary element" of their program. This element would be something that other doctoral programs might learn from and/or adapt for their needs. In response to this request, eight members (4 graduate students and 4 faculty/adminitrators) from the Department of Mathematics at the University of Southern California created this snapshot to share their new program in Applied Mathematics. The program more closely aligns with the interdisciplinary nature of the research that goes on in the department. The new format also allows for apprenticeship in ways that the previous model did not. Making public this element of their program enabled the department to share with others one way they cultivate "stewards" of the mathematics discipline--by providing students the chance to do research earlier and to apprentice with more experienced scholars.

Curtis Bennett

Loyola Marymount University

Curtis Bennett of Loyola Marymount University created this course portfolio to share his work with prospective secondary mathematics teachers. He wanted to create a document that he could pass on to other educators who might teach similar courses. The portfolio includes many aspects of the course--syllabus, student descriptions, assignments, history of the course, etc.--but focuses on his investigation of student learning. He was particularly curious about whether the research project helped students think more mathematically. This portfolio was developed as part of his fellowship with the Carnegie Academy for the Scholarship of Teaching and Learning in 2000.

Joanne da Luz

Carnegie Scholar

Carnegie Scholar, Joanne da Luz, documented a specific aspect of her teaching of a high school algebra class in San Francisco. She shares the ways that she helped her once math-fearful students articulate algebraic understandings through the "Colors and Algebra Project." She and her colleague, Justin Warren, developed the project which included the use of primary colors and geometric art to help her students understand, visualize, and present what they know. As part of her scholarship of teaching and learning work, Da Luz carefully listened to her students' development and use of mathematical language.


Anita Salem & Renee Michael

Rockhurst University

In 1999 Anita Salem and Renee Michael of Rockhurst University investigated the ways undergraduate students transfer mathematical concepts and methods to new situations. As part of a Carnegie Academy for the Scholarship of Teaching and Learning project, they created an activity through which they could help students transfer this knowledge and 'see' how it happened. The site documents their inquiry and results, which were not as expected, but did help them characterize aspects of student thinking and strategize for moving students to new understandings.


Timothy Boerst

Carnegie Scholar

Timothy Boerst explored the numerous learning opportunities that arise from focusing on one mathematical idea. He explores and shares how the attention to one idea, such as "The Rule of 3" can help teachers use particular mathematical representations to help students understand underlying mathematical ideas. This attention also may help teachers understand how different students solve mathematical problems. This site was produced in 2003 as part of his fellowship with the Carnegie Academy for the Scholarship of Teaching and Learning.

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