MAA Contributed Paper Session: Drawing on Our Students’ Thinking to Improve the Mathematical Education of Teachers
Design and
Implementation of Linking Courses: Connecting College Mathematics
with High School Mathematics for Pre-service Teachers
Paul Kehle
pkehle@indiana.edu
Dan Maki
Andy Norton
Dale Nowlin
January 2, 2005
Abstract
A collaborative effort among the Indiana University (IU) Mathematics Department, the IU School of Education, and local high school mathematics teachers resulted in the design of four one-credit courses that undergraduates take in conjunction with courses in calculus, abstract algebra, mathematical modeling, and probability & statistics. The primary objective of these linking courses is to help undergraduate pre-service mathematics teachers make connections among the content of their undergraduate mathematics courses, the content and pedagogy of the grades 6-12 mathematics curriculum, and standards for K-12 mathematics education. We seek to cultivate a fuller, more relevant understanding of the deeper mathematical ideas contained in the undergraduate courses. Examination of the undergraduates’ thinking about the concepts encountered in these courses leads to both content and pedagogical gains. This paper addresses the challenges, solutions, and pros and cons entailed in this approach to improving the quality of future mathematics teachers.
Motivation and Overview
Many undergraduates who are pursuing majors in mathematics and coursework that will lead to certification as high school mathematics teachers question the relevance of some of their mathematics courses (especially upper level ones) for their future careers. They are not always able to see the forest for the trees when it comes to appreciating how greater depth and breadth of mathematical understanding can inform the teaching of even the most elementary high school mathematics. Even in typical second- or third-semester calculus courses or first courses in analysis, when the emphasis on proof and theoretical underpinnings increases, some college students have trouble seeing how this material can and should inform the teaching of high school calculus courses (whether Advanced Placement or not).
In conjunction with the Indiana University School of Education, the Indiana University Mathematics Department developed and instituted a series of linked courses to address the relevancy issue and to improve generally the quality of high school mathematics instruction by better preparing new teachers. The primary goal of these linked courses is to help college students make connections between their college mathematics coursework and grades 6-12 mathematics teaching. Students take these one-credit linked courses, which count as education credits, concurrently with or shortly following completion of specific courses required for a major in mathematics.
Currently, linked courses are offered for four courses: third-semester calculus, abstract/linear algebra, probability & statistics, and mathematical modeling. Teaching certification students are now required to take any three linked courses prior to student teaching. Once these four courses are well established, we are considering creating others (e.g., for geometry or topology) to provide students with even more flexibility and opportunities for linking their current study of college mathematics to their future teaching of high school mathematics.
The term “linked” has a dual meaning in this context. A linked course is linked to and designed for a specific mathematics course, and in this linked course, links are made between the mathematics course and secondary school (grades 6-12) mathematics education.
History and
Although the concept of linked courses at Indiana University dates back several years to conversations involving faculty in the School of Education and the mathematics department, when the term “shadow” courses was used, it was the impetus provided by a National Science Foundation Mathematics and Science Partnership (MSP) grant that led to the development and implementation of the current linked courses. The development of these linked courses was a major focus of the pre-service secondary component of the MSP grant. Prior to this grant’s activity, an experimental version of a shadow course was taught for a linear algebra course. Some of the challenges faced in that initial attempt were scheduling, student motivation, and communication about the exact purpose of the course. Students seeking teaching certification have very full schedules not only semester to semester, but in terms of space for credits over the entire 4-year degree program. Some of those students who managed to find time to take this shadow course as an elective were expecting a course that would help them better understand the mathematical content of their course; in short, they were seeking additional instruction in linear algebra. This expectation was never the intent of the original shadow or current linked course concept.
Coinciding with
the MSP grant, the
Linking Concept
Linked courses help college students make both content and pedagogical connections between their college mathematics coursework and grades 6-12 mathematics education. These courses seek to generate the pedagogical-content knowledge and understanding that Shulman (1986) describes. In the same way that mathematics methods (or pedagogy) courses work to help students begin thinking more like teachers of mathematics than as students of mathematics, linked courses focus on this transition in very specific content domains (i.e., abstract algebra, calculus, modeling, and probability).
Central to such study is the mathematics itself, the ways the college students are thinking about and coming to understand this mathematics, and the keys to having this understanding effectively inform their teaching of secondary school mathematics. The teaching of secondary school mathematics is most effective when the teacher’s understanding of the mathematics is blended with an understanding of the cognitive development of adolescents and general pedagogical strategies adapted to particular content. The blending of content and pedagogy in effective mathematics instruction depends upon teachers (pre-service college students in this case) becoming more aware of their own understanding of the relevant mathematics. Becoming more metacognitively (Schoenfeld, 1985) aware of their own mathematical thinking will help them make the links needed to better inform their future teaching.
Logistical Issues
The original goal was to have students take the linked course concurrent with the mathematics course to which it is linked. However, in practice, finding a free time slot common to all students needing the course proved to be impossible, unless we were willing to have these 1-credit courses dictate which 3-credit courses the students could take and when. The packed nature of the students’ schedules led us to allow students to take linked courses during or after completing the course for which the linked course is designed. Students are advised strongly to take their three linked courses concurrently when possible or as soon after the relevant mathematics course as possible.
To accommodate students’ schedules and the schedules of local high school teachers who are expected to often teach linked courses, early evening time slots are used for scheduling them. Currently the algebra (abstract and linear) link and the modeling link are offered each fall, and the calculus link and the probability & statistics link each spring. Because there are several students taking these mathematics courses simultaneously, and because of the shortage of common free time in everyone’s schedules, we currently offer the two linked courses that meet each semester during the same time slot (e.g., Tuesday 7-8:30pm) on an alternating week basis. For example, in the fall the modeling link meets every other week, and in the intervening weeks, the algebra link meets at the same time in the same room. Those students taking both math courses attend every week and those taking just one of the math courses attend every other week when the link for their particular course meets.
The linked courses must dovetail with the typical secondary school mathematics pedagogy courses the students take because some students will be taking one or more linked courses during the semesters in which they are taking their 3-credit pedagogy (i.e., methods) courses. Because all of these courses focus on state and national standards, secondary school mathematical content, pedagogy, assessment (both course-based and standardized) care is taken not to duplicate assignments or activities. In practice, this redundancy is easily avoided as the linked courses are focused much more narrowly on specific college level mathematics content. Often times, when students encounter general principles of effective teaching in their pedagogy classes, they enjoy and value the opportunity to apply the general ideas to very specific concrete math content in their linked courses.
The large majority of the assignments in the linked courses are tied to useful products that the students can use during student teaching or in their first years in secondary school classrooms. Writing lesson plans, creating computer programs, developing other technology-based demonstrations, designing student projects and explorations, and accumulating files of relevant resources are the kinds of products the students produce in their linked courses. Currently in the algebra link, the students compile all of the material they generate into a website that is also burned onto a cd for the students to take with them.
The linked courses are taught by master teachers from area schools, by mathematics education faculty, or by teams of the above personnel. Initial feedback from students has been very positive with all of them happy with the practical material they take from these courses that will serve them during student teaching. The instructors of the linked courses are also satisfied to see that the students come to view the topics in their mathematics courses as much more than isolated exercises, theorems, proofs, and solutions to problems. At the end of the linked courses, the mathematical content has become part of a network of understanding that includes mathematics, pedagogy, and relevant practical issues in secondary school mathematics education. All of these gains are due to providing the students a structured opportunity to expand their thinking about college mathematics to include pedagogy and ties to secondary school mathematics content that they might not otherwise think of in relation to the advanced topics they are studying.
The grading of the linked courses is pass/fail. This decision was made to remove grading as a possible point of contention and increase the focus on developing the desired linkages between college and secondary mathematics education. In addition, the decision rested on the nature of a short 1-credit course not being well suited to multiple opportunities to assess students and their progress in meeting course objectives. This decision has been a sound one and there has been no evidence of a lack of engagement as a result.
Elements from Representative Syllabi
To realize the overall goal of the linked courses, each linked course utilizes a variety of more specific goals and assignments. In one recent algebra linked course the goals and related assignments were:
Course Objectives
1. To make connections between topics of higher algebra and secondary school curriculum and methods.
2. To produce a useful CD/website illustrating these connections.
3. To broaden our perspectives of secondary mathematics.
All assignments will be related to the production of the CD mentioned above. Each student is expected to attend class every week and contribute an additional 15 hours of work toward the CD outside of class.
A recent mathematical modeling course link’s objectives and assignments:
Objectives
The primary objective of this course is to help you make connections between the mathematical content of M447 and the teaching and learning of secondary school mathematics. Additionally, this course will enable you to prepare and collect mathematical modeling lesson plans for use during student teaching and as you begin your teaching career. By discussing mathematics modeling and mathematics pedagogy, you will also become more conversant with, and able to apply important concepts in the teaching of mathematics, (e.g., standards, curriculum planning, lesson design, open-ended problem solving, technology, assessment, and student-centered learning).
Assignments
1)
2) Design of 2 (depending on length) lesson plans driven by mathematical modeling; one of these will be a group project that may be tied to your M447 modeling project and one will be lessons you prepare individually
3) Participation in weekend workshops with secondary school teachers
Initial Student Feedback
Initial student feedback has been overwhelmingly positive. The students are continually engaged in not only making connections between their study of college mathematics and issues of secondary school mathematics education, but the connections they are making result in practical products that they will use during student teaching and beyond. The degree to which the connections, or links, get embodied in lesson plans, resources, or other products is the degree to which the connections will persist beyond the linked course. Our expectation is that students will be more open to making similar kinds of connections in their other mathematics course, and in their study of mathematics beyond their undergraduate majors. Here are a two students’ comments from recent course evaluations:
I guess that I made the most connections with the prime numbers and the factorizations. But more importantly all of the matrix activities that we did. I loved them and to be honest, did not think of matrices in that way at all before the class. Now I am thinking more creatively about activities that will create in-depth critical thinking about the concepts. To me, the class was one of the best (one of the most
inconvenient times) but also, probably one of the most helpful. I think that you did an excellent job and I really enjoyed the way you opened up a whole new way of thinking about the high school curriculum. —Algebra Link Student A Fall ‘04
I have enjoyed the class. I wish that my other math classes were a little more teacher—"think about this when you are teaching"—oriented. I do think that you have shown us how some of the topics are related and why we need to know the math behind the algorithms. —Algebra Link Student B Fall ‘04
Conclusion
Beyond the obvious gains in for the students in these linked courses and for their future secondary school students, we are excited about the linked courses as a site for research on secondary pre-service teachers’ development of pedagogical content knowledge. One of our conjectures is that because of the additional time spent reflecting on and using what they are learning about a particular domain of mathematics, students in linked courses will develop both a deeper and a broader understanding of mathematics and pedagogy that results in enhanced specific content-pedagogical knowledge (Schulman, 1986).
References
Schoenfeld, A. (1985). Mathematical problem solving.
Shulman, L. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2), 4-14.