Making Connections – Math from an Advanced Perspective

Connections

Connections are important. Preservice secondary mathematics teachers are expected to have made connections between university mathematics and secondary mathematics as a result of completing their program (Cuoco, 1998; Wu, 1999).  The development of understanding of mathematical connections ought to be a at least one of the major goals of secondary mathematics education (SME) programs.

I propose that many SME programs expect their students to make these connections through exposure to a variety of undergraduate mathematics courses but do not make the development of these connections an explicit goal of SME courses.  To investigate this claim, I describe an informal survey of SME programs in Michigan to see which make the development of mathematical connections an explicit goal in any SME courses.

One argument against making this an explicit goal might be that mathematical connections are not able to be explicitly taught. I investigate two available textbooks which have stated goals of developing a student’s understanding of mathematical connections.

I will use an informal survey to show that, although some are making the building of mathematical connections a visible goal, most SME programs in Michigan do not show connections as explicit goal of courses.

Connections

            In many preservice SME programs, students are required to complete mathematics requirements to satisfy a B.S. major in mathematics.  Mathematicians and mathematics education researchers agree that a strong mathematical background is a necessary, but not sufficient, condition of a successful high school mathematics teacher (Begle, 1979; MET, 2001; NCRTE, 1991; Wilburne & Long, 2010). Mathematicians and mathematics education researchers agree that there must be some mathematical pedagogical element included, but they are not in agreement on the specifics of that element.

Shulman (1986) recommended that teachers have a deep understanding of their subject matter content knowledge, which includes knowledge of the basic concepts in addition to the connections between areas within the discipline as well as to other disciplines.  This implies that preservice teachers do need a broad range of courses in their subject area, but also need to develop their understanding of the ways in which the basic concepts are connected.

Wu (1999) argued that mathematics courses engineered to meet the needs of preservice teachers do not need to be “watered down.” Instead, different concepts should be emphasized and they should be taught from an “advanced viewpoint” (p. 13).  Wu explained that by “advanced viewpoint” he meant looking at high school mathematics topics from the point of undergraduate mathematics courses – rigorously proving and explaining selected theorems and algorithms that are found in high school mathematics.

According to recommendations made in the NCTM standards and Common Core State Standards, school mathematics teachers are expected to make mathematical connections visible to their students (CCSS-M, 2010; NCTM, 2000). In order to do this successfully, SME programs are expected to make mathematical connections visible to preservice teachers (InTASC, 1995; MET, 2001; NCATE, 2007), that is, to teach the mathematics from an advanced viewpoint (Cuoco, 1998; Wu, 1999). Teacher educators are currently developing instructional methods, working to help SME preservice teachers develop their knowledge of connections between high school and undergraduate mathematics curriculum from different perspectives (Abramovich, 2009; Artzt, Sultan, Curcio, & Gurl, 2011; Fernández & Jones, 2006; Graham & Fennell, 2001; Smith, 2002; Stanley & Sundström, 2007; Usiskin, 2001; Wu, 1999).

Since knowledge of connections is expected of high school mathematics teachers and of preservice education programs, if the connections are not explicitly taught then they must be part of the hidden mathematics curriculum for preservice mathematics education programs.

Summary of Programs in Michigan

To determine the visibility of mathematical connections in SME programs in the United States, an extensive study would need to be conducted. As a first step, however, I conducted a brief, informal survey of each undergraduate SME program in Michigan. I used a list of colleges and universities (Wikipedia, 2011) and visited the website of each to determine if it offered an undergraduate SME program and, if so, what courses were required and what course goals were listed in the course descriptions.

I found 29 SME programs in Michigan, offered at both private and public universities and colleges and accredited through National Council for Accreditation of Teacher Education (NCATE) or Teacher Education Accreditation Council (TEAC)  (see Appendix for details). I scanned course descriptions and academic bulletins to compare course goals and to identify courses with a mathematics education emphasis, specifically those with making connections between high school and undergraduate mathematics curriculum.

Discussion

I tracked the accreditation of the programs to see if there were any interesting patterns in courses offered.  7 of the 29 are accredited by NCATE and 10 are accredited by TEAC, with 11 more listed as TEAC candidates. 18 identified themselves as having “Michigan Department of Education Approval.”  There were no clear patterns, which may indicate that the choice to emphasize mathematical connections as explicit goals is not related to accreditation by a specific agency.

I also tracked the mathematics and mathematics education courses that were required by each program, paying close attention to the course descriptions.  If a course seemed to fall into a mathematics category, but was being taught from an educational standpoint, I indicated that on the chart.  I also indicated courses with the goal of creating connections between secondary and tertiary mathematics courses or were being taught from an “advanced viewpoint.”

The majority of programs required between 10-14 mathematics courses and between 1-4 mathematics education courses.  11 programs required completion of a capstone course, although for 5 of these the capstone course seemed to be strictly mathematical.  Only seven programs of the 29 required a course that explicitly defined “connections between high school mathematics and undergraduate mathematics” or “high school mathematics from an advanced viewpoint” as goals of the course.

It is encouraging to note that there are programs with mathematical connections as stated goals. For example, Michigan State University requires a capstone course that explores “[h]igh school mathematics from an advanced viewpoint.”  As another example, Central Michigan University requires three mathematics courses for preservice teachers in which the mathematics is “explored through a problem-based and technology-enhanced approach connecting secondary mathematics curricula with undergraduate mathematics and pedagogical content.” These courses are: Algebra and Calculus, Probability and Statistics, and Geometry.

Fewer than a quarter of all secondary mathematics programs in Michigan, however, include mathematical connections between high school and undergraduate mathematics as explicitly stated goals in course descriptions for required courses.  It seems reasonable to conclude that there are too many programs in Michigan for which mathematical connections are not explicitly taught to the students, despite the recommendations in order to be successful students must make these connections. Admittedly, there may be many programs that focus on developing mathematical connections even though this focus is not reflected in the course descriptions.  For example, there may be universities that split mathematics courses into sections – one for preservice teacher mathematics majors and a second for other mathematics majors. On the other hand, programs that list mathematical connections or advanced viewpoints as goals in the course descriptions may not deliver. Further research is needed to discover how significant is the number of programs that do not include mathematical connections as a visible goal in program courses as well as how well-taught are mathematical connections in programs that include them as a stated goal.

Instructional Support

Wu (1999) mentions that one difficulty he found when planning to teach mathematics courses that were intended to help SME majors make mathematical connections between secondary and undergraduate mathematics courses was the lack of a good textbook. I show here that there are at least two textbooks available for a general course of this type. As previously mentioned, different programs attempt to teach mathematical connections from different perspectives.  I discuss the similarities and differences of the approaches of two textbooks that are available written specifically to address the need for SME majors to develop their understanding of mathematical  connections: Usiskin et al. (2002) and Sultan & Artzt (2010).

A capstone course in Michigan State University SME program uses Usiskin et al. (2002).  This text is written from “an advanced perspective,” that is, looking at certain elementary topics from a mathematician’s point of view.

Sultan & Artzt (2010) seems to be written from the opposite direction, that is, familiar topics from typical high school mathematics courses are presented and then linked with topics common in various undergraduate mathematics courses .  Artzt, a mathematics teacher educator, and Sultan, self-described as a traditional mathematics professor, originally collaborated around a project intended to positively affect the retention of SME majors at their university (Sultan & Artzt, 2005). They describe their text specifically as bridging “the gap between mathematics learned in college and the mathematics taught in secondary schools.”

The two textbooks have many similarities and differences.  First, both books share coverage of main topics. Both include many similar historical connections and some connections within mathematics topics.  Second, the textbooks differ in organization, focus and approach.  These similarities and differences appear throughout the books, but given limitations on space and time, I will focus on the chapter in each book that deals with equations.

Usiskin et al. begin by building the idea of equations through connections with advanced mathematical topics, develop these ideas and connections through several sections and then connect it all back to school mathematics by rigorously developing and explaining common algorithms and formulae.  Few numerical examples are provided and are not detailed. Historical connections are also discussed.

They use sets, equivalence and isomorphism to develop the student’s understanding of equality. They then give an overview of mathematical statements (e.g., existential and universal statements) and sets and group theory.  The concepts developed through these overviews are then used to develop and explain basic equation-solving algorithms first.  Finally, they end by discussing the development of the quadratic formula and other methods of solving higher-order polynomial equations.

Sultan & Artzt start with a familiar high school mathematics topic, and through the discussion of its properties and theorems, connect it to more advanced mathematical topics. Proofs and explanations are rigorous, but the writing is friendly and approachable.  Many examples are provided with detailed steps included. Historical connections are also discussed.

Sultan & Artzt begin the chapter immediately with a discussion of polynomials.  Instead of motivating mathematics through a description of the big idea, they begin with a problem the students have likely seen and can relate to – polynomials and use the quadratic formula as a solution method for a specific polynomial.

Usiskin et al. is a textbook that mathematicians would approve.  It is well-written, formal and rigorous.  High school mathematics is developed well, all with tools that appear throughout undergraduate mathematics courses.  Sultan & Artzt is closer to an actual high school mathematics text.  It is friendly and charming.  There is a strong level of rigor.  It is a book of higher mathematics written at the level of non-mathematicians.

In my opinion, neither book provides a good fit for general SME preservice teachers to develop connections between high school mathematics and undergraduate mathematics. Usiskin et al. be more appropriate for a graduate level course with a similar goal, while Sultan & Artzt might work well for an introductory course.  Similar to Goldilocks, one is too hot and one is too cold – there needs to be a middle ground.

Conclusion

Mathematical connections are a vital part of any mathematics lesson and preservice secondary mathematics teachers need to be prepared to include them. Preservice secondary mathematics teachers cannot learn the connections through straight mathematics courses alone, even though they will be expected to have this knowledge as a result of completing their program.  The development of understanding of mathematical connections must be a major goal of SME programs.

It is important to know if SME programs are hearing the strong recommendations that have been, and are continuing to be, made regarding the inclusion of development of mathematical goals in SME programs.  This survey of schools can only be used as a first step toward discovering whether schools are heeding these warnings. Further research is needed to discover how significant is the number of programs that do not include mathematical connections as a visible goal in program courses as well as how well-taught are mathematical connections in programs that include them as a stated goal.

These courses do need mathematicians and teacher educators to work together (MET, 2001), and concepts do need to be explained at the level of the students in the courses, but this does not mean that mathematical concepts should be “dumbed down” (Cuoco, 1998; Wu, 1999) .  Instead, concepts should be carefully selected to encompass the big ideas of high school mathematics and developed to include the historical and mathematical connections to advanced mathematical topics which should also be carefully selected.  It is easy to move too far in either direction.  I would suggest that in these days of standards-based curriculum, time should also be spent connecting the standards to the high school and undergraduate mathematics topics (Fernández & Jones, 2006; Senk & Thompson, 2003).

References

Abramovich, S. (2009). Hidden Mathematics Curriculum of Teacher Education: An Example. PRIMUS, 19(1), 39-56.

Artzt, A., Sultan, A., Curcio, F., & Gurl, T. (2011). A capstone mathematics course for prospective secondary mathematics teachers. Journal of Mathematics Teacher Education, 1-12. doi: 10.1007/s10857-011-9189-5

Beckmann, S. (2010). Mathematics for Elementary Teachers With Activity Manual: Addison-Wesley Longman, Incorporated.

Begle, E. G. (1979). Critical variables in mathematics education: findings from a survey of the empirical literature: Mathematical Association of America : National Council of Teachers of Mathematics.

CCSS-M. (2010). Common Core State Standards Initiative: Common Core State Standards for Mathematics.

Cuoco, A. (1998). What I wish I had known about mathematics when I started teaching: Suggestions for teacher-preparation programs. The Mathematics Teacher, 91(5), 372-374.

Fernández, E., & Jones, M. (2006). Emphasizing the NCTM Content Standards in Undergraduate Courses for Prospective Teachers. Mathematics and Computer Education, 40(3), 237-247.

Graham, K. J., & Fennell, F. (2001). Principles and standards for School Mathematics and teacher education: Preparing and empowering teachers. School Science and Mathematics, 101(6), 319-327.

InTASC. (1995). Model Standards for Beginning Teacher Licensing and Development: A Resource for State Dialogue.

Jackson, P. W. (1968). Life in Classrooms. New York: Holt, Rinehart and Winston, Inc.

Jackson, P. W. (1992). Conceptions of Curriculum and Curriculum Specialists. In P. W. Jackson & A. E. R. Association (Eds.), Handbook of research on curriculum: a project of the American Educational Research Association (Vol. 1, pp. 3-40): Macmillan Pub. Co.

MET. (2001). The Mathematical Education of Teachers. Issues in Mathematics Education. Volume 11 [and] Part I: Conference Board of the Mathematical Sciences,  Washington, DC 20036. .

NCATE. (2007). Professional Standards for the Accreditation of Teacher Preparation Institutions.

NCRTE. (1991). Findings from the Teacher Education and Learning To Teach Study. Final Report: National Center for Research on Teacher Learning, Michigan State University, East Lansing, MI.

NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.

Posner, G. J. (1995). Analyzing the curriculum: McGraw-Hill.

Senk, S. L., & Thompson, D. R. (2003). Standards-based school mathematics curricula : What are they? What do students learn? Mahwah, N.J.: Lawrence Erlbaum Associates.

Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2), 4-14.

Smith, J. C. (2002). Connecting Undergraduate Number Theory to High School Algebra: A Study of a Course for Prospective Teachers. Paper presented at the 2nd International Conference on the Teaching of Mathematics (at the undergraduate level), University of Crete, Hersonissos, Crete, Greece.

Stanley, D., & Sundström, M. (2007). Extended analyses: finding deep structure in standard high school mathematics. Journal of Mathematics Teacher Education, 10(4), 391-397. doi: 10.1007/s10857-007-9050-z

Sultan, A., & Artzt, A. F. (2005). Mathematicians Are from Mars, Math Educators Are from Venus: The Story of a Successful Collaboration. Notices of the AMS, 52(1), 48-53.

Sultan, A., & Artzt, A. F. (2010). The mathematics that every secondary school math teacher needs to know: Routledge.

Usiskin, Z. (2001). Teachers Mathematics: A Collection of Content Deserving to be a Field. The Mathematics Educator, 6(1), 85-98.

Usiskin, Z., Peressini, A. L., & Marchisotto, E. (2002). Mathematics for high school teachers: an advanced perspective: Pearson Education.

Wikipedia (Ed.). (2011). List of Colleges and Universities in Michigan: Retrieved from: http://en.wikipedia.org/wiki/List_of_colleges_and_universities_in_Michigan on 12/07/2011.

Wilburne, J., & Long, M. (2010). Secondary Pre-Service Teachers’ Content Knowledge for State Assessments: Implications for Mathematics Education Programs. Issues in the Undergraduate Mathematics Preparation of School Teachers, 1, 13.

Wu, H.-H. (1999). On the Education of Mathematics Majors. In E. A. Gavosto, S. G. Krantz & W. G. McCallum (Eds.), Contemporary issues in mathematics education: Cambridge University Press.

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