Middle-grades teachers are usually considered as teachers of 6th, 7th, and/or 8th grades (Conference Board of the Mathematical Sciences [CBMS], 2012). Despite the crucial transitioning role middle level mathematics education has for students during their K-12 mathematics education experiences, research indicates that many teacher education programs are either focused on elementary or secondary levels and not oriented toward developing content and pedagogy for preservice middle-grades mathematics teachers (PMMTs).
Indeed, the majority of middle grades teachers are likely prepared in a program designed as preparation to teach all academic subjects in grades K–8 or in a program to teach mathematics in grades 7–12 or 6–12. Moreover, programs that do offer specific preparation for middle grades often lead to multi-subject certification (such as a certificate to teach mathematics and science), making it challenging for future teachers to take all the mathematics recommended by this report (i.e., The Mathematical Education of Teachers II). (CBMS, 2012, p. 39)
Overlapping licensure and mixed grade programs make it challenging for both PMMTs and their teacher educators to clearly define the scope and goals of mathematics content courses for this population. Especially when a class has a mixed composition of preservice elementary teachers and PMMTs, instructors sometimes must make adjustments to the depth and breadth of the mathematics content to be able to meet the learning needs of all students. The Mathematics Teaching in the 21st Century Study (MT21), an international comparative study focused on the preparation of PMMTs across six countries, pointed out the widely varied mathematics learning opportunities provided for PMMTs in the U.S. compared to other countries, with PMMTs prepared in an elementary program having relatively weaker mathematics content knowledge (MCK), but stronger pedagogy, than those prepared in a secondary program (Schmidt et al., 2007).
Given the above challenges in the preparation of PMMTs’ MCK, described here are my reflections on the gap in PMMTs’ MCK, particularly about reasoning and proof, along with contributing factors and potential impact, drawn from my own teaching experience and related literature. I started teaching geometry content courses for PMMTs a couple of years ago, and from the beginning I have been challenged by their reluctance toward learning reasoning and proof, especially axiomatic reasoning and formal proof writing. As a teacher educator, I believe a mission of a mathematics content course for PMMTs is to equip them with solid MCK as suggested by teacher education standards. So, my initial reflection on the course design began with reexamining the alignment between the learning goals of the course and national teacher education standards and recommendations. According to the Council of the Accreditation of Educator Preparation Standards (National Council of Teachers of Mathematics [NCTM], 2012), PMMTs should be prepared “with depth and breadth” in geometry (p. 1), including “axiomatic reasoning and proof” (p. 2). The Mathematical Education of Teachers II (MET II) (CBMS, 2012) particularly emphasizes mathematical explanation and reasoning abilities in the preparation of PMMTs, such as “informally explaining and proving theorems about angles” (p. 44). In addition to developing solid MCK that PMMTs will teach, it is essential they acquire understandings of the connections to both elementary school and high school mathematical concepts (Association of Mathematics Teacher Educators [AMTE], 2017; CBMS, 2012). Although it is equally important for PMMTs to make connections to both grade level bands, it is more challenging for them to build solid understandings and proficient skills in upper grades mathematics. Because of this concern, I made sure the geometry content included in the course addressed a moderate level of axiomatic reasoning and proof, as they are important components of high school geometry with which PMMTs are supposed to make connections when teaching middle school geometry (NCTM, 2000; National Governors Association Center for Best Practices [NGA Center] & Council of Chief State School Officers [CCSSO], 2010).
I strategically devoted a small portion of the course to developing PMMTs’ knowledge and skills in writing simple proofs with applications of a set of triangle congruence theorems that are required in mathematics content standards for both middle school students and PMMTs (AMTE, 2017; NCTM, 2000, 2012; NGA Center & CCSSO, 2010). Knowing PMMTs’ varied prior knowledge about proofs, I started with an introduction on the basic elements in the axiomatic system of geometry (e.g., meaning of definitions, axioms, theorems, etc.), followed by a few basic laws of logic commonly used in deductive reasoning (e.g., Law of Detachment). To scaffold PMMTs’ proof writing, I started from structured proof tasks, such as filling blanks or matching the statement with the reason in a two-column proof, and then transitioned to simple paragraph proofs. I also tried to not emphasize form over content and encouraged them to write proofs in their favorite forms (Schoenfeld, 1994).
However, the idea of including a small portion of axiomatic reasoning and proof writing in the course seemed to conflict with PMMTs’ learning interests. Frequent feedback included their expectation of learning the geometry skills they will teach, but not concepts or ideas beyond that level. This perspective seemed to become a hurdle on PMMTs’ path to meeting the teacher education standards, specifically the goal of the course of being able to make connections between middle school geometry ideas and high school geometry ideas (AMTE, 2017; CBMS, 2012).
There are other factors that possibly could have contributed to PMMTs’ reluctance toward learning reasoning and proof. First, there was a noticeable lack of prior experience with this level of content in their K-12 mathematics learning experiences. Most of the PMMTs in my class recalled they had done some form of simple applications of geometry theorems but with no formal proof writing involved. Some PMMTs even mentioned that they had never seen formal proofs before this course, which made it difficult for them to conceptualize how an axiomatic system functions and to appreciate the important roles reasoning and proof play in such a system, including verification and explanation of the truth of mathematical statements (de Villiers, 1990). Relatedly, the CBMS asserts that PMMTs:
… may not be familiar with all of the expectations outlined in the CCSS [Common Core State Standards] for middle school students. Thus, they may question the need to learn things in their teacher preparation program that were not part of their own middle grades mathematics. (2012, p. 51)
In addition, as pointed out in MET II (CBMS, 2012), PMMTs usually have some confidence in their mathematical skills and learning abilities, since they have chosen mathematics teaching as a profession. Therefore, perhaps they are more willing to express and embrace their perspective on learning and understanding mathematics based on their prior learning experiences. On the other hand, PMMTs’ mathematics foundation may not be as solid as that of secondary preservice mathematics teachers, who are required to take a sequence of college level mathematics courses and may have better conceptual understandings of mathematics. PMMTs’ perspectives on knowing mathematics “may be based on their own success in learning facts and procedures rather than on understanding the underlying concepts upon which the procedures are based” (CBMS, 2012, p. 51).
PMMTs’ knowledge of and beliefs about mathematics can potentially impact their teaching practices, and thus their future students’ success in learning mathematics (Tchoshanov, 2011). When considered in a broader context, it is troubling that multiple international comparative studies have shown insufficient preparation for PMMTs in the U.S. when it comes to MCK. For example, through surveying a total of 2,627 PMMTs across six countries, the MT21 determined the range of percentage of instruction that focused on mathematics content during teacher preparation as 0%-60% across all countries. In the U.S., secondary and elementary programs required 30% and 20% focus on mathematics content on average, respectively, which are only higher than Mexico that “required no formal mathematics except what was provided in their mathematics pedagogy courses” (Schmidt, Blomeke, Tatto, & Hsieh, 2011, p. 3). Therefore, perhaps the 8th grade mathematics achievement gap between the U.S. and other leading Asian countries found in the Trends in International Mathematics and Science Study could possibly be linked to PMMTs’ preparation differences in the U.S. versus those countries (Schmidt et al., 2007). The MT21 also determined that in the U.S., PMMTs’ MCK has much room for improvement compared to teachers in Taiwan (in all five areas including algebra, functions, number, geometry, and statistics), Korea (in all areas), and Germany (in all areas but statistics). Although within the U.S. PMMTs prepared in a secondary program had stronger MCK when compared to those prepared in other types of programs, their performance was still “slightly below the international mean for algebra, functions and geometry” (p. 5). Another international comparative study involving 17 countries, Teacher Education and Development Study in Mathematics, showed that the opportunities to learn MCK (on both the tertiary and school levels) provided to PMMTs in the U.S. were below average in both the primary mathematics specialists and lower secondary groups, and not surprisingly their MCK performance level was also below average in each of the groups (Tatto & Senk, 2011).
Reasoning and proof play an important role in mathematics and mathematics education, and therefore should be integrated into students’ mathematical experiences across all grades and a wide range of content areas (Hanna, 2007; NCTM, 2000; Schoenfeld, 1994). To better prepare PMMTs’ MCK of reasoning and proof, it is crucial to understand PMMTs’ MCK gaps and the potential underlying causes, and to create sufficient and coherent learning opportunities throughout their K-16 mathematics education programs. My next step is to gain more knowledge about PMMTs’ conceptions of the nature, logic, and application of reasoning and proof in the context of geometry, with the aim of informing the broader mathematics education community about the design of PMMTs’ preparation programs and mathematics content courses.
Association of Mathematics Teacher Educators. (2017). Standards for Preparing Teachers of Mathematics. Retrieved from https://amte.net/sites/default/files/SPTM.pdf
Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II.
de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24(1), 17-24.
Hanna, G. (2007). The ongoing value of proof. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 3–18). Rotterdam, Netherlands: Sense Publishers.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author
National Council of Teachers of Mathematics. (2012). NCTM CAEP standards for mathematics teacher preparation. Retrieved from http://www.nctm.org/Standards-and-Positions/CAEP-Standards/
National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics.
Schmidt, W., Blomeke, S., Tatto, M. T., & Hsieh, F.-J. (2011). A study of middle school mathematics teacher preparation in six countries. In Teacher education matters (pp. 1–7). New York, NY: Teacher College Press, Columbia University.
Schmidt, W., Tatto, M. T., Bankov, K., Blömeke, S., Cedillo, T., Cogan, L., … Paine, L. (2007). The preparation gap: Teacher education for middle school mathematics in six countries. MT21 Report. East Lansing: Michigan State University, 32(12), 53–85.
Schoenfeld, A. H. (1994). What do we know about mathematics curricula? Journal of Mathematical Behavior, 13(1), 55–80.
Tatto, M. T., & Senk, S. (2011). The mathematics education of future primary and secondary teachers: Methods and findings from the Teacher Education and Development Study in Mathematics. Journal of Teacher Education, 62(2), 121–137.
Tchoshanov, M. A. (2011). Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 76, 141–164.