Applications of Teaching Secondary Mathematics in Undergraduate Mathematics Courses
Elizabeth G. Arnold, Elizabeth A. Burroughs, Elizabeth W. Fulton, James A. Mendoza ?lvarez
AAPPLICATIONS OF TEACHING SECONDARY MATHEMATICS INUNDERGRADUATE MATHEMATICS COURSES
Elizabeth G. Arnold, Elizabeth A. Burroughs, Elizabeth W. Fulton, James A. Mendoza ´AlvarezColorado State University, Montana State University, Montana State University, University ofTexas at ArlingtonPaper for TSG33 at the 14th International Congress on Mathematical Education (2020).Shanghai, China. [Delayed to July 2021 due to COVID-19 pandemic]
Robust preparation of future secondary mathematics teachers requires attention to the acquisitionof mathematical knowledge for teaching. Many future teachers learn mathematics content primarilythrough mathematics major courses that are taught by mathematicians who do not specialize inteacher preparation. How can mathematics education researchers assist mathematicians in makingexplicit connections between the content of undergraduate mathematics courses and the content ofsecondary mathematics? We present an articulation of five types of connections that can be usedin secondary mathematics teacher preparation and give examples of question prompts that mathe-maticians can use as applications of teaching secondary mathematics in undergraduate mathematicscourses.
INTRODUCTION
Secondary mathematics teacher preparation routinely includes the study of both mathematics con-tent and pedagogical strategies. It is well understood that secondary mathematics teachers musthave a strong understanding of the mathematics taught in secondary schools (Ferrini-Mundy &Findell, 2001). The Mathematical Education of Teachers II (MET II) Report (CBMS, 2012) rec-ommends that prospective secondary teachers complete the equivalent of an undergraduate major inmathematics with a focus on examining secondary school mathematics from an advanced perspec-tive, emphasizing the importance of making connections between the mathematics undergraduatesare learning and the school mathematics they will be teaching. In this vein, Wasserman (2018b)recommends that the content of secondary mathematics ought to inform how undergraduate math-ematics courses are taught. Despite these recommendations, many undergraduate mathematicsmajor programs in the United States do not meet these needs: “Evidence suggests that prospectivehigh school mathematics teachers, who earn a mathematics major or its equivalent, do not havesufficiently deep understanding of the mathematics of the high school curriculum” (Speer, King &Howell, 2015, p. 107). Zazkis and Leikin (2010) found that “many teachers perceive their under-graduate studies of mathematics as having little relevance to their teaching practice” (p. 1). Thereis a gap between the mathematics that prospective teachers are learning and the school mathemat-ics they will teach, pointing to the need to focus on connections to school mathematics within theundergraduate mathematics curriculum.Research in mathematics education has shown that it is not enough for future teachers tobe proficient with school mathematics; future teachers must also develop a kind of specializedknowledge for teaching. Mathematical knowledge for teaching (MKT) is a theory developed byBall, Thames, and Phelps (2008), focused on elementary mathematics teaching, that encompassesthe knowledge teachers require in order to “carry out the work of teaching mathematics” (p. 395)1 a r X i v : . [ m a t h . HO ] F e b nd plays an important role in teacher preparation programs. Stylianides and Stylianides (2014)articulate the need for teacher preparation programs to incorporate specific knowledge needed forteaching as well as opportunities for prospective teachers to apply this knowledge in context oftheir future practice. Yet, in the United States, it is often the case that prospective secondarymathematics teachers enroll in traditional mathematics major courses that are often taught bymathematicians who do not specialize in teacher preparation nor have a robust understanding ofmathematical knowledge for teaching.This research operationalizes mathematical knowledge for teaching to support mathematiciansin addressing it in their instruction. We present practical implications for prospective mathematicsteacher preparation that result from an articulation of five types of connections that can supportall instructors of mathematics content courses. THEORETICAL CONSIDERATIONS
Teaching is a complex and multifaceted skill that employs a wide range of knowledge. Shulman(1986) was among the first to propose that in order to teach mathematics effectively, teachersmust possess more than a separate understanding of content and pedagogy. He described anothercomponent of teaching, called pedagogical content knowledge, in which teachers combine theirpedagogical knowledge with their content knowledge. This idea has been influential in teachereducation research, and Ball et al. (2008) extended the idea of pedagogical content knowledgewhen developing the construct of mathematical knowledge for teaching (Figure 1).Figure 1: The components of mathematical knowledge for teaching (Ball et al., 2008)Ball et al. (2008) address two broad domains in the construct of MKT: subject matter knowledgeand pedagogical content knowledge. Subject matter knowledge comprises a teacher’s knowledgeof content within a course, how this content connects to previous and future topics, and differentways content can be interpreted to assist in teaching. Pedagogical content knowledge comprisesa teacher’s knowledge of the content in relation to teaching practices, student learning, and thecurriculum. Six kinds of knowledge relevant to teaching elementary mathematics make up thesetwo broad domains. • Common Content Knowledge is the mathematical knowledge and skills used in many sit-uations (teaching and non-teaching). It includes, among other things, being able to solve2athematical problems, recognize errors, and use terms and notation correctly. • Horizon Content Knowledge is knowledge of the mathematics that follows or could followthe mathematics being taught. It is an awareness of how the mathematical topics span acurriculum. • Specialized Content Knowledge is the mathematical knowledge and skills (beyond commoncontent knowledge) that are specific to teaching. • Knowledge of Content and Students is a knowledge about how students learn mathematicsand being able to anticipate common student errors. • Knowledge of Content and Teaching is knowledge about mathematical tasks and teaching,such as sequencing and selecting examples to help students learn. • Knowledge of Content and Curriculum is the knowledge of curriculum for a given topic. Thisincludes being familiar with materials and resources for teaching mathematics.Other researchers (e.g., Wasserman, 2018a; Wasserman et al., 2019) have addressed ways to de-velop mathematical knowledge for teaching in the preparation of secondary mathematics teachers.Wasserman et al. (2019) articulate four ways in which prospective teachers can engage in develop-ing mathematical knowledge for teaching and describe these as existing along a spectrum spanningfrom “mathematical in nature” to “pedagogical in nature” (p. 821).
Content tasks focus on the un-dergraduate mathematics that serves to deepen prospective teachers’ understanding of secondarymathematics.
Modeled Instruction consists of different modes of instruction implemented by aninstructor (e.g., inquiry-based learning) that can influence how prospective teachers will teach.Mathematical practices and mathematical habits of mind are the foundation for
Disciplinary Prac-tice ; Wasserman (2018a) describes this as focusing on what it means to “do” mathematics. Lastly,
Classroom Teaching tasks apply content to a specific teaching situation such as designing problems,sequencing activities, and describing, for example, what it means to multiply two fractions.
FIVE CONNECTIONS TO TEACHING SECONDARY MATHE-MATICS
Using Ball et. al.’s (2008) delineation of the six categories of MKT as our theoretical base, we de-fined five types of “connections for teaching” between undergraduate-level mathematics and knowl-edge for teaching secondary mathematics (Table 1). The MET II Report (CBMS, 2012) suggestsengaging prospective teachers in understanding “the connections” (p. 54) between undergraduatemathematics and school mathematics, but leaves these connections undefined. Though educationresearchers generally understand that the knowledge needed for teaching encompasses more thancontent knowledge, there is not such awareness among non-specialists, that is, among those whoteach the bulk of the undergraduate courses taken by prospective teachers but are not educationalresearchers, leaving a compelling need for an articulation of how to make such connections.The connections we formulate here complement the existing literature and extend the constructof MKT in a way that is useful for mathematicians who have a role in preparing secondary teachers.When developing our five connections, we first considered each broad domain of MKT separately.In considering how to translate subject matter knowledge for teaching secondary mathematics, we3sked ourselves the following question: what should undergraduate prospective teachers understandabout the undergraduate mathematics that is essential to the study of school mathematics? Wedetermined that understanding mathematics at this level requires being able to: 1) solve math-ematical problems foundational to school mathematics; 2) explain the reasoning supporting themathematical concepts; and 3) know what mathematics comes before and after the topic at hand.We used these three ideas to formulate our first three connections. In considering what it meansfor an undergraduate to develop pedagogical content knowledge in a mathematics course, we askedourselves: what do secondary students understand, and how can undergraduates practice helpingthese students with their understanding? We arrived at the importance of undergraduate prospec-tive teachers’ ability to: 4) look at student work and recognize and articulate what a student doesand does not understand mathematically; and then 5) think of questions that will help guide thatstudent’s understanding. These ideas comprise our last two connections.Table 1: Five connections to teaching secondary mathematics.
Connection Description Content Knowledge
Undergraduates use course content in applicationsor to answer mathematical questions in the course.2.
Explaining Mathematical Content
Undergraduates justify mathematical proceduresor theorems and use of related mathematicalconcepts.3.
Looking Back / Looking Forward
Undergraduates explain how mathematics topicsare related over a span of K-12 curriculumthrough undergraduate mathematics.4.
School Student Thinking
Undergraduates evaluate the mathematicsunderlying a student’s work and explain whatthat student may understand.5.
Guiding School Students’ Understanding
Undergraduates pose or evaluate guiding questionsto help a hypothetical student understand amathematical concept and explain how thequestions may guide the student’s learning.We note that we do not intend for these five connections to be a direct mapping to the kindsof knowledge described in the MKT framework. Rather, we formulate these connections in thisway to facilitate addressing them in curricular materials used by mathematicians when preparingfuture secondary teachers. Additionally, we do not claim that these five connections are disjoint;we fully expect that some problems and activities will engage undergraduates in more than one ofthese connections.We illustrate these five connections with examples. We created instructional materials thatincorporate these five types of connections into lessons on a selection of topics in undergraduatemathematics courses, and we have field tested these instructional materials in a variety of set-tings. We selected lessons in the undergraduate content areas of abstract algebra, calculus, discretemathematics, and statistics, because these areas are foundational for the mathematics studied insecondary schools. 4 ontent Knowledge
Content Knowledge connections require undergraduates to use content they learn in a course toanswer mathematical questions, where the undergraduate content directly relates to the content ofsecondary school mathematics. The following is a problem from a Statistics lesson on margin oferror:This problem engages undergraduates in using data from a random sample to estimate a popu-lation mean with the primary purpose of enhancing their understanding of the relationship betweensample size and margin of error, concepts they will teach their future students. These types of prob-lems are similar to those most mathematicians normally use in teaching undergraduate courses; weidentify them as Content Knowledge Connections because we have specifically chosen problemsthat focus on concepts that are firmly grounded in secondary content.
Explaining Mathematical Content
Explaining Mathematical Content connections require undergraduates to justify a mathematicalclaim, procedure, formula or theorem. Going beyond content knowledge, this type of connectionenforces deep understanding of content. An example from a lesson on groups of transformationsfor an Abstract Algebra course is:This problem focuses on undergraduates’ understanding of groups and the logical implicationsthat follow from a given algebraic structure. These types of problems involve important communi-cation skills which are particularly valuable for prospective teachers whose jobs will require themto explain mathematical concepts to others. The lessons we developed provide problems and ques-tions as a resource for mathematicians who may not normally require undergraduates to reflect anddiscuss why a mathematical approach is valid. 5 ooking Back/Looking Forward
Looking Back/Looking Forward connections require undergraduates to examine how the mathe-matics topics they are studying relate to topics within the secondary curriculum. The following isa problem from a Calculus lesson on derivatives of inverse functions:These types of questions situate mathematics topics in the context of secondary teaching andlearning. This connection differs from
Content Knowledge connections because, when developing
Content Knowledge problems, the relationship between the college mathematics to school mathe-matics is considered “behind the scenes”; the instructor identifies the mathematics in undergraduatecourses that is important for teaching secondary mathematics and creates problems that specifi-cally address the mathematics without explicitly referencing the connection. When writing
LookingBack/Looking Forward problems, the relationship between the college and secondary mathematicsis made explicit through the context in which the problem is posed. The lessons we developedsupport mathematicians who may not be familiar enough with the secondary curriculum to includesuch
Looking Back/Looking Forward connections on their own.
School Student Thinking
School Student Thinking connections require undergraduates to evaluate the mathematics presentedin a hypothetical student’s work and explain what that student may understand. The following isa problem from a lesson on the Foundations of Divisibility in an introduction to proofs course:Questions like these give prospective teachers opportunities to recognize mathematical under-standing in incomplete or incorrect solutions. Questions addressing
School Student Thinking alsoprompt undergraduates to explain why students may use different, yet correct, approaches to aproblem. The questions can use student thinking to guide undergraduates through unknown math-ematics. These types of questions provide opportunities for undergraduates to use or develop theirmathematical understanding by investigating other students’ perspectives. Our lessons provideexplicit curriculum support for mathematicians by providing examples of common school studenterrors with prompts for undergraduates to consider.
Guiding School Students’ Understanding
Guiding School Students’ Understanding connections require undergraduates to pose or evaluatequestions to help a hypothetical student understand a mathematical concept and to explain how the6osed questions may guide student learning. The following is a problem from a Binomial Theoremlesson in an introduction to proofs course:Posing mathematical questions that probe understanding or that advance understanding is animportant skill for future teachers to learn and practice (Smith and Stein, 2018). Mathematicianswho teach undergraduates may not be familiar enough with the school curriculum to provide suchopportunities without explicit curricular support.
DISCUSSION
Robust preparation of future secondary mathematics teachers requires attention to the acquisitionof mathematical knowledge for teaching. We offer these five connections as a means to tailor thecontent in undergraduate mathematics courses to the work of teaching. Our particular interestis how to incorporate connections into mathematics courses commonly included in a mathematicsmajor for prospective secondary mathematics teachers taught by non-specialists in mathematicseducation research. In turn, this will provide prospective secondary mathematics teachers with mul-tiple opportunities to build aspects of their own mathematical knowledge for teaching throughouttheir undergraduate mathematics studies.The five connections we describe align with Wasserman’s (2018a) delineation of the spectrum ofideas that help make the study of undergraduate mathematics content relevant to future secondaryteachers. Our first three connections, situated primarily in content knowledge, align with thekinds of curriculum decisions most mathematics instructors already make. Focusing on the contentof mathematics, both
Content Knowledge and
Explaining Mathematical Content connections aretypically incorporated into most undergraduate mathematics courses. The difference in our contextis that the content specifically relates to or arises from high school content in some way. Becausemany mathematicians consider ways to motivate the content of their course by relating the contentto previously studied mathematics concepts, the
Looking Back/Looking Forward connection mayenhance ways that mathematicians typically motivate concepts by providing more support forlinking to undergraduates’ high school experiences in mathematics or to foundational high schooltopics prospective secondary mathematics teachers will have to teach.Our articulation of the two pedagogically-focused connections (
Student Thinking and
GuidingStudent Understanding ) offers a novel perspective on teacher preparation in content courses. Theseconnections are complementary to, but distinct from, Wasserman’s (2018a)
Classroom Teaching activities. They are also distinct from the kinds of tasks that mathematics instructors wouldusually set for their undergraduate students. It is a teacher’s role to evaluate student work and7ose questions to assess and advance their students’ understanding. This is challenging work, andprospective teachers need exposure to these practices in their preparation. Problems incorporating
Student Thinking and
Guiding Student Understanding connections differ from traditional problemsfound in undergraduate mathematics courses. Nonetheless, they provide a novel way to furtherexplore undergraduates’ conceptual understanding via the ways in which undergraduates chooseto guide student understanding or pose questions to probe student thinking. These connections,in particular, extend the MET II Report’s (CBMS, 2012) call to make connections between themathematics prospective teachers are learning and the school mathematics they will be teaching byemphasizing how the mathematics in their undergraduate courses can help them understand theirfuture students’ mathematical work.These five connections extend the literature on mathematical knowledge for teaching and offerstructure for how we can support prospective teachers in content courses. We recommend the inclu-sion of these five connections across undergraduate courses as a means for mathematicians to makeexplicit connections between the content of undergraduate mathematics courses and the contentof secondary mathematics. We also envision that the five connections may promote broad use ofembedding applications of teaching secondary school mathematics into mainstream mathematicscourses for secondary mathematics teachers.
Additional Information
This material is based upon work supported by the National Science Foundation under Award No.DUE-1726624.
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