INFUSING PEDAGOGY IN MATHEMATICS CONTENT
IUMPST: The Journal. Vol 2 (Pedagogy), September 2020 [www.k-12prep.math.ttu.edu]ISSN 2165-7874INFUSING PEDAGOGY IN MATHEMATICS CONTENT COURSES FOR FUTUREELEMENTARY TEACHERSAlison S. MarzocchiCalifornia State University, FullertonFullerton, CAamarzocchi@fullerton.eduJoseph DiNapoliMontclair State UniversityMontclair, NJdinapolij@montclair.eduAbstractTo capitalize on limited time for mathematics teacher preparation, we suggest thatmathematics content courses are fruitful venues for infusing elements of effectivemathematics pedagogy. As content courses are typically taken before methods courses,doing so exposes pre-service teachers to mathematics teaching practices earlier intheir teacher preparation programs. In this paper, we share examples of how weprovide pre-service teachers the opportunity to build their mathematics contentknowledge while concurrently building their pedagogical knowledge. We invitemathematics teacher educators to find ways to provide similar opportunities in theirwork with pre-service teachers.IntroductionIn many teacher preparation programs, pre-service teachers learn mathematics content in aseparate setting from learning mathematics pedagogy (Steele and Hillen 2012). Contentinstruction and pedagogical instruction are often taught in different courses, separated bothtemporally and organizationally within teacher education systems. Yet “pedagogical knowledgeis neither discrete nor conceptually separable from the knowledge of the mathematics contentbeing taught” (p. 53). Like Steele and Hillen (2012), we wonder when and how teachers integratetheir knowledge of content and pedagogy if we continue to teach them separately. In this paper,we suggest that mathematics content and mathematics pedagogy should, whenever possible, betaught concurrently. We offer mathematics content courses for pre-service elementary teachers(PSTs) as fruitful settings for infusing elements of effective mathematics pedagogy early in theteacher preparation trajectory. We describe how the Mathematics Teaching Practices put forth bythe National Council of Teachers of Mathematics (NCTM) in Principles to Actions (NCTM2014) undergird our decisions to infuse pedagogy within mathematics content courses and callon our mathematics teacher educator colleagues to find methods by which to do the same.Though numerous textbooks exist (e.g., Beckmann 2018; Billstein, Libeskind, and Lott 2013;Van de Walle, Karp, and Bay-Williams, 2019) which are designed to increase PSTs’ conceptualunderstanding of mathematics, we suggest that mathematics content courses can extend PSTs’learning to also involve problems of practice. That is, rather than only solving the typicalmathematics tasks found in these textbooks, we can look for opportunities for PSTs to solvemathematics embedded in situations of everyday teaching. These types of situations includeassessing students’ mathematical work, using data to communicate to administrators and parents,designing classroom activities around particular mathematics concepts, etc. By includingcontexts such as these, PSTs are not only building their mathematics content knowledge, but alsobuilding their skills and confidence to enter the teaching profession.
A.S.Marzoccni, J.DiNapoli: Infusing Pedagogy in Mathematics Content Courses for Future Elementary TeachersConcurrently Learning Mathematics Content and Effective PedagogyPresently, many current and future teachers are expected to teach mathematics in ways thatdiffer from how they experienced mathematics as learners (Koellner et al. 2007; Thanheiser andJansen 2016). Koellner et al. (2007) caution that shifts to reform-based mathematics instructionwill require “a great deal of learning on the part of teachers” (p. 275). To help facilitate thisteacher learning, Thanheiser et al. (2010) posit a design principle for teacher preparationprograms, namely that classes for PSTs should effectively model teaching for mathematicalunderstanding. Thus, these programs should capitalize on multiple opportunities to expose PSTsto effective mathematics instruction. Though this exposure is likely to occur in mathematicsmethods courses (Amirshokoohi and Wisniewski 2018; Burton, Daane, and Giesen 2008; Steeleand Hillen 2012), we suggest an additional opportunity is available in mathematics contentcourses.One benefit of infusing pedagogy in mathematics content courses is that it introduces PSTs toeffective mathematics instruction earlier in their teacher preparation trajectories. Content coursestypically occur before methods courses in teacher preparation programs, allowing us to capitalizeon limited time for mathematics teacher preparation. In Building Support for Scholarly Practicesin Mathematics Methods (2017) – a volume containing contributions from over 40 mathematicsteacher educators on theoretical approaches to teaching mathematics methods – contributingauthors Harper, Sanchez, and Herbel-Eisenmann (2017) ask, “What can teacher educationprograms do to at least partially resolve the problem of time?” (p. 40). Their first suggestion is toimprove instruction in mathematics content courses. They claim, “If we are going to changeprospective teachers’ understanding and image of what it means to teach mathematicsconceptually, then we have to give them the opportunity to learn mathematics in compatibleways” (p. 40). They suggest that infusing effective mathematics pedagogy in content courses willimprove content knowledge, establish an early vision of effective mathematics instruction, andprovide PSTs with a solid foundation to do advanced work in mathematics methods courses.By linking mathematics content to effective mathematics pedagogy early in the teacherpreparation trajectory, we may be helping PSTs to shift their identities as future teachers ofmathematics. We are providing an earlier opportunity to “break the cycle” of mathematicsanxiety among elementary teachers and positively influence PSTs’ pedagogical contentknowledge, attitude, beliefs, and self-efficacy (Amirshokoohi and Wisniewski 2018). Rather thancontinuing their identities as mathematics learners from their K-12 experience, our mathematicscontent courses provide PSTs with opportunities to explore mathematics content through the lensof teaching. We have referred to this as having PSTs take off their ‘student hat’ and put on their‘teacher hat’ (DiNapoli and Marzocchi 2017). Battey and Franke (2008) provided two casestudies of teachers’ implementation of professional development and found that changes whichoccurred in the classroom were related to teacher identity and situation within communities ofpractice. They describe teaching as “a process of becoming a member in a defined group ofpractitioners with specific skills” (p. 128). Within the setting of a mathematics content course,our PSTs engage in tasks around the practice of teaching, providing opportunities to seethemselves as members of the teaching community.Others have infused mathematics content and pedagogy in settings such as a problemsolving-focused professional development program (Koellner et al. 2007), a content-focusedmathematics methods course around functions (Steele and Hillen 2012), and an elementarymathematics methods course (Amirshokoohi and Wisniewski 2018; Burton, Daane, and Giesen2008). In their problem solving-focused professional development program, Koellner et al.’s2
Issues in the Undergraduate Mathematics Preparation of School TeachersISSN 2165-7874(2007) participants were expected to draw on their specialized content knowledge to giveexplanations, make use of representations, evaluate ideas, identify misconceptions, makeconnections, and use language explicitly. PSTs in the content-focused mathematics methodscourse described by Steele and Hillen (2012) were given opportunities to move back and forthbetween positions of learner and teacher while analyzing student work, watching video, andconsidering multiple perspectives. Within the mathematics methods course described byAmirshokoohi and Wisniewski (2018), PSTs learned effective teaching strategies throughcontent-focused tasks structured in a student-centered lesson plan model.The work described above primarily focuses on infusing pedagogy within mathematicsmethods courses. As mentioned, we believe that exposure to effective mathematics pedagogy canoccur earlier in the teacher education trajectory, within the context of mathematics contentcourses for future elementary teachers. We extend the work of our colleagues in mathematicsmethods courses (Amirshokoohi and Wisniewski 2018; Burton, Daane, and Giesen 2008; Steeleand Hillen 2012) by providing classroom examples of infusing of pedagogy within a contentcourse guided by the Mathematics Teaching Practices put forth by NCTM in Principles toActions (NCTM 2014).ContextOur work is situated in the context of two different public universities on each coast of theUnited States. In both universities, the mathematics content courses in which we aim to infuseelements of effective mathematics pedagogy are intended for PSTs seeking certification to teachearly childhood and elementary school (P-3, K-6). As undergraduate students, PSTs complete asequence of two mathematics content courses prior to enrolling in mathematics methods coursesor participating in any field experiences. These mathematics content courses are designedsimilarly and taught by both authors, separately, at our respective universities. At the east coastuniversity, the first course focuses on counting and cardinality, number and operations, andoperations and algebraic thinking while the second course focuses on number and operations,ratios and proportional reasoning, geometry, and measurement. At the west coast university, thefirst course focuses on number and operations while the second course focuses on geometry,probability, and statistics.Throughout these two courses, PSTs are given opportunities to explore mathematics contentas participants in classroom activities that model and make explicit aspects of effectivestandards-based mathematics instruction. Course instruction is student-centered and inquirybased, and provides opportunities to individually and collaboratively grapple with mathematicalideas, communicate and share ideas with peers, and interact with different representations ofmathematical objects, especially representations that are concrete and visual. In these ways, ourmathematics content courses are designed to leverage individual and collective engagement withchallenging mathematics to develop conceptual understanding (Thanheiser and Jansen 2016).The course design emphasizes a social constructivist approach to learning by offering PSTsopportunities to make personal meaning of mathematical content and pedagogy throughindependent exploration, as well as through interaction and communication with peers and theinstructor. Though the entire course is designed to enact elements of effective mathematicspedagogy, in what follows we provide specific examples of activities that engage PSTs in the actof deliberate problem solving as teachers, as guided by Principles to Actions (NCTM 2014).3
A.S.Marzoccni, J.DiNapoli: Infusing Pedagogy in Mathematics Content Courses for Future Elementary TeachersExamples of Infusing PedagogyNCTM’s Principles to Actions (NCTM 2014) recommends actions for educationstakeholders to ensure high quality mathematics education. Contained in the document are eightstrongly recommended and research-informed Mathematics Teaching Practices (see Figure 1) toconsistently implement in every mathematics lesson. To provide readers with a sense of how weinfuse mathematics pedagogy within a content course, we focus on two Mathematics TeachingPractices and provide a classroom example for each:1. Support productive struggle in learning mathematics2. Elicit and use evidence of student thinkingThe provided examples serve to demonstrate the ways in which we strive to enactmathematics teaching practices in our own courses. With the motivation to infuse pedagogy inmathematics content courses, we encourage mathematics teacher educators to continue to findways to do the same.Figure 1. Eight Mathematics Teaching Practices from Principles to Actions (NCTM).Support productive struggle in learning mathematicsNCTM (2014) suggests providing opportunities for, and supporting student engagement with,productive struggle in the context of learning mathematics. Although PSTs in our contentcourses routinely engage with tasks that elicit productive struggle, we also seek to help PSTslearn about the nature of productive struggle: what it looks like, what it’s about, and why it’sproductive. Akin to Thanheiser and Jansen’s (2016) efforts to help PSTs share their exploratory4
Issues in the Undergraduate Mathematics Preparation of School TeachersISSN 2165-7874mathematical thinking, we consistently provide PSTs opportunities to describe how they arestruggling, the mathematical idea(s) that constitutes the object of their struggle, and how thisstruggle supports their learning. By making productive struggle explicit in their own learning, wehope the PSTs will recognize, support, and nurture productive struggle in their future students.An example of this occurred in a division of fractions task titled What Remains?, in whichPSTs needed to interpret the meaning of a remainder for the first time. The What Remains? taskwas inspired by Beckmann’s (2018) What to Do With the Remainder? class activity, located inthe Division and Fractions and Division with Remainder lesson, within the Division unit.Although Beckmann’s (2018) activity promotes conceptual understanding of fraction division,we extend this by asking pairs of PSTs to work collaboratively to solve a story problem bothconceptually and procedurally and to additionally participate in an online discussion forum. Theforum prompted PSTs to reflect on the ways in which they may have struggled with the problem.Sample PSTs’ work accompanied by excerpts of their reflections can be seen in Figures 2 and 3,respectively.The What Remains? task and reflection exercise was beneficial for PSTs because they couldmake clear connections between their problem-solving struggles and their mathematics learning(NCTM 2014). As illustrated in Figures 2 and 3, most PSTs encountered struggles reconcilingthe leftover 2/4 yards in the context of the problem, especially when juxtaposed with theprocedural answer of 5 2/3 badges. This pair of students specifically struggled with interpretingthe ‘leftover’ pieces in their conceptual solution as a portion of one badge. They conceptuallyarrived at an incorrect answer of 5 2/4 badges, and spent productive time thinking about whytheir conceptual and procedural answers were different, eventually relying on the repeatedsubtraction meaning of division to help them make meaning of the ‘leftover’ 2/4 yards. PSTscould read their peers’ articulation of their similar productive struggles with this problem in thediscussion forum, which helped them realize that including opportunities for struggle is a normaland necessary pedagogical practice for all mathematics teachers.Figure 2. Sample PSTs’ work on the What Remains? division of fractions task5
A.S.Marzoccni, J.DiNapoli: Infusing Pedagogy in Mathematics Content Courses for Future Elementary TeachersFigure 3. Sample PSTs’ productive struggle reflection for the What Remains? taskAs instructors, we routinely use these productive struggle reflections as catalysts fordiscussions about the mathematical ideas that were not immediately apparent to PSTs – such asinterpreting a remainder as a portion of the divisor – and how spending effort to wrestle withthose ideas can pay dividends in meaningful learning. In these ways, PSTs’ engagement in theselearning opportunities helps integrate their knowledge of mathematics content and the effectivepedagogy that helps develop it (Steele and Hillen 2012). In our content courses, PSTs seemoments of struggle as learning opportunities which guide our instruction, a practice we hopethey will enact in their future classrooms.Elicit and use evidence of student thinkingAccording to NCTM (2014), effective mathematics teaching involves using evidence ofstudent thinking to guide instruction. Busi and Jacobbe (2018), who quantitatively investigatedthe benefits for PSTs in courses that used student work samples as compared to courses that didnot, found positive shifts in beliefs about how mathematics should be taught. Though wefrequently make use of student data ourselves – by responding and adapting to PSTs’ currentunderstandings of the math content – we additionally strive to provide PSTs with the opportunityto enact this practice. Like other mathematics education instructors (e.g. Steele and Hillen 2012),we provide our PSTs with tasks and assessment items that involve interpreting and critiquingstudent work.As an example, PSTs were given a summative group assessment on probability calledProbability Quiz. The Probability Quiz assessment was inspired by Beckmann’s (2018) HowMany Keys Are There? class activity, located in the Counting the Number of Outcomes lesson,within the Probability unit. We extend these types of probability tasks by presenting PSTs with afictional elementary student’s probability quiz. PSTs were asked to assess the student’smathematical understanding of probability. Figure 4 shows an item from the fictional student’squiz, on which the fictional student is asked to determine the number of possible seatingarrangements in a classroom with 16 desks and 16 students, and to find the probability of astudent being seated in the front row. The fictional solution was intentionally designed to displaya variety of understandings and misconceptions around the content. Throughout the semester, a6
Issues in the Undergraduate Mathematics Preparation of School TeachersISSN 2165-7874norm is established that mathematics teachers must seek to find understanding within incorrectstudent solutions. PSTs are made aware that just because a student’s answer is wrong does notmean the student understands nothing about the concept, and that effective pedagogy is findingevidence of understanding and building from there. Figure 5 shows a sample group PSTassessment of what the fictional student might understand or misunderstand around the content.This group of PSTs correctly identified the fictional student’s understandings in how to set upthe problems but misunderstandings in which numbers to use.A task like Probability Quiz not only serves to build PSTs’ own mathematics contentknowledge but also provides them with the opportunity to enact NCTM’s (2014)recommendation of assessing students’ mathematical understanding. As mathematics teachereducators, we are able to enact this practice ourselves on a regular basis. Our work in infusingpedagogy in mathematics content courses extends this practice to our PSTs by providing them anearly opportunity to practice the complex skill of interpreting student work.Figure 4. An item from Probability Quiz: a fictional student’s probability quiz.7
A.S.Marzoccni, J.DiNapoli: Infusing Pedagogy in Mathematics Content Courses for Future Elementary TeachersFigure 5. A sample pre-service teacher assessment of a fictional student’s probability quiz.ConclusionOur work suggests that there are opportunities to infuse elements of effective mathematicspedagogy within mathematics content courses for future elementary teachers that extends beyondproblems included in mathematics content textbooks. Above we shared select examples of whatthis looks like in our classrooms. We encourage mathematics teacher educators to be mindful ofcontinued opportunities to infuse pedagogy in mathematics content courses. Doing so extendsthe work of researchers who have infused pedagogy within methods courses (Amirshokoohi and8
Issues in the Undergraduate Mathematics Preparation of School TeachersISSN 2165-7874Wisniewski 2018; Burton, Daane, and Giesen 2008; Steele and Hillen 2012). By infusingpedagogy in content courses, we are providing PSTs with additional and earlier opportunities toengage in the Mathematics Teaching Practices put forth by NCTM (2014). Infusing mathematicscontent and pedagogy within the same course is an improvement from typical models (whereinthey are taught separately), as the process of learning to teach mathematics is “less additive (e.g.,learn the content, then learn to teach it) and more iterative” (Steele and Hillen 2012, p. 53-54).An added benefit lies in providing PSTs with an earlier opportunity to shift their identities fromthat of a student to that of a teacher. By exposing PSTs to effective mathematics instructionearlier and more frequently, we hope they will ultimately implement effective planning andinstruction in their future classrooms, thus, benefitting all students (Amirshokoohi andWisniewski 2018; Thanheiser et al. 2010).ReferencesAmirshokoohi, Aidin, and Daniel P. Wisniewski. 2018. "Constructing Understanding in aMathematics Methods Course." Teaching Children Mathematics 24, no. 7 (May): 0442Battey, Dan, and Megan L. Franke. 2008. “Transforming Identities: Understanding Teachersacross Professional Development and Classroom Practice.” Teacher Education Quarterly 35,no. 3 (Summer): 127-149.Beckmann, Sybilla. 2018. Mathematics for Elementary Teachers with Activities. (5th ed).Boston, MA: Pearson Education.Billstein, Rick, Shlomo Libeskind, and Johnny Lott. 2013. A Problem Solving Approach toMathematics for Elementary School Teachers. (11th ed.). Boston, MA: Pearson Education.Busi, Rich, and Tim Jacobbe. 2018. "The Impact of Analyzing Student Work on PreserviceTeachers' Content Knowledge and Beliefs about Effective Mathematics Teaching." Issues inthe Undergraduate Mathematics Preparation of School Teachers 1, (August): 1-20.Burton, Megan, C. J. Daane, and Judy Giesen. 2008. "Infusing Mathematics Content into aMethods Course: Impacting Content Knowledge for Teaching." Issues in the UndergraduateMathematics Preparation of School Teachers 1, (May): 1-12.DiNapoli, J. & Marzocchi, A. S. (2017). Productive struggle: What we can learn from workingwith pre-service teachers. The ComMuniCator, 41(4), 10-13.Harper, Frances K., Wendy B. Sanchez, and Beth Herbel-Eisenmann. 2017. “Doing MathematicsAcross Languages: Exploring Possibilities for Supporting Emergent Bilinguals’Mathematical Communication and Engagement.” In Building Support for ScholarlyPractices in Mathematics Methods, edited by Signe E. Kastberg, Andrew M. Tyminski,Alyson E. Lischka, and Wendy B. Sanchez, 263-278. Charlotte, NC: Information AgePublishing, Inc.Koellner, Karen, Jennifer Jacobs, Hilda Borko, Craig Schneider, Mary E. Pittman, Eric Eiteljorg,Kim Bunning, and Jeffrey Frykholm. 2007. “The Problem-Solving Cycle: A Model toSupport the Development of Teachers’ Professional Knowledge.” Mathematical Thinkingand Learning 9, no. 3 (December): 273-303. https://doi.org/10.1080/10986060701360944National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: EnsuringMathematical Success for All. Reston, VA: NCTM.Steele, Michael D., and Amy F. Hillen. 2012. “The Content-Focused Methods Course: A Modelfor Integrating Pedagogy and Mathematics Content.” Mathematics Teacher Educator 1, no. 1(September): 53-70.9
A.S.Marzoccni, J.DiNapoli: Infusing Pedagogy in Mathematics Content Courses for Future Elementary TeachersThanheiser, Eva, C. A. Browning, M. Moss, T. Watanabe, and Gina Garza-Kling. 2010."Developing Mathematical Content Knowledge for Teaching Elementary SchoolMathematics." Issues in the Undergraduate Mathematics Preparation of School Teachers 1,(December): 1-13.Thanheiser, Eva, and Amanda Jansen. 2016. “Inviting Prospective Teachers to Share RoughDraft Mathematical Thinking.” Mathematics Teacher Educator 4, no. 2 (March): 145Van de Walle, John A., Karen S. Karp, and Jennifer M. Bay-Williams. 2019. Elementary andMiddle School Mathematics: Teaching Developmentally. (10th ed.). Upper Saddle River, NJ:Pearson Education.10
The work described above primarily focuses on infusing pedagogy within mathematics methods courses. As mentioned, we believe that exposure to effective mathematics pedagogy can occur earlier in the teacher education trajectory, within the context of mathematics
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CTET Mathematics include two important sections, CONTENT & PEDAGOGY and 50% questions are based on Pedagogy. The purpose of this eBook is to provide you quick revision notes on Mathematics Pedagogy. We have also provided questions from previous year’s exams to help yo
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National Institute for Japanese Language and Linguistics (NINJAL 1968) and Ebata (2013) in addition to data recorded in the field2. Section 2 presents an introduction to Owari dialect of Japanese and coalescence. I examine the Owari data in further depth and point out problems forced by synchronic analysis of coalescence. I examine simple and compound nouns as well as adjectival and verbal .
