2013 Conference Proceedings - University of Nevada, Las Vegas

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ROUTINES OF PRACTICE FOR SUPPORTING MATHEMATICAL CONNECTIONS:EARLY ALGEBRA CONTEXTJessie Chitsanzo StoreAlma Collegestorejc@alma.eduTeaching mathematics as a connected subject is very important. In fact, the Common Core StateStandards for Mathematics are meant to serve this purpose. However, several studies show thatthe intended curriculum often differs from the enacted curriculum. To realize the goal ofteaching mathematics as a connected subject then requires studies of pedagogical practices thatconnect different mathematical domains. This study discusses how elementary school teacherssupported students’ activities in connecting mathematical ideas within and across differentcontexts and how this practice supported students’ algebraic reasoning.Historically algebra has been viewed as manipulation of symbols and equation solving;therefore, algebra has normally been identified with symbolic thinking (Smith 2008). Earlyalgebra, on the other hand, views algebra as a human activity that involves doing, thinking, andtalking about mathematical ideas (Kaput, 2008). National Council for Teachers of Mathematics,(2000) defined early algebra as mathematical thinking that “emphasizes relationships amongquantities, including functions, ways of representing mathematical relationships, and the analysisof change” (p. 37). In the classroom, early algebra activities include expressing explicitgeneralizations as a description of systematic variations or relationships in pattern findingactivities (Mason, 2008).Algebra is identified as a gatekeeper for students’ future success in school (NCTM, 2000).Early algebra is included in elementary school curricular to enhance students’ algebraicreasoning and help them to be better equipped to succeed in formal algebra classes (Rivera,2006). Furthermore, early algebra is a tool for developing children's mathematical proficiencybecause it promotes conceptual understanding, procedural fluency, and positions students wellfor higher-level mathematics (Kaput, 1999; Blanton & Kaput, 2011; Carraher, Martinez, &Schliemann, 2008). Not only is developing students’ proficiency in early algebra a necessity fortheir current success, it is also necessary for their future success. However, while these potentialbenefits of early algebra are widely reported in literature, there is scarcity of research onclassroom activities that show how teachers may realize these benefits (Store, 2012).Emphasis on early algebra has simultaneously grown with emphasis on making connectionswhen teaching mathematics. Drawing on Hiebert and Carpenter (1992), mathematicalProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 179
connections are connections or relationships of mathematical ideas and representations fromtasks within and across mathematical domains and curricular areas. Moreover, makingconnections is the core of mathematical understanding. As Hiebert and Carpenter explained:The mathematics is understood if its mental representation is part of a network ofrepresentations. The degree of understanding is determined by the number andstrength of its connections. A mathematical idea, procedure, or fact is understoodthoroughly if it is linked to existing networks with stronger or more numerousconnections (p. 67).While potential benefits of early algebra and mathematical connections are widely reportedin literature, there is scarcity of research on classroom activities that show how teachers mayrealize these benefits and on how early algebra is a tool for mathematical connections. Thepurpose of this study is to contribute to research on creating classroom contexts for supportingstudents’ development of mathematical reasoning as recommended by Proulx and Berdnaz(2009). This paper addresses this research gap by reporting how elementary school teachersmade mathematical connections in their early algebra classes, thereby supporting students’beginning understanding of algebra, and understanding of other mathematical domains.Theoretical PerspectiveGreeno’s (1998) situative learning perspective informs this study. This perspective viewslearning as individuals’ construction of knowledge as they participate in communities ofpractice. Individuals’ constructed knowledge is a tool for and a product of participation incommunities. Since “individual mental structures certainly change as part of this learning”(Sawyer & Greeno, 2009; p.364), knowledge is transferrable from one community of practice toanother. That is, students’ impoverished understanding of algebraic ideas may reflect aspects ofthe communities in which they participated. “Situative approaches provide analyses focused oncoordination of actions of individuals with each other and with material and informationalsystems” (Anderson, Greeno, Reder & Simon, 2000; p.12). A more specific methodologicalimplication from this perspective is a focus on constraints and affordances of activity systems(Greeno, 2003). Affordances are aspects of an activity system that participants may use to reachtheir goals. Constraints structure the interaction between participants of a community andaffordances. Based on this framework, the situative research question for the current study is:What are the affordances and constraints for supporting understanding of mathematicalProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 180
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….where the Mathematicscomes swee
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THANK YOU TO OUR REVIEWERSKeith Ado
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Table of ContentsPreservice Teacher
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Support for Students Learning Mathe
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own problem solving, which is criti
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to get started and persistence. Tea
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Posamentier, A. S., Smith, B. S., &
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conceptual understanding, applicati
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Table 1Identified Mathematical Prac
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justify their statements, included
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Finally, engagement in MP.6 was ass
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PRESERVICE TEACHERS’ EMOTIONAL EN
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“experiences that are charged wit
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Number of journals containingEmotio
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ConclusionsStruggle and frustration
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Mathematics Teacher Candidates’ U
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function and applied the vertical l
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semester, about half of the course
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They further state that “the impo
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C. Laborde (Eds.) International Han
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(SCK), or knowledge of mathematics
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level of difficulty for each partic
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MKT Measures ScoresMathematics in G
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deep rooted belief in a single way
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THE INTERVIEW PROJECTAngel Rowe Abn
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involving addition and subtraction:
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6+7 4+9=6+(6+1) Substitution =4+(10
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We strongly believe that this inter
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AN INNOVATIVE APPROACH FOR SUPPORTI
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Practice throughout the investigati
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are expected to pursue. Teacher not
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students to organize their reports
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Slovin, H., Venenciano, L., Ishihar
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The research presented in this pape
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students’ confidence. Because bel
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triangulation necessitated examinat
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their ability to teach the mathemat
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SPATIAL REASONING IN UNDERGRADUATE
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journal prompt would be given as a
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given to the 33 students on the MPI
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to advance our way of life, then sp
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STUDENT CONCEPTIONS OF “BEST” S
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students are likely to interact wit
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opinion of the student body. This q
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At the highest level of reasoning a
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APPENDIXTo use two decks of cards t
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isolated and often occur in tandem
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with the CCSSM. Teachers read and d
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teachers’ role-play of SFMP #4. A
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Durkin, D. (1978-1979). What classr
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as well as the alignment between th
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Table 2Number of teachers per grade
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Table 4Classification Categories fo
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field so that research on the initi
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dynamic approach to learning conten
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Kindergarten Lesson FormatHow May W
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team’s goals? As much as possible
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6. While preparing the lesson, teac
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active learning and collective part
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classroom. “I would like to know
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I had never been brave enough to tr
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THE PATH OF REFORM IN SECONDARY MAT
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Our collaboration model was formed
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internal evaluator) were analyzed.
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DiscussionOn part I of the survey t
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whole department of secondary mathe
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discussion. Many texts include wild
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Page 139 and 140: could extend this inquiry to high s
Page 141 and 142: course titled Calculus with Busines
Page 143 and 144: no mathematical sense and should no
Page 145 and 146: Adopts the “111” (a term coined
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Page 149 and 150: algebraic expression is carried out
Page 151 and 152: Retrieved from http://secc.sedl.org
Page 153 and 154: furthermore, each model may result
Page 155 and 156: After instruction in the course, th
Page 157 and 158: Table 4The group’s categories and
Page 159 and 160: you choose three place values, æ 4
Page 161 and 162: APPENDIXTable 5Description of Combi
Page 163 and 164: of the presented number. Later, the
Page 165 and 166: Figure 1: Mean trajectories and MD
Page 167 and 168: Figure 2: Mean trajectories and MD
Page 169 and 170: Performance, 33, 1410-1419.Cohen Ka
Page 171 and 172: Moyer, 2007). At his or her own pac
Page 173 and 174: logged by the system and then retri
Page 175 and 176: curriculum. The nature of the onlin
Page 177 and 178: Cox, G., Carr, T., & Hall, M. (2004
Page 179 and 180: curriculum locally, within individu
Page 181 and 182: Popularity tallied whether or not a
Page 183 and 184: Amazingly, despite there being a fe
Page 185: ReferencesBlack, M. (1962). Models
Page 189 and 190: Table 1Instructional TasksSquareTab
Page 191 and 192: t-charts made it easier for student
Page 193 and 194: Figure 2. A Display of Student Stra
Page 195 and 196: ReferencesAnderson, J. R., Greeno,
Page 197 and 198: their parents in phenotype (observa
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Page 201 and 202: Figure 2. (A) P3. (B). Extension of
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Page 207 and 208: expected to learn and the inquiry a
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Page 211 and 212: hard for some children? The nature
Page 213 and 214: Lakoff and Nunez: specifically, tha
Page 215 and 216: Figure 5: Average hand trajectories
Page 217 and 218: Figure 6: Distributions of maximum
Page 219 and 220: ReferencesAnderson, J. R. (2005). H
Page 221 and 222: Social system perspectives view the
Page 223 and 224: urged students to think of some way
Page 225 and 226: Figure 1: The Discourse Patterns Du
Page 227 and 228: Figure 3 blow illustrates the devel
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2013 Conference Proceedings - University of Nevada, Las Vegas

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