2013 Conference Proceedings - University of Nevada, Las Vegas

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DEVELOPING DISCOURSE THAT PROMOTES REASONING AND PROOF:A CASE STUDY OF A CHINESE EXAMPLARY LESSONLianfang LuUniversity Of Arkansas At Little RockLxlu@Ualr.EduThomas E. RicksLouisiana State UniversityTomricks@Lsu.EduAs part of a larger study that investigated mathematics teaching reform in China (Lu, 2011), thisstudy focused on a high-quality model lesson that represented the recommended instructionalpractices in current Chinese mathematics education reform. We focused our analysis on thedesign of the lesson, the unfolding of discourse, and the development of students’ mathematicalreasoning and proof in the lesson.Theoretical perspectivesReasoning and proof are regarded as fundamental aspects in mathematics study for all grades(NCTM, 2000); however, research has shown that students have various difficulties when theywere engaged in reasoning and proofs. For example, students tend to rely on the authority fortruth; lack of understanding of proof; misuse empirical examples as arguments for proofs andlack of strategies of proofs (Harel& Sowder, 2007, Stylianou, Blanton, &Knuth, 2009;Thompson, Senk, Johnson, 2012).Mathematics educators have noticed the role of discourse in helping students overcome thesedifficulties. A significant challenge teachers face is how to develop discourse that promotesrobust mathematical understanding and abilities of reasoning (Stein, et al., 2008). Stein et al.(2000) highlighted that different tasks may engage students in different levels of mathematicalreasoning. They pointed out that tasks in discourse with higher-level cognitive demands, such asdoing mathematics activities provide opportunities to engage students in high-level thinking.Research also indicated that heuristic questions and suggestions asking for explanation andextension deepen students’ reasoning (Goldin, 1997). For instance, Martino and Maher (1999)found that the teacher’s posing of timely questions, while challenging learners to advance theirunderstanding assist students find successful solutions. Yackel and Cobb (1996) identifiedsociomathematical norms that encourage listening and explanations support students to developexplanation, justifications, and mutual understanding. More recent interest has pointed to“inquiry-based” instructional approach to improve reasoning and proof through engagingstudents in problem-solving (Smith, 2012).Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 213
Social system perspectives view the classroom as a learning system in which the teacher andstudents are participants of a mathematics communication network. Social system perspectivesmay add insights for promoting deep and rich mathematical understanding and reasoning indiscourse (Stein, et al., 2008). From social system perspectives, classroom discourse is nottransmission of information but correlated and recursive action among classroom members(Fleener, 2002; Luhmann, 1999; Maturana &Varela, 1987; Sfard, 2008). The generation ofmeaning is not located inside the mind of individuals, but in the discourse itself. Discourse growswith the contributions from individuals’ participation in the discourse, as Sfard highlighted,Discourse rules of the mathematics classroom, rather than being implicitly dictated by theteacher through her own discursive actions, are an evolving product of the teacher’s andstudents’ collaboration efforts. (p. 202)Such a perspective emphasizes a co-growth and a recursive process of developing reasoningand proof. It suggests that successful classroom mathematics discourse is self-generating, selfadapting,and self-maintaining; additionally, both participants and discourse cycles co-adapt andco-evolve. Sfard pointed out the richness and depth of understandings emerged in classroomdiscourse (the network of communications) reflect the quality of the discourse. Higher qualitymathematics instruction (usually conducted by more experienced teachers) manifests richerdiscourse.Data Sources and MethodsThis study investigated how a recognized expert mathematics teacher in urban Chinadesigned and facilitated classroom discourse that promoted meaningful understanding andreasoning in a mathematics lesson focusing on the minimum criteria for determining ifquadrilaterals are congruent (extending Side-Angle-Side and related postulates). The teacher,Mr. Wu (pseudonym) was both recognized as an outstanding mathematics teacher in the localcity and had participated in the development of China’s new national mathematics curriculum.Mr. Wu taught his exemplary model lesson to a random group of ninth graders during a districtlevelteaching development project as about 20 mathematics teachers from the district observed.The lesson lasted about one hour. Afterward, Mr. Wu held a debriefing session with the teachers.Data for this study included our observation and videotaping of the lesson and the debriefingsession, and interviews with Mr. Wu and the director of the district-level teaching developmentproject. Four video cameras were used to catch the teaching and learning from different angles.Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 214
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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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Data collection consisted of tests,
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I've not used children's literature
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could extend this inquiry to high s
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course titled Calculus with Busines
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no mathematical sense and should no
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Adopts the “111” (a term coined
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Specifically, clicking the “Click
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algebraic expression is carried out
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Retrieved from http://secc.sedl.org
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furthermore, each model may result
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After instruction in the course, th
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Table 4The group’s categories and
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you choose three place values, æ 4
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APPENDIXTable 5Description of Combi
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of the presented number. Later, the
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Figure 1: Mean trajectories and MD
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Figure 2: Mean trajectories and MD
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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
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Page 183 and 184: Amazingly, despite there being a fe
Page 185 and 186: ReferencesBlack, M. (1962). Models
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Page 189 and 190: Table 1Instructional TasksSquareTab
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Page 193 and 194: Figure 2. A Display of Student Stra
Page 195 and 196: ReferencesAnderson, J. R., Greeno,
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Page 211 and 212: hard for some children? The nature
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Page 215 and 216: Figure 5: Average hand trajectories
Page 217 and 218: Figure 6: Distributions of maximum
Page 219: ReferencesAnderson, J. R. (2005). H
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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