2013 Conference Proceedings - University of Nevada, Las Vegas

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Table 3Pictorial Representation (Growth Patterns)Item 16a 16b OverallScore Pre Post Pre Post Pre Post0 17% 17% 17%1 67% 86% 67% 57% 67% 71%2 17% 14% 17% 43% 17% 29%Table 4Table FormatItem 16c 16d 16e OverallScore Pre Post Pre Post Pre Post Pre Post0 17% 14% 6% 5%1 83% 86% 17% 33% 29%2 17% 14% 83% 86% 83% 100% 61% 67%The most improvement occurred in the area of verbal representations which may be due inpart to the fact that the participants were asked to not only solve a significant number of verbalproblems but also create a number of verbal scenarios to represent functions, disproportionate tothe other representations.The participants’ responses to the prompt “What is a function?” revealed that five of theseven participants possessed a process understanding of function while the other two indicated adependence and a correspondence understanding of function. Examples of the responses codedas process understanding are as follows: A function is an equation that modifies an input value to produce an output valuesuch that no two outputs are derived from the same input. A function is an equation or system that produces only one output value for eachinput value.The participant’s response that was coded as correspondence stated, “There is only onecorrespondence from each element in the domain to the range.” The participant’s response thatwas coded as dependence stated, “A function is a way of representing an equation where theoutput depends on the input.”DiscussionCooney et al. (2010) not only indicate that functions are challenging for students to learn andteachers to teach but go on to say that “students in grades 9-12 need to understand function wellif they are to succeed in courses that build on quantitative thinking and relationships” (p.1).Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 29
They further state that “the importance of understanding function and the challenge ofunderstanding them well make them essential for teachers of mathematics in grades 9-12 tounderstand extremely well themselves” (p. 1). In their book, Cooney et al. (2010) identifies fivebig ideas around which essential understanding of functions is developed. This study focusedprimarily on one of these big ideas, the notion of multiple representations of function. “Functionscan be represented in multiple ways, including algebraic (symbolic), graphical, verbal, andtabular representations. Links among these different representations are important to studyingrelationships and change” (Cooney, et al., 2010).The results of this study reveal that this group of secondary mathematics teacher candidateshas a limited understanding of the concept of function. Their ability to identify whether or not arepresentation was a function and provide a satisfactory explanation was not as strong as whatmight have been hoped for given they are near the end of their teacher preparation program.When presented with a graphical representation, the participants overwhelmingly used thevertical line test to determine if the graph represented a function or not. In line with Clements(2001) findings, the participants in this study also applied the vertical line test to the drawingincluded in one of the verbal representation problems indicating a lack of sophistication in theirunderstanding of function. Further, in their explanations on the pre/post-assessment, if theparticipants provided a reason other than the vertical line test they overwhelmingly failed tomention the “single-valuedness” of functions. They frequently indicated that one element fromthe domain should “produce” or “be aligned with one element from the range” but rarelyindicated that this should be a unique relationship or that for each element of the domain, there isexactly one element of the range.Although this group of teacher candidates has successfully completed a significant number ofcollege-level mathematics courses, their conception and understanding of function is limited andin some cases incorrect. Likewise, the limited nature of their explanations for why a particularrepresentation was or was not a function revealed that they may view mathematics as primarilyabout computation and rules. These findings align with Wilson’s (1994) suggestion thatalthough it is important for secondary mathematics teacher candidates to consider advancedmathematics topics, it may be more important that they are provided ample opportunity to reflecton their own conceptions and understandings while learning (or re-learning) mathematics theywill have to teach.Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 30
Page 1 and 2: ….where the Mathematicscomes swee
Page 3 and 4: THANK YOU TO OUR REVIEWERSKeith Ado
Page 5 and 6: Table of ContentsPreservice Teacher
Page 7 and 8: Support for Students Learning Mathe
Page 9 and 10: own problem solving, which is criti
Page 11 and 12: to get started and persistence. Tea
Page 13 and 14: Posamentier, A. S., Smith, B. S., &
Page 15 and 16: conceptual understanding, applicati
Page 17 and 18: Table 1Identified Mathematical Prac
Page 19 and 20: justify their statements, included
Page 21 and 22: Finally, engagement in MP.6 was ass
Page 23 and 24: PRESERVICE TEACHERS’ EMOTIONAL EN
Page 25 and 26: “experiences that are charged wit
Page 27 and 28: Number of journals containingEmotio
Page 29 and 30: ConclusionsStruggle and frustration
Page 31 and 32: Mathematics Teacher Candidates’ U
Page 33 and 34: function and applied the vertical l
Page 35: semester, about half of the course
Page 39 and 40: C. Laborde (Eds.) International Han
Page 41 and 42: (SCK), or knowledge of mathematics
Page 43 and 44: level of difficulty for each partic
Page 45 and 46: MKT Measures ScoresMathematics in G
Page 47 and 48: deep rooted belief in a single way
Page 49 and 50: THE INTERVIEW PROJECTAngel Rowe Abn
Page 51 and 52: involving addition and subtraction:
Page 53 and 54: 6+7 4+9=6+(6+1) Substitution =4+(10
Page 55 and 56: We strongly believe that this inter
Page 57 and 58: AN INNOVATIVE APPROACH FOR SUPPORTI
Page 59 and 60: Practice throughout the investigati
Page 61 and 62: are expected to pursue. Teacher not
Page 63 and 64: students to organize their reports
Page 65 and 66: Slovin, H., Venenciano, L., Ishihar
Page 67 and 68: The research presented in this pape
Page 69 and 70: students’ confidence. Because bel
Page 71 and 72: triangulation necessitated examinat
Page 73 and 74: their ability to teach the mathemat
Page 75 and 76: SPATIAL REASONING IN UNDERGRADUATE
Page 77 and 78: journal prompt would be given as a
Page 79 and 80: given to the 33 students on the MPI
Page 81 and 82: to advance our way of life, then sp
Page 83 and 84: STUDENT CONCEPTIONS OF “BEST” S
Page 85 and 86: 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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Performance, 33, 1410-1419.Cohen Ka
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Moyer, 2007). At his or her own pac
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logged by the system and then retri
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curriculum. The nature of the onlin
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Cox, G., Carr, T., & Hall, M. (2004
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curriculum locally, within individu
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Popularity tallied whether or not a
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Amazingly, despite there being a fe
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ReferencesBlack, M. (1962). Models
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connections are connections or rela
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Table 1Instructional TasksSquareTab
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t-charts made it easier for student
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Figure 2. A Display of Student Stra
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ReferencesAnderson, J. R., Greeno,
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their parents in phenotype (observa
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student learning calls for differen
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Figure 2. (A) P3. (B). Extension of
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the usual phenotypic assessments an
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teachers is the discrepancy between
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expected to learn and the inquiry a
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his partner about his observations,
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hard for some children? The nature
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Lakoff and Nunez: specifically, tha
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Figure 5: Average hand trajectories
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Figure 6: Distributions of maximum
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ReferencesAnderson, J. R. (2005). H
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Social system perspectives view the
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urged students to think of some way
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Figure 1: The Discourse Patterns Du
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Figure 3 blow illustrates the devel
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