2013 Conference Proceedings - University of Nevada, Las Vegas

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Related LiteratureSince the time “function” was established as an individualized concept in mathematics it hasplayed an important role throughout the mathematics curriculum. Cooney, Beckmann, and Lloyd(2010) purport that “functions compose a major area of school mathematics that is crucial forstudents to learn but challenging for teachers to teach” (p. 1). They continue by stating:Learners often have a narrow view of functions. On the basis of their frequent use oflinear and quadratic function, students tend to limit the concept of functions to equationsor orderly rules. They frequently overlook many-t-one correspondences or irregularfunctions that could be very useful in describing real-world phenomena (p. 1).Understanding that “the concept of function is central to students’ ability to describerelationships of change between variables, explain parameter changes, and interpret and analyzegraphs” (Clements, 2001, p. 745) is supported by the National Council of Teachers ofMathematics. They advocate in their Principles and Standards for School Mathematics (NCTM2000, p. 296) that instruction across all grade levels should “enable all students to understandpatterns, relations, and functions” (p. 296). Unfortunately, research studies indicate thatalthough, the concept of function is important in mathematics and should be developed overmany years with students, high school and college students have difficulty understandingfunction with any depth and flexibility.Studies conducted in the 70’s and 80’s well documented students’ difficulties withunderstanding function noting, among other things, that their mathematical understanding relatedto function often involved incorrect ideas and that the understanding they had developed wasoften narrow in scope (See Leinhardt, Zaslvasky, & Stein (1990) for a survey of the literature).The reasons cited for these challenges are plentiful and well documented, frequently pointing tothe notion that how students are introduced to functions and the problems they are asked to solverelated to functions are limiting (Buck, 1970; Dreyfus & Eisenberg 1982; Freudenthal,1982;Herscovics, 1982; Kaput, 1987; Lovell, 1971; Orton, 1970; Sierpinska 1992; Tall 1996; Vinner& Dreyfus, 1989). For example, a more recent study conducted by Clements (2001) includedthirty-five high school pre-calculus students. She found that only four could provide a definitionthat was consistent with or similar to the mathematical definition which includes the idea thatevery element in the domain must be mapped to a unique element in the range. Further, studentswho participated in this study seemed to focus primarily on graphical representations of theProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 25
function and applied the vertical line test in order to determine if a graph represented a functionor not.Since research has documented that mathematics students in both high school and collegelevelsettings have limited understandings of function this raises questions about the developedunderstanding of function of our high school mathematics teachers and high school mathematicsteacher candidates. Wilson (1994) examined the understanding of functions of one secondarymathematics teacher candidate and found her initial understandings were primarilycomputational (function machines, point plotting, vertical line test) and were in line with herpredominant view of mathematics as a collection of procedures. Further, Even (1993) found thatthe prospective teachers in his study had a limited conception of function and it influenced theirpedagogical thinking about functions. The implication of his study was that with only a limitedconception of function themselves, these teachers would have no choice but to provide their ownstudents with rules to follow for functions without concern for understanding. While there havebeen studies focused on mathematics teacher candidates understanding of functions, thoseparticularly focused on a multiple representations prospective of functions are not welldocumented.The National Council of Teachers of Mathematics has, for several decades, supported thedevelopment of a rich and flexible understanding of functions that includes student work withfunctions through multiple representations (NCTM, 1989; NCTM, 2000; Cooney, et al., 2010).In light of this emphasis and the adoption of the Common Core State Standards for Mathematics(2010) which includes functions as a primary strand for secondary mathematics students, thisstudy aimed to investigate one group of secondary mathematics teacher candidates’understandings of function. Thus, the research question was as follows:What are secondary mathematics teacher candidates’ understandings of function?MethodologyThis study, involving one class of secondary mathematics teacher candidates (N=7),employed qualitative research methods to examine the research question and analyze the data. Atwenty-three item pre/post-test was developed using items from several sources (see listfollowing the References) and was constructed such that participants were asked to identifywhether what was presented was a function and to explain their reasoning. Functions werepresented on the pre/post-test in multiple representations including, graphical, verbal, pictorial,Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 26
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: Mathematics Teacher Candidates’ U
Page 35 and 36: semester, about half of the course
Page 37 and 38: They further state that “the impo
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
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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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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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