2013 Conference Proceedings - University of Nevada, Las Vegas

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were to work with this child again), and to discuss any on-the-spot instructional decisions theymade while working with the child.Field experiences are repeatedly identified as the most significant part of teacher preparationprograms (McIntyre, Byrd, & Fox, 1996; Mewborn, 2000). However, more field experiences donot always lead to productive growth for PSTs (McIntyre, et. al., 1996). Experiences in the fieldcan simply be used to socialize PSTs into traditional ways of teaching mathematics, and PSTsrarely have opportunities to reflect critically about these experiences (Mewborn, 2000). Typicalfield experiences take place in a classroom full of students. In this setting, the focus tends to beon management concerns, and PSTs do not get to focus on mathematical content and makingsense of children’s mathematics (Mewborn, 1999). Our hope is that having them work with andreflect on one or two children’s mathematics, they will be more ready to focus on what studentsknow and how they learn while they are engaging in their field experiences each semester.Literature ReviewAs the PSTs are engaged in an interview with one or two students, we are aware that this isone of their first experiences in listening to and responding to children in mathematicallyproductive ways. “Clearly, the act of unpacking learners’ mathematics requires listening tostudents” (D’Ambrosio, 2004, p. 139). Davis (1996) suggests that while you can’t observelistening occur, you can infer how a teacher is listening through how they respond to students.You can also infer how a teacher is listening by what they are listening for and what they chooseto ignore. Questioning is one way that teachers respond to students. Questions are instructionaldecisions that can often occur on the spot or some may be planned in advanced. Questions can becategorized in three different ways: probing, prodding, and prompting. Probing involvesquestions to determine what or how a student is thinking, prodding questions are intended tokeep a student acting mathematically, and prompting questions attempt to elicit a specificresponse or strategy to a task (Abney, 2007).We also encourage the PSTs to construct a model of their child’s mathematics, where theycan both describe and analyze the child’s mathematics, we have found it helpful to provide themwith a conceptual framework of children’s whole number development. We discuss theCognitively Guided Instruction (CGI) framework for classifying story problems, countingstrategies, or other strategies. The CGI researchers identified several problem types through theirinteractions with children. We expose our PSTs to four basic structures for story problemsProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 43
involving addition and subtraction: join, separate, part-part-whole, and compare. The join andseparate problems involve an action and can have a result unknown, a change unknown or aninitial unknown. The part-part-whole and compare problems have no action, which tends tomake those more difficult for children to solve (Carpenter et. al, 1999).There are additional frameworks from the work of Steffe and von Glasersfeld (1983; 1988) tohelp teachers to consider how children’s mathematical thinking develops from direct modelingstrategies to more abstract and sophisticated strategies. These frameworks help teachers thinkabout how to use children’s current ways of operating to inform their instructional decisions.Children often begin solving story problems by directly modeling the story with counters orusing counting all strategies (Olive, 2001). It is not until children are able to see a group ofobjects as a unit that they are able to count on to solve story problems. Children at this level aresaid to be numerical and are counters of abstract unit items. Since counting is no longer rote forthem, they are said to have constructed their initial number sequence (INS). At the nextnumerical stage, which can be characterized as INS Plus, children are able to use the countingdown strategy more effectively and they have now determined that it is more efficient to solve anaddition problem by counting on from the largest number in the problem rather than the first.Thus, these children have constructed the commutative property of addition. Children who are atthe next level can solve all types of problems without the use of counting. They are able to takenumbers apart and put them back together in more convenient ways. They are said to beStrategic Additive Reasoners (SAR) (Steffe, et. al, 1983). The CGI researchers call thesestrategies using number facts.MethodsWe wanted to systematically study what PSTs get out of the interview project within thecontext of a subject matter preparation course. In particular, we wanted to know if the projectdesign meets the goals we set. For instance, did the PSTs learn to listen to children to informtheir mathematics and their instructional decisions? Were they able to identify counting schemesor strategic additive strategies that the child used when solving the CGI story problems? Werethey able to use their own mathematics to recognize the mathematical validity of a child’smethod?There were 26 participants in our study, all of whom were taking a Numbers and OperationsCourse designed specifically for Prospective Early Childhood PSTs. For all of the PSTs, this wasProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 44
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 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: THE INTERVIEW PROJECTAngel Rowe Abn
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
Page 87 and 88: opinion of the student body. This q
Page 89 and 90: At the highest level of reasoning a
Page 91 and 92: APPENDIXTo use two decks of cards t
Page 93 and 94: isolated and often occur in tandem
Page 95 and 96: with the CCSSM. Teachers read and d
Page 97 and 98: teachers’ role-play of SFMP #4. A
Page 99 and 100: 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
Page 105 and 106:
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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