2013 Conference Proceedings - University of Nevada, Las Vegas

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the first time that they worked with a child to learn to listen to and be responsive to the child’smathematics. This project involved PSTs working in pairs to interview an elementary-age childto allow them the opportunity to see how capable children are of solving problems. We analyzedthe work of thirteen pairs by coding instances of description, analysis, and instructionaldecisions. Sources of data came from PSTs’ written reports along with their presentations of theinterview, and their peers’ responses to their presentation. We were particularly looking for PSTswho seemed to be striving to understand their own teaching, their child’s mathematics, and theway in which it can inform their practice.FindingsDescription and AnalysisAll PSTs were able to describe the children’s mathematics using the language that theylearned in class. Their descriptions were informed by frameworks including CGI and Steffe et.al. They used language such as direct modeling, counting on, counting all, INS or SAR. In theirwritten reports the PSTs tended to simultaneously describe the child’s mathematics and analyzetheir descriptions. One PST wrote:…she was definitely numerical. Alex was able to count with the counters and had notrouble counting on. She also used strategic reasoning for many of the problems and wasable to explain to me how she worked it out….When I asked her how to solve six plusseven, she said she knew this “because six plus six is twelve and one more for seven.”This is a perfect example of “near doubles”. She used the basis of a double she knew andadded on one. When I asked her four plus nine, she knew this was thirteen because “tenplus four is fourteen and take away one”. She used the base of ten and added on fromthis.This PST was able to precisely describe what the child did to solve addition problems; she wasable to name these strategies, such as counting on and strategic reasoning. She was also able tocorrectly identify that the child’s mathematical actions indicated that the child was numerical.During the class presentation, the PST was able to write a series of equations and name themathematical properties in order to analyze the validity of the child’s strategic reasoning. This isthe series of equations that she wrote:Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 45
6+7 4+9=6+(6+1) Substitution =4+(10-1) Substitution=(6+6)+1 Associative Property =(4+10)-1 Associative Property=12+1 =14-1=13 =13Instructional DecisionsEven though this was the first time the PSTs had worked with children in this capacity, it wasevident that many instructional decisions were made, both planned and spontaneous. All reportsincluded at least one spontaneous instructional decision. One example was based on what a pairof PSTs observed their child do to confirm their belief that the child was at the INS+ level ofwhole number development. This pair of PSTs wrote the following in their report:Her approach proved to me that she can use the commutative property. She began withthe larger number although it appeared after the smaller number in the problem. Toreinforce this I asked her a similar question (I replaced the numbers with 6 and 9 in thatorder) and she still chose to start with the 9. I asked her why she chose to start with the 9instead of the 6 since it was first in the problem and she told me it was easier to start with9 because it was the bigger number. This supported my belief that she understood thecommutative property and was at least INS+.This excerpt shows the PSTs recognition of the mathematical properties that often were the focusof the content course on Numbers and Operations. It was evident that they were able to maketheir content knowledge usable in their work with children. They were also able to use theframework from class to analyze the child’s mathematics. However, in order for this analysis tooccur they had to use questions to probe the child’s mathematics. It was clear that these PSTswere intentional with their choice of ordering the numbers in the problem. It was clear that thesePSTs were listening for a particular strategy, specifically counting on from largest. They had ahypothesis, and used a specific task to test that hypothesis. We believe that this is an importantpart of the research process. Besides probing questions, which are information seeking, we alsosaw evidence that the PSTs used prompting questions to elicit a particular response.On the project description we asked the PSTs to respond to the question, “if you couldcontinue to work with this child, what concepts or kinds of problems do you think would beproductive work for her or him?” 12 out of 13 pairs were able to thoughtfully respond to thisProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 46
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 and 50: THE INTERVIEW PROJECTAngel Rowe Abn
Page 51: involving addition and subtraction:
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
Page 101 and 102: 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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2013 Conference Proceedings - University of Nevada, Las Vegas

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