2013 Conference Proceedings - University of Nevada, Las Vegas

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Figure 3. (A) Original drawing for 3 x 3. (B) Modified drawing for 4 x 4. Squares indicateadditions to original drawing.When asked to do the same thing for the problems 4 × 4, one teacher drew Figure 1(B), stating“I knew the answer to 3 × 3, and I knew how to count on. So I added one to every group [of 3],and I added a new group [of 4].” In this case, the parent genes were (a) her knowledge of 3 × 3(or of groups of dots in general) and (b) her counting on strategy. When asked why she used thisapproach, the teacher responded that “They [the problems] were close together, so it was easy todo.” In other words, the environment (problem) guided the notion of “fitness” of the idea. Thisnew approach was used later when this student took a geometric approach to this problem:Working in base-5, consider the sequence P 0 , P 1 , P 2 …. Find P 110 if:(a) P 0 = 0, and(b) P n+1 = P n + n + (n+1), for n ≥ 0This student calculated some terms by hand and recognized that P n are the square numbers. Shedemonstrated it as in Figure 2, which is very like Figure 1(B): in each case, one was added toeach (vertical) “group” in the original situation (P 3 ), and another new “group” (of size n+1) wasadded to get to the new situation (P 4 ). While not all genotypes will be used in multiple situations,the “transfer” of this genotype between situations indicates that this genotype is a thing in itself.It should be noted that it is often difficult to make these types of genotypic observationsbecause it needs a microgenetic (Kuhn, 1995) approach to observations. While this is possible inresearch and in classrooms, it requires a different stance on the part of the observer. It alsorequires a different approach to formative assessment, one in which the teacher looks not only atwhat problems can be solved, but also looks in detail at how they are solved.Multiplication in the LiteratureSteffe (1994) is an example of what happens when one focuses only on creating a taxonomyof the various aspects of learning think multiplicatively. The components identified (uniting,Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 193
Figure 2. (A) P3. (B). Extension of P3 to P4. (C) Final explanation for P4iterating composite units, etc.) are a step toward identifying what, in the GLP metaphor,constitute the genetic material of the thinking involved, but these do not address the way thisthinking evolves or comes into being. While we need these classifications, they do little to moveany student forward in his or her thinking without some concept for how new ideas can grow anddevelop. One of Steffe’s subjects, Tyrone, provides a powerful example for the way thinkingworks in our metaphor. Tyrone clearly has a sense of multiplication as repeated addition, anddemonstrates this through his initial step-counting by units of 20 to determine 20 × 20. Whenasked to calculate 30 × 20, he starts from his previous answer to 20 × 20 and step-counts foranother 10 groups of 20. The numbers he has encountered allow him to use an existing strategy(step-counting) to solve two very similar problems; he clearly understands that 30 × 20 = (20 +10) × 20 = 20 × 20 + 10 × 20 on an intuitive level. Tyrone is demonstrating problem solvingabilities that are inherent in the ability to mutate an existing solution strategy. Because this wassuccessful, Tyrone will now be encouraged to continue finding ways to adapt this strategy.Although all new student ideas are formed from prior understandings, like natural evolution,GLP does not always produce successful strategies. The fossil record shows events, like theCambrian Explosion (Gould, 1990), that have left their mark in the Burgess Shale, showing ahuge variety of evolutionary solutions, none of which are found in today’s natural world. Thesame is true for GLP. Van Dooren, deBock and Verschaffel (2010) provide copious examples ofstudents hybridizing existing strategies to produce new strategies that are then casually dropped.Their study looked at how 3 rd – 5 th graders handled problems that were either additive ormultiplicative in structure, having either integer ratios or non-integer ratios. Younger studentsProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 194
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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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Page 151 and 152: Retrieved from http://secc.sedl.org
Page 153 and 154: furthermore, each model may result
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Page 157 and 158: Table 4The group’s categories and
Page 159 and 160: you choose three place values, æ 4
Page 161 and 162: APPENDIXTable 5Description of Combi
Page 163 and 164: of the presented number. Later, the
Page 165 and 166: Figure 1: Mean trajectories and MD
Page 167 and 168: Figure 2: Mean trajectories and MD
Page 169 and 170: Performance, 33, 1410-1419.Cohen Ka
Page 171 and 172: Moyer, 2007). At his or her own pac
Page 173 and 174: logged by the system and then retri
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
Page 191 and 192: t-charts made it easier for student
Page 193 and 194: Figure 2. A Display of Student Stra
Page 195 and 196: ReferencesAnderson, J. R., Greeno,
Page 197 and 198: their parents in phenotype (observa
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Page 211 and 212: hard for some children? The nature
Page 213 and 214: Lakoff and Nunez: specifically, tha
Page 215 and 216: Figure 5: Average hand trajectories
Page 217 and 218: Figure 6: Distributions of maximum
Page 219 and 220: ReferencesAnderson, J. R. (2005). H
Page 221 and 222: Social system perspectives view the
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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2013 Conference Proceedings - University of Nevada, Las Vegas

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