2013 Conference Proceedings - University of Nevada, Las Vegas

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Table 3The groups’ categories and descriptions for permutation problemsType Problems Group 1 Group 2Arrangements(Ordered,withoutrepetition)Sequences(Ordered, withrepetition)ProblemsStatesNetflixPlane RoutesProblemsPasswordMC Exams1Gift Cards4-letter wordsVowelProblemsStates (Texas)NetflixPlane RoutesProblemsPasswordMC Exams1Gift Cards4-letter words[in Cases]MarblesDescription-Each thing canonly be in oneplace at a time (norepeats within set)-Order mattersDescription-Things beingdistributed todifferent positions-Order matters-One element canbe repeated (peoplegetting more thanone card)ProblemsStatesNetflixPlane RoutesM/F CommitteesProblemsPasswordMC Exams1Gift Cards4-letter words[missing]Description-The order ofchoices matters.-Choices cannotbe repeated.Description-Certain amountof spaces andeach space hasthe samenumber ofoptions.-Options can berepeated.The two focus groups had much more difficulty categorizing and describing combinationproblems (Subsets and Multisubsets). While an apparent difference exists between the twogroups in their ability to identify common structures between Subset problems (Group 2successfully accounted for 4 of the 5), both groups had particular difficulty with Multisubsetproblems (Table 4). Group 1 was unable to solve any of these problem types: indeed, theMarbles problem was incorrectly solved as a Sequence and the Hot Dogs problem, which wassolved in class, was connected to simple Subset problems (the group did not appreciate theunique characteristics of the original problem, which was then translated through a stars and barsmodel to a simple Subset problem). Group 2 solved the Skittles problem and was able to connectit to the Hot Dogs problem; however, rather than focusing on the similar characteristics of theseproblems, their description for this category was procedural (see Table 4), which demonstratesless sophisticated expertise (e.g., Schoenfeld & Hermann, 1982) and indicates that learners mayhave difficulty identifying common structures within combination problems.Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 149
Table 4The group’s categories and descriptions for combination problemsType Problems Group 1 Group 2Subsets(Unordered,withoutrepetition)Multisubsets(Unordered,withrepetition)ProblemsHandshakesSupreme CourtVotingMC Exams2M/F CommitteesProblemsHot DogsSummed DigitsSkittlesMarblesPizza ToppingsProblems[missing]Supreme CourtVoting[did not do][did not do]Hot DogsProblems[in Subset][did not do][did not do][in Sequence][did not do]Description-How manydifferentpositionseach elementcan occupy (9choose 6)ProblemsHandshakesSupreme CourtVotingMC Exams2[in Arrangement]ProblemsHot Dogs[did not do]Skittles[did not do][missing]Description-Take the groups andchoose a certain number.Using the “group”choose “number” getsrid of the duplicates. Theduplicates exist becauseorder does not matter.Description-Broke into groups toaccount for noduplicates. The barriersseparated into groups.Barriers made choosingeasy and allowed forchoosing all of one type.Lastly, Group 1 had an additional category; the two problems identified in this category,Vowel and MC Exams2, were characterized as a “Way to choose a minimum number ofoutcomes (1 vowel, 2 vowels, etc.). Elements have different characteristics, different groupings.Must look at characteristics as a subset of population.” In other words, they viewed problems assimilar if they were best solved by splitting into “cases,” which, for both problems, was anaccurate statement and approach. This gives an indication that, at times, participants madeconnections according to similar processes instead of structurally similar characteristics.DiscussionWhile many findings could be explored in more detail, we will focus on the insight gainedfrom two particularly difficult problems: Gift Cards and Pizza Toppings. These results from thefocus groups potentially shed some light on the learning and teaching of combinatorics problems.The Preferred Vantage PointThe Gift Cards problem (i.e., How many ways can you distribute a $1, $2, $5, $10, and $20gift card to 8 friends?) is a Sequence problem, which, overall, students were able to solve.However, as an individual case, this problem caused surprising difficulty. (Solving the problemalso caused over-generalization to other problems with repetition: “like the Gift Card problem,”Group 1’s reason for including the Marbles problem as a Sequence.) Both groups began bydrawing eight slots, one for each person. Their attempts to distribute the five gift cards to theseeight people included, among others, æ 8 (but then “a person could get more than one gift card”)öçè 5÷ øProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 150
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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
Page 105 and 106: Table 4Classification Categories fo
Page 107 and 108: field so that research on the initi
Page 109 and 110: dynamic approach to learning conten
Page 111 and 112: Kindergarten Lesson FormatHow May W
Page 113 and 114: team’s goals? As much as possible
Page 115 and 116: 6. While preparing the lesson, teac
Page 117 and 118: active learning and collective part
Page 119 and 120: classroom. “I would like to know
Page 121 and 122: I had never been brave enough to tr
Page 123 and 124: THE PATH OF REFORM IN SECONDARY MAT
Page 125 and 126: Our collaboration model was formed
Page 127 and 128: internal evaluator) were analyzed.
Page 129 and 130: DiscussionOn part I of the survey t
Page 131 and 132: whole department of secondary mathe
Page 133 and 134: discussion. Many texts include wild
Page 135 and 136: Data collection consisted of tests,
Page 137 and 138: I've not used children's literature
Page 139 and 140: could extend this inquiry to high s
Page 141 and 142: course titled Calculus with Busines
Page 143 and 144: no mathematical sense and should no
Page 145 and 146: Adopts the “111” (a term coined
Page 147 and 148: Specifically, clicking the “Click
Page 149 and 150: algebraic expression is carried out
Page 151 and 152: Retrieved from http://secc.sedl.org
Page 153 and 154: furthermore, each model may result
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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
Page 181 and 182: Popularity tallied whether or not a
Page 183 and 184: Amazingly, despite there being a fe
Page 185 and 186: ReferencesBlack, M. (1962). Models
Page 187 and 188: connections are connections or rela
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 201 and 202: Figure 2. (A) P3. (B). Extension of
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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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