2013 Conference Proceedings - University of Nevada, Las Vegas

More documents

Recommendations

Info

THE EVOLUTION OF STUDENT IDEAS: THE CASE OF MULTIPLICATIONKris H. GreenSt. John Fisher Collegekgreen@sjfc.eduBernard P. RiccaSt. John Fisher Collegebricca@sjfc.eduThe genesis of ideas, studied by Piaget and others, is important in understanding how studentslearn, which in turn, informs pedagogy. Using biological genetics as a metaphor for learning, werevisit previous work on multiplication, examine some teachers’ explanations of multiplicationand explore how these teachers use their explanations to generate new understandings ofmultiplication. This examination supports the use of the biological metaphor, and provides someinsight into how to teach multiplication with attention to flexible thinking.Taxonomies related to knowledge and learning abound (e.g., Anderson & Krathwohl, 2001;Kamii, Clark & Dominick, 1997). However, despite the large number of classifications ofknowledge and the recent understandings of constructivism, relatively little is known about thedynamics of how these understandings are constructed by learners. This is likely due to theprevailing metaphors regarding learning (Thelen 1997, 2005). In this paper, we propose ametaphor for learning – that learning is similar to biological evolution in many respects – thatenables a new look at the evolution of student ideas, benefitting researchers in student cognitionand teachers planning curriculum and instruction.Genetic Metaphor for LearningTheoretical FrameworkOn one hand, a biologically-inspired approach is nothing new, as Piaget’s geneticepistemology (Gallagher & Reid, 2002) followed his own training in zoology. Piaget’s notionsare insightful and influential when considering the genesis of knowledge, but do not take agenetic (in the biology sense) approach. Following Thelen (1997), we propose a new way to lookat some of these same issues, and we believe that this new lens can aid teachers in ways thatPiaget’s ideas cannot. In particular, this new metaphor can more clearly guide instruction thatincorporates existing student ideas, giving more guidance to constructivist approaches toteaching. Although discussing these individually is beyond the scope of this paper, this metaphoralso unifies a number of approaches to cognition.This piece is too short for a full explication of biological genetics, but the overview likely isknown to most readers, and only some details are necessary for the current work. Roughlyspeaking, in sexual reproduction biological offspring both resemble their parents and differ fromProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 189
their parents in phenotype (observable characteristics) and genotype (genetic makeup), withphenotypes being the result of genotypes growing in an environment. In reproduction of this sortthe genes of the offspring are usually some combination of two different genes from the parents;the dynamics of combining produces offspring whose genotypes are identical to their parents, orsome mixture or hybridization, etc. In addition, mutations, such as “mistakes” in copying genesor the result of external factors “interfering” with reproduction, can occur. In biologicalevolution, phenotypes - and hence, genotypes - in a population are “selected” in a probabilisticmanner through their fitness to the environment (e.g., “survival of the fittest”).Learning viewed as a genetic learning process (GLP) parallels the biological ideas ofphenotypes, genotypes, offspring, and selection. In this view, the ideas or concepts that a personholds are similar to the genes or alleles of biology. A GLP brings about new knowledge as anoffspring of previous understanding through the replication of a previous idea or the combiningof two (or more) existing ideas, or the mutation of a previous concept, and so on. The “fitness”of the offspring, however, is determined not so much by the environment as it is by thecoherence or efficiency of the new scheme as viewed by the learner. Because this view is one ofthe dynamics of learning, and not merely a taxonomy of approaches students can take, it canprovide additional insight into both teaching and learning.Genotype vs. Phenotype in GLPFirst, we note that there is an important difference between the genotype of knowledge andits phenotype. Let us consider two hypothetical students who answer the question “What is 12 ×23?” Each student (seemingly) follows the traditional algorithm, and gets the answer of 276:23× 124623_276Despite seeing the same phenotypic (observable) multiplication behavior in each student, thereare at least two quite different genotypic (but not easily observable) approaches to multiplication;often these differences become apparent only in later work. Verbal probing by the teacher mightdraw out differences between students (e.g., regarding place value or understanding of partialproducts) that could point to underlying genotypic variation.Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 190
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 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
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
Page 103 and 104:
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
Page 155 and 156: After instruction in the course, th
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
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: ReferencesAnderson, J. R., Greeno,
Page 199 and 200: student learning calls for differen
Page 201 and 202: Figure 2. (A) P3. (B). Extension of
Page 203 and 204: the usual phenotypic assessments an
Page 205 and 206: teachers is the discrepancy between
Page 207 and 208: expected to learn and the inquiry a
Page 209 and 210: his partner about his observations,
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
show all

2013 Conference Proceedings - University of Nevada, Las Vegas

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?