2013 Conference Proceedings - University of Nevada, Las Vegas

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HOW THE HAND MIRRORS THE MIND: THE EMBODIMENTOF NUMERICAL COGNITIONThomas J. FaulkenberryTexas A&M University - CommerceThomas.Faulkenberry@tamuc.eduMuch knowledge about how the mind does mathematics is based on the traditional, computerbasedmetaphor of cognition that assumes cognition is stage-based and independent of the motorcortex. In the present study, I provide evidence for an alternative view. I recorded participants’hand movements as they chose the correct parity (odd/even) for single-digit numerals.Distributional analyses of these movements indicated that responses resulted from competitionbetween parallel and partially-active mental representations rather than occurring in discretestages. Furthermore, this competition was carried through to the motor cortex, indicating thatnumerical representations are more tied to bodily affordances than previously thought.Researchers have been investigating mathematics learning for many years, particularlythrough the paradigm of cognitive psychology. From early attempts to understand howarithmetic facts are organized (Ashcraft & Battaglia, 1978) to formal models that specify thevarious cognitive processes involved in mathematical problem solving (Anderson, 2005), mostof these studies have made the implicit (albeit, metaphorical) assumption that the mind operateslike a computer. That is, perception informs cognition, and cognition informs action. In thisview, higher-level cognitive systems (memory, executive control, etc.) are thought to be quiteseparate from the lower-level systems (perception, motor action).This modular, discrete-systems approach to cognition has been the fundamental underpinningof most of our understanding of how the mind does mathematics. From the point of view of“mathematics is a collection of abstract ideas,” it makes sense that mathematical objects could belearned and operated on in a purely abstract fashion without any interaction with other (noncognitive)neural systems, such as the motor cortex. However, Lakoff and Nunez (2000)proposed the hypothesis that mathematics is learned through conceptual metaphor, a mechanismfor converting embodied (sensori-motor) reasoning to abstract reasoning. At the time,unfortunately, their view was almost entirely philosophical, and it elicited much debate betweencognitive scientists and mathematicians. Without behavioral evidence, the debate would be sureto stay within the realm of philosophy, and as such, not be widely accepted amongmathematicians and psychologists alike.In recent years, however, other cognitive scientists have proposed a view similar to that ofProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 205
Lakoff and Nunez: specifically, that the human mind is not a modular computer, but rather arich, dynamic system of parallel and partially-active representations (Spivey, 2007). In thisview, decisions are not made through modular “switches,” but instead are the result ofcompetition among many different partially-activated responses, simultaneously informed byfeedback from many other systems, including (especially) the motor systems.The canonical example of this view is found in the language-processing literature (Spivey,Grosjean, & Knoblich, 2005). In a language-comprehension task, they asked participants tolisten to words and, with a computer mouse, choose the picture that correctly represented thespoken word. During this task, they measured participants' hand positions by continuouslyrecording the (x,y) coordinates of their mouse. They found that when words were phoneticallysimilar (CANDY versus CANDLE, see Figure 1), the mouse tracks tended to deviate toward theincorrect alternative early in the response process, but eventually settle in to the correct answer.This is commonly taken as evidence for an embodied view of cognition, where responses resultfrom a dynamic competition between partially-active, unstable mental representations. In theclassic, modular view of cognition, the hand positions would not be so sensitive to influencefrom the decision process, as the motor system would not be called upon until the decision wasmade in the language center of the brain.Until now, no studies have investigated the processing of numerical information within sucha continuous, embodied-cognition framework. This is unfortunate, as the work of Lakoff andNunez (2000) has set the stage for such research to tease apart the contributions of differentcognitive and perceptual systems to numerical cognition. In the present study, I used the handtrackingparadigm of Spivey (2007) to capture the formation of numerical representations duringa parity judgment task. Participants quickly judged whether single digit numbers were even orodd. Responses were either consistent with spatial orientation of numbers (i.e., small numberson left side or large numbers on right side) or inconsistent (i.e., small numbers on right side,large numbers on left side. Two competing predictions were then tested. If numerical cognitionis indeed part of an embodied system, then hand trajectories in the inconsistent condition shouldshow a pull in the direction of the incorrect alternative, reflecting a settling of partial activationsof response alternatives during the response. If, on the other hand, numerical cognition isProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 206
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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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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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Page 169 and 170: Performance, 33, 1410-1419.Cohen Ka
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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
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Page 193 and 194: Figure 2. A Display of Student Stra
Page 195 and 196: ReferencesAnderson, J. R., Greeno,
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Page 211: hard for some children? The nature
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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
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