M3 AERA Paper F · 2014-04-25 · 2014 AERA National Conference: Mathematical Cognition:...

2014 AERA National Conference: Mathematical Cognition: Strategies, Working Memory, and Representations

Mobile Movement Mathematics: Exploring the gestures students make while explaining FrActions. Michael I. Swart, Ben Friedman, Sor Kornkasem, Sue Hollenburg , Susan Lowes and John B. Black Teachers College Columbia University Jonathan M. Vitale University of California Berkeley Sandra Sheppard and Frances Nankin WNET-‐ Thirteen: Cyberchase ABSTRACT Twenty 3rd and 4th grade students from an afterschool program in Harlem, New York City participated in an exploratory study about mathematical fractions. In a quasi-‐experimental post-‐test only design, researcher’s conducted clinical interviews to capture students’ gestures produced during explanations of fractions concepts. Using McNeill’s (1992) gestural dimensions taxonomy, student’s iconic (representative of an object or process) and metaphorical (of mathematical operations) gestures were significantly correlated to higher performance on fractions problems (r=.59, p< .0061). Granular analysis revealed students’ gestures enacting processes of shading, slicing, swiping, spanning, tracing, delineating, encircling, and drawing in the air among many. Educators can leverage these gestures as grounded and embodied (Lakoff & Johnson, 1980; Barsalou, 1999; 2008; Glenberg, 2000) representations for learning mathematical fractions. DESCRIPTORS: mathematics, gestures, fractions, embodiment, clinical interview, case study This work is supported by Teachers College Columbia University Institute for Learning Technologies and NSF Cyberlearning Exploratory Grant 1217093.

OBJECTIVE

There is a growing theme in mathematics education integrating cognition research into instructional design and curricular development. Understanding that mathematics is grounded in the world

around us (Dehaene, 1997) and its operations are embodied by our interactions therein (Glenberg, 2000; Lakoff & Núñez, 2001), investigators are probing deeper into how learners cognize mathematical concepts and, in turn, making mathematics more accessible to everyone.

Recent advents in technologies like smart phones, tablets, and gaming systems are creating new opportunities for learning in the forms of rich, robust and dynamic games, simulations and tools to help engage learners and embolden their conceptual understandings of mathematics. Consequently, the design processes for these new technologies should be informed both by theory as well as formative research (Samara and Clements, 2004). With this in mind, the current research explored the gestures associated with learners’ conceptualizations (Goldin-‐Meadow, Alibali, & Church, 1993; Goldin-‐Meadow, 2000; Roth, 2002) of mathematical fractions for the purpose of developing a user interface for a tablet-‐based gaming environment. THEORETICAL FRAMEWORK

Math is in the world around us. It’s that one red apple hanging, the basket we


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put it in and the delicious halves we split it into. These are the roots of our number sense (Dehaene, 1997). From infancy, people are endowed with a numerical sense that can subitize quantities up to 3 or 4 and approximate magnitudes and arithmetic operations (McCrink and Wynn, 2004; 2007; McCrink, Dehaene and Dehaene-‐Lambertz, 2007). In the brain, frontal and parietal areas quantify, meter, and compare -‐ a concert of visuospatial and motoric activity (Dehaene, 1997; Dehaene, Spelke, Pinel, Stanescu and Tsivkin, 1999; Siegler, Fazio, Bailey & Zhou, 2013). The number sense is the combination of the actions human perform, like traversing distances to collect, compare, sort, contain, carry or consume objects. It is the connection of our digits (on our hands) to the digits (numbers) we use to enumerate. These are the embodied sources of human mathematical understandings (Lakoff & Núñez, 2001)-‐ the anchors grounding human perceptual experiences (Glenberg, 2000; Barsalou, 1999). Just as Saxe (1988) found informal arithmetic amongst the young street traders in Brazil, mathematical understanding is in our nature and our nurture. Concepts formed in action are expressed through both verbal and non-‐verbal communications. Fortunately, non-‐verbal gesture represents an important, easily accessible bridge between action and communication that educators can use to bring the body and the mind together to better understand mathematics (Goldin-‐Meadow, Cook, and Mitchell, 2009; Meadow, Alibali & Church, 1993; Roth, 2002).

Human beings have communicated via gesture since long before formalized language developed (Hewes et al., 1973; Corballis, 1999). While most people conceive of gestures as physical movements that accompany speech, research has

demonstrated that gesture often precedes speech (Acredolo & Goodwyn, 1988; Iverson & Goldin-‐Meadow, 2005; Ozcliskan & Goldin-‐Meadow, 2005). Because gestures are integral in communication across languages and cultures, they represent a robust a means for educators to help learners reactivate (simulate) the perceptual states associated with the concept and pose the potential to reveal underlying strategies (Alibali & Nathan, 2007; 2009; Alibali & Goldin-‐Meadow, 1993; Goldin-‐Meadow, 1999).

In learning of mathematics, Goldin-‐Meadow, Cook, and Mitchell (2009) demonstrated that a pairing gesture could facilitate learners’ performances with arithmetic equations. Alibali, Bossok, Solomon, Syc and Goldin-‐Meadow (1999) posit that spontaneous gestures are often embodiments of their mental representations of math problems. A study by Segal (2011) effectively demonstrated that gestures that are congruent with the mathematical concepts being learned produce better learning outcomes than non-‐congruent gestures. Gestures determined to embody fraction concepts emphasize the congruency between one’s physical state and one’s mental state. This conforms with Hostetter’s and Alibali’s (2008) theory of Gestures as Simulated Action.

The current study explores the gestures that learners of fractions use when explaining their responses to fraction problems of identity, magnitude and equivalency (Schneider & Siegler, 2010; Lakoff & Nunez, 2000). The current research anticipated an array of responses ranging from relevant and irrelevant to fractions concepts. Moreover, amongst the gestures denoted as relevant to fractions, some were conceptually congruent to the process of fracturing and others were not. The goal is


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to develop game-‐based simulations that map students’ gestures onto the 2-‐dimensional touch-‐based interface to accompany simulated activities that represent real-‐world actions associated with fractions (Martin, Svihla & Smith, 2012) METHODS Participants. A total of twenty participants from grades 3 (n3rd=8) and 4 (n4th=12) of a public elementary after-‐school program in Harlem, New York provided parental consent to participate in an exploratory study that investigated student’s conceptions of fractions (Mean age = 9.7 [.61], 60% female). Procedure. Four researchers collaborated in conducting 35-‐55 minute scripted clinical interviews of elementary students working on fractions mathematics problems. Students were assigned to condition by grade level and completed either an individual interview (1-‐on-‐1) or a group interview as a part of triad (1-‐on-‐3) (see Lesh, 1981). All interviews were conducted in a specially designated classroom and were videotaped. Videos were subsequently reviewed and student’s gestures were coded. Upon completion of the interviews, students completed a 10-‐item post-‐test assessing basic knowledge of magnitude, identity and equivalency of fractions. MATERIALS. Data in the current study was both quantitative and qualitative from two primary sources: (1) scripted clinical interview (Appendix A) and (2) post-‐test (Appendix B). Students’ initial responses in the clinical interviews were coded as

correct or incorrect. After data collection was completed, researchers reviewed the videos and coded student’s gestures. Scripted Clinical Interview.

Section I: Open-‐Ended Assessment. A simple blank paper-‐pencil assessment in which students answered three questions about magnitude and identity of fractions and five conceptual questions about fractions. Researchers prompted students to “draw” their answers and subsequently probed for them to “explain why” and “show how”.

Section II: Scaffolded Assessment. A

second paper-‐pencil task prompted students to make equivalency judgments between fractions using a set of 6 equal-‐sized containers illustrated on a single sheet of 8.5” x 11” paper (see Figure 1). Section III: Proscriptive activity. As

math investigators, students used two sets of manipulatives (i.e., either partitioned strings or cups; see Figure 1) to answer more fractions problems about equivalency. Their duty was to order a given set of three fractions by determining comparatively if each fraction was less than, equal to or greater than the others. Section IV: Post-‐Test. After completing

the clinical interview, students were given a 10-‐item post-‐test to assess their knowledge of simple fractions (i.e., excluding improper or compound). The test comprised of three sections: (1) fractional magnitude (3 items), (2) identity (3 items); (3) equivalency (4-‐items) (see Hecht & Fischler, 2012).


Figure 1. Clinical Interview Materials: (a) the “tray of brownies” as depicted for the students for equivalency evaluations, (b) an illustrated example of the strings and (c) the cups that students used to make magnitude, identity and equivalency judgments. RESULTS Conceptual Models and Coding. McNeill’s (1992) taxonomy of gestures highlights that gestures cannot often be singularly defined as a particular kind of gesture; rather, they exist along a continuum (a.k.a., “Kendon’s Continuum”) and are more appropriately considered as dimensions. The current study uses McNeill’s quartet of semiotic gestural dimensions: (1) iconic -‐ representative of concrete entities and/or actions, (2) metaphoric -‐ represent of abstract entities or processes (e.g., spanning gesture to represent arithmetic summation), (3) deictic -‐ typically an index pointing to a referent, but can be performed with any part of the body, and (4) beat -‐ accompany the prosody of speech and can also be used for emphasis of time and context. Additionally, a fifth dimension, enactive, was also used to clarify when a gesture represented a process or procedure as well as a sixth dimension, symbolic, to denote when gesticulations were of actual abstract mathematical formalisms (i.e., drawing the number 3 in

the air, or writing a numerical fraction in the air). These six dimensions were combined to develop the gesture coding system used (see Table 1). Two raters coded 4 transcripts (8 students) in common and pooled the codes to establish inter-‐rater reliability that was high and statistically reliable (Cohen’s κ = .83, n = 10 codes, p < .01). Summary Variables. From the interview logs, researchers coded students’ gestures as either relevant or irrelevant to fractions concepts. Gestures that were iconic, enactive, or metaphorical of the process of fracturing or mathematical operations were considered relevant. The resulting percentage ratio of relevant gestures to total gestures was significantly positively correlated to student’s overall initial accuracy scores during the clinical interview (r = .49, p < .048). Moreover, students’ relevant gestures ratio was also significantly positively correlated to higher scores on clinical interview equivalency problems (r = .502, p < .025) (see Figure 2).


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GESTURE CODE DESCRIPTION BEAT B A beat gesture with no specific referent that accompanies speech

BEAT DEICTIC BD A beat gestures that points but to no entity in particular

BEAT ENACTIVE BE A beat gesture that accompanies speech about a process but with no specific referent or depiction

BEAT REFERENT BR A beat gesture that accompanies discussion about an object

DEICTIC REFERENT DR A pointing gesture that specifically references an entity, real or imagined

DEICTIC METAPHORICAL

DM A pointing gesture, the motion of which, is indicative of a mathematical operation

ICONIC I A gesture that depicts a specific entity or process ICONIC METAPHORICAL

IM A gesture that depicts a specific entity or process that represents a mathematical operation

SYMBOLIC S A gesture that depicts an actual abstract symbol (i.e., drawing the number 3) NON-‐GESTURAL ACTION NON An action performed by the student whereby they physically manipulate an object or

perform a process Table 1. Codes of Gestures

a) b) c) Figure 2. Three scatterplots contrasting (a) Total Gestures x Total Initially Correct (r = -‐.032, p = ns); to (b) Ratio of Relevant Gestures to Total Initially Correct (r = .49, p < .048; to (c) Ratio of Relevant Gestures to Clinical Interview Equivalency Problems (r = .502, p < .025)

Determining equivalency between

fractions is more complicated since it involves both judgments of magnitude and identity as well as arithmetic operations (Siegler et al, 2013; Hecht et al., 2012). As a result, 4th graders, having completed a curriculum that offered them more instruction and experience determining equivalency between fractions than 3rd graders, should demonstrate more advanced conceptual and procedural understanding of fractions. One-‐way

ANOVAs confirmed significant differences between 3rd (M = 3.22, SD = 1.48) and 4th (M = 4.82, SD =. 784) graders on equivalency problems from the clinical interview (F(1,18)=9.009, p < .009) as well as equivalency problems from the post-‐test (F(1,18)=5.79, p < .028) with 3rd (M = 1.11, SD = 3.33) and 4th (M = 1.91, SD =.944). Looking at the Gestures. Table 2 emphasizes how conceptually related gestures are associated with better


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understanding of fractions. Beyond the taxonomic classification of students’ gestures, a more detailed analysis reveals an extensive set of gesticulations that represent different actions, procedures and concepts (see Table 3).

GESTURES TOTAL % RANK BD 10 1.40% 7 BE 6 0.84% 9 BEAT 25 3.50% 5 BR 2 0.28% 10 DM 129 18.07% 3 DR 186 26.05% 2 ICONIC 9 1.26% 8 IM 283 39.64% 1 NON 51 7.14% 4 SYMBOLIC 13 1.82% 6

714 100.00%

CONCEPTUAL 427 59.80% TABLE 2. Rank order of total gestures and ratio of conceptual gestures. Qualitative Analysis: Findings from two Cases. A median split on initial score (Mdn=14) and ratio of relevant gestures (Mdn= .6125) revealed a significant difference between groups on (t(18) = -‐2.7154, p < 0.02). Given these two independent variables of interest, fracture problem accuracy and the ratio of relevant gestures – four possible categories of students may emerge: (1) those who make few gestures and demonstrate low initial accuracy, (2) those who make few gestures and demonstrate high initial accuracy, (3) those who make many gestures and demonstrate low initial accuracy, and (4) those who make many gestures and demonstrate high initial accuracy. The following two cases contrast low and high accuracy between two students producing many gestures.

Student DM (Low Accuracy, Many Gestures). This student demonstrated lower conceptual understanding (INITIAL CORR = 9, z=-‐1.22) with a very high rate of gesture (TOTAL GEST=45, z=1.42; RELEVANT GEST=22, z=.6;). For relevant gestures, this student had the lowest ratio amongst the participant pool (RATIO=.49, z=.41). Identifying this pattern presents an opportunity for teachers to utilize conceptually relevant gestures (Alibali & Nathan, 2007) and capitalize on any existing ones. Despite low accuracy, this student still employed conceptually relevant gestures included encircling, pointing to count, delineating boundaries, grasping objects and spanning gestures. Student AW (High Accuracy, Many Gestures) This student demonstrated high conceptual understanding (INITIAL CORR = 16, z=.58, ns) and a high rate of gesture (TOTAL GEST=40, z=.98; RELEVANT GEST=30, z=1.63;). A closer look reveals that this student had the second highest ratio of relevant gestures amongst the participant pool (RATIO=.75, z=.53). This student's gestural vocabulary was very robust, including aligning, chopping, cusping, delineating, denoting, drawing, encircling, grasping, measuring, pointing, shading, slicing, spanning, sweeping and swooping. Interestingly, this student, when waiting to answer, would engage in self-‐discussion that included conceptual gestures. Peculiarly enough-‐ this student, with so many gestures, never gestured to count, which may be another marker of conceptual development and mathematical understanding (Ginsburg, 1977; Gelman & Gallistel, 1978).


TABLE 3. Detailed List of Gesture with Descriptions, Frequencies and Rank Order. *Note: Table 3 contains overlap between non-‐gestural actions and the gestural action/functions they performed.

GESTURAL)CODE

GESTURAL)ACTION/FUNCTION EXEMPLAR TOTAL % RANK

DRPOINTING

"pointing0Individuated0Fingers0represent0different0parts" 248 33.74% 1

IM ENCIRCLING "encircles0the0entire0object0to0represent0the0whole" 73 9.93% 2

IMSHADING

"Simulate0shading0of0the0parts0to0represent0the0numerator" 52 7.07% 3

IM SLICING "Uses0pencil0as0tool0simulate0slicing0the0object" 50 6.80% 4

IEMDRAWING

"uses0middle0finger0to0draw0an0imaginary0object0denoting0a0whole0and0equality" 38 5.17% 5

IMSPANNING

"singleXhanded0finger0span0(thumb0and0index)0for0comparison0referencing0the0part" 34 4.63% 6

DMGRASPING

"grasps0w00index0and0thumb0to0ident0the0pieces0of0the0string" 32 4.35% 7

DMDELINEATING

"delineates0boundaries0of0each0quantity0for0comparison0to0express0difference0using0index0finger" 31 4.22% 8

BBEAT

"iterative0gesture0indicating0the0succession0of0counting0parts0but0with0do0definitive0referent" 31 4.22% 9

DMDENOTING

"uses0pen0as0tool0to0point0at0each0object0denoting0marks0for0comparison" 16 2.18% 10

NON ALIGNING "physically0aligns0cups0side0by0side0for0comparison" 15 2.04% 11

SSYMBOL

"scribes0the0formal0symbolic0representation0of01/20in0the0air" 12 1.63% 12

IM SPREADING"spreading0motion0using0both0hands0to0indicate0the0fracturing0thereof" 11 1.50% 13

IMOCCLUDING

"occludes0portions0of0the0whole,0leaving0the0remaining0part" 10 1.36% 14

NONPHYSICAL0ACT0(nonXgestural) "picks0up0the0strings0to0inspect0them" 9 1.22% 15

IM CHOPPING "two0handed0chopping0motion0to0break0up0whole" 9 1.22% 16DM COUNTING "pointing0to0count0the0tic0marks0on0the0cups" 8 1.09% 17

NONMOVE

"student0gets0low0to0have0a0level0view0of0the0waterlines0for0comparison" 8 1.09% 18

NON PICKSXUP"physically0picks0up0string0and0points0with0index0to0determine0number0of0parts0in0the0whole" 8 1.09% 19

IMTRACING

"traces0the0perimeter0of0the0rectangle0he0has0drawn0to0denote0the0whole" 8 1.09% 20

IMSWEEPING

"sweeps0using0hand0to0denote0removing0excess0in0order0to0isolate0a0piece" 7 0.95% 21

IMSWOOPING

"clockwise0SWOOPING0motion0DRAWING0a0curvilinear0line0across0all0the0parts0to0indicate0the0whole" 5 0.68% 22

IMCOMBINING

"combines0hands0together0from0far0apart0to0represent0synthesis0of0entire0object 4 0.54% 23

ICUSPING

"two0hands0cusp0together0gesture0to0signify0an0object 3 0.41% 24

IMMEASURING

"uses0two0hands0to00measure0of0the0pieces0to0compare0them" 3 0.41% 25

BESTRIKING

"Accompanying0speech0with0striking0forward0motion0indicating0equivalency0between0numerator0and0denominator" 3 0.41% 26

I ICONICATING"places0both0hands0open0flat0onto0the0table0top0to0represent0whole0fraction" 2 0.27% 27

IM PINCHING"pinches0a0thumb0and0index0together0to0measure0the0shaded0region" 2 0.27% 28

IMSWIPING

"swipes0across0using0index0to0moves0linearly0from0zero0point0to0qty0of0fraction" 2 0.27% 29

IM SCOOPING "scooping0gesture0away0from0an0imaginary0object" 1 0.14% 30735 100.00%


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SIGNIFICANCE Understanding how to use gestures

to access mathematical thinking (Alibali & Nathan, 2012) will help researchers develop contexts that are more engaging, informative and conducive to math learning. Since simply performing the actions (i.e., interacting with a manipulative) does note guarantee correct interpretations (Martin, Svihla & Smith, 2012), what is developed must be an effective combination of actions (i.e., gesture as simulated action) with proper interpretation of how gestures aid mathematical understanding.

Researchers in the current study explored students’ concepts of fractions through the gestures they made in their explanations. They found significant correlations between conceptual gestures and produced logs with detailed accounts of 3rd and 4th graders explaining how they conceive of mathematical fractions, the processes of making fractions, parts, wholes, numerators, denominators. This research will inform future work developing gesture-‐based learning technologies in hopes of creating new learning environments that can bridge between formal and informal settings, and between structured and unstructured curricula (Peppler, 2013).

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Appendix A: Clinical Interview Script (Questions not covered in the interviews are greyed out)

Interviews. Common Core Standards – 3.NF.A.3a 3.NF.A.3b 3.NF.A.3d _ Common Core Standards – 3.NF.A.1 3.NF.A.2a 3.NF.A.2b _ Paper & Pencil Pre-‐Test. This test consists of blank sheets of paper and a pencil. Below are the items and the script for the study. EXP: Ok. Here is a piece of paper and a pencil for you to draw with. Identity & Magnitude Exp: Principally, we are looking at the number of partitions, but if need, we can inquire about the comparable sizes of their partitions Use the following question sequence for each of the three sets below: EXP: Draw me a picture that shows… Why is that _________ (e.g. one-‐half)? Show me how you know? [Scaffolding] If they are unable to think of an object to draw, suggest a divisible object (i.e., an object, a collection of objects, a container, a path or a unitized container, examples include a pizza, skittle, a train, an ice tray, a glass) SET 1 (To evoke gestures in their explanations) 1. 1/2 of something; ❐ Correct ❐ Incorrect 2. 1/3 of something; ❐ Correct ❐ Incorrect 3. 1/4 of something. ❐ Correct ❐ Incorrect SET 2 4. 2/3 of something; ❐ Correct ❐ Incorrect 5. 3/4 of something. ❐ Correct ❐ Incorrect SET 3 6. 3/3 of something; ❐ Correct ❐ Incorrect 7. 4/4 of something; ❐ Correct ❐ Incorrect 8. What about 1/1? ❐ Correct ❐ Incorrect Paper & Pencil: General Questions. EXP: Have you done “fractions” in your math class? [YES] Okay then …what is a “fraction”? (This question is open ended enough to explore what they will include in their definitions)


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[Scaffolding] If they say no, then move on to next question below. EXP: What is a “whole” of something? EXP: What does it mean to have a “whole” of something? ❐ Correct ❐ Incorrect (This goal here is not complex, nor tricky-‐ simply for them to iterate something like “it’s the whole thing.”) EXP: What about a “part”? EXP: What does it mean to have a “part” of something? =❐ Correct ❐ Incorrect (These question is to explore their conceptions of parts to whole? If they do not have conception of parts being equal in size, this will be an important insight into their understanding) [Scaffolding] If they encounter difficulty with these questions, use a clay manipulative to facilitate these q’s, replace something with a divisible object (e.g., a pizza, a watermelon, a cake) rephrase the questions. EXP: What is a “numerator”? ❐ Correct ❐ Incorrect EXP: What is a “denominator”? ❐ Correct ❐ Incorrect (These questions are designed to see if the students can express in words, what they understand of the formal representation of a fraction as a mathematical symbol, If there is a discrepancy between their parts/whole knowledge and their formal understandings, then we can identify a source of contention in fractions learning) [Scaffolding] At this point, the experimenter can write a formal representation of a fraction (e.g., ½) if the student is unable to verbalize what these elements are, and then point to the fraction as written and ask the same questions, pointing to each as a referent as the question is asked. Paper & Pencil: Equivalency. (A group of fractions that they must evaluate their magnitude, identify their fraction representation, and compare all three fractions.) EXP: Here is a sheet of paper with five tray’s of brownies.


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The first one is a whole tray of brownies. The second is the same tray split for two people. The third is the same tray split for four people. The fourth is the same tray split for six people. The fifth and sixth trays are the same and both split for eight people. EXP: Are the parts all the same size for each dish/pan? ❐ Correct ❐ Incorrect EXP: Explain it to me? [Scaffolding] If they do not answer this correctly, point out to the learner that as you increase the number of parts, the size of each changes. Have them identify if the size is getting smaller or bigger? Next, we use their illustrations of the dishes/pans (A and B) and their resulting fractions {(2/2), (4/4), (6/6), (8/8)} to inquire the following: (Comparisons: (1) A pcs. > B pcs.; then (2) A pcs. < B pcs.; (3) A pcs = B pcs) EXP: So, using the all these trays of food, let’s see what we can figure out… EXP: If one person gets __ part(s) from this food (e.g., 2/2) and another person gets __ parts from this food (e.g., 8/8), who’s eating more? How do you know? Show me. 1. 1 part; (of the 2/2 tray) vs. 3 parts; (of the 8/8 tray)? ❐ Correct ❐ Incorrect 2. 4 parts; (of the 6/6 tray) vs. 3 parts; (of the 4/4 tray)? ❐ Correct ❐ Incorrect 3. 1 part (of the 4/4 tray) vs. 2 parts; (of the 8/8 tray)? ❐ Correct ❐ Incorrect General Questions. (These question are open ended to explore their conceptualizations without scaffolding their representations) EXP: What does it mean for two different fractions to be equal? ❐ Correct ❐ Incorrect (This is an open ended question to see what they know about equivalency.) [Scaffolding] The experimenter can use question 3 above as a reference if the student is unable to answer or understand this question.

Proscriptive Activity.


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In these activities, students are given numeric representations of fractions and allowed to choose from three different representations of numbers (strings (paths), cups (unitized containers) or clay bars (segmented object).

[Differentiation] If students demonstrate immediate proficiency, the researcher can move from denominators based in 2n to other denominators (indicated by the OR set)

Manipulatives: Unitized {1/1, 2/2, 3/3, 4/4, 5/5, 6/6, 7/7, 8/8, 9/9, 10/10, 12/12,14/14, 16/16, 18/18} a) Number line/Path – equally-‐lengthened segments of string, varying tic marks, to be colored with marker b) Containers – equal-‐sized clear containers, varying tic marks; to be filled with liquid GIVEN: the whole of any manipulative are the same size, such that x/x=1; GOALS: 1. demonstrate equivalency between various symbolic representations, such that (a/b) = (c/d) (e.g., 6/10 = 3/5 2. identify the whole; enumerate its parts

a) (b/b) , where b represents the whole b) (a/b) , where a represents the part (container/unitized container) c) (a/b) = (c/d) (e.g., 6/10 = 3/5)

PROBLEMS:

1. Set 1A {1/2, 3/4, 2/3}; Set 1B {5/8, 3/4, 7/14} 2. [Time Permitting] Set 2A {4/10, 3/9, 5/12}; Set 2B {4/10, 3/9, 5/12} 3. Set 3A {1/2, 7/14, 7/8}; Set 3B {2/3, 7/10, 8/12}

EXP: Okay, in this activity, we are going to be mathematicians.

We are going to get some numbers and figure out what they mean. As mathematicians, we can’t just know it ourselves; we have to explain it to our fellow mathematicians. We have to make sure our co-‐workers understand what we are thinking to find out if they agree or disagree with us.

Here is our first collection of numbers. EXP: [Within the Group] Okay, do we have all the fractions? Okay, before we even start, let’s look at the numbers.

Can you, as a group, make a prediction (smallest to biggest) about these fractions just from the numbers?

EXP: [To the group: choose a manipulative]


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Okay, which one of these (strings, cups, or clay) do you want to use? [Scaffolding] Suggest a manipulative if they are indecisive. EXP: [To the group: Determining the whole]

Now each of you pick the _________(string, cup, clay) from the set that you can use to represent each of the three fractions.

[Scaffolding] If they have trouble figuring out which to use, you can help each participant select and then continue. EXP: [To the group: Determining the parts] All right, everybody got his or her card with your fraction on it. Let’s find out about these numbers. EXP: [Ask each participant] (Researcher’s goal is to get the learner to discuss the whole and its part) How many parts are in your whole fraction? ❐ Correct ❐ Incorrect Show me how you know by using your (cup, string, clay)?

[Scaffolding] Point to the denominator of their fraction as written. Help them count the number of parts if they need assistance.

EXP: How many parts out of your whole fraction do you need? ❐ Correct ❐ Incorrect Show me how you know by using your (cup, string, clay)? [Scaffolding] Point to the numerator as a reference to determine the number they need and count the total number of partitions with the learner. Help them map the number onto the parts.

EXP: STRING: a) Color in the number of parts on the string U.CONT b) Fill in your cup with the right amount of liquid. ❐ Correct ❐ Incorrect

[Scaffolding] If they are unable to complete this task as a group, assist them in using their manipulative and determining their correct quantity.

[Scaffolding] If the students do not mention denominators or numerators, you can provide these labels for them and associate them to their quantities in each fraction.

EXP: Cool, now everyone has their fractions. For our research, we need to figure out how these fractions compare to one another. Lets use our (cups, strings, clay) to compare their sizes. Show me which is smallest, the biggest, and the one in between?


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[Scaffolding] If they are unable to complete this task as a group, assist them in using their manipulative and determining their correct quantity. Have them reference their fractions, show them the relations between the numbers and the size of their manipulative, and have them physically compare the manipulatives to one another. EXP: Which is the biggest? ❐ Correct ❐ Incorrect The smallest? ❐ Correct ❐ Incorrect Which one is in between? ❐ Correct ❐ Incorrect Are any of them equal? ❐ Correct ❐ Incorrect How do you know? [Open Questioning]

EXP: As a means for exploring more about the kids own conceptions of their gestures, at the end of this interview, the experimenter can ask open questions about any specific gestures that they learner provided. For example-‐ Exp: “I noticed that you use this gesture a lot. Tell me why you use that gesture when you talk about fractions?”


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Appendix B: Post-‐Test

Name: ______________________ Date of Birth: ______________ DIRECTIONS: Write the fraction that describes the shaded section of each figure below.

1.

2.

3.

DIRECTIONS: Shade each figure below to match the given fraction next to it.

4.

5.

6.


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DIRECTIONS: For each pair of fractions below, circle the one that is larger . If the fractions are equal, put an equals sign between them. OR, you can use the symbols < , > , =

7.

8.

9.

10.

M3 AERA Paper F · 2014-04-25 · 2014 AERA National Conference: Mathematical Cognition:...

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