Cravalho, Patrick

Slusser, Emily

English

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San Jose State University SJSU ScholarWorks Faculty Research, Scholarly, and Creative Activity 8-7-2020 Size vs. Number: Assigning Number Words to Discrete and Continuous Quantities Emily Slusser San Jose State University, emily.slusser@sjsu.edu Patrick Cravalho San Jose State University, patrick.cravalho@sjsu.edu Follow this and additional works at: https://scholarworks.sjsu.edu/faculty_rsca  Part of the Developmental Psychology Commons, and the Science and Mathematics Education Commons Recommended Citation Emily Slusser and Patrick Cravalho. "Size vs. Number: Assigning Number Words to Discrete and Continuous Quantities" Learning to count: New insights on the acquisition of symbolic numerical knowledge: Mathematical Cognition and Learning Society 2020 Conference (2020). This Presentation is brought to you for free and open access by SJSU ScholarWorks. It has been accepted for inclusion in Faculty Research, Scholarly, and Creative Activity by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu. Mathematical Cognition and Learning Society 2021 Conference   Symposium Title Learning to count: New insights on the acquisition of symbolic numerical knowledge  Symposium Abstract Humans have developed symbolic systems to represent and efficiently manipulate numerical information. Children gradually learn the rules and principles characterising the number system and how number words and numerals represent exact numerical quantities. The acquisition of symbolic number knowledge is a long and error-prone process that occupies children for several years, starting from the age of two until the first years of primary school, and it represents a crucial stepping stone for future mathematical achievement. In the symposium, we provide new insights on the acquisition of the symbolic numerical knowledge throughout development, including in how children initially learn number words, how they understand that they refer to number (as opposed to other dimensions of quantity), and how this achievement eventually leads to benefits in how number words relate to symbolic mathematics. The first talk (Slusser) explores the developmental trajectory of children’s understanding that number words refer to discrete numerical quantities, rather than continuous dimensions (e.g., “small”, “a lot”). Children fully grasp the discrete nature of number words when they are 3-knowers, at an intermediate stage toward the mastering of the cardinality principle.   The second talk (Krajcsi) revalidates the widely used Give-a-Number task, in which children create numerical sets according to a required number (e.g., “give me four apples”). The current literature describes a sudden change in performance and strategy implementation when children extend their cardinal knowledge beyond four (i.e., cardinal-principle knowers). Contrary to this prevailing view, the proposed re-evaluation highlights the presence of a smooth, rather than sudden, change in performance as children slowly acquire the numerical meaning of number words beyond four, challenging traditional accounts of number words being learned through a sudden insight into general counting principles.  The third talk (Sella) describes how the mastering of different numerical concepts relate to the understanding of the exact numerical magnitude represented by number words and Arabic numbers. The mastering of the predecessor knowledge (i.e., removing one item from a set leads to the preceding number word in the counting list) and the knowledge of the spatial order of numbers relate to the performance in number comparison tasks beyond the acquisition of the cardinality principle and the later-greater principle.  The fourth talk (Odic) shows how the eventual mastery of number words leads to other benefits in children’s numerical skills, most notably in children’s ability to associate symbolic number words with their intuitive sense of number (the Approximate Number System; ANS), with non-numeric dimensions (e.g., estimating the length of a line), and as a source of error detection in symbolic mathematics.  The combination of the four talks provides an overview of the development of symbolic numerical knowledge from children who have just begun learning number words to those who have largely mastered it while giving new insights challenging current theoretical views.    Organiser Francesco Sella Centre for Mathematical Cognition Loughborough University, UK Email: sella.francesco@gmail.com   Talk 1 Speaker: Emily Slusser Position: Associate Professor and Department Chair Institution: San Jose State University, US. Authors: Slusser, Emily1; Cravalho, Patrick1 1San Jose State University  Size vs. Number: Assigning Number Words to Discrete and Continuous Quantities Brief description of the talk: This talk explores when and how children come to understand that number words refer specifically to discrete numerosities (e.g., 1, 2, and 7), rather than various dimensions of continuous quantity (e.g., “small” in terms of surface area or “a lot” in terms of volume). Findings from a study of 2- to 4-year-old children indicate that number words are assigned exclusively to discrete numerosities at an intermediate stage of number word acquisition - after they have learned at least three number words (‘three-knowers’) - with evidence suggesting that executive functioning skills play an important role in this developmental trajectory.   Talk 2 Speaker: Attila Krajcsi Position: Professor Institution: ELTE Eötvös Loránd University, Hungary Authors: Krajcsi, Attila1 1 ELTE Eötvös Loránd University, Hungary  Revalidating the Give a Number task Brief description of the talk: The Give a Number task is an essential tool to measure preschoolers’ symbolic number knowledge. In a series of studies, reevaluating the original proposals by Wynn (1990, 1992), we found that contrary to the original findings there is no sudden increase in the task performance after understanding number 4. Additionally, strategy change between grabbing and counting can be observed, because only large numbers have to be counted; furthermore, the strategy does not change radically after understanding number 4. Based on these results, it might be reasonable to reevaluate the use of the task and former related findings of the literature.      Talk 3 Speaker: Francesco Sella Position: Lecturer Institution: Loughborough University, UK Authors: Sella, Francesco1; Lucangeli, Daniela2; Cohen Kadosh, Roi3; Zorzi, Marco4 1 Centre for Mathematical Cognition, Loughborough University, UK 2 Department of Developmental Psychology, University of Padova, Italy 3 Department of Experimental Psychology, University of Oxford, UK 4 Department of General Psychology, University of Padova, Italy  Learning the number sequence: preschool children make sense of numbers I will describe the acquisition of the cardinality principle, the mastering of successor and predecessor knowledge, the mapping between number words and external numerosities (ANS-to-word mapping), the spatial mapping of numbers and how all these abilities relate to the performance in number words and Arabic numbers comparison tasks. Our results revealed that, beyond the acquisition of the cardinality principle, the mastering of the predecessor knowledge and spatial order of digits relate to the understanding of the exact numerical magnitude represented by number words and Arabic digits.   Talk 4 Speaker: Darko Odic Position: Assistant Professor Institution: University of British Columbia, Canada Authors: Odic, Darko1; Dramkin, Denitza1; Wong, Harris1 1University of British Columbia, Canada  Developing the interface between number word and the intuitive number sense: challenges and benefits Young children have access to two distinct representations of number: an intuitive, but imprecise, perceptual number sense (ANS), and the slowly developing symbolic number words. In this talk, I show that the development of this interface is initially slow and protracted but, once children master the ability to translate between number words and the ANS, this gives them two key advantages for other numerical skills: (1) children can immediately form a novel number word interface with non-numeric dimensions, such as estimating area; (2) children can use their intuitive number sense as a source of error detection in formal mathematics.  Together, the talks argue that: a ) Children’s understanding of the discrete nature of number words only emerges at an intermediate stage toward mastering the cardinality principle; b) widely used tasks and their underlying models should be carefully reconsidered to obtain a more detailed understanding of the developmental stages children go through when learning number words; c) Learning the directional property of the counting list and the spatial order of numbers can scaffold the understanding of symbolic numerical magnitude; d) The ability to translate between number words and the ANS gives children crucial advantages for other numerical skills.  1Size vs. NumberAssigning Number Words to Discrete and Continuous QuantitiesEmily Slusser and Patrick CravalhoSan José State UniversityMCLS ConferenceAugust 7, 20202Overview• Background– Early Representations of Number– Number Knower-Levels– Quantity vs. Numerosity• Methods & Results– Give-N (Knower-Level)– Elephant Crocodile (Executive Functioning)– Transform Sets (Quantity v. Number)• Conclusions and Future Directions3Early Representations of NumberSmall-Exacte.g. Feigenson & Carey, 2003Large-Approximatee.g. Halberda et al. 20134Large, Exact Representations5Knower LevelsSarnecka, Goldman, & Slusser, 20146Sarnecka & Gelman (2004)Take Aways… The number word does not change when shaken or rotated, but does change when  an item is added or removed.7Slusser & Sarnecka (2011)8Number TrialsColor & Mood Trialsn=35 n=14** **Slusser & Sarnecka (2011)Take Aways… Match when prompted with color or mood (“find another picture with green turtles”). Do not match on numerositywhen prompted with a  number word until CP level. 9Size vs. Number• When and how do children come to understand that number words refer to discrete numerosity?• Do children think the number word changes as the individual items grow or shrink in size? • Influence of executive functioning (domain-general) and language and counting (domain-specific) skills?2- to 4-year-olds (n=45; 25 males; Mage= 3;10)10Give-N (Wynn 1990)Is that two fish?Can you put two            fish in the pond?Determine how many number words a child knows (Knower-Level)          and whether they understand the Cardinality Principle. 11Elephant-Crocodile (Leonard et al., 2014)When the elephant presses a button, press the samebutton that he presses. When the crocodile presses a button, press the other button, the button that he didn’t press. Measure children’s executive functioning skills (impulse control and cognitive flexibility). 12Transform Sets (Sarnecka & Gelman ‘04)I’m going to put six leaves in the box.Assess children’s understanding of number words –do number words refer to (discrete) numerosity?Now, are there five or six leaves in the box.16Transform Sets x Knower LevelThree/Four and CP Knowers understand that number labels change if, only if, an item is added or subtracted from a set. 17Number Trials x Knower LevelThis understanding is first applied to known numbers (“two” and “three” for Three-Knowers). 18Quantity Trials x Knower LevelOne Knowers may change number word labels in response to changes in continuous quantity (grow/shrink).19Number Trials x Executive FunctionEF correlates with performance on number trials.EF does not predict performance on quantity trials. 20ConclusionsChildren draw on AMS representations as they construct number concepts. AMS supports representation of discrete and continuous quantities. ∴ Children must figure out whether number words label numerosity or continuous quantity (often conflated).• Children appear to link number words to numerosity as they learn/induce the Cardinality Principle. • Executive functioning influences children’s developing understanding of number, but...? Thank You!emily.slusser@sjsu.eduMeganFosterAndreaMolinaAdrianaMontanoJulia DoanLeaWoodleyPatrick CravalhoGiovanna De Leon

Size vs. Number: Assigning Number Words to Discrete and Continuous Quantities

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Size vs. Number: Assigning Number Words to Discrete and Continuous Quantities

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