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While reading a science fiction book, Calculating God by Robert J. Sawyer, I came across the following lines:
“Well, if I showed you one object - one rock, say - you would not have to count it. You would just perceive its cardinality: you would know there was one object. The same thing happens with two objects. You just look at the pair of rocks and in a single glance, without any processing, you perceive that there are two of them present. You can do the same with three, four, or five items, if you are an average human. It is only when confronted with six or more items that you actually start counting them.”
“How do you know this?”
“I watched a program about it on the Discovery Channel.”
Assuming this is a real psychological phenomenon, is there a name or reference for it? I have contacted the author but have not received a response yet.
The author responded. It is called "Subitising" or "Subitizing" or "Perceiving cardinality at-a-glance". The term was coined by Kaufman in 1949.
The Mathematical Brain Across the Lifespan
Numerical cognition relies on interactions within and between multiple functional brain systems, including those subserving quantity processing, working memory, declarative memory, and cognitive control. This chapter describes recent advances in our understanding of memory and control circuits in mathematical cognition and learning. The working memory system involves multiple parietal–frontal circuits which create short-term representations that allow manipulation of discrete quantities over several seconds. In contrast, hippocampal–frontal circuits underlying the declarative memory system play an important role in formation of associative memories and binding of new and old information, leading to the formation of long-term memories that allow generalization beyond individual problem attributes. The flow of information across these systems is regulated by flexible cognitive control systems which facilitate the integration and manipulation of quantity and mnemonic information. The implications of recent research for formulating a more comprehensive systems neuroscience view of the neural basis of mathematical learning and knowledge acquisition in both children and adults are discussed.
Processing with Numbers Versus Letters
In this section, we focus on findings that describe how magnitude is calculated from letters versus numbers. Both letters and numbers are ordered left to right, follow the mental number line’s direction, are single digits, and are commonly used as labels. Specifically, research suggests that people process numbers and letters differently and that computing magnitude from letters is difficult. Numbers are readily associated with cardinal and ordinal properties (Kettle & Häubl, 2010 ), while letters have only readily available ordinal properties (Kara et al., 2015 ). Readily available cardinal properties enable inferring magnitude from items at once—for example, the magnitude between 9 and 16 is 7 units. Because numbers have a clear cardinal rule that each succeeding number is greater by 1 unit from the preceding number (Barrouillet & Fayol, 1998 ), calculating magnitude is easy.
By contrast, readily available ordinal properties only provide information on the position of the items in a sequence—for example, that A is in the first position, B is in the second position, and so on. Although people know that I comes before P in the letter sequence, they do not readily know how much further away P is from I (Jacob & Nieder, 2008 ). Instead, they must recall the entire sequence of the letters (i.e., the letters that succeed or precede), such as J, K, L, M, N, O, to calculate the magnitude of units from I to P (Jou, 2003 Klahr et al., 1983 ). We suggest that this lack of readily available cardinal information with letters makes inferring magnitude difficult. Next, we discuss how computational difficulty with letters affects distance perception in the preaction stage.
Preaction Stage: Naive Belief about Computational Difficulty
The preaction stage is defined as when a consumer perceives distance in a task before actually performing the task. For example, people perceive how far the airport gate is before actually starting to walk to the gate. Note that we do not suggest that people would be more or less accurate with numbers we are merely comparing people’s relative perception of distance when they use letter versus number cues.
The lack of readily available cardinality with letters makes computing magnitude from letters more difficult than numbers (Jou, 2003 Klahr et al., 1983 ). Previous research documents certain factors that can influence magnitude (or distance) between any two intervals (Dehaene et al., 1998 Restle, 1970 Whittlesea, 1993 Whittlesea & Williams, 2001 ). For example, research on the mental number line suggests that subsequent intervals of numbers are not always perceived as equally far apart but, instead, that numbers are represented by a spatial format (Dehaene et al., 1998 Restle, 1970 ), with the smaller magnitude being associated with the left side (negative numbers or numbers smaller in magnitude) and the larger magnitude with the right side (positive numbers or numbers larger in magnitude).
Thomas and Morwitz ( 2009 ) also found that because the closeness of stimuli makes magnitude perception difficult, the difficulty can be interpreted as a smaller magnitude. For example, when presented with two pairs of numbers, participants judged the magnitude of the price difference as smaller for pairs with greater computational difficulty (e.g., 4.97–3.96) than for pairs with lesser computational difficulty (e.g., 5–4), even though the magnitude of the computationally difficult pair was larger (e.g., 4.97–3.96, a magnitude difference of 1.01) than that of the computational easier pair (e.g., 5–4, a magnitude difference of 1.00).
Applied to our context of inferring distance from letter versus number cues, this naive belief would predict that people would perceive computationally difficult stimuli (i.e., letters) as closer (or shorter in distance) than relatively computationally easy stimuli (i.e., numbers). Thus, in the preaction stage, people would mistakenly attribute distance perception to factors rendered salient by naive theory about the computational difficulty. In the absence of actual feedback from the physical task, people would be forced to rely on this naive belief. Formally, we propose that in the preaction stage, the greater computational difficulty associated with letter cues will result in lesser distance perceptions than number cues.
This section discusses the theoretical mechanism underlying distance perception in the postaction stage. The postaction stage is defined as when people are performing or have already performed a task (e.g., walking, locating), using letter versus number cues. As mentioned previously, this research is confined to tasks that require physical performance. The actual physical experience and feedback received while performing the task would influence distance perception in this stage. Prior research has reliably associated feedback with the classic expectation–disconfirmation model (Oliver, 1977 , 1980 ). That is, feedback about the disconfirmation or violation of one’s expectations is received when one actually performs a task.
Prior research on the planning fallacy suggests that people are rather insensitive to contingencies and fail to incorporate them into their estimates (Buehler et al., 1994 ). Applied to current context, this research would predict that people start with an expectation of a shorter distance (with both letter and numbers cues) for the physical task because of the planning fallacy. However, unlike in the preaction stage, people receive ongoing feedback from the act of walking that informs them how much farther they need to walk in the postaction stage. Only after people start walking do they realize that the actual distance does not match their initial expectation. That is, there is a mismatch between their expectation and the actual distance. We propose that such expectation–disconfirmation is higher with letter cues than with number cues because the initial distance perception (as explained in the preaction stage) is lesser with letter cues than with number cues.
Therefore, applied to the context of inferring distance from letter cues in the postaction stage, we would predict a greater perception of distance than numbers. That is, people would perceive contrasting effects and a greater distance perception when their expectations are violated. By contrast, with number cues, expectation–disconfirmation is not as high, so they would assimilate and adjust their distance perception (relatively shorter). In summary, we suggest that the pattern of distance perceptions reverses from the preaction to the postaction stage because of the feedback about the violation of expectations received during the task performance. The metacognitive difficulty influences the preaction stage perception, whereas expectation–disconfirmation affects the postaction stage perception. Next, we empirically test our propositions to examine people’s perception of distance and the downstream influence on their decisions.
One popular doctrine in 20 th -century philosophy was conceptualism about perception. The core idea was that perceptual awareness is structured by concepts possessed by the perceiver. A primary motivation for conceptualism was epistemological: perception provides justification for belief, and this justificatory relation is only intelligible if perception, like belief, is conceptually structured (Brewer, 1999 McDowell, 1994 Sellars, 1956 ). We perceive that a is F, and thereby grasp perceptual evidence that justifies the belief that a is F and inferentially integrates with premises like If a is F then a is G to produce the belief that a is G.
Conceptualism is less popular today (cf. Bengson, Grube, & Korman, 2011 Mandelbaum, 2018 Mandik, 2012 ). The a priori justification for conceptualism has crashed face-first into a wall of empirical evidence. For instance, children and non-human animals possess perceptual capacities despite lacking many hallmarks of conceptual cognition (Bermudez, 1998 Burge, 2010a Block ms). Meanwhile, in adults, mental imagery and related phenomena implicate iconic rather than conceptual/propositional formats (Carey, 2009 Fodor, 2007 Quilty-Dunn, 2019a ). A growing contingent of theorists thus regard perception as a natural kind marked by its proprietary nonconceptual representations (Burge, 2014 Burnston, 2017a Carey, 2009 Kulvicki, 2015a Toribio, 2011 Block, ms see also Evans ( 1982 ) Hopp ( 2011 ) Peacocke ( 2001 ) for other nonconceptualist arguments).
Though opinion has shifted strongly in favor of nonconceptualism, it may be time for the pendulum to swing back. Putting the traditional normative motivations for conceptualism aside, it makes sense even from a purely descriptive, naturalistic perspective that at least some of the vehicles of perception should be conceptual. Many cognitive operations make use of concepts thus many cognitive responses to perception would be facilitated if some outputs of perception came prepackaged in a conceptualized format.
This point fits with modularity-based accounts of perception, and was fittingly made by Fodor in his discussion of input modules as “subsidiary systems” that must “provide the central machine with information about the world information expressed by mental symbols in whatever format cognitive processes demand of the representations that they apply to” (Fodor, 1983 , p. 40). Similarly, Mandelbaum argues that the outputs of modular perceptual systems ought to be conceptualized in order to “actually guide action by entering into other cognitive processes” ( 2018 , p. 271). It is an underemphasized explanatory virtue of modularity that it allows for a system to be distinctly perceptual (in virtue of its modularity) while outputting representations that are immediately consumable by cognition (in virtue of their format). Modularity-based versions of conceptualism thereby avoid full-fledged versions of the “interface problem” in interactions between perception, cognition, and action (Burnston, 2017b Butterfill & Sinagaglia, 2014 Mylopoulos & Pacherie, 2017 Shepherd, 2018 2019 ).
It is fully compatible with this modularity-based conceptualism that some perceptual processes output representations in nonconceptual (e.g., iconic) formats. Instead of insisting on conceptual structure as a transcendental epistemological requirement, modularity-based conceptualists can be pluralists about perceptual representation (Quilty-Dunn, 2019b ). As long as some significant component of perception is conceptual and feeds immediately into cognition, there is room for other perceptual representations to have other formats with other functional advantages. For example, perhaps iconic representations allow for richer, messier content to be encoded in perception, while sparse conceptual representations provide neatly packaged categorizations to central cognition.
However, perception is older than cognition. One might object that our perceptual systems evolved from creatures who lacked cognition, therefore there was no evolutionary pressure for concepts to figure in perception. In what follows, I'll sketch a version of conceptualism that posits concepts in perception independently of stimulus-independent cognitive abilities. In particular, I'll argue not only that adult humans have conceptually structured perceptual representations, but also that these conceptual outputs of perception constitute a natural representational kind found in children and animals alike. Perceptual object representations function to segment out particulars, track them, and predicate features of them, including conceptual categories. These object representations constitute an evolutionarily ancient and developmentally early source of predicate-argument propositional structure that is useful for (1) tracking individuals, (2) subsuming them under categories, and (3) distinguishing reference-guiding elements from pure attributions. These structures can function as evidential inputs to inferential processes in creatures that have the requisite inferential abilities.
I will first argue against stimulus-independence as a constitutive condition on conceptuality (Prinz, 2002 , p. 197 Beck, 2018 Burge, 2010b Camp, 2009 ) in favor of a Cartesian view that concepts are simply representations of a certain sort that, in principle, require no particular mental abilities for their instantiation in human and animal minds (Fodor, 2004 ). I'll then use empirical evidence to argue that, in fact, perceptual object representations are conceptualized propositional structures that develop (and likely evolved) prior to creatures’ abilities to use them in inference. The resulting picture preserves much of the letter—if not exactly the spirit—of traditional conceptualism.
The purpose of this brief article is to investigate the role of cardinality in children’s developing ability to interpret expressions of measurement. In recent decades, research on children’s developing understanding of number words has focused primarily on their knowledge of the cardinal principle, which says that the final tag in a count list carries special significance by indicating the cardinality of (or number of elements in) a set (Bloom & Wynn, 1997 Briars & Siegler, 1984 Carey, 2004, 2009a, b Fuson, 1988 Gelman & Gallistel, 1978 Gelman & Meck, 1983 Le Corre & Carey, 2007 Le Corre, et al., 2006 Syrett, Musolino, & Gelman, in press Wynn, 1990, 1992). There is a good reason for this: reference to exact set size represents a core component of number word meaning, differentiating it from lexical items with similar meaning and distribution (e.g., quantifiers such as some and other modifiers such as many, or several).
However, full mastery of number word meaning entails being able to correctly interpret phrases in which number words appear, which might not necessarily serve to pick out a set of objects in the real world where the number of objects, or cardinality, matches up with the exact number expressed. In fact, recent investigations of scalar implicatures in language acquisition has targeted instances where a number word does not necessarily pick out a set of that exact size, and may have an 𠆊t least’ interpretation (Hurewitz et al., 2006 Noveck, 2001 Musolino, 2004, 2009 Papafragou & Musolino, 2003). However, these cases still involve sets of discrete objects, and the research question concerns whether the grammar is structured in a way as to allow the sentence to be true when the cardinality expressed by the number word (e.g., two) is a proper subset of the cardinality corresponding to the relevant set of objects in the context (i.e., is more than two).
Number words can also appear in constructions in which the number word does not signal the cardinality of a set. A prime example is measure phrases (MPs). In a phrase such as 8-pound baby, the number word does not pick out the number of babies, but rather the total weight of the baby, which is not a set of objects in the world, the members of which can be verbally tagged. Thus, the road to becoming fully adult-like in the interpretation of natural language expressions with number words involves navigating through examples that seemingly diverge from the core aspect of number word meaning (the 𠆎xact’ interpretation) children strive so hard to master in the first four years.
In this article, I ask when children begin to correctly interpret such expressions of measurement, and what factors account for the instances when their interpretations diverge from those of adults. Here I focus on one factor in particular – cardinality – by manipulating the count-mass status of the target items. I further narrowed the focus to attributive MPs, such as 8-pound X. In such cases, the number word is prenominal and prosodically prominent, but does not necessarily serve to pick out a set of discrete objects with an exact cardinality. Moreover, the real-world referent is expressed by the second noun (X), rather than the one immediately following the number word. Such cases would thus appear to present a special challenge to the language learner. In two sets of experiments, I shed light on this challenging aspect of language development, demonstrating that while four-year-olds exhibit a developing command of the syntax-semantics mapping of attributive MPs, their performance in experimental tasks tapping into this knowledge is mediated by their tendency to interpret the number word in the MP as referring to cardinality of a contextually-relevant set of objects.
A Word From Verywell
While the cardinal traits are considered among the most dominant of characteristics, they are also quite rare. Few people are so ruled by a singular theme that shapes the course of their entire life.
The trait theories of personality suggest that each person’s personality is composed of a number of different characteristics. While early conceptualizations of the trait approach suggested hundreds or even thousands of traits existed (such as Allport’s approach), modern ideas propose that personality is composed of approximately five broad dimensions.
In our review, we argue that the development of infants’ communicative gestures is driven by infants’ perceptual sensitivities toward human motion, couplings between the vocal and motor systems and infants’ interactive experience. To demonstrate such complexity of the driving force, we need to broaden the concept of gestures. By definition, gestures are behaviors communicating intentions. However, the period of infancy is one of asymmetric communication between less and more competent partners. The infant, as the less competent partner, relies on resources in the physical and social environment that shape the meaning of communication. Hence, to be interpreted as communicative, infants’ gestures do not need to be intentional. They need to be, however, part of a social engagement toward a joint goal. For example, Reddy et al. (2013) demonstrated that 3-month-olds adjusted their posture in anticipation of being picked up when an adult was approaching. In our view, such adjustments as gaze direction, turning the head or lifting the arms within social interactions (Reddy et al., 2013) are gestures communicating the recognition of (and contribution to) the joint action of picking up. From our perspective, a still unsolved question in research on gesture development is how infants are guided to use more conventionalized communicative means that serve interactional purposes. In our review, we will focus on a gesture that is considered a milestone in communication development, namely pointing, review the production and comprehension development, and address its possible routes of conventionalization.
Musolino, J. (2015). The Soul Fallacy, Published by Prometheus Books, Amherst, NY and distributed by Random House.
“Beautifully written, impeccably researched, and compassionate” — Victor J. Stenger
Musolino, J., Sommer, J., and P. Hemmer (Eds) (under contract) The Science of Beliefs: A Multidisciplinary Approach. Cambridge University Press.
Persaud, K. &Hemmer, P. (2016). The Dynamics of Fidelity over the Time Course of Long-Term Memory. (Link to paper here.)
Hemmer, P., Tauber, S. & Steyvers, M. (2015). Moving beyond qualitative evaluations of Bayesian models of cognition. Psychonomic Bulletin & Review, DOI 10.3758/s13423-014-0725-z (Contact authors) (PDF)
Hemmer, P. & Persaud, K. (2014). Interactions between categorical knowledge and episodic memory across domains. Frontiers in Psychology 5 :584. doi: 10.3389/fpsyg.2014.00584. (PDF) Hemmer & Persaud Frontiers 2014
Hemmer, P. & Steyvers, M. (2009a). Integrating Episodic Memories and Prior Knowledge at Multiple Levels of Abstraction. Psychonomic Bulletin & Review, 16, 80-87. Hemmer_Tauber_Steyvers_PB&R 2014 Hemmer_Steyvers_PB&R
Hemmer, P. & Steyvers, M. (2009b). A Bayesian Account of Reconstructive Memory. Topics in Cognitive Science, 1, 189-202. Hemmer_Steyvers_topiCS
Hemmer, P., Steyvers, M., & Miller, B. (2010). The Wisdom of Crowds with Informative Priors. In S. Ohlsson & R. Catrambone (Eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society (pp 1130-1135). Austin, TX: Cognitive Science Society. http://www.cognitivesciencesociety.org/cogsci-archival-conference-information/ Hemmer_Steyvers_Miller_2010
Hemmer, P. & Steyvers, M. (2009c). Integrating Episodic and Semantic Information in Memory for Natural Scenes. In N.A. Taatgen & H. van Rijn (Eds.), Proceedings of the 31th Annual Conference of the Cognitive Science Society (pp. 1557-1562). Austin, TX: Cognitive Science Society. HemmerSteyversCogSci2009 HemmerSteyversCogsci2009_resubmit
Steyvers, M., Lee, M.D., Miller, B., & Hemmer, P. (2009d). The Wisdom of Crowds in the Recollection of Order Information. In J. Lafferty, C. Williams (Eds.) Advances in Neural Information Processing Systems, 23, (pp 1785-1793). MIT Press. NIPS_2009
Miller, B., Hemmer, P. Steyvers, M. & Lee, M.D. (2009e) The wisdom of crowds in rank ordering tasks. Proceedings of the 9th International Conference of Cognitive Modeling. MillerHemmerSteyversLeeICCM2009
Hemmer, P. & Steyvers, M. (2008). A Bayesian Account of Reconstructive Memory. In V. Sloutsky, B. Love, and K. McRae (Eds.) Proceedings of the 30th Annual Conference of the Cognitive Science Society. Mahwah, NJ: Lawrence Erlbaum. (pp. 327-332). [Award: Cognitive Science best paper on modeling higher-level cognition] HemmerSteyversCogSci2008 (2)
Hemmer, P. & Persaud, K. (2014). Interactions between Categorical Knowledge and Episodic Memory across Domains. Frontiers in Psychology, 5:584. doi: 10.3389/fpsyg.2014.00584.
Steyvers, M. & Hemmer, P. (2012). Reconstruction from Memory in Naturalistic Environments. In Brian H. Ross (Ed), The Psychology of Learning and Motivation, (126-144). Elsevier Publishing Bookchapter_published
Hemmer, P., Persaud, K., Venaglia, R.*, & DeAngelis, J.* (2014). What’s your source: Evaluating the effects of context in episodic memory for objects in natural scenes. In P. Bello, M. Guarini, M. McShane, & B. Scassellati (Eds.), Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp 2765-2770). Quebec City, CA: Cognitive Science Society. https://mindmodeling.org/cogsci2014/papers/478/ Hemmer_Persaud_Venaglia_DeAngelis_CogSci_2014
Hemmer, P., Tauber, S., & Steyvers, M. (2015). Moving beyond qualitative evaluations of Bayesian models of cognition. Psychonomic Bulletin & Review, 22, 614-628. Hemmer_Tauber_Steyvers_PB&R 2014
Robbins, T., Hemmer, P., & Tang, Y.* (2014). Bayesian Updating: A Framework for Understanding Medical Decision Making. In P. Bello, M. Guarini, M. McShane, & B. Scassellati (Eds.), Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp1299-1304). Quebec City, CA: Cognitive Science Society. https://mindmodeling.org/cogsci2014/papers/229/
Robbins, T. & Hemmer, P. (2017). Explicit predictions for illness statistics. In Gunzelmann, G., Howes, A., Tenbrink, T., & Davelaar, E. (Eds.), Proceedings of the 39th Annual Meeting of the Cognitive Science Society. London, UK: Cognitive Science Society, (pp. 2998-3003). Robbins & Hemmer 2017_Cognitive Science Proceedings
Robbins, T. & Hemmer, P. (2018). Lay Understanding of Illness Probability Distributions. In T.T. Rogers, M. Rau, X. Zhu, & C. W. Kalish (Eds.), Proceedings of the 40th Annual Conference of the Cognitive Science Society (pp. 2346-2351). Austin, TX: Cognitive Science Society. Persaud_Macias_Hemmer_Bonawitz19_FINALmanuscript distribution builder cog sci final
Kleinschmidt, D. F., & Hemmer, P. (2019). A Bayesian model of memory in a multi-context environment. In A. Goel, C. Seifert, & C. Freksa (Eds.), Proceedings of the 41st Annual Conference of the Cognitive Science Society. cogsci_Dave K_draft
Wall, D.G., Chapman, G., & Hemmer, P. (2018). Risky Intertemporal Choice with Multiple Outcomes and Individual Differences. In T.T. Rogers, M. Rau, X. Zhu, & C. W. Kalish (Eds.), Proceedings of the 40th Annual Conference of the Cognitive Science Society (pp. 2645-2650). Austin, TX: Cognitive Science Society.
Persaud, K., Hemmer, P., Kidd, C. & Piantadosi, S. (2017). Seeing colors: Cultural and environmental influences on episodic memory. i-Perception, 1-5. doi: 10.1177/2041669517750161 Seeing Colors_ Cultural and Environmental Influences on Episodic Memory
Persaud, K. & Hemmer, P. (2016). The Dynamics of Fidelity over the Time Course of Long-Term Memory. Cognitive Psychology, 88, 1-21.
Persaud, K. Macias, C., Hemmer, P., & Bonawitz, E. (2019). Age-related differences in the influence of category expectations on episodic memory in early childhood. In A. Goel, C. Seifert, & C. Freksa (Eds.), Proceedings of the 41st Annual Conference of the Cognitive Science Society. Persaud_Macias_Hemmer_Bonawitz19_FINALmanuscript
Hemmer, P., Persaud, K., Kidd, C., & Piantadosi, S. (2015). Inferring the Tsimane’s use of color categories from recognition memory. In Noelle, D. C., Dale, R., Warlaumont, A. S., Yoshimi, J., Matlock, T., Jennings, C. D., & Maglio, P. P. (Eds.), Proceedings of the 37th Annual Meeting of the Cognitive Science Society (pp 896-901). Austin, TX: Cognitive Science Society. https://mindmodeling.org/cogsci2015/papers/0161/paper0161.pdf HemmerPersaudKiddPiantadosi_CogSci2015
Persaud, K. & Hemmer, P. (2014). The influence of knowledge and expectations for color on episodic memory. In P. Bello, M. Guarini, M. McShane, & B. Scassellati (Eds.), Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp 1162-1167). Quebec City, CA: Cognitive Science Society. https://mindmodeling.org/cogsci2014/papers/206/
Persaud, K., McMahan, B., Alikhani, M., Pei, K.*, Hemmer, P., & Stone, M. (2017). When is Likely Unlikely: Investigating the Variability of Vagueness. In Gunzelmann, G., Howes, A., Tenbrink, T., & Davelaar, E. (Eds.), Proceedings of the 39th Annual Meeting of the Cognitive Science Society. London, UK: Cognitive Science Society, (pp. 2876-2881). Persaud McMahan Alikhani Pei Hemmer & Stone 2017_Cognitive Science Proceedings
Cox, G. E., Hemmer, P., Aue, W. R., & Criss, A. H. (2018). Information and processes underlying semantic and episodic memory across tasks, items, and individuals. Journal of Experimental Psychology: General, 147, 545-590. doi: 10.1037/xge0000407
Hemmer, P. & Criss, A. (2013). The Shape of Things to Come: Evaluating Word Frequency as a Continuous Variable in Recognition Memory. Journal of Experimental Psychology: Learning, Memory, & Cognition, 39, 1947-52. doi: 10.1037/a0033744 Hemmer & Criss 2013 JEP_LMC
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Musolino, J, and Landau, B. (2010). When theories don’t compete. Language Learning and development, 6(2), 170-178.
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Preschool number activities: How do you introduce numbers to preschoolers?
Preschool number activities often involve counting, but merely reciting the number words isn't enough.
Children also need to develop "number sense," an intuitive feeling for the actual amount associated with a given number.
Where does number sense come from?
Experiments suggest that even 6-month-old infants can tell the difference between 4 cookies and 8. And 14-month-old babies seem to grasp that counting tells us something about quantity (Wang and Feigenson 2019).
So that's a start. What's needed -- as kids get older -- are hands-on experiences. Inspired by research, the following games encourage kids to think about several key concepts, including
- The one-to-one principle of numerosity (two sets are equal if and only if their items can be placed in perfect, one-to-one correspondence)
- The principle of increasing magnitudes (the later number words refer to greater numerosities)
- The one-to-one principle of counting (each item is to be counted is counted once and only once)
- The stable order principle (number words must be recited in the same order)
- The cardinal principle (the last word counted represents the numerosity of the set)
As your child engages in these preschool number activities, keep in mind this advice (from my evidence-based guide to preschool math lessons ):
Start small. It's important to adjust the game to your child's attention span and developmental level. For beginners, this means counting tasks that focus on very small numbers (up to 3 or 4).
Keep it fun. If it's not playful and fun, it's time to stop.
Be patient. I t takes young children about a year to learn how the counting system works.
Six evidence-based preschool number activities
1. Matching sets: Teaching the one-to-one principle of numerosity
Matching items one-to-one is a surprisingly important mathematical concept. It's how we prove that two quantities are equal. Two sets contain the same number of items if the items in each set can be matched, one-to-one, with no items left over.
Researchers call this the "one-to-one principle of numerosity," and you can help kids master the concept with these simple, preschool number activities.
First, present kids with a small set of tokens arranged on a table or floor.
Then ask them to create an identical copy of this set using additional tokens. When finished, make a count of the items in each set -- the original and the copy.
Second, you can present kids with two sets at once.
In this case, make sure each set contains the same number of tokens, but arrange the tokens in different spatial patterns. Then have your child reproduce both of these sets, and do an end count to confirm that all sets are equal.
It can take almost a year for a two- or three-year-old to really understand how the counting system works, so don't be surprised if younger children have trouble counting beyond "1-2-3." Help kids with counting if needed, and challenge them to a greater number of tokens as their skills grow.
For another approach to these games, use printed cards, each with a picture depicting a set of dots or other small items.
The child views the card and creates a matching set of items using tokens. You can make the cards yourself, or buy some ready-made.
What should you use for tokens? For children under the age of three years, it's important to choose something that won't pose a choking hazard.
According to the U.S. Consumer Product Safety Commission, a ball-shaped object is unsafe for children under 3 years if the item is smaller than a 1.75" diameter golf ball. Other objects are unsafe if they can fit inside a tube with a diameter of 1.25" inches. Pieces from your toddler's building block set might do the job.
Also, try to stick with plain-looking tokens and card symbols.
You might think that little toy frogs or spiders would make counting more fun. But researchers have found that young children tend to get distracted by these details.
Kids learn more from preschool number activities when they manipulate simpler, more abstract items (Petersen and McNeil 2012). Plastic chips -- like those used for poker or bingo -- are a good choice for kids aged 3 and up.
2. Sharing at the tea party: Dividing up tokens into equal portions
Here is another activity to help kids practice one-to-one matching, inspired by the research of rian Butterworth and his colleagues (2008).
Choose three toy creatures to play the part of party attendees, and have your child set the table for them. Then give your child a set of "goodies" (tokens or real edibles) to share with the party guests. The total number of goodies should be a multiple of 3, so your child can distribute all the items equally and have no leftovers.
If your child makes a mistake and gives one creature too many tokens, you can play the part of another creature and complain.
You can also play the part of tea party host and deliberately make a mistake. Ask for your child's help? Did someone get too many tokens? Or not enough? Have your child fix it.
Once your child gets the hang of things, try providing him with one token too many and discuss what to do about this leftover.
One solution is to divide the remainder into three equal bits. But your child may come up with other, non-mathematical solutions, like eating the extra bit himself.
3. Sorting by quantity: Teaching the principle of increasing magnitudes
For these preschool number activities, use cards like those described in #1. You can use them in three ways.
Game one: Guess the right order.
To play this game, shuffle the cards, and then ask your child to place them, side by side, in a sequence of increasing magnitude.
For children who haven't yet learned to count, use cards that vary by a substantial amount, e.g., 3, 6, 10, and 15.
For children with emerging counting skills, use cards that differ by a single dot, and have kids guess first, then check their answers by counting.
What's the point of all this guesswork?
Experiments show that even babies can spot differences this large, and practicing these tasks may help children hone their estimation abilities -- abilities which are essential for future mathematics achievement.
For example, in a recent study, researchers tested five-year-olds with computer-based versions of these preschool number activities. The children weren't given enough time to count they simply took a quick look and answered based on their intuitive, visual impression.
Kids who practiced making progressively more difficult discriminations -- getting accurate feedback after each attempt -- experienced subsequent improvements in their ability to solve problems using symbolic numbers (Wang et al 2016).
Game two: Guess which card has more dots?
To play this game, select two cards, each displaying a different number of dots, and show them to your child. Which card has more dots?
Make sure you start with cards that differ by a ratio of at least 2:1. For instance, try 1 vs. 2, 2 vs. 4, and 2 vs. 5. You can also try larger numbers, like 6 vs. 12.
As your child gets practice with these easy-to-discriminate differences, you can present her with increasingly difficult choices (like 6 versus 8 or even 9 versus 10).
For a more playful variant of the game, you can use tokens instead of cards, and pretend they are something fun, like cakes. Dole out different amounts between you and ask, "Who has more?"
Be sure to give your child feedback about the correct answer.
Game three: Big guys eat more.
To play, use your cards, as well as three soft animal toys or dolls of varying size -- small, medium, and large.
Pretend the toys are party guests, and the items on the cards treats. Then
- line up the three toys in order of size,
- present your child with three cards, each card depicting a different number of dots, and
- ask your child to give the greatest number of treats to the largest toy, the second-greatest number to the second-largest toy, and the smallest number to the smallest toy.
Tell your child when he responds correctly ("That's right!"), and, if he makes a mistake, guide him to make another attempt ("That's not right -- try again!").
If you prefer, you can play the game with tokens instead of cards. And once you child learns to read and understand number symbols, you can use cards that display only Arabic numerals.
When researchers tested similar preschool number activities, they found that both dot-based and numeral-based games helped children develop better intuitions about quantity. But kids who played the Arabic numeral version of the game experienced greater growth in basic arithmetic skills (Honoré and Noël 2016).Image of preschool number activities from study by Honoré and Noël (2016).
4. Spot the goof: Teaching the one-to-one principle of counting and the principle of cardinality
Here's another "one-to-one" principle -- this time the one-to-one principle of counting. Kids need to learn that each item in a series is counted once and only once. And they also need to learn the principle of cardinality, the idea that the last word in our count represents the numerosity of a set.
Children learn these ideas through practice. But they might also learn by correcting others who make mistakes.
In one study, researchers asked preschoolers to watch--and help--a rather incompetent puppet count a set of objects (Gelman et al 1986). The puppet would occasionally violate the one-to-one principle by double-counting (e.g., “one, two, three, three, four. ). He also sometimes skipped an object or repeated the wrong cardinal value.
Kids ranging in age from 3 to 5 were pretty good at detecting these violations. So your child might have fun correcting your own goofball at home.
What if your child doesn't notice an error? Correct the goofball yourself. And either way, ask your child to explain what went wrong. In another, similar study, researchers found that preschoolers didn't make conceptual progress unless they were asked to explain the puppet's mistakes (Muldoon et al 2007).
For a discussion of how self-explanation can make preschool number activities and other educational experiences more valuable, see this Parenting Science review of the evidence.
5. One less / one more: Helping preschoolers develop intuitions about addition and subtraction
Young children have a long way to go before they are ready to perform basic arithmetic calculations like "2 +3 = 5," or "7 - 3 = 4." But research suggests we can help pave the way with preschool number activities like these.
Have a puppet or other toy character "bake cakes" (a set of tokens) and ask your child to count the treats. (You can count together if your child needs help.) Next, have the puppet bake one more cake and add it to the set.
Are there more cakes or fewer cakes now?ਊsk your child, and provide him with correct feedback afterwards.
Try the same thing with subtraction by having the puppet "eat" a cake. And vary the game by adding or subtracting other small amounts, like two or three.
Should we expect children to come up with accurate answers? Not necessary -- especially not if they are under the age of three years (Izard et al 2014).
But the experience of predicting and checking is valuable, and even when kids get the precise number wrong, they do a good job coming up with reasonable guesses. When researchers asked 3-, 4- and 5-year-olds to perform these tasks, they found that 90% of the guesses were in the right direction (Zur and Gelman 2004).
6. The Big Race: Increasing magnitudes and the number line
As your child begins to master the first few number words, you can also try these research-tested preschool number activities for teaching kids about the number line.
Copyright © 2006-2021 by Gwen Dewar, Ph.D. all rights reserved.
For educational purposes only. If you suspect you have a medical problem, please see a physician.
References: Preschool number activities
Butterworth B, Reeve R, and Lloyd D. 2008. Numerical thought with and without words: Evidence from indigenous Australian children. Proceedings of the National Academy of Sciences 105(35): 13179-13184.
Gelman R, Meck E, and Merkin S. 1986. Young children's numerical competence. Cognitive Development 1(1): 1-29.
Honoré N and Noël MP. 2016. Improving Preschoolers' Arithmetic through Number Magnitude Training: The Impact of Non-Symbolic and Symbolic Training. PLoS One. 11(11):e0166685.
Izard V, Streri A, Spelke ES. 2014. Toward exact number: young children use one-to-one correspondence to measure set identity but not numerical equality. Cogn Psychol. 72:27-53.
Muldoon KP, Lewis C, Francis B. 2007. Using cardinality to compare quantities: the role of social-cognitive conflict in early numeracy. Developmental Psychology 10(5):694-711.
Park J and Brannon EM. 2013. Training the Approximate Number System Improves Math Proficiency. Psychol Sci. 2013 Oct24(10):2013-9.
Petersen LA and McNeil NM. 2013. Effects of Perceptually Rich Manipulatives on Preschoolers' Counting Performance: Established Knowledge Counts.hild Dev. 84(3):1020-33.
Wang JJ and Feigenson 2019. Infants recognize counting as numerically relevant. Developmental Science 22(6): e12805.
Zur O and Gelman R. 2004. Young children can add and subtract by predicting and checking. Early childhood Research Quarterly 19: 121-137.
Content of "Preschool number activities" last modified 9/17
Image credits for "Preschool number activities"
title image of young children with number board by Ivan Radic / flickr
images of supplies for preschool number activities copyright Parenting Science
image of teddy bear tea party by Virginia State Parks
photograph of child looking at bears by Tom Page / flickr
image of bears and bags courtesy N. Honoré ਊnd MP Noël / PLos One 2016
Louis Guttman was born in Brooklyn, New York, on February 10, 1916, to Russian immigrant parents. When he was 3 years old the family moved to Minneapolis where he completed his formal education. His father was a self-taught amateur mathematician who published several papers in the American Mathematical Monthly.
Although skilled in mathematics he decided to major in sociology. His studies also included courses in psychology equivalent to a major in the field. Upon realizing the importance of statistics to social research, he returned to his study of mathematics, of which he had a solid knowledge. This led him to formalize—while still a graduate student—techniques for data analysis, some of which were published in 1941 and constitute the foundations of his later work on scale theory, factor analysis, and other topics (his first publication was in 1938 in Regression Analysis). Guttman attained his B.A. (1936), M.A. (1939), and Ph.D. (1942) degrees in sociology from the University of Minnesota. Guttman's doctoral dissertation constituted original formulations to the algebra of matrices in general and to the algebra of factor analysis in particular.
For 1 year he studied at the University of Chicago (a predoctoral fellowship), which was at that time the only place in the United States where factor analysis was taught. His ties with Samuel Stouffer began at that time. His theorems concerning communalities that he developed while a graduate student are still part and parcel of the basics of factor analysis theory. The issues associated with factor analysis and reliability theory were addressed by Guttman in a series of articles that appeared between 1941 and 1961. Some 15 years later, in the mid-1970s, the factor indeterminacy problem led to controversial publications that Guttman called “the Watergate of Factor Analysis.” His two then-unpublished papers concerning this issue can be found in Levy's 1994 compilation of his work.
In the 1970s, he also summarized his insights regarding statistics in a paper entitled “What Is Not What in Statistics” (1977). In this paper, he unmasks maltreatments of statistics and conceptualization in the social sciences by use of aphoristic and challenging headings such as “Partial Correlation Does Not Partial Anything,” “Proportion (or Percentage) of Variance Is Never ‘Explained’,” and “Nominal, Interval and Ratio Scales Are Not Scales.” With these, he not only exposed the naive lack of understanding of researchers, but indicated that most data analysis techniques, in particular statistical inference, do not provide answers to substantive problems and are irrelevant to replication, which is the “heart of cumulative science.”
Of special importance are the monotonicity coefficients developed by Guttman. Of these, the weak monotonicity coefficient, denoted by μ2 can be compared directly to Pearson's product-moment correlation coefficient. μ2 expresses the extent to which replies on one variable increase in a particular direction as the replies to the other variable increase, without assuming that the increase is exactly according to a straight line (as in Fig. 1 below). It varies between −1 and +1. μ2 = +1 implies a perfect monotone trend in a positive direction μ2 = −1 implies a perfect monotone trend in a negative or descending direction. By weak monotonicity is meant that ties in one variable may be untied in the order without penalty. Unlike Pearson's product-moment coefficient, μ2 can equal +1 or −1, even though the marginal distributions of the two variables differ from one another. Therefore, it is especially appropriate in conditions, as in the social sciences, in which marginal distributions differ from item to item.
Figure 1 . A simple function of variable y on variable x: the monotone trend.
The formula for weak μ2 is as follows:
When Pearson's coefficient equals + 1.00, 0, or − 1.00, the weak monotonicity coefficient μ2 for the same data will have the same values. In all other cases, the absolute value of μ2 will be higher than that of Pearson's coefficient, including the phi coefficient for dichotomies—in which case μ2 equals Yule's Q. Most similarity coefficients are special cases of Guttman's coefficient of monotonicity.
As for the problem of the misuse of statistical inference, Guttman developed a new kind of efficacy coefficient for comparing arithmetical means, based on the monotone concept. The coefficient is called a discrimination coefficient (DISCO). It is distribution-free, avoiding both unrealistic assumptions about the normality of population distributions and equality of variances within the populations. DISCO equals 1 if there is no overlap between the distributions, no matter how large the within variance. It equals 0 if there are no differences among the means.
He regarded none of the methods that he developed as sociology or psychology. It may come as a surprise to the uninitiated that theory, not method, was Guttman's primary interest. In fact, he saw the two as inseparable, arguing in 1982 that “A theory that is not stated in terms of the data analysis to be used cannot be tested.…the form of data analysis is part of the hypothesis.” Hence, his main interest lay in developing structural theories, which led to empirical lawfulness as evidenced in his definition of theory.
Guttman did not believe in mathematical psychology or mathematical sociology, and he rejected formalized axiomatic theories. The mathematics he used was not for the substantive theory but rather for technical features of data analysis. He stated that
Those who firmly believe that rigorous science must consist largely of mathematics and statistics have something to unlearn. Such a belief implies emasculating science of its basic substantive nature. Mathematics is contentless, and hence—by itself—not empirical science. … rather rigorous treatment of content or subject matter is needed before some mathematics can be thought of as a possibly useful (but limited) partner for empirical science. (Guttman 1991, p. 42)
During Guttman's service as expert consultant to the Secretary of War with the Research Branch of the Education Division of the U.S. Army in World War II, he developed scale analysis (1944) and techniques that were used for attitude research in many aspects of army life. These developments were documented and published in 1950 in the volume edited by Stouffer and colleagues, Measurement and Prediction (Vol. 4 of the American Soldier: Studies in Social Psychology).
Guttman's academic base during that period (beginning in 1941) was Cornell University, where he was associate professor of Sociology. There he met Ruth Halpern, a student at the upper campus, whom he married in 1943. In 1947 he received a postdoctoral fellowship from the Social Science Research Council (SSRC) of the United States, and they left for Palestine. They lived in Jerusalem, where their three children were born. Years later, his connections with Cornell University were renewed, and he served as a professor-at-large from 1972 to 1978.
The social science know-how accumulated during World War II at the Research Branch was useful in setting up a similar unit for the Hagana (the Jewish underground army in Palestine) while the British Mandate was expiring (1948). His collaboration with Uriel Foa began in this framework. The underground research group of volunteers developed into the Psychological Research Unit of the Israel Defence Forces, headed by Guttman. This unit provided the basis for the Behavioral Science Unit of the Israel Defence Forces. The Public Opinion Section of the army's Psychological Research Unit developed into the Israel Institute of Applied Social Research in 1955, of which Guttman was the scientific director. The history of the Institute, its unique continuing survey, and its scientific activities can be found in Gratch (1973) . In 1955, he became professor of Social and Psychological Assessment at the Hebrew University of Jerusalem. Louis Guttman passed away on October 25, 1987.
Guttman's work on factor analysis and other multivariate problems led him to nonmetric structural ideas such as the simplex, circumplex, and radex and to the idea of facets for definitional frameworks and structural theories. Guttman introduced the term circumplex in 1954 to indicate a particular kind of correlational pattern, having a circular arrangement. Since then, the circumplex model has been applied by many researchers to personal traits and interpersonal behaviors as an alternative to explanatory factor analysis.
The breakthrough for facet theory occurred in the mid-1960s when structural lawfulness in the area of intelligence tests was established, both with the advent of the electronic computer and with his collaboration with Lingoes, who was the first to program Guttman's nonmetric techniques. Only then did Guttman finally feel that he was “‘doing’ sociology and psychology,” as a flourishing facet theory helped to establish substantive structural laws of human behavior in a variety of areas such as well-being, adjustive behavior, values, intelligence tests, and involvement. Thus, he pioneered the road to a cumulative social science.
Even a brief inspection of his proposed tentative outline for his planned book on facet theory reflects the great variety of issues with which he was involved. A selection of a wide variety of Guttman's works, including the first two chapters from his unfinished book on facet theory, together with a full comprehensive and detailed bibliography (approximately 250 publications), is to be found in Levy's 1994 compilation.