Skip to main content
SpringerLink
Log in

Subitizing: The Neglected Quantifier

  • Chapter
  • First Online:
Constructing Number

Part of the book series: Research in Mathematics Education ((RME))

Abstract

We define and describe how subitizing activity develops and relates to early quantifiers in mathematics. Subitizing is the direct perceptual apprehension and identification of the numerosity of a small group of items. Although subitizing is too often a neglected quantifier in educational practice, it has been extensively studied as a critical cognitive process. We believe that subitizing also helps explain early cognitive processes that relate to early number development and thus deserves more instructional attention. We also contend that integrating developmental/cognitive psychology and mathematics education research affords opportunities to develop learning trajectories for subitizing. A complete learning trajectory includes three components: goal, developmental progression, or learning path through which children move through levels of thinking, and instruction. Such a learning trajectory thus helps establish goals for educational purposes and frames instructional tasks and/or teaching practices. Through this chapter, it is our hope that early childhood educators and researchers begin to understand how to develop critical educational tools for early childhood mathematics instruction. Through this instruction, we believe that children will be able to use subitizing to discover critical properties of number and build on subitizing to develop capabilities such as unitizing, cardinality, and arithmetic capabilities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aguirre, J., Herbel-Eisenmann, B. A., Celedón-Pattichis, S., Civil, M., Wilkerson, T., Stephan, M., … Clements, D. H. (2017). Equity within mathematics education research as a political act: Moving from choice to intentional collective professional responsibility. Journal for Research in Mathematics Education, 48(2), 124–147.

    Article  Google Scholar 

  • Antell, S. E., & Keating, D. P. (1983). Perception of numerical invariance in neonates. Child Development, 54, 695–701.

    Article  Google Scholar 

  • Ashkenazi, S., Mark-Zigdon, N., & Henik, A. (2013). Do subitizing deficits in developmental dyscalculia involve pattern recognition weakness? Developmental Science, 16(1), 35–46. https://doi.org/10.1111/j.1467-7687.2012.01190.x

    Article  Google Scholar 

  • Barendregt, W., LindstrÖm, B., Rietz-Leppänen, E., Holgersson, I., & Ottosson, T. (2012). Development and evaluation of Fingu: A mathematics iPad game using multi-touch interaction. Paper presented at the Proceedings of the 11th International Conference on Interaction Design and Children, Bremen, Germany.

    Google Scholar 

  • Baroody, A. J. (1986, December). Counting ability of moderately and mildly handicapped children. Education and Training of the Mentally Retarded, 21, 289–300.

    Google Scholar 

  • Baroody, A. J., Benson, A. P., & Lai, M.-l. (2003, April). Early number and arithmetic sense: A summary of three studies. Paper presented at the Society for Research in Child Development, Tampa, FL.

    Google Scholar 

  • Baroody, A. J., Lai, M.-L., & Mix, K. S. (2005, December). Changing views of young children’s numerical and arithmetic competencies. Paper presented at the National Association for the Education of Young Children, Washington, DC.

    Google Scholar 

  • Baroody, A. J., Lai, M.-l., & Mix, K. S. (2006). The development of young children’s number and operation sense and its implications for early childhood education. In B. Spodek & O. N. Saracho (Eds.), Handbook of research on the education of young children (pp. 187–221). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Baroody, A. J., Li, X., & Lai, M.-l. (2008). Toddlers’ spontaneous attention to number. Mathematical Thinking and Learning, 10, 240–270.

    Article  Google Scholar 

  • Beckwith, M., & Restle, F. (1966). Process of enumeration. Journal of Educational Research, 73, 437–443.

    Google Scholar 

  • Bonny, J. W., & Lourenco, S. F. (2013). The approximate number system and its relation to early math achievement: Evidence from the preschool years. Journal of Experimental Child Psychology, 114(3), 375–388. https://doi.org/10.1016/j.jecp.2012.09.015

    Article  Google Scholar 

  • Brownell, W. A. (1928). The development of children’s number ideas in the primary grades. Chicago: Department of Education, University of Chicago.

    Google Scholar 

  • Butterworth, B. (2010). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Sciences, 14, 534–541.

    Article  Google Scholar 

  • Cantlon, J. F., Brannon, E. M., Carter, E. J., & Pelphrey, K. A. (2006). Functional imaging of numerical processing in adults and 4-y-old children. PLoS Biology, 4, e125, 844–854.

    Article  Google Scholar 

  • Carper, D. V. (1942). Seeing numbers as groups in primary-grade arithmetic. The Elementary School Journal, 43, 166–170.

    Article  Google Scholar 

  • Chi, M. T. H., & Klahr, D. (1975). Span and rate of apprehension in children and adults. Journal of Experimental Child Psychology, 19, 434–439.

    Article  Google Scholar 

  • Chu, F. W., vanMarle, K., & Geary, D. C. (2013). Quantitative deficits of preschool children at risk for mathematical learning disability. Frontiers in Psychology, 4, 195. https://doi.org/10.3389/fpsyg.2013.00195

    Article  Google Scholar 

  • Clearfield, M. W., & Mix, K. S. (1999, April). Infants use contour length—not number—to discriminate small visual sets. Albuquerque, NM: Society for Research in Child Development.

    Google Scholar 

  • Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5, 400–405.

    Article  Google Scholar 

  • Clements, D. H., Battista, M. T., & Sarama, J. (2001). Logo and geometry. Journal for research in mathematics education monograph series (Vol. 10). Reston, VA: National Council of Teachers of Mathematics. https://doi.org/10.2307/749924

    Book  Google Scholar 

  • Clements, D. H., & Sarama, J. (1998). Building blocks—Foundations for mathematical thinking, pre-kindergarten to grade 2: Research-based materials development [National Science Foundation, grant number ESI-9730804; seewww.gse.buffalo.edu/org/buildingblocks/]. Buffalo, NY: State University of New York at Buffalo.

  • Clements, D. H., & Sarama, J. (Eds.). (2004a). Hypothetical learning trajectories [special issue]. Mathematical Thinking and Learning, 6(2), 81–89.

    Google Scholar 

  • Clements, D. H., & Sarama, J. (2004b). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6, 81–89. https://doi.org/10.1207/s15327833mtl0602_1

    Article  Google Scholar 

  • Clements, D. H., & Sarama, J. (2007). Building blocks—SRA real math teacher’s edition, grade PreK. Columbus, OH: SRA/McGraw-Hill.

    Google Scholar 

  • Clements, D. H., Sarama, J., & DiBiase, A.-M. (2004). Engaging young children in mathematics: Standards for early childhood mathematics education. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Clements, D. H., Wilson, D. C., & Sarama, J. (2004). Young children’s composition of geometric figures: A learning trajectory. Mathematical Thinking and Learning, 6, 163–184. https://doi.org/10.1207/s15327833mtl0602_1

    Article  Google Scholar 

  • Confrey, J. (1996). The role of new technologies in designing mathematics education. In C. Fisher, D. C. Dwyer, & K. Yocam (Eds.), Education and technology, reflections on computing in the classroom (pp. 129–149). San Francisco: Apple Press.

    Google Scholar 

  • Confrey, J., & Kazak, S. (2006). A thirty-year reflection on constructivism in mathematics education in PME. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present, and future (pp. 305–345). Rotterdam, The Netherlands: Sense.

    Google Scholar 

  • Davis, R. B., & Perusse, R. (1988). Numerical competence in animals: Definitional issues, current evidence, and a new research agenda. Behavioral and Brain Sciences, 11, 561–579.

    Article  Google Scholar 

  • Dawson, D. T. (1953). Number grouping as a function of complexity. The Elementary School Journal, 54, 35–42.

    Article  Google Scholar 

  • Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press.

    Google Scholar 

  • Demeyere, N., Rotshtein, P., & Humphreys, G. W. (2012). The neuroanatomy of visual enumeration: Differentiating necessary neural correlates for subitizing versus counting in a neuropsychological voxel-based morphometry study. Journal of Cognitive Neuroscience, 24(4), 948–964. https://doi.org/10.1162/jocn_a_00188

    Article  Google Scholar 

  • Dewey, J. (1938/1997). Experience and education. New York, NY: Simon & Schuster.

    Google Scholar 

  • Douglass, H. R. (1925). The development of number concept in children of preschool and kindergarten ages. Journal of Experimental Psychology, 8, 443–470.

    Article  Google Scholar 

  • Edens, K. M., & Potter, E. F. (2013). An exploratory look at the relationships among math skills, motivational factors and activity choice. Early Childhood Education Journal, 41(3), 235–243. https://doi.org/10.1007/s10643-012-0540-y

    Article  Google Scholar 

  • Feigenson, L., Carey, S., & Hauser, M. (2002). The representations underlying infants’ choice of more: Object files versus analog magnitudes. Psychological Science, 13, 150–156.

    Article  Google Scholar 

  • Feigenson, L., Carey, S., & Spelke, E. S. (2002). Infants’ discrimination of number vs. continuous extent. Cognitive Psychology, 44, 33–66.

    Article  Google Scholar 

  • Feigenson, L., Dehaene, S., & Spelke, E. S. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307–314.

    Article  Google Scholar 

  • Fitzhugh, J. I. (1978). The role of subitizing and counting in the development of the young children’s conception of small numbers. Dissertation Abstracts International, 40, 4521B–4522B.

    Google Scholar 

  • Freeman, F. N. (1912). Grouped objects as a concrete basis for the number idea. The Elementary School Teacher, 8, 306–314.

    Article  Google Scholar 

  • Fuhs, M. W., Hornburg, C. B., & McNeil, N. M. (2016). Specific early number skills mediate the association between executive functioning skills and mathematics achievement. Developmental Psychology, 52(8), 1217–1235. https://doi.org/10.1037/dev0000145

    Article  Google Scholar 

  • Fuson, K. C. (1992a). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R. Putman, & R. A. Hattrup (Eds.), Handbook of research on mathematics teaching and learning (pp. 53–187). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Fuson, K. C. (1992b). Research on whole number addition and subtraction. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243–275). New York, NY: Macmillan.

    Google Scholar 

  • Fuson, K. C., Carroll, W. M., & Drueck, J. V. (2000). Achievement results for second and third graders using the standards-based curriculum everyday mathematics. Journal for Research in Mathematics Education, 31, 277–295.

    Article  Google Scholar 

  • Gallistel, C. R., & Gelman, R. (2005). Mathematical cognition. In K. Holyoak & R. Morrison (Eds.), Cambridge handbook of thinking and reasoning (pp. 559–588). Cambridge: Cambridge University Press.

    Google Scholar 

  • Gebuis, T., & Reynvoet, B. (2011). Generating nonsymbolic number stimuli. Behavior Research Methods, 43(4), 981–986.

    Article  Google Scholar 

  • Gelman, R., & Butterworth, B. (2005). Number and language: How are they related? Trends in Cognitive Sciences, 9(1), 6–10.

    Article  Google Scholar 

  • Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Ginsburg, H. P. (1977). Children’s arithmetic. Austin, TX: Pro-ed.

    Google Scholar 

  • Glasersfeld, E. V. (1982). Subitizing: The role of figural patterns in the development of numerical concepts. Archives de Psychologie, 50, 191–218.

    Google Scholar 

  • Glasersfeld, E. V. (1995). Sensory experience, abstraction, and teaching. In L. P. Steffe & J. Gale (Eds.), Constructivism in education (pp. 369–383). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306, 496–499.

    Article  Google Scholar 

  • Hannula, M. M., Lepola, J., & Lehtinen, E. (2010). Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal of Experimental Child Psychology, 107, 394–406.

    Article  Google Scholar 

  • Hiebert, J. C. (1999). Relationships between research and the NCTM standards. Journal for Research in Mathematics Education, 30, 3–19.

    Article  Google Scholar 

  • Huntley-Fenner, G. (2001). Children’s understanding of numbers is similar to adults’ and rats’: Numerical estimation by 5-7-year-olds. Cognition, 78, 27–40.

    Article  Google Scholar 

  • Huntley-Fenner, G., Carey, S., & Solimando, A. (2002). Objects are individuals but stuff doesn’t count: Perceived rigidity and cohesiveness influence infants’ representations of small groups of discrete entities. Cognition, 85, 203–221.

    Article  Google Scholar 

  • Huttenlocher, J., Jordan, N. C., & Levine, S. C. (1994). A mental model for early arithmetic. Journal of Experimental Psychology: General, 123, 284–296.

    Article  Google Scholar 

  • Johnson-Pynn, J. S., Ready, C., & Beran, M. (2005, April). Estimation mediates preschoolers: Numerical reasoning: Evidence against precise calculation abilities. Paper presented at the Biennial Meeting of the Society for Research in Child Development, Atlanta, GA.

    Google Scholar 

  • Jordan, N. C., Hanich, L. B., & Uberti, H. Z. (2003). Mathematical thinking and learning difficulties. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 359–383). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Jordan, N. C., Huttenlocher, J., & Levine, S. C. (1992). Differential calculation abilities in young children from middle- and low-income families. Developmental Psychology, 28, 644–653.

    Article  Google Scholar 

  • Jordan, N. C., Huttenlocher, J., & Levine, S. C. (1994). Assessing early arithmetic abilities: Effects of verbal and nonverbal response types on the calculation performance of middle- and low-income children. Learning and Individual Differences, 6, 413–432.

    Article  Google Scholar 

  • Jordan, K. E., Suanda, S. H., & Brannon, E. M. (2008). Intersensory redundancy accelerates preverbal numerical competence. Cognition, 108, 210–221.

    Article  Google Scholar 

  • Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: MIT Press.

    Google Scholar 

  • Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62, 498–525.

    Article  Google Scholar 

  • Klahr, D. (1973a). A production system for counting, subitizing, and adding. In W. G. Chase (Ed.), Visual information processing (pp. 527–544). New York, NY: Academic.

    Chapter  Google Scholar 

  • Klahr, D. (1973b). Quantification processes. In W. G. Chase (Ed.), Visual information processing (pp. 3–31). New York, NY: Academic.

    Chapter  Google Scholar 

  • Klahr, D., & Wallace, J. G. (1976). Cognitive development: An information-processing view. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Koontz, K. L., & Berch, D. B. (1996). Identifying simple numerical stimuli: Processing inefficiencies exhibited by arithmetic learning disabled children. Mathematical Cognition, 2, 1–23.

    Article  Google Scholar 

  • Kühne, C., Lombard, A.-P., & Moodley, T. (2013). A learning pathway for whole numbers that informs mathematics teaching in the early years. South African Journal of Childhood Education, 3(2), 77–95.

    Article  Google Scholar 

  • Le Corre, M., Van de Walle, G. A., Brannon, E. M., & Carey, S. (2006). Re-visiting the competence/performance debate in the acquisition of counting as a representation of the positive integers. Cognitive Psychology, 52(2), 130–169.

    Article  Google Scholar 

  • Leibovich, T., Kadhim, S. A. R., & Ansari, D. (2017). Beyond comparison: The influence of physical size on number estimation is modulated by notation, range and spatial arrangement. Acta Psychologica, 175, 33–41.

    Article  Google Scholar 

  • Lester, F. K., Jr., & Wiliam, D. (2002). On the purpose of mathematics education research: Making productive contributions to policy and practice. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 489–506). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Levine, S. C., Jordan, N. C., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53, 72–103.

    Article  Google Scholar 

  • Libertus, M. E., Feigenson, L., & Halberda, J. (2013, May). Effects of approximate number system training for numerical approximation and school math abilities. Paper presented at the NICHD Mathematics Meeting, Bethesda, MD.

    Google Scholar 

  • Libertus, M. E., Feigenson, L., & Halberda, J. (2011b). Preschool acuity of the approximate number system correlates with math abilities. Developmental Science. https://doi.org/10.1111/j.1467-7687.2011.080100x.

  • Logan, G. D., & Zbrodoff, N. J. (2003). Subitizing and similarity: Toward a pattern-matching theory of enumeration. Psychonomic Bulletin & Review, 10(3), 676–682.

    Article  Google Scholar 

  • Lourenco, S. F., & Longo, M. R. (2011). Origins and development of generalized magnitude representation. Space, Time and Number in the Brain, 225–244. https://doi.org/10.1016/b978-0-12-385948-8.00015-3.

    Chapter  Google Scholar 

  • MacDonald, B. L. (2015). Ben’s perception of space and subitizing activity: A constructivist teaching experiment. Mathematics Education Research Journal, 27(4), 563–584. https://doi.org/10.1007/s13394-015-0152-0

    Article  Google Scholar 

  • MacDonald, B. L., & Shumway, J. F. (2016). Subitizing games: Assessing preschool children’s number understanding. Teaching Children Mathematics, 22(6), 340–348.

    Article  Google Scholar 

  • MacDonald, B. L., & Wilkins, J. L. M. (2016). Seven types of subitizing activity characterizing young children’s mental activity. In S. Marx (Ed.), Qualitative research in STEM (pp. 256–286). New York, NY: Routledge.

    Google Scholar 

  • MacDonald, B. L., & Wilkins, J. L. M. (2018). Subitising activity relative to units construction and coordination: A case study. Manuscript submitted for publication.

    Google Scholar 

  • Masataka, N., Ohnishi, T., Imabayashi, E., Hirakata, M., & Matsuda, H. (2006). Neural correlates for numerical processing in the manual mode. Journal of Deaf Studies and Deaf Education, 11(2), 144–152.

    Article  Google Scholar 

  • Mazzocco, M. M. M., Feigenson, L., & Halberda, J. (2011). Preschoolers’ precision of the approximate number system predicts later school mathematics performance. PLoS One, 6(9), e23749. https://doi.org/10.1371/journal.pone.0023749.t001

    Article  Google Scholar 

  • McCrink, K., & Wynn, K. (2004). Large number addition and subtraction by 9-month-old infants. Psychological Science, 15, 776–781.

    Article  Google Scholar 

  • Meck, W. H., & Church, R. M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9, 320–334.

    Google Scholar 

  • Mix, K. S., Huttenlocher, J., & Levine, S. C. (2002). Quantitative development in infancy and early childhood. New York, NY: Oxford University Press.

    Book  Google Scholar 

  • Mix, K. S., Sandhofer, C. M., & Baroody, A. J. (2005). Number words and number concepts: The interplay of verbal and nonverbal processes in early quantitative development. In R. Kail (Ed.), Advances in child development and behavior (Vol. 33, pp. 305–345). New York, NY: Academic.

    Google Scholar 

  • Moore, A. M., & Ashcraft, M. H. (2015). Children’s mathematical performance: Five cognitive tasks across five grades. Journal of Experimental Child Psychology, 135, 1–24. https://doi.org/10.1016/j.jecp.2015.02.003

    Article  Google Scholar 

  • Myers, M., Wilson, P. H., Sztajn, P., & Edgington, C. (2015). From implicit to explicit: Articulating equitable learning trajectories based instruction. Journal of Urban Mathematics Education, 8(2), 11–22.

    Google Scholar 

  • National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

    Google Scholar 

  • National Research Council. (2002). Scientific research in education (Committee on Scientific Principles for Educational Research Ed.). Washington, DC: National Research Council, National Academy Press.

    Google Scholar 

  • Nes, F. T. v. (2009). Young children’s spatial structuring ability and emerging number sense. (Doctoral dissertation). de Universtiteit Utrecht, Utrecht, the Netherlands.

    Google Scholar 

  • Nieder, A., Freedman, D. J., & Miller, E. K. (2002). Representation of the quantity of visual items in the primate prefrontal cortex. Science, 297, 1708–1711.

    Article  Google Scholar 

  • Olkun, S., & Özdem, S. e. (2015). The effect of conceptual subitizing training on calculation performance. Başkent University Journal of Education, 2(1), 1–9.

    Google Scholar 

  • Park, J., Bermudez, V., Roberts, R. C., & Brannon, E. M. (2016). Non-symbolic approximate arithmetic training improves math performance in preschoolers. Journal of Experimental Child Psychology, 152, 278–293. https://doi.org/10.1016/j.jecp.2016.07.011

    Article  Google Scholar 

  • Pepper, K. L., & Hunting, R. P. (1998). Preschoolers’ counting and sharing. Journal for Research in Mathematics Education, 29, 164–183.

    Article  Google Scholar 

  • Peterson, P. L., Carpenter, T. P., & Fennema, E. H. (1989). Teachers’ knowledge of students’ knowledge in mathematics problem solving: Correlational and case analyses. Journal of Educational Psychology, 81, 558–569.

    Article  Google Scholar 

  • Piaget, J. (1977/2001). Studies in reflecting abstraction. Sussex: Psychology Press.

    Google Scholar 

  • Piazza, M., Izard, V., Pinel, P., Le Bihan, D., & Dehaene, S. (2004). Tuning curves for approximate numerosity in the human intraparietal sulcus. Neuron, 44, 547–555.

    Article  Google Scholar 

  • Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306, 499–503.

    Article  Google Scholar 

  • Pinel, P., Piazza, D., Le Bihan, D., & Dehaene, S. (2004). Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron, 41, 983–993.

    Article  Google Scholar 

  • Potter, M., & Levy, E. (1968). Spatial enumeration without counting. Child Development, 39, 265–272.

    Article  Google Scholar 

  • Pylyshyn, Z. W. (2001). Visual indexes, preconceptual objects, and situated vision. Cognition, 80, 127–158.

    Article  Google Scholar 

  • Reigosa-Crespo, V., González-Alemañy, E., León, T., Torres, R., Mosquera, R., & Valdés-Sosa, M. (2013). Numerical capacities as domain-specific predictors beyond early mathematics learning: A longitudinal study. PLoS One, 8(11), e79711.

    Article  Google Scholar 

  • Revkin, S. K., Piazza, M., Izard, V., Cohen, L., & Dehaene, S. (2008). Does subitizing reflect numerical estimation? Psychological Science, 19(6), 607–614. https://doi.org/10.1111/j.1467-9280.2008.02130.x

    Article  Google Scholar 

  • Richardson, K. (2004). Making sense. In D. H. Clements, J. Sarama, & A.-M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 321–324). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Sandhofer, C. M., & Smith, L. B. (1999). Learning color words involves learning a system of mappings. Developmental Psychology, 35, 668–679.

    Article  Google Scholar 

  • Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.

    Book  Google Scholar 

  • Sarama, J., & Clements, D. H. (2011). Mathematics knowledge of low-income entering preschoolers. Far East Journal of Mathematical Education, 6(1), 41–63.

    Google Scholar 

  • Sayers, J., Andrews, P., & Boistrup, L. B. (2016). The role of conceptual subitising in the development of foundational number sense. In T. Meaney, O. Helenius, M. L. Johansson, T. Lange, & A. Wernberg (Eds.), Mathematics education in the early years (pp. 371–394). Switzerland: Springer.

    Chapter  Google Scholar 

  • Schaeffer, B., Eggleston, V. H., & Scott, J. L. (1974). Number development in young children. Cognitive Psychology, 6, 357–379.

    Article  Google Scholar 

  • Shuman, M., & Spelke, E. S. (2005, April). The development of numerical magnitude representation. Paper presented at the Biennial Meeting of the Society for Research in Child Development, Atlanta, GA.

    Google Scholar 

  • Silverman, I. W., & Rose, A. P. (1980). Subitizing and counting skills in 3-year-olds. Developmental Psychology, 16, 539–540.

    Article  Google Scholar 

  • Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. https://doi.org/10.2307/749205

    Article  Google Scholar 

  • Slusser, E. B., & Sarnecka, B. W. (2011). Find the picture of eight turtles: A link between children’s counting and their knowledge of number word semantics. Journal of Experimental Child Psychology, 110(1), 38–51.

    Article  Google Scholar 

  • Smith, C. L., Wiser, M., Anderson, C. W., & Krajcik, J. S. (2006). Implications of research on children’s learning for standards and assessment: A proposed learning progression for matter and the atomic-molecular theory. Measurement: Interdisciplinary Research and Perspective, 14(1&2), 1–98. https://doi.org/10.1080/15366367.2006.9678570

    Article  Google Scholar 

  • Solter, A. L. J. (1976). Teaching counting to nursery school children. Dissertation Abstracts International, 36(8-A), 5844B.

    Google Scholar 

  • Soto-Calvo, E., Simmons, F. R., Willis, C., & Adams, A.-M. (2015, December). Identifying the cognitive predictors of early counting and calculation skills: Evidence from a longitudinal study. Journal of Experimental Child Psychology, 140, 16–37. https://doi.org/10.1016/j.jecp.2015.06.011.

    Article  Google Scholar 

  • Starkey, G. S., & McCandliss, B. D. (2014). The emergence of “groupitizing” in children’s numerical cognition. Journal of Experimental Child Psychology, 126, 120–137. https://doi.org/10.1016/j.jecp.2014.03.006

    Article  Google Scholar 

  • Starkey, P., Spelke, E. S., & Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36, 97–128.

    Article  Google Scholar 

  • Starr, A., Libertus, M. E., & Brannon, E. M. (2013). Infants show ratio-dependent number discrimination regardless of set size. Infancy, 1–15. https://doi.org/10.1111/infa.12008.

    Article  Google Scholar 

  • Steffe, L. P. (1992). Children’s construction of meaning for arithmetical words: A curriculum problem. In D. Tirosh (Ed.), Implicit and explicit knowledge: An educational approach (pp. 131–168). Norwood, NJ: Ablex Publishing Corporation.

    Google Scholar 

  • Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–39). Albany, NY: SUNY Press.

    Google Scholar 

  • Steffe, L. P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. New York, NY: Springer.

    Book  Google Scholar 

  • Steffe, L. P., Thompson, P. W., & Richards, J. (1982). Children’s counting in arithmetical problem solving. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Tan, L. S. C., & Bryant, P. E. (2000). The cues that infants use to distinguish discontinuous quantities: Evidence using a shift-rate recovery paradigm. Child Development, 71, 1162–1178.

    Article  Google Scholar 

  • Titeca, D., Roeyers, H., Josephy, H., Ceulemans, A., & Desoete, A. (2014). Preschool predictors of mathematics in first grade children with autism spectrum disorder. Research in Developmental Disabilities, 35(11), 2714–2727. https://doi.org/10.1016/j.ridd.2014.07.012

    Article  Google Scholar 

  • Trick, L. M., & Pylyshyn, Z. W. (1994). Why are small and large numbers enumerated differently? A limited-capacity preattentive stage in vision. Psychological Review, 101, 80–102.

    Article  Google Scholar 

  • Tyler, R. W. (1949). Basic principles of curriculum and instruction. Chicago: University of Chicago Press.

    Google Scholar 

  • Vallortigara, G. (2012). Core knowledge of object, number, and geometry: A comparative and neural approach. Cognitive Neuropsychology, 29(1–2), 213–236. https://doi.org/10.1080/02643294.2012.654772

    Article  Google Scholar 

  • Wagner, S. W., & Walters, J. (1982). A longitudinal analysis of early number concepts: From numbers to number. In G. E. Forman (Ed.), Action and thought (pp. 137–161). New York, NY: Academic.

    Google Scholar 

  • Wang, J. J., Odic, D., Halberda, J., & Feigenson, L. (2016). Changing the precision of preschoolers’ approximate number system representations changes their symbolic math performance. Journal of Experimental Child Psychology, 147, 82–99. https://doi.org/10.1016/j.jecp.2016.03.002

    Article  Google Scholar 

  • Wang, M., Resnick, L. B., & Boozer, R. F. (1971). The sequence of development of some early mathematics behaviors. Child Development, 42, 1767–1778.

    Article  Google Scholar 

  • Wheatley, G. H. (1996). Quick draw: Developing spatial sense in mathematics. Tallahassee, FL: Mathematics Learning.

    Google Scholar 

  • Whelley, M. M. (2002). Subitizing and the development of children’s number knowledge (Doctoral dissertation). Retrieved from ProQuest dissertations and theses (3037768).

    Google Scholar 

  • Wickstrom, M. H. (2015). Challenging a teacher’s perceptions of mathematical smartness through reflections on students’ thinking. Equity & Excellence in Education, 48(4), 589–605. https://doi.org/10.1080/10665684.2015.1086242

    Article  Google Scholar 

  • Wright, R. J., Stanger, G., Cowper, M., & Dyson, R. (1996). First-graders’ progress in an experimental mathematics recovery program. In J. Mulligan & M. Mitchelmore (Eds.), Research in early number learning (pp. 55–72). Adelaide: AAMT.

    Google Scholar 

  • Wright, R. J., Stanger, G., Stafford, A. K., & Martland, J. (2006). Teaching number in the classroom with 4–8 year olds. London: Paul Chapman/Russell Sage.

    Google Scholar 

  • Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251.

    Article  Google Scholar 

  • Yun, C., Havard, A., Farran, D. C., Lipsey, M. W., Bilbrey, C. L., & Hofer, K. G. (2011, July). Subitizing and mathematics performance in early childhood. Paper presented at the Cognitive Science 2011 Conference Proceedings, Boston, MA.

    Google Scholar 

Download references

Acknowledgments

The research reported in this manuscript was supported by the National Science Foundation under Grant No. DRL-1222944. The findings and statements in this manuscript do not represent the views of the National Science Foundation.

Author information

Authors and Affiliations

  1. Morgridge College of Education, Marsico Institute for Early Learning and Literacy, University of Denver, Denver, CO, USA

    Douglas H. Clements

  2. Morgridge College of Education, University of Denver, Denver, CO, USA

    Julie Sarama

  3. School of Teacher Education and Leadership, Utah State University, Logan, UT, USA

    Beth L. MacDonald

Authors
  1. Douglas H. Clements
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julie Sarama
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Beth L. MacDonald
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Douglas H. Clements .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Anderson Norton

  2. Department of Psychology, University of Wisconsin–Madison, Madison, WI, USA

    Martha W. Alibali

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Clements, D.H., Sarama, J., MacDonald, B.L. (2019). Subitizing: The Neglected Quantifier. In: Norton, A., Alibali, M.W. (eds) Constructing Number. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-00491-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-00491-0_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-00490-3

  • Online ISBN: 978-3-030-00491-0

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation