Arthur Baroody, Professor of Curriculum & Instruction, University of Illinois at Urbana-Champaign
Parents, teachers, or others interested in early childhood mathematics education often ask: “On what should initial instruction focus?” My answer is verbal subitizing: recognizing small numbers (1 to 3 or 4 items) and labeling them with an appropriate number word. It often takes 2-year-olds many months to label collections of “one” and ‘two” accurately and reliably. Imagine my joy when my wife informed me that our younger daughter Arianne, 2 at the time, had learned “two.” I immediately went to the playroom, held up two fingers, and asked, “How many fingers Arianne?” “Two Daddy,” was her immediate reply. My heart swelled, but a scientist must be skeptical. So, I held up three fingers and asked the same question. Arianne again replied, “Two, Daddy.” The same reply was offered for 5 fingers or 10 fingers. Later that night, she bounced down the stairs to the playroom chanting “two” with each bounce. Ugh, apparently, “two” was merely her favorite sound for the day.
Children typically master verbal subitizing of one and two between the age of 2.5- and 3-years. Often “three” is then used to indicate “many” or “a lot.” Mastery of three and then four often comes after a child has turned 3.
Verbal subitizing is a key foundation for building number sense—providing a basis for meaningfully learning a variety of number, counting, and arithmetic concepts and skills (see Baroody & Purpura, 2017, for a detailed learning progression). Consider one important example. Children often learn to count collections in a one-to-one fashion without understanding the reason for doing so. That is, they do not understand the cardinality principle—that the last counting word used in counting process has special meaning: it represents the total number counted. Modeling the cardinality principle with collections a child can verbally subitize can significantly increase the likelihood of discovering the principle (Paliwal & Baroody, 2020). For example, if a child can “see” three, counting such a collection and emphasizing the last count word (“One, two, t-h-r-e-e”) or repeating the last number word used (e.g., “See three” or, perhaps even better, “See three cookies”) may enable children to understand the purpose of counting and discover the cardinality principle: Counting is another way of determining a collection’s total, which is indicated by the last number word.
Recently, I served on the advisory board for an NSF-funded project to investigate early mathematics learning. In response to the question of how to help 2- to 3-year-olds construct an understanding of small numbers, I was amazed that each of four other advisors offered a different opinion. Some researchers contend that such instruction should build on children’s innate number and counting knowledge or a pre-numerical core process by promoting verbal and object counting. Others suggests visual-spatial training is needed to enable children to systematically keep track of collection elements. Yet others believe linguistic cues, such as the plural rule for nouns, help children discern the exact meaning of small number words. An intervention study found that various approaches, save perhaps the visual-spatial training, were successful. Interestingly, the one thing all the successful conditions provided was a variety of examples and non-examples of a number (D. Hyde, personal communications, September 25, 2014; October 6, 2016). Clearly, further research is needed to answer the question about how to promote verbal subitizing of small numbers.
In the meanwhile, my money is on providing a variety of examples and non-examples, which can enable children to discover for themselves the critical or defining attribute(s) of a concept. Specifically, using a common label such as “two” with different-looking pairs (e.g., ● ●, ⚾ ⚾, □ □, ⇇ , △ △, ) can help children recognize that appearances (physical characteristics such as shape, color, or arrangement) are irrelevant and abstract the defining commonality (critical attribute) of a category represented by the number word (e.g., pairs of items in the case of “two”; Palmer & Baroody, 2011). Discerning what is common among different pairs of items, for example, can reinforce simultaneous attention to units (one thing, another thing) and the whole (the common label “two things”) necessary to construct an exact concept of a number. Using non-examples (e.g., “three is too many cookies, just take two”) can help define the “boundaries” of a number concept and eliminate its overgeneralization (e.g., using “two” or “three” to indicate “many”). Nonexamples, may prompt children to consider what is different between pairs such as ● ● and trios such as ●●●—the latter has one more thing/unit and is associated with the different number word “three.”
Baroody, A. J., with Purpura, D. J. (2017). Early number and operations: Whole numbers. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 308–354). Reston, VA: National Council of Teachers of Mathematics.
Paliwal, V., & Baroody, A. J. (2020). Cardinality principle understanding: The role of focusing on the subitizing ability. ZDM Mathematics Education, 52(4) 649–661. https://doi.org/10.1007/s11858-020-01150-0
Palmer, A., & Baroody, A. J. (2011). Blake’s development of the number words “one,” “two,” and “three.” Cognition & Instruction, 29(3), 265–296. https://doi: 10.1080/07370008.2011.583370