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8.6.3: Numerical Cognition

  • Page ID
    140102
    • Todd LaMarr
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    Understanding Numbers

    Evidence suggests that there is an innate, non-symbolic sense of quantity that gives rise to our basic numerical intuitions. The Approximate Number System (ANS) allows humans and non-human animals to estimate quantities without explicitly counting. For example, if you look at the two images in Figure \(\PageIndex{1}\), which image has more dots than the other? Can you figure it out by simply looking at the two images--without counting any of the dots? While this example is slightly challenging because the number of dots in the two images is similar, in experiments like this, using dots, infants have the innate ability to distinguish between various quantities. [1]

    two sets of images. A similar spacing of white dots on black background of both images.
    Figure \(\PageIndex{1}\): The Approximate Number System Test. ([2])

    The ANS is present at birth as newborns are able to discriminate object sets if the ratio is at least 1:3 (e.g., arrays of 4 vs.12 dots) (Izard et al., 2009). The ability to discriminate between displays of large numbers, known as number acuity (efficiency), improves throughout childhood (Halberda & Feigenson, 2008) with dramatic changes in number acuity observed within the first year of life. In a visual only comparison, four month olds can discriminate between a 1:4 ratio but not a change to a 1:3 ratio; however, if audio and visual information are both provided, four month olds can discriminate between a 1:3 ratio, but not a change to a 1:2 ratio (Wang & Feigenson, 2021). Six-month-old infants can reliably discriminate between sets with a ratio of 1:2 (e.g., arrays of 8 vs.16 dots) but cannot discriminate between sets with a 2:3 ratio (e.g., arrays of 8 vs.12 dots) (Feigenson, 2011; Xu & Spelke, 2000). By about 9 to 10 months of age, infants are able to discriminate 2:3 ratios (e.g., arrays of 8 vs. 12 dots) but fail with 4:5 ratios (e.g., arrays of 8 vs.10 dots) (Lipton & Spelke, 2003; Xu & Arriaga, 2007; Xu & Spelke, 2000).[3]

    Number acuity: 6 month olds can discriminate between sets with a ratio of 1:2 (e.g., arrays of 8 vs.16 dots) but cannot discriminate between sets with a 2:3 ratio (e.g., arrays of 8 vs.12 dots).
    Figure \(\PageIndex{2}\): Number acuity: 6 month olds can discriminate between sets with a ratio of 1:2 (e.g., arrays of 8 vs.16 dots) but cannot discriminate between sets with a 2:3 ratio (e.g., arrays of 8 vs.12 dots). ([4])

    Number acuity in infancy is related to later mathematical performance throughout childhood, adolescence and the adult years (Halberda et al., 2008; Mazzocco, Feigenson & Halberda, 2011; Libertus, Feigenson & Halberda, 2013; Starr, Libertus & Brannon, 2013). To illustrate, one study (Starr, Libertus & Brannon, 2013) placed two monitors in front of six month olds. One monitor always displayed the same number of dots (although the dots changed, varying in placement array) and the other monitor changed between various numbers of dots and placement array. Some infants spent more time looking at the monitor that displayed the changing numbers of dots than other infants. The researchers then followed the same children into preschool where they measured various math abilities. Results showed that the infants who spent more time looking at the monitor that displayed the changing numbers of dots had greater mathematical skills as preschoolers.

    While early number acuity may be inborn, other numerical abilities are not and develop overtime. Children come to understand the meaning of exact number words very slowly (Wynn, 1990, 1992). English-speaking children first learn the meaning of the word “one” around two-and-a-half years of age but lack knowledge of numbers larger than one. About four to five months after learning the meaning of “one,” children understand the word “two” but not larger numbers, such as “three” or “four.” It takes several more months for children to display knowledge of the word “three.” Not until children are three or four years of age do they fully grasp the cardinality principle—that each number word refers only to an exact set of that quantity with the last number in the count list referring to the total number of items in the set (Carey, 2009). [5]

    Toddler moving pieces of abacus with two fingers of left hand.
    Figure \(\PageIndex{3}\): Toddler using an abacus ([6])

    Before infants and toddlers may fully understand the meaning of specific number words, they show an early sensitivity to counting. Eighteen-month-old infants showed a preference for correctly ordered counting sequences; that is, although they were unable to recite the count list themselves, they recognized and preferred to listen to the correct order of the number words (Ip et al., 2018). Similarly, 14 to 18 month-old infants appear to be able to use their ability to recognize the count list to help them overcome typical memory limits (Wang & Feigenson, 2019). Infants generally display working memory capacity limits of three items and fail to remember the number of hidden items when it exceeds this limit (Feigenson & Carey, 2003). However, when objects are counted before being hidden, infants are able to overcome this memory limit (Wang & Feigenson, 2019). Thus, even though toddlers may not grasp the full meaning of number words, they may still be aware of the numerical nature of these words and may be able to use counting knowledge despite lacking precise representations of the quantities.

    stepping stones numbered 1 -7 laid on ground and spaced a few inches apart each in a "hop-scotch" pattern.
    Figure \(\PageIndex{4}\): CNumbered stepping stones. ([7])

    Although research suggests ANS to be innate, ANS is only an early foundation to support mathematical development. As mathematical knowledge continues to develop, caregivers play a critical role in supporting it. Infants and toddlers build early foundations for math during play and daily care routines. Young children naturally explore math concepts as they play, and caregivers support their math knowledge and vocabulary with the language they use. Caregivers can use math talk during care routines by discussing spatial concepts like, “I’m going to pick you up” or they can compare the size of their shoes as they get ready to go outside, “Your shoes are smaller than my shoes.” Caregivers can use math language during mealtimes: “How many blueberries do you have left? Do you need more?” The more math language children hear each day, the greater the growth of their math knowledge. [8]

    Additionally, infants and toddlers need time and space to play in open-ended ways with varied materials to boost their emerging math skills. Caregivers can introduce math concepts like size and shape, and spatial words like in, between, and under during any type of play or routine. Create and look for patterns in the environment and point them out to children. Find patterns in your clothing or around the environment like stripes on a rug and point them out to children while using math language to describe the patterns. Indoor and outdoor environments present unlimited ways that we can discuss math. When working with infants and toddlers, purposefully look around the environment and at the materials being used for math concepts you can discuss. [8]


    [1] ​​Ma et al., (2021). Approximate number sense in students with severe hearing loss: A modality-neutral cognitive ability. Frontiers in Human Neuroscience, 15, 296. CC by 4.0

    [2] Image by Todd LaMarr. CC by 4.0

    [3] Zorzi & Testolin (2018). An emergentist perspective on the origin of number sense. Philosophical Transactions of the Royal Society B: Biological Sciences, 373(1740), 20170043. CC by 4.0

    [4] Image by Todd LaMarr is licensed under CC by 4.0.

    [5] Silver et al., (2021). Measuring Emerging Number Knowledge in Toddlers. Frontiers in Psychology, 3057. CC by 4.0

    [6] Image from luis arias on Unsplash

    [7] Image from Eric Tompkins on Unsplash

    [8]Supporting Math Skills in Infants and Toddlers” from Head Start’s Early Childhood Learning and Knowledge Center is in the public domain.


    This page titled 8.6.3: Numerical Cognition is shared under a mixed 4.0 license and was authored, remixed, and/or curated by Todd LaMarr.