Introduction
Mathematics plays an important role in many of our everyday life experiences, from managing personal finances to measuring ingredients of a recipe. The process of understanding mathematical knowledge begins within the early stages of a child’s learning development. Mathematical concepts such as number sense and counting are skills that children begin developing in their early years through different life experiences i.e., language, symbols, play, and many other different learning experiences. A child’s ability to count is the stepping stone in their developmental process of learning more complex mathematical concepts such as recognizing different quantities in a set. This ability to recognize numbers in a set without counting is called subitizing (Penner-Wilger, n.d, p.1385). I will examine the literature on the process of subitizing and the different ways children subitize i.e., perceptual and conceptual. As well as the learning strategies used to foster this skill in an educational environment. Additionally, I will examine how children’s life experiences in their early years contribute to their mathematical knowledge and developmental process of mathematic concepts.
Review
According to Douglas et al. (1999) he describes that an individual must first learn how to count before they can subitize. This is because subitizing is a “developmental prerequisite to counting” (Douglas et al. 1999, p.1).A child that can immediately say the quantity shown in a set must first have the skill of counting in a sequential order i.e.,1,2,3. The skill of subitizing encompasses many of the skills related to the mathematical concept number sense such as composing and decomposing numbers. To compose means that a child can put numbers together; the number twenty-one is a two-digit number that is composed of a two (two tens) and one, (ones). To decompose means to break the number down into small numbers; the number twenty-one can be broken down into a (two) and a (one) (Macdonald and Shumway, 2016, p.343). Before a child can subitize they must be able to compose and decompose numbers because this skill is important for a child’s development for thinking and understanding how numbers work together and can be made up of different numbers and quantities. Having this skill is important because it will continue to build on more sophisticated mathematic knowledge of a child’s development i.e., addition and subtraction.
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Slaughter et al. (2011) describes that the skill of counting happens after a child turns the age of two. Up until then children are still learning and engaging in mathematic knowledge i.e., home environment. However, before a child can count they must be able to demonstrate these three principles outlined by Slaughter et al. (2011): One-to-one correspondence (i.e., matching an object to the appropriate number), counting in sequential order and cardinality (i.e., child counts out loud 1,2,3, the child repeats what the last number was, three). These three principals outlined by Slaughter et al. demonstrate that children have mathematical knowledge well before they enter the school environment. Slaughter et al. (2011) did a study where they examined infant’s watching a video to see their preference for correct versus incorrect counting. The results revealed that there was a positive relationship between infants preference with proper sequential counting. This indicates that during infancy, children are learning mathematic skills of counting even though a child may not have the oral language to count.
Children begin learning numbers during their infancy years and the concept of subitizing “begins perceptually when infants as young as six months old attend toward quantities as large as three but eventually changes to support early addition” (Macdonald and Shumway, 2016, p.341). The ability of being able to say the quantity of numbers in a set without counting each number is a skill that most children develop by the time they begin kindergarten. In the Ontario Ministry of Education’s Full Day Kindergarten Program (2016) an overall expectation outlined in the Demonstrating Literacy and Mathematics Behvaiour section describes that by the end of kindergarten a student should be able to demonstrate “An understanding of numbers, using concrete materials to explore and investigate counting, quantity, and number relationships” (Ontario Ministry of Education, 2016, p.216) In kindergarten, children use play-based activities to learn about different ways to count. A teaching strategy that helps support the concept of subitizing in the classroom is using dot plates, dice and cards. These types of games “elicit attention toward subgroups, with colour and space between items, further supports students’ ability to compose and eventually decompose number through subitizing activity” (Macdonald and Shumway, 2016, p.347). In the classroom it might not always be easy to observe every student’s learning abilities right away. By educators planning lessons, activities and games, this will allow students to practice these skills and provide educators with the opportunity to see students’ knowledge and understanding.
There are two ways in which children can subitize, this can either be through a perceptual or conceptual subitizing. Around the age of four some children may have the knowledge of being able to perceptual subitize, meaning that a child can identify immediately the quantity in the set with four or fewer numbers (Macdonald and Shumway, 2016, p.342). An example of perceptual subitizing is if a child were to look at a dot card that has four dots on it, they right away can recognize that there are four dots altogether. Conceptual subitizing however, refers to a child’s ability to categorize quantities and put them into subgroups to get the quantity i.e., using a dot card that has six dots on it, a child who can conceptual subitizewill see two sets of three and know that there are six dots all together. These two types of subitizing support children’s developmental learning in the concept of number sense and their ability to develop knowledge in recognizing different number patterns. Teaching subitizing in the classroom can be difficult for educators if they are unaware of strategies on how to teach it or if they don’t know all their students learning abilities. When examining the literature, a common strategy that was used when teaching children about subitizing were using dominions, dice and dot cards to help support children’s skills of subitizing. “Spatial patterns, such as those on dominoes, are just one kind. Creating and using these patterns through conceptual subitizing help children develop abstract number and arithmetic strategies” (Douglas et al, 1999, p.2). Using these tools in the classroom can be used to foster children’s knowledge and practice in conceptual subitizing, as well as being able to provide the educator with an opportunity to observe the student’s developmental skills. Since children are continuously growing their brains are also developing and gaining mathematical knowledge through lived experiences. This is because they are exposed to and surrounded by their family members that provide meaningful learning experiences. Pepper and Hunter (1998) describe this as having informal knowledge. This type of knowledge is not the same as the type of academic knowledge learned in school. The type of informal knowledge being described are the meaningful learning experiences during childhood that involve working with, hearing and seeing numbers in a variety of ways i.e., baking with parents, counting with fingers, going to the grocery store, outdoor environment and seeing number symbols and words. When a child begins school, the informal knowledge they learned in their early years will help when they are learning about other mathematical concepts i.e., counting, recognizing quantities and problem solving.
Theorist Piaget argues that before the age of seven, children do not have the developmental capabilities nor knowledge to understand a rational way of thinking (Baroody and Wilkins, 1999, p.49). However, Piaget has shown to be wrong because children have informal knowledge of mathematics from their early years as well as their past experiences of being exposed to mathematical literacy and symbols. “Knowledge of a mathematical domain begins with personal knowledge of specific examples (concrete knowledge) and gradually broadens into theoretical knowledge of generalities (abstract knowledge) (Baroody and Wilkins, 1999, p.50). In Ontario The Full Fay Kindergarten Program (2016) acknowledges the importance of play-based learning in the curriculum as this is a crucial way that children learn and develop new skills. When children are provided with the opportunity to explore and discover the world around them they are learning and developing ideas about different concepts in a concrete way i.e., using open-ended materials. Providing children with play-based math learning experiences that support concrete knowledge can help support their ability to think in a more abstract way.
The focus of mathematics in kindergarten are similar to the three principles Slaughter (2002) outlined; one-to-one correspondence, counting and cardinality. These three skills are a fundamental part of a child’s learning about the concept of number sense as they are interrelated and build onto many other math concepts. Baroody and Wilkins (1999) describe a model that consists of three phases: concrete knowledge, informal knowledge and formal knowledge, each one of these phases support children’s developmental skills regarding number sense. The first phase of concrete knowledge is how children learn one-to-one correspondence. An example described by Baroody and Wilkin (1999), there are five beads lined side by side in a row and the educator asks the child to count how many beads there are. After the child counts the number of beads and determines the quantity, the educator will then separate the beads by stretching them out leaving a little space between each one. Once the beads have been spaced out the educator will then ask the students how many beads there are in the row now. Even though there are the same number of beads but just separated, some children may think that the quantity has changed and need to recount how many there are. “Children’s ability to subitize different quantities changes due to perceptual and cognitive changes” (Macdonald and Shumway, 2016, p. 342). As children develop and gain more mathematic knowledge, their developmental process and ability to think about concepts such as counting and recognizing quantities improve. It is important that educators provide a variety of learning experiences i.e., verbal strategy’s, card and dice games to support such skills that allow children to practice these skills. The second phase described by Baroody and Wilkin (1999) is having concrete knowledge based on everyday experiences. As children develop their mathematic knowledge, their skills of counting, identifying numbers and symbols becomes more recognizable. They begin to learn about quantities and what the numbers are made up of in these quantities. “Counting collections in different arrangements, they can discover that appearances can be deceiving-that the number in a collection remains the same despite superficial changes in appearance” (Baroody and Wilkin, 1999, p.51). The one-to-one correspondence (bead example) is a good activity that educators can implement to help teach and support children’s learning about quantities and that the quantity of objects do not change even when the items are stretched out. The third phase is formal knowledge. This is when children in an educational institution “learn about mathematical symbols and manipulations of these symbols” (Baroody and Wilkin, 1999, p.51). As children progress into higher grades their development and process of these skills also increase, and they soon begin to learn more formal knowledge of mathematics, number representations and rules that need to be applied.
Conclusion
When teaching mathematics in kindergarten, it is important that educators equip themselves with their own knowledge and strategies of teaching mathematical concepts. The concept of subitizing is a fundamental skill for children’s learning of number sense. However, it is important that children first grasp the knowledge of being able to count, compose and decompose numbers to develop the ability to subitize. How these mathematical concepts are taught in the classroom is up to the educator to work alongside students while supporting each one of their learning needs and abilities. When children are motivated to learn it encourages confidence, curiosity and creativity.
Reflection
I found the process of writing this paper to be more challenging than I expected. When I began doing my research of the topic on subitizing I had found a lot of articles, blogs and webpages that provided a wealth of information. Many of these blogs and articles talked about teaching strategies and described different activities about this specific concept. I found many resources that explained different games and activities that can be used for teaching children about subitizing i.e., dot cards, dominos, card games and board games. When reading different articles about this topic and the activities used to support subitizing I thought they were relatable because I have seen some of these implemented within the classroom.
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After completing a placement in a kindergarten classroom, I had the opportunity to see how educators support different mathematical concepts in the classroom i.e., counting and subitizing. One lesson that stood out for me was seeing how the educator implemented using dot plates with the whole class. In this activity the educator had numerous plates each with different quantities and sets of dots on the plates. The educator would hold up a dot plate for a quick min to students and then hide the plate. The educator would then ask the students how many dots they saw on the plate. The educator would follow up by asking how they knew their answer and their thinking behind it. This was a good opportunity for the teachers to see where the children’s skill levels were when it came to subitizing. When I first saw the educators implement this activity I was unsure of how effective this learning experience was for students. However, as they continued to implement this activity throughout my placement, I saw how engaged children were in this activity and how much their counting and subitizing skills improved from the first time they implemented it.
When I was doing research on subitizing I came across numerous activities that used dot cards, dice games and card games to help students practice the skill of subitizing. Some research indicated how using those types of manipulatives provide students with the opportunity to practice working with different quantities and subgroups as well as their skills of composing and decomposing (Macdonald and Shumway, 2016, p.347). As an educator these are some tools and activity ideas that I would use in my own teaching lessons, I found the resource by Macdonald and Shumway particularly helpful when understanding the concept of subitizing and how it can be supported in the classroom learning.
Growing up I always had a hard time understanding and learning mathematical concepts, and I would find it difficult to follow the math lesson when my teacher was teaching. However, since I have been in the program of Early Childhood Education I have seen the importance and the role that math plays in our everyday lives. The concept of subitizing is an important skill for children to have because it allows children to group, count, identify numbers, and quantities. These mathematical concepts are what all individuals use throughout their lives, it is important that these skills are supported in the classroom, so all children can reach their full potential.
References
- Baroody, A., J., & Wilkins, J. L. M. (1999). The development of informal counting, number, and arithmetic skills and concepts. In J. V. Copley (Ed.), Mathematics in the early years (pp.48-65). Reston, VA: National Council of Teachers of Mathematic
- Clements, D. H. (1999). Subitizing: What is it? why teach it? Teaching Children Mathematics, 5(7), 400. Retrieved from http://ezproxy.lib.ryerson.ca/login?url=https://search-proquest.com.ezproxy.lib.ryerson.ca/docview/214140408?accountid=13631
- Macdonald, B. L., & Shumway, J. F. (2016). Assessing Preschoolers Number Understanding Reflect and Discuss. Teaching Children Mathematics,22(6), 340-348. doi:10.5951/teacchilmath.22.6.0340
- Ontario Ministry of Education. (2016). The kindergarten program. Retrieved from https://files.ontario.ca/books/edu_the_kindergarten_program_english_aoda_web_dec12. pdf
- Penner-Wilger, M., Fast, L., & LeFevre, J. (n.d). Subitizing, finger gnosis, and finger agility as precursors to the representation of number. 1385-1390. doi:10.22215/etd/2009-06553
- Pepper, K. L., & Hunting, R. P. (1998). Preschoolers Counting and Sharing. Journal forResearch in Mathematics Education,29(2), 183rd ser., 164. doi:10.2307/749897
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