“Number sense in preschool and kindergarten generally involves symbolic representations of numbers, both verbal (e.g., number names) and written (e.g., Arabic numerals), in contrast to more fundamental preverbal number knowledge that appears to develop without much input or instruction” (Jordan & Dyson, Paul H. Brookes Publishing Co.) Representing numbers in multiple forms is a key part of representing and communicating about numbers. Through their school years students learn many ways to represent numbers, with numerals and equations, as images, with objects, with standardized manipulatives and in words. Each representation builds on a student understanding. Traditionally students have learned the most concrete representations using objects or manipulatives first, moving to drawing pictures and finally to using numerals and words. Students need to learn that quantities can be represented in multiple ways and must be prepared to do so with accuracy and clarity.
With objects
Representing numerals with objects can take on one of two main forms, a collection of objects such as a counting collection or standardized manipulatives. Students just beginning their number sense journey are most familiar with a collection of objects such as beads, buttons, popsicle sticks, gems, small blocks, counters, tiles or other toys. As students build their familiarity with numbers and number sense, they are introduced to standardized manipulatives such as base ten blocks, fraction rods or circles, Cuisenaire rods, and algebra tiles. Students begin by counting the objects or putting them into groups of a specific number, they may then be introduced to five and ten frames to facilitate groupings of objects to allow them to be counted more efficiently. When students are being introduced to place value the base ten blocks as well as ten frames can be used to model numbers. Cuisenaire rods can be used to represent skip counting or other number relationships as well as fractions clearly.
Counting objects
Loose groups of buttons, tiles, beads, counters, and other small objects create opportunities for students to practice their counting skills as well as representing numbers in various ways. Students can count out individual items and small groups, practice grouping objects into groups of 5 and 10 for easier counting. These are often the best tools for students who are just beginning to build their number sense as they can be worked with more flexibly.
Standardized Manipulatives
As students build their number sense, they are introduced to manipulatives that have standardized or commonly agreed upon values. The most common is the base ten blocks, students are taught the meaning of the units, rods, flats, and cubes in the primary years and understanding that the value of the next size up is ten times higher than one before. Cuisenaire rods are another example of a standardized manipulative where each number from 1 to 10 is represented by both length and colour, with Cuisenaire rods the one is represented by a 1cm cube the following nine rods stretch to become rectangular prisms with each number gaining 1cm in lengths so the 10cm rod is 10cm in length. As these tools have agreed upon meanings, they allow students to work quickly and flexibly to compose and decompose larger numbers and communicate their understanding easily. The standardized manipulatives are a building block for the numerical language that is mathematics, creating tactile representations of ideas that students can interact with.
With images
Using images or drawings to represent numbers is a natural step after modeling numbers with objects. The images students use to represent quantities can be symbolic in the form of tally marks or dots or students may choose to draw the items themselves. Drawing allows students who are not yet able to write the numerals to show their understanding of quantities. The ability to draw images to represent quantities allows students to represent more complex ideas as they move into working with different operations. Tally marks are another form of drawing to represent quantity that aids in applying skip counting skills to determine quantity by working with the groups of 5. Students are frequently encouraged to draw to represent fractions to help them visualize the quantity. Representing numbers with images encourages students to visualize quantities when objects may not be appropriate or are not needed. Having students represent numbers with images allows them to communicate about quantities more flexibly and record their thinking for future reference or for others to see their thinking.
With words
“Another hurdle for young children is learning the names of the Arabic digits and numerals. For most adults, there is an automatic association between the written digit (5), the spoken word (“five”), and the written word (five), such that when someone hears or sees one of them, the others automatically come to mind. The translation from one format to another is referred to as transcoding.” (Krasa, Tzanetopoulos, & Maas, 2023) Young children’s first exposure to numbers is as spoken words typically by parents and care givers as they count aloud reciting the number sequence or practice counting skills with small groups of objects. As students develop their number sense, they connect written numerals with the spoken words and eventually to written words. Representing numbers with words is as much a literacy task as it is a numeracy task. English number words do not typically follow common phonics patterns and must be learned as sight words. When looking at the teen numbers this can cause confusion as they do not begin in a regular pattern and continue to say the ones place before the tens with fourteen, fifteen, sixteen etc. This requires a great deal of memorization as well as decoding and encoding skills. Teachers also need to be aware of not saying or writing the word “and” when saying or writing the number words for whole numbers, the word “and” is used to denote the decimal point in rational numbers. Being able to correctly write the number words also relies on a strong understanding of base ten number structure and using positional notation in word form, five hundred thirty-four (534) makes sense because the number words are in the correct positions whereas thirty, five hundred, four even though it’s the same number words does not encourage the reader to translate it to the correct numeral. When representing fractions with numbers of words specific words such as fourths, quarters, thirds, or halves may be used, it is also possible to say the denominator over the numerator.
With numerals
“It is one thing to know what a 6 looks like, say its name, and write it or point it out in a picture book. It is quite another to know what it means—that is, how many countable things it stands for, or symbolizes.” (Krasa, Tzanetopoulos, & Maas, 2023) Students are most familiar with representing numbers with the Hindu-Arabic digits and the base ten number system. Each digit is a symbol for a specific quantity from 0-9 while where the digit is located indicates it’s value. Students not only need to accurately assign digits to quantities, but they also need to use the correct place value.
Expanded form.
Understanding numbers in expanded form directly relates to students’ ability to use the additive property and understand the role of position in numbers. While 457 and 754 have the same digits they represent different quantities. Expanded form involves breaking numbers apart according to their positional notation so that the value of each number is clear. In expanded form 457 would be written as 400+50+7 this allows the reader to clearly see the value in each digit. Expanded form allows students to represent their understanding of a quantity as well as to break it down into its place value. Working with expanded form is a good tool for students who are just beginning to understand the role on place value in numeration by connecting the numerical representation to the base ten blocks they are using.