At first, children use concrete quantities to solve simple arithmetic problems and transition towards calculating with symbols.
Calculation becomes more fluent as children move on to calculating by reciting numbers in their head instead of counting with fingers or other concrete objects.
Through repeated practice, children begin to recall answers automatically, marking the emergence of arithmetic fact knowledge.
To successfully extend their arithmetic skills from small numbers to larger ranges, children need a solid understanding of the magnitude and structure of single- and multidigit numbers and number system logic.
Children gradually learn to map number words onto Arabic numerals, thereby increasing the integration between different numerical formats.
Understanding geometry starts with verbally naming common shapes, then developing the skills to use shape-related attributes and finally expressing geometric features using complex terminology.
Additional reading
Arithmetic skills
Counting quantities forms the basis for solving arithmetic calculations using symbols (i.e. numerals) (Sarnecka & Carey, 2008). Children in preprimary education practise addition and subtraction, mainly with concrete quantities or simple arithmetic stories. Story problems allow children to develop concepts of addition and subtraction without yet knowing the symbols + or – (e.g. ‘I have three balloons. One flies away. How many do I have now?’). Children start arithmetic calculations using small numbers and often resort to reciting numbers and counting with their fingers or objects. There is a gradual transition towards calculating with symbols, as the child learns how calculations with concrete objects and verbal problems are represented using numbers and signs (Knudsen et al., 2015). In other words, they are learning the basics of arithmetic. In the early stages of arithmetic development, children usually solve arithmetic problems through strategies based on reciting numbers. This often involves using their fingers or items for support (Dupont-Boime & Thevenot, 2018).
As children’s calculation skills become more fluent, they move on to calculating by reciting numbers mentally (Geary et al., 2012). To support these skills, it is important to understand how they calculate, such as the counting strategies they use (e.g. how many numbers they recite, where they begin counting). One characteristic of an efficient calculation strategy is the need to recite as few numbers as possible (Baroody, 1987). Examples of this include starting the calculation from the larger number when adding (e.g. for 2 + 6, the calculation starts from 6: 6, 7, 8) and when subtracting, using addition when the numbers to be calculated are close to each other. For example, in the calculation 9 – 7, it is more efficient to count forward from seven to nine (7, 8, 9, giving the result 2) than to count backwards from nine by taking out seven (8, 7, 6, 5, 4, 3, 2). A strong concept of numbers helps children perform calculations without needing to recite numbers. In the example above, a child can make use of their knowledge that nine is two greater than seven, so that when seven is taken away from nine, two remain. Through repeated practice, children begin to recall answers automatically, marking the emergence of arithmetic fact knowledge. This refers to the ability to retrieve basic number combinations—such as single‑digit addition, subtraction, multiplication and division facts—quickly and accurately from memory without relying on counting or other strategies.
At its best, calculation is a matter of a good grasp of conceptual knowledge, calculation procedures, factual information and their seamless combination. There is some variation between arithmetic operations in terms of the type of information emphasized when learning the skill. For example, multiplication relies most firmly on memorization (factual knowledge), so it can also be the most difficult for children who have difficulty learning combinations between numbers (arithmetic factual knowledge) (De Visscher & Noël, 2014).
Number-system knowledge
In addition to performing calculations with single‑digit numbers, children need a solid understanding of the magnitude and structure of multi‑digit numbers and the number system. Children in pre‑primary education typically have a good sense of the magnitude of single‑digit numbers and can identify the smallest and largest among them. As the number range expands to two‑digit numbers and beyond, children must increasingly understand place value and the base‑ten structure to interpret numerical magnitude—for example, distinguishing between 412, 421, 214 and 142.
Understanding the base-ten system is a key milestone in this: The value of a digit depends both on the digit itself and on its position. Each place value is ten times the one before it, forming the structure of whole numbers through ones, tens, hundreds and thousands. This understanding allows children to transfer arithmetic skills learned with small numbers to larger number ranges. For example, knowledge of number bonds to ten (e.g. 5 + 5, 9 + 1) becomes directly applicable to hundreds and thousands (e.g. 90 + ? = 100, 500 + 500 = ?), provided the child understands that the base of the system is ten (e.g. ten tens make a hundred, ten hundreds make a thousand). The same principle later extends to decimal numbers, where place values decrease by factors of ten into tenths, hundredths and thousandths Research suggests that children first understand that a multidigit number represents one whole number (Cheung & Ansari, 2021). After that, they begin to break the number into parts based on the position of each digit. For example, in the number 345, they may notice that the ‘3’, ‘4’ and ‘5’ each contribute something to the total. They also learn that positions have different values that decrease from left to right. However, at this stage, they often do not yet know exactly which position represents which value—for instance, that the second digit shows tens and the third shows hundreds. Only after they understand the general relationship between place and value can they learn the precise values of each position according to the base‑ten system.
Number transcoding refers to converting numbers from one format to another (e.g. transforming spoken number words into Arabic digits or vice versa). Children gradually learn to map number words onto their corresponding Arabic symbols, thereby increasing the integration between different numerical formats. Arabic digits form a relatively simple system of10 symbols and the place-value principle, whereas verbal number systems are more complex (Imbo et al., 2014). They rely on a limited lexicon organized into units, decades, hundreds and thousands and include irregular forms such as ‘eleven’ and ‘twelve’. Because only a small set of quantities can be expressed with single words, syntactic rules—both additive (e.g. one hundred and twenty) and multiplicative (e.g. five hundred)—govern the construction of larger numbers. Violations of these rules lead to characteristic errors. In dictated number-writing tasks, children most frequently produce syntactic errors, where the components are correct but the magnitude is wrong (e.g. one hundred thirty-four → 10034), whereas lexical errors (e.g. one hundred thirty-four → 124) involving incorrect number elements occur less often (Sullivan et al., 1996).
Geometry
Children’s understanding of shapes becomes more precise with age (Clements & Sarama, 2007). They first learn to name the common shapes verbally, gradually gain the ability to use shape-related attributes and express geometrical features by using complex terminology (Cross et al., 2009; Skoumpourdi, 2016). At age 3, children recognize simple shapes (i.e. triangles and squares); between ages 4 and 5, they recognize different kinds of rectangles; at age 5, they learn typical examples of other shapes (e.g. hexagons, rhombuses); and at age 6, they can distinguish between a rectangle and a parallelogram without right angles (Clements & Sarama, 2021). From age 4 and on, children not only recognize typical shapes (i.e. triangles, squares, circles) but also learn to recognize less typical triangles and quadrilaterals (e.g. different kinds of rectangles). The same progress applies to the depiction of shapes, starting from drawing a circle at the age of 3 to gaining skills to draw a star at the age of 6, and the progress is influenced by the maturation of fine motor skills (Payne & Isaacs, 2017; Sarama & Clements, 2009; Villarroel & Sanz, 2017).
From age 4 on, children are capable of composing two- and three-dimensional shapes to produce simple and familiar figures and structures that are several blocks high (Clements & Sarama, 2021). They can also compare two- and three-dimensional objects (Hawes et al., 2017; Jones & Tzekaki, 2016) and mentally turn objects, which means they understand principles of symmetry, asymmetry and rotation. Children’s skills in producing two- and three-dimensional shapes, however, become more advanced with age, meaning they can produce more complex familiar figures as their understanding of two- and three-dimensionality develops.
The aforementioned skills tend to become somewhat sophisticated by the age of 8 (Kaur, 2015); this is the time when children understand several geometric concepts and can depict a variety of shapes and figures in both two and three dimensions (Dagli & Halat, 2016; Skoumpourdi, 2016; Villarroel & Sanz, 2017). Understanding the two- and three-dimensionality of shapes and figures requires that children can compare and classify features but also understand part-whole relationships (Clements & Sarama, 2007; Hallowell et al., 2015).
References
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