Cultural tools refer to any tools that help us calculate, produce models, make predictions and understand the world more fully. Orthographies, writing systems, differ greatly. Alphabetic scripts are a writing system in which written symbols correspond to spoken sounds. In Chinese, each letter corresponds to a morpheme. Children find it difficult to realize that letters represent phonemes. Children get better at phonemes as they get older. Instruction is necessary for learning how to read and write. The environment plays a key role. The more children have learned about phonemes, the better they read and write.

Phonemic awareness is the idea that words consist of a sequence of phonemes. This idea does not come easily to young children. Phonological skills refer to the ability to detect and manipulate sounds at the phonetic, syllabic and intra-syllabic levels. Intrasyllabic units are units of speech that are smaller than syllables but larger than phonemes. An example of intrasyllabic units is rime (the unit that rhymes). Most children are aware of rimes from an early age. There is a positive relationship between sensitivity to rhyme and success in reading.

Conditional spelling rules are rules which determine that a letter or a group of letters represent one sound in one context and another sound in a different context. At first, children stick to a letter-sound association and don’t pay a lot of attention to conditional rules. Children only pass the pseudo-word test at the age of 10. In this test, children show that they understand the rule of the ‘silent-e’. Children’s success in reading determines how well they learn this particular spelling rule. In morpho-phonemic script, spelling rules are based on phonemes and on morphemes. Inflectional morphemes tell you something about the grammatical status of the word (e.g: tells you something about whether the word is plural or not). Derivational morphemes change the meaning of the word.

The difficulty in morphemic spelling rules could arise because of the fact that children are not at first aware of the morphemic structure of the words that they are trying to write. Young children may fail to use the conventional spellings for morphemes because they do not know enough about morphemes. The phenomenon of children’s spelling getting worse with some words at the same time as it gets better with others is widespread.

The overgeneralization of newly learned spelling patterns may be an essential part of learning and there may be an underlying three-step sequence:

1. Children start by spelling a particular sound (at the end of the word)
2. They learn another way of spelling this sound and begin to do it without knowing when it’s right
3. They learn the rule based on the feedback they receive

They use the rule first and learn about it later. It could also be that children write words correctly by rote and later infer the underlying rule for spelling inflexions on the basis of specific knowledge. Children’s word-specific knowledge is the basis for their learning of morphemic spelling rules. Instruction is an essential part of the process of learning letter-sound associations.

There are a set of principles of the number system:

1. Cardinality
   This is the principle that any set of items with a particular number is equal in quantity to any other set with the same number of items in it.
2. Ordinality
   This is the principle that numbers come in an ordered scale of magnitude.

There are also several counting principles:

1. One-to-one principle
   Each object must only be counted one and given a unique number tag.
2. Stable order principle
   Number words must be produced in a set order.
3. Last-number-counted principle
   The last number counted represents the value of that set.
4. Abstraction principle
   The number in a set is independent of any qualities of the members in that set.
5. Order irrelevance principle
   The order in which members of a set are counted does not affect the number of items in the set.

Words for numbers and base systems (e.g: decimal system) differ across languages and cultures. Infants distinguish small numbers. This suggests that infants have an innate understanding of number, although it could be that they were paying attention to the size and not the number. Wynn suggested that infants have a basic understanding of addition and subtraction because infants look longer at the impossible situation in Wynn’s research (the wrong answer to basic calculations). Criticism of Wynn’s work is that infants might just be responding to the numerosity of the situation and not because they have a basic understanding of simple arithmetic.

Gelman’s principle before skill hypothesis states that children grasp the numerical principles before they can put them into use effectively and consistently. Research has shown that violations of principles are more often the case with large number sets than with small number sets and this may be due to a problem with procedural skills, but this research leaves out several principles and the claim that numerical knowledge is innate is thus unimpressive.

Carey’s individuation hypothesis states that children have an innate parallel individuation system, which makes it possible for them to recognise and represent very small numbers exactly, but experience helps them acquire a better understanding of knowledge. As children age, they learn to distinguish sets of one item, to sets of two-items to sets of three-items. During this period, children also learn number words.

Carey states that children lift themselves up by their own intellectual bootstraps. They start to understand that the numerical system is a continuum and that numbers are in a constant order and this helps them learn about the ordinal properties of numbers. Experiments, in which children were asked to pick up a number of items, support this claim and shows that children are able to do this up until three items until they know counting principles.

There are several cultural tools of the numerical system, such as the decimal system. It is possible that English-speaking children have difficulties with the decimal system because the number words beyond 10 are vague. Eleven does not directly refer to 11.

Piaget is held back in understanding numbers by their lack of logic and has to acquire logical abilities in order to understand mathematics. Young children may know numbers very well, but do not actually understand what they’re doing when they’re counting. Piaget stated that children make a distinction between quantity and quotité. Quotité is the understanding of the children who realise that two sets of objects have the same number in the sense that counting each one leads to the same number word. This shows that children at first have a failure to use number as a comparative measure. Children have a poor one-to-one correspondence, the idea that two sets are in one-to-one correspondence if each object in one set has a counterpart if the other set. If they are in one-to-one correspondence they are equal in number.