Plato’s paradox refers to children acquiring language without appearing to be taught it. This raises the question of how children actually learn a language.

Word segmentation is one of the challenges of learning a language (i.e. where does one word start and where does it end in a stream of sounds). One proposed mechanism of learning is statistical learning. This holds that children learn from the regularities in language (e.g. picking up on strong vs. weak words). Children at 7.5 months of age are capable of word segmentation of 2-syllable words.

Chomsky states that children have an innate deep structure. This is an underlying understanding of language which enables the child to learn language. The specific language a child learns is the surface structure. Thus, according to Chomsky, people have a language acquisition device (LAD).

Phonology refers to the sounds of a language. Phonemic awareness (i.e. phonological awareness) refers to the knowledge that words consist of separable sounds and phonological recoding refers to understanding the sounds of phonemes. There are several stages of phonological development:

1. Reflexive crying and vegetative sounds (0 to 8 weeks)
2. Cooing and laughter (8 to 20 weeks)
3. Vocal play (16 to 30 weeks)
4. Reduplicated babbling (25 to 50 weeks) (e.g. bababa)
5. Jargon (9 to 18 months)

Morphology refers to the structure of words and to a system of rules for combining units of meaning into words (e.g. adding ‘ed’ to a word makes it past tense). A morpheme refers to the smallest unit of meaning in a language. Free morphemes can stand alone as words (e.g. fire; run) and bound morphemes cannot stand alone and are attached to free morphemes (e.g. the letter ‘s’). The mean length of utterance refers to the average number of morphemes a child uses in a sentence and is indicative of children’s linguistic development. Overregularization refers to overusing a rule (e.g. using ‘ed’ for everything that is past tense).

Syntax refers to a system of rules for creating phrases and sentences out of words. Holophrases refer to one-word sentences (i.e. single words with different intonations to convey different meanings). Telegraphic speech refers to using meaningful words and omitting words that make language easier to understand (e.g. daddy give milk).

Semantics refers to the meaning of language terms. Vocabulary is one indication of children’s semantic development. Children speak their first words at about 10 to 12 months of age and there typically is a word spurt at 18 months of age. Fast mapping refers to the ability to learn new words based on very little input and this may underlie the word spurt. Overextensions refer to stretching a familiar word beyond its correct meaning (e.g. ‘bird’ for everything that flies). This may lead adults to provide them with the correct word. Underextension refers to using a too strict category for a word (e.g. only one cat is a cat).

Word learning is influenced by lexical constraints:

1. Whole-object assumption
   This states that children assume that a new word refers to the whole object when they hear it.
2. Taxonomic constraint
   This states that children assume that words refer to things that are similar (e.g. two animals that look like rabbits are both rabbits).
3. Mutual exclusivity assumption
   This states that different words refer to different things.
4. Syntactic bootstrapping
   This states that children get an idea of the meaning of a word through its grammatical form.

Pragmatics refers to how language is used in a social context. Children need to learn that good messages have the right quantity of information (1), is at the proper level of description (2), are truthful (3) and relevant to the present context (4). Speech registers refer to different styles of speech (e.g. at home and school). A speech register at home that differs from one at school could lead to educational difficulties.

Communicative competence refers to the overall mastery of language. Collective monologues refer to egocentric exchanges where one speaks to somebody but not with somebody. Children typically have metacognitive deficits, meaning that they have a poor understanding of message quality and its role in determining the success or failure of communication. They also find it difficult to monitor their own speech. There is greater metacommunicative competence when messages are embedded in familiar scripts.

There is some evidence for a critical period of language acquisition. This means that (second) language learning is easiest to learn up to seven years of age and afterwards, it becomes more difficult to learn language. One proposed mechanism for the critical period is the less is more hypothesis. This holds that children have limited cognitive skills and this simplifies the body of language, allowing for easier learning of a language. There is thus first a simple language expression and a fast learning because a child is not distracted by the complexities of language due to the limited processing skills.

The gesture facilitation hypothesis states that the use of gestures facilitates the acquisition of spoken language (e.g. gestures can be used to get attention to something which is then paired with a verbal component). Gestures could lead parents to say words and sentences that children need to expand their speaking abilities. However, it is also possible that children who use gestures have better cognitive skills which may allow for easier language acquisition. Treating others as intentional beings (e.g. shared attention) is the social-cognitive foundation for language development.

Other factors which promote early language acquisition compared to late language acquisition is that children receive simpler input than adults (e.g. motherese). Adults may have a language acquisition support system (LASS) that responds to infants and young children by automatically altering speech to a more understandable form.

Children are also more motivated to learn language to be able to communicate with adults. There are several child factors that promote language acquisition:

• Motivation
• Personality (i.e. extravert child searches for language; introvert child observes well)
• Analytical ability.
• Phonological awareness.
• Working memory.

There are also several environmental factors that promote language acquisition:

• Security.
• Exposure to target language at home.

When learning a child a language, it is best to use the language that has been mastered best at home (i.e. native language) rather than the poorer target language (i.e. non-native, second language).

There is a difference between oral and written language. Oral language comes naturally whereas reading and writing does not and requires intensive instruction. Reading refers to recoding a grapheme to a sound and glueing together all the sounds of a word. Letter knowledge (1), motivation (2) and self-confidence (3) are important in learning to read. There is a bidirectional relationship between reading and working memory. The Matthew effect is important in explaining people’s later reading abilities.

Home literacy refers to the environment that is created by parents regarding literacy. This consists of formal literacy (e.g. aiding in learning how to read words) and informal literacy (e.g. reading to a child). The degree to which a parent engages in both forms of literacy influences a child’s reading abilities.

Bilingualism refers to being fluent in two languages with there being a distinction between simultaneous bilingualism and sequential bilingualism. Semi-lingualism refers to a lack of mastery in either language. Simultaneous bilinguals have a smaller vocabulary and a delayed syntactic development compared to monolinguals though this difference is minimized by the age of 8 or 9.

Gender differences in language development may exist due to cultural differences. It is also possible that prenatal exposure to testosterone influences later vocabulary or that girls are more vocal in infancy leading to an advantage in language acquisition.

Language is used to express thought but is not the same. The ability to use language may transform a child’s thinking and the use of language in social situations may facilitate this transformation. Thought and speech merge in development and become interdependent. Egocentric speech (i.e. private speech) may help children organize their thoughts as they cannot use language covertly (i.e. in their heads). Children rely more heavily on private speech when facing difficult rather than easy tasks.

Enumeration refers to determining the number of units in a set (e.g. how many people are in a group). Ordinality refers to determining how one number relates to another (e.g. is four more than five?). This is done through counting (1), estimation (2) and subitizing (3). This refers to the ability to quickly and accurately determine the number of a small set of units (i.e. immediately knowing that something is ‘three’). Innumeracy refers to a lack of deep understanding of what arithmetic principles are and what they mean. This puts one at risk of flawed reasoning which seems mathematically correct (e.g. two tubs of 25 degrees water makes one tub of 50 degrees water). Innumeracy may result from the difficulty controlling the activation of arithmetic schemas in multiple cerebral areas.

Language influences math acquisition as a more regular number system allows for easier learning of language (e.g. Cantonese vs. English). Children need to learn what quantity a number refers to and what that number means. Syntactic cues may teach children that number words are different from other words and semantic cues may teach children about the meaning of these words. The principle of contrast states that it is easier to recognize semantic cues when other words in a sentence (e.g. ‘big’; ‘nice’) are known and are linked to a known attribute whereas the number word is not yet linked to any known attribute.

Round numbers refer to approximate quantity and sharp numbers refer to a precise quantity. The two-number construction refers to an interval for numbers which indicates an approximation and a degree of uncertainty about the quantity (e.g. 10 or 20 people).

The frequency of numerals decreases systematically with number size. This means that the number three is more common than the number eight. However, there are elevated peaks in the frequency for round numbers (e.g. 10; 20). Benford’s law states that when one draws random numbers from any smooth distribution, the numbers will start more often with 1 than with 9. The human brain has a preference for smaller numbers and this may be because it is easier to represent smaller numbers. The frequency of numerals represents their importance in our mental lives.

There are several principles of counting:

1. 1-to-1 principle
2. There is a fixed, correct sequence of numbers.
3. Cardinality principle (i.e. the last number is the quantity).
4. Principle of abstraction (i.e. the rules of counting apply to any collection of units).
5. Order-irrelevance (i.e. the words may be counted in any order).

Counting may be innate while complex calculations are not. The minimum strategy refers to starting an addition equation with the larger quantity and count a number of times equal to the smaller of the two numbers (e.g. 4 + 2 = four, five, six, six!). Children tend to develop an intuitive understanding of commutativity (i.e. a+b=b+a) without any formal schooling.

People typically do not solve addition and multiplication problems by counting but retrieve results from a memorized arithmetic table. However, memory for this is flawed as all results are associated and human memory is associative (e.g. 7x5 and 7x6 are related). Next, the accuracy of mental representations may drop with number size (1), the order of acquisition may influence memory (2) and the amount of practice (3).

The automatization of arithmetic memory (e.g. seeing two numbers and automatically adding them up) starts at age seven. Calculation becomes tied to the language in which it is learned at school because arithmetic tables are learned verbatim. The brain calculates an estimate of the size and the exact result simultaneously. The meaning of computations is often ignored to have fast calculations which lead to interpretation errors (e.g. base rate fallacy).

Children’s subtraction algorithms have systematic errors (i.e. bugs) because textbooks do not cover all forms of subtraction and the child needs to come to one’s own conclusions following the teacher’s behaviour and examples.

The triple code model states that there is an analogue representation of numbers (i.e. representation that each number refers to a separate thing). There is a verbal label for this (e.g. the number three) and there is a visual representation (e.g. seeing three). The analogue representation is not symbolic and is learned naturally. The verbal and visual representation need to be learned and are taught at school.

There are several child factors in learning math:

1. Early math skills
   1. Approximate number system (ANS)
      This refers to the ability to estimate the number of units in a set and is non-symbolic. However, symbolic tasks are a more important predictor for math performance than non-symbolic tasks.
   2. Conservation of numbers
      This refers to comparing sets of units (e.g. density does not necessarily mean a larger number) and knowing where to place them on a number line.
2. Cognitive skills
   This includes working memory (1), inhibition (2) and ordering (3).

The home numeracy environment is important for learning math (e.g. helping the child learn math). Math anxiety refers to anxiety about math and this negatively influences one’s performance. A person’s math self-concept (i.e. self-esteem about math) is also important for one’s performance.

The higher a person’s success in math is the more this person practices with math. This, eventually, leads to higher improvement and performance in math. However, higher math success does not influence math anxiety and math self-concept. The experience of success is important for children.

Schools typically ignore one’s precocious math abilities (e.g. finger counting) and this can lead to a negative opinion on maths. The insistence on mechanical computation at the expense of meaning is detrimental for interesting children in mathematics. The goal of math curricula should be to improve children’s fluency in arithmetic and not perpetuate a ritual (e.g. memorize the arithmetic table).