As children develop cognitively from pre-lingual and pre-symbolic stages to the use of language and symbols to manipulate concepts, their abilities related to later mathematics learning are also developing. Some of the most critical cognitive abilities for mathematics learning are memory, language skills, and the ability to make mental representations of number and space.

Young children begin using their memory abilities as they interact with the environment and recall those experiences. Infants will respond to familiar faces and music. Children enjoy retelling stories and singing songs over and over again. As they begin noticing environmental print, children begin to understand the role of letters and numbers as abstract representations for familiar things. Names of streets, stores, candy, and numbers on houses and roads begin to take on meaning. Children ask to be taught to write their names and memorize the markings. Some children are so delighted with their new skills that they make markings everywhere—on books, walls, and under furniture.

With formal schooling, memory tasks become more challenging. Children must recall the written character for letters, numbers, and other symbols used in writing and mathematics. They are required to remember math facts and the sequences for performing operations with numbers. In problem solving, children are encouraged to recall a similar problem type or situation. Memory tasks are more successful when children learn through concept understanding rather than by rote memorization. New concepts should be connected to real-life experiences of children so that cognitive structures are formed in long-term memory.

Language development is critically integrated with mathematics development. Children use language to express relationships, assign labels, manipulate concepts, and communicate understandings with others. Language becomes the mediation tool for performing more difficult mathematics tasks, as can be seen in native language comparison studies (see, for example, Miura & Okamoto, 2003). Teachers use carefully selected language to ask questions and explain new concepts.

Children with language delays may have corresponding delays in mathematics development. Cognitive abilities of symbolic thinking, temporal–sequential organization, verbal memory, and rate of language processing are language abilities directly related to mathematics tasks (Jordan, Levine, & Huttenlocher, 1995). Nonverbal mathematics tasks, such as the manipulation of objects without verbal requirements, are generally developed earlier than verbal tasks (story problems) and are less sensitive to socioeconomic differences. Jordan, Levine, and Huttenlocher examined the calculation abilities of young children (kindergarteners and first graders) and found that children with language impairments performed as well as their peers on nonverbal calculations but significantly worse on story problems (presented orally) than the non-impaired group.

Younger children with language disabilities should concentrate on the cardinal aspects of number—number represents the number of objects in a set—and work with small quantities until those are firm concepts (Grauberg, 1998). They should not be discouraged from using compensatory strategies such as finger counting (Jordan, Levine, & Huttenlocher, 1995). Additional nonverbal activities can build concept understanding while the verbal aspects are strengthened. Some students may benefit from a delay in writing number symbols or using tallies of number symbols until number and quantity concepts are secure.

The ability to make mental representations is critical for later mathematics learning but very difficult to assess in young children. Infants can distinguish between sets of objects with one and two, three and four, and even eight and sixteen objects depicted (Sophian, 1998). This skill requires interpretation of two visual images as different, but researchers have not been able to determine whether a sort of nonverbal counting takes place or a visual subitizing. Toddlers (about age two) can distinguish between sets with different numbers of objects and relative magnitude of sets (e.g., a card with eight dots looks like it has more dots than a card with three dots), especially when the number of objects is not close enough to need counting (Ginsburg, 1982). Young children (about three years old) can recognize that adding objects to a set will result in more, and removing objects results in fewer.

The act of actually counting objects develops later because it becomes a bridge between the concrete (objects visualized) and abstract (objects matched with numbers). Although some children at 18 months can count up to three objects, it is not a firm one-to-one correspondence counting or view of magnitude, just labeling. Many five-year-old children can “count” to 20 but mostly by rote and often in number chunks. When asked for the number after 18, they may have to “hear” 15, 16, 17, 18, to recall “19.” Around age five children begin associating numbers by their relative magnitudes. Counting by tens may also be learned first by rote (without meanings of number or magnitude) and may not be a reflection of understanding until first or second grade.

Other important mental representations for children are those spatial aspects of mathematics that are, in fact, closely related to number concepts. Infants view quantities as overall amounts and relative amounts rather than discrete numbers (Mix, Huttenlocher, & Levine, 2002). Although some spatial tasks are closely related to number concepts (number lines, graphs, measurement in units), others seem to be less connected (flipping an object mentally, interpreting relative positions in space). Mental representations include the characteristics of objects (shape, size, color), relative positions of objects (distance, over, behind), rotations of objects as the same object (flips, turns), composition and decomposition of objects, recognition of symbols (equals sign, minus sign, numerals), spatial orientation, interpretation of drawings (map of classroom, graph of buttons), and even some concepts of time.

These concepts are developed through guided interaction with a rich environment and should not be required on a purely abstract level until concepts are formed—possibly not until the second grade in formal schooling. Some children will be ready and even eager by age three or four to use symbolic notations because older siblings do that, and parents may think it a sign of giftedness. But overemphasizing work at the abstract level using primarily paper and pencil tasks with four-, five-, and even six-year-olds may actually disrupt and delay stronger concept development that should develop from informal understanding at the concrete level (Pound, 1999).