### Early Childhood

The study of the developmental progression of mathematical abilities in humans shows that Humans have a pre-verbal number system, which is not language dependent. Their understanding of ordinality and cardinality is inherent and universal (Geary, 2000; Butterworth, 2005). Here are some highlights:

- Infants under one year show number sensitivity to groups of 3 to 4 objects
- At 18 months, babies show an understanding of simple and ordinal (i.e. less than, greater than) relationships
- Preschoolers show an understanding of ordinal relationships and counting
- Preverbal number system becomes integrated with emerging language through the use of number words, which they use for counting, adding, and subtracting
- Preschoolers develop an understanding of counting, ordinality, and cardinality (last number word in a set represents the total number in that set)

### Elementary School (ages 6 – 11)

By the end of elementary school, children have memorized basic addition, subtraction, and multiplication facts, as well as solve complex multi-column problems. Poor understanding of base-10 leads to errors in borrowing. (Geary, 2000) Here are some issues that affect mathematical learning in elementary school:

- The most difficult concept to learn is the base 10 structure of the Arabic number system
- Learning this system is dependent on educational practices and the number words used in the language. Example: English and French words do not correspond to the base 10 representation of the quantity.
- Failure to learn base 10 results in difficulties counting past 10 and learning to rename number groups

### Secondary School (ages 12-18)

A major expectation of these years is for students to learn to solve arithmetical and algebraic word problems. Several factors influence how difficult this process will be:

- Wording of the problem
- The ease with which sentences can be translated from verbal to mathematical representations
- The number of sentences that must be translated and integrated

The ability to solve problems is dependent on the development of a problem solving scheme for the problem type – i.e. a memorized sequence of problem solving steps for a class of similar problems. High quality instruction can act as a buffer against memory deficits and language-based difficulties (Geary, 2000).

### The Relationship Between Innate and Learned Mathematical Ability

Stanislas Dehaene, a French neuroscientist, has devoted much of his career to the study of number sense and which aspects of mathematical ability are innate and which are learned, as well as how the two areas overlap and affect each other. A basic problem with learning mathematics is that number sense is genetic, but exact calculation involves cultural tools, such as symbols and algorithms. These symbols and algorithms must be absorbed by parts of the brain that have evolved for other purposes. Nature provides the basic number sense, and culture provides numerals and number words. These three ways of thinking about numbers lie in three distinct areas of the brain. The human memory is largely associative, which makes it poorly suited to arithmetic, where associations can cause bits of knowledge to interfere with each other. For example, to retrieve the answer to 7 x 6, a reflex association to 7 + 6 can cause great trouble in the midst of problem solving. Therefore, multiplication facts have to be stored in a form that does not fit easily into the way human memory is organized. The result is that adults make errors in single digit multiplication 10 to 25 per cent of the time.

Language presents its own challenges to the development of mathematical abilities. English has special words for the numbers from 11 to 19 and for the decades from 20 to 90. The language itself challenges children learning to count and limits memory span of digits a person can hold. As a result of these discoveries, DeHaene believes that the teaching of mathematical concepts must be done in a way that takes into account the way human brains are organized for math, as well as the challenges posed by language and memory.

### References

Butterworth, B. (2005). The development of arithmetical abilities. *Journal of Child Psychology and Psychiatry, 46*(9), 3-18.

Geary, D.C. (2000). From infancy to adulthood: the development of numerical abilities. *European Child & Adolescent Psychiatry, 9*(2), 11–16.

Holt, J. (2008, March 10). Numbers guy. *The New Yorker, 84*(3), 42-47.