Multiplication and repeated addition

In typical mathematics curricula and standards, such as the Common Core State Standards Initiative, the meaning of the product of real numbers steps through a series of notions generally beginning with repeated addition and ultimately residing in scaling.

Once the natural (or whole) numbers have been defined and understood as a means to count, a child is introduced to the basic operations of arithmetic, in this order: addition, subtraction, multiplication and division. These operations, although introduced at a very early stage of a child's mathematics education, have a lasting impact on the development of number sense in students as advanced numeric abilities.

In these curricula, multiplication is introduced immediately after posing questions related to repeated addition, such as: "There are 3 bags of 8 apples each. How many apples are there in all? A student can do:

${\displaystyle 8+8+8=24,}$

or choose the alternative

${\displaystyle 3\times 8=24.}$

This approach is supported for several years of teaching and learning, and sets up the perception that multiplication is just a more efficient way of adding. Once 0 is brought in, it affects no significant change because

${\displaystyle 3\times 0=0+0+0,}$

which is 0, and the commutative property would lead us also to define

${\displaystyle 0\times 3=0.}$

Thus, repeated addition extends to the whole numbers (0, 1, 2, 3, 4, ...). The first challenge to the belief that multiplication is repeated addition appears when students start working with fractions. From the mathematical point of view, multiplication as repeated addition can be extended into fractions. For example,

${\displaystyle 7/4\times 5/6}$

literally calls for “one and three-fourths of the five-sixths.” This is later significant because students are taught that, in word problems, the word “of” usually indicates a multiplication. However, this extension is problematic for many students, who start struggling with mathematics when fractions are introduced . Moreover, the repeated addition model must be substantially modified when irrational numbers are brought into play.

Concerning these issues, mathematics educators have debated whether student difficulties with fractions and irrational numbers are exacerbated by viewing multiplication as repeated addition for a long time before these numbers are introduced, and relatedly whether it is acceptable to significantly modify rigorous mathematics for elementary education, leading children to believe statements that later turn out to be incorrect.

Multiplication can also be thought of as scaling. In the above animation, we see 3 being multiplied by 2, giving 6 as a result.

One theory of learning multiplication derives from the work of the Russian mathematics educators in the Vygotsky Circle which was active in the Soviet Union between the world wars. Their contribution is known as the splitting conjecture.

Another theory of learning multiplication derives from those studying embodied cognition, which examined the underlying metaphors for multiplication.

Together these investigations have inspired curricula with "inherently multiplicative" tasks for young children. Examples of these tasks include: elastic stretching, zoom, folding, projecting shadows, or dropping shadows. These tasks don't depend on counting, and cannot be easily conceptualized in terms of repeated addition.

Issues of debate related to these curricula include:

Multiplication is often defined for natural numbers, then extended to whole numbers, fractions, and irrational numbers. However, abstract algebra has a more general definition of multiplication as a binary operation on some objects that may or may not be numbers. Notably, one can multiply complex numbers, vectors, matrices, and quaternions. Some educators believe that seeing multiplication exclusively as repeated addition during elementary education can interfere with later understanding of these aspects of multiplication.