Multiplication and repeated addition

Comparing and contrasting multiplication and repeated addition sparked an extended mathematics education debate. While earlier references are available, the heated debate involving multiple mathematics educators started in the 1990s. At the heart of the debate is the deceptively simple question, "Is multiplication repeated addition?" Participants in the debate bring up multiple perspectives, including axioms of arithmetic, pedagogy, learning and instructional design, history of mathematics, philosophy of mathematics, neuroscience, and computer-based mathematics.

Background of the debate

In the early 1990s Leslie Steffe proposed the counting scheme children use to assimilate multiplication into their mathematical knowledge. Jere Confrey contrasted the counting scheme with the splitting conjecture. Confrey suggested that counting and splitting are two separate, independent cognitive primitives. This sparked academic discussions in the form of conference presentations, articles and book chapters.

Among teachers and parents, the debate originated with the wider spread of curricula that emphasized scaling, zooming, folding and measuring mathematical tasks in the early years. Such tasks both require and support models of multiplication that are not based on counting or repeated addition. Debates around the question, "Is multiplication really repeated addition?" appeared on parent and teacher discussion forums in the mid-1990s.

Keith Devlin wrote a Mathematical Association of America column titled, "It Ain't No Repeated Addition" that followed up on his email exchanges with teachers, after he mentioned the topic briefly in an earlier article. The column linked the academic debates with practitioner debates. It sparked multiple discussions in research and practitioner blogs and forums. Keith Devlin has continued to write on this topic.

Because the topic is still controversial and exciting for mathematics educators, it makes for good debate assignments for professional development of teachers of mathematics. It also generates many comments when it is brought online, and thus many bloggers and forum facilitators use it to spark conversations.

Pedagogical perspectives

From counting to multiplication

Consider the sequence of topics typical of many curricula and standards, for example, 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:

$8+8+8=24,$

or choose the alternative

$3\times 8=24.$

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

$3\times 0=0+0+0,$

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

$0\times 3=0.$

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,

$7/4\times 5/6$

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 positive irrational numbers are brought into play. If one regards repeated addition as “packaging the base into a new unit, and then getting more of those new units,” a logical geometric extension of repeated addition emerges. Think of $4\times 3$ as quadrupling 3 tennis balls. If the three balls were “packaged” into a single can, then we obtain 4 of these cans (the new units), and it results in our desired 12 tennis balls. This idea can be advanced, by way of the familiar geometry notion of similar triangles all the way through positive irrational numbers.

Debate questions related to these curricula include the following:

• Are student difficulties with fractions and irrational numbers caused by viewing multiplication as repeated addition for a long time before these numbers are introduced, or by some other features of curricula?
• Is it acceptable to significantly modify rigorous mathematics for elementary education, leading children to believe statements that later turn out to be incorrect?

From scaling to multiplication

Theories of learning such as the splitting conjecture, the work of the Russian mathematics educators in the Vygotsky Circle, and investigations of underlying metaphors for multiplication by those studying embodied cognition have inspired curricula with "inherently multiplicative" tasks for young children. Examples of the tasks include elastic stretching, zoom, folding, projecting or dropping shadows. These tasks don't depend on counting, and can't be easily conceptualized in terms of repeated addition. Debate questions related to these curricula include the following:

• Are the tasks accessible all young children, or only accessible to the best students and/or experimental teaching conditions?
• Can children achieve computational fluency if they see multiplication as scaling rather than repeated addition?
• Do children get confused by the two separate approaches to multiplication introduced closely together? Should scaling and repeated addition be introduced separately, and if so, when and in what order?

What can you multiply?

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.

Models and metaphors that ground multiplication

In the context of this article, models are concrete representations of abstract mathematical ideas that reflect some, or all, essential qualities of the idea. Models are often developed as physical or virtual manipulatives and curricular materials that accompany them. A part of the debate about multiplication and repeated addition is the comparison of different models and their curricular materials. Debate questions related to the models include the following:

• Does the model support multiplication of all types of numbers?
• Does the model support applications in some important areas? For example, combination models come up in probability and biology.
• How can we help students avoid confusion among multiple models of multiplication?

Set model presents numbers as collections of objects, and multiplication as the union of multiple sets with the same number of objects in each. This model is ubiquitous in math curricula, because its action is familiar to all students. The model is easy to set up with everyday materials or just fingers. It can be extended to multiplication of a fraction by a natural number. Some curricula extend the set model to negative numbers by introducing special "negative counters." However, the model can't be extended to multiplying a fraction by a fraction, or to multiplying real numbers.