The following table is not meant to be a complete list of ideas about the concept of multiplication. It is not meant to be definitive but it does include the basic concepts about multiplication for middle school learners. The inclusion of the last two columns about the definition of a prime number and whether or not 1 is considered a prime show that there are definitions adapted to teach school mathematics that teachers in the higher year levels need to revise. Note that branching and grouping which make 1 not a prime number can only model multiplication of whole numbers unlike the rest of the models. Multiplication as repeated addition has launched a math war. Formal mathematics, of course, has a definitive answer on whether 1 is prime or not. According to the Fundamental Theorem of Arithmetic, 1 must not be prime so that each number greater than 1 has a unique prime factorisation.
|If multiplication is …||… then a product is:||… a factor is:||… a prime is:||Is 1 prime?|
|REPEATED ADDITION||a sum (e.g., 2×3=2+2+2 = 3+3)||either an addend or the count of addends||a product that is either a sum of 1’s or itself.||NO: 1 cannot be produced by repeatedly adding any whole number to itself.|
|GROUPING||a set of sets(e.g., 2×3 means either 2 sets of three items or 3 sets of 2)||either the number of items in a set, or the number of sets||a product that can only be made when one of the factor is 1||YES: 1 is one set of one.|
|BRANCHING||the number of end tips on a ‘tree’ produced by a sequence of branchings(think of fractals)||a branching (i.e., to multiply by n, each tip is branched n times)||a tree that can only be produced directly (i.e., not as a combination of branchings)||NO: 1 is a starting place/point … a pre-product as it were.|
|FOLDING||number of discrete regions produced by a series of folds (e.g., 2×3 means do a 2-fold, then a 3-fold, giving 6 regions)||a fold (i.e., to multiply by n, the object is folded in n equal-sized regions using n-1 creases)||a number of regions that can only be folded directly||NO: no folds are involved in generating 1 region|
|ARRAY-MAKING||cells in an m by n array||a dimension||a product that can only be constructed with a unit dimension.||YES: an array with one cell must have a unit dimension|
The table is from the study of Brent Davis and Moshe Renert in their article Mathematics-for-Teaching as Shared Dynamic Participation published in For the Learning of Mathematics. Vol. 29, No. 3. The table was constructed by a group of teachers who were doing a concept analysis about multiplication. Concept analysis involves tracing the origins and applications of a concept, looking at the different ways in which it appears both within and outside mathematics, and examining the various representations and definitions used to describe it and their consequences, (Usiskin et. al, 2003, p.1)
The Multiplication Models (Natural Math: Multiplication) also provides good visual for explaining multiplication.
You may also want to read How should students understand the subtraction operation?