Tag: prime numbers

Posted in Algebra, Number Sense

The many faces of multiplication

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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?

Posted in Number Sense

What can the representations of numbers tell us?

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Numbers can be represented in different ways. The kind of representation we choose can highlight or de-emphasise the properties of the numbers.

Studies about understanding mathematics discuss about two kinds of representations of a mathematical idea: (1) transparent representations and (2) opaque representations. A transparent representation has no more and no less meaning than the represented idea(s) or structure(s). An opaque representation emphasizes some aspects of the ideas or structures and de-emphasizes others.

Examples:

  1. Representing  the number 784 as 28^2 emphasizes – makes transparent – that it is a perfect square, but de-emphasizes – leaves opaque – that it is divisible by 98.
  2. Representing the 784 as 13×60+4 makes it transparent that the remainder of 784 on dividing by 13 is 4, but leaves opaque its property of being a perfect square
  3. For a whole number k, 17k is a transparent representation for a multiple of 17, as this property is embedded or ‘can be seen’ in this form of the representation. However, it is impossible to determine whether 17k is a multiple of 3 by considering the representation alone. In this case we say that the representation is opaque with respect to divisibility by 3.
  4. An infinite non-repeating decimal representation (such as 0.010011000111. . .) is a transparent representation of an irrational number (that is, irrationality can be derived from this representation if the definition adopted is its being non-repeating, non-terminating decimal; It becomes an opaque representation for the definition of irrationals as numbers that cannot be expressed as quotient of two integers.)
  5. 2k+1 and 2k are transparent representations of odd and even numbers, respectively.

But what about prime numbers and irrational numbers in general? What are their representations? P for prime is not a representation.  In the article Representing numbers: prime and irrational, Rina Zaskis argued these two numbers have something in common: they both cannot be represented. Don’t we say irrational numbers are those that cannot be represented as a quotient and prime numbers are those that cannot be represented as a product? The examples I listed above were from the same paper. The author used them to argue the importance of representations and how the absence of it can become a cognitive obstacle to understand the concept.