This book covers the elementary school mathematics curriculum common in most parts of the world. Its aim is to serve educators (teachers and parents) as a guide for teaching mathematics at elementary school level. The book focuses both on content knowledge and on pedagogical content knowledge. It bridges the gap between fundamental mathematical principles and good teaching practices. It also offers the reader a glimpse on how mathematicians perceive elementary mathematics and presents ideas for specific mathematical activities.
Volume 2 focuses on content taught in the higher grades of elementary school. It covers the following topics: multiplication and division of multi-digit numbers, divisibility and primality, divisibility signs, sequences, fractions and their representations, and fraction arithmetic.
This chapter presents multiplication of rational numbersânatural numbers and fractions. Fraction multiplication is a deceivingly simple operation. At the stage where it is taughtâin fifth and sixth gradesâthe evaluation of products of fractions is a relatively easy task; one has to multiply numerator by numerator and denominator by denominator to obtain the desired product. However, as emphasized repeatedly throughout this book, knowing how to evaluate an arithmetic operation is far less important than knowing when to evaluate it, and why it is evaluated in this way.
Products of natural numbers can always be interpreted as repeated addition. Products of fractions, however, require a different interpretation. Thus, the multiplication concept must be extended and generalized in order to make sense of fraction multiplication. The main focus of this chapter is acquiring a unified image of multiplication, applicable to both natural numbers and fractions, and extending the scope of problems modeled by multiplication. Evaluation strategies are then derived based on our ability to correctly interpret products of fractions.
Hurdles:
(a)Children donât understand the meaning of fraction multiplication and are content with knowing how to evaluate products.
(b)Children are confused by the verbal interpretation of the multiplication sign as times in the context of whole numbers and of in the context of fractions.
(c)Children have a misconception whereby multiplication is always a magnifying operation.
Remedies:
(a)Explain the meaning of multiplication as a scaling operation; this interpretation is sensible for both whole numbers and for fractions.
(b)Practice fraction multiplication in the context of concrete word problems.
(c)Use visual aids to demonstrate fraction multiplication.
(d)Emphasize the relation between products of whole numbers, products of fractions and whole numbers and products of fractions.
(e)Use concrete models to demonstrate why multiplication by a fraction is not magnifying, but rather reducing.
31.1Numbers and multiplication revisited
The number concept was first introduced in this book in Chapter 1, and was further elaborated in subsequent chapters. Notably, fractions were introduced in Chapter 27, extending the number system, which at first consisted only of natural numbers.
Multiplication was first introduced in Chapter 7, building upon the notion of repeated addition. We encountered various settings in which multiplication models concrete problems, such as counting combinations and measuring area. In the context of fractions, products of natural numbers and a unit fractions were used to define general fractions. That is, the fraction
is defined as a repeated addition, m times, of the unit fraction
,
In this section we revisit both concepts of numbers and multiplication, showing that numbers and multiplication are intimately related concepts. Understanding this relation will help us build a coherent image of fraction multiplication.
31.1.1The number concept revisited
Numbers are abstract entities. They acquire a concrete meaning only once a whole, to which we attribute the value one, has been specified. For example, the natural number 5 is an abstract entity, but 5 elephants are concrete, and so are 5 fingers, 5 chicks and 5 continents. The common property of all those 5 âsomethingsâ is that they are sets of objects that can be matched to each other in a one-to-one manner (Volume 1, p. 2).
The concretization of fractions, just like the concretization of natural numbers, relies on the specification of a whole. The number ONE-HALF is an abstract entity, but ONE-HALF of a peach is concrete, and so are ONE-HALF of a pound of flour, ONE-HALF of an hour and ONE-HALF of a group of children. The common property of all those ONE-HALF of âsomethingâ is that if we join 2 such halves together, we obtain concrete objects, each of which designated as a whole. It is of primary importance to remember that the designation of a certain object as a wholeâwhether an elephant, a finger, a chick, a continent, a peach, a pound of flour, an hour or a group of childrenâis a matter of choice, and varies from one context to another. For example, a pound of flour can be a whole in one setting, and it can be 2 wholes or ONE-HALF of a whole in another settingâit all depends on which problem we are trying to solve.
We have already observed that natural numbers can be visualized as replicating machines, or in mathematical jargon, operators. As explained in Chapter 3, an operat...
