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Defining Mathematics Education

Defining Mathematics Education

The Seventy-fifth Yearbook is a celebration and reflection of the history of the NCTM yearbooks, and is a great resource on the key issues of mathematics education through the years.

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Part and wholes

In years 3 and 4, students use two dice, one labelled 1 to 6 (called the whole-number die) and the other labelled with fractions (called the fraction die). The fraction die has fractions appropriate to the year level (e.g. for year 3 the fractions could be \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{3}\), \(\frac{1}{5}\)).

Pairs of students roll the two dice to generate numbers to insert into the sentence:

(fraction roll) of my collection is (whole-number roll) so altogether I have ____

 

Two dice, one die displaying 1/4 and the other 5. One set of 5 cubes in one colour, three sets of 5 cubes in another colour to make a total of 20.

Example: One quarter of my collection is 5 so altogether I have __

Students model the problem, using counters to make a group of five.

They then make more groups of five until there are four equal groups, and calculate the total.

Teachers should:

  • encourage the formation of arrays to make the relationship with factors and multiples more obvious
  • discuss strategies for working out the total without having to build the array
  • ask what other fractions can be determined from the array, such as
    \(\frac{3}{4}\) of 20 = 15
    \(\frac{1}{5}\) of 20 = 4

In year 6, students solve problems to determine fractions of collections and multiples of those fractions. For example:

I lost \(\frac{1}{8}\) of my marbles. I have 42 left. How many did I start with?

Diagrammatic explanation of the problem posed with calculations and written responses.

Student work sample

Discuss with the class:

  • possible strategies for solving the problems
  • the effectiveness of various strategies.

For students not confident in factors and multiples, creating arrays on grid paper can provide scaffolding for finding strategies.