Activities designed to move students from steps 1-3 to step 4
Where are they now?
Model fractional language and concepts incidentally when dividing or sharing objects (e.g. an orange or a collection of berries) within small groups
Where to next?
Recognises and names halves of familiar objects (e.g. half an apple)
Building Fraction and decimal skills:
Even before they come to school many young children exhibit an awareness of fraction names such as half and quarter. During the first years of schooling, most will be able to halve a piece of paper, identify 3 quarters of an orange and talk about parts of recognised wholes (e.g. blocks of chocolate, pizza, Smarties etc.). While to an adult ear, this sounds like children understand the relationship inherent in fraction representatives, for many they are simply using these items to describe and/or enumerate well-known objects. Such children may not be aware of or even attending to the key ideas involved in a more general understanding of fractions; that is, that equal parts are involved, the number of parts names the parts, and that as the number of parts of a given whole are increased, the size of the parts (or shares) gets smaller.
The use of fraction words to name and recognize parts of recognized wholes lulls adults (teachers and parents) into thinking that many of these children are able to understand and use the fraction symbol in the early years of schooling despite the fact that most curriculum advice now advocates a delay in the formalisation of fractions. When children are introduced to the fraction symbol without a deep understanding of what each part of the symbol refers to they are inclined, quite naturally, to view both the numerator and the denominator as counting or ‘how many numbers’. This leads ultimately to such misconceptions as ‘3/12 is bigger than 3/8 because 12 is bigger than 8’. It also leads to the well-known ‘Freshman’s error’, that is the tendency to add denominators when adding fractions (Siemon, 2003).
Imagine a student encountering the symbols we use to record fractions. She is told that ¾ is the same as three out of four. The student then demonstrates her understanding of fraction notation by stating that three people out of four people is the same as ¾, two people out of five people is the same as 2/5 and five people out of nine people is the same as 5/9. All appears well until this student surprises you by writing ¾ + 2/5 = 5/9. For this student and many others, fractions are “two numbers” rather than a single thing.
The rapid transition from modeling fractions to recording fractions in symbolic form, numerator over denominator, can contribute to many students’ confusion. Recording fractions in symbolic form needs to build on an underpinning conceptual framework of units (or parts) and collections of parts that form new units. Emphasizing numeric rules too soon without underlying meaning discourages students from attempting to see rational numbers as something sensible (CMIT LFN, 2007, p,47).
Activities and Assessments
Beehives – 1:1 Correspondence
Adapted from ‘Beehive’, Developing Efficient Numeracy Strategies Stage 1 page 34. NSW Department of Education and Training, Professional Support and Curriculum Directorate 2003
Focus: Matching the results of a count with a numeral, while reinforcing the idea of one count per item.
How: See Beehives – 1:1 Correspondence sheet and Video Example below
Beehives – Matching with manipulatives (Video Example)
Adapted from ‘Beehive’, Developing Efficient Numeracy Strategies Stage 1 page 34. NSW Department of Education and Training, Professional Support and Curriculum Directorate 2003
Focus: Students determine the number of a collection of counters, checking the count against a numeral
How: See Beehives – Counting with manipulatives sheet as well as Video Example below
Spin ‘n’ Cover
Focus: Identifying numerals and matching them wih collections, building the idea that numbers are not just about order, but about quantity.
How: For small groups of around 4 students. Each student is given a collection of counters of a unique colour, for example one student will have green counters, another student blue counters and so on. On each turn, a student spins the spinner provided (or throws a 10 sided die) and locates a square on the game board that matches the numeral indicated on the spinner. The student then places one of their counters on to that square, then passes the spinner to the next player. If there are no unoocupied squares matching the number indicated on the spinner, the student misses a turn. The aim is to be the first to place three counters (of the same colour) in a row, either horizontally, vertically or diagonally.
Questions to ask students during this activity: “What is that number?”, “Can you find a square with that many dots in it?”, “How many dots in that square?”, “Can you count them?”, “Is that number of dots the same as the number on your spinner?”
Numerals and Collections Memory
Focus: Visualising collections for given numerals helps to consolidate the idea of each numeral representing a specific quantity