I am over the moon to be writing a blog post about the basic ideas of mathematics! During this module we have been encouraged to look past the simple ideas and working on developing a relational understanding between different mathematical concepts. SO, to be able to take a step back and explore the advantages of basic ideas is music to my ears! Basic maths is definitely my type of maths!
Ma (2010) describes the property of basic ideas as being ‘ideas that recur throughout mathematics learning creating a solid foundation for future learning’ (p.121). As teachers, this is an essential element which we need to encourage pupils to identify when approaching different mathematical situations. The basic/simple concepts link together individual topics and allow students to see patterns between the different strands, exploring the interconnectedness of mathematics. Rutherford (1990) states ” A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced.” Once these basic ideas have been learned, pupils can use this existing knowledge to help them solve more complex calculations.
I have chosen the topic of fractions to further explore the notion of basic ideas in mathematics.
What are the basic ideas of fractions? I perceive the basic ideas to be:
- Fractions represent a number of equal parts of a unit/whole number
- Fractions can represent a point on a line
- Numerator (top number) represents how many sections of the whole part you have
- Denominator (bottom number) represents how many sections are in the whole.
What are the more complex ideas of fractions? I see the more complex ideas to be:
- They can represented as a proportion of a set ( leaners need to build on the basic idea of fractions representing a number of equal parts of a unit to extend the meaning to include a number of equal parts of a set.) (Haylock, 2014)
- They can be used to model division – 3/8 can also be read as 3 divided by 8.
- Knowledge of fractions can be extended to knowledge of ratio – 4/7 can be see as 4:7 E.g. For every 4 females, there are 7 males.
- Improper fractions (top-heavy fractions) -leaners need to understand the basic idea of numerator and denominator to be able to solve problems involving improper fractions. For example; knowing that 6/6 is equal to 1 whole, 12/6 is equal to two, 18/6 is equal to three.
- Mixed fractions – using the knowledge of numerators and denominators to know that 3 1/3 represents 3 wholes and then 1 section of 3 out of a whole.
- Addition and subtraction with mixed fractions – need knowledge of common factors to be able to convert both denominators into the same times table. (links to the idea of connectedness)
“Knowing about a key idea in mathematics, such as fractions, involves knowing how fractions relate to whole numbers, where they belong on a number line, how they link to ideas of ration and proportion, the connection between fractions and the division operation, the links between a range of modes of representing fractions, and a host of other points” (Haylock, 2014,p.9).
Understanding the basic ideas and concepts of fractions creates a web from which we can build on to develop our knowledge and relational understanding of more complex fractional calculations.
The following video is a good resource for introducing the bass ideas and concepts of fractions.
Links and extra reading
Fractions – The basics – Good resource starting with the basic ideas and concepts involved in fractions.
Sources
Haylock, D. (2014) Mathematics Explained for Primary Teachers (5th edn.). London: SAGE
Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge
Rutherford, J.F (1990) “The Nature of Mathematics” in Science for all americans. Oxford: Oxford University Press. Available at: http://www.project2061.org/publications/sfaa/online/chap2.htm Accessed: 31/10/15
