Part – Whole Comparisons

Single Representation

Representation

  

Formal

A jar holds 50 ml of orange when full. The jar has 40 ml of orange left in it after Mike uses some. What fraction of the jar has orange left in it?

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Four Operations

Having noted the importance of the equivalence of fractions it would be remiss not to ground the teaching of the four-arithmetic operations on this important concept.

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Glossary

Cognitive conflict is a psychological state involving a discrepancy between cognitive structures and experience, or between various cognitive structures. For explanations relating to mathematics see the research paper by David Tall (1977) https://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot1977a-cog-confl-pme.pdf

Manipulatives

Manipulatives can be any or all the practical apparatus that we use in classrooms such as Multilink Cubes, Cuisenaire Rods, Faction tiles, Area tiles. This list is not exhaustive, and I am sure you can add your own particular favourites. They are all practical bits of equipment that learners can pick up and manipulate and the concrete connection to the abstract concept (fractions) helps bridge the gap in conceptual understanding. There is a huge quantity of academic research-based learning theory work on how practical equipment supports

Further Reading

Academic

(Again this is not exhaustive the order being alphabetical)

 

Bruce, C., Chang, D., Flynn, T. and Yearley, S., (2013). Foundations to learning and teaching fractions: Addition and subtraction. Retrieved July, 4, p.2014.

 

Websites

There are a huge number of websites that can aid the mathematics teacher (in no particular order).

https://thinkforwardeducators.org/blog/five-ways-to-start-teaching-fractions

https://global.oup.com/education/content/primary/series/making-numbers/?region=uk

References

 

Ball, D. L. (1993). Halves, pieces, and twoths: Constructing representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: An integration of research pp. 157–196. Hillsdale, NJ: Erlbaum.

Behr, M. J., Post T., Silver E., Mierkiewicz D., (1980). Theoretical foundations for instructional research on rational numbers. Proceedings PME 1 pp. 60-67.

Equivalent Fractions

Some of these misconceptions all have a basis in the concept of equivalence, and we should think carefully about two significant fraction properties. Basing the teaching of fractions on two fundamental fraction properties might alleviate some of the above misconceptions. Every fraction is part of two special groups or families

  1. A family of fractions with the same name (denominator) but different numerator values

  2. A family of equivalent fractions with different denominators.

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Errors or Misconception

Before we start thinking about misconceptions* in the teaching and learning of fractions let us think about the difference between an error and a misconception

an error is an inaccuracy, but a misconception is the result of misunderstanding

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Visualisation

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