The circular or pizza model is often cited as the most common form of fractions modelling in primary school classrooms. Here I use a Year 6 SATS question to demonstrate why an oblong-rectangular area model may be a more useful tool in helping children to succeed.
Fractions can cause anxiety – not just for pupils, but for those teaching the concept too. Of course, at their most basic, fractions are simply the expression of a mathematical relationship between quantities. Yet as teachers, we know that fractional thinking is one of the most mathematically complex areas of learning in the primary classroom. There are five different meanings of the fraction notation (see Keiran, 1976), and therefore five different ways that the concept may be presented to children.
Yet, despite the wide variety in the way fractions can be constructed, one fraction model seems to dominate presentation in the classroom – shaded circles, or the so-called pizza fraction. In one study by Castro-Rodríguez et al. (2016), almost half of the models used by participating prospective teachers to demonstrate fractions were figures of circular area models.
I believe area models have an important role to play (amongst a variety of other visual models) in constructing meaning around fractional thinking. However, having a predominant, yet only-partial area concept image focused around a circular model may inhibit the visual fluency that supports fractional problem-solving. Research by Moss and Case (1999), Witherspoon (2002) and Clarke (2006) demonstrates that the usefulness of circular models is particularly limited in this respect, especially given the difficulty in the equipartitioning of a circle and the subsequent interpretation on the result.
The key to the successful use of area models in fractional thinking may therefore be the variation in their use. This means building on what the children should already have learned about the essential features of the circular model (the idea of an enclosed area representing one whole unit; the equipartitioning of that space into parts of equal size; the relationship between the number of parts identified, in relation to the number of equal parts in the whole) by varying the non-essential features of the model – in this instance, its shape. This could give teachers and pupils alike the fluency to tackle increasingly complex fraction problems, using a wider range of area-model tools.
To demonstrate, this is a question from the 2017 KS2 National Curriculum Tests (Paper 2: Reasoning) which uses a visual representation of a circular model with 1/4 and 1/2 to demonstrate a fractional problem.
Typically, a pupil with high levels of conceptual fluency may approach this problem by calculating a common denominator to create the equivalent fractions 3/12 and 2/12, thereby determining that the non-shaded portion remaining is 7/12.
However, pupils with the ability to reconceptualise this problem as a rectangular area model may not need to complete any calculating to find a common denominator or equivalent fractions at all. The steps to achieving this are detailed below.
Step 1: Reconceptualise the circular area model as an oblong-rectangular area model.
Step 2: Represent 1/6 of the area model by dividing the oblong-rectangle into 6 equal parts and shading one of them.
Step 3: Recognise how to represent 1/4 of an oblong rectangle by dividing the shape in half again and shading 1/4 of the area.
Step 4: Count the number of equal parts in the area model (12) and recognise this as the denominator.
Step 5: Count the number of unshaded parts in the area model (7) and recognise this as the numerator.
Step 6: 7/12’s of the whole is not shaded.
Oblong-rectangular area models can be used to help children both visualise and understand other processes, such as the addition and multiplication of fractions. I purport that children may need a carefully planned progression of fractions models from their very earliest years in school, to equip them with the tools they need to assist the transition into more sophisticated fractional thinking. Moving beyond pizzas to development real fluency and variation in the use of area models may be a first step.
Marie Birchall is a practising primary school teacher and postgraduate student specialising in primary school mathematics. She can be found on Twitter @marie_birchall.
References and further reading
CASTRO-RODRIGUEZ, E., PITTA-PANTAZI, D., RICO, L. and GOMEZ, P., 2016. Prospective teachers’ under- standing of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics. 92, pp. 129–146.
CLARKE, D., 2006. Fractions as Division: The forgotten notation. Australian Primary Mathematics Classroom. 11(3), pp. 4-10.
GU, L., HUNAG, R., and MARTON, F., 2004. Teaching with Variation: A Chinese Way of Promoting Effective Mathematics Learning.
KIEREN, T.E.,1976. On the mathematical, cognitive, and instructional foundations of rational numbers. In R. LESH, ed., Number and Measurement: Papers from a Research Workshop ERIC/SMEAC. Columbus, OH. pp. 101–144.
KILPATRICK, J., SWAFFFORD, J. and FINDELL, B., eds, 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.
MOSS, J. and CASE, R., 1999. Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education. 30, pp. 122–147.
NUNES, T., BRYANT, P., HURRY, J. and PRETZLIK, U., 2006. Fractions: difficult but crucial in mathematics learning. In Teaching and Learning Research Briefing 13. Accessed: 21st March 2021 https://highlandnumeracyblog.files.wordpress.com/2015/01/nunes-et-al-fractions_difficult-but-crucial-in-mathematics-learning.pdf
WITHERSPOON, M. L., 2002. Fractions: In search of meaning. In D. L. CHAMBERS, Ed., Putting Research into Practice in the Elementary Grades. Reston, VA: National Council of Teachers of Mathematics. pp. 133-136.