Teaching Mathematics: not wrong, just differently right

In the usual order of presentation of fractions, the concept of ratios is left until the upper primary years.  I am not sure that approach is wise.

I have noticed that a fair proportion of those learning fractions  (even at 11 years old)   if asked to write the fraction shown by  [X X X X X O O O] (really by a corresponding set of black / white circles, an image which I am too lazy to insert here)  will write either 3/8 or 3/5.

A computer would mark both of these as incorrect.

Many teachers explain to the class that they expect students to count the coloured circles as the fractional part, and accept the 3/8 at least once, but mark the 3/5 as wrong.   Few explain that the student has seen the ratio relationship – in the time constraints of class they say  just that it is not the fraction.  The unintended lesson hits – the ratio-perceivers’ perception is flawed, they do not “see” maths.

Consider a different approach:  ask the class to “write the fractions shown by the image”.  Touch on the darkened ones traditionally being the fraction numerator, and thus the one they should use for teachers.  Welcome the 3/5,  or 5/3,and explain that the student has noticed the ratio relationship – but that it should be written as 3:5, not 3/5 (we traditionally write the smaller first.)   By using the vinculum we are saying that the bottom represents one set  divided into that many parts and the top shows how many of these parts are in the subset we are examining, whereas the ratio (:)  form says that the colon-separated sides add to make the whole.  (This format allows for cooking  ratios such as 1:1:2, basic biscuit and cake weights of butter / sugar/ flour.)

Aside: a topic for another time – like the technical terms “phone” ,”phoneme”, and “morpheme”, does the term “subset” belong in class before upper – primary?

Ask the class whether they want to investigate  ratios as well as fractions, even if they are not on the curriculum for the year (a conspiracy of learning [1]).

What can this mean to the students? The non-standard forms are seen as mathematically sound, but not the traditional (and thus  preferred) form  – just a matter of presentation, and thus not a big error.    The ratio perceivers have opened up an option for the class to explore – their perception is affirmed as being of an important mathematical relationship, though it is not the fractional one.  They just have to learn which label we use for which relationship – again, a matter of presentation – and they can go home saying “I saw ratios, and most of the others hadn’t noticed them!” .  Isn’t that better than “I suck at Maths!”?

In addition, there is the opportunity to re-emphasise that the key to fractions is that they are the relation of a current subset to the theoretical unit set, related by division of the unit set (into the number of parts shown in the denominator) and multiplication of the unit fraction by the numerator.     This is a difficult concept, but can be examined using the excellent physical approach to fractions reported by Doug Clarke [2], explicitly forming fractions by sharing (i.e. division and multiplication).  This is the key to grasping equivalent fractions  rather than “doing the sums” without understanding.

Aside:  this links to the concept of fractions as pure numbers  (How big is a quarter?  How big is one?  One what? 1/4 cup is larger than 1/2 teaspoon.)   Eventually, it also links to the concept of  percentages as special fractions where the unit set is divided into a hundred parts, so we can have fractions or decimals as numerators.  The excellent and widely used First Steps in Mathematics/Number [3],  says that percentages are special ratios – and indeed, being a subset of the special ratios called “fractions”, they are – but I think it makes more sense to link them in the first place to the immediate superordinate set.

1. Louden, W, Rohl, M, & Hopkins, S. (2008).  Teaching For Growth:  Effective teaching of literacy and numeracy.  Department of Education and Training, Western Australia http://www.uwa.edu.au/__data/assets/pdf_file/0005/615686/TFG_Final_Report.pdf.

2.  Clarke, D.  Fractions as division: The forgotten notion?  Australian Primary Mathematics Classroom 11 (3) 2006

3.  Willis,S., Jacob,L., Powell,B., Tomazos,D. & Treacy,K. (2004).
“First Steps in Mathematics: Number – understand whole and decimal numbers, understand fractional numbers.” Port Melbourne, Victoria: Rigby / Harcourt Education, 2004



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