Common Misunderstandings - Level 6 Generalising

While considerably more is expected of students at this level in relation to their understanding and use of number than is expected at earlier levels (eg, see VELS, 2005, p.36), most students are able to work with rational numbers to some extent and have an emerging appreciation of the real numbers. However, this is not necessarily the case when these numbers are represented by pro-numerals or used in expressions containing pro-numerals. For many, the very power and density of algebraic text can be the feature that renders it impenetrable.

There is an extensive body of research which has examined the various difficulties students experience with algebraic text, ranging from misunderstanding of the equal sign, and assigning literal meanings to letters (eg, 3a interpreted as 3 apples) to viewing expressions as instructions to operate, rather than as objects that can be operated on in their own right (eg, that 4x-7 is an object that can be multiplied by any other number or pro-numeral).

While reading, interpreting, and working with algebraic text is one issue, constructing algebraic text to describe relationships is another area of difficulty for many students. A range of external representations (eg, balances, concrete materials, graphs, diagrams, or tables of values) are typically used to explore patterns and relationships in school mathematics. Referred to as intermediate sign systems by Filloy and Sutherland (1996), they variously serve to facilitate the construction of meaning for the conventional mathematical sign system, in this case, the “algebra code” (p.143). One of the difficulties here is that different conceptions arise from different representations and these may inhibit students’ capacity to make connections between representations, generalise, or indeed, recognise when a previously learnt representation is inappropriate. For example, while it is meaningful to interpret 5 ×  = 20 as ‘find the number which 5 must be multiplied by to equal 20’, this interpretation (or intermediate sign system) cannot usefully replace x in the equation, 5x + 9 = 3x. Nor is it appropriate to expect that strategies that work for the former equation, such as ‘back-tracking’, will work with equations like the latter where the unknown appears on both sides of the equation.

The difficulties experienced in making the transition from arithmetic to algebra may be due to/associated with:

  • naïve understanding of the equal sign in terms of ‘makes’ or the ‘answer is…’;
  • different interpretations of letters (Booth, 1988) and/or a lack of knowledge about the conventions used to record generalised expressions (eg, that multiplication is recorded as 3a not a3 or 3 × a);
  • limited understanding of the properties of numbers and operations (eg, multiplication only understood in terms of groups of, division not seen as the inverse of multiplication);
  • an inadequate understanding of arithmetic and/or an over-reliance on procedural solution strategies aimed at getting numerical answers;
  • little/no experience in communicating mathematical relationships in words and/or translating relationships described in words into symbolic expressions, for example, “s is 8 more than t” or the “Niger is three times as long as the Rhine” (MacGregor, 1991, pp.95-97); and
  • limited access to multiplicative thinking and proportional reasoning more generally which restricts students’ capacity to recognise and describe relationships in terms of factors.

By the end of Level 6 students are expected to be able to work meaningfully with a wider range of numbers and mathematical relationships in whatever form they appear, including equations, identities, inequalities, functions and relations.

A key indicator of the extent to which students are ready to engage with these curricula expectations is their capacity to deal with equivalent forms of expressions, recognise and describe number properties and patterns, and work with the complexities of algebraic text.