Student Understandings of Numeracy Problems: Semantic Alignment and Analogical Reasoning
Davis, James, Australian Mathematics Teacher
Despite compulsory mathematics throughout primary and junior secondary schooling, many schools across Australia continue in their struggle to achieve satisfactory numeracy levels. Numeracy is not a distinct subject in school curriculum, and in fact appears as a general capability in the Australian Curriculum, wherein all teachers across all curriculum areas are responsible for numeracy. This general capability approach confuses what numeracy should look like, especially when compared to the structure of numeracy as defined on standardised national tests. In seeking to define numeracy, schools tend to look at past NAPLAN papers, and in doing so, we do not find examples drawn from the various aspects of school curriculum. What we find are more traditional forms of mathematical worded problems.
In mathematics, worded problems tend to be associated with teaching contextualised mathematics, aiming towards relational understanding in mathematics, rather than superficial instrumental understanding (Skemp, 1989). Given the heavy content of the Australian curriculum, the use of rich, context based mathematics is not a common approach to teaching mathematics. Teaching the contexts, as well as the mathematics, is often too time consuming for most schools. In cases where context, relational understanding and the use of analogies have been studied, the involvement students have in the analogical reasoning processes have been limited (Richland, Holyoak & Stigler, 2004). This limited student control over the reasoning processes is a possible reason why students misunderstand worded problems (Richland, Holyoak & Stigler, 2004). As a consequence, the teaching of mathematical worded problems is not something that happens in great depth in mathematics classrooms, and it certainly does not happen in the form of numeracy, in other curriculum areas.
Identifying the difficulty students experience with worded problems, one school has sought to address this by explicitly teaching students to interpret and understand numeracy worded problems. The aim of this paper is to present a theoretical rationale for this school's approach, and to describe the methodology and teaching experiences of an innovative extra numeracy program for Year 9 students.
Study method and context
This study is an analysis of one schools approach to addressing low levels of numeracy. The school is a state high school in regional southern Queensland with approximately 1000 students. The primary barrier to students achieving better numeracy outcomes was identified in terms of the specific literacy requirements for solving worded problems. These worded problems appear in NAPLAN tests, and as such they define the nature of required numeracy skills in our schools.
Numeracy: A cognitive sciences perspective
A cognitive sciences perspective may view numeracy worded problems as simplistic, contrived situations to represent context, involving relationships between concrete objects or illustrations that require resolution using quantitative methods. Understanding the language of the problem and relating the language to mathematics requires a unique set of cognitive skills. These cognitive skills may be defined conceptually, in terms of semantic alignment and analogical reasoning.
Semantic alignment refers to the alignment of the text of worded problems with an intended mathematical structure that will lead to a solution. This results in a unique textual structure that requires students' to possess an ability to follow a specialised set of rules to comprehend the language of worded problems. These rules are often not self evident to the student, nor to the teacher. These semantic rules tend to be embedded in worded problems and appear to be taken for granted by educators (Bassok, 2001). For this reason, worded problems adopt unique text characteristics, constructed by educators to omit irrelevant text. …