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Inside Teaching : August 2010
Inside Teaching | August 2010 CURRICULUM & ASSESSMENT 34 long-answer questions – in the future to address this issue. The limitations of multi-choice questions include their inability to ascertain the meta cognitive analysis underpinning student choices of options, which is essential if we’re to reliably test the student’s confdence and disposition to use mathematics. It’s impossible to know whether students have used the reasoning strategies that we want students who are numerate to have, or if they’ve merely guessed, even if they select the correct option. We can only assume they’ve reasoned and that their choices are underpinned by numerate dispositions. If, though, students are being taught deep understandings of concepts – as distinct from mathematics methods, procedures and algorithms that tend to dominate many mathematics lessons – they’ll have the confdence and ability to apply their mathematical understandings to a range of situations and contexts, resulting in numerate behaviours. A not inconsiderable spin-off is that they’ll be able to successfully complete NAPLAN numeracy questions. What does this mean for teaching? The depth of knowledge needed for mathematical numeracy as I’ve described above is based on informed reasoning that enables students to select correct responses for NAPLAN questions without undertaking specifc computation in most cases. Students are best prepared to be successful with the NAPLAN numeracy tests by being taught to draw on the numeracy skills that best serve them in meeting the numeracy demands of life: to do with estimation, based on deep understandings of mathematical concepts; visualisation and ‘imaging’ of context; and common sense. Some might argue that computational skills and routines are just as appropriate as these skills, but for this particular test genre and under these test conditions, students will be more effcient if they use estimation skills, since the questions rarely demand, or indeed require, computational skill beyond mental computation needed for estimation. This has profound implications for the teaching and learning of mathematics in our schools. Our primary aim is to teach the deep understandings that result in numerate behaviours; the direct spin-off from this is improvement in NAPLAN numeracy results. Let’s look at the following questions, frst from the 2009 NAPLAN numeracy test, then from the 2008 test, with results from one state which seem to indicate a lack of deep understanding of numbers and how they work. Consider Year 7 Question 14, on the previous page, and it’s clear that while we need to account for literacy skills, which are part of numeracy, large proportions of Year 7 students don’t understand fractional numbers. Question 25 on the same test validates this conclusion. Considering the curriculum content of mathematics and other subject areas that builds on the understanding of fraction – for example, percentage, scale, rates and proportion – in subsequent years, this is alarming to say the least. Do we understand what the misunderstandings are that these students have? Are these misunderstandings being corrected through targeted teaching following the release of this data? Do we know how to do this? The results from Question 4 of the 2008 Year 7 NAPLAN numeracy test reveal a great deal about the degree of student understanding of the concept of average. Clearly, the students who selected 40, 65 or 260 minutes don’t understand the concept of average or they would have known at a glance that these three responses couldn’t possibly be correct. Students who selected these three responses are likely to have attempted to calculate the correct solution using an algorithm rather than using some visualisation of the fve daily times and their position on a number line with respect to each other – the most appropriate