Senior secondary mathematics in Queensland

Ian Cronk, Queensland Board of Senior Secondary School Studies

The nature of the criteria in the Mathematics A, B, C syllabuses in Queensland has encouraged some innovative thinking by mathematics teachers who are now compelled to consider aspects of assessment that traditionally had not been seen as significant. Queensland mathematics teachers are responding positively to these challenges, ensuring that students’ results are a valid and reliable indication of performance in the general objectives identified in the syllabuses.

Student achievement and school-based assessment

For a student to be awarded a Level of Achievement upon exit from a course of study in mathematics in Queensland it is necessary to reach a specified standard in each of the three criteria in the Senior Mathematics syllabuses.

The three assessment criteria, Communication, Mathematical Techniques, and Mathematical Applications are addressed through Queensland’s system of school-based assessment. Hence schools are free to develop their own assessment programs, provided that these are an accurate reflection of the syllabus and are articulated in the schools’ work programs.

To assist teachers of mathematics, annual reports from each of the three mathematics State Review Panels are published and circulated to all schools. These outline the findings and trends observed by State and District Panels over the year and give advice to schools that may assist in the interpretation of syllabus intent and requirements. Also the Review Officers communicate and negotiate with schools on behalf of the State Review Panel and run district meetings to assist teachers and clarify issues for them. Further clarification is offered from time to time via Information Statements issued to all schools by The Board.

The nature and structure of the mathematics syllabuses have led teachers to critically evaluate their own teaching strategies and methods of assessment in order to responsibly assess students in the subject. Some challenges and (re)solutions are identified and discussed.

Assessing students in the communication criterion

The syllabus specifies that information on student achievement against this criterion should be collected by a global consideration of the communication skills evident in student responses to tasks used to assess student performance in Mathematical Techniques and Mathematical Applications. It is therefore essential that mathematics teachers consider carefully the implications for students who may not engage well in the more challenging Mathematical Techniques and Mathematical Applications assessment items. Guidance given to teachers suggests that a student’s performance in Communication should be assessed only on work that has been attempted adequately in the other criteria.

The perception of Communication assessment has established it as an enabling criterion, used to qualify a student for a Level of Achievement but not necessarily as significant as the other two criteria for distinguishing student performance, certainly at high levels of achievement.

For teachers to adequately compare student performance with syllabus standards in Communication, there is a need for an ongoing concerted effort to inform teachers of the relationship between the syllabus objectives and the exit descriptors, and help them to consistently consider expectations in student work.

Introducing the use of technology in Senior Mathematics

Incorporating today’s technology into the mathematics courses in a meaningful and practical way and assessing the use of such instruments and associated techniques has presented a challenge in assessment.

The introduction of the graphic calculator and computer software for use in mathematics has prompted teachers to consider the syllabus objectives in the second dimension of Mathematical Techniques, Use of Instruments. This has created a need for teachers to become innovators of imaginative learning experiences in the classroom and assessment tasks that require students to gain mastery over technology as a tool to assist in problem solving. Student tasks are now able to take on a more realistic nature with respect to the challenging numerical and algebraic manipulation required. This produces more realistic tasks dependent on the student’s selection of appropriate techniques and innovative problem-solving strategies, rather than the laborious, repetitive and time consuming calculations previously experienced with any substantial task. This technology has also provided the opportunity for more open-ended research and investigations with students able to use the ‘what if’ approach.

There is, however, some disparity between schools regarding the availability of such technology. School-based assessment addresses this through its school-specific assessment where an external assessment system may not. Indeed, the introduction of technology has provided a ‘playground’ for those imaginative, creative right-brainers among mathematics teachers in Queensland with such projects as ‘AToMIC’ (Applications to Mathematics Incorporating Calculators), a publication compiled by The Queensland Association of Mathematics Teachers. This consists of a collection of senior mathematics projects involving the use of graphic calculators and developed by mathematics teachers in Queensland.

Providing a range of assessment situations

Teachers have responded actively to the challenge of presenting a range of objectives to consider in each assessment task so that students with varying abilities in different objectives may demonstrate performances in each. To achieve maximum opportunities for all students, teachers have introduced a variety of assessment situations, such as written tests, assignments, practical investigations, oral presentations and projects.

Assessing performance in Mathematical Applications by provision of solutions

The provision of solutions for grading is a significant departure from traditional ‘marking’ of student work in mathematics and has been the subject of much discussion among Queensland mathematics teachers since its introduction into the syllabus.

Clarification of ‘solution’ has been offered to teachers from State Review Panels via Information Statements: ‘The provision of a solution involves selecting a technique or combination of techniques and proceeding to conclusion, allowing for minor mechanical as opposed to conceptual errors.’

In the assessment of Mathematical Applications the syllabuses require the provision of solutions in simple unfamiliar situations and complex unfamiliar situations to varying degrees: for example, to achieve an ‘A’ grade in Mathematical Applications a student must provide solutions ‘consistently’ in simple situations and ‘generally’ in complex situations. Embedded in the standards matrix is the understanding that the verbal descriptors recognize that if a student has satisfied an exit descriptor, then performance in the syllabus objectives has been demonstrated. The terms ‘consistently’, ‘generally’, ‘occasionally’, ‘rarely’ are used to describe the extent to which students are expected to provide solutions.

This notion has invited purists to try to make a literal quantitative translation to numerical ‘cut-offs’ and this has produced challenges in interpretation. Many mathematics teachers have translated these terms into percentage measures to use as numerical benchmarks, for example, ‘consistently’ has been equated to ‘more than 75% of the time’; ‘generally’ has been equated to ‘more than 50% of the time’; ‘occasionally’ has been equated to ‘more than 25% of the time’. The advice to teachers on this issue has been that these benchmarks should not be ‘set in stone’. They should not be used mechanistically or rigidly. Thus students whose performances in Mathematical Applications are near the threshold must be compared carefully with syllabus descriptors since student work may give detailed information about student performance in syllabus objectives, while results will not render as much detailed information.

This requirement of ‘provision of solutions’ does not recognize credit for progress towards a solution when allocating grades. Hence a student may typically receive a grade based purely on the number of solutions provided. It becomes important, then, for teachers to ensure that a range of complexity from simple to complex must be presented to students so that they have the opportunity to provide solutions at all levels. Equally important is the need for teachers to ensure that adequate assessment items are presented to students so that valid judgements may be inferred from the syllabus descriptors. Teachers need to monitor assessment opportunities to satisfy themselves that opportunities have been adequately provided for all students.

Table 1 Extract from Assessment Analysis Sheet

Gauging complexity in Mathematical Techniques and Mathematical Applications

The notion of complexity has proved one of the biggest challenges for teachers. Since situations must provide for a range from simple to complex, teachers are required to gauge the level of complexity of each item. Advice to teachers emphasizes that it is reasonable to consider that the level of complexity ranges in a continuum (simple to complex) rather than a clear dichotomy. The responsibility of gauging complexity is scrutinized by review panels and hence there needs to be some consistency and consensus in the notion of complexity of an assessment task. Guidance is offered to teachers and review panellists in this respect by way of advice offered by State Review Panels through the assistance of the Review Officers.

The notion of complexity has been approached along the lines of ‘components of complexity’ (Clarke & Williams, 1997). The Annual State Review Panel Report (BSSSS, 1996) and an Information Statement (BSSSS, 1998), identified four components.

‘The level of complexity of an item may be influenced by

the number of techniques required to solve the problem

the level of difficulty of each of these techniques in isolation

the obviousness of the appropriate selection of techniques and suitable combination to lead to solution

the obviousness of an appropriate pathway to solution in the light of the students’ learning experiences’.

These four aspects may be gauged individually and then considered collectively to arrive at an overall level of complexity for each item. This guidance has encouraged healthy teacher discussion on complexity issues and brought about some common understandings. The discussion has also emphasized to teachers that levels of complexity may be partly dependent upon students’ school-specific learning experiences. Therefore reviewers and teachers should exercise caution when commenting on item complexity from their own particular perspective. Suggestions have been offered to schools on ways that they may assess complexity, while setting assessment items, and these have been demonstrated by State Review Officers in workshops designed to highlight these issues.

Instituting unfamiliarity in assessment opportunities in Mathematical Applications

The crucial difference between an item used to assess Mathematical Techniques and one used to assess Mathematical Applications is the notion of ‘unfamiliarity’ in Mathematical Applications. For an item to be used for assessment in Mathematical Applications it must contain an aspect of unfamiliarity (BSSSS, 1998). This unfamiliar aspect then requires a student to demonstrate the appropriate selection of techniques to solve the problem, a Mathematical Applications syllabus objective. As with the notion of complexity, it is important for reviewers and teachers to be aware of the specific school-based perspective when considering the aspect of familiarity in assessment from other schools. For this reason it is also incumbent upon teachers to be able to identify the aspects of unfamiliarity present in their assessment items to satisfy requirements of the syllabus.

The search for effective Mathematical Applications assessment items has brought teachers together to share assessment ideas and pool their resources. The idea of shared item banks has been a useful resource although care needs to be taken to ensure that the aspect of unfamiliarity is not eroded by over-distribution.

Assessing justification as an integral part of Mathematical Applications

Since justification is a dimension of Mathematical Applications it is essential to address any concerns that teachers may have regarding its assessment. There has been some confusion over the nature of justification and its perceived similarity to the Communication criterion. It is clear upon inspection that the Communication objectives as stated in the syllabus are quite different to the justification objectives.

Review Officers have been working with teachers to clarify the differences. The main distinction is that justification involves the development of logical arguments and the evaluation of the validity of arguments, while communication is more about conveying ideas and arguments. There is some debate about the place of justification as an integral part of solution since it is argued that some syllabus justification objectives may be demonstrated without the student providing a solution. This is a consideration that may be resolved by healthy ongoing debate.

Mathematics teachers respond to challenge

It is true to say that criteria and standards as stated in the Queensland Mathematics syllabuses are presenting challenges to the mathematics teachers of Queensland. The challenges and (re)solutions presented in this paper may serve to indicate the measures that teachers throughout Australia are prepared to take in order that, ultimately, students realize their full mathematical potential, enhance their understanding of the world, and improve the quality of their participation in a rapidly changing, technological society.

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