Art of problem solving geometry pdf
This article considers how the principled design of interactive, computer-delivered tasks can enable the assessment of problem solving and process art of problem solving geometry pdf in ways that would not be possible in a conventional test. 9 and 13 year-old students who were not being challenged by the regular curriculum. The economies of large-scale computer-based testing are biased in favour of this model, despite its weakness in testing problem-solving. The World Class Tests project had the freedom largely to define its own syllabus and, specifically , to focus on problem solving skills without the usual obligation to assess the wider mathematics curriculum.
On the other hand this project was required, after an initial research phase, to deliver externally marked assessments in quantity. These were published and administered by an awarding body. Early in the tendering process, this was altered to include a mix of computer- and paper- based tests, sat twice a year. Although aimed at more able students, a key constraint of the design was that the tasks should not require above-average curriculum knowledge, but should focus on more sophisticated reasoning and insight. A sample balancing sheet is shown in Figure 1b.
Task type This attempted to summarise the main purpose of the task, and to justify why someone might be faced with it in the real world. The limited time allowed for assessment and lack of emphasis on curriculum knowledge precluded any fine-grained coverage within the science or technology domains. National Curriculum for England and Wales for the level which the candidates were already expected to have attained. Any knowledge above this level had to be introduced by the task itself. This needed to be balanced to ensure that the overall test was relevant to the experience of students. Less familiar contexts would tend to make the task more challenging, even if the underlying principles were familiar.
This is atypical of many problems that occur in real life. However, any problem solving task requires an open middle where some non-routine search for solution strategies has to be made. Tasks may also ask for multiple solutions which experience has shown to be challenging for students. In addition to the pencil-and-paper-only tests, the computer-based tests would also be accompanied by a paper workbook. For the mathematics tests, these were used purely to provide space for rough working. In the case of problem solving, however, some on-screen questions would instruct the students to write the response in their workbook.
It was also the only way that tasks could be trialled in the early stages of the project, before the data collection infrastructure was in place. Towards the end of the project, as experience was gained by the designers, the dependence on the answer books was waning. The availability of the written paper-based tests meant that the computer tests did not have to waste effort replicating tasks that were known to work well on paper, and could concentrate on ideas that exploited the computer to the full. The answer booklet for the computer test meant that the computer could be used to present contexts and information in an interactive format without sacrificing the ability to ask less structured, investigative questions. The main constraint was that the test was to be assembled from self-contained, 5 to 15-minute tasks. Although such tasks are long compared to those typically found on current mathematics tests, it is quite short for the sort of open-ended investigations suggested by the criteria above.
This limited the amount of software development effort that could be devoted to an individual task. All the vegetables and fruits that sinks overflow less than they weigh. All the food that float overflow how much they weigh. Has gone to 2 decimal places. Now describe the strategy for always winning the game. The student’s ability to formulate a strategy was put to the test by challenging them to beat the computer by the greatest margin possible. As a follow up, their understanding of the strategy was probed by asking them to imagine a variant of the game with 50 cards instead of 10 and to suggest the best opening moves.
One advantage of computer-based tasks is that the student can be offered a substantial database, rather than the dozen-or-so cases feasible in a paper test. This allows assessment of the important processes of choosing appropriate data, representing, summarising and interpreting it. As can be seen from the example screens, where questions required a substantial written answer, students were directed to answer in the paper workbook. While this could have been replaced by a type-in text box, this would have placed a constraint on the type and format of answers possible. Figure 9 is a purely written answer, but the formatting provides valuable evidence of a systematic approach.
Figure 10 shows a tabulated response, also providing clear evidence of systematic work and good choice of representation. Again, this would have been complicated for the candidate to replicate on computer, and providing a pro-forma table to fill in would have distorted the question by guiding the response. Figure 11 uses sketches which would obviously have been difficult to capture on a computer. It can be seen from these examples that each student went on to produce a purely verbal answer to the second part of the question, where they are asked to draw a hypothesis from the data.
2 whether systematic records had been kept. However, there are two disadvantages with that approach. Small-scale school trials of the tasks took place at an early stage to ensure that students could engage with the task and demonstrate progress. This approach differs from most test development, which is typically centred on detailed, but abstract, specifications of the curriculum areas to be covered, around which the tasks are constructed. The above context-led technique would be impractical if applied universally, so some tasks were inevitably written to address gaps in coverage or balance as the test was assembled. Ideas for computer-based tasks arose in various ways. It was then up to the computer task designer to develop the ideas into a workable specification.