Search

Cryptography is a stimulating way to introduce and consolidate ideas in statistics, computational linguistics, combinatorics and modular arithmetic. Two of the authors have been carrying out didactical experiences starting back in 1989 at a primary school level, without any special technology. A game is set up which involves cryptographers and cryptanalysts. Simple substitution ciphers are broken by building letter frequency hystograms by parallel work, so as to achieve what is being felt as statistical significance. Pupils quickly discover Markov models and the slight non-stationarity of the linguistic process. We have initiated a new round of experiments at a different level of age, 14-16, and technology. We take advantage of computer software to deepen our analysis of cipher systems and Markov models. The friendly (and cheap) technology of graphing calculators is used to analyse perfect and pseudo-perfect ciphers and to discuss the elusive notion of randomness.

This paper concerns a study of the performance of students in a recent linear algebra examination. We investigated differences in performance in tasks requiring understanding of the concepts with those that required only the use of routine procedures and factual recall. Central to this study was the use of a taxonomy, based on Bloom's Taxonomy, for characterising assessment tasks, which we have described in previous publications. The ful1 taxonomy has 8 categories, which fall into 3 broad groups. The first group (A) encompasses tasks which could be successfully done using a surface learning approach, while the other two (B and C) require a deeper learning approach for their successful completion. Tasks on the examination paper were put into one of the three groups and comparisons were made concerning the performance of individual students in each of these areas. There are several interesting areas to investigate. The first is to identify those students whose performance in group A was markedly different to their performance on groups B and C. There is considerable disquiet amongst mathematics lecturers at tertiary level as to the routine algebraic skills of incoming students and of students studying mathematics at university (see for example the ICMI Study into the Teaching and Learning of Mathematics at University Level, 2001). There is a conjecture that students who have poor technical skills are not able to succeed in university mathematics. The contrapositive conjecture that good technical skills (such as algebraic dexterity) are necessary for success in university mathematics is often taken for granted. The taxonomy allows us to test this hypothesis as we can compare performance in group A tasks (routine) with performance in higher level B and C tasks. We have also investigated whether or not the data supports any systematic effect of differences in sex or language background in the performance on the three groups. The sample contained a large cohort of students with who had a home language other than English. We tested the hypothesis that such students would have difficulty with the conceptual aspects of the course, since these normal1y require greater language facility. This proved not to be the case.

Seventy students from a first semester calculus course ranked 8 mathematics tasks as to perceived difficulty before attempting these tasks and actual difficulty after completing the tasks. Students also completed two examinations, one based on facts and procedures and the other based on applications and concepts. The tasks were designed to fit into a taxonomy of mathematical skills. We have found that students perceive questions to be difficult for a number of reasons. In general, questions requiring conceptual understanding ae regarded as more difficult that those which require factual recall or the use of routine procedures. There was not a strong link between familiarity with the question type and ranking of difficulty. Students were sufficiently familiar with the some typ es of question to be able to perceive inherent difficulties, such as a complex differentiation. We found that in five out of eight cases, students' perceptions of the difficulty did not change after they had done the task. In one case they found the question to be more difficult than expected and in two cases to be easier. It is not clear to us why students found one question to be more difficult than expected. It may be that some of the complexities (such as the use of the intermediate value theorem) were not immediately apparent. It is also significant that NESB students rated this question as easier than ESB students. This was the case both before and after attempting the question. Student comments are also presented.

The last decade has seen a substantial increase in the cultural and academic diversity of commencing tertiary education cohorts. The challenge for mathematics and statistics educators is the development of curriculum measures which address the language related difficulties of language minority students (Cocking & Mestre, 1988) and improve learning outcomes for all students. Our focus in this paper is on enhancing language and communication skills in culturally diverse undergraduate statistics cohorts. Most students have difficulty adjusting to the formal language requirements of academia. Non-English speaking background (NESB) students can have particular difficulty with the reading and assessment demands of Western universities if they are not adequately supported (e.g. Ballard & Clanchy, 1997). This is especially problematic when discrepancies between verbal and written expression and true intellectual ability result in assessment penalties. What is required are curriculum models which focus on what students do as opposed to deficit models which focus on who students are (Biggs, 1999). We describe curriculum development in two subjects designed to teach language skills in statistics. Both subjects require students to engage with academic language and to develop statistical discourse skills relevant to modern professionals in the quantitative sciences. Methods used to encourage this include explicitly teaching academic reading techniques, and group research projects that are peer assessed. The projects are designed to develop statistical concepts within the context of professional practice and to address key competency requirements of relevant professional associations. We will present data that suggest that NESB students have more difficulty than ESB students on "traditional" statistics assessment tasks and describe curricula interventions that assist those students to achieve their academic potential. The reaction of students to these developments has been very positive. The quality of the work is impressive and students improve both their statistical knowledge and their reading and writing skills.