Julian Williams

Diagnostic assessment of children’s proportional reasoning

Abstract A proportional reasoning item bank was created from the relevant literature and tested in various forms. Rasch analyses of 303 pupils’ test results were used to calibrate the bank, and data from 84 pupils’ interviews was used to confirm our diagnostic interpretations. A number of sub-tests were scaled, including parallel ‘without models’ and ‘with models’ forms. We provide details of the 13-item ‘without models’ test which was formed from the ‘richest’ diagnostic items and verified on a further test sample ( N =212, ages 10–13). Two scales were constructed for this test, one that measures children’s ‘ratio attainment’ and one that measures their ‘tendency for additive strategy.’ Other significant errors — ‘incorrect build-up,’ ‘magical doubling/halving,’ ‘constant sum’ and ‘incomplete reasoning’ — were identified. Finally, an empirical hierarchy of pupils’ attainment of proportional reasoning was formed, incorporating the significant errors and the additive scale.

Using Intuition From Everyday Life in 'Filling' the gap in Children's Extension of Their Number Concept to Include the Negative Numbers

We report here an instructional method designed to address the cognitive gaps in childrens mathematical development where operational conceptions give rise to structural conceptions (such as when the subtraction process leads to the negative number concept). The method involves the linking of process and object conceptions through semiotic activity with models which first record processes in situations outside mathematics and subsequently mediate activity with the signs of mathematics. We describe two experiments in teaching integers, an interesting case in which previous literature has focused on the dichotomy between the algebraic approach and the modelling approach to instruction. We conceptualise modelling as the transformation of outside-school knowledge into school mathematics, and discuss the opportunities and difficulties involved.

The association between mathematics pedagogy and learners' dispositions for university study

We address the current concerns about teaching‐to‐the‐test and its association with declining dispositions towards further study of mathematics and the consequences for choice of STEM subjects at university. In particular, through a mixed study including a large survey sample of over 1000 students and their teachers, and focussed qualitative case studies, we explored the impact of ‘transmissionist’ pedagogic practices on learning outcomes. We report on the construction and validation of a scale to measure teachers’ self‐reported pedagogy. We then use this measure in combination with the students’ survey data and through regression modelling we illustrate significant associations between the pedagogic measure and students’ mathematics dispositions. Finally, we discuss the potential implications of these results for mathematics education and the STEM agenda.

Measuring students’ transition into university and its association with learning outcomes

Previously we showed how we measured pedagogy and revealed its association with learning outcomes of sixth-form college mathematics students. In this project we followed a similar approach to the study of university transition. We particularly sought to identify the students’ perceptions of the transitional experience, and measure the association with learning outcomes. We drew on longitudinal surveys of students entering different programmes in five universities. Following them into their first year or so, allowed us to track their ‘disposition to complete the course’ and their ‘disposition to study more mathematics’, inter alia. We developed and validated two ‘fit-for-purpose’ measures of students’ perception of their transition, one we call ‘perception of the transitional gap/jump’ and one we call ‘degree of positive feeling about the transition’. We report some statistically and educationally significant associations between these and the students’ developing dispositions, and discuss the prospects fo...

School or college mathematics and workplace practice: an activity theory perspective

This paper reports on a case study in which we detail how a college mathematics and chemistry student struggles to make sense of the graphical output of an experiment in an industrial chemistry laboratory. The students attempts to interpret unfamiliar graphical conventions are described and contrasted with those of college mathematics. Our analysis of this draws on activity theory to assist in understanding the position of the student in both the college and the workplace. This highlights the limitations of the experience of the student at college and we question how the mathematics curriculum might be adapted to assist students in making sense of workplace graphical output.

Mathematics students’ aspirations for higher education: class, ethnicity, gender and interpretative repertoire styles

This paper reports how students talk about their aspirations in regard to higher education (HE) and their mathematics, what ‘repertoires’ they use to mediate this discourse, and how students’ predominant ‘repertoire style’ relates to their cultural background. Our analyses draw on an interview sample (n=40) of students selected because they are ‘on the cusp’ of participation or non‐participation in mathematically demanding programmes in further and higher education. The interviews explored the students’ aspirations for their future in general and HE in particular, influences on these choices, and the place of mathematics in these. Thematic analysis revealed four interpretative repertoires commonly in use, which we call ‘becoming successful’, ‘personal satisfaction’, ‘vocational’, and ‘idealist’ repertoires. Most of the sample was found to use a single, predominant repertoire, which we call their repertoire ‘style’: what is more, this style is found to be strongly related to background factors independently obtained. The implications for policy and practice are discussed.

Teachers telling tales: the narrative mediation of professional identity

This paper draws on the biographical narratives of two mathematics teachers who describe themselves as ‘traditional’ and ‘connectionist’ teachers respectively. Holland et al.s amalgam of Bourdieu, Vygotsky and Bakhtin, including ‘figured worlds’, ‘positionality’, ‘self-authoring’, and ‘world-making’ is used to examine these narratives. Differences between the two narratives include (i) their histories of compliant or oppositional identities as learners, and subsequently as teachers; (ii) their different experiences of ‘understanding’ and ‘tricks’, and (iii) their different use of figures as role models or anti-heroes in their self-authoring as teachers. It is argued that these narratives might ‘make worlds’ and provide future teachers in turn with figures for their own professional identity work.

Handling missing data: analysis of a challenging data set using multiple imputation

Missing data is endemic in much educational research. However, practices such as step-wise regression common in the educational research literature have been shown to be dangerous when significant data are missing, and multiple imputation (MI) is generally recommended by statisticians. In this paper, we provide a review of these advances and their implications for educational research. We illustrate the issues with an educational, longitudinal survey in which missing data was significant, but for which we were able to collect much of these missing data through subsequent data collection. We thus compare methods, that is, step-wise regression (basically ignoring the missing data) and MI models, with the model from the actual enhanced sample. The value of MI is discussed and the risks involved in ignoring missing data are considered. Implications for research practice are discussed.

Enrolment, achievement and retention on ‘traditional’ and ‘Use of Mathematics’ pre-university courses

This paper investigates enrolment, attainment and drop-out rates for two different English pre-university advanced mathematics, AS-level, courses, a ‘traditional’ and an innovative ‘Use of Mathematics’ pre-university course. Very different student profiles were found for those enrolled on each course, and a model of attainment at the pre-university level showed a relatively complex relationship with prior achievement at the end of compulsory schooling. Although those pupils who had relatively high prior achievement tended also to achieve relatively highly on the pre-university courses, this relationship was not evident for lower scores. Those pupils with ‘mid-range’ prior attainment tended to make the smallest gains. Taking prior attainment into account, the difference in attainment outcomes between the two courses is small. However, these courses do differ with respect to the number of students retained, with the ‘Use of Mathematics’ course retaining a significantly higher proportion of the students. Contextual factors are discussed, suggesting implications for policy and practice in mathematics education.

The association of classroom interactions, year group and social class

We investigate differences in the teacher–learner interactions in Reception, Year 2 and Year 4 science and literacy classrooms through three measures: (i) the proportion of open questions asked by the teacher, (ii) the rate of successful responses, and (iii) wait‐times. A regression analysis of data from 20 schools and 102 lessons suggests that classrooms in socio‐economically disadvantaged areas offer distinctive patterns of interaction, i.e., typically associated with those approximately two years younger in the more affluent school districts. We then closely examine the quality of two contrasting dialogues from reception science classes of schools in poor and affluent areas. We see how teachers’ questions can work or fail to work to achieve the expected quality in scientific dialogue, and thus how effective use of open questioning might be indicative of quality. In conclusion, we discuss quality of talk as an explanation of class differences in learning outcomes of schooling.