Learning to think spatially : what do students 'see' in numeracy test items?

Learning to think spatially in mathematics involves developing proficiency with graphics. This paper reports on 2 investigations of spatial thinking and graphics. The first investigation explored the importance of graphics as 1 of 3 communication systems (i.e. text, symbols, graphics) used to provide information in numeracy test items. The results showed that graphics were embedded in at least 50 % of test items across 3 year levels. The second investigation examined 11 – 12-year-olds’ performance on 2 mathematical tasks which required substantial interpretation of graphics and spatial thinking. The outcomes revealed that many students lacked proficiency in the basic spatial skills of visual memory and spatial perception and the more advanced skills of spatial orientation and spatial visualisation. This paper concludes with a reaffirmation of the importance of spatial thinking in mathematics and proposes ways to capitalize on graphics in learning to think spatially.

Mathematical modelling of muscle recruitment and function in the lumbar spine

Low back pain is an increasing problem in industrialised countries and although it is a major socio-economic problem in terms of medical costs and lost productivity, relatively little is known about the processes underlying the development of the condition. This is in part due to the complex interactions between bone, muscle, nerves and other soft tissues of the spine, and the fact that direct observation and/or measurement of the human spine is not possible using non-invasive techniques. Biomechanical models have been used extensively to estimate the forces and moments experienced by the spine. These models provide a means of estimating the internal parameters which can not be measured directly. However, application of most of the models currently available is restricted to tasks resembling those for which the model was designed due to the simplified representation of the anatomy. The aim of this research was to develop a biomechanical model to investigate the changes in forces and moments which are induced by muscle injury. In order to accurately simulate muscle injuries a detailed quasi-static three dimensional model representing the anatomy of the lumbar spine was developed. This model includes the nine major force generating muscles of the region (erector spinae, comprising the longissimus thoracis and iliocostalis lumborum; multifidus; quadratus lumborum; latissimus dorsi; transverse abdominis; internal oblique and external oblique), as well as the thoracolumbar fascia through which the transverse abdominis and parts of the internal oblique and latissimus dorsi muscles attach to the spine. The muscles included in the model have been represented using 170 muscle fascicles each having their own force generating characteristics and lines of action. Particular attention has been paid to ensuring the muscle lines of action are anatomically realistic, particularly for muscles which have broad attachments (e.g. internal and external obliques), muscles which attach to the spine via the thoracolumbar fascia (e.g. transverse abdominis), and muscles whose paths are altered by bony constraints such as the rib cage (e.g. iliocostalis lumborum pars thoracis and parts of the longissimus thoracis pars thoracis). In this endeavour, a separate sub-model which accounts for the shape of the torso by modelling it as a series of ellipses has been developed to model the lines of action of the oblique muscles. Likewise, a separate sub-model of the thoracolumbar fascia has also been developed which accounts for the middle and posterior layers of the fascia, and ensures that the line of action of the posterior layer is related to the size and shape of the erector spinae muscle. Published muscle activation data are used to enable the model to predict the maximum forces and moments that may be generated by the muscles. These predictions are validated against published experimental studies reporting maximum isometric moments for a variety of exertions. The model performs well for fiexion, extension and lateral bend exertions, but underpredicts the axial twist moments that may be developed. This discrepancy is most likely the result of differences between the experimental methodology and the modelled task. The application of the model is illustrated using examples of muscle injuries created by surgical procedures. The three examples used represent a posterior surgical approach to the spine, an anterior approach to the spine and uni-lateral total hip replacement surgery. Although the three examples simulate different muscle injuries, all demonstrate the production of significant asymmetrical moments and/or reduced joint compression following surgical intervention. This result has implications for patient rehabilitation and the potential for further injury to the spine. The development and application of the model has highlighted a number of areas where current knowledge is deficient. These include muscle activation levels for tasks in postures other than upright standing, changes in spinal kinematics following surgical procedures such as spinal fusion or fixation, and a general lack of understanding of how the body adjusts to muscle injuries with respect to muscle activation patterns and levels, rate of recovery from temporary injuries and compensatory actions by other muscles. Thus the comprehensive and innovative anatomical model which has been developed not only provides a tool to predict the forces and moments experienced by the intervertebral joints of the spine, but also highlights areas where further clinical research is required.

Communicating assessment criteria is not sufficient for influencing students' approaches to assessment tasks – perspectives from a differential equations class

This report presents the findings of an exploratory study into the perceptions held by students regarding the use of criterion-referenced assessment in an undergraduate differential equations class. Students in the class were largely unaware of the concept of criterion referencing and of the various interpretations that this concept has among mathematics educators. Our primary goal was to investigate whether explicitly presenting assessment criteria to students was useful to them and guided them in responding to assessment tasks. Quantitative data and qualitative feedback from students indicates that while students found the criteria easy to understand and useful in informing them as to how they would be graded, the manner in which they actually approached the assessment activity was not altered as a result of the use of explicitly communicated grading criteria.

Primary students' performance on map tasks : the role of context

This study investigated the longitudinal performance of 583 students on six map items that were represented in various graphic forms. Specifically, this study compared the performance of 7-9-year-olds (across Grades 2 and 3) from metropolitan and non-metropolitan locations. The results of the study revealed significant performance differences in favour of metropolitan students on two of six map tasks. Implications include the need for teachers in non-metropolitan locations to ensure that their students do not overly fixate on landmarks represented on maps but rather consider the arrangement of all elements encompassed within the graphic.

Complex learning through cognitively demanding tasks

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by “continuing education as usual” (The National Academies, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualization. These technologies have led to significant changes in the forms of mathematical thinking that are required beyond the classroom. This paper argues for the need to incorporate future-oriented understandings and competencies within the mathematics curriculum, through intellectually stimulating activities that draw upon multidisciplinary content and contexts. The paper also argues for greater recognition of children’s learning potential, as increasingly complex learners capable of dealing with cognitively demanding tasks.

Understanding graphicacy : students’ making sense of graphics in mathematics assessment tasks

The ability to decode graphics is an increasingly important component of mathematics assessment and curricula. This study examined 50, 9- to 10-year-old students (23 male, 27 female), as they solved items from six distinct graphical languages (e.g., maps) that are commonly used to convey mathematical information. The results of the study revealed: 1) factors which contribute to success or hinder performance on tasks with various graphical representations; and 2) how the literacy and graphical demands of tasks influence the mathematical sense making of students. The outcomes of this study highlight the changing nature of assessment in school mathematics and identify the function and influence of graphics in the design of assessment tasks.

Jabberwocky : the complexities of mathematical English

Mathematical English is a unique language based on ordinary English, with the addition of highly stylised formal symbol systems. Some words have a redefined status. Mathematical English has its own lexicon, syntax, semantics and literature. It is more difficult to understand than ordinary English. Ability in basic interpersonal communication does not necessarily result in proficiency in the use of mathematical English. The complex nature of mathematical English may impact upon the ability of students to succeed in mathematical and numeracy assessment. This article presents a review of the literature about the complexities of mathematical English. It includes examples of more than fifty language features that have been shown to add to the challenge of interpreting mathematical texts. Awareness of the complexities of mathematical English is an essential skill needed by mathematics teachers when teaching and when designing assessment tasks.

Repeating patterns : strategies to assist young students to generalise the mathematical structure

This paper focuses on very young students' ability to engage in repeating pattern tasks and identifying strategies that assist them to ascertain the structure of the pattern. It describes results of a study which is part of the Early Years Generalising Project (EYGP) and involves Australian students in Years 1 to 4 (ages 5-10). This paper reports on the results from the early years' cohort (Year 1 and 2 students). Clinical interviews were used to collect data concerning students' ability to determine elements in different positions when two units of a repeating pattern were shown. This meant that students were required to identify the multiplicative structure of the pattern. Results indicate there are particular strategies that assist students to predict these elements, and there appears to be a hierarchy of pattern activities that help students to understand the structure of repeating patterns.

Early mathematical learning: Number processing skills and executive function at 5 and 8 years of age

This research investigated differences and associations in performance in number processing and executive function for children attending primary school in a large Australian metropolitan city. In a cross-sectional study, performance of 25 children in the first full-time year of school, (Prep; mean age = 5.5 years) and 21 children in Year 3 (mean age = 8.5 years) completed three number processing tasks and three executive function tasks. Year 3 children consistently outperformed the Prep year children on measures of accuracy and reaction time, on the tasks of number comparison, calculation, shifting, and inhibition but not on number line estimation. The components of executive function (shifting, inhibition, and working memory) showed different patterns of correlation to performance on number processing tasks across the early years of school. Findings could be used to enhance teachers’ understanding about the role of the cognitive processes employed by children in numeracy learning, and so inform teachers’ classroom practices.