Relationships between mathematical knowledge for teaching and teaching practice: the case of proof

Teachers of mathematics orchestrate opportunities for interactions between learners and subject matter. Therefore, mathematics teachers need rich, multidimensional content knowledge for teaching mathematics, which incorporates knowledge of the subject matter, students, and teaching. Studying this mathematical knowledge for teaching (MKT) necessitates more than a unidirectional assessment. In this study, the mathematical knowledge for teaching reasoning and proving of two secondary mathematics teachers was investigated through classroom observations and clinical assessments. Results indicate that using MKT as a frame for examining classroom practice, in addition to assessing the MKT a teacher possesses in a clinical setting, provides an in-depth and innovative method for investigating MKT. The comparison of the two cases also identifies student positioning as a key mediating factor between MKT and opportunities to learn. Implications for using MKT as a lens for examining practice in teacher education are discussed.     

USING CONSULTATION TO SUPPORT ENGLISH LEARNERS: THE EXPERIENCES OF BILINGUAL SCHOOL PSYCHOLOGISTS

Through semi‐structured interviews, this study explored 11 bilingual school psychologists’ (BSPs) consultation experiences with teachers of English learners (EL) to determine referral concerns, recommendations made, challenges encountered, preparation experiences, and skills most needed. The most common referral issue concerned students’ academic performance and teachers’ attempts to parse out language acquisition difficulties from learning problems. Principal consultation challenges included the scarcity of trained professionals, teacher bias, and the stress of competing school priorities. BSPs relied most on their interpersonal skills and relationship building skills, as well as their knowledge of EL‐focused interventions and their problem identification skills. Recommendations to their monolingual school psychology colleagues included the need to actively seek out a) language and culture resources, b) relevant professional development opportunities, and c) consultations with fellow BSPs. Study limitations and implications are discussed.     

Defining Social Goods and Distributing Them Fairly: The Development of Conceptions of Fair Testing Practices

Conceptions of fair educational practices develop differently for testing and learning situations. After about age 8, children judged peer tutoring to be an unfair practice during a test. (Most 6–8-year-olds judged it fair). Yet, peer tutoring was seen as the most fair way to help students learn until at least age 17. In addition, 4 levels of conceptions of fair testing practices were found. Adult-like conceptions of fair testing practices emerged at about age 11, which contrasts with about 18 for conceptions of fair learning practices. Conceptions of fair testing practices parallel the development of the concepts of ability and effort as factors limiting immediate performance; whereas conceptions of fair learning practices parallel the development of concepts concerning the long-term acquisition of intelligence. These findings support the conclusion that children's understanding of different types of situations is reflected in their conceptions of social justice.     

Teaching Mathematics through Play to Primary School Children

The use of ‘play’ as a means of teaching elementary mathematics is discussed. The games used are described and the underlying theory explained. Results obtained from work with over 150 children are presented and the case for ‘Playway’ argued in the light of these.     

Ocular dominance,reading and spelling: a reassessment of a measure associated with specific reading difficulties

The association of reading disability with perceptuomotor problems has had a chequered history. This study examines the relationship between a new test of ocular dominance and reading and spelling ability in a group of 323 schoolchildren. Similar rates for unfixed ocular dominance were found for poor readers as in comparable studies. The rates of unstable ocular dominance did not differ between those with reading and spelling difficulties and controls. These findings in a ‘blind’ study question the role of this test of eye dominance as a diagnosis of ‘dyslexia’.     

Mobile learning in higher education: A comparative analysis of developed and developing country contexts

The purpose of this paper is to compare and contrast characteristics of use and adoption of mobile learning in higher education in developed and developing countries. A comparative case study based on a survey questionnaire was conducted with 189 students (undergraduate and postgraduate) from Makerere University in Uganda and the University of Adelaide in Australia. The Unified Theory of Acceptance and Use of Technology (UTAUT) was employed as the theoretical framework. The results indicated that higher education students in developed and developing countries use a range of technologies for learning, with major differences between Uganda and Australia. The study concludes that mobile learning in higher education in developed and developing country contexts is still at an experimental stage with students using mobile devices in pedagogically limited ways.     

Choice behavior in transition: Development of preference in a free-operant procedure

Twenty acquisition curves were obtained from each of 8 pigeons in a free-operant choice procedure. Every condition began with a phase in which two response keys had equal probabilities of reinforcement, and, as a result, subjects’ responses were divided fairly evenly between the two keys. This was followed by a phase in which one key had a higher probability of reinforcement than the other, and the development of preference was observed. In all but a few cases, response proportions increased for the key with the higher probability of reinforcement. In most conditions, the two probabilities differed by .06, but the actual probabilities varied (from .16 and .10 in one condition to .07 and .01 in another). Development of preference for the key with the higher probability of reinforcement was slower when the ratio of the two reinforcement probabilities was small (.16/.10) than when it was large (.07/.01). This finding is inconsistent with the predictions of several different quantitative models of acquisition, including the kinetic model (Myerson & Miezin, 1980) and the ratio-invariance model (Horner & Staddon, 1987). However, the finding is consistent with a hypothesis based on Weber’s law, which states that the two alternatives are more discriminable when the ratio of their reinforcement probabilities is larger, and, as a result, the acquisition of preference is faster.     

FROM DOUBLE JEOPARDY TO DOUBLE INDEMNITY: SUBTLETIES OF TEACHING INTERDISCIPLINARY GERIATRICS

Each of the fields of geriatrics and interdisciplinary practice intensifies the usefulness and effectiveness of the other. Combining geriatrics and interdisciplinary practice also magnifies the complexity of two singularly complex fields. However, the subtle reasons for their complexity may escape the understanding of clinicians, educators, policy makers, and administrators. Attempting to treat older patients who have complex problems while ignoring common principles of geriatrics and teamwork can cause a kind of double jeopardy. It can potentially harm the patient and decrease the provider's sense of mastery. Training either students or health care providers to appreciate the interactions and the complexities of these two fields could produce a double indemnity for the health care system and for older persons who need health care. It requires recognizing subtle factors like the barrier of presumed knowledge, the interpretation of the meaning of function, frailty, the lure of medical technology, professional hierarchies, diverse views of disease, and the myth of insufficient evidence.