The What and Why of Whole Number Arithmetic: Foundational Ideas from History,Language and Societal Changes

After more than five years of collaboration on whole number arithmetic (WNA), we summarise our experiences, focusing on the process, the merits and the limits of the ICMI Study 23, together with the potential for future activity and for addressing different kinds of audience. We have not worked alone. A very knowledgeable and helpful International Program Committee (IPC) shared the whole process of preparation of this volume. We wish to thank them all for their long-lasting (and not yet finished) collaboration; although, obviously, the responsibility for some delicate choices and possible mistakes and misunderstandings is left to the two of us. The two epigraphs above, from the French philosopher and sinologist François Jullien and from an ancient Chinese saying, summarise our attitude now. This international study has offered us the opportunity to increase our knowledge and start two complementary processes: –– Becoming aware of some deep values of our own culture (our ‘unthought’) which we may have considered in the past the only possible choice or, at least, the most suitable choice for an ideal ‘human nature’. –– Considering the possibility of introducing into our own practices, beliefs and values (our ‘jade stone’) the processes of innovation, not copied from but influenced by practices, beliefs and values of another culture. The Study Volume is an account of the collective memory of participants offered to the wider community of primary mathematics educators, including researchers, teachers, teacher educators and policymakers. It is a product of fruitful collaboration between mathematicians and mathematics educators, in which, for the first time in the history of ICMI, the largely neglected issue of WNA in primary school has been addressed. The volume reports all the activities of the Conference. Many co-authors, who were involved in a collective co-authorship, are listed at the end of the volume

Inheritance and genetic mapping of resistance to Alternaria alternata f. sp. lycopersici in Lycopersicon pennellii

The fungal pathogen Alternaria alternata f. sp. lycopersici produces AAL-toxins that function as chemical determinants of the Alternaria stem canker disease in the tomato (Lycopersicon esculentum). In resistant cultivars, the disease is controlled by the Asc locus on chromosome 3. Our aim was to characterize novel sources of resistance to the fungus and of insensitivity to the host-selective AAL-toxins. To that end, the degree of sensitivity of wild tomato species to AAL-toxins was analyzed. Of all members of the genus Lycopersicon, only L. cheesmanii was revealed to be sensitive to AAL-toxins and susceptible to fungal infection. Besides moderately insensitive responses from some species, L. pennellii and L. peruvianum were shown to be highly insensitive to AAL-toxins as well as resistant to the pathogen. Genetic analyses showed that high insensitivity to AAL-toxins from L. pennellii is inherited in tomato as a single complete dominant locus. This is in contrast to the incomplete dominance of insensitivity to AAL-toxins of L. esculentum. Subsequent classical genetics, RFLP mapping and allelic testing indicated that high insensitivity to AAL-toxins from L. pennellii is conferred by a new allele of the Asc locus.

Teaching and learning about whole numbers in primary school

This book offers a theory for the analysis of how children learn and are taught about whole numbers. Two meanings of numbers are distinguished – the analytical meaning, defined by the number system, and the representational meaning, identified by the use of numbers as conventional signs that stand for quantities. This framework makes it possible to compare different approaches to making numbers meaningful in the classroom and contrast the outcomes of these diverse aspects of teaching. The book identifies themes and trends in empirical research on the teaching and learning of whole numbers since the launch of the major journals in mathematics education research in the 1970s. It documents a shift in focus in the teaching of arithmetic from research about teaching written algorithms to teaching arithmetic in ways that result in flexible approaches to calculation. The analysis of studies on quantitative reasoning reveals classifications of problem types that are related to different cognitive demands and rates of success in both additive and multiplicative reasoning. Three different approaches to quantitative reasoning education illustrate current thinking on teaching problem solving: teaching reasoning before arithmetic, schema-based instruction, and the use of pre-designed diagrams. The book also includes a summary of contemporary approaches to the description of the knowledge of numbers and arithmetic that teachers need to be effective teachers of these aspects of mathematics in primary school. The concluding section includes a brief summary of the major themes addressed and the challenges for the future. The new theoretical framework presented offers researchers in mathematics education novel insights into the differences between empirical studies in this domain. At the same time the description of the two meanings of numbers helps teachers distinguish between the different aims of teaching about numbers supported by diverse methods used in primary school. The framework is a valuable tool for comparing the different methods and identifying the various assumptions about teaching and learning.</p