Problem solving in a 21st century mathematics curriculum

Research on problem solving in the mathematics curriculum has spanned many decades, yielding pendulum-like swings in recommendations on various issues. Ongoing debates concern the effectiveness of teaching general strategies and heuristics, the role of mathematical content (as the means versus the learning goal of problem solving), the role of context, and the proper emphasis on the social and affective dimensions of problem solving (e.g., Lesh & Zawojewski, 2007; Lester, 2013; Lester & Kehle, 2003; Schoenfeld, 1985, 2008; Silver, 1985). Various scholarly perspectives—including cognitive and behavioral science, neuroscience, the discipline of mathematics, educational philosophy, and sociocultural stances—have informed these debates, often generating divergent resolutions. Perhaps due to this uncertainty, educators’ efforts over the years to improve students’ mathematical problem-solving skills have had disappointing results. Qualitative and quantitative studies consistently reveal mathematics students’ struggles to solve problems more significant than routine exercises (OECD, 2014; Boaler, 2009)...

Basic components in the scienctific didactical training of the secondary school mathematics teachers

Secondary mathematics teacher training in Spain is currently the subject of a heated revision debate. The speed of social, cultural, scientific and economic changes have left a hundred years old teacher training model well behind. However, academical inertia and professional interests are impeding a real new training of the mathematics teacher as an autonomous mathematical educator. Teachers of Didactic of Mathematics and the Spanish Associations of mathematics teachers have recently been discussing the issue. Their conclusions are included here.

How mathematics is defined and measured in PISA 2003

In this report I present a summary of the three dimensions used in PISA 2003assessment in mathematics: Content, Process and Situation, and I includesome examples of items.

Issues in second language proficiency

Benati provides clarity about the characteristics and notion of language proficiency in the field of second language acquisition. He looks at four areas of research paradigmatically related to the role of proficiency: theorizing and measuring second language proficiency; the dimensions of L2 proficiency; factors contributing to the attainment of L2 proficiency and attaining L2 proficiency in the classroom. It also contains a variety of research accounts about the specific factors which have an effect on proficiency together with a theorised measurement of proficiency in second language research. It will be required reading for researchers in applied linguistics and second language acquisition.

The impact of integrated programming on student attitude and achievement in grade 9 academic mathematics and science

This research studioo the effect of integrated instruction in mathematics and~ science on student achievement in and attitude towards both mathematics and science. A group of grade 9 academic students received instruction in both science and mathematics in an integrated program specifically developed for the purposes of the research. This group was compared to a control group that had received science and mathematics instruction in a traditional, nonintegrated program. The findings showed that in all measures of attitude, there was no significant difference between the students who participated in the integrated science and mathematics program and those who participated in a traditional science and mathematics program. The findings also revealed that integration did improve achievement on some of the measures used. The performance on mathematics open-ended problem-solving tasks improved after participation in the integrated program, suggesting that the integrated students were better able to apply their understanding of mathematics in a real-life context. The performance on the final science exam was also improved for the integrated group. Improvement was not noted on the other measures, which included EQAO scores and laboratory practical tasks. These results raise the issue of the suitability of the instruments used to gauge both achievement and attitude. The accuracy and suitability of traditional measures of achievement are considered. It is argued that they should not necessarily be used as the measure of the value of integrated instruction in a science and mathematics classroom.

The Educational Experience of Students of Chinese Origin in a French-Speaking Context : the role of school, family, and community

Cette recherche vise à documenter l’expérience scolaire des élèves québécois d’origine chinoise à l’école secondaire de langue française et à examiner les dynamiques qui influencent la réussite scolaire de ces élèves. Elle s’intéresse plus précisément aux impacts des facteurs relatifs à l’école, à la famille immigrante, et à ceux de la communauté ethnique sur l’intégration de ces jeunes dans un contexte francophone. Les données ont été principalement recueillies à travers des entretiens semi-structurés approfondis auprès d’élèves d’origine chinoise et de différents acteurs du paradigme éducatif (parents, acteurs scolaires et intervenants communautaires). D’autres instruments, tels que l’analyse du contenu de documents et de médias, ont également été utilisés afin de fournir des informations contextuelles et d’enrichir les données d’entrevues. Les données ont été analysées selon un cadre théorique ouvert et inclusif où la réussite scolaire des élèves issus de l’immigration est mesurée en mettant l’accent sur l’influence de la maîtrise de la langue d’enseignement, du capital culturel et social de la famille et de la communauté immigrante, ainsi que des facteurs systémiques au niveau de l’école. Les résultats de cette étude dans trois écoles cibles montrent qu’en général les élèves d’origine chinoise connaissent une expérience positive, surtout en ce qui concerne leur performance scolaire en mathématiques et sciences. Cependant, les nouveaux arrivants ont tendance à éprouver des difficultés dans l’apprentissage du français et pour leur intégration sociale. En effet, le processus d’intégration socioscolaire des jeunes chinois est sous l’influence des différents milieux qu’ils fréquentent. À propos de l’influence des dynamiques scolaires, les résultats de la recherche indiquent qu’une relation maître-élève positive joue un rôle important dans la réussite éducative de ces élèves. Toutefois, l’insuffisance du soutien à l’apprentissage défavorise l’intégration linguistique et sociale des élèves nouvellement arrivés. Les données de cette étude soulignent notamment le rôle de la famille immigrante et de la communauté ethnique dans l’expérience scolaire de ces jeunes. D’une part, sous l’impact des dynamiques familiales, notamment ce qui à trait au projet migratoire, à la culture chinoise et à l’expérience pré- et post-migratoire, les parents immigrants chinois s’impliquent activement dans les études de leurs enfants, malgré des barrières linguistiques et culturelles. D’autre part, afin de surmonter les effets négatifs des faibles liens entretenus avec l’école de langue française, les parents chinois ont largement recours aux ressources au sein de la communauté ethnique, tels que les médias de langue chinoise, les organismes ethnospécifiques de services aux immigrants, l’école du samedi et les institutions religieuses ethniques. Ces institutions sociales ethniques contribuent à soutenir les valeurs culturelles, échanger des informations, établir des modèles pour les jeunes et à fournir des services appropriés en matière culturelle et linguistique.

Embracing the Crisis in the Foundations of Mathematics

The crisis in the foundations of mathematics is a conceptual crisis. I suggest that we embrace the crisis and adopt a pluralist position towards foundations. There are many foundations in mathematics. However, ‘many foundations’ (for one building) is an oxymoron. Therefore, we shift vocabulary to say that mathematics, as one discipline, is composed of many different theories. This entails that there are no absolute mathematical truths, only truths within a theory. There is no unified, consistent ontology, only ontology within a theory.

Using spreadsheets in chemical education to avoid symbolic mathematics.

Traditionally, quantum theory has traditionally relied heavily on the use of  mathematics. However, there is a significant cohort of students who are  weak in mathematics, for example, students who are majoring in   biochemistry, biological sciences, etc. This paper reports on the use of  spreadsheets to generate approximate numerical solutions and visual  (graphical) descriptions as a method of avoiding or minimizing symbolic  manipulations, mathematical derivations and numerical computation. A  specific example from quantum theory is provided. Some aspects of  educational pedagogy of spreadsheet usage are discussed.

The nature of spontaneity in high quality mathematics learning experiences

Spontaneity has been linked to high quality learning experiences in mathematics (Csikszentmihalyi & Csikszentmihalyi, 1992; Williams, 2002).This paper shows how spontaneity can be identified by attending to the nature of social elements in the process of abstracting (Dreyfus, Hershkowitz, & Schwarz, 2001). This process is elaborated through an illustrative example—a Year 8 Australian male student who scaffolded his learning by attending to images in the classroom that were intended for other purposes. Leon’s cognitive processing was not ‘observable’ (Dreyfus et al., 2001) in classroom dialogue because Leon ‘thought alone’. Post-lesson videostimulated reconstructive interviews facilitated study of Leon’s thought processes and extended methodological techniques available to study thinking in classrooms.

Implementation of student active learning in primary mathematics in Indonesia

Twenty years after the first pilot projects began to develop Student Active Learning (SAL) in Indonesia, and four years since it was adopted for use in the last provinces, this research investigates the implementation of Student Active Learning in Indonesian primary mathematics classrooms. A study of the relevant literature indicates that teaching based on constructivist principles is unlikely to be implemented well in mathematics classrooms unless there are high quality teachers, readily available manipulative materials, and a supportive learning environment. As Indonesian schools often lack one or more of these aspects, it seemed likely that Student Active Learning principles might not be ‘fully’ implemented in Indonesian primary mathematics classrooms. Thus a smaller scale, parallel study was carried out in Australian schools where there is no policy of Student Active Learning, but where its underlying principles are compatible with the stated views about learning and teaching mathematics. The study employed a qualitative interpretive methodology. Sixteen primary teachers from four urban and four rural Indonesian schools and four teachers from two Victorian schools were observed for four mathematics lessons each. The twenty teachers, as well as fourteen Indonesian headteachers and other education professionals, were interviewed in order to establish links between the background and beliefs of participants, and their implementation of Student Active Learning. Information on perceived constraints on the implementation of SAL was also sought. The results of this study suggest that Student Active learning has been implemented at four levels in Indonesian primary mathematics classrooms, ranging from essentially no implementation to a relatively high level of implementation, with an even higher level of implementation in three of the four Australian classrooms observed. Indonesian teachers, headteachers and supervisors hold a range of views of SAL and also of mathematics learning and teaching. These views largely depended on their in-service training in SAL and, more particularly, on their participation in the PEQIP project Typically, participants’ expressed views of SAL were at the same or higher level as their views of mathematics learning and teaching, with a similar pattern being observed in the relationship between these latter views and their implementation of SAL principles. Three factors were identified as influencing teacher change in terms of implementation of SAL: policy, curricular and organisational, and attitudes. Recommendations arising from this study include the adoption of reflection as an underlying principle in the theory of SAL, the continuation and extension of PEQIP type projects, changes in government policy on curriculum coverage and pre-service teacher training, and more support for teachers at the school and local authority levels.

Using everyday experiences in teaching secondary mathematics in Malawi : possibilities and constraints for change

Proponents of socially and culturally oriented mathematics education have argued that teaching approaches which value and connect with the learner's prior knowledge and everyday experience are more likely to promote active, meaningful, relevant and liberatory learning than approaches which rely on transmission and abstract presentation of mathematical content. In Malawi, proposals to reform the outdated secondary mathematics curriculum have been made with the aim of aligning mathematics instruction with the social and political changes in the current Malawian society. Using a case study approach, this study investigated the extent to which everyday experiences could be used as a vehicle for changing the learning and teaching of secondary mathematics in Malawi. The study was collaborative, taking place over a period of five months in severely overcrowded and poorly resourced classes in two schools. It involved three mathematics teachers in a cycle of planning and teaching mathematics lessons based on the use of everyday experiences, and observation of and reflection on these lessons, in order to document the effects of using everyday experiences on student learning and teachers' teaching practices. The data was collected through student questionnaires; classroom observations and fieldnotes; interviews and reflective meetings with teachers; and informal meetings with key education officials in Malawi. Mathematics examination results from students involved in this study and a corresponding group from the previous year were collected. A reflective and critical approach was adopted in the interpretation and discussion of the data. Teachers' participation in this study resulted in heightened awareness of their teaching roles and the value of linking school mathematics with everyday experience. The study also shows that students found mathematics interesting and important to learn despite their lack of success in it. In addition, the study documented a number of constraints to change in mathematics instruction such as teachers' focus on mathematics content and examination requirements, and students' resistance to inquiry learning. It also recorded possibilities and barriers to collaboration both between teachers and researchers, and teachers themselves. The findings of this study are timely since they could serve to inform the reform of the Malawian secondary mathematics curriculum currently being undertaken, which began without a critical examination of the classroom conditions necessary to accommodate a socio-politically relevant mathematics education.

Fostering meaningful learning by using word problems in post-secondary mathematics

This research is a case study of action research investigating the effects of using word problems with Chinese Malaysian post-secondary students studying mathematics as an enabling science. A variety of word problems, a discussion method, reflective and peer learning, specific values in mathematics, and relevant socio-cultural and language issues were explored.

Gender differences in motor skill proficiency from childhood to adolescence : a longitudinal study

Students' proficiency in three object control and three locomotor skills were assessed in 2000 (M age = 10.06 years, SD = 0.63) in New South Wales, Australia and in 2006-07 (M age = 16. 44 years, SD = 0. 64). In 2006-07, 266 students, 138 girls (51.9%) and 128 boys (48.1%), had at least one skill reassessed. Boys were more object control proficient than girls. Childhood object control proficiency significantly predicted (p =. 001) adolescent object control proficiency ([r.sup.2] = .39), and, while gender was significant (p = .001), it did not affect the relationship between these variables (p = .53). Because childhood object control proficiency is predictive of subsequent object control proficiency, developing skills in childhood is important.

Computers are taking mathematics into the next century : gender differences in attitudes of secondary mathematics students to the use of computers

Secondary students who participated in a computer enhanced mathematics program expressed positive attitudes about the use of computers. They viewed computers as a source of pleasure, success, relevance and/or power in mathematics. Girls were more likely than boys to qualify their support for the use of computers and more likely to view computers as a source of success in mathematics. Boys were more likely to claim that computers brought pleasure or relevance to mathematics learning.