Interleaving in Mathematical Category Learning

It has been observed that Asian, in particular Chinese students frequently outperform their Western counterparts in international academic comparisons. However, it is still open to doubt as to whether they actually possess a deeper conceptual understanding than their Western counterparts and as to whether they would perform as well in non-routine mathematical problems (such as open-ended problems). The reason for such queries owes as much to the students' as to teachers' conceptions of mathematics. When mathematics is regarded as an absolute truth or a set of rules governing symbols, students tend to consider doing mathematics as the memorisation of algorithms and learning mathematics as a process of transmission. We posed certain hypothetical situations to students in Hong Kong and China and found that they possess a relatively restricted conception of mathematics. Later, we investigated this phenomenon at greater depth by the use of open-ended problems. We found that students usually approached a mathematical problem by searching for a rule that identifies what is given, what is being asked and the category of topic for the problem. Evidence has also shown that this approach to mathematical problems is largely shaped by the way they experience learning, their response to task demands, and the classroom environment. In other words, such a restricted conception of mathematics which exists both within the students and in the classroom culture has led students to tackle mathematical problems by searching for rules rather than approaching them through a conceptual understanding of the context. It is well known that students' conception of mathematics is closely related to their problem-solving behaviour. This conception is shaped by a number of factors, in particular, by the space of learning in which they 'live'. Of particular significance is the finding that the teachers' conception of mathematics could contribute to the shaping of this 'lived space'. These inter-relationships are the focus of a series of studies which will be reported in the present paper.

Mathematics as a school subject is designed to solve the everyday problem created by human activities; these problems are domesticated in the different cultures of man all over the world. However, the performance of students keeps dwindling due to the westernize approach of teaching and learning mathematics. In other to solve this precarious situation this paper advocated the use of mother tongue in the teaching of mathematics. Mother tongue approach here refers to first-language education as a medium of instruction. The use of mother tongue in the teaching and learning of mathematics is meant to address the high functional illiteracy of Nigeria where language plays a significant factor. Using mother tongue to teach the basic concepts of numbers, operations and other concepts of mathematics helps build a strong foundation for the understanding and learning of higher mathematics. This approach is effective not only in getting the interest of students in the lesson but as a springboard in teaching new mathematical concepts, principles and in deepening student understanding on how mathematical operations or processes work. The mother tongue concepts of teaching and learning mathematics in this work shall be based on the following sub-topics: Numeration/Counting System, Basic Arithmetic Operations, Rhymes and Geometric Concepts. The paper completely reduced the Eurocentric mathematics beliefs of students and make them discover how best mathematics can be learning from their mother tongue. The paper recommended amongst others that teachers of mathematics should adopt the use of mother tongue in teaching mathematics in order to improve students understanding of the subject.

The lived space of mathematics learning

For a long period, some students in schools thought that Physics and Mathematics were difficult subjects. These two subjects are a scourge for students in most schools, not only in the NTB area but also in various other regions throughout Indonesia. For this reason, multiple efforts need to be made to overcome these learning difficulties. Learning Physics and Mathematics should emphasize providing direct experience to develop student competencies, directed to "find out" and "do" to help students gain understanding. It will be significantly easier for students if learning Physics and Mathematics invites students to learn to formulate concepts inductively, based on empirical facts, first through the inquiry method. Teachers are faced with the low necessary abilities of students. Not a few teachers have complained about their students' low abilities, especially in understanding the basic concepts of Physics and Mathematics. Many factors influence this, one of which is the input factor (students). Students from different schools have different levels of understanding, both in Physics and Mathematics. It is a fact that, in general, students who graduated from SMP / M.Ts. There are still many basic concepts of Physics and Mathematics which are relatively low. When comparing students who come from the city and out of town schools, public and private schools, they have quite significant differences in graduates' ability. This condition also contributes to the low quality of the input. Based on this condition, we hold community service activities to strengthen understanding of the basic concepts of Physics and Mathematics for students as their initial provision for studying other material in more depth. The approach used is an empirical approach with guided inquiry methods. The results obtained are that the learning method of this model is very popular with students and teachers, and with the help of Guided Inquiry-based worksheets, it is straightforward for students to improve their understanding of the basic concepts of physics and mathematics, because the guidance in the LKS is very structured.

This research departs from the phenomenon that occurs in schools that the teacher's low understanding of the concept of Mathematics in the learning process. Therefore, a school principal needs to consider a strategy to increase teacher competency skills so that they can improve their abilities. This study aims to determine the effectiveness of the implementation of in-house training in improving teacher competence on the concept of Mathematics, to describe the process of increasing teacher competence about the concept of Mathematics through the implementation of in-house training activities, to measure the magnitude of the increase in teacher competence on the concept of Mathematics through the implementation of in-house training. The research was conducted at SMP Negeri 10 Bogor in the first semester of the 2019/2020 academic year. The results of this study indicate that by implementing in-house training can motivate teachers to improve their competence in understanding mathematical concepts. Before carrying out the in-house training, of the 5 Mathematics teachers, almost all of the teachers did not have the correct understanding of the concept of Mathematics learning. After the in-house training on the concept of learning Mathematics, there was an increase in the first cycle as many as 3 teachers (60%) became aware of the concept of Mathematics, as many as 1 person (20%) understood enough, and as many as 1 person (20%) did not understand. In the second cycle there was an increase to all teachers, as many as 5 people (100%) understood the concept of learning Mathematics. That the implementation of in-house training can improve the competence of teachers about the concept of Mathematics. Therefore, researchers suggest that the implementation of In house training in improving teacher competence should be socialized and used as an alternative in understanding Mathematics concepts for teachers in the Bogor City Education Office.

Students’ performances in mathematics in Ghana have not been as good as they should be (Ministry of Education Science and Sports, 2007; Ministry of Education Youth and Sports, 2004a). Concepts such as fractions and measurement have often been cited as being problematic to students but these are the very concepts which students have much prior knowledge of, through their engagement in out-of-school/everyday mathematical practices in Ghanaian society. In other contexts researchers have highlighted the role that culture plays in mathematics teaching and learning (Bishop, 1991; Seah, 2004). This study therefore sought to investigate cultural influences on primary school students’ mathematical conceptions and practices in Ghana, as they move between contexts of out-of-school mathematical practices (OOSM) in the home and in-school mathematical practices (ISM) in the school. Three theoretical lenses were drawn to illuminate the problem. These included the cultural nature of mathematical knowledge by Bishop (1988), sociocultural theories on learning by Vygotsky (1978) and students’ transitions between contexts of mathematical practices by Abreu, Bishop and Presmeg (2002). Two main research issues and seven research questions were posed to guide the study. The main research issues were: 1) What are the sociocultural influences on Ghanaian students’ mathematics learning? 2) What are Ghanaian students’ transition experiences between the home and school contexts and how do these affect their learning in school? Questionnaires were administered and responded to by 137 primary school teachers and 24 of their headteachers, from 25 (out of all 74) primary schools, selected through stratified random sampling in the Cape Coast Metropolitan area in Ghana. The selection was followed by interviews with 32 primary school students (four each from grade 4 and grade 6), their teachers and headteachers from four (out the 25) schools. Documentary evidence of how teachers handled culture differences was also collected. The data gathered from the closed ended items in the questionnaire survey were analysed quantitatively through the use of frequency counts and descriptive statistics (means) whilst the open ended items were analysed qualitatively, as were the focus group interviews as well as interviews with teachers and headteachers. The study revealed that exposure to school mathematical culture influences the use of OOSM in ISM in some cases; students’ perceptions about mathematics reflected those of the headteachers and teachers; students identified with school mathematical culture (ISM) despite their recognition of OOSM also as a form of mathematics; evidence of cultural influences was observed especially in students’ conceptions and practices in identification of arithmetic fractions and division (as in sharing) in real life problems; students mistakes appeared to largely depend on out-of-school logic; generally practical activities evoked out-of-school thinking whilst paper and pencil activities evoked in-school thinking; teachers generally ignored cultural differences and rather concentrated on teaching school mathematics. The study recommends the need for teachers to see beyond students’ mistakes, as their mistakes could be based on a different logic system. In order to make mathematics more realistic to students, a three-tier teaching strategy is proposed to gradually expand students’ mathematics schema to include ISM.

In this chapter, we put forward the claim that any specific view of internationalisation corresponds to a particular orientation for learning and teaching in mathematics. We use a critical discourse perspective to explore variation in the intentions and outcomes of an ‘internationalised curriculum’ and apply the results to the discipline of mathematics. We support the discussion with reference to several components of our research: in particular, a study on students’ conceptions of mathematics and learning in mathematics, and another study reporting on lecturers’ understanding of the intersections between teaching and sustainability, an important correlate of internationalisation. Our aim is to consider the way in which internationalisation contributes as a ‘value’ orientation for our students’ approaches to their study and indeed to their whole lives. We then apply our model to a practical discussion of the construction of learning environments that support a focus on students’ professional formation and the development of their global perspectives

Identifying Taiwanese junior-high school students' mathematics learning profiles and their roles in mathematics learning self-efficacy and academic performance

In this chapter we investigate students’ views about learning mathematics. We continue our analysis of the interviews that we conducted with undergraduate mathematics majors at an Australian university. Here we move the focus to their ideas about how they go about studying mathematics. We present a theoretical model based on our research findings, aiming to build on and expand earlier descriptions of students’ learning approaches. While the small convenience sample that we have used for our interviews could limit the generalisability of our findings to the larger group of mathematics students as a whole, our results seem consistent with previous literature on student learning, and also with our findings about students’ conceptions of mathematics. We found that students’ conceptions of learning mathematics could be described using three hierarchical levels. At the narrowest level, students learned by focusing on the disparate techniques of mathematics. A broader level was represented by a focus on the subject of mathematics itself. In the most comprehensive view, students considered the role of mathematics in their lives and talked about developing a mathematical way of thinking and viewing the world, satisfying their intellectual curiosity and helping them to grow as a person. The hierarchical nature implies that students with the broadest ‘life’ conception of learning mathematics were also aware of the ‘subject’ and ‘techniques’ aspects, but those with the narrowest techniques conception did not seem to aware of the broader ideas. There seems to be an obvious parallel between these conceptions of learning mathematics and the conceptions of mathematics itself that we identified previously.

There is widespread belief that computers should be used for the teaching and learning of mathematics. Research indicates that computers are primarily used in mathematics classes: (1) to reinforce previously taught concepts, (2) to allow students to construct computer programs to simulate mathematical techniques known to the student and (3) to explore mathematical microworlds encompassing mathematical ideas and concepts normally known to the student. Furthermore, it is said that pre‐service teachers should experience the learning of mathematical ideas and concepts of which they had no prior experience in environments in which computers are just one of the resources available for exploring and experimenting with these ideas and concepts. How should these learning environments be constructed so that pre‐service teachers are sensitised to the value of doing mathematics in such environments? Is a student's understanding of novel mathematical concepts enhanced when s/he explores it in a computer‐enriched environment? An experiment with pre‐service teachers was carried out in a college of education for blacks in South Africa. This article describes the insights gained from this experiment.

What do our students think of mathematics and how do we help prepare them for the workplace? This article reports results from an open-ended questionnaire with engineering students in five countries. We investigate how they conceive of mathematics and how they perceive their university study has prepared them for the workforce. The results show that students have different ideas of mathematics itself and how they will use it. Some view mathematics as a tool to be used in their professional lives, others think they will use the analytical and problem solving skills. There were also those who do not believe they will use mathematics at all---that computers will do it all for them. Since notions about future utility can influence engagement with learning, we as teachers need to proactively expand students' perceptions of the fundamental importance of mathematics to working as an engineer. I provide an example of an assessment task that will help students expand their ideas of how they will use mathematics in their careers. References Graduate Careers Council of Australia. Website, 2007. http://www.gradlink.edu.au O. Bringslid, G. Rodriguez, and A. de la Villa. dMath: a European project for the restatement of mathematics teaching. Eur. J. Eng. Ed., 32(1):9--20, 2007. A framework for qualifications of the European higher education area. Technical report, Ministry of Science, Technology and Innovation, Denmark, 2005. http://www.bologna-bergen2005.no/Docs/00-Main_doc/050218%QF_EHEA.pdf H. Hult, M. Abrandt Dahlgren, L.O. Dahlgren, and H. Hard af Segerstad. Freshmen's and seniors' thoughts about education, professional identity and work. Proc. Aust. Ass. Res. Ed., 2003. http://www.aare.edu.au/03pap/dah03725.pdf P. Kent and R. Noss. Mathematics in the university education of engineers. Technical report, Ove Arup Foundation, London, 2003. http://www.ioe.ac.uk/rnoss/REMIT/Kent-Noss-report.pdf L. Kirkup, L.N. Wood, G. Mather, and P. Logan. Are you being serviced? Promoting quality service teaching. In Uniserve Science, pages 37--42, Sydney, 2003. I. Markes. A review of literature on employability skill needs in engineering. Eur. J. Eng. Ed., 31:637--650, 2006. R. Martin, B. Maytham, J. Case, and D. Fraser. Engineering graduates' perceptions of how well they were prepared for work in industry. Eur. J. Eng. Ed., 30:167--180, 2005. P. Petocz, A. Reid, L.N. Wood, G.H. Smith, G. Mather, A. Harding, J. Engelbrecht, K. Houston, J. Hillel, and G. Perrett. Undergraduate students' conceptions of mathematics: An international study. Int. J. Sc. Math. Ed., 5:439--459, 2007. A. Reid, L. Wood, G. Smith, and P. Petocz. Intention, approach and outcome: university mathematics students' conceptions of learning mathematics. Int. J. Sc. Math. Ed., 3:567--586, 2005. L. Wood and A. Reid. Conversations with graduates: reflections on learning mathematics. In Proc. 3rd Int. Conf. Teaching of Mathematics at the Undergraduate Level, Istanbul, Turkey, 2006. Turkish Mathematical Society. Issued on compact disc. L. Wood, G. Smith, G. Mather, A. Harding, J. Engelbrecht, K. Houston, G. Perrett, J. Hillel, P. Petocz, and A. Reid. Student conceptions---understanding performance. In Proc. 3rd Int. Conf. Teaching of Mathematics at the Undergraduate Level, Istanbul, Turkey, 2006. Turkish Mathematical Society. Issued on compact disc. L.N. Wood. Research in mathematics education in Australasia 2000--2003, chapter University mathematics teaching and learning. Flaxton: Post Pressed, 2004. Editors B. Perry, G. Anthony and C. Diezmann.

This study reports the findings of a study which investigated junior secondary school students’ conceptions of and approaches to learning mathematics and their relationships in Mainland China. Two questionnaires, conceptions of learning mathematics (COLM) and approaches to learning mathematics (ALM), were administered to 1590 students. Descriptive analysis results suggest that Chinese junior school students tend to hold (1) higher-level conceptions of learning mathematics rather than lower-level conceptions, and (2) deep approaches to learning mathematics with a rather mixed surface and deep motive in learning mathematics. Correlation and regression analysis results confirm a structural relationship between students’ conceptions of and approaches to learning in the subject of mathematics. Two factors of students’ lower-level conceptions of learning mathematics, “memorizing” and “testing”, were the strongest predictor for the surface approaches to mathematics learning, while students’ higher-level conceptions of learning mathematics, such as “applying” and “understanding and mathematical thinking”, had a noticeable effect on their deep approaches to learning mathematics. However, under the pressure of examination in Mainland China, “understanding and mathematical thinking” was also found to exert quite a strong influence on students’ “surface motive”.

The Badjaos are known to be shy, ridiculed, discriminated, and cannot interact with other people or races thus affects their behavior in the community and performance in schools. The study aimed at describing the processes and techniques of the individualized learning mechanisms; determining their pre-test and post-test scores; and ascertaining if there was a significant difference in the pre-test and post-test scores of the participants in the individualized learning. We introduced an academic intervention program in Mathematics through tutorials by implementing individualized and personalized interactions. The Mathematical topics and contents included are the results of the needs analysis survey during the focus group discussion (FGD) and key informant interviews (KII). By utilizing the quasi-experimental setup, the pre-test and the post-test results showed some significance regarding mean differences. Qualitative data was also collected. The research activities had developed a harmonious learning environment among stakeholders. The levels of interest of the Badjao children became very promising as manifested by their gestures and behavior in the learning environment. Supporting these results are their performance inside the classroom and the revelation of their teachers. The tutorial program has impacted on the lives of the Badjao children. They appreciated the activity. The Badjao children in the study now can count and comprehend numerical figures without difficulty. They said that it is not difficult to engage these concepts in Mathematics in their everyday living. There is a desire to continue this program to other cultures in Mindanao as well. The research program had developed a sense of cultural understanding and eventually, it was discovered that the learning and tutoring in Mathematics via individualized learning was found to be an efficient tool for learning in Mathematics with the specific cultural background.

This research is motivated by the lack of student learning activity that affects the understanding of student concepts in mathematics learning in class XI F 4.2 SMA Veteran 1 Sukoharjo. The importance of student activity in learning needs to be done so that learning becomes interesting and students have the curiosity to learn more deeply related to the knowledge taught by the teacher, in addition, understanding the concept is one of the important factors that supports student activity in the learning process. This study aims to improve the activity and understanding of student concepts in mathematics learning using PBL based on lesson study. The subjects in this study were 27 students of class XI F4.2 majoring in social sciences. The study results showed that using the PBL learning model based on lesson study can improve the activity and understanding of student concepts in learning. Based on the study's results, the percentage of student activity increased from pre-cycle 15% and cycle 1 by 59% then cycle 2 by 100%. This is the same as the increase in student's conceptual understanding from the pre-cycle stage with a percentage of 37%, then cycle 1 by 59% to cycle 2 by 100%, so it can be concluded that using the lesson study-based PBL learning model can increase students' activeness and conceptual understanding.

The selection of the appropriate learning model is very important because it can make more meaning in the process of teaching and learning activities. Teacher Professional Education Program expects the practitioner to be able to choose appropriate learning model to the class conditions. Even in schools that have implemented a zoning system, namely public junior high school 5 Surakarta. This paper aims to find out the description and comparison of the four learning models used by Teacher Professional Education Program at public junior high school 5 Surakarta towards understanding students' mathematical concepts. The learning model used in the research were Number Head Together (NHT), Discovery Learning (DL) with recitation, Problem Based Learning (PBL) and Teams games Together (TGT). The methodology of this research uses descriptive and inferential statistics. Thus, inferential statistical analysis using one-way variance analysis. Data are understanding students' mathematical concepts were obtained by the tests. From the results of the study, it can be found that the average understanding of mathematical concepts of students using DLwith recitation has a better effect on understanding students' mathematical concept. In contrast, TGTs' class produce the lowest understanding students' mathematical concepts while the most varied understanding students' mathematical concepts are in the NHTs' class.