Students' difficulties, conceptions and attitudes towards learning algebra : an intervention study to improve teaching and learning

The skills necessary to identify and analyse errors and misconceptions made by students are needed by teachers of all levels especially at the lower secondary school level in Malaysia. If students are to be successful in tackling mathematical problems later in their schooling, the one prerequisite is the mastery of basic concepts in algebra. Despite the best efforts of the teachers, students still develop algebra misconceptions. Is it possible to reduce or eliminate these misconceptions? The research involved a survey of 14 year-old students in Form 2 (Grade 8) in the Penampang district of Sabah, East Malaysia. The focus of this study lies in students’ difficulties, conceptions and attitudes towards learning algebra in the framework of conceptual change. A possible way to help students overcome their learning difficulties and misconceptions is by implementing diagnostic teaching involving conflict to foster conceptual change. The study involved evaluating the efficacy of a conceptual change instructional programme involving cognitive conflict in (1) facilitating Form 2 students’ understanding of algebra concepts, and (2) assessing changes in students’ attitudes towards learning mathematics, in a mixed quantitativequalitative research design.A 24-item Algebra Diagnostic Test and a 20-item Test of Mathematics-Related Attitudes (TOMRA) questionnaire were administered as a pretest and a posttest to 39 students in each of a heterogeneous high-achieving class and a below-average achieving class. In addition 9 students were purposefully selected to participate in the interview.The results of the study indicated that students’ difficulties and misconceptions from both classes fell into five broad areas: (1) basic understanding of letters and their place in mathematics, (2) manipulation of these letters or variables, (3) use of rules of manipulation to solve equations, (4) use of knowledge of algebraic structure and syntax to form equations, and (5) generalisation of rule for repetitive patterns or sequences of shapes.The results also showed that there was significant improvement in students’ achievement in mathematics. Further, students’ attitude towards inquiry of mathematics lessons showed significant positive improvement. Enjoyment remained high even though enjoyment of mathematics lesson showed no change. Also, changes in students’ understanding (from unintelligible to intelligible, intelligible to plausible, plausible to fruitful) illustrated the extent of changes in their conceptions.Different pedagogies can affect how conceptual change and challenge of misconceptions occurs. Therefore, knowledge of the origin of different types of misconceptions can be useful in selecting more effective pedagogical techniques for challenging particular misconceptions. Also, for teachers to create an effective learning experience they should be aware of and acknowledge students’ prior knowledge acquired from academic settings and from everyday previous personal experiences. Since all learning involves transfer from prior knowledge and previous experiences, an awareness and understanding of a student’s initial conceptual framework and/or topic can be used to formulate more effective teaching strategies. If this idea is taken a step further, it could be said that, because misconceptions comprise part of a conceptual framework, then understanding origins of misconceptions would further facilitate development of effective teaching strategies.Further research is needed to help teachers to understand how students experience conflict, how students feel when they experience conflict, and how these experiences are related to their final responses because cognitive conflict has both constructive and destructive potential. Thus, by being able to interpret, recognise and manage cognitive conflict, a teacher can then successfully interpret his/her students’ cognitive conflict and be able to make conceptual change more likely or help students to have meaningful learning experiences in secondary school algebra

Students' difficulties, conceptions and attitudes towards learning algebra : an intervention study to improve teaching and learning

The skills necessary to identify and analyse errors and misconceptions made by students are needed by teachers of all levels especially at the lower secondary school level in Malaysia. If students are to be successful in tackling mathematical problems later in their schooling, the one prerequisite is the mastery of basic concepts in algebra. Despite the best efforts of the teachers, students still develop algebra misconceptions. Is it possible to reduce or eliminate these misconceptions? The research involved a survey of 14 year-old students in Form 2 (Grade 8) in the Penampang district of Sabah, East Malaysia. The focus of this study lies in students’ difficulties, conceptions and attitudes towards learning algebra in the framework of conceptual change. A possible way to help students overcome their learning difficulties and misconceptions is by implementing diagnostic teaching involving conflict to foster conceptual change. The study involved evaluating the efficacy of a conceptual change instructional programme involving cognitive conflict in (1) facilitating Form 2 students’ understanding of algebra concepts, and (2) assessing changes in students’ attitudes towards learning mathematics, in a mixed quantitativequalitative research design.A 24-item Algebra Diagnostic Test and a 20-item Test of Mathematics-Related Attitudes (TOMRA) questionnaire were administered as a pretest and a posttest to 39 students in each of a heterogeneous high-achieving class and a below-average achieving class. In addition 9 students were purposefully selected to participate in the interview.The results of the study indicated that students’ difficulties and misconceptions from both classes fell into five broad areas: (1) basic understanding of letters and their place in mathematics, (2) manipulation of these letters or variables, (3) use of rules of manipulation to solve equations, (4) use of knowledge of algebraic structure and syntax to form equations, and (5) generalisation of rule for repetitive patterns or sequences of shapes.The results also showed that there was significant improvement in students’ achievement in mathematics. Further, students’ attitude towards inquiry of mathematics lessons showed significant positive improvement. Enjoyment remained high even though enjoyment of mathematics lesson showed no change. Also, changes in students’ understanding (from unintelligible to intelligible, intelligible to plausible, plausible to fruitful) illustrated the extent of changes in their conceptions.Different pedagogies can affect how conceptual change and challenge of misconceptions occurs. Therefore, knowledge of the origin of different types of misconceptions can be useful in selecting more effective pedagogical techniques for challenging particular misconceptions. Also, for teachers to create an effective learning experience they should be aware of and acknowledge students’ prior knowledge acquired from academic settings and from everyday previous personal experiences. Since all learning involves transfer from prior knowledge and previous experiences, an awareness and understanding of a student’s initial conceptual framework and/or topic can be used to formulate more effective teaching strategies. If this idea is taken a step further, it could be said that, because misconceptions comprise part of a conceptual framework, then understanding origins of misconceptions would further facilitate development of effective teaching strategies.Further research is needed to help teachers to understand how students experience conflict, how students feel when they experience conflict, and how these experiences are related to their final responses because cognitive conflict has both constructive and destructive potential. Thus, by being able to interpret, recognise and manage cognitive conflict, a teacher can then successfully interpret his/her students’ cognitive conflict and be able to make conceptual change more likely or help students to have meaningful learning experiences in secondary school algebra

Difficulties in understanding mathematics: an approach related to working memory and field dependency

It is commonly agreed that learning with understanding is more desirable than learning by rote. Understanding is described in terms of the way information is represented and structured in the memory. A mathematical idea or procedure or fact is understood if it is a part of an internal network, and the degree of understanding is determined by the number and the strength of the connections between ideas. When a student learns a piece of mathematical knowledge without making connections with items in his or her existing networks of internal knowledge, he or she is learning without understanding. Learning with understanding has progressively been elevated to one of the most important goals for all learners in all subjects. However, the realisation of this goal has been problematic, especially in the domain of mathematics where there are marked difficulties in learning and understanding. The experience of working with learners who do not do well in mathematics suggests that much of the problem is that learners are required to spend so much time in mathematics lessons engaged in tasks which seek to give them competence in mathematical procedures. This leaves inadequate time for gaining understanding or seeking how the procedures can be applied in life. Much of the satisfaction inherent in learning is that of understanding: making connections, relating the symbols of mathematics to real situations, seeing how things fit together, and articulating the patterns and relationships which are fundamental to our number system and number operations. Other factors include attitudes towards mathematics, working memory capacity, extent of field dependency, curriculum approaches, the classroom climate and assessment. In this study, attitudes, working memory capacity and extent of field dependency will be considered. The work will be underpinned by an information processing model for learning. A mathematics curriculum framework released by the US National Council of Teachers of Mathematics (NCTM, 2000) offers a research-based description of what is involved for students to learn mathematics with understanding. The approach is based on “how learners learn, not on “how to teach”, and it should enable mathematics teachers to see mathematics from the standpoint of the learner as he progresses through the various stages of cognitive development. The focus in the present study is to try to find out what aspects of the process of teaching and learning seem to be important in enabling students to grow, develop and achieve. The attention here is on the learner and the nature of the learning process. What is known about learning and memory is reviewed while the literature on specific areas of difficulty in learning mathematics is summarised. Some likely explanations for these difficulties are discussed. Attitudes and how they are measured are then discussed and there is a brief section of learner characteristics, with special emphasis on field dependency as this characteristic seems to be of importance in learning mathematics. The study is set in schools in Nigeria and England but the aim is not to make comparisons. Several types of measurement are made with students: working memory capacity and extent of field dependency are measured using well-established tests (digit span backward test and the hidden figure test). Performance in mathematics is obtained from tests and examinations used in the various schools, standardised as appropriate. Surveys and interviews are also used to probe perceptions, attitudes and aspects of difficulties. Throughout, large samples were employed in the data collection with the overall aim of obtaining a clear picture about the nature and the influence of attitudes, working memory capacity and extent of field dependency in relation to learning, and to see how this was related to mathematics achievement as measured by formal examination. The study starts by focussing on gaining an overview of the nature of the problems and relating these to student perception and attitudes as well as working memory capacity. At that stage, the focus moves more towards extent of field dependency, seen as one way by which the fixed and limited working memory capacity can be used more efficiently. Data analysis was in form of comparison and correlation although there are also much descriptive data. Some very clear patterns and trends were observable. Students are consistently positive towards the more cognitive elements of attitude to mathematics (mathematics is important; lessons are essential). However, they are more negative towards the more affective elements like enjoyment, satisfaction and interest. Thus, they are very realistic about the value of mathematics but find their experiences of learning it much more daunting. Attitudes towards the learning of mathematics change with age. As students grow older, the belief that mathematics is interesting and relevant to them is weakened, although many still think positively about the importance of mathematics. Loss of interest in mathematics may well be related to an inability to grasp what is required and the oft-stated problem that it is difficult trying to take in too much information and selecting what is important. These and other features probably relate to working memory overload, with field dependency skills area being important. The study identified clearly the topics which were perceived as most difficult at various ages. These topics involved ideas and concepts where many things had to be handled cognitively at the same time, thus placing high demands on the limited working memory capacity. As expected, working memory capacity and mathematics achievement relate strongly while extent of field dependency also relates strongly to performance. Performance in mathematics is best for those who are more field-independent. It was found that extent of field dependency grew with age. Thus, as students grow older (at least between 12 and about 17), they tend to become more field-independent. It was also found that girls tend to be more field-independent than boys, perhaps reflecting maturity or their greater commitment and attention to details to undertake their work with care during the years of adolescence. The outcomes of the findings are interpreted in terms of an information processing model. It is argued that curriculum design, teaching approaches and assessment which are consistent with the known limitations of the working memory must be considered during the learning process. There is also discussion of the importance of learning for understanding and the problem of seeking to achieve this while gaining mastery in procedural skills in the light of limited working memory capacity. It is also argued that positive attitudes towards the learning in mathematics must not only be related to the problem of limited working memory capacity but also to ways to develop increased field independence as well as seeing mathematics as a subject to be understood and capable of being applied usefully

The Influence of Teaching Instruction and Learning Styles on Mathematics Anxiety in the Developmental Mathematics Classroom

In the US, an estimated 25% of four-year college students and up to 80% of community college students suffer from a moderate to high degree of mathematics anxiety (MA) (Chang & Beilock, 2016). Many scholars have noted that mathematics anxiety can be regarded as a significant factor in determining a student's achievement and mathematics related jobs. In the existing literature body, many researchers noted that MA may stem from teaching methods that are more conventional and rule-bounded such as lecture-style classroom models. On the other hand, MA can be mitigated by inquiry-based learning classroom models where students construct knowledge through inquiry, communication, critical thinking, and group work. However, the current literature has not built the connection between different teaching styles and students' individual differences with respect to MA. The individual differences are associated with the personality of the learner, learning styles, learning speed, and needs and interests of the learner. Depending on a student's learning style and a compatible teaching style, the student may actively participate in their own learning with less mathematics anxiety. Thus, the purpose of this study is to determine the influence of different teaching styles on MA, when interacted with Kolb’s and Gregorc’s (1984) four different learning and thinking styles. The research questions investigated in this study are: 1) What is the difference between a lecture classroom model (LCM) and an inquiry-based learning classroom model (IBL) on students’ mathematics anxiety levels over a fifteen-week semester of a college-level remedial mathematics course?; 2) What is the difference between a lecture classroom model (LCM) and an inquiry-based learning classroom model (IBL) on mathematics anxiety levels for students with different learning and thinking styles (as defined by Kolb’s and Gregorc’s learning styles) over a fifteen-week semester?; and 3) What aspects of instructional approaches (LCM and IBL) do students with different learning and thinking styles report as being related to mathematics anxiety? The abbreviated version of the mathematics anxiety rating scale (A-MARS), Kolb’s learning styles inventory, Gregorc’s thinking styles, and Written questionnaire were used to measure students’ MA levels and identify their learning and thinking styles. The results provided evidence that IBL instruction is beneficial for the students with MA, especially with mathematics test anxiety and mathematics course anxiety. Only numerical task anxiety was not significant. Thus, student-centered learning pedagogies turned out to be an effective and engaging method for lowering MA. However, there was no evidence to support the overall relationship between the constructs of learning and thinking styles and MA levels, above and beyond the instructional approaches. Classifying students according to learning and thinking styles did not influence students’ MA levels in this study over the 15 academic weeks. Moreover, after a 15 academic weeks, students in both LCM and IBL classes responded positively to key components of LCM and IBL classroom models. This implies that both LCM and IBL approaches still are important models regardless of students’ MA levels

Utilising artificial neural networks (ANNs) towards accurate estimation of life-cycle costs for construction projects

This study aimed to establish a new model of Life Cycle Cost (LCC) for construction projects using Artificial Neural Networks (ANNs). Survey research and Costs Significant Items (CSIs) methods were conducted to identify the most important cost and non-cost factors affecting the estimation of LCC. These important factors are considered as input factors of the model. The results indicated that neural network models were able to estimate the cost with an average accuracy between 91%-95%

Applied Cognitive Sciences

Cognitive science is an interdisciplinary field in the study of the mind and intelligence. The term cognition refers to a variety of mental processes, including perception, problem solving, learning, decision making, language use, and emotional experience. The basis of the cognitive sciences is the contribution of philosophy and computing to the study of cognition. Computing is very important in the study of cognition because computer-aided research helps to develop mental processes, and computers are used to test scientific hypotheses about mental organization and functioning. This book provides a platform for reviewing these disciplines and presenting cognitive research as a separate discipline

Evaluation of topic-based adaptation and student modeling in QuizGuide

This paper presents an in-depth analysis of a nonconventional topic-based personalization approach for adaptive educational systems (AES) that we have explored for a number of years in the context of university programming courses. With this approach both student modeling and adaptation are based on coarse-grained knowledge units that we called topics. Our motivation for the topic-based personalization was to enhance AES transparency for both teachers and students by utilizing typical topic-based course structures as the foundation for designing all aspects of an AES from the domain model to the end-user interface. We illustrate the details of the topic-based personalization technology, with the help of the Web-based educational service QuizGuide—the first system to implement it. QuizGuide applies the topic-based personalization to guide students to the right learning material in the context of an undergraduate C programming course. While having a number of architectural and practical advantages, the suggested coarse-grained personalization approach deviates from the common practices toward knowledge modeling in AES. Therefore, we believe that several aspects of QuizGuide required a detailed evaluation—from modeling accuracy to the effectiveness of adaptation. The paper discusses how this new student modeling approach can be evaluated, and presents our attempts to evaluate it from multiple different prospects. The evaluation of QuizGuide across several consecutive semesters demonstrates that, although topics do not always support precise user modeling, they can provide a basis for successful personalization in AESs

Mathematical transfer by chemistry undergraduate students

This thesis reports on a study of the transfer of mathematical knowledge by undergraduate chemistry students. Transfer in this research refers to the students’ ability to use mathematical concepts, previously experienced within a mathematics course, within chemistry contexts. A pilot study was undertaken with a sample of second-year undergraduate chemistry students in order to determine their ability to transfer mathematical knowledge from a mathematics context to a chemistry context. The results showed that, while certain students could transfer (i.e., answer mathematical items correctly in a mathematics context and then in a chemistry context), many students were unable to transfer due to insufficient mathematical knowledge. These results motivated the main study, in which students’ ability to transfer mathematical concepts was investigated and analysed in two respects. These were the degree to which transfer was present, and the degree to which a particular characteristic, namely students’ ability to correctly explain their mathematical reasoning, underpinned successful transfer. It was found that students who evidenced an ability to explain their reasoning in a mathematics context associated with transfer. An intervention programme was designed which focused on the development of student understanding of mathematical concepts, both in terms of symbolic actions and linking these symbolic actions with mathematical referents/objects. This intervention programme was informed by current mathematics-educational theories. The evaluation of the intervention programme involved determining students’ mathematical understanding, their ability to transfer, and their opinions as to its usefulness. While the majority of the students found the intervention programme beneficial, students’ competency in respect of linking mathematical actions with referents/objects varied over the different concepts studied. Students’ ability to transfer also varied from one concept to another. The systematic process adopted in this study, of both determining students’ ability to transfer and the factors influencing transfer, and using this information together with mathematics-educational theories in developing intervention programmes, is applicable to transfer studies across other disciplines