Knowledge Of Mathematical Concepts Research Articles

Junior high school students’ mathematical creative thinking ability based on gender differences in plane and solid geometry subjects

Mathematical creative thinking is the ability of students to find solving ways that are unusual, unique, and have never been found by others. The development of the concept of mathematical creative thinking can make students despair and give up, so that another ability is needed so that they are able to survive and turn their difficulties to be opportunities for themselves to advance. This study aims to determine the mathematical creative thinking ability of class VIII junior high school students in one of the junior high schools in Bandung Barat on the subject to construct plane and solid geometry based on gender differences. The samples are 4 students consisting of two men and two women. Besides that, the discussion of mathematical creative thinking skills for junior high school students is viewed from gender differences, between male and female students in terms of their creative thinking abilities. The instrument of this research is a test of the ability to think creatively and interview. This study is a descriptive qualitative research that aims to describe the ability of creative thinking mathematically in solving problems of plane and solid geometry. The results show that the ability difference between male and female students is that the female students have much better mathematical creative thinking. In addition to gender factors, other factors can also influence the achievement of creative thinking skills; namely, students and teachers’ readiness to learn, students’ basic abilities and knowledge of mathematical concepts. Male students are more detailed in explaining the shape of each space given to the problem.

Preparing pre-service mathematics teachers for STEM education: an analysis of lesson proposals

Based on an investigation of research literature on the role of mathematics within the context of STEM education and twenty-first century skills, this explorative study presents and analyses 19 pre-service teachers’ lesson proposals for innovative STEM activities in secondary mathematics classrooms. Drawing on a categorisation of twenty-first century skills and key concepts from Realistic Mathematics Education and Basil Bernstein’s writings, the analysis indicated a use of detailed instructions regarding what students should do but unspecified criteria for their expected knowledge productions. Rather than serving as a digital tool for problem solving, programming was used for the purpose of generalising students’ conceptual knowledge in mathematics. At the same time a focus on modelling and applications in the STEM activities tended to weaken the disciplinary character of mathematics. The learning of various twenty-first century skills was promoted to different extents. The study raises a discussion of affordances and constraints regarding students’ access to mathematical knowledge through different modes of integration of mathematics and other STEM subjects aiming to support the development of twenty-first century skills.

A Framework for Investigating Qualities of Procedural and Conceptual Knowledge in Mathematics—An Inferentialist Perspective

This study introduces inferentialism and, particularly, theGame of Giving and Asking for Reasons(GoGAR), as a new theoretical perspective for investigating qualities of procedural and conceptual knowledge in mathematics. The study develops a framework in which procedural knowledge and conceptual knowledge are connected to limited and rich qualities of GoGARs. General characteristics of limited GoGARs are their atomistic, implicit, and noninferential nature, as opposed to rich GoGARs, which are holistic, explicit, and inferential. The mathematical discussions of a Grade 6 class serve the case to show how the framework of procedural and conceptual GoGARs can be used to give an account of qualitative differences in procedural and conceptual knowledge in the teaching of mathematics.

Compute mindlessly. Not! Map consciously

This paper utilizes concept mapping as a tool for conscious and deliberate knowledge building in mathematics and its extension to algorithms. Currently, alleged defects in mathematics education are obvious: instead of conceptual elaboration, everyday praxis relies on routine computations that are likely to lead into alienated concepts with weak connections to prior knowledge. A concept map visualizes the existing conceptual structure, and whenever new information is brought in, it will be placed in the map by clearly explicating its linkage to the previous concepts. In the Finnish mathematics education, such new knowledge is programming content that is integrated into elementary school mathematics in 2014 Finnish National Curriculum. This content is crystallized as the requirements of computational and algorithmic thinking, the utilization of respective data structures, and adequate amount of hands-on practice to internalize good coding conventions. This study examines secondary (N = 19) and higher education students (N = 10) and their conceptual knowledge of mathematics concentrating on the domain of algorithms in particular. The concept maps drawn by the students are evaluated using the SOLO taxonomy. To conclude, a consensus map of algorithms is represented and linked to the elementary mathematics syllabus.

Pedagogical Content Knowledge of Mathematics Pre-service Teachers: Do they know their students?

The aim of this paper is to describe pre-service mathematics teachers’ knowledge about students. In this paper, knowledge about students refers to teachers’ knowledge of mathematical concepts that are difficult for students, common misconceptions that frequently experienced by students, possible sources of students’ errors, and methods to eliminate the difficulties and misconceptions. Twenty-four preservice mathematics teachers enrolled in micro-teaching course participated in the study. Interviews and written documents were employed to collect data. The data indicated that preservice teachers had problem in recognizing the source of students’ misconceptions and error. Pre-service teachers also found difficulty in finding effective ways to minimize those misconceptions and error. Interview data revealed that pre-service teachers frequently failed to recognize students’ misconception due to lack of conceptual knowledge of mathematics. Moreover, pre-service teachers had issues in generating effective teaching strategies as they had limited knowledge of teaching methods. The results suggest that mathematics pre-service teachers should have sufficient knowledge of both mathematics knowledge and pedagogic knowledge so that they can help their students in learning mathematics effectively.

Using argumentation as a learning strategy to improve student performance in engineering Statics

ABSTRACTResearch suggests that a significant reason that a large number of students earn low grades in the fundamental engineering science course Statics is that they may be entering the course with incorrect conceptual knowledge of mathematics and physics. The self-explanation learning approach called collective argumentation helps k-12 students to understand their misconceptions of mathematical principles that often appear abstract to them. This study investigated collective argumentation as an instructional approach that helps engineering students identify and correct their misconceptions of topics taught in Statics. Results suggest that argumentation improves student performance as measured by grades earned on semester exams. Survey and focus group results suggest that students did not understand the argumentation process. Therefore, the students did not like using it as a learning approach.

The Conceptual Mathematical Knowledge and Analytical Thinking for the First Stage Students at Math Sciences Department, Faculty of Education for Pure Sciences, IBN Alhaithem, University of Baghdad

The Conceptual Mathematical Knowledge and Analytical Thinking for the First Stage Students at Math Sciences Department, Faculty of Education for Pure Sciences, IBN Alhaithem, University of Baghdad

Preservice and inservice teachers' knowledge, beliefs, and instructional planning in primary school mathematics

We studied relationships between mathematical knowledge, beliefs about teaching and learning, and instructional planning. We contrasted beliefs, mathematical knowledge, and skills of preservice teachers (n = 47) with those of inservice teachers (n = 31) at the primary school level. For both groups, participants were generally more constructivist than traditional in beliefs, and showed evidence of low conceptual knowledge in mathematics. Multiple regression analyses indicated different relationships between conceptual knowledge and constructivist beliefs for inservice versus preservice teachers. Conceptual knowledge played a strong role in instructional planning in mathematics. Implications for future research and teacher education are discussed based on these findings.

Serial-order learning impairment and hypersensitivity-to-interference in dyscalculia

In the context of heterogeneity, the different profiles of dyscalculia are still hypothetical. This study aims to link features of mathematical difficulties to certain potential etiologies. First, we wanted to test the hypothesis of a serial-order learning deficit in adults with dyscalculia. For this purpose we used a Hebb repetition learning task. Second, we wanted to explore a recent hypothesis according to which hypersensitivity-to-interference hampers the storage of arithmetic facts and leads to a particular profile of dyscalculia. We therefore used interfering and non-interfering repeated sequences in the Hebb paradigm. A final test was used to assess the memory trace of the non-interfering sequence and the capacity to manipulate it. In line with our predictions, we observed that people with dyscalculia who show good conceptual knowledge in mathematics but impaired arithmetic fluency suffer from increased sensitivity-to-interference compared to controls. Secondly, people with dyscalculia who show a deficit in a global mathematical test suffer from a serial-order learning deficit characterized by a slow learning and a quick degradation of the memory trace of the repeated sequence. A serial-order learning impairment could be one of the explanations for a basic numerical deficit, since it is necessary for the number-word sequence acquisition. Among the different profiles of dyscalculia, this study provides new evidence and refinement for two particular profiles.

Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics

There is a long-standing and ongoing debate about the relations between conceptual and procedural knowledge (i.e., knowledge of concepts and procedures). Although there is broad consensus that conceptual knowledge supports procedural knowledge, there is controversy over whether procedural knowledge supports conceptual knowledge and how instruction on the two types of knowledge should be sequenced. A review of the empirical evidence for mathematics learning indicates that procedural knowledge supports conceptual knowledge, as well as vice versa, and thus that the relations between the two types of knowledge are bidirectional. However, alternative orderings of instruction on concepts and procedures have rarely been compared, with limited empirical support for one ordering of instruction over another. We consider possible reasons for why mathematics education researchers often believe that a conceptual-to-procedural ordering of instruction is optimal and why so little research has evaluated this claim. Future empirical research on the effectiveness of different ways to sequence instruction on concepts and procedures is greatly needed.

Heterogeneous Peer-Tutoring: An Intervention That Fosters Collaborations and Empowers Learners

IntroductionIn this scenario, Nick, an accelerated algebra student who is taking algebra a year early, was the assigned tutor and Heather, a pre-algebra student, was the intended tutee. They were among the 14 students who participated in a Heterogeneous Peer Tutoring (HPT) program designed by the classroom mathematics teacher.Thirty minutes remain until the end of the school day. Fourteen eighth grade students, in an academic assistance period known as advisory, are seated in pairs dispersed throughout a classroom. The students are working on a problem set of classifying linear and nonlinear functions. The classroom teacher, Jake, is circulating the room monitoring students' progress. The teacher frequently questions the students to gauge their understanding of linear and nonlinear functions. While some students are able to explain the key characteristics of linear and nonlinear functions, many others are unable to do so. Two students, Nick and Heather (pseudonyms), are working on a problem that requires them to determine if a given function is linear or nonlinear. Nick is having trouble comprehending this topic and seeks Heather's help. Collaboratively, they discuss possible solution processes to the problem. A few minutes before the end of the advisory, Jake asks Nick, Do you know which of these functions are nonlinear and which are linear? Nick correctly identifies the functions. Jake poses a follow up question: How can you tell the difference between the two, especially with these problems where there is no graph provided? Calm and collected, Nick explains that the functions containing variables with exponents not equal to one are nonlinear. Nick extends his response by referring to a graph, clearly identifying the piece that makes the function non-liner. The teacher is impressed. Nick's partner Heather leans forward and says, He learned that from me, beaming with pride. Nick happily acknowledges.The goal of the program was to help pre-algebra students (tutees) deepen their knowledge of mathematical concepts by providing them an opportunity to collaborate with their peers-accelerated algebra students (tutors). The above snippet provides one instance of students' learning process. Evident also in this example is a role reversal between a tutor and a tutee. Over the duration of the program, the tutees and tutors gained a mutual respect for one another as they realized that they could both learn from each other's distinctive learning experiences. In this paper, we outline key features of an HPT program that was implemented with a group of middle school students, and we describe how this program impacted both the tutors' and the tutees' learning experiences. In so doing, we outline how this model could be used to enact some of the essential attributes of a middle level education program as outlined in the position paper of the Association for Middle Level Education (formerly National Middle School Association), This We Believe: Keys to Educating Young Adolescents (NMSA, 2010).Peer tutoring: An overviewPeer tutoring is a strategy in which, People from similar social groupings who are not professional teachers [help] each other to learn and [learn] themselves by teaching (Topping, 1996, p. 322). There are many forms of peer tutoring such as class-wide peer tutoring, homogeneous peer tutoring, and heterogeneous peer tutoring. Class-wide peer tutoring enables students in a class to tutor one another (Allsopp, 1997). In a homogeneous tutoring environment tutees and tutors are of the same ability level. In contrast, in a heterogeneous tutoring environment, tutees are taught by tutors of a higher ability level (Stenhoff 8c Lignugaris/Kraft, 2007). Many teachers have used an HPT technique as an intervention strategy to enhance student learning in reading, science, and mathematics (Robinson, Schofield, 8c SteersWentzell, 2005). In an academic setting, the use of peer tutoring in heterogeneous classrooms allows students of different learning backgrounds to collaborate and learn together, which often helps remove stigmas associated with receiving tutoring (Allsopp, 1997). …

Adolescents' understanding of inversion and associativity

The present study investigated when an understanding of the mathematical concepts of inversion and associativity matures and whether the application of these concepts to problem solving requires the interruption of computational strategies (e.g., Siegler & Araya, 2005). In the study, 40 adolescent participants per grade from Grades 7, 9, and 11 and 40 adult participants solved multiplication and division inversion and associativity problems and completed a task that measured whether the execution of the inversion shortcut or associativity strategy prevents the execution of competing computational strategies. Inversion shortcut use approached adult levels in Grade 9. Associativity strategy use significantly increase in early adulthood. Also, there was considerable individual variability in strategy use. Finally, the execution of both conceptually-based strategies interrupted computational strategies. Thus, adolescence is an important developmental period for understanding multiplicative concepts and applying conceptual mathematical knowledge to problem solving may require the interruption of procedural knowledge.

Taking Math Students From “Blah” to “Aha”: What can we do?

In many post-secondary, introductory mathematics courses failure and withdrawal rates are reaching as high as 50% and average GPA is steadily decreasing. This is a problem that has been witnessed across the globe. With widespread reforms taking place in K-12 mathematics education, many innovative teaching strategies have been created, implemented, and analyzed in an attempt to increase the conceptual knowledge of mathematics students at all levels. However, most of these strategies are not appropriate for large, lecture-style classrooms as are usually seen in large universities. This paper outlines some teaching and assessment strategies that have been modified to accommodate the needs and restrictions of such classrooms.

Preservice Elementary Teachers’ Perspectives about the Roles of Conceptual Understanding and Factual/Procedural Knowledge in Learning and Teaching Mathematics and Science

Preservice teachers often enter teacher education programs with naive views about mathematics and science as well as the various domains of knowledge needed for effective teaching. A significant body of research has demonstrated that teacher knowledge is multifaceted and consists of both subject matter knowledge and pedagogical content knowledge. In this study, a cohort of preservice elementary teachers participated in an activity involving two components of subject matter knowledge (factual/procedural and conceptual). Analysis compares the preservice elementary teachers’ confidence in their understanding of and ability to teach factual/procedural and conceptual knowledge in mathematics and science. Qualitative analysis of a discussion following the activity provides insights into the quantitative findings.

The power of professional learning communities

Abstract In this paper I share some key principles and examples from the Data Informed Practice Improvement Project. The project develops an innovative model of mathematics teacher development, with three main strands: teachers work with data from their classrooms, they use this data to understand and engage with learner errors in mathematics, and they do this collectively in professional learning communities, with facilitation from members of the project team. I describe each of these strands and present three examples that show how the strands work together to support teacher learning in professional communities. These examples illuminate a key achievement of the project: a focus on learner errors in professional learning communities can develop powerful conceptual knowledge of mathematics among teachers at the same time as developing teachers’ knowledge for teaching and teaching practices.

Articulating syntactic and numeric perspectives on equivalence: the case of rational expressions

Our study concerns the conceptual mathematical knowledge that emerges during the resolution of tasks on the equivalence of polynomial and rational algebraic expressions, by using CAS and paper-and-pencil techniques. The theoretical framework we adopt is the Anthropological Theory of Didactics (Chevallard 19:221–266, 1999), in combination with semiotic aspects from the instrumental approach to tool use. The analysis we present is based on interviews carried out with a 10th grade student who participated in our research. Our findings highlight the mathematical knowledge (technological discourse) constructed in the process of confronting, differentiating, and articulating the several mathematical techniques and theoretical ideas (pertaining to the numeric perspective and the syntactic perspective on algebraic equivalence) related to the designed equivalence tasks.

Preservice Secondary Mathematics Teachers’ Knowledge of Students

The aim of this paper is to present the nature of preservice secondary mathematics teachers’ knowledge of students as emerged from a study investigating the development of their pedagogical content knowledge in a methods course and its associated field experience. Six preservice teachers participated in the study and the data were collected in the forms of observations, interviews and written documents. Knowledge of students is defined as teachers’ knowledge of what mathematical concepts are difficult for students to grasp, which concepts students typically have misconceptions about, possible sources of students’ errors, and how to eliminate those difficulties and misconceptions. The findings revealed that preservice teachers had difficulty in both identifying the source of students’ misconceptions, and errors and generating effective ways different than telling the rules or procedures to eliminate such misconceptions. Furthermore, preservice teachers’ knowledge of students was intertwined with their knowledge of subject matter and knowledge of pedagogy. They neither had strong conceptual knowledge of mathematics nor rich repertoire of teaching strategies. Therefore, they frequently failed to recognize what conceptual knowledge the students were lacking and inclined to address students’ errors by telling how to carry out the procedure or apply the rule to solve the given problem correctly

Accounting for individual variability in inversion shortcut use

This study investigated whether children's inversion shortcut use (i.e., reasoning that no calculations are required for the problem 4 × 8 ÷ 8, as the answer is the first number) is related to their analogical reasoning ability, short-term memory capacity, and working memory capacity. Children from Grades 6 and 8 solved multiplication and division inversion problems and classical analogy word problems and completed memory tasks. Analogical reasoning ability and working memory functioning both accounted for individual variance in inversion shortcut use. These findings suggest that the ability to understand relationships and executive functioning may enable children to internally represent and manipulate mathematical problems, facilitating the application of conceptual mathematical knowledge to generate the inversion shortcut.

Conceptual knowledge and mathematics achievement of matriculation students

The purpose of this study was to determine matriculation students’ conceptual knowledge of mathematics and its relationship to their mathematics achievement. This study involved 250 students from a matriculation college in northern Malaysia. The instrument used was a conceptual knowledge test of sequences and series. Data were analyzed by descriptive and inferential statistics using mean, standard deviation, t-test, and correlation. The findings revealed that conceptual knowledge of accounting students was low when compared to that of physical and life science students. The results of the t-test indicated that there was a significant mean score difference in conceptual knowledge between science and accounting students. Science students had a significantly higher mean score than accounting students did. In addition, a relationship was found to exist between conceptual knowledge and achievement in mathematics.

The Effects of Problem Based Learning on Mathematics Performance and Affective Attributes in Learning Statistics at Form Four Secondary Level

The challenge of globalisation today requires students to acquire problem solving and communication skills besides good procedural and conceptual knowledge of mathematics. This study was conducted to explore the effects of Problem Based Learning (PBL) as an alternative instructional strategy in the teaching and learning of mathematics effectively in Malaysian secondary schools. Previous research had discovered that that PBL enhances student's team work, problem solving skills and communication skills. In addition, interest in the subject significantly improved. A quasi experimental study with non-equivalent control group posttest only design was conducted to investigate the effects of PBL on form four Malaysian student's mathematics performance, instructional efficiency and affective attributes. The experiment was carried out for six weeks involving 53 Form Four students randomly selected from the district of Port Dickson. The experimental (PBL) group (n=29) were exposed to the PBL instruction whereas the control (CT) group (n=24) were taught conventionally. There were five instruments used in this study namely, mathematical learning worksheets, a posttest, Paas Mental Effort Rating Scale, a questionnaire on perception towards group work, interest in mathematics and perception towards mathematics learning experience and a rubric evaluating student's effective use of Polya's problem solving procedures, mathematical communication and teamwork. The data were analysed using analysis of covariance (ANCOVA) and independent t-test. student's response to the questionnaire and researcher's observations based on the rubric were described using means, standard deviations and percentages. The findings of the study indicated that PBL is just as efficient as the conventional teaching strategy in enhancing Form Four student's mathematics performance. Even though both groups of students showed positive perception towards group work, interest in mathematics and perception towards the learning experience they went through, the PBL group used the Polya's problem solving procedures more effectively, displayed better mathematical communication skills and showed stronger teamwork compared to the CT group.