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

2020 ◽  
Vol 51 (5) ◽  
pp. 574-599
Author(s):  
Per Nilsson
Keyword(s):  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Theoretical Perspective ◽  
Mathematical Discussions ◽  
Teaching Of Mathematics

This study introduces inferentialism and, particularly, the Game 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.

scholarly journals Maths concepts in teaching: Procedural and conceptual knowledge

Pythagoras ◽  
2005 ◽  
Vol 0 (62) ◽  
Author(s):  
Caroline Long
Keyword(s):  
Prospective Teachers ◽  
School Environment ◽  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Teaching Practice ◽  
Teaching Of Mathematics

In teaching a general course on mathematics for prospective teachers, I have found the theoretical distinction between conceptual knowledge and procedural knowledge (Hiebert & Lefevre, 1986) a useful focus for teaching practice. The constructs provide a scaffold for the learning of mathematics by the students and for thinking about the teaching of mathematics in the school environment. These theoretical insights uncover in part the processes for acquiring knowledge and provide a tool for addressing problematic areas of learning.

scholarly journals Advantages of discussion and cognitive conflict as an instructional method in contemporary teaching of mathematics

2012 ◽  
Vol 44 (1) ◽  
pp. 92-110
Author(s):  
Irena Misurac-Zorica ◽  
Maja Cindric
Keyword(s):  
Conceptual Knowledge ◽  
Teaching And Learning ◽  
Procedural Knowledge ◽  
Descriptive Analysis ◽  
Cognitive Conflict ◽  
Construction Of Knowledge ◽  
Teaching And Learning Mathematics ◽  
Starting Point ◽  
Teaching Of Mathematics ◽  
Decimal Numbers

Contemporary theories of teaching and learning mathematics emphasise the importance of learner?s active participation in the teaching process, in which discovery and logical reasoning lead to the construction of student?s knowledge. In this form of teaching, it is important to detect students? misunderstandings and errors that can occur during learning. Uncovered tacit and false conceptions of students? knowledge can greatly contribute to the opposite effect in the construction of knowledge. In teaching mathematics, there are many situations which leave students with ambiguities and misunderstandings, and create an impression in children that teaching of mathematics and mathematical knowledge itself is something that is not possible. Discussion and cognitive conflict are methods which have their starting point in the theory of constructivism. The aim of our study has been to determine whether application of the method of discussion and cognitive conflict in learning to divide decimal numbers leads to the enhancement of student?s procedural knowledge and conceptual knowledge about the division of decimal numbers. Longitudinally, we monitored two groups of 117 pupils of the fifth grade. In the first group, which was taught according to the guidelines of contemporary mathematics education, students engaged in discussion, discovering their misunderstandings and errors, and the cognitive conflict resulted in correct concepts. The second group of students were taught traditionally, learning the procedure and then practicing it. The paper presents a descriptive analysis of the process of teaching and quantitative analysis of the performance based on the comparison of conceptual and procedural knowledge of both groups. Results of our work show that the application of contemporary methods of discussion and cognitive conflict affects the increase of procedural and conceptual knowledge of the division of decimal numbers.

scholarly journals The Effect of Geogebra on Students’ Conceptual and Procedural Knowledge: The Case of Applications of Derivative

2017 ◽  
Vol 7 (2) ◽  
pp. 67 ◽  
Author(s):  
Mehmet Fatih Ocal
Keyword(s):  
Conceptual Knowledge ◽  
Teaching And Learning ◽  
Procedural Knowledge ◽  
Dynamic Geometry ◽  
The Other ◽  
Control Group ◽  
Other Hand ◽  
Significant Difference ◽  
Before And After ◽  
Post Test

Integrating the properties of computer algebra systems and dynamic geometry environments, Geogebra became an effective and powerful tool for teaching and learning mathematics. One of the reasons that teachers use Geogebra in mathematics classrooms is to make students learn mathematics meaningfully and conceptually. From this perspective, the purpose of this study was to investigate whether instruction with Geogebra has effect on students’ achievements regarding their conceptual and procedural knowledge on the applications of derivative subject. This study adopted the quantitative approach with pre-test post-test control group true experimental design. The participants were composed of two calculus classrooms involving 31 and 24 students, respectively. The experimental group with 31 students received instruction with Geogebra while the control group received traditional instruction in learning the applications of derivative. Independent samples t-test was used in the analysis of the data gathered from students’ responses to Applications of Derivative Test which was subjected to them before and after teaching processes. The findings indicated that instruction with Geogebra had positive effect on students’ scores regarding conceptual knowledge and their overall scores. On the other hand, there was no significant difference between experimental and control group students’ scores regarding procedural knowledge. It could be concluded that students in both groups were focused on procedural knowledge to be successful in learning calculus subjects including applications of derivative in both groups. On the other hand, instruction with Geogebra supported students’ learning these subjects meaningfully and conceptually.

scholarly journals Three levels of naturalistic knowledge

2019 ◽  
Author(s):  
Andreas Stephens
Keyword(s):  
Semantic Memory ◽  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Dual Process ◽  
Long Term Memory ◽  
Cognitive Ethology ◽  
System 2 ◽  
System 1 ◽  
Level Of Analysis

A recent naturalistic epistemological account suggests that there are three nested basic forms of knowledge: procedural knowledge-how, conceptual knowledge-what, and propositional knowledge-that. These three knowledge-forms are grounded in cognitive neuroscience and are mapped to procedural, semantic, and episodic long-term memory respectively. This article investigates and integrates the neuroscientifically grounded account with knowledge-accounts from cognitive ethology and cognitive psychology. It is found that procedural and semantic memory, on a neuroscientific level of analysis, matches an ethological reliabilist account. This formation also matches System 1 from dual process theory on a psychological level, whereas the addition of episodic memory, on the neuroscientific level of analysis, can account for System 2 on the psychological level. It is furthermore argued that semantic memory (conceptual knowledge-what) and the cognitive ability of categorization are linked to each other, and that they can be fruitfully modeled within a conceptual spaces framework.

Why Is the Midpoint an Average?

2013 ◽  
Vol 106 (7) ◽  
pp. 514-519 ◽  
Author(s):  
Lingguo Bu
Keyword(s):  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Learning Mathematics ◽  
Mathematics For Teaching ◽  
The Relationship

The relationship between a midpoint and an average showcases the interplay between procedural knowledge and conceptual knowledge in learning mathematics for teaching.

Knowledge Acquisition in Computer-Based Learning Environments among Elementary Children

1997 ◽  
Vol 26 (1) ◽  
pp. 19-25 ◽  
Author(s):  
Andrew S. Chirwa
Keyword(s):  
Learning Environments ◽  
Learning Strategies ◽  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Learning Strategy ◽  
High Order ◽  
Factual Knowledge ◽  
Computer Based Learning ◽  
Computer Based ◽  
Computer Based Learning Environments

The need to understand how children acquire knowledge in computer-based learning environments led the researcher to undertake this study. The purpose was to develop a conceptualization of what learning strategies children frequently use to process conceptually demanding material. The goal was to expose children to different categories of courseware that featured multimedia, drill and practice, simulations, tutorials, spreadsheets, and databases; and to determine learning strategies including elaboration, organization, integration, and recall. The object was to compare the types of learning strategy and nature of knowledge forms acquired during the process of learning the given material in a subject area. The study was conducted at Washington Elementary School; and participants were children in the third through sixth grades. Data was collected by using surveys, formal observations, and formative and summative evaluation procedures. Results show that 80 percent of the time the students had attention focused on the learning material and gained an elevated level of awareness. The learning strategies imagery, exemplifying, and networking were used 70 percent of the time as means to gain conceptual knowledge, factual knowledge, procedural knowledge, and develop high order thinking. The learning strategies covert practice, overt practice, and identifying key ideas were used 60 percent of the time to gain conceptual knowledge, factual knowledge, procedural knowledge, and rules in the subject areas. The learning strategy categorization was used 40 percent of the time as means to gain conceptual knowledge, factual knowledge, procedural knowledge, and rules. The learning strategies sentence elaboration and anticipation were used 30 percent of the time to gain conceptual knowledge, factual knowledge, procedural knowledge, rules, high-order rules, and develop high order thinking. These findings have implications to learning and knowledge acquisition in computer-based learning environments, instructional design, program development and improvement, and technology and teacher education.

scholarly journals THE EFFECT ON STUDENTS’ ARITHMETIC SKILLS OF TEACHING TWO DIFFERENTLY STRUCTURED CALCULATION METHODS

2020 ◽  
Vol 78 (2) ◽  
pp. 167-195
Author(s):  
Margareta Engvall ◽  
Joakim Samuelsson ◽  
Rickard Östergren
Keyword(s):  
Mathematics Teaching ◽  
Conceptual Knowledge ◽  
Decomposition Method ◽  
Intervention Study ◽  
Procedural Knowledge ◽  
Empirical Support ◽  
Factual Knowledge ◽  
Calculation Methods ◽  
Traditional Algorithm ◽  
Development Of Students

Mastering traditional algorithms has formed mathematics teaching in primary education. Educational reforms have emphasized variation and creativity in teaching and using computational strategies. These changes have recently been criticized for lack of empirical support. This research examines the effect of teaching two differently structured written calculation methods on teaching arithmetic skills (addition) in grade 2 in Sweden with respect to students’ procedural, conceptual and factual knowledge. A total of 390 students (188 females, 179 males, gender not indicated for 23) were included. The students attended 20 classes in grade 2 and were randomly assigned to one of two methods. During the intervention, students who were taught and had practiced traditional algorithms developed their arithmetic skills significantly more than students who worked with the decomposition method with respect to procedural knowledge and factual knowledge. These results provided no evidence that the development of students' conceptual knowledge would benefit more from learning the decomposition method compared to traditional algorithm. Keywords: arithmetic skills, decomposition method, intervention study, mathematics education, traditional algorithm, written calculation.

scholarly journals Conceptual Knowledge OR Procedural Knowledge or Conceptual Knowledge AND Procedural Knowledge: Why the Conjunction is Important to Teachers

2021 ◽  
Vol 46 (2) ◽  
pp. 57-71
Author(s):  
Derek Hurrell ◽  
Keyword(s):  
Mathematics Teaching ◽  
Conceptual Knowledge ◽  
Teacher Educators ◽  
Procedural Knowledge ◽  
Insight Into

The terms conceptual knowledge and procedural knowledge are often used by teachers and never more so than when discussing how teachers teach, and children learn mathematics. This paper will look at literature regarding conceptual and procedural knowledge and their place in the classroom, to offer teachers and teacher educators’ advice on some of the more pressing issues and understandings around them. A thorough synthesis of extant and seminal literature will provide advice to teachers and teacher educators on how a deeper insight into conceptual and procedural knowledge could improve the quality of mathematics teaching.

scholarly journals Improving the Conceptual and Procedural Knowledge of Prospective Teachers through Multisensory Approach: Experience from Indonesia

2018 ◽  
Vol 3 (2) ◽  
pp. 106 ◽  
Author(s):  
Yurniwati Yurniwati
Keyword(s):  
Prospective Teachers ◽  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Research Study ◽  
Learning Activities ◽  
State University ◽  
Mathematics Problems ◽  
Hands On ◽  
Mathematics Understanding ◽  
Concrete Materials

Abstract. In mathematics, there is conceptual and procedural knowledge. Conceptual knowledge is about ideas or mathematics understanding but procedural knowledge is about procedure to solve mathematics problems. Multisensory approach involve many senses like kinaesthetic,  visual and auditory to gain knowledge. This research aims to find information about how to apply multisensory approach to improve conceptual and procedural knowledge of prospective teacher in Jakarta State University. This action research study used Kemmis and Taggart model and implemented in two cycles. The data were collected through questionnaires and observation sheets. Then, the data was analyzed descriptively.  The research results showed that the multisensory approach can enhance the conceptual and procedural knowledge of the prospective teachers. The Kinaesthetic approach was implemented in hands-on activity using concrete materials while the visual using images. The concrete materials and image provide different presentation but it helped to constructed concepts and abstraction. Furthermore, the auditory approach was developed along learning activities trough discussion to produce and clarify the ideas. Keywords: Conceptual knowledge, Procedural knowledge, Multisensory approach  

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