Conceptual Knowledge, Procedural Knowledge, and Metacognition in Routine and Nonroutine Problem Solving

2021 ◽  
Vol 45 (10) ◽  
Author(s):  
David W. Braithwaite ◽  
Lauren Sprague
Keyword(s):  
Problem Solving ◽  
Conceptual Knowledge ◽  
Procedural Knowledge

Teaching Toward the Big Ideas of Algebra

2000 ◽  
Vol 6 (4) ◽  
pp. 226-231
Author(s):  
Sonia Woodbury
Keyword(s):  
Problem Solving ◽  
Conceptual Knowledge ◽  
Middle Grades ◽  
Procedural Knowledge ◽  
Conceptual Structure ◽  
Algebraic Manipulation ◽  
Relational Understanding ◽  
The Past ◽  
Starting Point ◽  
Big Ideas

IN WHAT WAYS DO WE WANT MIDDLE-GRADES STUDENTS TO UNDERSTAND ALGEBRA? Hiebert and Carpenter (1992) describe the need for students to gain both procedural knowledge and broadly connected conceptual knowledge to understand mathematics. A knowledge of rules and procedures provides students with tools for efficient problem solving. However, in learning the procedures of algebraic manipulation, for example, students often develop what Skemp (1978) calls an “instrumental understanding” of algebra. He explains, “It is what I have in the past described as ‘rules without reasons,’ without realizing that for many pupils… the possession of such a rule, and the ability to use it, was what they meant by ‘understanding’ ” (p. 9). Skemp contrasts instrumental understanding with “relational understanding,” which “consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point” (p. 14).

scholarly journals THE TAXONOMY OF THE QUESTIONS IN THE ITALIAN NATIONAL ASSESSMENT OF KNOWLEDGE OF MATHEMATICS

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Daniel Doz
Keyword(s):  
Problem Solving ◽  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
National Assessment ◽  
Language Of Instruction ◽  
Taxonomic Level ◽  
The Past ◽  
Closed Type ◽  
Level Problem ◽  
Official Data

In Italy, all grade 10 students are required to take the national assessment of knowledge of mathematics, which is prepared by the INVALSI Institute. No official data about any taxonomic level of the questions in these assessments has been published on the Institute’s website. In the present work, we analyzed seven INVALSI examinations for school with Slovene as language of instruction and we focused on the Gagne’s taxonomic level of each individual question in the assessments. The most frequent category in national assessments is “Routine procedural knowledge”, followed by “Basic and conceptual knowledge”. We found that, even though the interest in problem-solving activities has increased in the past years, the taxonomic level “Problem-solving knowledge” is the less frequent. Moreover, we wanted to analyze the distribution of the different taxonomic levels among the question typologies (open- and closed-type questions) and we found that questions from lower taxonomic levels are more likely to be closed-type, while “Problem-solving questions” are more likely to be open-type. Furthermore, we were interested in analyzing the distribution of taxonomic levels among the four topic dimensions “Geometry”, “Data and prevision”, “Numbers and quantities” and “Relations and functions”. We found that the taxonomic level “Problem-solving knowledge” are more present in the categories “Number and quantities” and “Relations and functions”.

scholarly journals Thinking Process of Naive Problem Solvers to Solve Mathematical Problems

2016 ◽  
Vol 10 (1) ◽  
pp. 1 ◽  
Author(s):  
Jackson Pasini Mairing
Keyword(s):  
Problem Solving ◽  
Procedural Knowledge ◽  
Mental Image ◽  
Research Subjects ◽  
Maximum Score ◽  
Mathematical Problems ◽  
Problem Solving Strategies ◽  
Thinking Process ◽  
Problem Solvers ◽  
Holistic Rubric

Solving problem is not only a goal of mathematical learning. Students acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations by learning to solve problems. In fact, there were students who had difficulty in solving problems. The students were naive problem solvers. This research aimed to describe the thinking process of naive problem solvers based on heuristic of Polya. The researcher gave two problems to students at grade XI from one of high schools in Palangka Raya, Indonesia. The research subjects were two students with problem solving scores of 0 or 1 for both problems (naive problem solvers). The score was determined by using a holistic rubric with maximum score of 4. Each subject was interviewed by the researcher separately based on the subject’s solution. The results showed that the naive problem solvers read the problems for several times in order to understand them. The naive problem solvers could determine the known and the unknown if they were written in the problems. However, they faced difficulties when the information in the problems should be processed in their mindsto construct a mental image. The naive problem solvers were also failed to make an appropriate plan because they did not have a problem solving schema. The schema was constructed by the understanding of the problems, conceptual and procedural knowledge of the relevant concepts, knowledge of problem solving strategies, and previous experiences in solving isomorphic problems.

scholarly journals The tip of the iceberg in organic chemistry classes: how do students deal with the invisible?

2015 ◽  
Vol 16 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Nicole Graulich
Keyword(s):  
Organic Chemistry ◽  
Problem Solving ◽  
Cognitive Skills ◽  
Conceptual Knowledge ◽  
Chemistry Education ◽  
Scholarly Work ◽  
Related Research ◽  
Research Areas ◽  
Research Activities ◽  
Problem Solving Behavior

Organic chemistry education is one of the youngest research areas among all chemistry related research efforts, and its published scholarly work has become vibrant and diverse over the last 15 years. Research on problem-solving behavior, students' use of the arrow-pushing formalism, the investigation of students' conceptual knowledge and their cognitive skills have shaped our understanding of college students' understanding in organic chemistry classes. This review provides an overview of research efforts focusing on student's perspectives and summarizes the main results and pending questions that may guide subsequent research activities.

Diagramming and Algebraic Word Problem Solving for Secondary Students With Learning Disabilities

2018 ◽  
Vol 54 (4) ◽  
pp. 212-218
Author(s):  
Gloria A. Carcoba Falomir
Keyword(s):  
High School ◽  
Learning Disabilities ◽  
Problem Solving ◽  
Secondary Students ◽  
Word Problem ◽  
Conceptual Knowledge ◽  
The United States ◽  
Students With Learning Disabilities ◽  
Word Problem Solving ◽  
Problem Solving Skills

Algebra is considered an important high school course because it is recognized as the gateway to higher mathematics, college opportunities, and well-paying jobs. In the United States, most secondary schools require students to be proficient in algebra to be able to graduate from high school. One major component of algebra is word problem solving, which is used in algebra courses to teach students mathematical modeling and applied problem-solving skills. However, word problem solving is often a significantly challenging area for students with learning disabilities because it involves computing mathematical equations and implementing a myriad of cognitive processes that require conceptual knowledge. Diagrams are considered an effective and powerful visualization strategy because they help students see the hidden mathematical structure of the problem. The use of diagrams is recommended as students work toward more complex math concepts in middle school and high school.

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.

scholarly journals Proses Berpikir Mahasiswa dalam Memecahkan Masalah Fungsi Pembangkit

2018 ◽  
Vol 3 (1) ◽  
pp. 29-39
Author(s):  
Ninik Mutianingsih ◽  
Lydia Lia Prayitno ◽  
Agus Prasetyo Kurniawan
Keyword(s):  
Problem Solving ◽  
Generating Function ◽  
Procedural Knowledge ◽  
Think Aloud ◽  
Thinking Process ◽  
Low Ability

This study aims to describe students’ thinking process in solving the problems of generating function. This is a case study that classifies students into three categories: high, medium, and low. Subjects were asked to solve the problems then use think aloud to reveal their thinking process. The results show that in understanding the problem, using conceptual and procedural knowledge related to the problems, strategies, and experiences they have in solving similar problems between the three subjects is different. Subjects with high and medium capability are able to reveal the problem-solving component of Polya in detail, whereas low-ability subjects use only some of the problem-solving components of Polya.

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