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”.

Encouraging Mathematical Thinking

1997 ◽  
Vol 3 (1) ◽  
pp. 66-72
Author(s):  
Julianne C. Turner ◽  
Karen Rossman Styers ◽  
Debra G. Daggs
Keyword(s):  
Problem Solving ◽  
Mathematics Instruction ◽  
Middle Grades ◽  
Mathematical Thinking ◽  
Challenging Problem ◽  
The Past ◽  
Past Experiences ◽  
Constructing Meaning

With these words, the NCTM (1989, 65) portrays a dilemma familiar to many middle-grades teachers. Although many teachers strive to involve their students in active and challenging problem-solving activities, students' past experiences may have instilled preconceptions that mathematics is mechanical, uninteresting, or unattainable. In addition, many teachers lack models and examples of how to design mathematics instruction so that it fosters students' engagement. Because the middle grades are crucial years for developing students' future interest in mathematics, middle-grades teachers must take seriously the challenge of presenting mathematics as an exciting discipline that is relevant and accessible to all students. For the past two year, we have been experimenting with approaches that will inte rest students in challenging mathematics while supporting them in constructing meaning.

Mathematics Teachers' Conceptual Knowledge of Algebra: A Literature Review/ Pengetahuan Konseptual Algebra Guru Matematik: Satu Kajian Literatur

2019 ◽  
Vol 12 (1) ◽  
Author(s):  
Nor Adibah Abdullah
Keyword(s):  
Literature Review ◽  
Mathematics Teachers ◽  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Secondary School Teachers ◽  
Past Research ◽  
Knowledge Level ◽  
School Teachers ◽  
The Past ◽  
Research Findings

Conceptual knowledge is one of the mathematical knowledge areas which should be mastered by mathematics teachers besides factual, procedural, and metacognitive knowledge. Teachers are prone to cast conceptual knowledge aside and prioritise procedural knowledge without understanding the concepts which influence the options of mathematical strategies and models in the context of problem solving. A literature review was conducted to discuss past research findings in identifying secondary school teachers' conceptual knowledge in the algebra topic. 68 past research articles were referred to and chosen based on the research purpose which was to investigate mathematics teachers' conceptual knowledge level and its implementation in several mathematical topics quantitatively and qualitatively in local and international contexts.  There have been a lot of researches conducted on teachers' conceptual knowledge in several mathematical topics, however there is still a lack of research in investigating teachers' conceptual knowledge on the algebra topic. Most of the past research findings showed that the teachers' conceptual knowledge level was at a low level in several mathematical  topics therefore a similar research is also needed in the algebra topic. This study can be extended to a more improved research in spite of the research sample, design or research methodology, instruments, and other factors. 

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.

Innovation in Curriculum: Fostering Student Discourse: Don't Ask Me! I'm Just the Teacher!

2001 ◽  
Vol 7 (4) ◽  
pp. 218-221
Author(s):  
Jeffrey A. Frykholm ◽  
Mary E. Pittman
Keyword(s):  
Critical Thinking ◽  
Problem Solving ◽  
Middle Grades ◽  
Learning Theories ◽  
Significant Shift ◽  
Mathematical Communication ◽  
The Past ◽  
Mathematical Connections ◽  
Teachers And Students ◽  
Mathematics Curricula

Throughout the past several years, middle-grades mathematics curricula have undergone a significant shift. Recently developed curriculum programs based on both recommendations of the NCTM and contemporary learning theories now emphasize problem solving, critical thinking, mathematical connections, and mathematical communication in ways that they did not before. As these powerful curriculum programs continue to find a stronghold in our middle schools, new implications and roles for both teachers and students are becoming clear.

Holism, Chinese Medicine and Systems Ideologies: Rewriting the Past to Imagine the Future

2018 ◽  
Author(s):  
Volker Scheid
Keyword(s):  
Eighteenth Century ◽  
Chinese Medicine ◽  
Medical Humanities ◽  
Science And Technology Studies ◽  
Human Beings ◽  
The Past ◽  
Starting Point ◽  
The Us ◽  
Shared Objective ◽  
Definition Of

This chapter explores the articulations that have emerged over the last half century between various types of holism, Chinese medicine and systems biology. Given the discipline’s historical attachments to a definition of ‘medicine’ that rather narrowly refers to biomedicine as developed in Europe and the US from the eighteenth century onwards, the medical humanities are not the most obvious starting point for such an inquiry. At the same time, they do offer one advantage over neighbouring disciplines like medical history, anthropology or science and technology studies for someone like myself, a clinician as well as a historian and anthropologist: their strong commitment to the objective of facilitating better medical practice. This promise furthermore links to the wider project of critique, which, in Max Horkheimer’s definition of the term, aims at change and emancipation in order ‘to liberate human beings from the circumstances that enslave them’. If we take the critical medical humanities as explicitly affirming this shared objective and responsibility, extending the discipline’s traditional gaze is not a burden but becomes, in fact, an obligation.

Truth Matters: Cognitive Integrity as an Intervention for Self-Deception

2017 ◽  
Author(s):  
Eugenia Isabel Gorlin ◽  
Michael W. Otto
Keyword(s):  
Decision Making ◽  
Problem Solving ◽  
Autonomous Agents ◽  
Therapeutic Interventions ◽  
Empirical Validation ◽  
The Past ◽  
Adaptive Problem

To live well in the present, we take direction from the past. Yet, individuals may engage in a variety of behaviors that distort their past and current circumstances, reducing the likelihood of adaptive problem solving and decision making. In this article, we attend to self-deception as one such class of behaviors. Drawing upon research showing both the maladaptive consequences and self-perpetuating nature of self-deception, we propose that self-deception is an understudied risk and maintaining factor for psychopathology, and we introduce a “cognitive-integrity”-based approach that may hold promise for increasing the reach and effectiveness of our existing therapeutic interventions. Pending empirical validation of this theoretically-informed approach, we posit that patients may become more informed and autonomous agents in their own therapeutic growth by becoming more honest with themselves.

Cyber Peacekeeping: Critical Evaluation of Digital Blue Helmets Program

2020 ◽  
pp. 17-27
Author(s):  
Fahad Nabeel
Keyword(s):  
United Nations ◽  
Critical Evaluation ◽  
Threat Detection ◽  
Information And Communications Technologies ◽  
The Past ◽  
Research Article ◽  
Un Peacekeeping ◽  
Starting Point ◽  
Communications Technologies ◽  
Future Potential

In 2016, the United Nations (UN) launched the Digital Blue Helmets (DBH) program under its Office of Information and Communications Technologies (OICT). The launching of DBH was a continuation of a series of steps that the UN and its related agencies and departments have undertaken over the past decade to incorporate cyberspace within their working methodologies. At the time of inception, DBH was envisioned as a team capacitated to act as a replica of a physical peacekeeping force but for the sole purpose of overseeing cyberspace(s). Several research studies have been published in the past few years, which have conceptualized cyber peacekeeping in various ways. Some scholars have mentioned DBH as a starting point of cyber peacekeeping while some have proposed models for integration of cyber peacekeeping within the current UN peacekeeping architecture. However, no significant study has attempted to look at how DBH has evolved since its inception. This research article aims to examine the progress of DBH since its formation. It argues that despite four years since its formation, DBH is still far away from materializing its declared objectives. The article also discusses the future potential roles of DBH, including its collaboration with UN Global Pulse for cyber threat detection and prevention, and embedding the team along with physical peacekeepers.

‘I AM’: Indigenous consciousness for authenticity and leadership

Leadership ◽  
2021 ◽  
pp. 174271502199959
Author(s):  
Chellie Spiller
Keyword(s):  
True Self ◽  
The Past ◽  
Starting Point ◽  
The Future

This article encourages a move away from the excessively inward gaze of ‘to thine own self be true’ and explores ‘I AM’ consciousness as a starting point. An I AM approach encourages a move from the measurable self to the immeasurable expansiveness and mystery of our own becoming. It is to step beyond the lines drawn around the ‘true self’ or the lines that others would have us draw. I AM consciousness reflects an ancient Indigenous thread that echoes through millennia and reminds humans that we are a movement through time, and each person is a present link to the past and the future, woven into a fabric of belonging.

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