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Full-Text Articles in Mathematics
Developing Problem Solving Experiences In Practical Action Projects, François Pluvinage
Developing Problem Solving Experiences In Practical Action Projects, François Pluvinage
The Mathematics Enthusiast
Problem solving doubtless is an essential element of mathematical learning, so that mathematics educators often are satisfied when finding situations that lead their students to such activity. But in many cases, the chosen situations and the ways to guide students' works are not sufficiently analyzed from a didactic point of view. Our goal in the present analysis is to underline the possible ways for managing the situations, and to exhibit the parameters that educators have at their disposition within their role as mediator between students and mathematical knowledge and know-how.
Framing The Use Of Computational Technology In Problem Solving Approaches, Manuel Santos-Trigo, Matías Camacho Machín
Framing The Use Of Computational Technology In Problem Solving Approaches, Manuel Santos-Trigo, Matías Camacho Machín
The Mathematics Enthusiast
Mathematical tasks are key ingredient to foster teachers and students’ development and construction of mathematical thinking. The use of distinct computational tools offers teachers a variety of ways to represent and explore mathematical tasks which often extends problem solving approaches based on the use of paper and pencil. We sketch a framework to characterize ways of reasoning that emerge as result of using computational technology to solve a task that involves dealing with variation phenomena.
Thoughts About Research On Mathematical Problem- Solving Instruction, Frank K. Lester Jr.
Thoughts About Research On Mathematical Problem- Solving Instruction, Frank K. Lester Jr.
The Mathematics Enthusiast
In this article, the author, who has written extensively about mathematical problem solving over the past 40 years, discusses some of his current thinking about the nature of problem-solving and its relation to other forms of mathematical activity. He also suggests several proficiencies teachers should acquire in order for them to be successful in helping students become better problem solvers and presents a framework for research on problem-solving instruction. He closes the article with a list of principles about problem-solving instruction that have emerged since the early 1970s.
Developing The Art Of Seeing The Easy When Solving Problems, Alfinio Flores, Jaclyn Braker
Developing The Art Of Seeing The Easy When Solving Problems, Alfinio Flores, Jaclyn Braker
The Mathematics Enthusiast
For Leonardo da Vinci “saper vedere”, that is, knowing how to see, or having the art to see, was the key to unlocking the secrets of the visible world. Saper vedere included a precise sensory intuitive faculty as well as artistic imagination (Heydenreich, 1954) which were at the root of Leonardo’s inventiveness and creativity. According to Leonardo, to understand, you only have to see things properly (Bramly 1994, p. 264). Knowing how to see is also important in mathematics. The Italian mathematician Bruno de Finetti (1967) stresses this importance in his book on “Saper vedere” in mathematics. He highlights several …
Inquiry- Without Posing Questions?, Helle Alrø, Marit Johnsen-Høines
Inquiry- Without Posing Questions?, Helle Alrø, Marit Johnsen-Høines
The Mathematics Enthusiast
This article discusses what inquiry conversations could mean when learning mathematics.3 Referring to Gadamar’s distinction of true and apparent questions it is discussed what it takes to be inquiring and if this attitude necessarily includes posing questions. Which qualities are expressed in inquiring questions, and what other ways of communicating may have an inquiring function in learning conversations? The intention is to develop and frame the concept of ’inquiry’ in learning conversations, and this is the focus of analysis of an authentic classroom situation, where teacher and pupils are exploring the concept of ’volume’. Further, this analysis informs a …
Investigating Engineers’ Needs As A Part Of Designing A Professional Development Program For Engineers Who Are To Become Mathematics Teachers, Ragnhild Johanne Rensaa
Investigating Engineers’ Needs As A Part Of Designing A Professional Development Program For Engineers Who Are To Become Mathematics Teachers, Ragnhild Johanne Rensaa
The Mathematics Enthusiast
The average age for teachers in mathematics and physics in Norway is about 60 (Sørensen, 2003) and few students taking a masters degree in a science subject choose a career as a teacher (KD, 2006). This implies that there may be a critical lack of science teachers in schools in the near future (Næss, 2002). At the same time, rather many graduated engineers – who have worked as engineers sometimes several years – want to contribute to the educational setting by taking part time jobs as teachers; lector-II (Vassnes, 2008). To combine this supply and demand for qualified teachers in …
The Role Of Theory When Studying Epistemological Characterizations Of Mathematics Lecture(R)S, Magnus Österholm
The Role Of Theory When Studying Epistemological Characterizations Of Mathematics Lecture(R)S, Magnus Österholm
The Mathematics Enthusiast
The study presented in this paper is a contribution to the scientific discussion about the role and use of theory in mathematics education research. In particular, focus is here on the use of and comparison between different types of theories and frameworks, which is discussed primarily through the example of an empirical study examining what types of messages about mathematics are conveyed in lectures. The main purpose of this paper is to examine how different types of theories and frameworks can affect different parts of the research process.
University Mathematics Teachers' Views On The Required Reasoning In Calculus Exams, Ewa Bergqvist
University Mathematics Teachers' Views On The Required Reasoning In Calculus Exams, Ewa Bergqvist
The Mathematics Enthusiast
Students often use imitative reasoning, i.e. copy algorithms or recall facts, when solving mathematical tasks. Research shows that this type of imitative reasoning might weaken the students' understanding of the underlying mathematical concepts. In a previous study, the author classified tasks from 16 final exams from introductory calculus courses at Swedish universities. The results showed that it was possible to pass 15 of the exams, and solve most of the tasks, using imitative reasoning. This study examines the teachers' views on the reasoning that students are expected to perform during their own and others mathematics exams. The results indicate that …
Beyond Post-Modernity In Mathematics Education, Ole Skovsmose
Beyond Post-Modernity In Mathematics Education, Ole Skovsmose
The Mathematics Enthusiast
A radical form of postmodernity is presented with reference to Nietzsche’s ideas with respect to truth, knowledge, sciences, progress, democracy, and ethical values in general. Thereafter is presented Foucault’s archaeology of knowledge. This brings us forward to the notion of genealogy, which is a defining idea for the postmodern conception of critique. However, it is emphasised that a critique can address the generativity of mathematical rationality by considering mathematics-based fabrications. Finally, imagination is presented as yet another feature of a critical enterprise. It is illustrated how such a three dimensional critical enterprise is relevant for both mathematics and mathematics education. …
Loss Of Dimension In The History Of Calculus And In Student Reasoning, Robert Ely
Loss Of Dimension In The History Of Calculus And In Student Reasoning, Robert Ely
The Mathematics Enthusiast
Research indicates that calculus students often imagine objects losing a dimension entirely when a limit is taken, and that this image serves as an obstacle to their understanding of the fundamental theorem of calculus. Similar imagery, in the form of “indivisibles”, was similarly unsupportive of the development of the fundamental theorem in the mid-1600s, unlike the more powerful subsequent imagery of infinitesimals. This parallel between student issues and historical issues suggests several implications for how to provide students with imagery that is more productive for understanding the fundamental theorem, such as the imagery of infinitesimals or the more modern quantitative …
Demographics Of Self-Selected Participants In Mathematical Tracks In Swedish Upper Secondary School, Linda Mattsson
Demographics Of Self-Selected Participants In Mathematical Tracks In Swedish Upper Secondary School, Linda Mattsson
The Mathematics Enthusiast
This study investigates the three demographic factors gender, geographical origin and parents educational level among self-selected students in mathematical tracks in Swedish Upper Secondary School (n=147). Comparisons are made, through statistical tests of significance, with corresponding student distributions on the closely related Natural Science program (n = 1528). Data is obtained from Statistics Sweden. Results confirm that females are under-represented in mathematical tracks (p<0,01) and that parents of students in mathematical tracks have higher educational level than parents of students in the comparison group (p<0,05). Results do not show that students with foreign background are underrepresented in mathematical tracks, but raise questions about under-representation of Swedish students. Possible implications of these findings for these and similar programs are discussed.
Foundations For Success In Mathematical Competitions: A Study Of Best Praxis In Lower Secondary Schools In Norway, Steinar Thorvaldsen, Lars Vavik
Foundations For Success In Mathematical Competitions: A Study Of Best Praxis In Lower Secondary Schools In Norway, Steinar Thorvaldsen, Lars Vavik
The Mathematics Enthusiast
The purpose of this study is to investigate the following questions: What factors leads to success in mathematics, and how can these success factors and qualities be described? Will the teacher’s education and pedagogical praxis have an impact on good learning results? We report results from a case-control association study on among high achievement classes in mathematics in Norway. The data were collected from matched pairs of schools, paired on the basis of location and socioeconomic status. The questionnaire was first distributed to teachers in 38 Norwegian secondary schools at grade 9, which have had repeated success in the annual …
From Designing To Implementing Mathematical Tasks: Investigating The Changes In The Nature Of The T-Shirt Task, Claire Vaugelade Berg
From Designing To Implementing Mathematical Tasks: Investigating The Changes In The Nature Of The T-Shirt Task, Claire Vaugelade Berg
The Mathematics Enthusiast
From looking at research literature it is possible to see that research on design, implementation and analysis of mathematical tasks is an actual theme: there is a special issue of the Journal of Mathematics Teacher Education (2007) with Anne Watson, John Mason and Orit Zaslavsky as editors (Watson and Mason, 2007), a book published by Clarke, Grevholm and Millman (2009) concerning “Tasks in primary mathematics teacher education” and under ICME 11 in Mexico (2008) the title of one of the Topic Study Groups was “Research and development in task design and analysis”. In addition several substantial research projects conducted in …
Values In Effective Mathematics Lessons In Sweden: What Do They Tell Us?, Aihui Peng, Mikaela Nyroos
Values In Effective Mathematics Lessons In Sweden: What Do They Tell Us?, Aihui Peng, Mikaela Nyroos
The Mathematics Enthusiast
This study aims to examine values in effective mathematics lessons in Sweden from the perspectives of students in different groups and their teachers. By using methods with lesson observations, student focus group interviews and teacher interviews, it shows that instructional explanation and classroom atmosphere with quietness are shared-values of students and their teachers. The findings propose some crucial issues which related to how mathematics teaching could be adjusted to different students’ learning conditions and whether it needs more instructional explanation in mathematics teaching in Sweden.
Evaluating Mersenne Primes Using A Single Quadrant Expanding Square, Cory P. Daignault, Douglas Sprague, Joe L. Mott
Evaluating Mersenne Primes Using A Single Quadrant Expanding Square, Cory P. Daignault, Douglas Sprague, Joe L. Mott
The Mathematics Enthusiast
By forming a table of sequential odd numbers using a single quadrant expanding square pattern it was observed that multiple Mersenne primes fall into the first column. We prove that the Mersenne primes which fall into the first column will be of the form p = 3 mod 4, where p is the Mersenne prime exponent. Focusing the search of primes to those which fall into the first column of this square may be a means to increase the speed at which large primes are discovered.
From Conic Intersections To Toric Intersections: The Case Of The Isoptic Curves Of An Ellipse, Thierry Dana-Picard, Nurit Zehavi, Giora Mann
From Conic Intersections To Toric Intersections: The Case Of The Isoptic Curves Of An Ellipse, Thierry Dana-Picard, Nurit Zehavi, Giora Mann
The Mathematics Enthusiast
Starting from the study of the orthoptic curves of parabolas and ellipses, we generalize to the case of isoptic curves for any angle, i.e. the geometric locus of points from which a parabola or an ellipse are viewed under a given angle. This leads to the investigation of spiric curves and to the construction of these curves as an actual intersection of a self-intersecting torus with a plane. The usage of a Computer Algebra System facilitated this investigation.
Student Enrolment In Classes With Frequent Mathematical Discussion And Its Longitudinal Effect On Mathematics Achievement, Karl W. Kosko
Student Enrolment In Classes With Frequent Mathematical Discussion And Its Longitudinal Effect On Mathematics Achievement, Karl W. Kosko
The Mathematics Enthusiast
Mathematical discussion has been identified as being beneficial to students’ understandings of mathematics (Goos, 1995; Lee, 2006). Students in classrooms with more effective math discussion have been observed to engage more frequently in discussion (e.g. Hiebert & Wearne, 1993), but the converse is not necessarily true (e.g. Manouchehri & St. John, 2006). Utilizing hierarchical linear modeling, the present study examined student enrollment in classes with more and less frequent discussion and such enrollment’s effect on mathematics achievement over time. Results indicated that students enrolled in classes that discuss math “almost every day” consistently have higher math achievement than students enrolled …
Mathematical Competitions In Hungary: Promoting A Tradition Of Excellence & Creativity, Julianna Connelly Stockton
Mathematical Competitions In Hungary: Promoting A Tradition Of Excellence & Creativity, Julianna Connelly Stockton
The Mathematics Enthusiast
Hungary has long been known for its outstanding production of mathematical talent. Extracurricular programs such as camps and competitions form a strong foundation within the Hungarian tradition. New types of competitions in recent years include team competitions, multiple choice competitions, and some exclusively for students who are not in a special mathematics class. This study explores some of the recent developments in Hungarian mathematics competitions and the potential implications these changes have for the very competition-driven system that currently exists. The founding of so many new competitions reflects a possible shift in the focus and purpose of competitions away from …
Some Reflections On Mathematics And Mathematicians: Simple Questions, Complex Answers, Juan Eduardo Nápoles Valdes
Some Reflections On Mathematics And Mathematicians: Simple Questions, Complex Answers, Juan Eduardo Nápoles Valdes
The Mathematics Enthusiast
In this work we present some reflections on mathematics and mathematicians. Special emphasis is placed on the questions (1) what is mathematics? And (2) what is a mathematician? Some reflections and open questions are posed at the end of the work.
Visualizations And Intuitive Reasoning In Mathematics, Kajsa Bråting
Visualizations And Intuitive Reasoning In Mathematics, Kajsa Bråting
The Mathematics Enthusiast
On the basis of historical and didactical examples we consider the role of visualizations and intuitive thinking in mathematics. Examples from the 17th and 19th century have been used as well as smaller empirical studies at upper secondary school level and university level. We emphasize that a mathematical visualization does not reveal its intended meaning. With experience we can learn to interpret the visualization in different ways, depending on what is asked for.
If Not, Then What If As Well- Unexpected Trigonometric Insights, Stanley Barkan
If Not, Then What If As Well- Unexpected Trigonometric Insights, Stanley Barkan
The Mathematics Enthusiast
In performing an exercise of “What if not”, one can end up with a paucity of structure. Adding alternative structure can be a rich source of discovery, as we present here. The framework of this presentation is the original voyage of discovery, from a trivial geometric problem to the derivation of some unexpected trigonometric formulae based on regular polygons. The original “voyage” has been changed only sufficiently to make the text readable.
Reflections On Teaching With A Standards-Based Curriculum: A Conversation Among Mathematics Educators, Jill Newton, Ricki Geller, Lindsay Umbeck, Lisa Kasmer
Reflections On Teaching With A Standards-Based Curriculum: A Conversation Among Mathematics Educators, Jill Newton, Ricki Geller, Lindsay Umbeck, Lisa Kasmer
The Mathematics Enthusiast
Many teachers and researchers have written about the challenges inherent in adopting new teaching practices in mathematics classrooms (e.g., Chazan, 2000; Clarke, 1997; Heaton, 2000). The authors of this article, all with secondary mathematics teaching experience, are convinced by research suggesting that Standards-based mathematics curricula are beneficial for student learning.1 However, the first three authors had not used such curriculum materials in their own classrooms, and we desired experience using a Standards-based mathematics curriculum with secondary students. To this end, we taught a week-long summer course with a focus on linear functions to high school students who had previously struggled …
Increasing The Use Of Practical Activities Through Changed Practice, Frode Olav Haara, Kari Smith
Increasing The Use Of Practical Activities Through Changed Practice, Frode Olav Haara, Kari Smith
The Mathematics Enthusiast
This study sets out to examine the influence of a value-based intervention on two elementary school teachers’ use of practical activities in mathematics teaching. The intervention was a “Values and Knowledge Education” (VaKE)-based in-service course that introduced the two teachers to a value-based approach to mathematics teaching. The introduction included examples that were supported by use of practical activities. Interviews prior to the intervention made the teachers aware of an inconsistency between the desired and actual practice of their own teaching. The intervention provided them with a possibility of narrowing the gap between vision and practice by changing practice. Qualitative …
Number Theory And The Queen Of Mathematics, Megan Wagner
Number Theory And The Queen Of Mathematics, Megan Wagner
The Mathematics Enthusiast
Geometry is an integral component of secondary mathematics curriculum. From my experience in a mathematics teacher preparation program, I have seen a real push to connect geometry to other areas of mathematics. Secondary geometry can often be presented without clear or any connections to other areas of mathematics. One main purpose of this paper is to explore geometry and its rightful connection to other areas of mathematics, specifically number theory. Such strong emphasis is placed on drawing connections to number theory because of its intrinsic value in enhancing understanding of mathematical concepts. Learning number theory has positive ramifications for students, …
Translating And Adapting The Mathematical Knowledge For Teaching (Mkt) Measures: The Cases Of Indonesia And Norway, Dicky Ng, Reidar Mosvold, Janne Fauskanger
Translating And Adapting The Mathematical Knowledge For Teaching (Mkt) Measures: The Cases Of Indonesia And Norway, Dicky Ng, Reidar Mosvold, Janne Fauskanger
The Mathematics Enthusiast
This paper examines issues of translating and adapting an instrument that aims at measuring mathematical knowledge for teaching in Indonesia and Norway. The instrument was created for use in the U.S., and we discuss problematic and challenging issues of translation and adaptation. Two items from the released items pool were translated using a common framework modified from a previous study to exemplify critical issues that need to be resolved prior to using such instrument in another country. Themes identified in this study include a) minor challenges due to cultural differences; b) the use of technical language in schools; c) incommensurable …
Was Pythagoras Chinese- Revisiting An Old Debate, Ross Gustafson
Was Pythagoras Chinese- Revisiting An Old Debate, Ross Gustafson
The Mathematics Enthusiast
Two of the great ancient civilizations were those of the Greeks and the Chinese. Many great works of art, architecture, philosophy and literature have been produced by both of these civilizations. When it comes to mathematics in the Western World, the Greeks have also been credited with many contributions to the field, especially geometry. Anyone who has completed a standard high school mathematical curriculum has been introduced to the names of Pythagoras, Euclid and Archimedes and their methods. Not as widely credited in the Western world are the mathematical contributions of the ancient Chinese civilization. An examination into Chinese mathematics …
Research On Practical Rationality: Studying The Justification Of Actions In Mathematics Teaching, Patricio Herbst, Daniel Chazan
Research On Practical Rationality: Studying The Justification Of Actions In Mathematics Teaching, Patricio Herbst, Daniel Chazan
The Mathematics Enthusiast
Building on our earlier work conceptualizing teaching as the management of instructional exchanges, we lay out a theory of the practical rationality of mathematics teaching—that is, a theory of the grounds upon which instructional actions specific to mathematics can be justified or rebuffed. We do that from a perspective informed by what experienced practitioners consider viable but also in ways that suggest operational avenues for the study of instructional improvement, in particular for improvements that enable students to do more authentic mathematical work. We show how different kinds of experiments can be used to engage in theory building and provide …
Complex Learning Through Cognitively Demanding Tasks, Lyn D. English
Complex Learning Through Cognitively Demanding Tasks, Lyn D. English
The Mathematics Enthusiast
The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by “continuing education as usual” (The National Academies, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualization. These technologies have led to significant changes in the forms of mathematical thinking that are required beyond the classroom. This paper argues for the need to incorporate future-oriented understandings and competencies within the mathematics curriculum, through intellectually stimulating activities that draw upon multidisciplinary content and contexts. The paper also argues for greater recognition of children’s …