Open Access. Powered by Scholars. Published by Universities.®
Articles 31 - 60 of 411
Full-Text Articles in Mathematics
The Secret Life Of 1/N: A Journey Far Beyond The Decimal Point, Christopher Lyons
The Secret Life Of 1/N: A Journey Far Beyond The Decimal Point, Christopher Lyons
The Mathematics Enthusiast
The decimal expansions of the numbers 1/n (such as 1/3 = .03333..., 1/7 = 0.142857...) are most often viewed as tools for approximating quantities to a desired degree of accuracy. The aim of this exposition is to show how these modest expressions in fact have much more to offer, particularly in the case when the expansions are infinitely long. First we discuss how simply asking about the period (that is, the length of the repeating sequence of digits) of the decimal expansion of 1/n naturally leads to more sophisticated ideas from elementary number theory, as well as to …
The History Of Algorithmic Complexity, Audrey A. Nasar
The History Of Algorithmic Complexity, Audrey A. Nasar
The Mathematics Enthusiast
This paper provides a historical account of the development of algorithmic complexity in a form that is suitable to instructors of mathematics at the high school or undergraduate level. The study of algorithmic complexity, despite being deeply rooted in mathematics, is usually restricted to the computer science curriculum. By providing a historical account of algorithmic complexity through a mathematical lens, this paper aims to equip mathematics educators with the necessary background and framework for incorporating the analysis of algorithmic complexity into mathematics courses as early on as algebra or pre-calculus.
Integral Of Radical Trigonometric Functions Revisited, Natanael Karjanto, Binur Yermukanova
Integral Of Radical Trigonometric Functions Revisited, Natanael Karjanto, Binur Yermukanova
The Mathematics Enthusiast
This article revisits an integral of radical trigonometric functions. It presents several methods of integration where the integrand takes the form 1+/- sin x or 1+/- cos x. The integral has applications in Calculus where it appears as the length of cardioid represented in polar coordinates.
Mathematical Problem-Solving Via Wallas’ Four Stages Of Creativity: Implications For The Undergraduate Classroom, Milos Savic
Mathematical Problem-Solving Via Wallas’ Four Stages Of Creativity: Implications For The Undergraduate Classroom, Milos Savic
The Mathematics Enthusiast
The central theme in this article is that certain problem-solving frameworks (e.g., Polya, 1957; Carlson & Bloom, 2005) can be viewed within Wallas’ four stages of mathematical creativity. The author attempts to justify the previous claim by breaking down each of Wallas’ four components (preparation, incubation, illumination, verification) using both mathematical creativity and problem-solving/proving literature. Since creativity seems to be important in mathematics at the undergraduate level (Schumacher & Siegel, 2015), the author then outlines three observations about the lack of fostering mathematical creativity in the classroom. Finally, conclusions and future research are discussed, with emphasis on using technological advances …
Plato On The Foundations Of Modern Theorem Provers, Ines Hipolito
Plato On The Foundations Of Modern Theorem Provers, Ines Hipolito
The Mathematics Enthusiast
Is it possible to achieve such a proof that is independent of both acts and dispositions of the human mind? Plato is one of the great contributors to the foundations of mathematics. He discussed, 2400 years ago, the importance of clear and precise definitions as fundamental entities in mathematics, independent of the human mind. In the seventh book of his masterpiece, The Republic, Plato states “arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument” (525c). In the light of this …
Aesthetics In School Mathematics: A Potential Model And A Possible Lesson, Hartono Tjoe
Aesthetics In School Mathematics: A Potential Model And A Possible Lesson, Hartono Tjoe
The Mathematics Enthusiast
Earlier studies on improving classroom practice in mathematics have suggested a closer attention to nurturing an aesthetic appreciation for mathematics in students’ learning experiences. Recent evidence nonetheless reveals little indication of its presence. This article offers a potential model of the case for aesthetics in school mathematics. Central to this model is the harmonious hierarchy of necessity, existence, and uniqueness without any of which the case for aesthetics in student learning might be suboptimal, if not untenable. This article offers an example of the proposed model using a possible lesson designed to engage students aesthetically in the learning of mathematics. …
Assessing Mathematical Knowledge For Teaching: The Role Of Teaching Context, Geoffrey Phelps, Heather Howell
Assessing Mathematical Knowledge For Teaching: The Role Of Teaching Context, Geoffrey Phelps, Heather Howell
The Mathematics Enthusiast
Assessments of mathematical knowledge for teaching (MKT), which are often designed to measure specialized types of mathematical knowledge, typically include a representation of teaching practice in the assessment task. This analysis makes use of an existing, validated set of 10 assessment tasks to both describe and explore the function of the teaching contexts represented. We found that teaching context serves a variety of functions, some more critical than others. These context features play an important role in both the design of assessments of MKT and the types of mathematical knowledge assessed.
Guest Editorial: Mathematical Knowledge For Teaching: Developing Measures And Measuring Development, Reidar Mosvold, Mark Hoover
Guest Editorial: Mathematical Knowledge For Teaching: Developing Measures And Measuring Development, Reidar Mosvold, Mark Hoover
The Mathematics Enthusiast
No abstract provided.
Interview Prompts To Uncover Mathematical Knowledge For Teaching: Focus On Providing Written Feedback, Yeon Kim
The Mathematics Enthusiast
One area of study that has been gathering enthusiastic attention and interest is mathematical knowledge for teaching (MKT). How to research MKT, however, is still unsettled despite the plethora of unexamined areas of practice. As one of ways to unearth and measure MKT, this study uses interview prompts designed to providing written feedback, as a target area of practice. This study specifies in what ways the interview prompts are used in order to provide a comprehensive method to researching MKT. From interviews across professional communities with different kinds of mathematical expertise, the author develops a conceptual model based on the …
Why Defining The Construct Matters: An Examination Of Teacher Knowledge Using Different Lenses On One Assessment, Chandra H. Orrill, Allan S. Cohen
Why Defining The Construct Matters: An Examination Of Teacher Knowledge Using Different Lenses On One Assessment, Chandra H. Orrill, Allan S. Cohen
The Mathematics Enthusiast
What does it mean to align an assessment to the domain of interest? In this paper, we analyze teachers’ performance on the Learning Mathematics for Teaching assessment of Proportional Reasoning. Using a mixture Rasch model, we analyze their performance on the entire assessment, then on two different subsets of items from the original assessment. We consider the affordances of different conceptualizations of the domain and consider the implications of the domain definition on the claims we can make about teacher performance. We use a single assessment to illustrate the differences in results that can arise based on the ways in …
Tme Volume 13, Numbers 1 And 2
Knowledge For Equitable Mathematics Teaching: The Case Of Latino Ells In U.S. Schools, Aaron T. Wilson
Knowledge For Equitable Mathematics Teaching: The Case Of Latino Ells In U.S. Schools, Aaron T. Wilson
The Mathematics Enthusiast
This paper reports the exploration of an aspect of knowledge needed for equitable mathematics teaching. Pedagogical Content Knowledge for Teaching Mathematics to English Language Learners (PCK-MELL) was proposed as a theoretical knowledge construct, a subdomain of MKT, and the construct was investigated through a process of survey instrument development and administration. The survey contained items intended to measure teachers’ knowledge of the obstacles encountered by ELLs in math classes, of the resources that ELLs draw upon, and of instructional strategies for teaching ELLs. Analysis of middle school mathematics teachers’ responses (N = 42) offered insights into how to improve the …
Making Progress On Mathematical Knowledge For Teaching, Mark Hoover, Reidar Mosvold, Deborah L. Ball, Yvonne Lai
Making Progress On Mathematical Knowledge For Teaching, Mark Hoover, Reidar Mosvold, Deborah L. Ball, Yvonne Lai
The Mathematics Enthusiast
Although the field lacks a theoretically grounded, well-defined, and shared conception of mathematical knowledge required for teaching, there appears to be broad agreement that a specialized body of knowledge is vital to improvement. Further, such a construct serves as the foundation for different kinds of studies with different agendas. This article reviews what is known and needs to be known to advance research on mathematical knowledge for teaching. It argues for three priorities: (i) finding common ground for engaging in complementary studies that together advance the field; (ii) innovating and reflecting on method; and (iii) addressing the relationship of such …
Use Of Mathematical Tasks Of Teaching And The Corresponding Lmt Meaures In The Malawi Context, Mercy Kazima, Arne Jakobsen, Dun N. Kasoka
Use Of Mathematical Tasks Of Teaching And The Corresponding Lmt Meaures In The Malawi Context, Mercy Kazima, Arne Jakobsen, Dun N. Kasoka
The Mathematics Enthusiast
We discuss the adaptation and piloting of the previously developed U.S.-specific measures of mathematical knowledge for teaching to the Malawi context. The purpose is to produce measures that can be used to evaluate changes in mathematical knowledge for teaching gained through primary teacher education, thus informing teacher educators on the most effective evidence-based practices. By interviewing 14 teachers, we first examine whether the 16 recurrent mathematical tasks of teaching tasks identified in the U.S. are applicable to the Malawi context. This is followed by the discussion of the adaptability of the U.S. developed number concept and operations LMT measures. Next, …
In-Service Teachers' Reasoning About Scenarios Of Teaching Mathematics To English Language Learners, Sultan Turkan
In-Service Teachers' Reasoning About Scenarios Of Teaching Mathematics To English Language Learners, Sultan Turkan
The Mathematics Enthusiast
The student population in the U.S. and worldwide is becoming increasingly diverse, creating a need to support all learners, especially linguistically and culturally diverse subpopulations such as English language learners (ELLs). From a social equity standpoint, the need to support these learners is critical especially in mathematics classrooms. In the U.S, the demand for mathematics teachers who are adequately prepared to teach ELLs has in fact risen. Yet, little is known about what knowledge base is essential to teach mathematics to ELLs. Driven by the need to explore this knowledge base, in this paper I explore what is involved in …
What Does It Take To Develop Assessments Of Mathematical Knowledge For Teaching?: Unpacking The Mathematical Work Of Teaching, Sarah Kate Selling, Nicole Garcia, Deborah L. Ball
What Does It Take To Develop Assessments Of Mathematical Knowledge For Teaching?: Unpacking The Mathematical Work Of Teaching, Sarah Kate Selling, Nicole Garcia, Deborah L. Ball
The Mathematics Enthusiast
In the context of the increased mathematical demands of the Common Core State Standards and data showing that many elementary school teachers lack strong mathematical knowledge for teaching, there is an urgent need to grow teachers’ MKT. With this goal in mind, it is crucial to have research and assessment tools that are able to measure and track aspects of teachers’ MKT at scale. Building on the concept of “mathematical tasks of teaching” (Ball et al., 2008), we report on a new framework that unpacks the mathematical work of teaching that could serve as a scaffold for item writers who …
Teachers And Their Educators - Views On Contents And Their Development Needs In Mathematics Teacher Education, Mika Koponen, Mervi A. Asikainen, Antti Viholainen, Pekka E. Hirvonen
Teachers And Their Educators - Views On Contents And Their Development Needs In Mathematics Teacher Education, Mika Koponen, Mervi A. Asikainen, Antti Viholainen, Pekka E. Hirvonen
The Mathematics Enthusiast
Finland has scored well in international assessments (e.g. PISA, TIMSS), and the pressure to attain excellent scores has activated a drive toward even more effective mathematics teacher education. This article presents the results of a qualitative assessment of the mathematics teacher education provided by the University of Eastern Finland. In this study, the views held by practicing teachers (N=101) and teacher educators (N=19) are compared so that the outstanding development needs of mathematics teacher education in terms of their contents can be revealed. The data was gathered via an electronic survey and was mainly analyzed using data-driven methods. In addition, …
Guest Editorial: Risk – Mathematical Or Otherwise, Egan J. Chernoff
Guest Editorial: Risk – Mathematical Or Otherwise, Egan J. Chernoff
The Mathematics Enthusiast
No abstract provided.
What Can Education Learn From Real-World Communication Of Risk And Uncertainty?, David Spiegelhalter, Jenny Gage
What Can Education Learn From Real-World Communication Of Risk And Uncertainty?, David Spiegelhalter, Jenny Gage
The Mathematics Enthusiast
Probability is a difficult topic to teach, not least because it is rather unclear what it actually means. Modern risk communication has tackled general public incomprehension of probability statements by using the metaphor of ‘expected frequencies’ – for example, “of 100 people like you, we would expect 10 to have a heart attack or stroke in the next 10 years.” We show how these ideas can be taken into the classroom as the basis for teaching probability, using frequency tree diagrams as the fundamental representation. Empirical frequency trees can be used to summarise a series of classroom experiments, and then …
Developing Strategic And Mathematical Thinking Via Game Play: Programming To Investigate A Risky Strategy For Quarto, Peter Rowlett
Developing Strategic And Mathematical Thinking Via Game Play: Programming To Investigate A Risky Strategy For Quarto, Peter Rowlett
The Mathematics Enthusiast
The Maths Arcade is an extracurricular club for undergraduate students to play and analyse strategy board games, aimed at building a mathematical community of staff and students as well as improving strategic and mathematical thinking. This educational initiative, used at several universities in the U.K., will be described. Quarto is an impartial game played at the Maths Arcade, in that there is one set of common pieces used by both players, and one where stalemates are a common outcome. While some students play without apparent direction until a winning opportunity appears, others adopt a more risky strategy of building the …
Risky Research Business: Mathematics Education Research On The Margins, Erika C. Bullock
Risky Research Business: Mathematics Education Research On The Margins, Erika C. Bullock
The Mathematics Enthusiast
Although we would like to believe that decisions about what research to conduct and how to conduct it are based solely on researcher interest and societal need, the reality is that external political and disciplinary factors do play a role. Scientifically based research (SBR) is one example of external political pressures that shape researcher choice both directly and indirectly. Additionally, disciplines like mathematics education operate under hidden curricula that have the potential to marginalize particular research foci. The purpose of this paper is to consider the implications of such a narrow focus on a young mathematics education researcher’s choices about …
Teaching Risk In School, Andreas Eichler, Markus Vogel
Teaching Risk In School, Andreas Eichler, Markus Vogel
The Mathematics Enthusiast
Although risk is an important topic for society it is seldom addressed when teaching statistics and probability. In this paper we refer to this discrepancy identifying three obstacles for teaching risk in school regarding the mathematical and the situational aspect of risk. Based on two educational constructs, i.e. probability literacy and modelling, we discuss existing approaches for teaching risk in school and propose two strategies for promoting risk as a valuable issue for students based again on the distinction of the mathematical and situational aspect of risk.
Risk—A Fundamental Condition Of Doing Mathematics, Wolff-Michael Roth, Jean-François Maheux
Risk—A Fundamental Condition Of Doing Mathematics, Wolff-Michael Roth, Jean-François Maheux
The Mathematics Enthusiast
The theme of this special issue is risk. But risk is not a common topic of investigation in mathematics education, lest it be an occasional interest in “at risk” students, generally defined as those who likely will fail at school. In this study, we are not interested in this rather limited use of the risk concept. Instead, we show that risk not only is a condition of human life generally, but also a necessity for teaching and learning mathematics. To show this, we develop the concept of risk with materials from a second-grade mathematics unit on geometry. Implications are drawn …
Making Decisions About Gambling: The Influence Of Risk On Children’S Arguments, Annie Savard
Making Decisions About Gambling: The Influence Of Risk On Children’S Arguments, Annie Savard
The Mathematics Enthusiast
This article presents results from a study on decision-making towards eventual participation to gambling activities by grade 4 students. For this study, six learning situations were proposed in a fourth grade classroom. The researcher, who was also the teacher, proposed some extra activities in order to define gambling. Students learned about probability and developed, at the same time, the ability to think critically about gambling. She then proposed three fictional situations of gambling to the students, and asked them if and why they would (or would not) participate in the situation. By studying the arguments that students provided, she explored …
Students’ Language Repertoires For Prediction, David Wagner, Joseph Dicks, Paula Kristmanson
Students’ Language Repertoires For Prediction, David Wagner, Joseph Dicks, Paula Kristmanson
The Mathematics Enthusiast
Communication about prediction is complex in a number of ways. First, language is by nature recursive — language is an indicator of meaning as well as a force that shapes meaning. Second, the same language used to communicate prediction in uncertain environments is used for other purposes. In this article, we describe how the recursive nature of language impacted the choices we made in a cross-sectional longitudinal study aimed at gaining insight into children’s language repertoires relating to conjecture. We then explore some Grade 6 students’ communication about prediction to develop insight into their meaning and meaning-making with prediction language. …
Levels Of Reasoning Of Middle School Students About Data Dispersion In Risk Contexts, Ernesto Sánchez, Antonio Orta
Levels Of Reasoning Of Middle School Students About Data Dispersion In Risk Contexts, Ernesto Sánchez, Antonio Orta
The Mathematics Enthusiast
The aim of this research study is to explore students’ reasoning concerning variation when they compare groups and have to interpret dispersion in terms of risk. In particular, we analyze in this paper the responses to two problems from a questionnaire administered to 82 ninth-grade students. The problems consist of choosing between two and three groups of data by comparing them. The first one composed of losses and winnings coming from a hypothetical game; the second is about medical treatments. The results show the difficulty students had in interpreting variation in a risk context. Although they identify the data group …
Pedagogy Of Risk: Why And How Should We Teach Risk In High School Math Classes?, Nenad Radakovic
Pedagogy Of Risk: Why And How Should We Teach Risk In High School Math Classes?, Nenad Radakovic
The Mathematics Enthusiast
Risk is everywhere yet the concept of risk is seldom investigated in high school mathematics. After presenting arguments for teaching risk in the context of high school mathematics, the article describes a case study of teaching risk in two grade 11 classes in Canada- an all-boy independent school (23 boys) and a publicly funded religious school (19 girls and 4 boys). The findings suggest that the students possessed intuitive knowledge that risk of an event should be assessed by both its likelihood and its impact. Following and amending pedagogic model of risk (Levinson, R., Kent, P., Pratt, D., Kapadia, R., …
Risk Intuitions And Perceptions: A Case Study Of Four Year 13 (Grade 12) Students, Stephanie Budgett, Lorraine O'Carroll, Maxine Pfannkuch
Risk Intuitions And Perceptions: A Case Study Of Four Year 13 (Grade 12) Students, Stephanie Budgett, Lorraine O'Carroll, Maxine Pfannkuch
The Mathematics Enthusiast
In the New Zealand school statistics curriculum, year 12 students (aged 16-17) are required to solve problems that involve interpreting risk and relative risk within a range of meaningful contexts. In a small exploratory study we investigate the risk conceptions of four year 13 students who performed at the excellence level in their year 12 externally-assessed examination on this topic. Through questionnaires and interviews we investigate the ways in which these students perceive and express risks associated with a variety of everyday activities and also how they compare the risks of several adverse outcomes. We also explore the strategies they …