Insight into the fractional calculus via a spreadsheet

David A. Miller and Stephen J. Sugden have written an article called Insight into the Fractional Calculus via a Spreadsheet. The article was published in the most recent issue of electronic journal Spreadsheets in Education. The article is freely available as a pdf download, but here is a copy of the abstract:
Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of  fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.



Comments

Post a Comment

Popular posts from this blog

Special issue on early childhood mathematics teaching and learning

Journal für Mathematik-Didaktik had a special issue on early childhood mathematics teaching and learning in their latest issue . In addition to the editorial by Andrea Peter-Koop and Petra Scherer, the issue included the following articles: Fostering Early Mathematical Competencies in Natural Learning Situations—Foundation and Challenges of a Competence-Oriented Concept of Mathematics Education in Kindergarten, by Hedwig Gasteiger  Attitudes of Kindergarten Educators about Math, by Christiane Benz  Non-numerical and Numerical Understanding of the Part-Whole Concept of Children Aged 4 to 8 in Word Problems, by Petra Langhorst, Antje Ehlert, Annemarie Fritz  Young Children’s Structure Sense, by Miriam M. Lüken  First-Graders’ Development of Calculation Strategies: How Deriving Facts Helps Automatize Facts, by Michael Gaidoschik The “Non-canonical” Solution and the “Improvisation” as Conditions for Early Years Mathematics Learning Processes: The Concept of the “Interac...
Read more

Students' fraction comparison strategies

Doug M. Clarke and Anne Roche have written an article entitled Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction . The article was published online in Educational Studies in Mathematics on Friday. Here is a copy of the article's abstract: As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that predicted, with the exception of fractions with the same numerators and different denominators, which proved surprisingly difficult. Students who demonstrated the greatest success were likely to use benchmark (transitive) and residual thinking. It is hypothesised that the methods of these successful students could form the basis of instr...
Read more

How to develop mathematics for teaching and understanding

Image
Susanne Prediger has written an article about How to develop mathematics-for-teaching and for understanding: the case of meanings of the equal sign . The article was published online in Journal of Mathematics Teacher Education on Thursday last week. Point of departure in her article is the very important question about what mathematical (content) knowledge prospective teachers need. A main claim which is raised already in the introduction is: "Listen to your students!" In the theoretical background, Prediger takes Shulman's classic conceptualization of three main categories of content knowledge in teaching as point of departure: Subject-matter knowledge Pedagogical-content knowledge Curricular knowledge She continues to build heavily on the work done by Hyman Bass and Deborah Ball (e.g. Ball & Bass, 2004), and she goes on to place her own study in relation to the work of Bass and Ball: Whereas Bass and Ball (2004) concentrate on the first part of their program, namel...
Read more