Carolyn Yackel

MANAGEMENT, MOTIVATION AND STUDENT-CENTERED INSTRUCTION I: ANALYTICAL FRAMEWORK

ABSTRACT We describe a research-based analytical framework (MALA, for motivation, activity, learning and attribution) for the examination of classroom management issues in student-centered instruction. We review research in classroom management for university and college-level classrooms and describe the research base for the MALA framework.

Student Learning Objectives and Mathematics Teaching.

ABSTRACT The current work is the first article in a two-paper series exploring the role of explicit learning objectives in undergraduate mathematics instruction. A definition of student-learning objective (SLO) is introduced. We give examples of SLOs for topics from introductory college and university mathematics courses. We list potential advantages of a general program of grounding mathematics instruction in a set of such explicit SLOs. In the second paper in this series, an explicit, step-by-step algorithm for creating sets of student learning objectives will be described and its use illustrated.

MANAGEMENT, MOTIVATION AND STUDENT-CENTERED INSTRUCTION II: APPLICATIONS

ABSTRACT We apply a research-based analytical framework (MALA, for motivation, activity, learning and attribution) to better understand two student-centered college mathematics classes. We demonstrate ways in which the framework can be used to systematically identify problems with classroom management, to generate meaningful suggestions for improvement of teaching, and to identify specific and effective classroom management practices.

Introduction to the PRIMUS Special Issue on Mathematics and the Arts in the Undergraduate Classroom

Abstract In this editorial introduction to the Special Issue on Mathematics and the Arts in the Undergraduate Classroom, we explore how the explosion of interest in connections between math and the arts over the past several years has moved into the classroom. We describe papers in this special issue that report on a range of classroom experiences combining math and the arts in ways that enhance student learning and expand student appreciation of both mathematical and artistic ways of thinking.

Partial Fractions in Calculus, Number Theory, and Algebra

In second-semester calculus, most students view partial fractions as either a miracle or a torture, sometimes both. Yet, students rarely understand why rational functions can always be written in terms of partial fractions. The rationale comes from abstract algebra, but a simple analogue in the natural numbers reveals many of the intricacies. If you have an interest in examining partial fractions beyond accepting their existence for use in a technique of integration, read on. We begin by recalling the general theorem on the existence of partial fractions in calculus, after which we investigate the corresponding version in the natural numbers. Then we move to a discussion of the calculus version: rational functions over the reals. Finally, we generalize the theorem to rational functions in one variable over an arbitrary field. (For a discussion of the general case of Euclidean domains, see Packard and Wilson [4].)

The 2015 Joint Mathematics Meetings exhibition of mathematical art

Amidst the frenetic pace of the Joint Mathematics Meetings, each year one can find intellectual and visual excitement blended with tranquility by stepping inside the Mathematical Art Exhibition. This year the juried exhibit featured 124 pieces by 80 artists. Using twoand three-dimensional digital printing, yarn, paint, wood, and beads, these artists sought to communicate mathematical ideas though visual media. Heightened awareness of the mathematics encoded in an artwork enhances the depth of the piece, helping to place it in context and rendering it more significant and enjoyable. After all, beauty may be in the eyes of the beholder, but mathematics is in the mind of the beholder. The three main avenues for obtaining the relevant mathematical context for a particular artwork are to read the statement accompanying the piece, attend a talk by the mathematician/artist, or query the mathematician/artist. Attending talks, especially those in the Mathematics and the Arts session, and having one-on-one discussions regarding a piece bring the work to life. As most exhibit goers do not have the opportunity to discuss work with the artists or to attend presentations expounding upon the work, it is incumbent upon each artist to thoughtfully write the statement accompanying his or her piece. Interestingly, both mathematics and art are considered difficult fields requiring a great deal of skill and creativity. Frequently, outside observers will mention the idea of combining mathematics and art as a wonderful device for allowing an easier entry into the field; however, it is always important to ask to which field mathematics or art one will gain easier entry, as the intent of the conversant depends on their natural bent. Indeed, one wonders what sort of mathematics or art would make approaching the area easier. For this reason, explanations of the interplay between the mathematics and the art become ever more important so that viewers can see beyond surface triviality. Conversations with exhibitors often reveal that the field of mathematical art allows them to blend their passions and to develop talents and interests beyond a hobby level. Forays into the field often involve using a craft to visualize mathematics or using a mathematical object within a piece of art. Ideally, a mathematical artwork would push forward both the domains of mathematics and of art. However, that bar is too high for all but the most significant and creative work. Mathematical art that enriches both mathematics and art is, in fact, a great achievement, perhaps by making visual an abstract mathematical construct in an aesthetically pleasing way or by developing artistic techniques so that a mathematical piece arises naturally as an outcome. Another form of research in the field of mathematical art can involve deep understanding of the mathematics underpinning the artistic process, which can in turn allow for intricate design through artwork. Explaining these complicated concepts in statements accompanying pieces is certainly a challenge; however, the mathematical artist who wishes to give the public at large an opportunity to connect with the mathematical content must attempt some sort of mathematical explanation. A description that gives the viewer a small window into the mathematical process undergirding the work is part of what renders the piece mathematical art rather than just visual art for many viewers. In the remainder of the report, we examine especially striking examples of work in which the mathematical and the artistic aspects inform each other. For examples, we first turn to those selected for awards, as chosen by a panel of judges, who use the following criteria to determine prize recipients: (1) Mathematical depth and sophistication (2) Craftsmanship (3) Aesthetic appeal (4) Originality and innovation (5) Overall interest. Two of the three prize categories focus essentially on twoand three-dimensional art: ‘Best Photograph,

(k)not cables, braids

Cable patterns are the hallmark of Irish fisherman knits, and stitches that cross and weave are also found in some traditional Estonian knitting [3] as well as in designs from Bavaria, Tyrol, Alsace, Norway, and Denmark. It makes sense that these patterns would be popular in cold climates, because cabling uses more yarn than flat (e.g., stockinette) knitting, so the resulting garments are thicker and warmer.

MENTAL MAPS AND LEARNING OBJECTIVES: THE FAST-SLO ALGORITHM FOR CREATING STUDENT LEARNING OBJECTIVES

ABSTRACT We review the processes of goal identification employed in teacher planning. A new type of graphical organizer (Fact-Action-Schema-Transcript or FAST) is introduced and its applications to teacher planning discussed. We describe a pair of algorithms for (a) creating FASTs for undergraduate mathematics and (b) using FASTs to create sets of student learning objectives (SLOs). Application of these algorithms to lesson planning are illustrated with several examples from introductory college mathematics courses.