Educational Mathematics Seminar

Mondays, 4 - 5:30, in Malott Hall 224.

This week and all weeks whatever presentation there is will be less than one hour leaving significant time for discussions and questions.

30 August:
David Henderson & Daina Taimina (Cornell)

"A New View of the History of Geometry and How it Can Be Used to Clarify Misconceptions"

Abstract:

We have found that students and even mathematicians are often confused about the history of geometry. In addition, many expository descriptions of geometry (especially non-Euclidean geometry) contain confusing and sometimes-incorrect statements. This is true even in expositions written by well-known research mathematicians. These include answers to questions such as: What and when was the first non-Euclidean geometry? Can you trisect any angle? What does "straight" mean in geometry?

We introduce the notion that the main aspects of geometry today emerged from four strands of early human activity, which seemed to have occurred in most cultures: art/patterns, navigation/stargazing, motion/machines, and building structures. These strands developed more or less independently into varying studies and practices that from the 18-th and 19-th century on were woven into what we now call geometry.

We will then discuss how these four strands can help us to clear up common misconceptions and increase people's interest, both in geometry and its history.

6 September:
David Henderson

"APOS Theory: Preparing for Dubinsky's Visit"

Abstract: Ed Dubinsky is a the developer of APOS Theory ("an extension of Piaget's theory of reflective abstraction applied to the undergraduate mathematics curriculum"). In order to take full advantage of Ed Dubinsky's visit on September 13, we will introduce and discuss some of his papers that he suggests we look at. Please look at the papers before the seminar. The three papers are:

1. (with M. McDonald), APOS: "A Constructivist Theory of Learning in Undergrad Mathematics Education Research", 2001. In D. Holton et. (Eds.), The teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publishers, 273-280. Download as PDF file http://www.math.kent.edu/~edd/ICMIPaper.pdf (73K, 22 pages)

2. (with Kirk Weller, Anne Brown, Michael McDonald, and Cynthia Stenger), Intimations of Infinity, Notices of the AMS, August 2004.
http://www.ams.org/notices/200407/fea-dubinsky.pdf

3. "at least skim" (with Weller, Clark, Loch, McDonald and Merkovsky) "An examination of Student Performance Data in Recent RUMEC Studies". In Review. Download as PDF File http://www.math.kent.edu/~edd/Performance.pdf (1278K, 33 pages)

13 September:
Ed Dubinsky (Kent State University)

"APOS Theory and the Issue of Encapsulation"

Abstract: In the seminar, I will give a very brief introduction (less than 30 minutes) and we will spend all of the remaining time with questions and discussions. I will assert that in coming to an object conception of some mathematical concept, one ALWAYS makes an encapsulation, even when the object is understood visually. I would be very interested to explore the view which I think David Henderson has expressed that there are cases in which a geometric approach can be an alternative to encapsulation.

[The seminar will assume some acquaintance with Dubinsky's work as will be introduced and discussed in the seminar on 6 September.]

The seminar will be followed by an informal dinner party at the Henderson/Taimina home for all participant.

14 September:
Dubinsky will also give a colloquium talk "SOME REMARKS ON UNDERSTANDING THE CONCEPT OFINFINITY" at the Mathematics Department's Oliver/Olivetti Club. Tuesday, 14 September, 4:15-5:15.

Abstract: The concept of infinity has intrigued philosophers, mathematicians and historians for millenia.  There are paradoxes, dichotomies and many questions.  Some of these questions have been answered to the satisfaction of mathematicians and other scholars, but often these answers are not very helpful to students.  And some of the questions remain unanswered.

In this talk I will describe some of the issues mentioned above and discuss how a certain theory of learning, called APOS Theory, can propose new answers to some of these questions, answers which have a potential for pointing to pedagogical strategies that might help students develop a deeper understanding of this concept which is all-important in mathematics.

Other than a reasonble understanding of infinity and undergraduate mathematics the talk will not depend on any specific background either in mathematics or in cognition.

20 September:
Prema Junius (Mansfield University)
Title: Development of students’ understanding of straight lines on plane and sphere

    Abstract:

I study was conducted qualitatively using naturalistic inquiry to gain insight into the complex cognitive processes involved in integrating Euclidean and non-Euclidean geometry. As part of the study I examined how notions of “straightness” developed in a senior level modern geometry class. The text used was Experiencing Geometry on Plane and Sphere, by David Henderson. In Chapter two, Henderson asks, “What is straight on the sphere?” This problem will be the focus of my talk.

There was conflict with prior knowledge of Euclidean geometry when the students were exposed to non-Euclidean geometry; the conflict involved images and definitions. I will discuss the conflict and show how it was resolved by motion, mental rotation and analogy; the analogy having its source in individual experience. (I sat with the students for seven semesters -- words and strange metaphors describing motion cropped up all the time as the students explained things to each other in groups, but I could not record everything as it was too much for me.)   

The research supports the assertions of Henri Poincaré: motor space is the space and motion is the basis of geometry. According to Poincaré visual space is only one part of space and in the notion of this there is something artificial; the real space is motor space. He also made other bold statements: One does not learn a new geometry; one gets used to it. One geometry is not truer than another geometry, only more convenient, and Euclidean geometry will always be the most convenient geometry!   

27 September:
Maria Terrell (Cornell)

Good Questions for Deeper Learning of Mathematics

Abstract: Since Spring 2003 a team of faculty and graduate students has been developing interactive teaching and learning methods for teaching calculus at Cornell. The essence of the approach is to engage students each day during class in peer discussions about thought-provoking, deeply conceptual mathematics questions. The approach relies on two types of questions: web based pre-class warm up questions tied to reading assignments, and in-class discussion generating questions. The questions help instructors stimulate students’ thinking before and during class, and technology helps instructors monitor students’ understanding through frequent assessments of what they are learning. This presentation will include samples of the materials we are developing, a demonstration of the web based software and electronic classroom polling system, and a summary of the results of our preliminary trial.

4 October:
Florence Fassanelli (AAAS)

The Relation between the History of Art and History of Mathematics For Pedagogical Purposes

Abstract: How can knowing the history of art help us to better understand or be more interested in mathematics? How can knowing the history of mathematics help us to better understand or be more interested in art? How can this help our teaching and learning?

This seminar is about the art that affected the progress of mathematics and mathematics that informed the progress of art. A painting, print, or sculpture will be presented along with the historical background of the work of art and the artist and the history of the mathematics related to the art. Mathematicians will include Albrecht Dürer, Helaman Ferguson, and Piero Della Francesca. Artists will include, Max Bill, Salvador Dali and Naum Gabo.

   

11 October:  Fall break -- no seminar

18 October:
Kristin Camenga (Cornell)
Variation in Writing Assignments in Mathematics Classes

Abstract:  Writing in mathematics classrooms has been gaining momentum in the last fifteen years. Some of the reasons teachers use writing are to get all students actively involved in learning, to give the teacher feedback on student understanding, to help students reflect on their own thinking, and to prepare students for standardized tests which now require written answers. Many different forms of writing are being used, from note-taking to project reports to journals, but assignments with the same name may vary widely between teachers. I am currently beginning research to categorize the different types of writing assignments currently used in secondary math classrooms and to provide a vocabulary for the different elements of the writing tasks. This research is an attempt to give mathematics teachers more specific ideas about how they can use writing in their classrooms and help them communicate specifically how they are using writing, as well as providing a detailed basis for further research on the effects of different kinds of writing on student learning. I will give some background on the writing to learn and writing across the curriculum movements and why I think that the use of writing in mathematics classrooms requires specific attention. Also, I will share some assignments I have collected from teachers and share some preliminary thoughts on how they might be analyzed and categorized. References, contacts, and feedback on the research so far will be solicited.

 

25 October:
Marita Hyman (Cornell, Anthropology)

MATHEMATICS AND THE ABORIGINAL IMAGINATION: CONVERGENCES AND CONFLICTS IN NORTHEASTERN ARNHEM LAND

Abstract: Government agencies in the United States and Australia are presently confronting a similar dilemma with regard to the teaching and learning of mathematics in multi and bicultural settings. To what extent does the common assumption of mathematical universality undermine efforts to provide a culturally responsive classroom experience for multi-cultural learners of mathematics? In both of these countries there is a growing awareness that the translation of mathematical ideas to schoolchildren remains fraught with challenges due, in part, to standard curricula and assessment techniques. These techniques largely ignore the cultural, social and historical construction of mathematical ideas. Faced with similar problems, educational organizations in both the United States and Australia are turning to similar means to study this question. Currently, in both countries, government agencies are openly supporting a burgeoning interdisciplinary dialogue among mathematicians, sociologists, anthropologists, and practicing educators working in multi or bicultural settings to explore the embeddedness of math within specific sociocultural contexts. For example, past work with Yolngu speakers from Arnhem Land, Australia suggests that notions of some mathematical ideas may be similar to mainstream conceptions that form the basis for the Northern Territory school curricula, but may be expressed differently. Specifically, there appears to be a similarity in the recursive logic in both conventional number and Yolngu kinship systems. Aboriginal ideas may also differ dramatically from mainstream conceptions, as for example the perceptions of patterns, space and spatial relationships between objects. My goals for this presentation will be to stimulate conversation on the nature of mathematical ideas and raise possible implications of this discussion for teachers and learners in Indigenous settings, particularly the specific sociocultural context in which I am working in northeastern Arnhem Land.


1 November: 
Susan Piliero (Cornell)

Discussion of Schoenfeld's paper on the "Math Wars"

Abstract: Participants are requested to read: Alan H. Schoenfeld (2003). Math Wars.

Most individuals believe that "math is math", a discipline both universally correct and culture-free. However, notions such as what "counts" as mathematics, how to best teach mathematics, how  to assess  "mathematical competence" etc. are value-laden and controversial. In this paper Schoenfeld describes the historical context for today's "math wars" between proponents of Standards-based reform mathematics (pejoratively referred to as "fuzzy math") and more traditional anti-reformists, analyzes the assumptions underlying each position, and attempts to define a middle ground between these two seemingly dichotomous positions, for the good of America's children, who are caught in the crossfire.

8 November:   no seminar, see 11 November

Thursday, 11 November. 10:10am-11:00am, Malott 205
Deborah Loewenberg Ball and Hyman Bass, University of Michigan
Teaching Mathematical Reasoning

Participants are requested to read two book chapters in preparation for the seminar:

Ball, D. L. , & Bass, H. (2003). Making mathematics reasonable  in school. In J. Kilpatrick, W. G. Martin,  and D. Schifter (Eds.), A Research Companion to Principles  and Standards for School Mathematics, (pp.  27-44). Reston, VA: National Council of Teachers of Mathematics. pdf


Ball, D. L., and Bass, H. (2000). Making believe: The collective  construction of public mathematical knowledge in the elementary classroom. In  D. Phillips (Ed.), Yearbook of the National Society for the Study of Education,  Constructivism in Education, (pp. 193-224). Chicago: University of Chicago  Press. pdf

15 November:  
Sarah Manning (NYC),
MATH DIFFICULTIES:  What we know, what we do not know, & what we can do.

Abstract:  Two-thirds of our nation's 8th graders perform below the basic level of proficiency in mathematics.  Sub-populations of students within this performance bracket are struggling with learning disabilities, emotional-behavioral disorders, attention issues, and inadequate instruction.  Most instruction and assessment methods lack the necessary differentiations to address these learning conditions. Additionally, there is a shortage of math and special education instructors. 

Recent changes in federal funding have targeted math cognition research.  This field is in a nascent stage. Nonetheless, there is a growing body of research that begins to provide guidance with regard to the construction of effective intervention programs for these students, their teachers, and their caregivers.

In this presentation, I will present a working definition of math difficulties, briefly describe the different factors that contribute to math difficulties, and provide concrete examples of how these issues are/are not being addressed in classrooms today.   Additionally, I will describe our research - in particular the computer-based tools and services currently in development to address this area of math education. 

22 November:   
David Henderson
"Culturally Responsive Mathematics Curriculla" -- Report from the NSF conference

ABSTRACT: I will report on my reactions to the recent NSF conference and bring up for discussion some of the issues that the conference raised in me. This will include:


29 November:    TBA

6 December:   TBA