Deep Thoughts on the River Crossing Game
Dan Canada and Dave Goering
The Power of Investigative Calculus Projects
John Robert Perrin and Robert J. Quinn
Why Aren't They Called Probability Intervals?
Thomas F. Devlin
The condition of transparency refers to the intricate dilemma in the teaching of mathematics about how and how much to focus on various aspects of proof and how and how much to work with proof without a focus on it. This dilemma is illuminated from a theoretical point of view as well as from teacher and student perspectives. The data consist of university students’ survey responses, transcripts of interviews with mathematicians and students as well as protocols of the observations of lectures, textbooks and other instructional material. The article shows that the combination of a socio-cultural perspective, Lave and Wenger’s and Wenger’s social practice theories and theories about proof offers a fresh framework for studies concerning the teaching and learning of proof.
A couple of new articles have been published online in International Journal of Mathematical Education in Science and Technology:
Plenary lecture: Michèle Artigue
Title: Didactical Design in Mathematics Education
Increasing interest in design issues. Reflection on the value of the outcomes of didactical research, and impact of research on educational practices.
Motivation: external and internal
A bit late, but here are my notes from the plenary lecture from the third day at Norma08:
Plenary lecture - Eva Jablonka
PART 1 - "Mathematics for all. Why? What? When?"
Math as a core subject in compulsory education (empirical fact). Industrialised countries provide basic maths for all (in school). BUT - many children don't go to school in several countries around the world. It varies between countries when children can stop taking mathematics courses.
Mathematics for all, beyond primary level - why?
Goals as an apologetic discourse.
Common list of justifications:
S.L. Stockero has written an article that has recently been published in Journal of Mathematics Teacher Education. The article is entitled: Using a video-based curriculum to develop a reflective stance in prospective mathematics teachers. Here is the abstract of the article:
Plenary lecture - P. Drijvers
Title: "Tools and tests"
Drijvers starts off giving some introductory notes about the Freudenthal Institute.
"Tools" = technological tools in this connection.
Why use tools and tests? The teaching and learning should be reflected in the assessment, and assessment should be driven by teaching and learning.
What are we actually assessing? Tools skills or mathematical skills?
Tests with tools, why would we do it?
Plenary - J. Skott
The Norma 08 Conference takes place in Copenhagen this week, and I am attending. I will therefore have a focus on this conference this week. The first plenary lecture was presented by Danish researcher Jeppe Skott, and here are my notes from the presentation (which was very interesting by the way). I also plan on covering the conference on twitter, so take a look there as well for live reports!
Title: "The education and identity of mathematics teachers"
Research on mathematics teachers has grown tremendously during the past 20-30 years. Skott starts with a presentation about publications, journals, monographs, etc.
Two main concerns:
In the mathematics education literature, there is currently a debate about the mechanisms by which group discussion can contribute to mathematical learning and under what conditions this learning is likely to occur. In this paper, we contribute to this debate by illustrating three learning opportunities that group discussions can create. In analyzing a videotaped episode of eight middle school students discussing a statistical problem, we observed that these students frequently challenged the arguments that their colleagues presented. These challenges invited students to be explicit about what mathematical principles, or warrants, they were implicitly using as a basis for their mathematical claims, in some cases recognize the modes of reasoning they were using were invalid and reject these modes of reasoning, and in other cases, attempt to provide deductive support to justify why their modes of reasoning were appropriate. We then describe what social and environmental conditions allowed the discussion analyzed in this paper to occur.Interestingly enough, they use Toulmin's model of argumentation as a part of the theoretical framework for their analyses. The research that they report and discuss in this article occurred in the context of a research project called "Informal Mathematics Learning", which is a project supported by the NSF.
Through examples we explore the practice of mathematical pursuit, in particular on the notion of proof, in a cultural, socio-political and intellectual context. One objective of the discussion is to show how mathematics constitutes a part of human endeavour rather than standing on its own as a technical subject, as it is commonly taught in the classroom. As a “bonus”, we also look at the pedagogical aspect on ways to enhance understanding of specific topics in the classroom.The other article is called "Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework", and it is written by Susanne Prediger, Angelika Bikner-Ahsbahs and Ferdinando Arzarello. The article has a focus on the diversity of theories in mathematics education research, and how we can deal with that. Here is the abstract:
The article contributes to the ongoing discussion on ways to deal with the diversity of theories in mathematics education research. It introduces and systematizes a collection of case studies using different strategies and methods for networking theoretical approaches which all frame (qualitative) empirical research. The term ‘networking strategies’ is used to conceptualize those connecting strategies, which aim at reducing the number of unconnected theoretical approaches while respecting their specificity. The article starts with some clarifications on the character and role of theories in general, before proposing first steps towards a conceptual framework for networking strategies. Their application by different methods as well as their contribution to the development of theories in mathematics education are discussed with respect to the case studies in the ZDM-issue.
The article goes on to discuss views on statistics, before the author describes one of her favorite tasks about Minard's map (a famous combined map, graph and chart that documents the losses suffered
What do you think of when you hear the word ‘statistics’?
reading any further, give an instant view on how statistics makes you
feel and how your learners may feel. Why do you think the way you do
This paper concerns a study analysing cognitive continuities and distances between argumentation supporting a conjecture and its algebraic proof, when solving open problems involving properties of numbers. The aim of this paper is to show that, unlike the geometrical case, the structural distance between argumentation and proof (from an abductive argumentation to a deductive proof) is not one of the possible difficulties met by students in solving such problems. On the contrary, since algebraic proof is characterized by a strong deductive structure, abductive steps in the argumentation activity can be useful in linking the meaning of the letters used in the algebraic proof with numbers used in the argumentation. The analysis of continuities and distances between argumentation and proof is based on the use of Toulmin’s model combined with ck¢ model.Algebra is used in several different domains in mathematics, but this article has a focus on the algebra that is taught and learned in secondary school (Grade 12 and 13). After having elaborated and presented a theoretical framework for her analysis of proofs, Pedemonte presents some data that has been collected from prospective primary school teachers. These students were attending a course at the University, and their solutions to two open problems were analyzed according to the theoretical framework (the solutions of 7 students' solutions to each of the two problems were analyzed).
Wireless networks are fast becoming ubiquitous in all aspects of
society and the world economy. We describe a method for studying the
impacts of combining such technology with dynamic,
representationally-rich mathematics software, particularly on
participation, expression and projection of identity from a local to a
public, shared workspace. We describe the types of mathematical
activities that can utilize such unique combinations of technologies.
We outline specific discourse analytic methods for measuring
participation and methodologies for incorporating measures of identity
and participation into impact studies.
In this paper we tackle the issue of an eventual stability of teachers’
activity in the classroom. First we explain what kind of stability is
searched and how we look for the chosen characteristics: we analyse the
mathematical activity the teacher organises for students during
classroom sessions and the way he manages the relationship between
students and mathematical tasks. We analyse three one-hour sessions for
different groups of 11 year old students on the same content and with
the same teacher, and two other sessions for 14 year old and 15 year
old students, on analogous contents, with the same teacher (another
one). Actually it appears in these two examples that the main
stabilities are tied with the precise management of the tasks, at a
scale of some minutes, and with some subtle characteristic touches of
the teacher’s discourse. We present then a discussion and suggest some
inferences of these results.
For many years, the author has been involving his students in classroom
teaching of their own classes. The day-to-day practice is described,
and the advantages and disadvantages for both the instructor and the
students are discussed. Comparisons with the Moore Method of teaching
In this article we derive all salient properties of analytic functions,
including the analytic version of the inverse function theorem, using
only the most elementary convergence properties of series. Not even the
notion of differentiability is required to do so. Instead, analytical
arguments are replaced by combinatorial arguments exhibiting properties
of formal power series. Along the way, we show how formal power series
can be used to solve combinatorial problems and also derive some
results in calculus with a minimum of analytical machinery.
NCTM journal: Teaching Children Mathematics has published the April issue of this year, and it has the following contents (articles):
Alice in Numberland: Through the Standards in Wonderland by Donna Christy, Karen Lambe, Christine Payson, Patricia Carnevale and Debra Scarpelli
Learning Our Way to One Million by David J. Whitin
Problem-Solving Support for English Language Learners by Lynda R. Wiest (free preview article)
Educational Studies in Mathematics recently published an article called: "From static to dynamic mathematics: historical and representational perspectives". The article is written by Luis Moreno-Armella, Stephen J. Hegedus and James J. Kaput. The point of departure for this article is the issue of new digital technologies, their capacities, issues concerning design and use of them, etc. They build upon one of Kaput's works on notations and representations, in order to:
We explore some key constructs and research themes initiated by Jim
Kaput, and attempt to illuminate them further with reference to our own
research. These ‘design principles’ focus on the evolution of digital
representations since the early 1990s, and we attempt to take forward
our collective understanding of the cognitive and cultural affordances
they offer. There are two main organising ideas for the paper. The
first centres around Kaput’s notion of outsourcing of processing power,
and explores the implications of this for mathematical learning. We
argue that a key component for design is to create visible, transparent
views of outsourcing, a transparency without which there may be as many
pitfalls as opportunities for mathematical learning. The second
organising idea is Kaput’s notion of communication and the importance
of designing for communication in ways that recognise the mutual
influence of tools for communication and for mathematical expression.
In 2003 (in the U.S.), the National Assessment of Educational Progress (NAEP) administered assessments in reading and mathematics for grades 4 and 8. Representative samples of students were made from about 100 public schools in each state. A research report called "Comparison Between NAEP and State Mathematics Assessment Results: 2003" now focus on the question whether these results are comparable to the results published by individual state testing programs. The entire report is available online (only!), and can be downloaded in PDF format (Vol I and II).
The introduction contains some interesting historical remarks about achievement testing in the U.S., and this might be interesting to non-Americans (like myself).
International Journal of Mathematical Education in Science and Technology has published their third issue (of 8) this year. In the table of contents, we find the following original articles:
Authors: K. Renee Fister; Maeve L. McCarthy
Author: Betty McDonald
Author: Stan Lipovetsky
Authors: Modestina Modestou; Iliada Elia; Athanasios Gagatsis; Giorgos Spanoudis
How does one assess the accuracy of academic success predictors? ROC analysis applied to university entrance factors
Authors: Juana-Maria Vivo; Manuel Franco
Author: I. S. Jones
Author: Jesper Rydén