Mathematics Teacher, May 2008

The May issue of Mathematics Teacher has also arrived. The list of contents presents the following articles, whereas the last one is a free preview article:

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

Teaching Children Mathematics, May 2008

The May issue of Teaching Children Mathematics has also appeared, and it contains the following articles:

Instructional Strategies for Teaching Algebra in Elementary School: Findings from a Research-Practice Collaboration

Darrell Earnest and Aadina A. Balti

Insights into Our Understandings of Large Numbers

Signe E. Kastberg and Vicki Walker

The first article is a free preview article.

Mathematics Teaching in the Middle School, May 2008

The May issue of Mathematics Teaching in the Middle School has arrived, and it contains the following articles:

Teaching and Learning Mathematics through Hurricane Tracking

Maria L. Fernandez and Robert C. Schoen

The Importance of Equal Sign Understanding in the Middle Grades

Eric J. Knuth, Martha W. Alibali, Shanta Hattikudur, Nicole M. McNeil and Ana C. Stephens

Exploring Segment Lengths on the Geoboard

Mark W. Ellis and David Pagni

What Do Students Need to Learn about Division of Fractions?

Yeping Li


MTL, Issue 2, 2008

Issue 2 of Mathematical Thinking and Learning has appeared with the following articles:

The issue also includes an editorial and a book review of the new book on the KappAbel mathematics competition by Tine Wedege and Jeppe Skott.

ESM, May 2008

Students' encounter with proof

Kirsti Hemmi from Stockholm University has written an article that was recently published (online first) in ZDM. The article is entitled: "Students’ encounter with proof: the condition of transparency". Here is the abstract of the article:
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.

New ESM-articles

A couple of new (online first) articles have been published by Educational Studies in Mathematics:
  • David Tall has written an obituary of Jim Kaput: "James J. Kaput (1942–2005) imagineer and futurologist of mathematics education". Abstract: Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his ‘big ideas’ as he lived through the most amazing innovations in technology that have changed our lives more in a generation than in many centuries before. His vision continues as is exemplified by the collected papers in this tribute to his life and work.
  • Roberta Y. Schorr and Gerald A. Goldin have written an article called "Students’ expression of affect in an inner-city simcalc classroom". Abstract: This research focuses on some of the affordances provided by SimCalc software, suggesting that its use can have important consequences for students’ mathematical affect and motivation. We describe an episode in an inner-city SimCalc environment illustrating our approach to the study of affect in the mathematics classroom. We infer students’ development of new, effective affective pathways and structures as they participate in conceptually challenging mathematical activities. Our work highlights the roles of dignity and respect in creating an emotionally safe environment for mathematical engagement, and makes explicit some of the complexity of studying affect.
  • Richard Lesh, James A. Middleton, Elizabeth Caylor and Shweta Gupta have written an article entitled: "A science need: Designing tasks to engage students in modeling complex data". Abstract: In this information age, the capacity to perceive structure in data, model that structure, and make decisions regarding its implications is rapidly becoming the most important of the quantitative literacy skills. We build on Kaput’s belief in a Science of Need to motivate and direct the development of tasks and tools for engaging students in reasoning about data. A Science of Need embodies the utility value of mathematics, and engages students in seeing the importance of mathematics in both their current and their future lives. An extended example of the design of tasks that require students to generate, test, and revise models of complex data is used to illustrate the ways in which attention to the contributions of students can aid in the development of both useful and theoretically coherent models of mathematical understanding by researchers. Tools such as Fathom are shown as democratizing agents in making data modeling more expressive and intimate, aiding in the development of deeper and more applicable mathematical understanding.


New articles

A couple of new articles have been published online in International Journal of Mathematical Education in Science and Technology:

  • "Improving senior secondary school students' attitude towards mathematics through self and cooperative-instructional strategies" by S. A. Ifamuyiwa and M. K. Akinsola. Abstract: This study investigated the effects of self and cooperative-instructional strategies on senior secondary school students' attitude towards Mathematics. The moderating effects of locus of control and gender were also investigated. The study adopted pre-test and post-test, control group quasi-experimental design using a 3 × 2 × 2 factorial matrix with two experimental groups and one control group. Three hundred and fifty SSS II students from six purposively selected secondary schools in Ijebu-North Local Government Area of Ogun State were the subjects. Three instruments were developed, validated and used for data collection. Analysis of Covariance (ANCOVA) and Scheffé post hoc analysis were the statistics used for data analysis. Findings showed that the treatments had significant main effect on students' attitude towards Mathematics. The participants exposed to self-instructional strategy had the highest post-test mean attitude score. The study found no significant main effects of locus of control and gender on the participants' attitude towards Mathematics. It was concluded that Mathematics teachers should be trained to use self and cooperative learning packages in the classroom, since the strategies are more effective in improving students' attitude towards Mathematics than the conventional method.
  • "Algorithmic contexts and learning potentiality: a case study of students' understanding of calculus" by Kerstin Pettersson and Max Scheja. Abstract: The study explores the nature of students' conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students' understandings of the two concepts. Intentional analysis of the students' written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very 'operations' of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.


Norma 08 - Day 4

Plenary lecture: Michèle Artigue

Title: Didactical Design in Mathematics Education

Current context
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

  • math education is a sensitive domain for our societies
  • increasing pressure of international evaluations, tests, etc.
  • increasing debates about curriculum reforms and the supposed influence of didactical research on these
External side (Burkhardt and Schoenfeld, 2003)
  • Start from evidence that educational research does not often lead directly to practical advances
  • Development of "engineering research"
  • Design experiments - promising model of interaction
Internal side (Cobb, 2007)
  • Multiplicity of theoretical frame
  • Two criteria proposed
  • Multi-level vision of design
  • Experimental design has to be its unique methodology
Didactical design - mathematics education
Diversity of perspectives
  • Didactical design as research tool
  • Didactical design as development tool
  • Math education as design experiment
Didactical engineering (emerge in the early eighties)
Initial distinction between phenomenotechnique and didactical engineering

Didactical engineering as a research tool, shaped by theoretical foundations
- Influence of the theory of didactical situations (Brousseau)
    Learning processes as adaptation processes (Piaget)
    Focus on situation and milieu
    Distinction between different functionalities of mathematics knowledge (acting, expressing and communicating, proving)
    The teacher role

Didactical engineering - the predominant research methodology in the French didactic culture (esp. in the eighties)

Relationships between research and practice
  • Relationship that is not under theoretical control
  • Products communicated in different arenas (publications, teacher formation, etc.)
Relationships between research and practice
  • Relationship that is not under theoretical control
  • Products communicated in different arenas (publications, teacher formation, etc.)
  • Results reproduced, used in textbooks, etc.
Subsequent evolution
  • Better understanding of teachers' practices
  • Development of less invasive research methodologies
  • New theoretical constructions
  • Substantial body of research that impacts the vision of didactical design
Didactical design today
  • Still a tool widely used
  • Same epistemological sensitivity
  • Importance of interaction with the milieu, more sophisticated vision of the teacher role
  • Same importance to the a priori analysis
  • (Differences on the view of the teacher in France and Italy, for instance)
  • Didactical engineering still a research tool
  • Practical part - type of task (technique)
  • Theoretical part - technicological discourse (theory)

Norma 08 - Day 3

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:

  • Skills for everyday life and activities for workplaces (useful)
  • Sharing cultural heritage
  • Learning to think critically (formative goal)
Examples of critical thinking in classrooms (Harols Fawcett, 1938)
  • Selecting significant words and phrases, careful definition
  • Require evidence to support conclusions
  • Analyzing evidence
  • Recognize hidden assumptions
  • Evaluate the argument itself
  • etc.
"Everybody counts" (National Academy of Sciences, 1989)
Help develop critical habits of mind, understand chance, value proof etc. (p. 8).

The notion of "thinking critically" - what is it?
Fawcett - precision of language
Swedish example - relation to environment, etc. (global view, more vague)

Is there an epistemic quality of mathematics that is linked to thinking critically? (interesting question!)

Recent descriptions - renaissance of formative and methodological goals
- Communicating mathematically (discuss, advantages, disadvantages, etc.)

    Communicating freely and critical thinking takes place in some sort of an ideal democratic environment.

    Are mathematics classrooms ideal speech communities?

- Learning to model and solve problems mathematically
    Danger of overemphasizing utility (connections with engineering, social science departments, etc.)

- Recruitment into the mathematics, science and engineering pipeline as justification (economic development in a country, etc.)
    There has to be a "critical mass" from which to select future mathematicians. (similar argument to sports, being successful in sports)

How successful are the students in compulsory mathematics courses for all?
International tests (PISA, TIMSS, etc.) - only a small percentage will reach the highest level. Discussions of "average achievements", comparisons between countries.

Compulsory mathematics, not for all. Global failure of math education?
Which groups of students are successful/less successful? (interesting question)

PART 2 - "Mathematics for all!" (mission statement)
  • Demographic development (declining number of students, in many industrialised countries)
  • "Learning to leave?" - Successful students often end up moving away (from their country, local area, etc.) - How can a mathematics curriculum serve the local needs of local communities?
  • Organization of participation - students' choices. Why do so many students choose not to pursue further studies in mathematics after the compulsory course? To what extent should we "force" them to choose mathematics?
  • Changes in social contexts
  • Increased stress on instrumental knowledge and of the marketability of skills. Danger of oppositions between rationales for mathematics and liberal arts for instance.
  • Professional groups fighting against the "contamination of mathematical knowledge". Consequence of shift towards process skills in the curricula. (Back to basics movement, math wars, etc.)
  • A widening gap of mathematical knowledge between constructors and consumers of mathematics (Skovsmose, 2006) - threat to democracy (you have to rely on the experts).
  • The "de-mathematizing" and restricting effects of mathematical technology. Use of technology liberates us from the details of mathematics.
  • Confrontations of local knowledge and mathematical knowledge acquired at school. (Students don't appear to use the mathematics they learned in school outside the classroom)
Research is addressing some of these challenges:
  • Classroom research looking into these speech communities
  • Concern about "mathematical literacy"
  • Empirical studies of local mathematical practices at work-places (and local communities)
  • Students' goals and motives
  • Consequences of changes in students' backgrounds
  • Problem of transition between different tracks of mathematics education
Jablonka doesn't think there will be a universal curriculum for all.


Video-based curriculum

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:

Although video cases are increasingly being used in teacher education as a means of situating learning and developing habits of reflection, there has been little evidence of the outcomes of such use. This study investigates the effects of using a coherent video-case curriculum in a university mathematics methods course by addressing two issues: (1) how the use of a video-case curriculum affects the reflective stance of prospective teachers (PTs); and (2) the extent to which a reflective stance developed while reflecting on other teachers’ practice transfers for reflecting on one’s own practice. Data sources include videotapes of course sessions and PTs’ written work from a middle school mathematics methods course that used a video-case curriculum as a major instructional tool. Both qualitative and quantitative analytical methods were used, including comparative and chi-square contingency table analyses. The PTs in this study showed changes in their level of reflection, their tendency to ground their analyses in evidence, and their focus on student thinking. In particular, they began to analyze teaching in terms of how it affects student thinking, to consider multiple interpretations of student thinking, and to develop a more tentative stance of inquiry. More significantly, the reflective stance developed via the video curriculum transferred to the PTs’ self-reflection in a course field experience. The results of this study speak to the power of using a video-case curriculum as a means of developing a reflective stance in prospective mathematics teachers.

Norma 08 - Day 2

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?

  • Prepare students
  • Allows for different types of questions
  • Assessments should reflect learning
  • etc.
Drijvers goes on to present some examples from other countries (France, Germany, etc.) of tasks where technological tools are involved. The use of tools in the tasks is often questionable (or non-existent). In some examples, graphing calculators are allowed, but the tasks do not indicate any usage of these tools. Drijvers also presents some examples that are interesting to discuss from the point of view of "realism" and "authenticity", and he takes up this discussion in a few cases. Ends the section of examples with an example from the Netherlands, and he makes a humorous comment about this being the perfect example of a really good task. In discussing this example, Drijvers continually come back to the issue that this is something that you can imagine. And in the Dutch vocabulary, "realism" means something that you can imagine. Within a Dutch context, a realistic task is therefore a task that the students can imagine.

He then brings the discussion to a meta-level, introducing concepts like artifacts and instruments, and goes on with a presentation of what is called instrumental genesis.

Conclusions so far:
  • Assessment with technology is an issue in many countries
  • Reasoning, interpretation and explanation is also asked about (not just ICT-output)
  • Different ways of dealing with technology (discusses some trends)
Tools for digital assessments. Why digital assessment?
Discusses some of the limitations of software, types of feedback, etc.

All in all, an interesting presentation with several important issues being raised.

Norma08 - Day 1

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:

  • Teachers' knowledge
  • Teachers' beliefs
In the 1980s - a shift in the view of learning, mathematics, etc. changed the whole field of school mathematics (fallibilism, social constructivism). Teachers placed in a new role, as opposed to before. Teachers supposed to understand what students are doing, and to guide their learning. New role: planned unpredictability (interesting concept!)

Teachers' knowledge
Displays a couple of examples from the literature that displays teachers' (lack of) knowledge about mathematics (for teaching). Perhaps pre-service education is not what it should have been?
The importance of Shulman's work. The article "Those who understand..." A main idea: content matters! Two of Shulman's concepts important:
  • Content knowledge
  • Pedagogical content knowledge
What is it that teachers' should know about? (content knowledge)
What is it that makes a topic difficult? (pedagogical content knowledge)

The mathematics of the classroom - the mathematics of the mathematician.
Liping Ma - asked teachers in China and the US lots of questions concerning basic mathematics. Many teachers (esp. the US teachers) weren't able to solve the problems. A basic question for her - What is the relevant knowledge needed by teachers? American teachers - list of disconnected procedures. Chinese teachers - alle these procedures were related. "Understanding with bredth."
D. Ball, H. Bass et al. Classroom based approach. Mathematical challenges from the classroom. (Elements from the LMT measurements) D. Ball calls it "unpacking mathematical knowledge" - digging deeply into the conceptual issues.

A shift in the area of developing a knowledge base for teaching:
  • From - number of courses
  • to - knowledge of school mathematics (L. Ma)
  • to - knowing in action (D. Ball)
Beliefs research in math education
In order for any reform to have an impact there needs to be a change in the teachers' beliefs.
Developing and changing beliefs. Several suggestions and attempts (see points in slide).
Relationship between beliefs and practice.

A moral so far: There is a need for contextualizing mathematics education to the act of teaching.

Discussion of the relationship (or expected relationship) between development of curriculum and curriculum material and teaching practice.
As researchers, a main issue is the one of theorizing practice.

Poses an interesting question: In what sense is mathematics education an applied field?
Points at an interesting quote by P. Cobb about the issue of mathematics education (research).
Interesting model about the dimensions of research (by Stokes).

A main issue for research in math education is maybe not about theorizing, but about having impact on practice.

The end of the talk filled with intriguing questions and interesting metaphors. (Thaetetus' ship - if you replace a plank, and then another plank, when is it no longer Thaetetus' ship, but a new one?)

All in all a very interesting presentation! Hopefully these notes could be deciphered by others as well...


Learning from group discussions

Keith Weber, Carolyn Maher, Arthur Powell and Hollylynne Stohl Lee has written an article called "Learning opportunities from group discussions: warrants become the objects of debate" that has recently been published online by Educational Studies in Mathematics. The article deals with the interesting issues concerning discourse and learning opportunities in group discussions. Here is the abstract of the article:
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.

New ZDM-articles

Two new articles has recently been published (online first) by ZDM. The first article is written by Man-Keung Siu, and it is entitled "Proof as a practice of mathematical pursuit in a cultural, socio-political and intellectual context". Here is the abstract of the article:
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.


Teaching Statistics, May 2008

The May issue of Teaching Statistics has arrived. This is not a journal I have followed in the past, I must admit, but there are some interesting articles in this issue. One article is entitled: "Inspired by Statistics?" The introduction to the article at least made me think:

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
about statistics?

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
by Napoleon's army in his disastrous Russian campaign of 1812
). She describes the way she planned and worked with this task in her teaching, and then finishes off with a discussion about inspiration for future tasks.

Argumentation and algebraic proof

B. Pedemonte has written an article that has recently been published (online first) in ZDM. The article has a focus on a "core activity" in mathematics, and it is called: "Argumentation and algebraic proof". Here is the abstract of the article:
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).


JRME, May 2008

The May issue of Journal for Research in Mathematics Education (JRME) has already arrived, and it contains the following articles:

ZPC and ZPD: Zones of Teaching and Learning

Anderson Norton and Beatriz S. D'Ambrosio

The Impact of Middle-Grades Mathematics Curricula and the Classroom Learning Environment on Student Achievement

James E. Tarr, Robert E. Reys, Barbara J. Reys, Óscar Chávez, Jeffrey Shih and Steven J. Osterlind

Learning to Use Fractions: Examining Middle School Students' Emerging Fraction Literacy

Debra I. Johanning

The Linear Imperative: An Inventory and Conceptual Analysis of Students' Overuse of Linearity

Wim Van Dooren, Dirk De Bock, Dirk Janssens and Lieven Verschaffel

Teaching With Games of Chance: A Review of The Mathematics of Games and Gambling

Laurie Rubel

NOMAD, March 2008

The first issue of NOMAD this year has finally arrived, at least the web page has finally been updated to indicate that. Unfortunately, the articles are not available online, but you can read the abstracts (and the editorial in its entirety). The issue contains the following articles:


Studying new forms of participation

Stephen J. Hegedus and William R. Penuel wrote an article that was recently published online in Educational Studies in Mathematics. The article is called "Studying new forms of participation and identity in mathematics classrooms with integrated communication and representational infrastructures", and here is the abstract of the article:
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.


Rounded fractals

International Journal of Computers for Mathematical Learning has a column called "Computational Diversions". Michael Eisenberg recently wrote an article/entry in this column called "Rounded Fractals". The article is both practical and interesting, and it provides several examples concerning the generation of fractal designs. In the beginning of the article, he mentions turtle geometry (Logo), but the examples are made by making use of the method of iterated function systems. The article also contains a challenge, so anyone interested in fractals might want to take a look.


NCTM Annual Meeting

The NCTM Annual Meeting started today in Salt Lake City, Utah. The theme for the conference is "Becoming Certain About Uncertainty". The conference has lots of interesting sessions and exhibitions. The program is downloadable as a pdf, but if you want the full program, it is 17,3 MB! You might also want to take a look at the rather impressive list of featured speakers.

Stability of teachers' classroom activity

M. Pariès, A. Robert and J. Rogalski recently published an article called "Analyses de séances en classe et stabilité des pratiques d’enseignants de mathématiques expérimentés du second degré" in Educational Studies in Mathematics. The article is in French, but here is the abstract in English:
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.

Student presentations in the classroom

David L. Farnsworth has written an article called Student presentations in the classroom. The article was published in International Journal of Mathematical Education in Science and Technology today. Here is the abstract:
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
are made.

Analyticity without differentiability

A new article has appeared in International Journal of Mathematical Education in Science and Technology. The article is written by E. Kirillova and K. Spindler, and it is entitled: Analyticity without differentiability. Her is the abstract of the article:
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.


Teaching Children Mathematics, April 2008

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)

Mathematics Teaching in the Middle School, April 2008

The April issue of Mathematics Teaching in the Middle School has arrived, and it presents the following articles:

By Way of Introduction: Developing Mathematical Understanding through Representations

Developing Mathematical Understanding through Multiple Representations
by Preety N. Tripathi (free preview article)

Promoting Mathematics Accessibility through Multiple Representations Jigsaws
by Wendy Pelletier Cleaves

Oranges, Posters, Ribbons, and Lemonade: Concrete Computational Strategies for Dividing Fractions
by Christopher M. Kribs-Zaleta

Student Representations at the Center: Promoting Classroom Equity
by Kara Louise Imm, Despina A. Stylianou and Nabin Chae

Analyzing Students' Use of Graphic Representations: Determining Misconceptions and Error Patterns for Instruction
by Amy Scheuermann and Delinda van Garderen

Developing Meaning for Algebraic Symbols: Possibilities and Pitfalls
by John K. Lannin, Brian E. Townsend, Nathan Armer, Savanna Green and Jessica Schneider

Sense-able Combinatorics: Students' Use of Personal Representations
by Lynn D. Tarlow

The Role of Representations in Fraction Addition and Subtraction
by Kathleen Cramer, Terry Wyberg and Seth Leavitt

After the Math Panel

A little less than a month ago, the National Mathematics Advisory Panel published their final report on the future of mathematics education in the U.S. The report has raised much discussion in the U.S., and today I came across an interesting blog called After the Math Panel. In this blog, an educator and mom gives us her opinions and analyses of the report. The blog contains some interesting and readable summaries of the report, and I think it is worth reading!


From static to dynamic mathematics

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:

(...) present new theoretical perspectives on the design and use of digital technologies, especially dynamic mathematics software and “classroom networks.”
In the article they present some interesting perspectives on the historical development on media, from static to dynamic, and they discuss some dynamical perspectives related to variation and geometry (dynamic geometry, like Cabri, Geometer's Sketchpad, etc.). Here is the abstract of this interesting article:

The nature of mathematical reference fields has substantially evolved with the advent of new types of digital technologies enabling students greater access to understanding the use and application of mathematical ideas and procedures. We analyze the evolution of symbolic thinking over time, from static notations to dynamic inscriptions in new technologies. We conclude with new perspectives on Kaput’s theory of notations and representations as mediators of constructive processes.


Awards and medals

According to the Math Forum, the following people have been given awards in our field recently:
  • Anna Sfard has received the Hans Freudenthal Medal for 2007 (see this post for more information)
  • Jeremy Kilpatrick has received the Felix Klein Medal for 2007 (see this post for more information)
Both news were posted at the request of Mogens Niss, who is Chair of the ICMI Awards Committee. The posts linked above give a nice overview of the research efforts of these two distinguished scholars.

Aztec math

Both National Geographic and Scientific American published articles about Aztec mathematics yesterday. The article in National Geographic focused on a specialized arithmetic that Aztec mathematicians developed to measure tracts of taxable land. In this arithmetic they used symbols like hearts, hands and arrows, which probably had a relation to the human body. The article refers to a study that was reported in this week's issue of Science. Science covers the topic in a news story as well as the research article. The Scientific American article also focus on the hearts and arrows, and they also refer to another article (in Science) about the Aztec number system. So, for those interested in Aztec mathematics in particular, and history of mathematics in general, there are lots of interesting and up to date articles to read here!


Implementing Kaput's research programme

Celia Hoyles and Richard Noss recently published an article called "Next steps in implementing Kaput's research programme" in Educational Studies in Mathematics. These two distinguished professors have written a multitude of books and articles together in the past, so you might have come across something written by "Hoyles and Noss" before. In this particular article, they explore and discuss some key ideas from Jim Kaput and connect them to their own research. Here is the abstract of the article:
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.


Testing, testing and comparing test results...

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).

"Joined Up Mathematics"

The Annual Conference 2008 of the MA/ATM, entitled "Joined Up Mathematics" starts today at Keele University, UK. The conference is closing on Saturday. The opening speaker of today's program is Anne Watson. Her presentation has been given the title: "Fragments and Coherence". Other keynote speakers are John Mason, Rob Eastaway and Mike Askew.

IJMEST, vol. 39, issue 3

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
DOI: 10.1080/00207390701690303

Author: Betty McDonald
DOI: 10.1080/00207390701688141

Author: Stan Lipovetsky
DOI: 10.1080/00207390701639532

Authors: Modestina Modestou; Iliada Elia; Athanasios Gagatsis; Giorgos Spanoudis
DOI: 10.1080/00207390701691541

Authors: Juana-Maria Vivo; Manuel Franco
DOI: 10.1080/00207390701691566

Author: I. S. Jones
DOI: 10.1080/00207390701734523

Author: Jesper Rydén
DOI: 10.1080/00207390701639508


Excellent math blog

There are many academic journals in our field, and there are many articles to read if you want to keep up. On some occasions though, a couple of days might pass by without any new publications from the major journals. On instances like that, you might want to take a look at some of the mathematics related blogs on the internet. One of my favorites is Wild About Math! by Sol Lederman. This blog presents several interesting articles about mathematics and how to learn "to get wild about Math", and a regular feature of the blog is the "Monday Math Madness contest" (Sol loves mathematical problems and puzzles). You can also find a list of links to other web pages with mathematical problems and puzzles.