Tools of American Mathematics Teaching, 1800-2000

The last issue of TCRecord includes a review of a book that I wasn't aware of before, but that certainly looks interesting: "Tools of American Mathematics Teaching, 1800-2000", by Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts. The book was published last year. Here is a taster of Alexander P. Karp's excellent review of the book:

In today’s classrooms graphing calculators have become routine, yet thousands of teachers can easily recall a time when they did not exist. Not so with the blackboard, which seems to us something that is almost as old as the idea of education itself. This, however, is by no means the case. Two hundred years ago, and for several decades afterwards, blackboards were a novelty in American classrooms and their use was regarded as a particular feature of teaching style. And indeed, the transition from small tablets made of slate to a large blackboard for the whole class went along with a transition to working simultaneously with a large group of students—a transition that can hardly be viewed as anything other than fundamental.

Science & Education, February 2009

The February issue of Science & Education has been published. None of the articles in this issue are directly related to mathematics education, and the theme of the issue is "Politics and philosophy of science". Still, the issue might be worth checking out, especially if you are interested in the above mentioned theme. 



The CERME 6 conference starts today in Lyon, France. The conference is organized by ERME, which is the European society for Research in Mathematics Education. The main aims of ERME are to:

(...)to promote communication, cooperation and collaboration in research in mathematics education in Europe

Unfortunately, I am not attending the conference myself, so I am not going to report from it. If you want to learn more about the scientific program for the conference, you can find it here. Below is the location of the conference venue:

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In search of an exemplary mathematics lesson in Hong Kong

Ida Ah Chee Mok has written an article that was published in ZDM on Thursday. The article is entitled In search of an exemplary mathematics lesson in Hong Kong: an algebra lesson on factorization of polynomials. The theoretical perspectives for this article are mathematical enculturation and the theory of learning through variation (variation theory). The study which is described in the article is part of the Learner's Perspective Study (LPS). This study
(...) has engaged researchers in the investigation of mathematics classrooms of teachers in Australia, China, the Czech Republic, Germany, Israel, Japan, Korea, the Philippines, Singapore, South Africa, Sweden and the USA.
Here is the article abstract:
The author here describes an exemplary grade-8 algebra lesson in Hong Kong, taken from the data of the learners’ perspective study. The analysis presents a juxtaposition of the researcher’s analysis of the lesson with the teacher and students’ perspectives of the lesson. The researcher’s perspective applies the theory of variation for which the main concern of learning is the discernment of the key aspects of the object of learning and that the description of variations delineates the potential of the learning space. Some persistent features were illustrated, namely, the teacher talk was a major input in teaching; the technique of variation was used in the design of the mathematical problems and the dimensions of variation created in the class interaction provided a potential learning environment; the teacher taking seriously the student factor into account in his philosophy and practice. From the standpoint of enculturation, the teacher’s influence as an enculturator is intentional, significant and influential.

Problem-solving and cryptography

Tobin White has written an interesting article about cryptography and problem solving. The article is entitled Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography, and the article was published online in Educational Studies in Mathematics on Thursday. Those of you who don't have a subscription to this journal will be interested to know that the article is an Open Access article, and it is therefore available to all! (Direct link to pdf download) Here is the abstract of the article:
This paper introduces an applied problem-solving task, set in the context of cryptography and embedded in a network of computer-based tools. This designed learning environment engaged students in a series of collaborative problem-solving activities intended to introduce the topic of functions through a set of linked representations. In a classroom-based study, students were asked to imagine themselves as cryptanalysts, and to collaborate with the other members of their small group on a series of increasingly difficult problem-solving tasks over several sessions. These tasks involved decrypting text messages that had been encrypted using polynomial functions as substitution ciphers. Drawing on the distinction between viewing functions as processes and as objects, the paper presents a detailed analysis of two groups’ developing fluency with regard to these tasks, and of the aspects of the function concept underlying their problem-solving approaches. Results of this study indicated that different levels of expertise with regard to the task environment reflected and required different aspects of functions, and thus represented distinct opportunities to engage those different aspects of the function concept.

Using history in mathematics education

Uffe Thomas Jankvist has written an article about using history in mathematics education. The article was published in Educational Studies in Mathematics last week, and it is entitled: A categorization of the “whys” and “hows” of using history in mathematics education. Here is the abstract of his article:
This is a theoretical article proposing a way of organizing and structuring the discussion of why and how to use the history of mathematics in the teaching and learning of mathematics, as well as the interrelations between the arguments for using history and the approaches to doing so. The way of going about this is to propose two sets of categories in which to place the arguments for using history (the “whys”) and the different approaches to doing this (the “hows”). The arguments for using history are divided into two categories; history as a tool and history as a goal. The ways of using history are placed into three categories of approaches: the illumination, the modules, and the history-based approaches. This categorization, along with a discussion of the motivation for using history being one concerned with either the inner issues (in-issues) or the metaperspective issues (meta-issues) of mathematics, provides a means of ordering the discussion of “whys” and “hows.”

Pursuing excellence

Rongjin Huang and Yeping Li have written an article called Pursuing excellence in mathematics classroom instruction through exemplary lesson development in China: a case study. The article was published online in ZDM on Friday. To me, this article is interesting for a few reasons:
  • It has a focus on teaching mathematics
  • It has a focus on how to develop exemplary lessons
  • It has a focus on learning from "master teachers"
  • It provides a nice insight into chinese mathematics teaching
Several aspects in this study remind me of the Lesson Study approach and theories related to Mathematical Knowledge for Teaching (MKT), both of which are among my main research interests. Here is an abstract of their article:
In this article, we aim to examine the features of mathematics classroom instruction excellence valued in China. The popular approach to pursuing mathematics classroom instruction excellence through exemplary lesson development is also investigated to demonstrate the nature of teaching culture that has been advocated and nurtured in China. Features of an exemplary lesson are analyzed in detail, and the practicing teacher’s experience through participating in the development of the exemplary lesson is examined as well. Finally, the implications of developing exemplary lessons for pursuing excellence in mathematics classroom instruction as a culturally valued approach in China are also discussed.


Re-mythologizing mathematics

David Wagner and Beth Herbel-Eisenmann have written an article entitled Re-mythologizing mathematics through attention to classroom positioning. the article was published online in Educational Studies in Mathematics on Tuesday. Here is their article abstract:
With our conceptualization of HarrĂ© and van Langenhove’s (1999) positioning theory, we draw attention to immanent experience and read transcendent discursive practices through the moment of interaction. We use a series of spatial images as metaphors to analyze the way positioning is conceptualized in current mathematics education literature and the way it may be alternatively conceptualized. This leads us to claim that changing the way mathematics is talked about and changing the stories (or myths) told about mathematics is necessary for efforts to change the way mathematics is done and the way it is taught.


Gem #4: Hardy's Apology

This gem from the history of mathematics is more recent. It was published in 1940 by British mathematician G.H. Hardy. The book/essay was written when Hardy (then 62) felt that he no longer had the ability to contribute to the field of mathematics. A main theme in the book is concerning mathematical beauty, and he believed that the most beautiful mathematics was that, which had no application! Luckily, this book is also in the public domain, and you can read it in below (or download the pdf):

A Mathematician's Apology


Students' use of technological tools

Ioannis Papadopoulosa and Vassilios Dagdilelis have written an article that was published online in the Journal of Mathematical Behavior yesterday. The article is entitled Students’ use of technological tools for verification purposes in geometry problem solving. Here is a copy of the article abstract:

Despite its importance in mathematical problem solving, verification receives rather little attention by the students in classrooms, especially at the primary school level. Under the hypotheses that (a) non-standard tasks create a feeling of uncertainty that stimulates the students to proceed to verification processes and (b) computational environments – by providing more available tools compared to the traditional environment – might offer opportunities for more frequent usage of verification techniques, we posed to 5th and 6th graders non-routine problems dealing with area of plane irregular figures. The data collected gave us evidence that computational environments allow the development of verification processes in a wider variety compared to the traditional paper-and-pencil environment and at the same time we had the chance to propose a preliminary categorization of the students’ verification processes under certain conditions.


JRME, January 2009

Journal for Research in Mathematics Education (JRME) has released the January issue of 2009 (vol. 40, issue 1). It contains the following list of articles:

Unfortunately, only the editorial is freely available for all to read. You might also be interested in looking up the issue as listed in the ProQuest database.

Students' perceptions

Mashooque Ali Samo has written an article called Students' Perceptions Abouth the Symbols, Letters and Signs in Algebra and How Do These Affect Their Learning of Algebra: A Case Study in a Govenrment Girls' Secondary School, Karachi. This article pays attention to misconceptions that arise in Algebra, and it has been published in International Journal for Mathematics Teaching and Learning. Here is the article abstract:
Algebra uses symbols for generalizing arithmetic. These symbols have different meanings and interpretations in different situations. Students have different perceptions about these symbols, letters and signs. Despite the vast research by on the students‟ difficulties in understanding letters in Algebra, the overall image that emerges from the literature is that students have misconceptions of the use of letters and signs in Algebra. My empirical research done through this study has revealed that the students have many misconceptions in the use of symbols in Algebra which have bearings on their learning of Algebra. It appears that the problems encountered by the students appeared to have connection with their lack of conceptual knowledge and might have been result of teaching they experience in learning Algebra at the secondary schooling level. Some of the findings also suggest that teachers appeared to have difficulties with their own content knowledge. Here one can also see that textbooks are also not presenting content in such an elaborate way that these could have provided sufficient room for students to develop their relational knowledge and conceptual understanding of Algebra. Moreover, this study investigates students‟ difficulty in translating word problems in algebraic and symbolic form. They usually follow phrase- to- phrase strategy in translating word problem from English to Urdu. This process of translating the word problem from English to their own language appears to have hindered in the correct use of symbols in Algebra. The findings have some important implications for the teaching of Algebra that might help to develop symbol sense in both students and teachers. By the help of symbol sense, they can use symbols properly; understand the nature of symbols in different situations, like, in functions, in variables and in relationships between algebraic representations. This study will contribute to future research on similar topics.

Preservice teachers' subject matter knowledge of mathematics

Ramakrishnan Menon has written an article entitled Preservice teachers' subject matter knowledge of mathematics. The article has been published in International Journal for Mathematics Teaching and Learning. Here is the abstract of the article:
Sixty four preservice teachers taking a mathematics methods class for middle schools were given 3 math problems: multiply a three digit number by a two digit number; divide a whole number by a fraction; and compare the volume of two cylinders made in different ways from the same rectangular sheet. They were to a) solve them, explaining their solution, b) classify them as easy, of medium difficulty, or difficult, explaining the rationale for their classification, and c) explain how they would teach/help children to solve them. Responses were classified under three categories of subject matter knowledge, namely traditional, pedagogical, and reflective. Implications of these categories to effective math teaching are then discussed.


Intuitive vs analytical thinking

Uri Leron and Orit Hazzan have written an article called Intuitive vs analytical thinking: four perspectives. The article was recently published online in Educational Studies in Mathematics. Here is the abstract of their article:
This article is an attempt to place mathematical thinking in the context of more general theories of human cognition. We describe and compare four perspectives—mathematics, mathematics education, cognitive psychology, and evolutionary psychology—each offering a different view on mathematical thinking and learning and, in particular, on the source of mathematical errors and on ways of dealing with them in the classroom. The four perspectives represent four levels of explanation, and we see them not as competing but as complementing each other. In the classroom or in research data, all four perspectives may be observed. They may differentially account for the behavior of different students on the same task, the same student in different stages of development, or even the same student in different stages of working on a complex task. We first introduce each of the perspectives by reviewing its basic ideas and research base. We then show each perspective at work, by applying it to the analysis of typical mathematical misconceptions. Our illustrations are based on two tasks: one from statistics (taken from the psychological research literature) and one from abstract algebra (based on our own research).

Using graphing software in algebra teaching

Kenneth Ruthven, Rosemary Deaney and Sara Hennesy have written an article that was published online in Educational Studies in Mathematics on Saturday. It is entitled: Using graphing software to teach about algebraic forms: a study of technology-supported practice in secondary-school mathematics. Besides having a focus on the use of graphing software, the article also discusses issues related to classroom teaching practice, teacher knowledge and teacher thinking. Here is the abstract of their article:
From preliminary analysis of teacher-nominated examples of successful technology-supported practice in secondary-school mathematics, the use of graphing software to teach about algebraic forms was identified as being an important archetype. Employing evidence from lesson observation and teacher interview, such practice was investigated in greater depth through case study of two teachers each teaching two lessons of this type. The practitioner model developed in earlier research (Ruthven & Hennessy, Educational Studies in Mathematics 49(1):47–88, 2002; Micromath 19(2):20–24, 2003) provided a framework for synthesising teacher thinking about the contribution of graphing software. Further analysis highlighted the crucial part played by teacher prestructuring and shaping of technology-and-task-mediated student activity in realising the ideals of the practitioner model. Although teachers consider graphing software very accessible, successful classroom use still depends on their inducting students into using it for mathematical purposes, providing suitably prestructured lesson tasks, prompting strategic use of the software by students and supporting mathematical interpretation of the results. Accordingly, this study has illustrated how, in the course of appropriating the technology, teachers adapt their classroom practice and develop their craft knowledge: particularly by establishing a coherent resource system that effectively incorporates the software; by adapting activity formats to exploit new interactive possibilities; by extending curriculum scripts to provide for proactive structuring and responsive shaping of activity; and by reworking lesson agendas to take advantage of the new time economy.

Measuring teachers' beliefs about mathematics

M.A. Lazim and M.T. Abu Osman have written an article called Measuring Teachers' Beliefs about Mathematics: A Fuzzy Set Approach. The article was published in the current issue of International Journal of Social Sciences. Here is the abstract of their article:
This paper deals with the application of a fuzzy set in measuring teachers' beliefs about mathematics. The vagueness of beliefs was transformed into standard mathematical values using a fuzzy preferences model. The study employed a fuzzy approach questionnaire which consists of six attributes for measuring mathematics teachers' beliefs about mathematics. The fuzzy conjoint analysis approach based on fuzzy set theory was used to analyze the data from twenty three mathematics teachers from four secondary schools in Terengganu, Malaysia. Teachers' beliefs were recorded in form of degrees of similarity and its level of agreement. The attribute 'Drills and practice is one of the best ways of learning mathematics' scored the highest degree of similarity at 0.79860 with level of 'strongly agree'. The results showed that the teachers' beliefs about mathematics were varied. This is shown by different levels of agreement and degrees of similarity of the measured attributes.


Mathematics in Early Childhood (book)

A new and interesting book has been published (or is about to be published) by the National Academies Press: "Mathematics in Early Childhood: Learning Paths Toward Excellence and Equity". The book has 560 pages, and it costs $51.26 when ordered online. So far, the book appears to be available for pre-order only.


Gem #3: Newton's Principia

Isaac Newton is arguably one of the greatest scientists (and mathematicians) of all times, and his Principia is one of the great works from the history of mathematics. Together with Leibniz, Newton is normally acknowledged as the founder of differential and integral calculus. If you want to download Principia to your computer, you can head over to the Internet Archive. The original was in Latin, but you can read an English translation below:

Newton's Principia

epiSTEME 3

A little more than a month ago, Mumbai (India) was the venue for a three-day massacre that caught the world's attention (see for instance this Newsweek article). This week, a far more peaceful event takes place in Mumbai, namely the 3rd International conference to review research on Science, TEchnology and Mathematics Education (epiSTEME 3). The conference presents a number of interesting main speakers, but unfortunately there appears to be little or no live coverage. As far as I can tell, none of the presentations are put online, but you can still get an impression by reading the extensive list of abstracts.


The cost of poor math skills

The National Centre for Excellence in the Teaching of Mathematics (UK) presents the news of a new report about "The long term cost of numeracy difficulties". The report concludes that poor skills in mathematics ends up costing the society an enormous amount of money. BBC reports:
Children who are bad at maths at school end up costing the taxpayer up to £2.4bn a year, a report suggests.
Head of distrubution and product at Barclays, Mike Amato said to BBC:
We are very conscious that every child needs basic numeracy skills for survival.
This is also discussed in The Times and other sources. A key message is that spending money on mathematics education will save us a lot of money in the future.

If you have more information on this, links to other sources, similar studies in other countries, etc., feel free to leave a comment!

Gem #2: Hilbert's "The Foundations of Geometry"

David Hilbert (1862-1943) was one of the most important mathematicians of last century. He worked most of his life in Göttingen, which had a very important mathematics center at the time. Here, Hilbert was surrounded by excellent mathematicians like Felix Klein, John von Neumann, Ernst Zermelo, Emmy Noether and more.

One of Hilbert's achievements was to initiate a shift towards a more modern axiomatic method in mathematics, and in particular in geometry. In relation to this, he proposed a research project, called "Hilbert's program", which aimed at formulating a solid and complete logical foundation for mathematics. Hilbert's "The Foundations of Geometry" is therefore one of the most important modern works in mathematics, although his program did not succeed. The book is therefore a natural follow-up for Gem #1: Euclid's "The Elements" (which is regarded as one of the most important mathematics texts ever, and in particular related to geometry). If you want to download the book in pdf format, you can go to the Gutenberg Project. Otherwise, you can read it here:

David Hilbert - The Foundations of Geometry


Gem #1: Euclid's Elements

When I was a student, I was lucky enough to study in a school which had a very good library of books related to mathematics and mathematics education. Nowadays, you can study many of the great classical texts online. In 2009, I am going to share with you several gems that I have found online. In my quest for these texts on mathematics/mathematics education, a natural first stop is with one of the greatest mathematical texts of all times: The Elements, by Euclid.

Here is the text:
Euclid Elements

You can also download (or read online) this great book in Google Books. See these two links for two versions of the text.

Happy new year, and happy reading!