JMTE - October 2009

Journal of Mathematics Teacher Education has just released the October issue (Volume 12, Number 5). This issue contains 5 interesting articles:

• Conditions of progress in mathematics teacher education, by João Pedro da Ponte
• Teachers’ innovative change within countrywide reform: a case study in Rwanda, by Alphonse Uworwabayeho
• Alignment, cohesion, and change: Examining mathematics teachers’ belief structures and their influence on instructional practices, by Dionne I. Cross
• Multiple representations as sites for teacher reflection about mathematics learning, by Amy E. Ryken
• Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge, by Jon D. Davis

Alphonse Uworwabayeho's article abat the Rwandan case study is an Open Access article.

ZDM, November 2009

ZDM - The International Journal on Mathematics Education - has published the November issue of 2009 (Volume 41, Number 6). The issue contains the following nine articles:

• Curriculum research to improve teaching and learning: national and cross-national studies, by Gerald Kulm and Yeping Li
• Mathematics teachers’ practices and thinking in lesson plan development: a case of teaching fraction division, by Yeping Li, Xi Chen and Gerald Kulm
• Approaches and practices in developing school mathematics textbooks in China, by Yeping Li, Jianyue Zhang and Tingting Ma
• Mathematics curriculum: a vehicle for school improvement, by Christian R. Hirsch and Barbara J. Reys
• School mathematics curriculum materials for teachers’ learning: future elementary teachers’ interactions with curriculum materials in a mathematics course in the United States, by Gwendolyn Monica Lloyd
• How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable, by Bikai Nie, Jinfa Cai and John C. Moyer
• Cross-cultural issues in linguistic, visual-quantitative, and written-numeric supports for mathematical thinking, by Karen C. Fuson and Yeping Li
• Conceptualizing and organizing content for teaching and learning in selected Chinese, Japanese and US mathematics textbooks: the case of fraction division, by Yeping Li, Xi Chen and Song An
• Cross-national comparisons of mathematics curriculum materials: what might we learn? by Edward A. Silver

The article by Karen C. Fuson and Yeping Li is an Open Access article, so that one should be available to all, even non-subscribers :-)

Seminar with Bharath Sriraman

On Thursday, we had a very nice seminar with Bharath Sriraman, or rather we had him as a visitor at University of Stavanger for the whole day! Bharath is professor of mathematical sciences at the University of Montana. He is also founder and editor of The Montana Mathematics Enthusiast, a great online journal within the field of mathematics education research, and editor of numerous other journals, book series, etc.

When he visited us on Thursday, he held a lecture with a focus on gifted students, one of his specialties. Here are my notes from the lecture:

Gifted students - presentation by Bharath Sriraman

How do we figure out if a student is gifted? Nature versus nurture - is it genetic, or is it due to upbringing. Why is it okay for a child to be talented in sports and not so much so in a subject like mathematics?

When it comes to funding, little money is spent on gifted education. (Less than 1% of the funding for special needs education - giftedness is viewed as a special need!)

In the U.S. there is an east versus west debate. Why are they doing so much better in the eastern systems? The western system is viewed as fostering creativity and freedom, but why is it that so many of the prodigies are from the eastern part of the world?

In the U.S., public schools are poorly funded, teachers are not held in high regard or paid well, etc.

Interesting fact:  U.S. has the highest prison population proportion in the western, developed world - 30% of the prisoners are high school dropouts.

In the Asian countries, there is a lot of focus on moral, hard work, perseverance, etc. Exams are very competitive! In the East, the point of an exam is to stratify the society. Late bloomers do not have a chance within the Eastern system! The U.S. (and Western) system, however, allows for a second chance.

As a teacher, there is always the potential conflict between equity and excellence! This could be seen as a false dichotomy! Alternative perspectives:

• The Hamilton tradition stressed elitism,
  
• whereas the Jacksonian tradition suggests that everyone is equal no matter what
• The Jeffersonian tradition stresses that you should give people equal opportunities, and then it is up to them to use these opportunities

Mathematical intelligence is considered as:

• a strong indicator of general intelligence
• numerical and spatial reasoning is part of the IQ score
• ...
  

The construct of intelligence is controversial in many ways since the psychometric testing. History of IQ-test: Binet-Simon tests --> Stanford-Binet tests, etc. The modern Stanford-Binet test is very much like a mathematical test, and you don't take into account socio-cultural and environmental variables. Still, it is very much used as a test for giftedness.

There are, however, some alternative views when it comes to discussing giftedness. Usiskin (Uni. Chicago) tried to classify the mathematical talent in the world in a hierarchy of Level 0 to Level 7.

• Level 0 - no talent. Adults who know very little mathematics
• Level 1 - culture level. Adults who have some number sense (comparable to grades 6-9), and they have learned it through usage
• Level 2 - represent the honors high school student
• Level 3 - the "terrific" student, those who score 750-800 on the SAT.
• Level 4 - the "exceptional" students, those who excel in math competitions
• Level 5 - represents the productive mathematician
• Level 6 - the exceptional mathematician
• Level 7 - the all-time greats, Fields medal winners in mathematics

Krutetskii's book is very nice - about gifted students!

Problem: a pole is 15 meters tall, another one is 10 meters tall. You have a rope from the top of one to the bottom of the other, and vice versa. How tall is the crossing point of the ropes from the ground?

There is a difference between Creativity and creativity (everyone has the latter, the former is related to being creative within a certain field).

There are lots of way to adapt the curriculum so that the gifted students get what they need.

Research shows that there are no harmful effect on early college admission - the students manage well, and they adapt well.

In the U.S. there is a lot of emphasis on the modeling-based curricula nowadays, and this gets a lot of funding. Several programs are made which are based on real-world situations. (one from Montana!)


After this interesting lecture, he gave a presentation of a new book that he has been editing together with Lyn English: Theories of Mathematics Education: Seeking new frontiers. The book is published by Springer, and has just been released. Bharath told that the book took him five years to finish, and it is definitely going to become an important contribution to our field!

Thanks a lot for the visit, Bharath, and for sharing this day with us! Hopefully, this is only going to be the first in a series of visits to Stavanger!

Mathematics curriculum: a vehicle for school improvement

Christian R. Hirsch and Barbara J. Reys have written an article entitled Mathematics curriculum: a vehicle for school improvement. This article was recently published online in ZDM. Here is a copy of their article abstract:

Different forms of curriculum determine what is taught and learned in US classrooms and have been used to stimulate school improvement and to hold school systems accountable for progress. For example, the intended curriculum reflected in standards or learning expectations increasingly influences how instructional time is spent in classrooms. Curriculum materials such as textbooks, instructional units, and computer software constitute the textbook curriculum, which continues to play a dominant role in teachers’ instructional decisions. These decisions influence the actual implemented curriculum in classrooms. Various curriculum policies, including mandated end-of-course assessments (the assessed curriculum) and requirements for all students to complete particular courses (e.g., year-long courses in algebra, geometry, and advanced algebra or equivalent integrated mathematics courses) are also being implemented in increasing numbers of states. The wide variation across states in their intended curriculum documents and requirements has led to a historic and precedent-setting effort by the Council of Chief State School Officers and the National Governors Association Council for Best Practices to assist states in the development and adoption of common College and Career Readiness Standards for Mathematics. Also under development by this coalition is a set of common core state mathematics standards for grades K-12. These sets of standards, together with advances in information technologies, may have a significant influence on the textbook curriculum, the implemented curriculum, and the assessed curriculum in US classrooms in the near future.

CAS calculators in algebra instruction

S. Aslι Özgün-Koca has written an article called Prospective teachers’ views on the use of calculators with Computer Algebra System in algebra instruction. This article has recently been published online in Journal of Mathematics Teacher Education. Here is the abstract of the article:

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This study investigated the views of Turkish prospective secondary mathematics teachers on the use of advanced calculators with CAS in algebra instruction. An open-ended questionnaire and group interviews revealed prospective teachers’ views and beliefs about when and why they prefer three possible uses of CAS—black box, white box, or Symbolic Math Guide (SMG). The results showed that participants mainly preferred the white box methods and especially SMG to the black box method. They suggested that while the black box method could be used after students mastered the skills, the general white box method and SMG could be used to teach symbolic manipulation.

MTL, Volume 11, Issue 4

A new issue of Mathematical Thinking and Learning has appeared: Volume 11, Issue 4, 2009. The issue contains three research articles (along with an editorial by Lyn D. English and a book review):

• Learning Mathematics via a Problem-Centered Approach: A Two-Year Study, by Candice L. Ridlon 
• Efficacy of Different Concrete Models for Teaching the Part-Whole Construct for Fractions, by Kathleen Cramer; Terry Wyberg 
• Reasoning-and-Proving in School Mathematics Textbooks, by Gabriel J. Stylianides
  

Teachers' perceptions about the purpose of student teaching

Keith Leatham from Brigham Young University in Utah, U.S., is one of the scholars who have made important contribution to research of teachers' beliefs in mathematics education research in the last couple of years. I very much like his proposed framework for viewing teachers' beliefs as sensible systems (from his 2006 article in Journal of Mathematics Teacher Education). Now he has written a new article with focus on beliefs (or this time it is referred to as perceptions), and he has co-written this article with a colleague from Brigham Young University: Blake E. Peterson. Their article is entitled Secondary mathematics cooperating teachers’ perceptions of the purpose of student teaching, and it was published online in Journal of Mathematics Teacher Education last week. Here is their article abstract:

This article reports on the results of a survey of 45 secondary mathematics cooperating teachers’ perceptions of the primary purposes of student teaching and their roles in accomplishing those purposes. The most common purposes were interacting with an experienced, practising teacher, having a real classroom experience, and experiencing and learning about classroom management. The most common roles were providing the space for experience, modeling, facilitating reflection, and sharing knowledge. The findings provided insights into the cooperating teachers’ perceptions about both what should be learned through student teaching and how it should be learned. These findings paint a picture of cooperating teachers who do not see themselves as teacher educators—teachers of student teachers. Implications for mathematics teacher educators are discussed.

A case study in Rwanda

I haven't read many scientific articles in mathematics education from or about Rwanda, but here is one! Alphonse Uworwabayeho from Kigali Institute of Education in Rwanda, and University of Bristol, UK, has written an article entitled Teachers’ innovative change within countrywide reform: a case study in Rwanda. The article was published online in Journal of Mathematics Teacher Education on Wednesday. This is even an Open Access article, so everyone should have full access to it! Here is the abstract of the article:

This article presents practical perspectives on mathematics teacher change through results of collaborative research with two mathematics secondary school teachers in order to improve the teaching and learning of mathematics in Rwanda. The 2006 national mathematics curriculum reform stresses pedagogies that enhance problem-solving, critical thinking and argumentation. Teachers need to use new teaching strategies. This article is a case study looking at issues around developing teachers’ use of interactions in mathematics classrooms independently of the national programme. Outputs of the study include teachers’ awareness of the need for change and their increased flexibility to accept learners’ autonomy in shifting from teacher-centred to learner-centred pedagogy. Geometer’s Sketchpad challenged teachers’ practice and then provoked reflection to improve student learning.

Teachers' use of representation

An article called Prospective elementary teachers use of representation to reason algebraically has recently been published online in The Journal of Mathematical Behavior. The article was written by Kerri Richardson, Sarah Berenson and Katrina Staley. Here is the abstract of their article:

We used a teaching experiment to evaluate the preparation of preservice teachers to teach early algebra concepts in the elementary school with the goal of improving their ability to generalize and justify algebraic rules when using pattern-finding tasks. Nearly all of the elementary preservice teachers generalized explicit rules using symbolic notation but had trouble with justifications early in the experiment. The use of isomorphic tasks promoted their ability to justify their generalizations and to understand the relationship of the coefficient and y-intercept to the models constructed with pattern blocks. Based on critical events in the teaching experiment, we developed a scale to map changes in preservice teachers’ understanding. Features of the tasks emerged that contributed to this understanding.

Students’ perceived sociomathematical norms

Esther Levenson, Dina Tirosh and Pessia Tsamir (all from Tel Aviv University in Israel) have written an article that was recently published in The Journal of Mathematical Behavior. The article is entitled Students’ perceived sociomathematical norms: The missing paradigm. Here is the article abstract:

This study proposes a framework for research which takes into account three aspects of sociomathematical norms: teachers’ endorsed norms, teachers’ and students’ enacted norms, and students’ perceived norms. We investigate these aspects of sociomathematical norms in two elementary school classrooms in relation to mathematically based and practically based explanations. Results indicate that even when the observed enacted norms are in agreement with the teachers’ endorsed norms, the students may not perceive these same norms. These results highlight the need to consider the students’ perspective when investigating sociomathematical norms.

ESM - November issue

A new issue of Educational Studies in Mathematics has been released, Volume 72, Number 2 / November, 2009. This issue contains seven interesting articles:

• Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context, by Mariana Montiel, Miguel R. Wilhelmi, Draga Vidakovic and Iwan Elstak
• Changing practice, changing minds, from arithmetical to algebraic thinking: an application of the concerns-based adoption model (CBAM), by Jeanne Tunks and Kirk Weller
• Conditional inference and advanced mathematical study: further evidence, by Matthew Inglis and Adrian Simpson
• Didactical designs for students’ proportional reasoning: an “open approach” lesson and a “fundamental situation”, by Takeshi Miyakawa and Carl Winsløw
• Bridging the macro- and micro-divide: using an activity theory model to capture sociocultural complexity in mathematics teaching and its development, by Barbara Jaworski and Despina Potari
• Proof constructions and evaluations, by Andreas J. Stylianides and Gabriel J. Stylianides
• Researchers’ descriptions and the construction of mathematical thinking, by Richard Barwell
  

Maths week in Ireland

The 4th annual Maths Week Ireland takes place this week, and it is presented as an "all island celebration of mathematics" at the official website. The Irish Times published an interesting article about this yesterday. There are several activities going on during the week, and it is all "timed to coincide with the annual Hamilton Day" on October 16th. William Rowan Hamilton (1805-1865) was one of Ireland's most famous mathematicians, and he was also a physicist and astronomer.

Make sure to check out the official website for a list of events!

Sudoku: Strategy versus structure

"You don't have to know any mathematics to solve a Sudoku puzzle," one of the major Norwegian newspaper claimed when the Sudoku wave started a couple of years ago. I have always disagreed strongly with this, and it seems that Provan J. Scott, who has written an article about this in American Mathematical Monthly, has a similar opinion. The article is entitled Sudoku: strategy versus structure. Here is the article abstract:

Sudoku puzzles, and their variants, have become extremely popular in the last decade. They can now be found in major U.S. newspapers, puzzle books, and web sites; almost as pervasive are the many guides to Sudoku strategy and logic. We give a class of solution strategies-encompassing a dozen or so differently named solution rules found in these guides-that is at once simple, popular, and powerful. We then show the relationship of this class to the modeling of Sudoku puzzles as assignment problems and as unique nonnegative solutions to linear equations. The results provide excellent applications of principles commonly presented in introductory classes in finite mathematics and combinatorial optimization, and point as well to some interesting open research problems in the area.

Teachers' difficulties during problem-solving instruction

Alexander Karp has written an article that was published online in Journal of Mathematics Teacher Education on Saturday. The article is called Analyzing and attempting to overcome prospective teachers’ difficulties during problem-solving instruction. Here is the abstract of Karp's article:

This article analyzes the experiences of prospective secondary mathematics teachers during a teaching methods course, offered prior to their student teaching, but involving actual teaching and reflexive analysis of this teaching. The study focuses on the pedagogical difficulties that arose during their teaching, in which prospective teachers lacked pedagogical content knowledge and skills. It also analyzes the experience of the course itself, which was aimed at scaffolding the work of prospective teachers on developing their pedagogical content knowledge and skills.

Curriculum research to improve teaching and learning

Gerald Kulm and Yeping Li have written an article entitled Curriculum research to improve teaching and learning: national and cross-national studies. The article was recently published online in ZDM. Here is the abstract of their article:

Curriculum, as a cultural and system-specific artifact, outlines mathematics teaching and learning activities in school education. Studies of curriculum and its changes are thus important to reveal the expectations, processes and outcomes of students’ school learning experiences that are situated in different cultural and system contexts. In this article, we aim to propose a framework that can help readers to develop a better understanding of curriculum practices and changes in China and/or the USA that have been reported and discussed in articles published in this themed issue. Going beyond the selected education systems, further studies of curriculum practices and changes are much needed to help ensure the success of educational reforms in the different cultural and system contexts.

Mathematics instruction for students with learning disabilities

Russel Gersten and colleagues have written an article called Mathematics Instruction for Students With Learning Disabilities: A Meta-Analysis of Instructional Components. This article was published in the recent issue of Review of Educational Research. Here is the abstract of their article:

The purpose of this meta-analysis was to synthesize findings from 42 interventions (randomized control trials and quasi-experimental studies) on instructional approaches that enhance the mathematics proficiency of students with learning disabilities. We examined the impact of four categories of instructional components: (a) approaches to instruction and/or curriculum design, (b) formative assessment data and feedback to teachers on students' mathematics performance, (c) formative data and feedback to students with LD on their performance, and (d) peer-assisted mathematics instruction. All instructional components except for student feedback with goal-setting and peer-assisted learning within a class resulted in significant mean effects ranging from 0.21 to 1.56. We also examined the effectiveness of these components conditionally, using hierarchical multiple regressions. Two instructional components provided practically and statistically important increases in effect size–teaching students to use heuristics and explicit instruction. Limitations of the study, suggestions for future research, and applications for improvement of current practice are discussed.

Multiple solution methods and multiple outcomes

Pessia Tsamir, Dina Tirosh, Michal Tabach and Esther Levenson have written an article about Multiple solution methods and multiple outcomes—is it a task for kindergarten children? This article was recently published online in Educational Studies in Mathematics. Here is a copy of their article abstract:

Engaging students with multiple solution problems is considered good practice. Solutions to problems consist of the outcomes of the problem as well as the methods employed to reach these outcomes. In this study we analyze the results obtained from two groups of kindergarten children who engaged in one task, the Create an Equal Number Task. This task had five possible outcomes and five different methods which may be employed in reaching these outcomes. Children, whose teachers had attended the program Starting Right: Mathematics in Kindergartens, found more outcomes and employed more methods than children whose teachers did not attend this program. Results suggest that the habit of mind of searching for more than one outcome and employing more than one method may be promoted in kindergarten.

How syntactic reasoners can develop understanding

Keith Weber has written an article that was recently published in The Journal of Mathematical Behavior. The article is entitled How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Here is the abstract of Weber's article:

This paper presents a case study of a highly successful student whose exploration of an advanced mathematical concept relies predominantly on syntactic reasoning, such as developing formal representations of mathematical ideas and making logical deductions. This student is observed as he learns a new mathematical concept and then completes exercises about it. The paper focuses on how Isaac developed an understanding of this concept, how he evaluated whether a mathematical assertion is true or false, how he generated counterexamples to disprove a statement, and the general role examples play for him in concept development and understanding.

What's sophisticated about elementary mathematics?

The current issue of American Educator includes a very interesting article about mathematics teaching in the elementary grades. The article is entitled What's Sophisticated about Elementary Mathematics? Plenty—That's Why Elementary Schools Need Math Teachers, and it is written by Hung-Hsi Wu, who is professor emeritus of mathematics at UC Berkeley. I should probably say a lot more about him, because his list of merits is impressive, but I will leave it to you to find out more.

It appears to be a rather common impression that teaching elementary mathematics is ... well, rather elementary. I mean, the mathematics is quite simple, so how hard can it be? In this article, Wu provides a very nice introduction to how challenging it can actually be. In the introductory part of the article, he claims: "The fact is, there's a lot more to teaching math than teaching how to do calculations." In the article, he provides examples of how hard it can actually be to teach something as "elementary" as place value and fractions.

I am tempted to quote more or less the entire article, because so many interesting issues are presented here, but I will not. I am, however, going to recommend that you take the time and read this excellent article. If you are somewhat interested in teaching mathematics, I am sure you will find this interesting!

Thanks a lot to Assistant Editor Jennifer Dubin for telling me about this article, by the way! I appreciate it :-)

Developing school mathematics textbooks in China

Yeping li, Jianyue Zhang and Tingting Ma have written an article entitled Approaches and practices in developing school mathematics textbooks in China. This article was published online in ZDM a couple of days ago. This article gives a very interesting overview of how Chinese textbooks are developed, but it also provides nice insights into how mathematics teaching has developed (or is developing) in China. Here is the abstract of this article:

In this study, we aim to examine and discuss approaches and practices in developing mathematics textbooks in China, with a special focus on the development of secondary school mathematics textbook in the context of recent school mathematics reform. Textbook development in China has its own history. This study reveals some common practices and approaches developed and used in selecting, presenting and organizing content in mathematics textbooks over the years. With the recent curriculum reform taking place in China, we also discuss some new developments in compiling and publishing high school mathematics textbooks. Implications obtained from Chinese practices in textbook development are then discussed in a broad context.

The productive notion of mathematics laboratories

Michela Maschietto and Luc Trouche have written an article called Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories. The article was published online in ZDM on Monday. Although they start with a glimpse from a babylonian clay tablet, their main focus is on the development of tools and use of tools in the last century. In the main part of their article, they have a strong focus on the so-called mathematics laboratories. Here is the abstract of their article:

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.