New IJMEST articles

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A couple of new articles have been published online in International Journal of Mathematical Education in Science and Technology. Several are so-called classroom notes (see this link for full list), and then there are these two original articles:

  • Pilot study on algebra learning among junior secondary students, by Kin-Keung Poon and Chi-Keung Leung. Abstract: The purpose of the study reported herein was to identify the common mistakes made by junior secondary students in Hong Kong when learning algebra and to compare teachers’ perceptions of students’ ability with the results of an algebra test. An algebra test was developed and administered to a sample of students (aged between 13 and 14 years). From the responses of the participating students (N = 815), it was found that students in schools with a higher level of academic achievement had better algebra test results than did those in schools with a lower level of such achievement. Moreover, it was found that a teacher’s perception of a student’s ability has a correlation with that student’s level of achievement. Based on this finding, an instrument that measures teaching effectiveness is discussed. Last but not least, typical errors in algebra are identified, and some ideas for an instructional design based on these findings are discussed.
  • Student connections of linear algebra concepts: an analysis of concept maps, by Douglas A. Lapp,  Melvin A. Nyman and John S. Berry. Abstract: This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud’s theory of conceptual fields and Pirie and Kieren’s model for the growth of mathematical understanding. In addition to the existing techniques for analysing concept maps, two new techniques are developed for analysing qualitative data based on student-constructed concept maps: (1) temporal clumping of concepts and (2) the use of adjacency matrices of an undirected graph representation of the concept map. Findings suggest that students may find it more difficult to make connections between concepts like eigenvalues and eigenvectors and concepts from other parts of the conceptual field such as basis and dimension. In fact, eigenvalues and eigenvectors seemed to be the most disconnected concepts within all of the students’ concept maps. In addition, the relationships between link types and certain clumps are suggested as well as directions for future study and curriculum design.

Insight into the fractional calculus via a spreadsheet

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David A. Miller and Stephen J. Sugden have written an article called Insight into the Fractional Calculus via a Spreadsheet. The article was published in the most recent issue of electronic journal Spreadsheets in Education. The article is freely available as a pdf download, but here is a copy of the abstract:

Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of  fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.

JMTE – October 2009

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ZDM, November 2009

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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:

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

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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

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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

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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

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100 open lectures

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Yesterday, a very nice blogpost appeared over at onlinecollege.org: 100 Incredible Open Lectures for Math Geeks. In this post, they have collected and organized 100 sets of lectures within several fields of mathematics, like Algebra, Calculus, Geometry, etc. Towards the very end of the list, there is even a collection of lectures related to teaching! The lectures are from some of the best universities in the world, like Stanford, MIT, Princeton, Harvard, Berkeley, UCLA, etc. The lecturers are of course also extremely good. For instance, you’ll find a lecture that includes Professor Terrence Tao from UCLA. Professor Tao is arguably one of the best mathematicians of our days, and he has received numerous awards, like the Fields Medal (in 2006). I also recommend the lectures of Professor Gilbert Strang (MIT) in Linear Algebra. Several of the lectures from the list are from Academic Earth, so you might want to check out that site as well.

This is a very useful list, and as it is said in the original post:

You can indulge your love of mathematics in these great lectures and lecture series. Some are meant to review the basics and others will keep you on the cutting edge of what renowned researchers are doing in the field, but all will help you expand your knowledge and spend a few hours enjoying a topic you love.

Teachers’ perceptions about the purpose of student teaching

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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.