Education Week has an interesting article about the uncertainties about the skills that are needed to be a successful mathematics teacher. The point of departure for the article is the recent report by the National Mathematics Advisory Panel in the U.S. The report has several suggestions about the curriculum, cognition, instruction, etc. When it comes to the skills that are needed to become a good mathematics teacher, though, the answers were fewer:
A new article has been published online at Educational Studies in Mathematics. The article is entitled: "The role of scaling up research in designing for and evaluating robustness", and it is written by J. Roschelle, D. Tatar, N. Shechtman and J. Knudsen. Here is the abstract of the article:
The full title of this new ZDM article is: "When, how, and why prove theorems? A methodology for studying the perspective of geometry", and it is written by P. Herbst and T. Miyakawa.
Every theorem has a proof, but not every theorem presented in schools (not only in the U.S., although this is the focus of the article). Why is that? Here is the abstract of the article, which truly raises some important questions:
Megan E. Staples wrote an article called: "Promoting student collaboration in a detracked, heterogeneous secondary mathematics classroom". The article was published online in Journal of Mathematics Teacher Education on Wednesday. Here is the abstract of the article:
Several researchers have addressed the issue of collaboration and group work, and Staples analyzes the role of one teacher in this respect. Staples observed 39 lessons in the study, and data was collected through field notes, reflective memos, and 26 lessons were also video-taped. She also conducted interviews with most of the students and the teacher, and she collected curriculum documents, etc. During the data analysis, four categories emerged that were critical for understanding the teacher's role (p. 8):
- Promoting individual and group accountability
- Promoting positive sentiment among group members
- Supporting student–student exchanges with tools and resources
- Supporting student mathematical inquiry in direct interaction with groups
The classroom is a complex system, and this is something Staples discuss a lot in the article. Understanding this complexity and being able to analyze it, is something she emphasizes as being important for both future and current teachers.
And interesting article. In the theoretical foundations, she refers (among others) to the works of researchers like E. Cohen and J. Boaler.
The final program of the Norma 08 conference has arrived (download as pdf). I am not going to repeat the entire program here, but I will point at the plenary lectures that will be presented at the conference:
- Monday, April 21, 16:30-17:30 - Jeppe Skott (Theme B)
- Tuesday, April 22, 11:00-12:00 - Paul Drijvers (Theme C)
- Wednesday, April 23, 11:00-12:00 - Eva Jablonka (Theme D)
- Thursday, April 24, 11:00-12.00 - Michèle Artigue (Theme A)
Jon R. Star and Sharon K. StricklandDevelopment of a performance assessment task and rubric to measure prospective secondary school mathematics teachers’ pedagogical content knowledge and skills
In the U.S., the National Mathematics Advisory Panel (on request from the President himself) has delivered a report to the President and the U.S. Secretary of Education. This final report was delivered on March 13, and is freely available for anyone to download (pdf or Word document). I know this is old news already, but I will still present some of the highlights from the report here. Be also aware that there will be a live video webcast of a discussion of the key findings and principle messages in the report. The webcast will be held tomorrow, Thursday March 26, 10-11.30 a.m. Eastern Time. This discussion will be lead by Larry R. Faulkner (Chair of the Panel) and Raymond Simon (U.S. Deputy Secretary of Education).
A key element of the report is a set of "Principal Messages" for mathematics education. This set of messages consists of six main elements (quoted from pp. xiii-xiv):
- The mathematics curriculum in Grades PreK-8 should be streamlined and should emphasize a well-defined set of the most critical topics in the early grades.
- Use should be made of what is clearly known from rigorous research about how children learn, especially by recognizing a) the advantages for children in having a strong start; b) the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic (i.e., quick and effortless) recall of facts; and c) that effort, not just inherent talent, counts in mathematical achievement.
- Our citizens and their educational leadership should recognize mathematically knowledgeable classroom teachers as having a central role in mathematics education and should encourage rigorously evaluated initiatives for attracting and appropriately preparing prospective teachers, and for evaluating and retaining effective teachers.
- Instructional practice should be informed by high-quality research, when available, and by the best professional judgment and experience of accomplished classroom teachers. High-quality research does not support the contention that instruction should be either entirely "student centered" or "teacher directed." Research indicates that some forms of particular instructional practices can have a positive impact under specified conditions.
- NAEP and state assessments should be improved in quality and should carry increased emphasis on the most critical knowledge and skills leading to Algebra.
- The nation must continue to build capacity for more rigorous research in education so that it can inform policy and practice more effectively.
- Conceptual knowledge and skills
- Learning processes
- Instructional practices
- Teachers and teacher education
After having presented their principle messages, the panel present 45 main findings and recommendations for the further development of mathematics education in the U.S. These 45 findings and recommendations are split in the following main groups (strongly resembling the list of task groups above):
- Curricular content
- Lesson processes
- Teachers and teacher education
- Instructional practices
- Instructional materials
- Research policies and mechanisms
- Digital Images + Interactive Software = Enjoyable, Real Mathematics Modeling by Andy Ventress
- Investigating the Mathematical Process with Nonlinear Asymptotes by Michael J. Bossé, Karen A. DeUrquidi, David L. Edwards and N.R. Nandakumar
- Using Technology to Promote Mathematical Discourse Concerning Women in Mathematics by Lyn Phy
Linda Pilkey-Jarvis and Orrin H. Pilkey have written an article in Public Administration Review about the use of mathematical models in environmental decision making. Mathematical models are used extensively in the context of environmental issues and natural resources, and when these methods were first used, they were thought to represent a bridge to a better and more foreseeable future. There has also been much controversy in this respect, and the authors pose the question whether the optimism about the use of these models were ever realistic. In this article, they review the two main types of such models: quantitative and qualitative.
After a review of these types of models, they provide a list of ten lessons that policy makers should learn when it comes to quantitative mathematical modeling:
- The outcome of natural processes on the earth’s surface cannot be absolutely predicted.
- Examine the excuses for predictive model failures with great care and skepticism.
- Did the model really work? Examine claims of past "successes" with the same level of care and skepticism that "excuses" are given.
- Calibration of models doesn’t work either.
- Constants in the equations may be coefficients or fudge factors.
- Describing nature mathematically is linking a natural flexible, dynamic system with a wooden, inflexible one.
- Models may be used as "fig leaves" for politicians, refuges for scoundrels, and ways for consultants to find the truth according to their clients’ needs.
- The only show in town may not be a good one.
- The mathematically challenged need not fear models and can learn how to talk with a modeler.
- When humans interact with the natural system, accurate predictive mathematical modeling is even more impossible.
In the wrapping up of the article, they clarify their main argument:
Pilkey-Jarvis, L. & Pilkey, O.H. (2008). Useless Arithmetic: Ten Points to Ponder When Using Mathematical Models in Environmental Decision Making. Public Administration Review 68 (3) , 470–479 doi:10.1111/j.1540-6210.2008.00883_2.x
Yesterday, NCME (National Council on Measurement in Education) started their annual meeting. NCME's mission is among other things to "Advance the science of measurement in the field of education", so the focus is not on mathematics education solely. There are, however several presentations that deal with mathematics in the program. Here are the ones that I could find:
- Shelley Ragland, James Madison University, Christina Schneider, CTB/McGraw- Hill, Ching Ching Yap, University of South Carolina, Pamela Kaliski, James Madison University: The Effect of Classroom Assessment Professional Development on English Language Arts and Mathematics Student Achievement: Year 2 Results
- Carol Parke, Duquesne University, Gibbs Kanyongo, Duquesne University, Steven Kachmar, Duquesne University: Examining Relationships among Large-Scale Mathematics Assessment Performance, Grade Point Average, and Coursework in Urban High Schools
- Michelle Boyer, CTB/McGraw-Hill, Enrique Froemel, Office of Student Assessment, Evaluation Institute, Supreme Education Council, State of Qatar, Richard Schwarz, CTB/McGraw-Hill: Obtaining Comparable Scores for Arabic and English Tests of Mathematics and Science Administered under the Qatar Comprehensive Educational Assessment Program
- Catherine Taylor, University of Washington, Yoonsun Lee, Washington State Department of Education: Analyses of Gender DIF in Reading and Mathematics Items from Tests with Mixed Item Formats
- Saw Lan Ong, Universiti Sains Malaysia: Effects of Test Language on Students’ Mathematics Performance
- Bryce Pride, University of South Florida, Yi-Hsin Chen, University of South Florida, Teresa Chavez, University of South Florida, Corina Owens, University of South Florida, Yuh-Chyn Leu, National Taipei University of Education: An Exploration of Cognitive Skills and Knowledge underlying the TIMSS-2003 Fourth Grade Mathematics Items
- Richard Sudweeks, Brigham Young University, Maria Assunta Forgione, Brigham Young University, Robert Bullough, Brigham Young University, Damon Bahr, Brigham Young University, Eula Monroe, Brigham Young University, Scott Thayn, Brigham Young University: Constructing Vertically Scaled Mathematics Tests for Tracking Student Growth in Value-Added Studies of Teacher Effectiveness
- Samantha Burg, Metametrics, Inc.: An Investigation of Dimensionality across Grade Levels for Grades 3-8 Mathematics Achievement Tests
Yesterday, the 2008 annual meeting of AERA started. Although this is not only a mathematics education conference, it has a lot of interesting presentations for our field as well. A brief search through the searchable program gave 353 hits on individual presentations with the word "math" in the title. There are also several paper sessions with themes related to mathematics education. Today, for instance, there is a session entitled "Addressing Mathematics Education in Special Education", which has the following participants:
Beyond Either/Or: Enhancing the Computation and Problem-Solving Skills of Low-Achieving Adolescents
*Brian A. Bottge (University of Kentucky), Jorge Enrique Rueda-Sarmiento (University of Wisconsin - Madison), Ana C. Stephens (University of Wisconsin - Madison)Calculators, Friend or Foe? Calculators as Assessment Accommodations for Students With DisabilitiesInterventions to Enhance Math Problem Solving and Number Combinations Fluency for Third-Grade Students With Math Difficulties: A Field-Based Randomized Control Trial
*Lynn S. Fuchs (Vanderbilt University), *Sarah Rannells Powell (Vanderbilt University), *Pamela M. Seethaler (Vanderbilt University), *Rebecca O'Rand Zumeta (Vanderbilt University), Douglas Fuchs (Vanderbilt University)The Effects of Conceptual Model-Based Instruction on Solving Word-Problems With Various Contexts: “Transfer in Pieces”The Effects of Two Manipulative Devices on Hundreds Place-Value Instruction
Ferdinando Arzarello, Marianna Bosch, Josep Gascón and Cristina Sabena have written an article called "The ostensive dimension through the lenses of two didactical approaches", that has recently been published (online first) in ZDM. Here is the abstract.
This article by Gila Hanna and Ed Barbeau was published online two days ago in ZDM. The article examines a main idea from an article by Yehuda Rav in Philosophia Mathematica, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”. An interesting theme of an article, with strong implications. Here is the entire abstract:
The aim of this dissertation is to describe and analyze how discourse as a theoretical and didactical concept can help in advancing knowledge about the teaching of mathematics in school. The dissertation has been written within a socio-cultural perspective where active participation and support from artefacts and mediation are viewed as important contributions to the development of understanding. Discourse analysis was used as a theoretical point of departure to grasp language use, knowledge construction and mathematical content in the teaching practises. The collection of empirical data was made up of video and audio tape recordings of the interaction of teachers and pupils in mathematics classrooms when they deal with problem-solving tasks, as well as discussions between student teachers as they engage in planning a teaching situation in mathematics. Discourse analysis was used as a tool to shed light upon how pupils learn and develop understanding of mathematics.
The results of my studies demonstrate that discussions very often are located in either a mathematical or in an every-day discourse. Furthermore, the results demonstrate how change between every-day and mathematical language often takes place unknowingly. Also the results underline that a specific and precise dialogue can contribute towards teachers’ and pupils’ conscious participation in the learning process. Translated into common vocabulary such as speak, think, write, listen and read teachers and pupils would be able to interact over concepts, signs, words, symbols, situations and phenomena in every-day discourse and its mathematical counterpart. When teachers and pupils become aware of discursive boundary crossing in mathematics an understanding of mathematical phenomena can start to develop. Teachers and pupils can construct a meta-language leading to new knowledge and new learning in mathematics.
Christer Bergsten has wrote an article called "On the influence of theory on research in mathematics education: the case of teaching and learning limits of functions", which was recently published (online first) by ZDM. Here is the abstract of the article:
Regular papers theme B: Education and identity of mathematics teachers
IS THERE ALWAYS TRUTH IN EQUATION? Iiris Attorps and Timo Tossavainen
Journal of Mathematics Teacher Education (JMTE) recently published an (online first) article by A.J. Stylianides and Deborah L. Ball entitled "Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving". The article has a particular focus on knowledge about proof:
This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof (quoted from the abstract).
M. Kaldrimidou, H. Sakonidis and M. Tzekaki have written an article that has recently been published online in ZDM. The article is entitled "Comparative readings of the nature of the mathematical knowledge under construction in the classroom", and it makes an attempt to:
(...) empirically identify the epistemological status of mathematical knowledge interactively constituted in the classroom. To this purpose, three relevant theoretical constructs are employed in order to analyze two lessons provided by two secondary school teachers. The aim of these analyses was to enable a comparative reading of the nature of the mathematical knowledge under construction. The results show that each of these three perspectives allows access to specific features of this knowledge, which do not coincide. Moreover, when considered simultaneously, the three perspectives offer a rather informed view of the status of the knowledge at hand (from the abstract).
- "I would rather die": reasons given by 16-year-olds for not continuing their study of mathematics by M. Brown, P. Brown and T. Bibby
- The capacity of two Australian eighth-grade textbooks for promoting proportional reasoning by S. Dole and M. Shield
- "If you can count to ten you can count to infinity really": fostering conceptual mathematical thinking in the first year of primary school by P. Iannone and A.D. Cockburn
- Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis by I. Biza, C. Christou and T. Zachariades
- The role of affect in learning Real Analysis: a case study by K. Weber
International Electronic Journal of Mathematics Education published their first issue this year a while ago (see my post about it). Now, the articles and abstracts are finally available as well! The abstracts are available in plain HTML format, whereas the articles can be freely downloaded in PDF format. I find one of the articles particularly interesting, as it concerns the same area of research as I am involved in myself (teacher thinking and teacher knowledge). The article was written by Donna Kotsopoulos and Susan Lavigne, and it is entitled: Examining “Mathematics For Teaching” Through An Analysis Of Teachers’ Perceptions Of Student “Learning Paths”
I enclose a copy of the abstract here:
The March issue of Mathematics Teacher is out, with several interesting articles:
- Teaching Algebra and Geometry Concepts by Modeling Telescope Optics by Lauren M. Siegel, Gail Dickinson, Eric J. Hooper and Mark Daniels
- Tangent Lines without Calculus by Jeffrey M. Rabin
- The Dreaded "Work" Problems Revisited: Connections through Problem Solving from Basic Fractions to Calculus by Felice S. Shore and Matthew Pascal (Free preview)
- Developing Knowledge of Teaching Mathematics through Cooperation and Inquiry by Maria Lorelei Fernández
This study examined conceptions of algebra held by 30 preservice
elementary teachers. In addition to exploring participants’ general
“definitions” of algebra, this study examined, in particular, their
analyses of tasks designed to engage students in relational thinking or
a deep understanding of the equal sign as well as student work on these tasks. Findings from this study suggest that preservice elementary
teachers’ conceptions of algebra as subject matter are rather narrow.
Most preservice teachers equated algebra with the manipulation of
symbols. Very few identified other forms of reasoning – in particular,
relational thinking – with the algebra label. Several participants made comments implying that student strategies that demonstrate traditional
symbol manipulation might be valued more than those that demonstrate
relational thinking, suggesting that what is viewed as algebra is what
will be valued in the classroom. This possibility, along with
implications for mathematics teacher education, will be discussed.
(See also this list of interactive geometry software!)
A main element in this thesis is the perspectives on learning mathematics through collaborative problem solving. This perspective has received attention by several of Carlsen's colleagues in Agder in the past (see e.g. Bjuland, 2004; Borgersen, 1994; Borgersen, 2004). Carlsen presents an analysis of how upper secondary students engage in problem-solving processes in order to achieve mathematical understanding, and he presents four separate studies within this field.
Bjuland, R. (2004). Student teachers' reflections on their learning process through collaborative problem solving in geometry. Educational Studies in Mathematics, 55(1):199-225.
Borgersen, H. E. (1994). Open ended problem solving in geometry. Nordisk Matematikkdidaktikk, 2(2): 6-35.
Borgersen, H. E. (2004). Open ended problem solving in geometry re-visited. Nordisk Matematikkdidaktikk, 9(3), 35-65.
Carlsen, M. (2008). Appropriating mathematical tools through problem solving in collaborative small-group settings. PhD thesis, University of Agder, Faculty of Engineering and Science, Kristiansand, Norway.
- "Recruiting and retaining secondary mathematics teachers: lessons learned from an innovative four-year undergraduate program", is a JMTE-article written by A.F. Artzt and F.R. Curcio. They describe some of the innovative aspects of a NSF funded program (TIME 2000), that was started as a response to the critical shortage of qualified mathematics teachers in the U.S.
- "Imagination as a tool in mathematics teacher education" was written by O. Chapman for JMTE. Chapman describes some of the theory within this field, and he also makes a description of some of his own experiences with a class of prospective mathematics teachers, before he makes connections between other related articles in this forthcoming issue.
- "How are theoretical approaches expressed in research practices? A report on an experience in comparing theoretical approaches with respect to the construction of research problems" is an article written for ZDM by S. Prediger. She explores the idea that theoretical approaches might be usefully compared in terms of the ways in which they lead researchers to construe commonsense classroom problems (quote from the abstract).
- "Toward networking three theoretical approaches: the case of social interactions" was written by I. Kidron et al. and published online (in ZDM) Tuesday, March 4 (all four articles were published at the same date). The discussions in this article was initiated at CERME4 and continued at CERME5, and the focus is on comparing, contrasting and combining different theoretical frameworks that are currently used in mathematics education.
A core component of the program of the symposium is five work groups, where several of the participants have posted interesting articles for download. The themes of the working groups are:
- WG1 - Disciplinary mathematics and school mathematics
- WG2 - The professional formation of teachers
- WG3 - Mathematics education and society
- WG4 - Resources and technology throughout the history of ICMI
- WG5 - Mathematics education: an ICMI perspective
- PL0: Moments of the life of ICMI [Opening Plenary]
- PL1: The development of mathematics education as an academic field
- PL2: Intuition and rigor in mathematics education
- PL3: Perspectives on the balance between application & modelling and 'pure' mathematics in the teaching and learning of mathematics
- PL4: The relationship between research and practice in mathematics education: international examples of good practice
- PL5: The origins and early incarnations of ICMI
- PL6: ICMI Renaissance: the emergence of new issues in mathematics education
- PL7: Centres and peripheries in mathematics education
- PL8 (Closing Plenary): ICMI: One century at the interface between mathematics and mathematics education – Reflections and perspectives
- Bingolbali, E. & Monaghan, J. (2008). Concept image revisited. Educational Studies in Mathematics. Published online 29 February 2008.
- Norton, A.H. & McCloskey, A. (2008).
Teaching experiments and professional development. Journal of Mathematics Teacher Education. Published online 29 February 2008.
- Schur, Y. & Galili, I. (2008).
Thinking Journey - a New Mode of Teaching Science. International Journal of Science and Mathematics Education. Published online 29 February 2008.
- Bleicher, R.E. (2008).
Variable Relationships among Different Science Learners in Elementary Science-Methods Courses. International Journal of Science and Mathematics Education. Published online 29 February 2008.