Experts’ strategy flexibility for solving equations

Jon R. Star and Kristie J. Newton have written an article about The nature and development of experts’ strategy flexibility for solving equations. The article was published online in ZDM last week. Algebra is an area of mathematics in which many pupils struggle. There is also an agreement among many researchers that proficiency in algebra includes understanding as well as skills. This study aims at investigating the flexibility of experts’ strategies when solving algebraic equations. Eight experts in school algebra were participating in the study, and their flexibility was measured using a researcher-designed algebra test as well as semi-structured interviews. These interviews were conducted immediately after the participants had completed the test.

Here is the abstract of their article:

Largely absent from the emerging literature on flexibility is a consideration of experts’ flexibility. Do experts exhibit strategy flexibility, as one might assume? If so, how do experts perceive that this capacity developed in themselves? Do experts feel that flexibility is an important instructional outcome in school mathematics? In this paper, we describe results from several interviews with experts to explore strategy flexibility for solving equations. We conducted interviews with eight content experts, where we asked a number of questions about flexibility and also engaged the experts in problem solving. Our analysis indicates that the experts that were interviewed did exhibit strategy flexibility in the domain of linear equation solving, but they did not consistently select the most efficient method for solving a given equation. However, regardless of whether these experts used the best method on a given problem, they nevertheless showed an awareness of and an appreciation of efficient and elegant problem solutions. The experts that we spoke to were capable of making subtle judgments about the most appropriate strategy for a given problem, based on factors including mental and rapid testing of strategies, the problem solver’s goals (e.g., efficiency, error-free execution, elegance) and familiarity with a given problem type. Implications for future research on flexibility and on mathematics instruction are discussed.

Changing practice, changing minds

I like the title of a new article written by Jeanne Tunks and Kirk Weller, especially the first part of it! Here is the entire title: Changing practice, changing minds, from arithmetical to algebraic thinking: an application of the concerns-based adoption model (CBAM). This article was published online in Educational Studies in Mathematics on Saturday, and it discusses the results of a yearlong innovation program called “Teacher Quality Grant”. And, just to avoid any misunderstandings: it is not only the title of the article I find interesting. The article itself is very interesting, and the program described also appears to be quite interesting. Here is the abstract of the article:

This study examines the process of change among grade 4 teachers (students aged 9–10 years) who participated in a yearlong Teacher Quality Grant innovation program. The concerns-based adoption model (CBAM), which informed the design and implementation of the program, was used to examine the process of change. Two questions guided the investigation: (1) How did teachers’ concerns about and levels of use of the innovation evolve during the course of the project? (2) What changes in teachers’ perceptions and practices arose as a result of the innovation? Results showed that several of the teachers’ concerns evolved from self/task toward impact. With continued support, several participants achieved routine levels of use, which they sustained beyond the project.

Hidden lessons

Amy B. Ellis and Paul Grinstead have written an article that was published in The Journal of Mathematical Behavior last week. The article is entitled Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. Here is a copy of their article abstract:

This article presents secondary students’ generalizations about the connections between algebraic and graphical representations of quadratic functions, focusing specifically on the roles of the parameters a, b, and c in the general form of a quadratic function, y = ax2 + bx + c. Students’ generalizations about these connections led to a surprising finding: two-thirds of the students interviewed identified the parameter a as the “slope” of the parabola. Analysis of qualitative data from interviews and classroom observations led to the development of three focusing phenomena in the classroom environment that inadvertently supported a focus on slope-like properties of quadratic functions: (a) the use of linear analogies, (b) the rise over run method, and (c) viewing a as dynamic rather than static.

Algebra: Use it or lose it?

Yesterday, there was an interesting article in The Spectrum. The title of the article is “Algebra: Use it or lose it?“, and the claim that is put forth by author Sarah Clark was that algebra teachers all over the world are lying when they tell students that algebra is important because they’ll use it in their daily life.

Clark (32) describes herself as a non-traditional student:

(…) who hasn’t taken an algebra class in 15 years. If, for the past 15 years, I had been using algebra in my everyday life, I would be blowing through my algebra homework with ease, thinking, “Hey! I just did this yesterday while I was washing laundry,” or, “I’m so glad I’ve known this all along. I’d never be able to drive anywhere without it!” or “Wow! I just used this formula last week to calculate the ratio of jazz to classical music on my iPod.

Apparently, this is not what she has experienced. On the contrary, she has never experienced using algebra in her daily life, and she now finds herself uncapable of doing it. She also proposes an algebra revolution, where we should share the truth with every student who is struggling with algebra: these skills will not be crucial for you in adult life.

There are lots of things to comment on these statements, for sure. And lots of people did comment on it already (so be sure to read the comments below the article as well!). Deb Peterson at made an interesting (external) comment to the article, that might be worth reading.

Myself, I think all these claims about how mathematics is/can be useful in your everyday life is a mixed bag. I think Clark’s article illustrates a common issue as well: when teachers claim that mathematics is useful in everyday life, it might be their own everyday life they think of rather than their students’. (Lots of people have written about the connections with everyday life, and if you are interested, you might want to take a look at my own PhD thesis: Mathematics in everyday life: a study of beliefs and actions.)

Overcoming Algebra

Next Tuesday, there is going to be a free live “webinar” over at Presenters in this web-based seminar are Jon R. Star and Mary Jo Tavormina. Star is educational psychologist and assistant professor of education at Harvard University, whereas Tavormina is elementary mathematics manager in the Chicago Public Schools. Here is a description of the webinar:

One of the biggest challenges in K-12 education today is how to help students overcome their struggles in introductory algebra. Many students fail or are barely able to keep up in their first algebra course, typically taught in 8th or 9th grade. In response, state and school district officials are trying to solve this problem in several ways, such as by encouraging better teacher preparation, including an emphasis on algebra, and by revamping courses and curricula to help struggling students, such as through the creation of “algebra readiness” classes aimed at girding students for the challenges of that class. In addition, policymakers at all levels have called for an improved, more streamlined approach to teaching elementary and middle-grades math as a way of preparing students for algebra.

This webinar will bring together a number of experts who have examined students’ experiences with algebra. One of the goals is to explore the fundamental question: Why do so many students find algebra so difficult? The webinar will then examine efforts by districts and private curriculum-developers to help these students. It will also touch on major developments at the national level in this area, such as the release last year of a report of the National Math Advisory Panel, which called for more coherent math curricula at early grades as a foundation for algebra.

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.

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.

Building intellectual infrastructure

James Kaput wrote an article that was published online in Educational Studies in Mathematics on Friday. The article is entitled: Building intellectual infrastructure to expose and understand ever-increasing complexity. Here is the abstract of the article:

This paper comments on the expanded repertoire of techniques, conceptual frameworks, and perspectives developed to study the phenomena of gesture, bodily action and other modalities as related to thinking, learning, acting, and speaking. Certain broad issues are considered, including (1) the distinction between “contextual” generalization of instances across context (of virtually any kind—numeric, situational, etc.) and the generalization of structured actions on symbols, (2) fundamental distinctions between the use of semiotic means to describe specific situations versus semiosis serving the process of generalization, and (3) the challenges of building generalizable research findings at such an early stage in infrastructure building.

Book review: "Algebra in the Early Grades"

The latest issue of Teachers College Record includes a book review of “Algebra in the Early Grades”. This important book was edited by late James J. Kaput together with David W. Carraher and Maria L. Blanton, and it was published by Lawrence Erlbaum Associates in 2007. David Slavit provides a thorough review, which gives a nice insight into the main parts of the book.

If you are interested, you might want to check out the information about the book in Google Books (which includes links to where you can buy the book), and you might also be interested in taking a look at this page about Early Algebra.

New journal: Educational Designer

A new journal for educational research has seen the light of day: Educational Designer! The journal is an online journal, and it was established by the International Society for Design and Development in Education. One of the articles in the first issue is written by Malcolm Swan, mathematics education researcher from the University of Nottingham. The article is concerned with Designing a Multiple Representation Learning Experience in Secondary Algebra. Here is the abstract of Swan’s article (but the entire article is available online!):

This paper describes some of the research-based principles that I use when designing learning experiences to foster conceptual understanding. These principles are illustrated through the discussion of one type of experience: that of sorting multiple representations. I refer to learning experiences rather than tasks, because tasks are only one component of the design. Close attention is also paid to the role of the teacher in creating an appropriate climate for learning to take place.

After a brief excursion into my own theoretical framework, I describe the educational objectives behind my design and provide a detailed explanation of it in one topic, that of algebraic notation.  This is followed with an explanation of the principles that informed the design and the evolution of the task. Finally, I briefly indicate how the design might be generalised to include other topics.