Arthur B. Powell

Expanding Participation in Problem Solving in a Diverse Middle School Mathematics Classroom.

In this paper, we discuss our experiences with an after-school program in which we engaged middle-school students with low socioeconomic status from an urban community in mathematical problem solving. We document that these students participated in many aspects of problem solving, including the posing of problems, constructing justifications, developing and implementing problem-solving heuristics and strategies, and understanding and evaluating the solutions of others. We then delineate what aspects of our environment encouraged the students to take part in these activities, particularly emphasising the proactive role of the teacher, the tasks the students completed, and the social norms of our after-school sessions. Finally, we discuss the relationship between our study and the literature on equity research in mathematics education.

Understanding and Overcoming “Disadvantage” in Learning Mathematics

Past research has largely characterized disadvantage as an individual or social condition that somehow impedes mathematics learning, which has resulted in the further marginalization of individuals whose physical, racial, ethnic, linguistic and social identities are different from normative identities constructed by dominant social groups. Recent studies have begun to avoid equating difference with deficiency and instead seek to understand mathematics learning from the perspective of those whose identities contrast the construction of normal by dominant social groups. In this way of thinking, “understanding” disadvantage can be discussed as understanding social processes that disadvantage individuals. And, “overcoming” disadvantage can be explored by analyzing how learning scenarios and teaching practices can be more finely tuned to the needs of particular groups of learners, empowering them to demonstrate abilities beyond what is generally expected by dominant discourses. In this chapter, we consider theoretical and methodological perspectives associated with the search for a more inclusive mathematics education, and how they generally share a conceptualization of the role of the teacher as an active participant in researching and interpreting their students’ learning. Drawing from examples with a diverse range of learners including linguistic, racial and ethnic minorities, as well as deaf students, blind students, and those with specific difficulties with mathematics, we argue that by understanding the learning processes of such students we may better understand mathematics learning in general.

Researching Teachers' Knowledge for Teaching Mathematics

This report theorizes and provides empirical evidence of how researchers and educators might recognize categories of teachers’ knowledge for teaching as teachers teach and discuss with peers their student’s mathematical behavior and their practice. Its theoretical orientation engages work by Shulman on pedagogical content knowledge, Ball and Bass on mathematical knowledge for teaching, and Steinbring on teachers’ epistemological knowledge. The empirical evidence emerges from the practice of teachers working with working class African American and Latino students in a poor, urban school district in the United States of America. The results of this investigation, part of larger, broader inquiry, suggest that the categories of teachers’ knowledge implicate each other. Key words : teaching and learning, teacher training, teaching practice, teaching of mathematics

Inscriptions, Mathematical Ideas and Reasoning in VMT

In this chapter, we trace collaborative problem solving as an interactive, layered building of meaning among learners working as a small group. Our analytic aim is to investigate how students through their inscriptive signs collaboratively build mathematical ideas, heuristics and lines of reasoning in the VMT environment.

Challenging Tasks and Mathematics Learning

In this chapter, we present a view of didactical goals of challenging mathematical problems and the cognitive importance of problem-solving schemas. We distinguish between mathematical tasks, exercises and challenging problems and discuss how challenging problems promote the construction of problemsolving schemas. Similar in purpose to the nine case studies presented in Chapter 5, we offer six diverse examples of challenging mathematics problems from varied cultural and instructional contexts. For each example, we examine issues related to its mathematical, cognitive and didactical aspects. Two examples are research-based and accompanied by analysis and discussion of students’ work, while the other examples are informed by considered reflection on their use in practice. In the aggregate, the examples illustrate how challenging mathematics problems are suitable for a range of learners and diverse didactical situations; how such problems can be instruments to stimulate creativity, to encourage collaboration, and support the formation of problem-solving schemas; and, finally, how the use of challenging problems invite educators to study learners’ emergent mathematical ideas, reasoning and schemas.

So Let’s Prove It!

In previous chapters, we observed students throughout middle school and high school working on and making sense of two isomorphic problems in combinatorics – the towers problems and the pizza problems. In this chapter, we see how students just finishing high-school work on another isomorphic problem, demonstrating the application of techniques and ways of thinking that they developed throughout their previous years in the study. We further address the challenge that Davis (1992a) proposes to mathematics education researchers to investigate the emergence among learners of what lies at the core of mathematics: mathematical ideas. Here, a cohort of four high-school seniors – Brian, Jeff, Mike, and Romina – elaborates mathematical ideas and reasoning through work on the Taxicab Problem. They display criteria and techniques for justifying claims and an awareness of the power of generalizing, particularly as an aid to respond to special cases.

Student agency and counter-narratives in diverse multilingual mathematics classrooms : Challenging deficit perspectives

Mathematics classrooms around the world serve students who are learning the dominant language of instruction. These students’ forms of participation in mathematical activity have often been examined from deficit perspectives. Mathematics education research is in great need of counter-narratives to such prevailing deficit assumptions so that we can see how such learners productively use existing resources to engage in mathematics. In this chapter we examine potentially fruitful ways of framing identity and learning centered on student agency that can be brought to bear on the analysis of emergent multilinguals’ mathematical activity. We then illustrate the utility of agency-centered framings with vignettes of student interactions that focus on how emergent bilinguals used multiple linguistic resources in powerful ways. The vignettes are drawn from a variety of international mathematics classroom contexts and focus on students as creative users of linguistic resources in ways that serve a variety of functions during mathematical activity.

Interlocuções e saberes docentes em interações on-line: um estudo de caso com professores de matemática

Integrando matematica, educacao, comunicacao, TIC e ciencia cognitiva, este artigo e resultado de um projeto de pesquisa em Educacao Matematica que, atraves de estudos empiricos, tem como objetivo analisar situacoes cognitivas e condicoes pedagogicas que favorecam a aprendizagem em ambientes virtuais. O enquadramento teorico da investigacao e baseado em estudos sobre aprendizagem matematica que combinam a comunicacao e o pensamento, bem como as interacoes e as interlocucoes. Aqui analisamos reflexoes on-line entre os professores de Matematica dentro de um ambiente chamado Virtual Math Teams (VMT), colaborando para resolver um problema de geometria do taxi. Ilustramos diferentes tipos de interlocucao (informativa, negociativa, avaliativa e interpretativa), bem como dominios de conhecimentos (epistemologia, didatica e mediacao) identificados com o conhecimento profissional dos professores. Nossos resultados indicam que interlocucoes interpretativas e negociativas tem maior potencial para aprimorar o pensamento matematico dos interlocutores. O estudo tambem destaca que, por meio da identificacao e da analise de propriedades de interlocucao, os pesquisadores podem obter insights sobre o conhecimento profissional dos professores.

Supporting Students’ Productive Collaboration and Mathematics Learning in Online Environments

Digital technologies provide a wide range of tools and functions that can support students’ learning of mathematics as well as the development of their mathematical and collaborative practices. Bringing such technologies to mathematics classrooms often do not parallel students’ previous classroom experiences, especially when collaborative practices are emphasized. When facilitating mathematics learning, discrepancies between students’ previous classroom experiences and their expected engagement with new collaborative technologies result in challenges to which teachers need to attend. In this chapter, we describe how a high school mathematics teacher engaged his students in an online collaborative environment, Virtual Math Team with GeoGebra (VMTwG), and how he addressed students’ technological and collaborative challenges to support growth in their geometrical understanding. From a cultural historical perspective, we present a model of how teachers can support students’ instrumentation of collaborative environments and mathematical understanding. In our model, during a mathematical activity, teachers progressively decentralize their role and, simultaneously, support students’ development and performance of collaborative practices. This model informs the theory of instrumental orchestration (Trouche L, Interact Comput 15(6):783–800, 2003; Trouche L, Int J Comput Math Learn 9(3):281–307, 2004; Trouche L, Instrumental genesis, individual and social aspects. The didactical challenge of symbolic calculators. Springer, New York, pp 197–230, 2005) by providing a pedagogical intervention trajectory that supports students’ instrumental genesis (Rabardel P, Beguin P, Theor Issues in Ergon Sci 6(5): 29–461, 2005) of collaborative mathematical environments and shifts students’ focus from their teacher to their peer collaborators.