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Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics

  • January 1992
  • In book: NCTM Handbook of research on mathematics teaching and learning (pp.334-370)
  • Publisher: Macmillan
  • Editors: Grouws, D.A.
Authors:
Alan H Schoenfeld at University of California, Berkeley
Alan H Schoenfeld
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Learning to Think Mathematically_A
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Learning to Think Mathematically_A
HS.pdf
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All content in this area was uploaded by Alan H Schoenfeld on Sep 12, 2016
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... Se identifica la resolución de problemas como una actividad importante para el aprendizaje de las matemáticas (Schoenfeld, 1992;Trigo, 1997;Ministerio de Educación Nacional, 2003;NCTM, 2000;Benítez, 2006;English & Sriraman, 2010). ...
... Desde la mirada de (Schoenfeld, 1992), las creencias acerca de las matemáticas son la perspectiva desde la cual cada persona se acerca al mundo matemático y pueden determinar la forma, los procedimientos, el tiempo e intensidad del esfuerzo con que abordará un problema dado. Un aspecto que señala también Schoenfeld es que las creencias se abstraen de las propias experiencias y de la cultura en la que uno está inmerso, y recalca que los estudiantes construyen sus creencias sobre las matemáticas formales, su sentido de la disciplina, en gran medida de las experiencias vividas en el aula. ...
... Entre las creencias más comunes frente a la educación matemática encontramos entre otras: la matemática es una actividad difícil y aburrida (Martínez Padrón, 2008;Gómez Chacón, 2003); la matemática es una actividad solitaria, realizada por individuos de forma aislada, solo hay una forma correcta de resolver cualquier problema matemático, las matemáticas que se aprenden en la escuela tienen poco o nada que ver con el mundo real (Schoenfeld, 1992). ...
Aportes para la Enseñanza de la Matemática
Article
  • Mar 2023
R. Este artículo tiene como objetivo contribuir al conocimiento de las creen-cias que tienen los estudiantes de educación secundaria sobre las matemáticas. El análisis se ha enfocado en dar respuesta a las siguientes preguntas: ¿Desde qué perspectivas teóricas entendemos los problemas de matemáticas?; ¿Cuál es el concepto de creencia en el marco de la educación matemática?; ¿Cuáles son las creencias más comunes que tienen los estudiantes de secundaria sobre la re-solución de problemas?; ¿Cómo inciden algunas creencias de los estudiantes en la resolución de problemas de matemáticas? Identificamos a partir de la literatu-ra, algunas creencias personales e idiosincráticas que pueden incidir en la forma como los estudiantes reaccionan frente al aprendizaje de las matemáticas y deter-minan la manera en que abordan la solución de un problema. Concluimos que los maestros deben generar estrategias que permitan identificar las creencias que tie-nen los estudiantes sobre las matemáticas, reforzar las que inciden positivamente y transformar las que sean necesarias para favorecer el análisis y la resolución de problemas. Palabras clave: Creencias, Resolución de problemas, Educación Matemática.
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... Coming to understand mathematics means not only developing a grasp of facts, procedures, and concepts, but learning to think in and with the discipline, internalizing disciplinary habits of mind and the practices of doing mathematics (Schoenfeld, 2017). As discussed in the first main section of this chapter, robust mathematical thinking involves: understanding and enacting both the content and practices of the discipline; being able to employ a wide range of problem-solving strategies; employing monitoring and selfregulation and other metacognitive processes effectively; and having belief systems that support the use of one's mathematical knowledge and understandings (Schoenfeld, 1985(Schoenfeld, , 1992. There is evidence that all of K-12 mathematics (and at least some mathematics at the college level) can be taught as a sensemaking enterprise; moreover, when students experience it as such, they retain/regenerate their understandings longer and more deeply (Bell, 1993;Swan, 2006). ...
... The importance of content and processes, problem-solving, and metacognition and belief systems has been recognized for at least 35 years (Schoenfeld, 1985(Schoenfeld, , 1992, and a "standards movement" emphasizing mathematical processes began in 1989. Yet, little of that sense making has made its way into classrooms. ...
... Thanks to the advent of cognitive science in the 1970s and 1980s, new methods became available to explore the nature of mathematical thinking and problemsolvingspecifically, taking a process view of problem-solving (researching what people did that enabled them to be successful at solving problems) rather than focusing largely on test scores as evidence of successful interventions. The framework discussed below was available by the mid-1980s (see Schoenfeld, 1985Schoenfeld, , 1992. ...
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... The large amount and diversity of research has generated a range of meanings associated with the terms "problem" and "problemsolving" that sometimes detract from the clarity of the studies conducted (Mason, 2015). At the present time, there is a broad consensus that "problem" refers to a mathematical task that generates in someone a sense of problematicity, of not knowing a direct way to solve it and desire to find it (Mason, 2015;Schoenfeld, 1992). Thus, for a mathematical task to be a problem, it must have the potential to generate an intellectual challenge to the person trying to solve it, in a way that enhances its mathematical development, promotes its conceptual understanding and mathematical reasoning, and the competence to communicate mathematical ideas (Cai & Lester, 2010). ...
... Problem-solving is one of the main processes involved in mathematical thinking (Drijvers et al., 2019;Schoenfeld, 1992), but it is also fundamental in technology, engineering, biology, physics, or medicine. Industries are exhibiting a tendency to perform routine tasks in an automated way, increasingly valuing their employees' problem-solving skills (Chan & Clarke, 2017). ...
Peer assessment processes in a problem-solving activity with future teachers
Article
Full-text available
  • Mar 2023
Assessing problem-solving remains a challenge for both teachers and researchers. With the aim of contributing to the understanding of this complex process, this paper presents an exploratory study of peer assessment in mathematical problem-solving activities. The research was conducted with a group of future Secondary mathematics teachers who first were asked to individually solve an open-ended problem and then, to assess a classmate's answer in pairs. We present a study of two cases involving two pairs of students, each of whom assessed the solution of a third classmate. The analysis was carried out in two interrelated phases: (a) individual solutions to the mathematical problem and (b) the peer assessment process. The results show that, in both cases, the assessors were strongly attached to their own solutions, which directly influenced the assessment process, focused on aspects that involve the general problem-solving process and the results. The main difference between the evaluation processes followed by the two pairs lies in the concept of assessment. While the first pair focuses on assessing the resolution process and errors, the other focuses its discussion on giving a numerical grade.
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... According to Flavell [4], metacognition is the process of an individual consciously aware of, considering, and controlling cognitive processes and strategies. Various interpretations of the term have developed since then, according to Du Toit and Kotze [7], where several definitions of metacognitive processes, including Schoenfeld's [8] definition, attribute importance to monitoring and regulating cognitive processes. Metacognition was defined by Flavell [9] as knowledge that adapts the cognitive processes implicated in an activity so that it promotes effective understanding. ...
Mathematics Preservice Teachers’ Preparation in Designing Mathematics-Based Programming Activities Rich in Metacognitive Skills
Article
Full-text available
  • Mar 2023
Programming problems enrich the environment of mathematics learning, adding the flavor of technology to these problems. This is especially true when this programming is Scratch based, where Scratch is being used to make students’ learning of mathematics more meaningful. This role of programming in the mathematics classroom points at the importance of preparing mathematics teachers for designing mathematics-based programming problems activities. The present research describes one attempt to prepare mathematics preservice teachers in designing mathematics-based programming problems activities that could be used in the classroom to teach both programming and mathematics concepts. Twenty-three preservice teachers participated in the research, where they worked in eight groups of 2-3 members in each group. Data collected through observations based on video recordings of the sessions in which the preservice teachers discussed with the pedagogical supervisors the designed mathematics activities. The preparation model comprises of five stages related to the educational environment and to the design notions. The results show special importance for the concepts of struggle and devolution in designing this kind of activities, in addition to the concept of equilibrium between the creative and imitative thinking. The results also show the useful application of metacognitive skills when designing the activities, especially when designing the directions given to the students for solving each of the programming activities.
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... • The nature and structure of problem-solving expertise (Davidson & Sternberg, 2003;Schoenfeld, 1985Schoenfeld, , 1992Silver & Marshall, 1990;Elgrably & Leikin, 2021). • Effective ways of teaching the solving of mathematical problems (Kilpatrick, 1985;Schoenfeld, 1985;Silver & Marshall, 1990). ...
Creativity and Challenge: Task Complexity as a Function of Insight and Multiplicity of Solutions
Chapter
  • Mar 2023
Mathematical problem solving is the heart of mathematical activities at all levels. Problem-solving is both the means and the ends of the development of mathematical knowledge and skills as well as of the advancement of mathematics as a science. Researchers distinguish between problem-solving algorithms, problem-solving strategies and heuristics and problem-solving insight. Insightful and divergent thinking are at the base of mathematical creativity. This chapter analyzes the mathematical challenge embedded in problem-solving tasks from the point of view of evoked mathematical insight and the use of multiple solution strategies. While a variety of variables (such as conceptual density, level of concepts, length of solution or use of different presentations) determine the complexity of mathematical problems, the insight component and the requirement to solve problems in multiple ways increase the mathematical challenge of the task. Researchers distinguish between different types of mathematical insight as they relate to the distinction between mathematical expertise and mathematical creativity. In this chapter, we introduce a distinction between mathematical tasks that allow insight-based solutions and tasks that require mathematical insight. We provide empirical evidence for our argument that tasks that require mathematical insight are of a higher level of complexity.KeywordsInsightful solutionMultiple solution strategiesInsight-requiring tasksInsight-allowing tasks
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... We found that students who were more versatile in playing the reverse game, strategically separating certain points and examining how other points vary with dragging (i.e., higher S), were better able to identify the properties of the figure (G), and generated counterexamples (F) and conjectures (CQ) of higher quality. A possible explanation can be derived from the vast literature on problem solving (e.g., Leikin, 2007;Schoenfeld, 1992), which points to the flexible use of alternative strategies as an indication of higher order reasoning skills. ...
Computer-Supported Assessment of Geometric Exploration Using Variation Theory
Article
Full-text available
We report on an innovative design of algorithmic analysis that supports automatic online assessment of students’ exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic activity format for if–then propositions and implemented the activity on a platform that collects and analyzes students’ work. Finally, we searched for ways to use variation theory to analyze ninth-grade students’ recorded work. We scored and classified data and found correlation between patterns in exploration stages and the conjectures students generated. We demonstrate how automatic identification of mistakes in the early stages is later reflected in the quality of conjectures.
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The cops model for collaborative problem-solving in mathematics
Article
Full-text available
  • Mar 2023
While the importance of collaborative problem-solving has been highlighted as an important skill by the OECD (Organisation for Economic Co-operation and Development), no specific model exists for mathematics facilitators to implement when engaging their students in group problem-solving. In this paper, we introduce the CoPs model for Collaborative Problem-solving in mathematics, a structured approach to facilitating group-work of this kind. We outline the theoretical foundations of the CoPs model as well as its implementation in practice. We then consider its usage in a case study with 89 highly-achieving post-primary school students from numerous mixed schools around Ireland and discuss student feedback on the experience of the CoPs model as an approach to collaborative problem-solving in mathematics. Students were positive about this approach to collaborative problem-solving, suggesting that this model may provide an approach that teachers could introduce into classroom settings when teaching problem-solving in mathematics.
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Supporting metacognitive talk during collaborative problem solving: a case study in Scottish primary school mathematics
Article
  • Mar 2023
This small-scale study explored the presence of collaborative metacognitive talk (CMT) during collaborative problem solving (CPS) in a Scottish primary mathematics class. Content analysis was conducted on student and teacher group interactions during CPS (n = 12 students in 3 groups × 3 sessions). The largest proportion of CMT was teacher–student, suggesting the use of a dyadic teaching approach between teacher and student as opposed to the teacher facilitating student–student CMT. A teacher focus group provided some tentative explanations as to why the dyadic approach might be adopted over a facilitative role to support student–student CMT, including focussing on strategy use, school culture around CPS, and teacher beliefs about learners’ abilities. Implications for practice are discussed.
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The Influence of Royaumont on Mathematics Education in the USA
Chapter
  • Mar 2023
The influence of the 1959 Royaumont Seminar on US mathematics education is described. Five periods are discussed: The 1960s New Math, the post-New Math of the 1970s and 1980s, the 1990s reform, the 2000s standards-based era, and the 2010s Common Core State Standards. The landmarks are initially described by tracing the teaching of geometry, but then other key Royaumont themes are developed: The use of set theory, logic, and mathematical structure; the use of problem solving and inquiry; and the role of research mathematicians. The views of three European mathematicians, Jean Dieudonné, René Thom, and Hans Freudenthal, whose writings in US publications stimulated important discussions, provide a second lens for examining the streams of thought that emerged. Examples from US mathematics education research, state policy documents, and instructional materials are given to illustrate the impact of the Royaumont meeting on the USA.
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Organising Committee E VA A DA MKO PHD CSABA GA BOR KE ZI PHD ERIKA PERGE PHD ATTILA VA MOSI Conference on Problem-based Learning
Conference Paper
Full-text available
  • May 2022
Az építőmérnök képzésben BSc és MSc szakon is szerepel a környezeti hatásvizsgálat témaköre, amit mint környezetvédelmi szakmérnök és hatásvizsgálat végzésére jogosult szakértő oktatok. Az oktatás módszereit az évek során szerzett oktatói és mérnöki tapasz talat és az egyetemen a velünk szemben érvényesített követelményrendszer alakítja, pedagógiai, oktatásmódszertani képzés nélkül. A cikkben azt fejtem, ki, hogy ezzel a háttérrel az alkalmazott módszerek és tapasztalatok mennyiben felelnek meg a problémaala pú tanulás ismérveinek és hol lehetne ebben az irányban módosítani, illetve, ha ez elvárás, akkor javítani a módszereken.
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