ChapterPDF Available

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

Authors:
Alan H Schoenfeld at University of California, Berkeley
Alan H Schoenfeld
Download full-text PDFDownload full-text PDF
Read full-text
Download citation
Learning to Think Mathematically_A
HS.pdf
Content uploaded by Alan H Schoenfeld
Author content
All content in this area was uploaded by Alan H Schoenfeld on Sep 23, 2016
Content may be subject to copyright.
Learning to Think Mathematically_A
HS.pdf
Content uploaded by Alan H Schoenfeld
Author content
All content in this area was uploaded by Alan H Schoenfeld on Sep 12, 2016
Content may be subject to copyright.
... Definitions and discussion of key concepts Problem Both "problem" and "problem solving" have had different, and sometimes contradictory meanings through the history, and different understandings of what constitute these concepts still exist today (Lester, 1994;Stanic & Kilpatrick, 1989;Schoenfeld, 1992;Pehkonen, 1995;Fan & Zhu, 2000). In general there are two different definitions of problem found in the literature. ...
... According to Schoenfeld (1992) the scientific status of heuristic strategies à la Polya (1945) has been problematic, especially in the 1970s, when referring to studies by Wilson (1967) and Smith (1973), which concerned the classical transfer-problem of learning. Both studies indicated that the heuristics that students were taught did not, despite their seemingly generality, transfer well into new domains. ...
... Problems have occupied a central place in mathematics curriculum since antiquity (Stanic & Kilpatrick, 1989). In contrast, the importance of problem solving just started to be recognized in the late 1970s (Schoenfeld, 1992), and in terms of research it has been a focus for three decades (Zhu, 2003). Since the late 1970s the importance of developing pupils' competence in problem solving has been widely recognized. ...
What characterises the heuristic approaches in mathematics textbooks used in lower secondary schools in Norway?
Article
Full-text available
  • Dec 2011
In this paper I present findings of an analysis of how mathematics textbooks treat heuristic approaches. The aim of this analysis is to give a characterisation of the occurrence of nine well-known heuristic approaches by analysing 740 examples presented in six ninth grade textbook series. The findings show that many of the problems in the examples are being solved by using one or more heuristic approaches, but the characteristics of the examples and the textbooks’ lack of discussion of the approaches themselves make it challenging to teach and learn these in school. The heuristic approaches seem to be used rather incidentally, which is supported by the fact that none of the textbooks explicitly treat or mention problem solving.
View
... It also includes the ways this information is represented, stored, organized and accessed (e.g. Schoenfeld, 1992). Another important part of mathematical knowledge base for problem solving, and, apparently, for problem posing, is a set of rules and norms that exist in the particular domain about legitimate and prototypical connections between different pieces of mathematical information (Schoenfeld, 1992). ...
... Schoenfeld, 1992). Another important part of mathematical knowledge base for problem solving, and, apparently, for problem posing, is a set of rules and norms that exist in the particular domain about legitimate and prototypical connections between different pieces of mathematical information (Schoenfeld, 1992). This kind of knowledge is constructed through continuous exposure to various mathematical problems, elaboration on a part of them and storage of problems in the knowledge base. ...
... Schemas are referred to as organized structures of mental actions for associating new information with already existing one (e.g. Schoenfeld, 1992). They are used for making a personal sense of information, coding and storing it in the long-term memory as well as for recalling and decoding it back (e.g. ...
Feeling of innovation in expert problem posing
Article
Full-text available
  • Nov 2012
This paper is one of the reports on a multiple-case study concerned with the intertwining between affect and cognition in the mechanisms governing experts when posing new mathematical problems. Based on inductive analysis of a single case of an expert poser for mathematics competitions, we suggest that the desire to experience the feeling of innovation may be one of such mechanisms. In the case of interest, the feeling was realized through expert’s reflections on the problems he created in the past, by systematically emphasizing how a new problem was innovative in comparison with other familiar problems based on the same nesting idea. The findings are discussed in light of past research on expert problem posers and expert problem solvers.
View
... This study is part of the LCM-project (Jaworski et al., 2007) within the Norwegian Research Council's KUL 1 programme and it addressed the question of the nature of Norwegian students' beliefs, attitudes, and emotions in mathematics (and its teaching and learning). Many studies have provided evidence that students' learning outcomes in mathematics are strongly related to their beliefs about mathematics (Furinghetti & Pehkonen, 2000;Pehkonen, 2003;Schoenfeld, 1992;Thompson, 1992). Pehkonen (1995) notes that beliefs do influence not only how students learn mathematics but they "may also form an obstacle for effective learning in mathematics" (p. ...
... Comparing my sixth factor items with these two results I decided to adopt the wording from Brew and colleagues (2006), and call the factor a MAD-factor, an abbreviation of Mathematics as an Absolute Discipline as this factor have a resonance with the factor described in Brew et al. (2006) and Schoenfeld's (1992) statements that are as well related to mathematics as a discipline. ...
... Controlling the relationship between the statements "it is innate to be good" and "mathematics is difficult" (factor Self-confident) it appeared that 70 % of students who thought mathematical abilities to be innate agreed that mathematics is difficult (only 32 % of these students agreed that "mathematics is easy" A14). Looking closely at items B2, B3, B4, B6 and statements from the Typical student beliefs about the nature of mathematics presented by Schoenfeld (1992) (statements like "mathematics problems have one and only one right answer"; "there is only one correct way to solve any mathematics problem", etc.) it seemed that students in my study did not consider time as an important factor when solving problems, nor were they sure that there is only one answer to a mathematics task (c 70 % disagreed with the item B3). More than 66 % of students expressed that it does not show their lack of mathematical knowledge when they make mistakes and almost 3 out of 4 valued the procedure for finding an answer more than the answer itself. ...
An investigation of Norwegian students’ affective domain in mathematics
Article
Full-text available
  • Dec 2009
After decades of research in the affective domain in mathematics education, and search for ways to enhance students’ positive attitudes towards the discipline, the perception that to be able to do mathematics is innate remains a widespread belief. Already twenty years ago the Fourth NAEP study concluded that students believe mathematics to be important, difficult and based on rules, and theses attributions also characterise the view of mathematics even two decades later. As a relationship exists between the claims ”mathematics is difficult” and ”mathematics is boring” one could assume that students lack interest towards mathematics. The conclusions about the present situation are based on a study carried out in Norway in 2005. This paper documents and analyses the data from the study. Six factors are identified and analysed in relation to students ́ affective domain in mathematics. The six factors are: interest, hard-working, self-confidence, usefulness, insecurity, and MAD (Mathematics as an Absolute Discipline).
View
... In the literature, one can find several overviews on mathematics-related belief research (e.g. Op't Eynde et al., 2002;Pehkonen, 1994Pehkonen, , 2004Pehkonen & Törner, 1996;Philipp, 2007;Rösken, Törner & Pepin, 2011;Schoenfeld, 1992;Thompson, 1992;Underhill, 1988aUnderhill, , 1988b. About a decade ago the first book on mathematical belief research (Leder et al., 2002) was published, in order to give an overview on different research perspectives and on research done. ...
... In his review of research on affect and mathematics, McLeod (1992, p. 575) noted: "Although affect is a central concern of students and teachers, research on affect in mathematics education continues to reside on the periphery of the field". Additionally Schoenfeld (1992) states that there is much research done on students' beliefs, but not so much on teachers' beliefs. The situation now appears to be changed and teachers' beliefs are more studied. ...
Research on mathematical beliefs – the birth and growth of the MAVI group in 1995–2012
Article
Full-text available
  • Nov 2012
In the 1980s the meaning of beliefs for teaching and learning aroused also to the consciousness of mathematics educators. Therefore, here it is firstly sketched the research field of mathematical beliefs, in order to understand why belief research has been a topic for an international research group for more than 30 years. The aim of the paper is to have a look at the history of MAVI and to describe its development within the years 1995–2012. A special look is given at the birth of the MAVI group in the middle of the 1990s and then its development through the last 18 years will be described. Also some statistics on the MAVI meetings and their participants are presented. And in the appendix the total list of the MAVI proceedings is documented.
View
... The format may also unwittingly involve "greater implications than intended by the developers" (Hill, Sleep et al., 2007, p. 150). The MC format may solidify the misconception that mathematical competence is demonstrated by quick solutions (Schoenfeld, 1992) among some teachers, and teachers who do not think of mathematics as quick solutions to routine problems may feel marginalized by the format. Even if Hill, Dean and Goffney (2007) conclude that their work on validation corrects for common problems of MC items, the aspects mentioned above should be taken into consideration when translating and adapting measures. ...
... In addition to these more MKT-specific challenges with the MC format, the teachers' reflections also indicated several issues concerning the item format that are not specific to the MKT items. Some of the more experienced teachers' reflections on the format indicate an anticipation that mathematical competence is demonstrated by quick solutions, as reported by Schoenfeld (1992). The teachers also raised some issues related to more general test-taking strategies in their reflections. ...
Does the format matter? How the multiple-choice format might complicate the MKT items
Article
Full-text available
  • Dec 2011
In order to design appropriate professional development programs for teachers, an instrument has been developed in the U.S. to measure teachers’ mathematical knowledge for teaching. The process of translating and adapting these measures for use in other countries involves several challenges. This article focuses on issues related to the multiple-choice format of the items. Analyses of focus-group interviews reveal that the multiple-choice format may complicate the items. The teachers’ reflections about the format in this Norwegian case contribute to the understanding of this important challenge.
View
... In addition, the notion of problem solving has different meanings in research literature in mathematics education (Lesh & Zawojewski, 2007;Lester, 1994;Stanic & Kilpatric, 1989). Already Schoenfeld (1992) stated that " 'problems' and 'problem solving' have had multiple and often contradictory meanings through the years -a fact that makes interpretation of the literature difficult" (p. 337). ...
An analysis of mathematical modelling in Swedish textbooks in upper secondary school
Article
Full-text available
  • Oct 2013
A new national curriculum has recently been implemented in the Swedish upper sec-ondary school where one of the goals to be taught is modelling ability. This paper presents a content analysis of 14 ”new” mathematical textbooks with the aim to investigate how the notion of mathematical modelling is presented. An analytic scheme is developed to identify mathematical modelling in the textbooks and to analyse modelling tasks and instructions. Results of the analysis show that there exist a variety of both explicit and implicit descriptions, which imply for teachers to be attentive to complement the textbooks with other material.
View
... Op 't Eynde, De Corte & Verschaffel, 2002). The papers by Schoenfeld (1985Schoenfeld ( , 1992 and Pehkonen (1995) contributed concerning students' beliefs about mathematics in general; McLeod (1992) and Kloosterman (1996) described the connection to affect and motivation; Lester, Garofalo and Kroll (1989) were concerned with self-concept beliefs related to mathematics problem solving. McLeod (1992) made an important contribution to organizing the field. ...
View of mathematics – an investigation of Estonian university students
Article
Full-text available
  • Jun 2014
This study reports on first-year Estonian university students’ view of mathematics. The data was collected from 970 university students of different disciplines. The participants filled out a Likert-type questionnaire that was developed using previously published instruments. This paper documents and analyses the data from the study. In this study students agreed that mathematics is an important and valuable subject. Female students have a more positive view of mathematics than male students. Science students have a more positive view of mathematics than non-science students.
View
... non-routine tasks, exploration of concepts, processes, relations etc., or when different strategies are endorsed when solving the task) are, in this study considered an invitation to the pupil to engage in and experience problem-solving strategies and methods (cf. Schoenfeld, 1992). ...
Mathematics teachers’ knowledge-sharing on the Internet: pedagogical message in instruction materials
Article
Full-text available
  • Sep 2016
This article reports on a study of teacher-shared documents containing mathematical tasks published on the Internet. The aim was to identify the goals, methods and pedagogical justifications presented in the documents and what was needed to solve the tasks. Content analysis was used to define their pedagogical message. The results show that the documents mainly involve content goals for younger pupils that are not consistent with the explicit descriptions. The conceptual goals are communicated to a great extent, but are not supported by task features. The reasons for why the tasks given are expected to lead to a certain goal are very often implicit, and, as a result, the content of the documents and the quality of the tasks are somewhat unclear to other teachers.
View
... The knowledge demonstrated by teachers has been previously examined to gain an understanding of the kind of knowledge base that makes a good teacher. It has been argued, for example, that the quality of a teacher's knowledge has a strong influence on how the teacher is able to link and use his/her knowledge in both the preparation of lessons and also in teaching (Schoenfeld, 1992). Shulman introduced the term Subject matter knowledge to describe the extent and organization of a teacher's knowledge of the subject to be taught (e.g. ...
Finnish entry-level students’ views of teacher knowledge and the characteristics of a good mathematics teacher
Article
Full-text available
  • Dec 2017
This paper reports a study of the views held by Finnish students at the start of their university studies concerning their understanding of the knowledge and characteristics of a good mathematics teacher. A total of 97 students following a basic university course responded to a questionnaire. The results showed that a knowledge of teaching mathematics was more often used to describe the good mathematics teacher than a knowledge of mathematics. According to the students’ views, mathematics teachers need to be able to take the level of understanding of individual students into account in their teaching. Good mathematics teachers were also considered to be skilled in explaining, simplifying and transforming mathematical contents for their students. A good mathematics teacher was often described by the respondents as a patient, clear, inspiring and consistent person. On the other hand, characteristics such as humorous, likeable, empathetic, or fair were seldom used in the students’ responses to describe a good mathematics teacher. Those respondents who planned to become teachers demonstrated a more learner-centred concept of a good mathematics teacher than did those who were aiming at some other subject specialist profession than that of teaching.
View
... Various authors consider problem-solving to be what mathematics and computer science do. DS has developed at the intersection of these fields (Schoenfeld, 1992;Gallopoulos, Houstis, Rice, 1994;de St. Germain, 2008;Lanthier, 2011). The Polish DSTs indicated that DS is a different way of solving problems than software development (Kuncewicz, 2019), and that defining the problem to be solved is the initial and most crucial stage of a DS project (Alekseichenko, 2019a). ...
Are They Doing Artificial Intelligence? (Re)Constructing the Primary Activity in Data Science
Article
Full-text available
  • Nov 2024
Data science (DS) is concerned with building so-called artificial intelligence, i.e., computer systems that automate tasks based on historical data. This article is the first attempt to examine DS using Adele E. Clarke's framework of social worlds. The main goal of this paper is to show the (re)construction of primary activity based on the example of the social world of DS in Poland. Methodological reflection on this (re)construction is an underdeveloped element in the study of social worlds; therefore, this paper strives to make this process explicit. The empirical background is a three-year ethnographic study, following Clarke's situational analysis approach. The methodological results demonstrate the indispensability of collaborative ethnography in (re)constructing primary activity and the importance of finding palpable elements as those being crucial to understanding primary activity. The substantive results focus on the idea that data scientists do not refer to their activity as doing artificial intelligence.
View
ResearchGate has not been able to resolve any references for this publication.