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Structural understanding in advanced mathematical thinking
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... The most comprehensive book about it is in Czech (Hejný, 2014). Elements of the theory are described in various publications such as Hejný (2012), Kuřina (2009) andStehlíková (2004). In brief, the theory describes concept development in mathematics as consisting of several levels, beginning with motivation, through the stage of isolated models (concrete cases of future knowledge) and the stage of generic models (which comprise all isolated models and can substitute for them) up to the abstract knowledge level. ...
... Within this theory, insufficient understanding is captured by the term mechanical understanding, which means knowledge that is not supported by generic models and is mostly grasped by memory only. This theory has been successfully used by researchers in CZ, SK and Poland for the description of the construction of knowledge from different fields of mathematics for pupils and students of different ages (e.g., Jirotková & Littler, 2002;Jirotková & Slezáková, 2013;Krpec, 2016;Robová, 2012;Stehlíková, 2004;Vaníček, 2009). 15 On the one hand, the theory has practical applications, the most prominent being a new approach to teaching called scheme-based education . ...
... Finally, we will mention intervention studies of an experimental versus control group type, which are not frequent in published CZ and SK research 17 (e.g., Sedláček, knowledge of university students has also been studied (Simpson & Stehlíková, 2006;Stehlíková, 2004). For example, the former study documented that in the development of an examples-to-generality pedagogy, an emphasis on the guidance of joint attention is needed rather than the free-for-all of unguided discovery, that is, on teachers and learners making sense of structures together, with the teacher able to explicitly guide attention to those aspects of the structure that will be the basis of later abstraction and to the links between the formal and general with specific examples. ...
This open access book discusses several didactic traditions in mathematics education in countries across Europe, including France, the Netherlands, Italy, Germany, the Czech and Slovakian Republics, and the Scandinavian states. It shows that while they all share common features both in the practice of learning and teaching at school and in research and development, they each have special features due to specific historical and cultural developments. The book also presents interesting historical facts about these didactic traditions, the theories and examples developed in these countries.
... The most comprehensive book about it is in Czech (Hejný, 2014). Elements of the theory are described in various publications such as Hejný (2012), Kuřina (2009) andStehlíková (2004). In brief, the theory describes concept development in mathematics as consisting of several levels, beginning with motivation, through the stage of isolated models (concrete cases of future knowledge) and the stage of generic models (which comprise all isolated models and can substitute for them) up to the abstract knowledge level. ...
... Within this theory, insufficient understanding is captured by the term mechanical understanding, which means knowledge that is not supported by generic models and is mostly grasped by memory only. This theory has been successfully used by researchers in CZ, SK and Poland for the description of the construction of knowledge from different fields of mathematics for pupils and students of different ages (e.g., Jirotková & Littler, 2002;Jirotková & Slezáková, 2013;Krpec, 2016;Robová, 2012;Stehlíková, 2004;Vaníček, 2009). 15 On the one hand, the theory has practical applications, the most prominent being a new approach to teaching called scheme-based education (Hejný, 2012). ...
... For example, problem solving and the diversity of pupils' solutions (Kaslová, 2017), geometric ideas (Kuřina, Tichá & Hošpesová, 1998), and children discovering mathematical concepts and strategies in a learning environment (Jirotková & Slezáková, 2013). On the other hand, the development of mathematical knowledge of university students has also been studied (Simpson & Stehlíková, 2006;Stehlíková, 2004). For example, the former study documented that in the development of an examples-to-generality pedagogy, an emphasis on the guidance of joint attention is needed rather than the free-for-all of unguided discovery, that is, on teachers and learners making sense of structures together, with the teacher able to explicitly guide attention to those aspects of the structure that will be the basis of later abstraction and to the links between the formal and general with specific examples. ...
This chapter presents the emergence of research in didactics of mathematics in the former Czechoslovakia and gives a glimpse at its present state. It is done against the background of the history of schooling in the area and with respect to international influences such as the New Math movement. Due to a limited access to international research prior to the Velvet Revolution in 1989, Czechoslovak research developed relatively independently, yet its character was similar to that of the West. An overview of research after the Revolution is divided into four streams: development of theories, knowledge and education of teachers, classroom research, and pupils’ reasoning in mathematics. Each stream is described by relevant work by Czech and Slovak researchers (with a focus on empirical research) and illustrated by publications.
... Number -3 could have simply been chosen because his/her concept image of inverse is a negative number. It is widely accepted that students tend to rely on their images from number theory when studying and applying group theory (e.g., Hazzan, 1999, Stehlikova, 2004. They often hold a deeply rooted image of the additive identity in numerical contexts necessarily being 0 and the additive inverse a negative number. ...
... Example 2: A student displays SSP-5 if he/she is solving an equation x + 50 = 5 in structure (Z 99 ,+) (see above) and he/she says: "I will subtract 50 from both sides of the equation which means that I will add the additive inverse of 50, that is 49, to both sides." (Stehlikova, 2004) ...
... There is another way of interpreting some problems students have with understanding binary operations, their properties and objects (identity, inverse). Stehlikova (2004) in her research on structuring mathematical knowledge in advanced mathematics described a student coming to know a particular arithmetic structure as a process of development from dependence of the new structure on ordinary arithmetic to gradual independence. ...
Building on some research on structure sense in school algebra, this contribution focuses on structure sense in university algebra, namely on students' understanding of algebraic operations and their properties. Two basic stages of this understanding are distinguished and described in detail. Some examples are given on student teachers' insufficient structure sense and interpreted in terms of various stages of structure sense.
... The term structure is widely used and most people feel no need to explain what they mean by it. In different contexts the term structure can mean different things to different people (e.g., Dreyfus & Eisenberg, 1996;Hoch & Dreyfus, 2004;Stehlíková, 2004). The term algebraic structure is usually used in abstract algebra and may be understood to consist of a set closed under one or more operations, satisfying some axioms. ...
... The number -3 could have simply been chosen because their concept image of inverse element is a negative number. It is widely accepted that students tend to rely on their images from number theory when studying and applying group theory (e.g., Stehlíková, 2004). They often hold a deeply rooted image of the additive identity element in numerical contexts necessarily being 0, and the additive inverse element a negative number. ...
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of
algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra
structure sense components are sub-components of some university algebra structure sense components, and that several components
of university algebra structure sense are analogies of high school algebra structure sense components. We present a theoretical
argument for these hypotheses, with some examples. We recommend emphasizing structure sense in high school algebra in the
hope of easing students’ paths in university algebra.
... The case study below is a part of long-term research on university students' ability to structure mathematical knowledge (Stehlikova, 2004). The mathematical research tool is a finite arithmetic structure created originally by M. Hejny and called restricted arithmetic (or RA). ...
... It is not our purpose to present the development of Molly's understanding of RA (see Stehlikova, 2004). Rather, we will summarise the influence of the above process on her professional development. ...
The paper deals with the education of teachers to be prepared for the demands of the educational reform in our schools. The tool we use, among others, is engaging student-teachers in their own mathematical research thus providing them with experience of 'doing' mathematics. This is hoped to influence their future teaching towards using more investigative ways of teaching. A long-term case study of a university student is presented which looks into the influences investigations have on a solver.
... Activity (Stehlíková, 2004): Notation: N is the set of natural numbers, Z is the set of integers, R is the set of real numbers. The mapping []: R → Z, x a [x] is called the integer part ([x] is the integer such that ...
This report is a part of ongoing research on prospe ctive teachers' mathematical content knowledge. One of the main objectives of te acher training is to determine the balance between theoretical and practical knowledge and ski lls, i.e. the knowledge of mathematics (mathematical concepts and procedures, methodology, relationship with other areas etc.) and the knowledge of learning/teaching mathematics, bel iefs and attitudes towards mathematics and practical skills. This report deals with the following question: "Sho uld a future mathematics teacher meet during his/her professional training non-stand ard mathematical structures which he/she will never use in school practice?'. We claim that the answer is positive; the reasons are illustrated by examples of non-standard structures.
Kapitola popisuje historii a současný stav výzkumu v didaktice matematiky v České republice. Jsou shrnuty významné mezníky v historii tohoto vědního oboru v mezinárodním prostoru a jejich vliv na národní výzkum. Pozornost je věnována zejména metodologii výzkumu v didaktice matematiky a některým současným vlivným teoriím (např. teorii didaktických situací a teorii generických modelů). Současné problémy vzdělávání z pohledu výzkumu didaktice matematiky jsou rozděleny do čtyř oblastí: Znalosti a vzdělávání (budoucích) učitelů matematiky, Výzkum ve školní třídě, Výzkum matematického uvažování žáků a Technologie ve výuce matematiky. Každá oblast je ilustrována národními výzkumy. Na závěr jsou shrnuty perspektivy vývoje v didaktice matematiky v ČR.
The chapter aims to give an overview of the development and state of the didactics of the expressive disciplines and discuss the prospects for their future. We provide summaries of the development in the particular disciplines in Czech Republic and internationally and also we reflect current issues of didactics of literature education, art education, music education and drama education.
Abstract algebra courses tend to take one of two pedagogical routes: from examples of mathematics structures through definitions
to general theorems, or directly from definitions to general theorems. The former route seems to be based on the implicit
pedagogical intention that students will use their understanding of particular examples of an algebraic structure to get a
sense of those properties which form the basis of the fundamental definitions. We will explain the transition from examples
to abstract algebra as a series of shifts of attention and in this paper we will use a case study to examine the initial shift,
which we will call apprehending a structure, and examine how one student came to apprehend the structure of the commutative ring Z99.
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The human consciousness appears to he a duality. On the one hand it is ever creating, and on the other hand it is ever holding fast to that which it has created. These two activities belong, respectively, in the realm of art and in the realm of knowledge. For the purpose of discussion, let us define art as the emergence of new life-forms of the human consciousness, and knowledge as the more or less permanent system of invariants which the human consciousness retains.
the goal of many research and implementation efforts in mathematics education has been to promote learning with understanding / drawing from old and new work in the psychology of learning, we present a framework for examining issues of understanding / the questions of interest are those related to learning with understanding and teaching with understanding / what can be learned from students' efforts to understand that might inform researchers' efforts to understand understanding
the framework we propose for reconsidering understanding is based on the assumption that knowledge is represented internally, and that these internal representations are structured / point to some alternative ways of characterizing understanding but argue that the structure of represented knowledge provides an especially coherent framework for analyzing a range of issues related to understanding mathematics (PsycINFO Database Record (c) 2012 APA, all rights reserved)
One part of what a teacher has to do when preparing his lessons is to consider whether to use a certain text and, if so, how to use it. He must, then, have a good understanding of what the text contains and how its contents are being presented to the future student-reader. This chapter is concerned with a way to make this understanding possible, and, by providing a list of focuspoints, it offers an instrument for analyzing a text in order to obtain a good insight into its contents.