In this paper, we present an empirical study carried out in a country where STEM education does not have a tradition, and the content and didactics of mathematics are typically studied separately from the content and didactics of biology. In that context, we focus on analogies in the conceptual structure of mathematics and biology, and study how prospective mathematics and biology teachers at the end of their professional training perceive them. Specifically, we investigate to what extent these teachers reason analogously when solving analogous mathematics and biology problems and to what extent they are aware of the analogies represented by the problems. The results are presented in the form of a structured system of qualitative categories that can help indicate opportunities for the development of the perception of such analogies and, consequently, for the development of conceptual knowledge in both school subjects made through mutually effective knowledge transfers.

Badatelsky orientovaná výuka a formativní hodnocení jsou představiteli moderních metod a postupů ve výuce. Aby mohly být smysluplně a efektivně používány, je nutné vytvořit příležitosti k tomu, aby se s nimi učitelé seznamovali, promýšleli je a propojovali je s cíli a obsahy výuky. Tento text si klade za cíl představit online multimediální vzdělávací prostředí Hyperspace, které formou videozáznamů, textových a obrázkových popisů výuky představuje badatelsky orientovanou výuku a formativní hodnocení, a vede uživatele k jejich reflexi. Cílovou skupinou pro toto prostředí jsou (budoucí) učitelé i vzdělavatelé učitelů. Text nejprve představuje teoretická východiska, na kterých je prostředí postaveno, poté jeho design a vývoj. Detailně jsou popsány jednotlivé komponenty i možnosti práce s Hyperspace pro různé cílové skupiny. Na závěr jsou diskutovány výzvy pro budoucí rozvoj Hyperspace. Text je určen především vzdělavatelům učitelů, kteří chtějí využívat multimediální prostředí pro podporu (budoucích) učitelů.

In this contribution, we address the gap that has appeared in mathematics education research and practice with the emergence of dynamic geometry environments and build on the opportunities these environments offer to school geometry. In our qualitative empirical study, we investigate how to elaborate on the general model of conceptual knowledge to make it applicable to dynamic geometry tasks, specifically to tasks including dynamic geometric constructions. We present a design of dynamic constructions of quadrilaterals that comply with Euclidean constructions, derive an assessment instrument based on them, and study what information the instrument can provide about the quality of students’ conceptual knowledge. We present the results in the form of an assessment framework consisting of an example of the assessment instrument and an ordered system of qualitative categories serving as an assessment codebook for interpreting students’ responses in terms of the quality of conceptual knowledge. To clarify the relations between the assessment framework and the general model of conceptual knowledge, we establish a system of subdimensions of conceptual knowledge that indicates how conceptual knowledge can be understood in the context of dynamic geometric constructions and identifies the conceptual knowledge needed to achieve individual categories of the assessment framework.

This contribution focuses on reasoning about quadrilaterals provided by lower secondary school students when working with pre-prepared dynamic constructions. On this topic, we present an exploratory empirical qualitative study carried out within a GeoGebra Classroom environment, and our diagnostic instrument consists of a set of dynamic constructions of quadrilaterals that are based just on a composition of lines and circles. The dynamic constructions consist of the same construction steps as with a straightedge and a compass on paper, without any relational or measurement information provided by the software. The hierarchy in the dynamic constructions is tied to properties of diagonals and takes on various levels and structure (one level, two consecutive levels, two parallel pairs of consecutive levels). For each of the constructions, the participants of the study reasoned which shapes could be found in the construction and why. Various levels of reasoning as well as various levels of students' understanding of quadrilaterals appeared in data. The findings indicate that, at least for the lower secondary school students, the combination of dynamic manipulations and geometric constructions could form a significant space for scaffolding the identification of the features of quadrilaterals and the comprehension of the inclusive relations between them.

THE DESIGN AND EVALUATION OF A VIGNETTE-BASED COURSE FOR ENHANCING PRE-SERVICE TEACHERS’ NOTICING OF STUDENTS’ GEOMETRICAL THINKING

Teachers' analysis of vignettes can be a key for connecting specific classroom situations with mathematics education theories. As vignettes are representations of practice with relevance for professional requirements of the mathematics classroom, vignettes also represent or portray meaningful theoretical elements. The use of vignettes in pre-service and in-service teacher professional development needs, however, conceptual and evidencebased exploration. Building on prior work with video, text, and cartoon vignettes, the project coReflect@maths aims at exploring the potentials of vignette-based work both for supporting professional learning and for research into aspects of mathematics teachers' expertise. Key aspects of the project work will be presented.

This contribution belongs to a larger empirical study that focuses on issues related to the implementation of inquiry-based learning and formative assessment in science and mathematics education, while it also refers to the issue of STEM education. Here, we discuss the two topics from the perspective of professional preparation of primary school teachers. We employ an educational tool called Concept Cartoons and perceive it as a common diagnostic tool for investigating modes of reasoning about general statements in arithmetic, geometry and biology. The presented qualitative exploratory empirical study maps and codes various kinds of reasoning that can be identified with the tool and investigates possibilities of a joint coding procedure. As a result, it provides a conversion table between various modes of reasoning in the three subject domains. The arisen code categories cover the field of generic examples, including the initial stages so that they can be used for scaffolding the process of learning the foundations of deductive reasoning. The joint approach to reasoning in mathematics and biology shows how argumentation and formative assessment can be understood equally and developed simultaneously in both school subjects. It helps us to see how the two school subjects can be integrated didactically.

Předkládaná teoretická studie se zaměřuje na otázky související s propojením formativního hodnocení a badatelsky orientované výuky ve školní praxi a k této problematice přistupuje společně pro matematiku a přírodovědné předměty (konkrétně přírodopis). Na základě společných rysů badatelsky orientované výuky v obou předmětech a s využitím teoretického modelu pro popis interakcí při poskytování okamžité zpětné vazby učitelem jsme vytvořili nový teoretický model pro popis interakcí při formativním hodnocení realizovaném během badatelsky orientované výuky. Výsledný teoretický model pro formativní hodnocení při badatelsky orientované výuce představujeme prostřednictvím sady schémat, která obecně popisují průběh výukového bloku s badatelskou úlohou z pohledu formativního hodnocení, a prostřednictvím nového tzv. double:ESRU modelu pro kódování různých typů interakcí, ke kterým dochází při formativním hodnocení. Možnosti teoretického modelu ilustrují podrobné analýzy dvou virtuálních výukových bloků s badatelskou úlohou (matematického a přírodopisného). Prezentovaný model nabízí pedagogickému výzkumu nástroj pro podrobnou přiléhavou analýzu formativního hodnocení při badatelsky orientované výuce, učitelům a budoucím učitelům nabízí nástroj pro podporu implementace formativního hodnocení a badatelského přístupu do jejich vlastní výuky. Vizualizace propojení obou přístupů a nezávislost modelu na školním předmětu by měly pomoci porozumět možnostem, které formativní hodnocení při badatelsky orientované výuce nabízí.

This contribution addresses issues related to the use of formative assessment in mathematics education. It shows how an educational tool called Concept Cartoons may be used in professional preparation of primary school teachers to mediate their conceptual understanding of a videorecorded lesson and thus prepare them for a future implementation of formative assessment into their own school practice. The text introduces a classroom experiment carried out with future primary school teachers during a video club organized within their professional teacher training, and a Concept Cartoon picture created on the motives of one of the observed video recordings. The study focuses on aspects related to various ideas provided by primary school students during the video-recorded mathematical lesson and various responses to them provided by future teachers before and after working with the Concept Cartoon. The mathematical topic in the focus this paper is the topic of fractions, namely a double discount expressed by a fraction.

This paper introduces an exploratory empirical qualitative study that has been carried out with two diverse groups of future primary school teachers (before vs after the attendance of a course on didactics of mathematics). The study uses an educational tool called Concept Cartoons accompanied by a set of six indicative questions as a means of collecting data on pedagogical content knowledge (PCK) in mathematics. In particular, the study focuses on future teachers’ written responses to virtual pupils’ opinions in a virtual classroom situation related to the algorithm of written addition of natural numbers. The findings reveal ten different code categories of displays of PCK related to knowledge of pupils (three categories), knowledge of tasks (one category) and knowledge of instruction (six categories), some of them related to strong PCK, others to weak PCK. According to the findings, all the categories related to knowledge of pupils occurred in the post-didactic group only, all the categories that appeared only in the pre-didactic group are connected to weak PCK, and all the categories that occurred only in the post-didactic group are connected to strong PCK.

The article focuses on an educational tool called Concept Cartoons and its possible use in professional preparation of future primary school teachers. In particular, it presents the method of how Concept Cartoons can be employed as a tool for diagnosing subject matter knowledge and pedagogical content knowledge in mathematics. The first part of the contribution introduces the concept of teachers' knowledge that is used in the article, and the Concept Cartoon tool. It describes the original Concept Cartoons method for classroom use at primary and secondary school levels that was established by Keogh and Naylor, and then it introduces our work on diagnostic Concept Cartoons method, including a commented summary of our recent research on qualitative diagnosis of subject matter knowledge and pedagogical content knowledge in mathematics. The second part of the contribution introduces another step in the methodology, a mixed approach to the issue that enables to enrich the qualitative results with quantitative characteristics. The mixed method is illustrated through a small empirical study that shows how exactly the quantitative enrichment might be provided.

The contribution focuses on issues related to the implementation of formative assessment methods into inquiry based teaching, by means of issues related to solving twelve multiple-step arithmetic word problems based on operations with natural and rational numbers. These word problems have multiple correct solution procedures and the presented qualitative exploratory empirical study investigates how varied and how usual might be correct solution procedures provided by diverse groups of solvers – future primary school teachers attending diverse university mathematics courses of diverse forms and/or time extent. According to written data collected from 149 solvers, six notions are introduced in the paper: majority, minority and even solution procedures, and majority, minority and mixed solvers. Issues regarding minority solvers are recognized as an important element for implementing formative assessment methods. All the six notions are illustrated in the paper by samples of solution procedures and diagrams of relative frequency. Implications are given for formative assessment within any kind of education involving multiple-step word problems, regardless of the extent of implemented inquiry.