Participants' knowledge was identified using prepositional analysis. The strongest evidence of a relationship between teacher pedagogical content knowledge. Only pedagogical content knowledge seems to have an impact on the quality of long history of discussion and debate around the connection between teacher. teaching and on pedagogical content knowledge in particular. given level, the variety of instructional materials available in relation to those programs, and.
Often the same or at least overlapping information is displayed in varying representations e. The flexible use of such representations plays an important role in learning mathematics Heinze et al.
Both, American and German standards of education stress the importance of multiple representations e. Students should be able to flexibly use representations, communicate mathematical ideas through them, select, apply, and translate among them, and to interconnect them. Although multiple external representations can support learning Ainsworth, ; Eitel et al.
Especially the transition and integration processes between representations pose a crucial obstacle to learners Duval, In fact, many learners tend to use multiple external representations suboptimally in that they only use a few familiar ones or fail to integrate them Schwonke et al. The present study analyzed the effects of providing knowledge about multiple representations in an integrated or separated way. More specifically, we addressed the following four research questions. Firstly, we tested whether the integration of knowledge types would enhance the applicability of PPK concepts.
At the meantime, we did not expect integration to yield beneficial effects in applying aspects of PCK, because we assumed that due to them already being content-related, no major inertia problem exists which could be overcome by combining different knowledge types. Secondly, we assumed that an integrated presentation facilitates switching back and forth between content-specific and content-independent considerations when thinking about teaching problems.
Teachers' pedagogical content knowledge and its relation with students' understanding
Accordingly, we predicted that the integrated condition is superior in applying both perspectives simultaneously i. Thirdly, as participants with low prior knowledge or low working memory capacity might be overwhelmed by an integrated presentation, we assumed aptitude—treatment interaction effects: The positive effects of integration are moderated by prior knowledge Hypothesis 3 and working memory capacity Hypothesis 4.
The participants volunteered to participate and received a compensation of 15 Euros for participating. All participants were aware of taking part in research.
Before starting, a standardized explanation about ethical guidelines was read out loud and participants provided verbal informed consent. Participants who declined to provide the verbal informed consent were offered the possibility to withdraw from the experiment and still receive the financial compensation. All participants provided written informed consent allowing us to use their collected data anonymously for publications. All data was anonymously collected and analyzed. The incentive was 15 Euros, disbursed immediately after participation.
Their average age was Most of them were in their second or fourth semester of teacher training They were recruited in various mathematics lectures and via posters at the Institute of Mathematics over a period of one month. After registration, the participants were randomly assigned to one of the two learning conditions and participated in a group session where they worked individually on computers. In the second condition, the knowledge types i. Dependent variables comprised two scores, one for the application of PPK and one for the PCK aspects, as well as measures of the combined use of knowledge types.
This task was designed in accordance with the reading span task described by Unsworth et al. The participants read and subsequently classified sentences to be either sensical or nonsensical. While doing so, they were asked to remember a set of unrelated letters presented at the end of each sentence e. The amount of sensical and nonsensical sentences was balanced and the sentences 10—15 words long.
Nonsensical sentences were created by replacing a single word in an otherwise sensical sentence. The sentences were presented in sets, varying from two to five sequenced sentences. Following each set, participants were asked to recall the presented letters in their correct order of appearance.
There were three trials per set size with different sentences altogether 42 Items. One point was awarded for each correctly retrieved letter provided that it had also been recalled in correct position. The total score was computed by adding up the awarded points. Three questions tapped on each knowledge type e. Thus the crucial difference between conditions was that PPK and PCK aspects on multiple external representations were treated either interrelated or apart from one another.
The learning environments were based on several mathematical and psychological book chapters and journal articles. We based the mathematics education stance i. We addressed several didactic aspects worth consideration when working with multiple external representations e. We presented translation pitfalls between different representations of fractions e. We presented psychological functions of multiple external representations e.
In the separated condition, the two learning environments i. Unlike the separated condition, the integrated condition encompassed just one learning environment. It was established by using the contents of the single learning environments and combining them in a thematically coherent sequence. In order to smooth transitions between stances and achieve integration of different topics i.
These phrases did not, however, contain additional information on teaching with multiple representations e. The number of words and basic information on the two experimental conditions were kept constant. Through a range of textual devices, paper articles, film or video, software programs, Luke adds, texts construct and position an ideal and so on—present particular versions of the reader.
While speaking about visual Hall, ; cf. Audiences readersMaster- Any and all of the authors and producers of man explains, those texts are first and foremost storytellers. They are compelled to occupy a particular physical tell selected stories in well-established and com- position by virtue of the positioning of the prehensible codes of a world that is reconstructed, camera. Identifying and being conscious of this represented, packaged, and shaped in ways that physical position should quickly reveal that we inform students, pedagogically, what and how to are also being invited to occupy a social space.
Ellsworth knowledge intended to make possible. Emphasiz- concludes that in order to make sense of a film ing the need to re-conceptualize the existing or, any text on its own terms, more appropriately, non-existing relationship between content and pedagogy at the time the the viewer must be able to adopt—if only midsShulman and his colleagues high- imaginatively and temporarily—the social, po- lighted the centrality of the relationship between litical, and ideological interests that are the content and pedagogy both in and to a meaningful conditions for the knowledge it constructs.
But while the idea of edge from a particular social, political, and pedagogical content knowledge continues to re- ideological point of view. Conceiv- than simply re-present it. To that extent, subject-area where one ends and one begins, what and who texts are, and must be considered, inherently activates them, how they operate together to direct pedagogical. Showing that and how the relationship be- tween content and pedagogy is more complex, the 4.
Integrating pedagogical content knowledge and pedagogical/psychological knowledge in mathematics
What, then, might of education, must encourage the reunion broadening the definitions of pedagogy and con- of pedagogy and content. How, for example,p. And how can the idea of relevancy of the concept of pedagogical content pedagogical content knowledge itself be used to knowledge itself.
Instead, it examines the defini- incorporate and address elements of the pedago- tions of content and pedagogy underlying current gical act which currently are, by and large, uses of pedagogical content knowledge and raises excluded from consideration? Nor does it What it Ferguson, but neglects to address the human does mean, however, is that whether teachers relations they attempt to produce as readers recognize it or not, the pedagogies they design are engage them.
What we have, then, are representa- not and should therefore not be considered as tions whose functions as pedagogical devices isolated from the pedagogical invitations for remain, by and large, obscure to teachers—those learning embedded in the materials they bring who perhaps most need to understand those into their classrooms.
Consequently, thinking pedagogical functions and make them transparent about pedagogy in teacher education must incor- to others. In materials and who those stories privilege and why other words, thinking of the pedagogies enacted by something many teachers already do. That To conceive of pedagogical content knowledge is, we might disagree with what such texts tell us in terms of what has been suggested in this paper, yet believe the way they tell it has little to do with we must regard it more than a matter of teachers the stories being told or the opportunities we, as knowing their subject matter and using that readers, have to make meaning of and in them.
While the considering the degree to which one ought to work representations McDiarmind et al. Such a possibility is already instructional and instructing.
That is, did not escape McDiarmind et al. Indeed, each of those ideas and theories in classrooms manages them—as any other text brought into the class- those who attempt to engage them. Take, for exam- to know something in the subject area and how ple, the pedagogical nature of representation. In other questions—ones which, in line with critical peda- words, representations of subject matter are gogy, pertain to issues of power, knowledge, and pedagogical; they do not only tell students some- the politics of representation—than those nor- thing about the world, they also position them to mally asked in teacher education, a focus on the know of and be in the world in some ways rather pedagogical nature of representation is not in- than others.
In that sense, strategies used by herently antithetical to some of the more recent authors, just like those used by teachers, are discussions within the literature on pedagogical intended to teach. As pedagogical devices, both content knowledge that have moved beyond the sets of strategies, to use Popkewitzp. In organization to their consciousness. And Grossman et al. Teaching in pre- should know in order to make effective instruction service education classes then requires a double is the instructional nature of knowledge?
Teacher education, in that case, is not Admittedly, such an approach would require a only about teaching students how to make content double epistemological shift; one which pertains instructional; it is also about examining the not only to how teacher educators conceive of the instructional nature of content and what that idea of pedagogical content knowledge itself but means for what teachers need to do with that one which pertains to the entire project of teacher knowledge as they plan their own pedagogical preparation.
To think of pedagogical content layer to be added to it. As in the case of Focusing on the pedagogical nature of content in many other conversations about the relationship teacher education means going beyond indeed between what student teachers get outside and against the current division of labor between inside colleges of education, the discussion should colleges of education and colleges outside of address the substance of what prospective teachers education; a division in which prospective teachers receive and do not receive in their disciplinary acquire content area knowledge in colleges outside courses, not the number of credits they received.
If we think of pedagogical content knowledge Although this poses various epistemological not simply as learning how to make content and subsequently, pedagogical, perhaps even instructional but as examining the instructional organizational challenges, these challenges should nature of content, where should such a concept be not be a reason to abandon the project.
As addressed in the current division of labor?
Should pedagogic devices, subject-area texts regulate the it be part and parcel of studying subject-area relations between the possible, the potential and content? Or should it be addressed together with the actual. They provide readers with a selectively an overly crowded list of issuesafter the fact, in constructed social reality—social knowledge and colleges of education?
Poststructural gogical content knowledge to include such issues investigations in education. The concept of a knowledge base. Minnea- and teachers who use them are necessarily silent. University of Minnesota Press. A study of practical At issue, then, are finding ways for preservice knowledge.
Educational films against critical peda- how to organize knowledge for students to gogy. Hidden curriculums in the knowledge is organized to be experienced Giroux, classroom pp. From preparation to practice; Incorporating a broader notion of pedago- designing a continuum to strengthen and sustain teaching.
Perspectives on relation of the learner to the text contentbut learning to teach. Jossey- what contentand the how processthat Bass. To foster those levels of learning among its Routledge. The order of discourse.
A post-structuralist reader borrow once more from Gore, to be more publicly pp. Schooling and the struggle for public life: Identity politics and the new cultural criticism.
Integrating pedagogical content knowledge and pedagogical/psychological knowledge in mathematics
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