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Contributing to research on students’ multifaceted meanings for angles (e.g., angles as ray pairs, as regions, and as turns), we report on three undergraduate students’ meanings for central and inscribed angles in circles. Specifically, we characterize how these meanings govern their mathematical activities when engaging in a circle geometry task,...
Contexts in source publication
Context 1
... student completed these tasks back-to-back individually during an hour-long session. In the pre-/post-test, we provided each student with a handout that included a graphical definition of central and inscribed angle (see Figure 4a) and asked them to complete a proof (see Figure 4b). We collected each student's written solutions for analysis. ...
Context 2
... student completed these tasks back-to-back individually during an hour-long session. In the pre-/post-test, we provided each student with a handout that included a graphical definition of central and inscribed angle (see Figure 4a) and asked them to complete a proof (see Figure 4b). We collected each student's written solutions for analysis. ...
Context 3
... researchers (e.g., Hardison, 2018;de Matos, 1999) have noted that students tend not to assimilate ray pairs as reflex angles. Our inclusion of a reflex angle in the pre-and post-test (see Figure 4b and Figure 5d) afforded us an understanding of the extent to which the students' meanings were applicable to different angle contexts and the affordances and limitations of those meanings. The diagram we included in the pre-and post-test was not a prototypical illustration for central and inscribed angles in curricula and instruction, and we conjecture it was a novel representation to the students. ...
Context 4
... they are both inside, so then I figured out that can't be right." By "one has to be outside, one has to be inside," Joanna was referring to the figure shown in Figure 4a, in which the central angle was outside the quadrilateral ABOC, and the inscribed angle was inside the quadrilateral. As she applied this idea to assimilate her initial solution (see Figure 6a), where the central angle was inside the quadrilateral enclosing the inscribed angle, she decided to revise her solution to be the reflex central angle (see Figure 7a) so that her meaning was compatible. ...
Context 5
... I looked at this [∠1 in Figure 4b], well, that's obtuse angle, so this one [∠2 in Figure 7a] has to be like…really big. So that's what I learned and kept going with this. ...
Context 6
... alternative interpretation of Jack's "flipped over" was that before C passed A, the inscribed angle ACB was constructed by AC on left and BC on the right (facing into ∠C from the bottom of the circle; see Figure 14a); after C passed A, the angle was constructed by BC on the angle's left and AC on the right (facing into ∠C from the top of the circle; see Figure 14b). Since the ray pairs of the inscribed angle and its "flipped" angle had different orientations, the central angle AOB should correspondingly "flip" to its reflex angle with BO on the left and AO on the right (viewing from the top of the circle; see Figure 14b). ...
Context 7
... alternative interpretation of Jack's "flipped over" was that before C passed A, the inscribed angle ACB was constructed by AC on left and BC on the right (facing into ∠C from the bottom of the circle; see Figure 14a); after C passed A, the angle was constructed by BC on the angle's left and AC on the right (facing into ∠C from the top of the circle; see Figure 14b). Since the ray pairs of the inscribed angle and its "flipped" angle had different orientations, the central angle AOB should correspondingly "flip" to its reflex angle with BO on the left and AO on the right (viewing from the top of the circle; see Figure 14b). ...
Context 8
... alternative interpretation of Jack's "flipped over" was that before C passed A, the inscribed angle ACB was constructed by AC on left and BC on the right (facing into ∠C from the bottom of the circle; see Figure 14a); after C passed A, the angle was constructed by BC on the angle's left and AC on the right (facing into ∠C from the top of the circle; see Figure 14b). Since the ray pairs of the inscribed angle and its "flipped" angle had different orientations, the central angle AOB should correspondingly "flip" to its reflex angle with BO on the left and AO on the right (viewing from the top of the circle; see Figure 14b). It is also possible that Jack considered CA and OA to be the initial sides of the angles and CB and OB to be the terminal sides, and they switched after the angles are "flipped over." ...
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Citations
... Facing this fact, we identified the studies relative to conceptual knowledge, specifically of Geometry, tend to focus in higher degree on future teachers, as seen in the investigations by Maia and Proença (2016), Zuya (2017), Castilho and Proença (2018) and Yurniwat and Soleh (2019), Liang and Castillo-Garsow (2020). In the case, researches about teachers, as those of Steele (2013) and Sunzuma and Maharaj (2019), previously cited, indicate the need to widen and deepen the investigation in this subject with active teachers. ...
The aim of the article is to analyze the conceptual knowledge of a group of mathematics teachers about quadrilaterals and the way they practice their teaching. We conducted exploratory research with 23 teachers who answered an online questionnaire, whose results we analyzed descriptively and categorically. The best-known characteristic of quadrilaterals was that it is a plane figure and the least known was that it is a simple figure. Notable quadrilaterals were mentioned more than irregular shapes. In addition, the presence of irrelevant attributes such as thick line and being rotated made it difficult to recognize some figures. In teaching quadrilaterals (n = 17), four teachers would act as expositors of their ideas. Two teachers would not address the non-examples. Eleven teachers would deal with examples and non-examples. In conclusion, training is needed to understand other examples, non-examples and irrelevant attributes to teach in a way that promotes conceptual development.
