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Structures of Inquiry and Corpus-Relevant Skills
Published in Stephen Hester, David Francis, Eric Livingston, Ethnographies of Reason, 2016
Stephen Hester, David Francis, Eric Livingston
Let me give a brief, but only a brief, sense of this ignored work. As in Figure 10, given ΔABC, let the line EC⃡ contain side AC¯ with A lying between E and C. Then the angle bisector AD⃗ of ÐEAB is called an external bisector of the triangle. Similar to the theorem about internal angle bisectors, a theorem of Euclidean geometry claims that an external angle bisector of a triangle divides the opposite side of the triangle in the same ratio as the adjacent sides. I was aware that the theorem could be proved by calculating the areas of triangles; I wanted to see if I could find such a proof.
Preservice teachers’ experiences with digital formative assessment in mathematics
Published in International Journal of Mathematical Education in Science and Technology, 2022
The activity emphasized two meta-pedagogic mathematic issues: (a) the need for proof when the PSTs discover that two statements are equivalent (the first and the third statements); and (b) providing scaffolding for PSTs to prove the angle bisector theorem (if AD is an angle bisector, then ); this theorem is used extensively in the geometry curriculum in secondary school. The second statement is always true because of the common altitude from vertex A; if AD is an angle bisector, it follows from the definition of the angle bisector that the altitudes DG and DH are equal (first statement), which leads to the third statement because of the equivalence between the first and third statements.
Various solution methods, accompanied by dynamic investigation, for the same problem as a means for enriching the mathematical toolbox
Published in International Journal of Mathematical Education in Science and Technology, 2018
From the point E, we drop a perpendicular EN to the side AB. We denote by M the point of intersection of this perpendicular with the angle bisector AF. By using the angle bisector theorem in the triangle AEN, we obtain: The triangles AMN and AFB are similar; therefore, we can write the proportion: We substitute the relations (**) and (***) in (*) to obtain: a2 = x (y + z).
High school mathematics teachers’ content knowledge of the logical structure of proof deriving from figural-concept interaction in geometry
Published in International Journal of Mathematical Education in Science and Technology, 2020
In the interviews, two teachers (teacher-1 and teacher-4) stated that the angle bisector and the perpendicular bisector needed to intersect in the outside of the triangle. Two teachers (teacher-2 and teacher-3) stated that they should not intersect at all, and one teacher (teacher-5) mentioned that there should be an intersection point in the outside any triangle. Even though s/he could show by making a drawing that in obtuse and right-angled triangles, the angle bisector and perpendicular bisector need to intersect outside the triangle, s/he could not achieve this in acute-angled triangles. Moreover, although all the teachers indicated that the drawing was erroneous, when they were asked to prove their claims, they found it sufficient to make intuitive and experimental explanations based on the drawings. An interview excerpt from an interview with teacher-4 who stated that the angle bisector and perpendicular bisector intersects in the exterior of the triangle is as follows:Researcher:What do you think about the given proof? What would you say to a student who came to you with such a proof?Teacher-4Of course, there is an error in the proof. I would tell the student that there is an error deriving from the drawing of the triangle. I would state that the intersection of the angle bisector and the perpendicular bisector was not point ‘D’.ResearcherWhere should it be?Teacher-4Now, if we were to draw it here (see Figure 10). Well, they didn’t intersect within the triangle. It would be wrong if they did … ResearcherCould you prove this assumption of yours?Teacher-4I did, here, if the base were 7 units (Figure 10), for example, while the angle bisector is here, the perpendicular bisector would be here.As can be understood from the given interview excerpt, the teacher-4 showed that there was an error deriving from the drawing of the triangle by drawing a triangle him/herself with side lengths of 6, 7 and 8 units. When the teacher was asked to prove his/her claim, s/he found it sufficient to provide an intuitional and experimental explanation based on a sample drawing. The teacher thinks that using an example to validate the conjecture is sufficient for a mathematical proof.