Abstract
What mathematical knowledge is required for teaching has been researched by many, with Ball et al. (J Teach Educ 59(5):389–407, 2008) and their practicebased theory of mathematical knowledge for teaching (MKT) primary among them. However, what is required in terms of mathematical knowledge for teacher educator training has been researched much less. The present study builds on an emerging framework on mathematical knowledge for teaching teachers (MKTT) by Zopf (Mathematical knowledge for teaching teachers: the mathematical work of and knowledge entailed by teacher education, University of Michigan, Ann Arbor, 2010) augmented with more recent developments in the field. The empirical basis for the study stems from an implementation of a short course on history in mathematics education at the Danish School of Education, and in particular on two case studies involving three teacher educator students participating in that course. Based on analyses of their developments in terms of MKTT, we point to the potential of using history in both teacher and teacher educator training, and in particular historical source material. As part of the analyses, we also consider the teacher educator students’ development of disciplinary knowledge of mathematics, especially in relation to developing and making use of knowledge of the epistemology of mathematics and mathematical work. We assert that the cases we present here address Zopf’s call for case studies from different teacher education contexts as well as investigations of novice teacher educators, and further contribute to the developing theory of MKTT.
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Notes
 1.
HPM refers to the International Study Group on the Relations between the History and Pedagogy of Mathematics, which is affiliated with the International Commission on Mathematical Instruction (ICMI), and concerns the role history of mathematics in mathematics education.
 2.
 3.
 4.
Alternatively, the educators may hold a master’s degree in mathematics. Note that teacher educators in Denmark are not required to possess a PhD.
 5.
In 2016 the three other short courses were: “The Relevance Problem of Mathematics Education” taught by Tomas Højgaard; “Mathematics for All” by Lena Lindenskov; and “Technology in Mathematics Education” by guest professors Morten Blomhøj and Morten Misfeldt.
 6.
As supplemental literature for this lesson, students were encouraged to examine Jankvist and Kjeldsen (2011).
 7.
 8.
Supplementary literature for session 4 included Smestad et al. (2014).
 9.
In her early work, Ball (1988) also included PCK, and in particular the ability to represent mathematics in different ways in order for other to understand as part of knowledge of mathematics.
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Appendix: Center text from Fig. 1 (partial translation)
Appendix: Center text from Fig. 1 (partial translation)

VII.10 Excerpt from Viète’s theory of equations

Third sentence in another way

If, when B is greater than half of D:

If A cube − B square thrice into A is equated to B square into D,

but

B square [thrice] into E − E cube will be equated to B square into D,

and there are two right angled triangles with equal hypotenuse B, such that the acute angle subtended by the perpendicular of the first, is triple the acute angle subtended by the perpendicular of the second, while double the base of the first, is D, making the double of the base of the second be A…

[Let] 1C − 300N be equated to 432;

or even let

300N − 1C be equated to 432.

There are two right triangles, of which the common hypotenuse is 10 [radius B] … Moreover double the base of the first is \(\frac{432}{100}\) [chord D] and … the base becomes 2 \(\frac{16}{100}\) … when it is said that 300N − 1C is equated to 432, 9 + R57 [9 + \(\sqrt {57}\)] or 9 − R57 [9 − \(\sqrt {57}\)] will be made to be 1 N [chord A]. (adapted from Nickalls 2006, pp. 204–205)
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Jankvist, U.T., Clark, K.M. & Mosvold, R. Developing mathematical knowledge for teaching teachers: potentials of history of mathematics in teacher educator training. J Math Teacher Educ 23, 311–332 (2020). https://doi.org/10.1007/s1085701809424x
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Keywords
 Mathematical knowledge for teaching
 Mathematical knowledge for teaching teachers
 Teacher educator students
 History of mathematics
 Primary historical sources