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Natural Numbers
Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020
Nita H. Shah, Vishnuprasad D. Thakkar
In normal algebra working, like factorization and equation solving, we use certain properties of operations such as addition and multiplication. For example, in solving x − 7 = 3 we add 7 to both the sides to make it (x − 7) + 7 = 3 + 7, which in turn gets transformed to x + (−7 + 7) = 10, which ultimately gives solution x = 10. Some people transfer −7 from the left side to the right side with a sign change and solve the equation. In this example, we have used the associative property of addition (a + b) + c = a + (b + c). Two more concepts, existence of identity and inverse, are used, which go unnoticed.
Transferring specialized content knowledge to elementary classrooms: preservice teachers’ learning to teach the associative property
Published in International Journal of Mathematical Education in Science and Technology, 2018
SCK is a key component of ‘mathematical knowledge for teaching’ (MKT, [5]). MKT is developed from Shulman's [13] Pedagogical Content Knowledge (PCK) by complementing it with two major components on subject matter knowledge: common content knowledge (CCK) and SCK. Consider, for example, the associative property of multiplication. Knowing that (3 × 2) × 4 = 3 × (2 × 4) is an instance of the associative property of multiplication that can be considered as CCK, which may hold for many educated adults. However, being able to use specific representations to illustrate this property so that elementary students can make sense of it demands SCK. MKT also contains two main categories of PCK, that is, knowledge of content and students (KCS) and knowledge of content and teaching (KCT). Using the example of the associative property of multiplication, knowing students’ common misconceptions of this property is deemed as KCS while having ready strategies to deal with misconceptions belongs to KCT. According to Ball et al. [5], among these four components, SCK is a unique type of content knowledge specifically needed for teaching (as opposed to CCK) and it does not demand knowledge of students and of teaching context (as opposed to KCS and KCT). Due to these merits, Morris et al. [2] argued that SCK is a viable candidate for teacher education and thus should draw increasing attention of teacher education. In fact, Leavy [14] found that obtaining knowledge in one of the four major subcomponents of MKT can motivationally impact learning in the other subcomponents. Thus, by impacting SCK in this study, it is likely that the other components of knowledge will also be influenced.