In their paper titled *Forms of Knowledge of Advanced Mathematics for Teaching*, Stockton and Wasserman (2017) identified five forms of knowledge of advanced mathematics for teaching. In their study, they attempted to explain and argue why teachers training to be teachers need to know more than what they would need to teach the K-12 content. By “knowledge of advanced mathematics for teaching”, Stockton and Wasserman refer to a teacher’s knowledge and ways of thinking, which can then be translated into appropriate teaching actions for conveying the related K-12 mathematics content to students. They are not suggesting that the teacher should be introducing his/her K-12 students to the advanced mathematical material directly, but rather that such knowledge might inform their practices for teaching school mathematics. **Five Forms of Advanced Mathematical Knowledge for Teaching** The five forms of knowledge of advanced mathematics for teaching are:

*peripheral knowledge*,

*evolutionary knowledge*,

*axiomatic knowledge*,

*logical knowledge*, and

*inferential knowledge.*The author described these knowledge in terms of forms of knowing rather than what mathematical content it includes.

- Peripheral knowledge:
*How simple things become complicated later on*

2. Evolutionary knowledge: *How mathematical ideas evolve(d)*

This category highlights mathematics as an ongoing process, driven by the desire to answer open questions and resolve unsettled issues. When teachers understand the process by which a particular mathematical idea developed over time, or how an elementary idea is drawn to completion at advanced levels of mathematics, they are better prepared to set students along a fruitful mathematical path that paves the way for both the development of mathematical questions of their own and the resolution of those questions later on.

3. Axiomatic knowledge: *How mathematical systems are rooted in specific axiomatic foundations*

A necessary precursor to teachers’ development of students’ understanding of mathematical ideas is the ability to see the development of an idea from its base principles (axioms). This process occurs quite visibly in the study of geometric systems, for example, when the ideas of perpendicular or parallel lines are developed from the building block notions of point, line, plane, distance, etc. Since teachers are required to guide students in the development of these richer mathematical ideas, they should therefore have the knowledge themselves of the process via which more complex definitions and concepts are constructed from those foundational building blocks.

4. Logical knowledge: *How mathematical reasoning employs logical structures and valid rules of inference*

Given how regularly students are asked to explain why an algorithm works or how they solved a problem, it is clear that teachers need to have a deep understanding of different processes for mathematical proof and the ability both to generate their own logically- sound explanations and to interpret and respond to arguments provided by students. All of these tasks are supported by a teacher’s knowledge of valid logical rules of deductive inference and ability to apply logical structures to mathematical scenarios (such as use of precise mathematical language and definitions, for example).

5. Inferential knowledge: *How statistical inference differs from other forms of mathematics reasoning*

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