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*

Teachers’ in-depth understanding of how future developments “complicate the picture” can support a more-nuanced treatment of the what they teach. In practice, teachers regularly abridge content in their explanations to help crystalize the essence of an idea. This practice can sometimes lead to false oversimplifications and misstatements. For example, common misstatements from teachers (or students) may include “multiplication makes things larger”, “you can’t subtract a larger number from a smaller number”, “anything to the zero power is one” or “the even numbers make up half of the whole numbers so there are half as many of them.” Thus, it would be beneficial for teachers to have some awareness of the advanced versions of the topic and how that eventual complexity is related to the simplified cases under consideration.

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 TME, vol. 14, nos1,2&.3, p. 593

language and definitions, for example).

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

Although mathematics teachers are frequently charged with teaching probability and statistics, statistics educators argue that the two fields are fundamentally different. As a field of study, mathematics has historically inquired into and made conclusions about certainties. In fact, rigorous deductive proof provides absolute certainty about these conclusions. Statistics however has to do with the non-deterministic nature of conclusions drawn from data, due to the omnipresence of variability.

The inherent intermingling between both statistical and mathematical reasoning requires teachers to be particularly attentive to how statistical ideas are presented and treated. Statistics are largely about understanding and interpreting events of chance, whereas mathematical reasoning aims to conclude about events of certainty. The duality present with regard to interpreting probabilistic and statistical ideas and inferences represent an important form of knowledge of advanced mathematics for teaching.

Reference:

Stockton, Julianna Connelly and Wasserman, Nicholas H. (2017) “Forms of Knowledge of Advanced Mathematics for Teaching,” The Mathematics Enthusiast: Vol. 14 : No. 1 , Article 30.