Homework Slip: Arbitrary and Necessary

    Hewitt distinguishes between two categories of mathematical knowledge: necessary and arbitrary. Arbitrary knowledge involves information, such as terminology, or symbols, that students must accept as granted since they cannot logically justify them. On the other hand, the necessary knowledge consists of ideas and concepts that students may freely research and understand. Hewitt highlights that students gain through actively using reasoning to get an understanding when it comes to necessary knowledge. Instead of focusing on retention, this method helps students understand and retain topics more deeply by allowing them to feel the natural logic of mathematics.

    Hewitt claims that arbitrary knowledge can be taught directly since it incorporates established norms. Here, the emphasis is on communicating effectively to ensure that students grasp established these norms (for example, that "5" denotes a certain amount) without having to come to an understanding, freeing up time to thoroughly examine the relevant information. Here, the emphasis is on communicating effectively to ensure that students grasp these concepts (for example, that "5" denotes a certain amount) without having to deduce them, freeing up time to thoroughly examine the relevant information.

    Hewitt's terms on necessary and arbitrary knowledge provides a framework that might significantly influence my approach to lesson and unit preparation as a future math instructor. This viewpoint motivates me to consider the information I want pupils to absorb and the most effective ways to support this. Practically speaking, I may create courses that allow students to absorb random aspects without spending too much time on them by presenting common themes in a straightforward and succinct manner. This might include brief mini-lessons or the use of visual aids to explain vocabulary and symbols at the start of a course.

**EDIT**

    When it comes to essential information, the emphasis switches to giving students the chance to use reasoning skills and discover mathematical facts on their own. For instance, while teaching the idea of quadratic equations in a high school classroom, I can lead students in exploring the process of completing the square instead of teaching them the quadratic formula. I chose this example because this was my topic for my Micro Presentation. Students can understand the basic reasoning and necessity that leads to the formula by being guided through exercises that need them to reorder and change quadratic equations step-by-step. A spirit of discovery is fostered by having students work in groups to find patterns and try various approaches. As a future teacher, I would also have students observe how these changes occur in real time by using visuals and dynamic.