What are some of the theoretical frameworks and methodologies that inform mathematics education research?

This is spot on. In my teacher training, we were exposed to the power of math talks. In a nutshell, students were given a math problem that could be correctly solved using a variety of approaches. Students were given time to solve the problem as far as they could independently, and then were asked to come together in groups to discuss their answers and approaches. This provided rich conversation and opportunities for students to use vocabulary to communicate and explain their answers. In this way, students became immersed in each other's processes and evaluated the quality and respective levels of fluency.

One theory of cognition in mathematics education is APOS Theory: Action, Process, Object, Schema. These are cognitive stages students go through when learning a new topic. For example, with functions, students often begin with the action of evaluating functions, as formulas or tables. With practice, this becomes a process: They can imagine the result of evaluating a function at many values, and represent this as a graph, or work backwards to figure out possible inputs from outputs. The transition to "object" involves transforming and inverting functions. Continuing to revisit and incorporate new perspectives forms a "schema." Determining cognitive stages for a specific concept is called finding a "genetic decomposition" of that concept.

I love the aspect of problem solving where it invloves those tasks that require students to draw on their abstract thinking, logical reasoning, intuition, and creativity. Educators can enhance students' problem-solving skills by providing authentic and contextually relevant problems, fostering a problem-solving mindset.

It has been shared with me, and I do believe that there is generational trauma around math in school. Unknowingly, families can hand down deficit and fixed mindsets to their children around learning math through sayings like "I was never good at math." "Math has always been hard for us." There is a lot of healing that has to be done and restructuring our language around the process of learning math if for no other reason than the life skills math has the potential to reinforce in our students. Learning math teaches discipline, meticulousness, and precision. While these skills are taught in other subjects, they lend themselves well to be emphasized in a math classroom.

This makes me think of stereotype threat and studies that have shown that seemingly impartial language and word choice in physics classrooms can have tractable impacts on student performance based on gender. Learning does not occur in a vacuum and our societal context still applies in cases of who we as educators believe can and cannot do math.

In discoursing about mathematics; it should be made clearer the difference between the principles of mathematics and the use and interpretation of mathematics. Mathematics is the only discipline, beside Christianity, that require the use and interpretation be based on first principles. Foundational principles and properties do not change. The uses might! Any person should be able to go back to the chain of interpretation to discover the first foundational bedrock principles. If they are not there, it is a false interpretation and a false use.

Although we say that learning and teaching are two facets of mathematics education, instructional aspect of mathematics education is often overlooked. All the perspectives mentioned are very challenging to control and can be improved in the long term since they are macro factors. However, instructional aspect, which is a micro factor, is way more controllable and improvable in the short term. Also, research has indicated that a significant portion of issues in mathematics education is related to instructional approaches, methods, and materials. Therefore, instructional aspect of mathematics education deserves, at least, as much attention as other perspectives.