One pause I had was when the comparison between conservative and progressive approaches to math education. I have never thought about how teaching methods reflect different beliefs on what math is? The conservative approach focuses on more fluency, facts, procedures, and applying algorithms correctly (I think this is how I was mostly taught?) On the other hand, the progressive approach focuses more on understanding, inquiry, and problem solving. I think both approaches have their places in math education, and good teaching is not necessarily choosing one approach only.
Another pause was during the discussion of math anxiety and how negative experiences with math can actually keep going for generations. Students who learn math mainly through conservative approaches, and struggles, may grow up believing that they are simply bad at math, and these negative attitudes can then be passed down into later generations. This reminds me: being a math teacher, explaining how math works is essential, not just something to add on to a lesson.
Lastly, I was very surprised about how politics and national events play their roles in math education. After the Soviets launched Sputnik 1, concerns about the Soviets led to the New Math movement, which introduced more advanced topics like Calculus to high school classrooms, hoping to help future scientists. Later debates about this "Math Wars" also sparked wider debates about school standards, teaching methods, and education as a whole. I had always though math was a politically neutral subject (since when 1+1 involved politics), but turns out I was wrong!
(edit) To add on to last paragraph, I think China had something similar back in the 1960s-1970s as well. During the cultural revolution, normal schooling was heavily disrupted and curriculum was replaced in part by political study and manual work.
research: https://www.sciencedirect.com/science/article/abs/pii/S030438780700003X
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