I have enjoyed listening through the beginning half of this course. It seems that ensuring that grammar, logic, and rhetoric are present throughout the whole length of the math curriculum is key. Giving children early experiences with mathematical concepts before they need to work with paper, pencil, and mathematical symbols is a way to draw them into mathematical conversations and thinking. Andrew gave great examples to show how to a teacher can help a child to grow in their understanding of "making" and "breaking" groups of numbers to help a child develop an tangible understanding of what place value is. My daughter (1st grade) was able to grasp multi-digit addition and subtraction more deeply after a few experiences of playing with groups in the way that Andrew demonstrated. In the classical tradition, students should sometimes move from theory to practice, but should also be challenged to move from practice (problem) to theory (or proof). Learning about what problems gave rise to these theorems and proofs provides a historical connection within the curriculum. The lecture about three proofs demonstrated an important point that students should be shown how mathematical proofs are beautiful, they were actually so clear and enjoyable that my daughters joined me in watching a few. While students at all stages may not be able to derive these proofs, they can follow that they are made logically, given that they have sufficient background in the subject.