This page includes some framing thoughts about algorithms that may not be included in the video lectures:
Everyone's question: "What is the best way to teach multidigit arithmetic?"
This is a hotly debated question in math education.
There are (intelligent, well regarded people) who claim that children should develop their own strategies that make sense to them for working with multidigit numbers. Some of these people go on to claim that you should not teach (and expect all children to learn) the standard algorithms. Their reasoning is that children need learn to rely on their ability to think through and make sense of numbers and mathematics, and that teaching the standard algorithm discounts childrens' own sense making strategies. In order to make good decisions in guiding children to understand multidigit strategies that they and their classmates have invented, it is helpful for teachers to be familiar with some of the common strategies that students develop, and to have some ways to represent student invented strategies on the white board. The assignments about non-standard algorithms are intended for you to think about some of the common student-invented strategies, and practice representing them on paper. I highly reccomend the book "Number Talks" as an addition source of this information)
There are other (intelligent, well regarded people) who claim that children should be explicitly taught the standard algorithm. These people also emphasize that children also need to be taught why the standard algorithm makes sense. Different people have different strategies for helping children make sense of the standard algorithm. I, personally, am heavily influenced by the Montessori process where children learn the standard algorithm with 3 sets of sequentially more abstract manipulatives before learning the standard algorithm with numbers. My observations are that children can be taught to use their understanding of how and why the manipulative algorithm works to understand how and why the algorithm works with numbers. The video explanation assignments are intended for you to practice showing students how to make the connections between a base-ten manipulative algorithm and the standard pencil and paper algorithm.
The Common Core State Standards has language that implies that we should be doing both.
The second grade standards:
CCSS.Math.Content.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
CCSS.Math.Content.2.NBT.B.9 Explain* why addition and subtraction strategies work, using place value and the properties of operations (*Explanations may be supported by drawings or objects.)
implies a method of teaching that emphsizes students making sense of 2-digit addition and subtraction. It leaves open whether that 2-digit work should involve base 10 manipulatives or be primarily mental math work. There are many good examples available of teachers effectively teaching in both ways. It is clear that the emphasis is intended to be on making sense and understanding; it is a reasonable interpretation that the standards expect that at second grade, the emphasis would be on student-invented strategies for addition and subtraction.
The fourth grade standard:
CCSS.Math.Content.4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
states that by fourth grade students are to learn and become fluent with the standard (US standard) algorithms for addition and subtraction. Fluency implies that students practice the standard algorithms to where they become automatic. Since it is the standard algorithm that children are to learn, this standard implies that there be direct instruction by the teacher in this algorithm. Direct instruction (when done well) involves modeling the process, explaining the process (and involving students in explaining the process), and providing guided practice for students as they learn the process.
Because we have both of these teaching approaches implied in the standards at these two different grade levels, we can infer that the writers of the standards did not intend for these to be mututally exclusive, but instead they intended for there to be a larger emphasis on student reasoning at first, and that the teacher build on student developed understanding in explaining and teaching the standard algorithm. (There are also many intelligent, well regarded people who would advocate for this approach).
Language
A lot of people dislike the words "borrow" and "carry" when talking about subtraction and addition. Bassarear in the chapter on subtraction explains that having two different words can mask the understanding that the processes of borrowing and carrying are the same but reversed. I dislike the words "borrow" and "carry" because they are short-hand references to a fairly complicated process. When adding or subtracting with regrouping, the carrying or borrowing step is generally the most problematic--it is the place where students are most likely to make a mistake. When you are recording yourself explaining the standard algorithm, I want you to substitute for the words borrow or carry a longer, more descriptive sentence or phrase. I want you to slow down and explain what's really going on. "Carrying a 1" is something you do without thinking about the process, in contrast "grouping 10 ones to make a ten and writing 1 in the tens column to show that ten" or even "trading 10 ones for a ten" tells more about what's going on with the numbers and the amounts. As a teacher, you need to be able to unpack processes that are very fast and efficient for you, and explain them in small, detailed steps to the learners. So don't use borrow and carry. Make a longer, more detailed explanation.