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"Students can learn through activities focused upon big ideas and when they encounter a
need for a new method they learn it within the activity. When students do this, their brains are
primed to learn the new method as they are curious and they need the method."

Creating the need for the math prevents the "when are we ever going to learn this" conundrum. Teachers will need a lot more support to develop and implement activities where the need is created for the accompanying math.

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"This teaching of methods to students when they encounter a need for them has been shown in
research to be the most effective method for teaching mathematics (Schwartz & Bransford,
1998)."
When students see a need or have a purpose for learning something, the interest and desire to do so is self-generated. No one is telling them they have to learn it just because. Rather, the students want to learn the mathematics so that they may continue on their path of discovery. Furthermore, as teachers, we are providing the just-in-time supports rather that teaching content that students already know or can figure out on their own.

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"When teaching
through the big ideas in mathematics most of the smaller ideas and methods naturally arise and
students can learn them in a meaningful purposeful way. The ideas that never come up are
probably not very important to learn!"

This is scary and a little freeing at the same time. After decades of being told explicitly what is "important," it feels a bit risky to leave some of these topics To chance. However, teaching this way could also be a discovery process of what is important and essential for learning.

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"Yet when we ask students what they think math is, most will say that it is a
lot of different rules and methods...One of the reasons is
that teachers are given sets of standards to teach and no matter how good the standard writers
are, they all cut mathematics up into small pieces...This
often leads teachers to skim through content quickly...The connections between
ideas are invisible and students do not develop one of the most important insights into
mathematics that they can develop – that mathematics is a set of connected, big ideas."

These lines are from the first paragraph. I've known some teachers to teach math really well, but they taught algorithms and rules and methods in an effort to boost their test scores to look good. It wasn't for the sake a student seeing connections in math, or growing students' understanding of the big ideas. When I had opportunities to work with some of these students, they were stuck on finding the correct answer, and they were unwilling to deviate from the given method that taught to them by their teacher. On paper it appeared these students had an understanding of math, but not necessarily on a deeper level.

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"When teachers work on identifying and discussing big ideas, they become attuned to the mathematics that is most important and that they may see in tasks, they also develop a greater appreciation of the connections that run between tasks and ideas."

I had a few lines that I loved (Well I loved this whole article!) but I wanted to include this one since I did not see it listed yet.

It is so important, that time is taken to talk through, collaborate on, and digest these big ideas as a team of teachers.  Many of us learned math initially as "a set of methods to be remembered" (pg. 1) until that one (or, more than one) special teacher inspired us to make the connections.  Like the article said, depending on the concept, or material, many of us need to experience and discover the connections ourselves before we can dream to support our students in their journey of discovery.

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"This approach of teaching to big ideas and teaching smaller ideas when they arise, has the advantage of students always wanting to learn the smaller methods as they have a need for them to help solve problems."

This line stood out to me the most because I teach math 3 and a course a 4th year math course. My students often do not understand why they need to learn the concepts that we cover.  Having my students see the need for a skill and want to learn it would be amazing.

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"Many teachers
were themselves taught mathematics as a list of content, missing understanding of the big
ideas that make mathematics a cohesive whole."

This resonated with me because it receives such little attention but plays a critical role in how teachers themselves experience math and how they, in turn, view math instruction for their students.

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"This teaching of methods to students when they encounter a need for them has been shown in
research to be the most effective method for teaching mathematics (Schwartz & Bransford,
1998)."
This is essentially creating the need/desire to know more. Creating purpose for what students are to learn.

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Another way to pare down the standards...

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Another way to pare down the standards...

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"When teaching
through the big ideas in mathematics most of the smaller ideas and methods naturally arise and students can learn them in a meaningful purposeful way. The ideas that never come up are
probably not very important to learn!"

I am back to the fallacy that I have to know the procedure in order to do the math.  Granted, students might get stuck in places not knowing the math so starting with the big ideas they now have a need to know.  Often though, you can connect/build new learning on what they do know, which is the true definition of scaffolding.

Related to this is the "deconstructing of standards" I have no problem with understanding a standard, but the standard needs to be RE-CONSTRUCTED and combined with other standards to be effective.

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"This approach of teaching to big ideas and teaching smaller ideas when they arise, has the
advantage of students always wanting to learn the smaller methods as they have a need for
them to help solve problems."
This line stands out to me because it helps students make the connections to what they are learning and makes students want to learn in order to solve a problem.