What is it that they don't get? They don't get the chain of reasoning involved in mathematics. Imagine being in a class in a subject where each new year you are presented with refinements and extensions of last year's reasoning, that you didn't get. Anxious? Yeah, you are going to be more than a little anxious. And this goes on for years.
The instant that the mathematics has exceeded their ability to follow this chain of reasoning is their departure point. Unfortunately, that instant becomes apparent when they start algebra however by simple reasoning it must have occurred much earlier. I haven't looked enough at pre-algebra and arithmetic but whatever you see on day one in algebra was already fated in those previous 8 years. Algebra classes generally start off with a lot of new notation and language and that is fine IF the pre-algebra course brought the students into the kiddy pool of algebraic reasoning worthy of all these new refinements to mathematical reasoning. If however, these kids are showing up with nothing more than a class schedule then you are obviously going to have a problem. How can you expect to refine reasoning that doesn't even exist yet?
Filling them with motivation by telling them about all the jobs they can do if they learn algebra is hardly addressing the problem. All you are doing is telling them all the jobs they cannot do. Engaging them in activities that have math in them may sound nice but still does not directly address the problem of not getting it. And if you are not careful with the activities and design them poorly, all you will do is make "not getting it" less stressful. These activities then become nothing more than palliative care for a dying patient. I have done enough tutoring to look at these activities and tell when one author is actually still trying to win the game and another has given up and chosen the palliative route.
And Jonathan will make this exact same choice many times over, every semester, with the students in his class, probably more so than others because of the nature of his students. Blaming all the teachers before him that were faced with the very same situation as he is being faced with now is a foul in my book. He should think deeply after he has tried to reach a student and failed about what actually failed. And he will find that no matter how much he talked about algebra, talked about great mathematicians or did fun activities, at the end of the day when he presents an actual algebra problem requiring nothing but algebraic reasoning, the student still does not get it.
I understand the reasons for these motivational and engagement methods but let's be real, now that you got their attention, you need to get them reasoning. And in all of my studies of this problem, they seem to be showing up for algebra class, but they are showing up entirely in the wrong frame of mind without prerequisite reasoning experience. And reasoning is so personal that it is very difficult to keep up with each and every student to see if they are using their head enough. As a father I can monitor my son much more closely and in a more personal manner than any teacher but even I have to keep on my toes. Since I am very mathematical it's easy for me to spot reasoning flaws.
Unless you only teach off of AMC exams (which is impossible in the beginning), no matter how you teach math, there will always be a rote path of some sort available to the student. You have to make sure that the student doesn't rely on this rote-ness and instead connects to the mathematical reasoning. Why are there rote paths? As Jonathan stated, there are techniques and algebraic procedures and you cannot escape that. But those techniques and procedures are supposed to be the culmination of chains of reasoning, not the steps to bake a cake. The learning cycle is supposed to bring the student to the edge of solving a particular type of problem and the "technique" is actually someone else's solution and the student is supposed to go "So that's how they did it!" They are not supposed to memorize the technique they are supposed to get the reasoning behind the technique and own it for themselves. Lou asked me to critique traditionally paced curriculums and I said "They expect the student to run a perfect race." As they move through the material they expect the student to keep up with the reasoning and by doing this you are not teaching technique, you are developing their reasoning skills. Once you lose that connection then the student has no choice but to fall back to rote learning and while baking a cake may only require one to memorize a sequence of steps, problem solving requires reasoning strategy and you will fail without it, unless of course the test is "fixed" to pass them anyways.
So the number one reason why kids fail math is very simple and any "mathematical" teacher knows this. The kid fails get the chain of reasoning. And all these new "proxy" methods of teaching fail to address this problem or they avoid it entirely with intentional deliberateness because they don't want to face reality.
Math is about mathematical reasoning. Everything else that is discussed here are refinements to that reasoning. Word walls and journals are so secondary to the problem at hand that I will not even comment on these things any more. Teaching kids algebra when they are 6 years old is retarded but typical of many approaches to teaching math that appear to be very non-mathematical in their reasoning. In fact, it is very evident that the people coming up with these "theories" are not very mathematical at all. Kids do not fail algebra when they are 14 because they didn't learn algebra when they were 6. They fail algebra when they are 14 because they didn't learn arithmetic when they were 6. They fail because their ability to reason mathematically isn't even in the ballpark when they step into the class.
I do not preach one curriculum for all students. I would love that the world be so nice. But the fact is that many students do follow the traditional pace and their reasoning keeps up with the math. And it is a fact that many do not. This trend seems to appear very early but I do not put any weight into any trend before third grade because everyone develops on a different timeline. After third grade though, if you want to lick this thing, then you need to focus on reasoning ability and not be shy about it. The test can be done in person and be free response in format and it should consist entirely of problems with a range of reasoning difficulty appropriate for that age. No preconceptions of how reasoning works, just problems that require reasoning. And once you get the results, take it from there. If you don't do that then you are not being sincere about this at all. I would never treat a cancer patient with a drug that I thought might work, and walk away feeling good as if I did something, without ever knowing if it did work.
Every teacher with any mathematical ability and reasonable experience knows the problem and we most certainly have the ability to create tests of mathematical reasoning ability (I am not talking MC tests). We are just not in the right mindset to accept the results and deal with them as best we can. Instead we come up with an endless series of theories with some slight obfuscated measure of success in some slight obfuscated and ridiculous measure of math and announce to the world that we have licked it. It would be comical if it wasn't connected to such a serious subject.