- Multiplication tables
- Algorithms for addition, multiplication, long division, etc.
It's particularly aggravating that several of the issues cut to the very heart of why our writing systems in numbers and words are originally designed as they were: to support simple adding, multiplying, and division procedures by hand; to easily convert spoken sounds to writing by way of phonics. Lose sight of that, and you lose the very essence of those systems of writing.
So doesn't this explain why remedial algebra is the single-most devastating course in the university curriculum, preventing about half of all community college students from ever graduating? After all, it's now the first time in their academic career that students are finally forced, inescapably, it attend to the detail and structure of things. And they've been set up for failure; when our students cannot actually parse the structure of a sentence (don't know what a verb is, what clause connects to what, how to diagram a sentence, etc.), then it's impossible for them to translate word/application problems to math (to say nothing of actually solving them). And it's very sad and heart-wrenching to watch.
What is particularly galling about all this is that this has occurred precisely at the same time as the world around has become more driven by machines, computers, technology, and an increasingly technocratic government structure. All of our young people carry computing devices at all times as a very intimate part of their lifestyle, but their understanding is at about the same level as a cargo-cult. This is why "STEM" academic careers are held out as some kind of bizarre alien life-form that the normals cannot hope or imagine crossing into. To the extent that we have removed the capacity to understand structure from our students, we are making them unavoidably victims of the highly technocratic society that controls their lives, without any hope of understanding it.
Obviously this painful disconnect between prior classes and the introductory algebra course cannot last -- and all signs are that, long term, the algebra class will likely be removed as a requirement even for a college degree (my prediction). And thus a mile-high iron wall will be put in place between the unlearned masses and the elite who are educated in the "real deal" of structure, mathematics, language, and computer skills.