That sounds like what we are doing. We have a native speaker nearby who provides tutoring, and Mittelpunkt is immersion-style. The Goethe Institut requires it, and it seems to be one of the few which move into C1. We just thought it would be nice to add some additional content. We'll probably continue to use Deutsche Welle online for that (it's free, and has a good bit of variety).

Guessing not. Shame - no advanced German learners here?

Nor at all beneficial - universities don't need any additional paperwork than they already have, and a 1-off certification doesn't help them evaluate the student. The checklist is still nice, but really, you just need to do the basic program, and you're fine.

Bueller?

Very little review - it's mostly additional depth. The topics are similar, but the content is different.

We pull problems from different textbooks. The only issue we've had is that the problems tend to be too easy, so we often have to "go up" - eg, for Intermediate Algebra, we pull "challenge" problems from a regular precalculus text. They're frequently still too easy.

Bueller?

Is anyone aware of online or distance course options which use the Mittelpunkt texts (specifically B1+ through C1 levels)? We're looking for content beyond Deutsch Aktuell or Komm Mit, which we have already used, and roughly only reach B1 level (third year HS) competency. Thanks!

Right there with you. I believe breadth adds perspective in math, and can convert a math-tolerant individual into a math lover. "Gifts" may not appear for a while, yet. Personally, I wouldn't artificially slow progress. What you are describing - providing space between learning cycles leading to better comprehension - is a well-known and real phenomenon. However, that doesn't improve by extending the space between cycles beyond a certain point. What we have chosen to do (and it works!) is add additional math threads to our curriculum. At first, it slows things down a little, but within a few months, you start to see acceleration, as concepts begin to tie into one another. So, we haven't done the other books as the only math content. We've used multiple books at all times. It hasn't slowed anything. Younger DS is on his second full year of algebra and third year of geometry, and is quite advanced on both fronts. Older DS did 3 precalculus texts and 2 calculus texts, and is finishing AP calculus as a freshman. Some of that may be talent, but some of it is simply method. We're absolutely certain of that.

Geometry and Intermediate Algebra are clearly full-year courses, but the others can all be done in a half-year, or even in a summer.

Maybe there was more to the story? DYS has been very good for us and quite a few 2E kids... Regardless, for the OP, no - the portfolio is not needed. It was mainly used for borderline cases. We submitted some samples of work, but that was about it. Doubt it mattered at all, to be honest.

Was that DYS or DA? I don't believe DYS filters based on 2E. We have a number of "quirky" DYS friends...

It doesn't really split out well. For Intermediate Algebra, we pull the majority of test questions from the challenging problems from precalculus texts. It doesn't fully cover precalculus, though - more like College Algebra. The intro book doesn't quite cover a typical algebra 2 course, either. If you were to stick with AOPS, you could reasonably call it algebra 1, 2, and 3, followed by trig and/or precalculus.

As far as I can tell, pitting one against the other serves one and only one purpose: to satisfy the personal needs of the proponent. Those on the "for" side (whichever that is) are already close to 100% set. Therefore, I find such arguments to be almost pointless beyond affirmation (which some people honestly do need). I have never heard a single argument which has influenced my religious beliefs nor my thought processes regarding science. My experiences have been like yours - arguments were cherry-picked, playing to a common logical fallacy. I find it much more fruitful to consider how little we know and understand, and marvel in the wonder and splendor we are given to live in.

Can't disagree with that logic. If it follows the same guidelines and curricular content, then it would make sense in that state. Here, we'd see 0.5 Algebra IA, 0.5 Algebra 1B, 1 Consumer / home economics, 1 math models or algebra 2, and it would be sufficient.

Sorry, but I have to disagree here. Algebra 1 is one credit, no matter how long it takes. However, it doesn't mean doom-and-gloom - she can still complete two more credits (including algebra 2) in time. Around here, it is not unusual to do mathematical modeling, consumer math, or something else along those lines prior to algebra 2.

:) This is what we had to do, as well. We actually had older DS run through three books of precalc, after having already done it in geometry and algebra 2. The third precalc was AoPS, which is just a *tad* bit more advanced, but also much more interesting!

Algebra 2 gives a basic intro only. Though it could conceivably be skipped before precalculus, I would not recommend it. I haven't seen any algebra 2 text (certainly not Forester) adequately cover trig to consider skipping prwcalculus' coverage. A dedicated trig course might be sufficient, though.

Do NOT skimp on trig - particularly if calculus is anywhere in the future! Trigonometric substitutions need to be as comfortable as basic algebra. They will be called upon extensively. I strongly recommend letting any repetition stay in scope. Even three rounds of trig will not hurt.

We've got friends who have had similar experiences - some straight through college. I was that way, and DW was that way. Unfortunately, there's really nothing for it but to hang in.

AoPS has a nice book, or you can find an old "Theory of Arithmetic" textbook. It's sort of "fun with math" but with a serious purpose.

Math facts are overrated. You can have fun by playing multiplication war or having timed "races" through the deck, but honestly, there will be so much repetition in the various levels of math that the facts will eventually come on their own. Number theory is also a good way to learn ways to decompose numbers into easier-to-manipulate parts.

Only public school kids. It's almost trivial to reach that threshold, for what it's worth.

Great to hear! One cautionary note: make sure he is aware that the pace will slow when the challenge increases. The point of BA / AoPS is to push through difficult challenges. By starting at a review level (not a bad thing), he may get a sense of overconfidence, and when he hits "the wall," may not be prepared to push himself as much as he will need to. As he progresses, he will have to stretch himself beyond the current level of effort.

There are not many that are both curricula-worthy and accessible. "Ghost in the Atom" is not bad, but certainly not worthy of a curriculum. Brian Greene and Sean Carroll have some nice pop-science books. We tried looking for something in the vein of "A Very Gentle Introduction to the Mathematics of Relativity," to no avail. That book was very good, in that it described the mathematics bit-by-bit. It allowed the reader to actually understand the theory, even without a real command of the math (you'll know what tensor calculus is, but you may not be able to manipulate tensors yourself). Quantum physics really needs something similar in order to describe the various states and behaviors. Otherwise, you just have supplemental reading. One thing you might try is a good quality chemistry book (e.g., Atkins' Chemical Principles). It will teach just enough to make it worthwhile. MIT-OCW's "Principles of Chemical Science" opens with several weeks of it, and would make a legitimate supplemental course (going through ~lecture 15).