Why would replacing “standard high school” algebra, with “considerably more abstract topics”, enable dyscalculia students “to see things faster”?

The best person to answer your question is the author of the quote, of course, but I think I can give it a shot too.

The short answer is as usual: any deviation from the so called "standard" brain functioning short of complete shutdown creates all of the three: disadvantages, advantages, and quirks. Which ones will you expect with discalculia? The official symptom description reads (a few sources combined and the meanings distilled):

People who have dyscalculia struggle with numbers and math because their brains don’t process math-related concepts like the brains of people without this disorder. It involves core weaknesses in number sense, memory for math facts, and calculation, often leading to challenges with time, money, and quantity estimation. However, their struggles don’t mean they’re less intelligent or less capable than people who don’t have dyscalculia.

If this is correct (and I hope the Cleveland Clinic webpage and similar sources are reasonably reliable), I can see the point immediately: the traditional curriculum involves many "drill to kill" and "let's remember this for now without full explanation" techniques and just as many routine computations. So, if your ability to keep numbers in your head and manipulate them speedily is limited, you'll struggle a lot. One of the middle school kids I helped with math could follow the full explanation of why the total number of different ways to get dressed for rollerskating (the conditions are that you have 3 operations: putting the shoe on, tying the laces, and buckling the shoe latch. The last two can be done in any order, but only on the shoe that is on the foot already; the left and the right foot are distinguishable and you can switch between them as many times as you want in the process; try to figure the answer out in your head before reading further to evaluate the level of conceptual difficulty here) is what it is, but was getting C's or even F's on the multiplication tests because she was still in the middle of the exercise sheet when the whole class finished and because she occasionally had trouble remembering if five times three is twenty five or tventy three and had to resort to adding 5+5+5 to get the correct 15.

Removing all that entirely or almost entirely and just asking if a big number times a big number is big or small, if the sum of a few positive numbers can be negative, if the number of leaves in a healthy deciduous forest in the summertime is bigger or smaller than the number of tree trunks, how to organize the river crossing for a wolf, a goat, and a cabbage head, etc. (the questions many "normal" students have severe trouble with because no memorized algorithm helps much with them, forget about graphing calculators) in addition to clearly defining all terms and operating at the level requiring nothing beyond common sense (but, at least occasionally, a fairly refined and sophisticated common sense) can make these students feel like the robot character in Azimov's "Stranger in Paradise" when he finally gets deployed to Mercury after being trained and tuned up in the Earth conditions (reread it if you forgot the plot).

Note that I do not say that we should avoid the drill to kill approach entirely. Nothing of the kind. Keep it and insist on memorizing the multiplication tables, trig functions of 30, 45, 60, 90 degree angles, the factorization of $a^3-b^3$, etc., but give these kids triple amount of time for multiplying 123 and 456, quadruple amount of time for memorizing which one is the median, which one is the bisector, and which one is the altitude in a triangle, and ten times more time for remembering which theorem is called Pythagorean, which one is called Leibnitz, and which one is called Stokes. Like myself, they do not rely on their memory when doing math at all, only for me it was a conscious choice rather than a medical necessity. All that I or they are capable of memorizing is one key idea, one main twist, one counterintuitive step in the proof. The rest has to be figured out anew every time. Abstraction for them is not a difficult artificial mental process, but a natural way to organize the information.

When a normal person processes the sentence

"John Lennart Smale, a Texas farmer in the 9th generation, whose grand-grand-grandparents escaped Ireland after the Portadown massacre and found a new home in the faraway continent of North America, was sitting in the town saloon drinking rum and talking to the sheriff about the strange disappearance of several cows from his herd a couple of weeks ago, the cows that have never been found despite John's whole family was searching for them day and night."

(that's pretty much how

"For every positive epsilon and every point $x$, there exists a positive delta such that for every $y$ whose distance to $x$ is less than delta, the deviation of the value of the function at $y$ from that at $x$ is less than epsilon."

sounds to our students) and tries to reproduce it, he goes word by word from the beginning to the end and usually manages fairly well. The natural for me way to reproduce this very sentence is

Cows: disappeared. Belonged to John. Search useless, Sheriff getting ivolved
John: farmer, 9th generation, Irish ancestors. Ancestors: immigrants, escaping religious persecution. Location: saloon; drink: rum State: Texas. Persecution: Portadown massacre. Johns family: searching for cows. Search duration: 2 weeks. John's full name: John Lennart Smale.

That is all I could recover after 5 minutes without looking at the sentence, and I placed it exactly in the order it arose in my head when trying. Now try to run this experiment on yourself and see how you think.

The point is that abstracting and establishing logical cross-references seems to work for people with discalcula as a primary compensation mechanism for the normal linear memory handicap when processing any information in their life from the early age, and they get as good at it as some blind people with their senses of hearing and touch, which, in the absence of the normal sight become the primary navigation means. So when they deal with highly abstract math, they just find themselves in the situation when their natural way of thinking is exactly what is needed for the maximal efficiency (in both processing and communicating) after many years of suffering in the environment that forces them to convert

:information source: personal experience :degree of certainty: high :competence: questionable

into

"I'm not an expert, of course, but after 25+ years of teaching of students of all levels of brilliance and observing their struggles and successes, I am pretty sure that what I'm saying now may have some bearing on what's really going on here"

every time they need to communicate their thoughts and feelings to others.

Just my two cents as always :-)

I'm going to propose two plausible explanations. But first some basis.

Calculation and Abstract Mathematics are not the same kind of mathematics.

Calculation is the memorization and application of algorithms on numbers. It is about speed, accuracy, and is best done by getting your conscious mind out of the way.

Abstract Mathmatics is understanding logical relationships between descriptions of ways to think about patterns and the implications. You can get to the point where your conscious mind is not in the way here, but that is a long way down the road.

Dyscalculia is a learning disorder with being able to learn how to calculate, at least as taught in school. Schools for the most part teach calculation via memorization, cargo cult tricks, and practice. You memorize addition and times tables.

Students are traditionally tested on your speed and accuracy of adding numbers together. How well they can produce the numerical result, and then problems that stack multiple such calculations on top of each other. How this relates back to a formal or abstract definition of addition is unimportant to the testing criteria.

Only rarely are logical relationships tested, like "if you add up positive numbers, can the result be negative?", or "you have a large number of negative numbers you are multiplying together. Is there an easy way to tell if the result will be negative or positive?"

The even more abstract topics -- set theory, groups, rings, fields -- also aren't mainly about calculation. There are very few algorithms you have to memorize and practice and get perfect in order to reason about these topics.

And while you can memorize facts and arguments (or proofs), abstract mathmatics evaluation isn't a series of "produce a proof for this fact" repeated dozens or hundreds of times, where each proof is nearly identical and can be mechanically produced from the fact. The strategies used in calculation aren't very effective.

Instead, what you get in abstract mathematics are patterns of arguments, logical connections. Each abstraction is supported from both above and below - from above, what the abstraction connects to more concretely, and from below by the arguments (proofs) and definitions about how it works

Things you know from above (the concrete uses) and below (the axioms and rules) are related logically and can be used to fill in gaps in your knowledge (error correction). Arguments and proofs are related to this, in that producing an argument or proof is like filling in the holes in a story.

So abstract mathematics is full of reasoning; but none of it is "how fast can you learn an algorithm and reliably execute it". The relation to dyscalculia (and calculation) is quite remote.

Based on this, we shouldn't expect one's skills at Calculation and Abstract Mathematics to be super tied to each other. There might be a common skill that boosts both, or a general intelligence factor.

Now, the students are in the same level of mathematics education. This means neither has been thrown out of class and placed in remedial education, and possibly students that fell behind where given extra attention to catch up.

This acts as a filter on the sample set of people; the kids have a "minimal level of math skill" to be in a class.

Berkson's paradox means that if there are two ways to qualify, they will anti-correlate within such a population. Being skilled at running algorithms and having the ability to handle math manually are two ways to solve math problems; so in a math classroom with any filtering at all, they'll be anti-correlated if they are uncorrelated without that filter.

The second possibility is an actual advantage for the dyscalculic. Students who are good at Calculation are going to learn how to train their own brain to shut off the conscious mind when doing these tasks. The initial "drill" period is the only part where they have to consciously work at the task, and these initial "drill" periods are designed to reduce cognitive load for the learning student. Then the next level of calculation is added, which assumes the previous level has been in-brain automated.

Students who have difficulty automating the calculation tasks are instead forced to do them manually in their conscious mind. This is going to be a lot more work in practice. But the practice should give them a larger working memory and teach them how to handle lots of facts and information at once. The error rate is still going to be much higher.

Faced with abstract mathematics, the calculation skilled students will discover that their ability to learn algorithms fails them. Their approach won't work. They are forced to keep using their conscious minds, something they are not used to having to do in their math education, and on problems that aren't spoon-fed for ideal practice.

The students not skilled at calculation are also forced to use their conscious minds, but they have been doing so for all of their math work already. They are used to having to think instead of shutting their brain off and letting the numbers fall out. Their working memory may be better, their coping strategies work better.

Before, their error rate and slow speed when doing 7234*23+234 meant that they evaluated poorly. But in this new context, making a small logical error in a 9 input factor logical argument is evaluated as a good job. And if their error correction strategy was to look around and see if it makes sense, they can find that error and remove it.

The calculation skilled student is used to 7234*23+234 type problems, and solves them by following the rigid set of rules for solving them, and does so fast and reliably. Their conscious mind is only observing the steps. When faced with a 9 input factor logical problem there is no such rigid set of rules to follow to solve it, and they haven't practiced the same argument 100s of times before so they can't just grind it out.

So it is plausible that they could have an advantage, given the different way to approach problems.