Math = Numeracy, logic, geometry. Those are the three reasons I learned math. All three benefit from memorization. All three are practical. All three speed up future thinking. All three are scalable. All three are intuitive. All three are thinks you can't live without.
The reason why math is 'hard' is the same reason 'exercise' is hard. It takes a combination of skill and practice (a/k/a discipline) to make it 'easy'. And in a similar way to the endless "how to lose weight" fads, we see "how to do make math easy," when <easy> is not 'in their nature'.
The main problem with math education is that it is falsifiable. Math teachers can be measured, because math performance can be measured. This is why it makes educators nervous. And, I suspect, why they mostly suck at teaching it. They are the ultimate risk-averse sub-population. Subconsciously, 'educators' <hate> the idea of accountability, because it is predicated on the notion of potential failure. But the 'educational' establishement is <built> on the foundation of unquestionable <authority>. so math is for these guys an existential threat.
TLDR: To be good at teaching math, you have to be open to measuring your failure, which makes people feel bad. Including the teachers.
No. No no no. Yes, arithmetic math does benefit from repetition and memorization, is useful, and your comments on why getting good at arithmetic is hard are spot on.
But math is so much more than arithmetic. Math is about learning to apply a variety of available solutions and processes to solve a problem. Math is about understanding relationships. It's why I get better at math in a combination of my physics and programming courses than I ever did in algebra or calculus.
In fact, the biggest problem with mathematics education is that it is not falsifiable. Measuring reasoning, logic, critical thinking, and problem solving skills is almost as hard as teaching them. Writing a generally applicable test that measures whether a teacher taught you to actually use and solve problems with math is nigh-impossible, made worse that students (and most of all, parents) don't want to be held accountable for problem solving, because it's too fuzzy, or because you just "have to be smart". The fact that real math skills are tough to teach and tough to measure is the problem, not the reverse.
Math is much more than arithmetic as it is taught. Understanding arithmetic through the lens of relationships (as you stated) is no different than algebra, geometry, trigonometry, calculus, differential equations...
Arithmetic is important, especially in applications. But I wouldn't say it forms the backbone of all mathematics. You can do symbolic calculations just fine without any actual numbers. And a number of fields don't even involve any numbers at all, not even in applications.
Another problem with math education is that many of the people in leadership positions in public school systems are, when it come to math and related skills, functionally illiterate. Those people see themselves as successful and are respected as leaders but have little use for math themselves. It's hard, then, for them to believe that an understanding of math is genuinely essential and worth fighting for.
Your exercise analogy is perfect. I think your concept of Math is in direct opposition to the author's (and mine).
The author is arguing that math is about logically solving problems based on a small set of simple rules. That students memorizing a formula are learning a formula but not learning Math. Like you said, Math is like exercise, where you challenge yourself with harder problems and over time you become a better problem solver. The shortcoming with American public schools, or whatever we're talking about here, is that the teacher's incentive is to get their student to achieve a competency in repeating memorized instructions, something that can and is tested. This leads to numerous students who don't enjoy or understand Math for what it really is, and fail to develop the part of their brain that solves problems logically. Instead, they learn a bunch of formulas that are totally pointless unless they happen to need that exact formula in the future. So, evaluating teachers based on the performance of their students at applying different formulas is part of the problem, not the solution.
I'm not sure we disagree too much. My view is that you need to teach strong fundamentals so you don't need to think (ie, worry) about them later. This reduces degrees of freedom, and frees up your mind. The reason you do this is precisely that it lets you focus on higher level abstractions. It might just be me (probably is) but I found that math got more intuitive this way. Also, if your aim is to make the basic building blocks both reliable and intuitive, it will get used more in daily life. If you need to reference a text, it will ~never happen. [1,2]
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[1] What are <legos>, but simple geometric objects?
[2] Life is an open book test, there is just not time or the bandwidth. And I'm not talking about memorizing a calculus text, etc just for fun. But more basic - the bulding blocks. But if you want to play for high stakes later on this is the Ante, IMHO. That probaly wasn't clear in the initial comment I made (as several people have pointed out logic~abstraction, etc). So I'm not saying this should be the limit or ceiling of an education. Its just the opposite. But its shocking the kinds of things that happen otherwise. And its not fair to the kids who never get a shot at doing serious work, because their fundamentals are unsound. "Subtle bigotry of low expectations" and all that.
It's so unmathematical to use the equal sign there... and of course there are plenty of people who live their entire lives without any of those three things.
Math is also about finding structure and patterns that don't arise in any of those three contexts. It's about learning how to generalize and specify. It can be about isolating the relevant parts of a real-world problem so you can solve it in a way that matters.
None of those skills involve or benefit from memorization, but they're damn well more important than memorizing a multiplication table or knowing facts about convex polygons.
There is a lot of value in this comment. For example, the notion of abstraction as seperate from logic. Isolation and prioritization. etc. The importance of basic structures.
But: I'd be carful with language.
Example (1)
None of those skills involve or benefit from memorization
But consider <Memory> management is important from the perspective of <cognitive science>. Memorization helps you process and prioritize. Consider the notion of abstracting structure from data, using pattern matching. How does one get better at this? (1) you have empirical data; or (2) you process it efficiently. Memorization facilitates data processing. It also provides for important empirical data-sets and a means of prioritization both of which are crucial for heuristic reasoning. Not to mention formal modes of analysis.
Example (2)
Math is also about finding structure and patterns that don't arise in any of those three contexts.
Where do you think the "pattern" in pattern matching comes from (if not Geometry)? It's the basic things that are most often overlooked. Consider the most elementary use of applied statistics: Linear regression. The abstraction Y=mX+B? That has nothing to do with geometry, rigt? The point is that these most basic things are <critical> to more advanced concepts. Things you seem to take for granted, but are things of great value to the novice and advanced practitioner alike. And if they are not learned (so they can be "forgotten" in memory) they are <impediments> to progress for many students.
If you are innumerrate, you're going to have trouble with your bills and interest rates and such; if you can't apply logic then you'll have trouble holding a job where you have to make a lot of decisions; but geometry? How many people are concerned with geometry beyond the most rudimentary concepts like interpreting a street map. I just think statements like 'you can't live without geometry' further the 'disconnect' (as they say in America) between the people who 'get' maths and the people who don't. It's almost like a religious person telling you that 'you need Jesus in your life' or something - it doesn't actually persuade in any way, just signals that this person maintains some sort of alien, abstract belief system which they cannot relate to the lives of ordinary people.
Also, Physics: The arch. A triangular truss. A lever.
Also, Architecture: Lines, planes, volumes.
These are some very basic (perhaps 'unremarkable') everyday uses of geometry. I don't think at all the fact that they are rudimentary is a problem. Because they are so pervasive, it is all the more important to have mastery of the subject. IMHO.
With that example we should be teaching our high school students graph theory. And I'm completely in support of that. Nobody uses geometry to read a map, because geometry is about shapes and angles, not routes.
The reason why math is 'hard' is the same reason 'exercise' is hard. It takes a combination of skill and practice (a/k/a discipline) to make it 'easy'. And in a similar way to the endless "how to lose weight" fads, we see "how to do make math easy," when <easy> is not 'in their nature'.
The main problem with math education is that it is falsifiable. Math teachers can be measured, because math performance can be measured. This is why it makes educators nervous. And, I suspect, why they mostly suck at teaching it. They are the ultimate risk-averse sub-population. Subconsciously, 'educators' <hate> the idea of accountability, because it is predicated on the notion of potential failure. But the 'educational' establishement is <built> on the foundation of unquestionable <authority>. so math is for these guys an existential threat.
TLDR: To be good at teaching math, you have to be open to measuring your failure, which makes people feel bad. Including the teachers.