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TBF "what calculus is" -- i.e., the level of rigor at which topics are taught -- varies a lot even among college courses called "calculus"...

In particular, IMO the distance between AP calculus and honors calculus at an ivy is larger than the difference between this material and AP calculus.

Up to a point, but does it not all grow out of the concept of limits?

I would say that the fundamental idea to understand is the concept of changing ratios.

Limits seems to me to be more a tool in order to calculate the ratios than a fundamental concept.

I agree with that. You can hand limits as a tool to someone already comfortable with the concepts and present them as a way to quantify their intuitive understanding.

But without limits, you cannot actually do much with the concepts. Whether you call limits a concept or a tool, they are essential to the practice of calculus, and, historically, to its emergence (and not trivially so, either.)

Yeah but you can do well on the AP calculus exam without understanding limits, if you can calculate derivatives and solve some word problems related to derivatives.

Hence, the gap between "5 year old calculus" and "high school calculus" is smaller than the gap between "high school calculus" and "actual calculus"

> They simple do not have the neural matter to comprehend fairly advanced ... abstractions like the relationship between a value on an x and a y axis.

I don't think that's true. Its wholly dependent on what they've been focusing on thus far. To use the dismissive "they simply" is a misrepresentation of the plethora of candidates. Did you mean "most"? Also its not a question of brain matter but the current development of that matter towards these sorts of ideas.

Its also a completely different subject to emotion control which you brought up somewhere else. This is about understanding, not control.

In order to bring our arguments to a common frame of reference, I think that I can reword my point as follows: to do the sort of abstract reasoning that is required for what is commonly (so not like the OP, who redefines calculus as 'abstract(ish) concepts that will be useful in real math later on') considered 'calculus', one needs a part of the brain that is not just 'grown' by training, but that actually needs to grow to a certain baseline before it can function at all - after which it can be further developed by training/practice.

So in other words, some brain functions just do not physically 'exist' (the matter needed for those functions) at very early ages. You can't take any 2 year old, or 3 or 5 year old, and practice until they get it. You have to wait until that part grows naturally. And growing (or maybe just 'activating') that part earlier doesn't say anything about overall intelligence, either.

And yes, it's different from emotion control, but not dissimilar - the mechanisms is the same.

All of this, of course, for the general population. I can't deny there are child prodigies who understand these things years before their peers; they exist, it's a simple observation. My point is: some things you cannot train in children. You have to wait for the brain to get there; and children not being there yet is not (necessarily) because of lack of training.

Hopefully this clears up my position, and contrasts it with yours. If it does in the way I think it does, it becomes a matter of the state of the art in neurology and neural development. My information comes from having read some popular literature (books, not the 'parenting' section of Marie Claire, but still...) on child development; not a very authoritative position, I'll freely admit. I never got the impression anywhere that this position is controversial or even just not universally accepted, but if it is, then our (apparent) disagreement can be reframed in these terms. (to preempt - at that point I don't have anything to add except the tried and proven 'trawl google scholar until I find something that I can bend into supporting my position, to make it appear that I have Science(TM) on my side although I don't really know anything about it'; but maybe I'm getting ahead of myself here :) )

oh that's fine. :) I'll disagree on outlook though. I believe that at least neural-net-wise a brain develops according to what one practices the most. If that's reasoning then its reasoning.

I think its a bit frivolous to just assume that abstract concepts are beyond children of that age, I think if anything its more a failing of our means of explaining them correctly. Of course its ultimately possible when the three year old I chat to claims to understand that they are lying to me or are just seeing their response as a hoop to jump through, however I would posit that the vast majority of the required components are formed by this point. The remainder just being a process of dispelling the various foggy bits through practice and exploration. TL;DR; I fancy the brain to be more a thing that just improves over time. Even if incorrect I certainly doubt that "reasoning" or "abstraction" are just modules that are either IN or OUT of the developing brain. I feel like these sorts of distinctions are unproductive and too coarse to make.

Like I get that there are bits that are somewhat nonfunctional. She struggles to remember but she can if she really tries or events had impact, she can process-of-elimination to find the location of one of our tribe, she can understand that a word has double or triple meaning and double checks with us when we give her an old label with new meaning. I feel like there is plenty of stuff going on there that could be fashioned into "reasoning" and "abstraction", hence my issue with your assertion.

The map is not the territory - charts are just one way to represent an independent and a dependent variable.

True; but do you have anything in mind that is less abstract than a graph and still expresses the continuous (or not, depending on what you're looking at...) nature of a relationship between two variables? Because I don't know of any way that is easier and more intuitive to understand than that. But maybe I'm just projecting based on my own preferences.

I think you have to understand how two things can change "with" each other before you can understand how to read a chart. Personally I think seeing something vary with time or distance is more immediately understandable than reading a chart.

Reading a chart yes, but that's because (IMO) reading is harder than making one. For example, no matter how much someone would explain, I could never understand the relationship between an angle and a sine wave, until I had a teacher made me draw a circle and an x/y axis alongside each other and project the line in the circle onto that axis. I still don't think one can really explain this without resorting to describing what happens visually. But again, maybe that's just me.

"Children understand that things change over time and can interpolate to some degree" would be a non-clickbaitey, but also very boring title.

> abstractions like the relationship between a value on an x and a y axis.

That's entirely the point. You can understand calculus without understanding graphs.

It may sound crazy, but it's true. You can understand a subject without all of the dependencies academia tends to categorize Maths into.

> They simple do not have the neural matter to comprehend fairly advanced (from an evolutionary point of view)

Do you have some links to research on this? Sounds interesting.

Most books on child development has (a) chapter(s) on neuroplasticity and neural development, and how that relates to actual skills. I've only read one and skimmed a few, the one I read was a Dutch language one called 'Oei ik groei' so that's probably not much use to you. If you start reading any book on this I'm sure you'll find plenty on it, and they'll probably point you in the direction of dozens if not hundreds of others.

But if you're referring specifically to the 'evolutionary point of view' part in my sentence, I don't even remember what I meant by that - it doesn't make much sense, re-reading my sentence. I guess I just meant that brains develop in a way that makes them, early on, incapable of doing some things that adults can. Similar to how until the frontal lobe is fully grown (18/20 years old), children/youth can't control their impulses the same way someone with a grown, healthy frontal lobe can. So it's useless punishing children for some things because they literally can't change some behaviors; it's like beating a quadriplegic until he walks because 'he should just try harder'. (This is obviously a very simplistic and probably in many ways wrong description; I'm just using this as an analogy for what I was getting at).