Five-Year-Olds Can Learn Calculus (2014) • Hacker News Board

Re: Five-Year-Olds Can Learn Calculus (2014)

#81

Post by sp332 » Tue, Sep 05, 2017, 1:31 PM UTC

roel_v wrote: post #71

sp332 wrote: post #64

Earlier quoted context omitted.

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.

Re: Five-Year-Olds Can Learn Calculus (2014)

#82

Post by roceasta » Tue, Sep 05, 2017, 1:35 PM UTC

iovrthoughtthis wrote: post #60

roceasta wrote: post #21

Earlier quoted context omitted.

The fact that school math is thought to be 'scary' and 'hard' is not the issue. Plenty of video games are known to be scary and hard. For instance, they contain monsters and are difficult to complete. They even have objective grades (scores). What makes the difference is that video games aren't compulsory. So they represent true play . Which is what makes them educational, unlike a school curriculum, however much it…

Hard and scary carry differently meanings in those contexts though, surely? Video games are safe e.g you can always try again, you can do incrementally well. Video games are scary in an exhilarating way, you blood will pump and your pupils will dilate but at the end you can put it down, turn the lights on and watch funny cat videos to recover. Math (at school) is hard because you get one shot. Finite lives. After eno…

Nicely illustrated. As you imply, the scariness is a property of schooling rather than of mathematics itself. And where game play is scary, it's at a level chosen by and controlled by the player, for optimal learning I suspect.

Re: Five-Year-Olds Can Learn Calculus (2014)

#83

Post by jasongrout » Tue, Sep 05, 2017, 1:35 PM UTC

One of the best curriculums I've seen for children (grades 2-5) learning math is the Beast Academy series: https://beastacademy.com/

They have challenging problems that emphasize basic and advanced concepts in a variety of ways, often posed as games or puzzles.

Re: Five-Year-Olds Can Learn Calculus (2014)

#84

Post by rockostrich » Tue, Sep 05, 2017, 1:38 PM UTC

jamesrcole wrote: post #59

jonnybgood wrote: post #2

I have no doubt they can. There's so much negative cultural conditioning towards mathematics by eliciting negative emotions that it's ridiculous (i.e. math is scary, math is hard, math is for smart people, etc.). I believe kids who excel at math is not due to them being smarter but due to the bypassing of the cultural conditioning. The bypassing can be due to the parents, teachers, some mentor, or their particular fa…

> cultural conditioning I suspect that mathematics being poorly taught is also a major contributor.

This is the biggest contributor. Teachers focus on making sure they follow precise steps whenever trying to work through a problem. While that's helpful for certain things, if the student can explain why they are doing what they are doing then they won't be able to apply what they've learned to solving new problems.

It's definitely not an easy problem to solve and I certainly don't envy elementary/middle school teachers.

Re: Five-Year-Olds Can Learn Calculus (2014)

#85

Post by mannykannot » Tue, Sep 05, 2017, 1:48 PM UTC

eecc wrote: post #65

Ntrails wrote: post #58

Earlier quoted context omitted.

Of course understanding fourier transforms isn't reliant on my ability to do 6x7 - but arithmetic as a lifeskill is actually valuable. It's like saying that grammar and spelling aren't important because spellcheck will sort it out. Splitting out arithmetic from "Maths" doesn't seem terribly sensible to me. Times tables aren't really problematic, and they do provide opportunities to teach valuable concepts (5x6 = 6x5)…

No, you're straying away from the original point which was: there's no need to make rote memorization of arithmetic tricks and multiplication tables a hard prerequisite to everything else about math. Nobody claimed that arithmetic is absolutely useless

I think it is undeniable that both a certain level of working memory and of basic algebra is necessary to understand mathematics at the next level. Someone who could actually understand an "entire college-level course in linear algebra or calculus or graph theory or even combinatorics perfectly well", yet be unable to remember a basic multiplication table, or figure out how to calculate 11*15, would have a very interesting pathology.

Re: Five-Year-Olds Can Learn Calculus (2014)

#86

Post by mannykannot » Tue, Sep 05, 2017, 1:54 PM UTC

throwawayjava wrote: post #51

roel_v wrote: post #31

Anyone commenting on this (whether saying 'yes' or 'no'), please first read the article. What it considers 'calculus', none of us here would consider calculus. Because very few 5 year olds are capable of understanding even something as basic as a line plot (or '2d graph' or however you want to call it). They simple do not have the neural matter to comprehend fairly advanced (from an evolutionary point of view) abstra…

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?

Re: Five-Year-Olds Can Learn Calculus (2014)

#87

Post by Quarrelsome » Tue, Sep 05, 2017, 1:58 PM UTC

roel_v wrote: post #75

Quarrelsome wrote: post #72

Earlier quoted context omitted.

> 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…

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 g…

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.

Re: Five-Year-Olds Can Learn Calculus (2014)

#88

Post by aethertap » Tue, Sep 05, 2017, 1:59 PM UTC

jasongrout wrote: post #83

One of the best curriculums I've seen for children (grades 2-5) learning math is the Beast Academy series: https://beastacademy.com/ They have challenging problems that emphasize basic and advanced concepts in a variety of ways, often posed as games or puzzles.

I can second this, with a minor caveat. My daughter (third grade) loves these books, and we get a lot of good problem-solving into the curriculum as a result. The caveat, in my opinion, is that they don't provide enough cyclic review in their default configuration.

Each chapter includes 80-100 problems, divided in to usually between 4-8 sections. The problems are great, but once a section is complete they're weak on later refreshes. I've been working around this by doing even-numbered problems the first time through a section, then half of the odds a few days later when we're a couple of sections downstream, then selecting randomly from all of the unfinished problems in the entire curriculum for just a couple of extra "old stuff" problems each day throughout the year. We also supplement with a number of other great resources, if you're looking to implement a more problem- and exploration-oriented math curriculum:

1. Kitchen Table Math is great for selecting concepts to lead number talks with (for building number sense - this is the first part of our day)

2. Saxon has excellent spaced-repetition exercises for shoring up the calculation side of things, and giving the student some easy wins for confidence building (we typically use Saxon's material as a warmup before Beast Academy)

3. Thinking Mathematically (the one by J. Mason and L. Burton) has a unique and useful mental process for attacking hard problems when you're not handed a nice formula to plug things into. Once a week, we work through a hard problem using the method in this book.

4. I haven't worked it in yet, but Arthur Benjamin's "Secrets of Mental Math" has a lot of stuff in it that will solidly connect arithmetic and algebraic thinking later.

Re: Five-Year-Olds Can Learn Calculus (2014)

#89

Post by nogbit » Tue, Sep 05, 2017, 2:04 PM UTC

hyperpallium wrote: post #35

jonnybgood wrote: post #2

I have no doubt they can. There's so much negative cultural conditioning towards mathematics by eliciting negative emotions that it's ridiculous (i.e. math is scary, math is hard, math is for smart people, etc.). I believe kids who excel at math is not due to them being smarter but due to the bypassing of the cultural conditioning. The bypassing can be due to the parents, teachers, some mentor, or their particular fa…

One thing that's difficult about math is all the "mathematical maturity " required, much of which is notation and... abuse of notation. Math seems like an Arts subject, where you must learn all the cultural background, as in English Literature. I'm not sure it's necessary, given it's not logical nor self-contained, but is arbitary and historical. Then again, I'm not absolutely convinced it's not necessary...

It is necessary but could be improved. If all we had was roman numerals we would still be in the dark ages.

Re: Five-Year-Olds Can Learn Calculus (2014)

#90

Post by 5040 » Tue, Sep 05, 2017, 2:11 PM UTC

jonnybgood wrote: post #2

I have no doubt they can. There's so much negative cultural conditioning towards mathematics by eliciting negative emotions that it's ridiculous (i.e. math is scary, math is hard, math is for smart people, etc.). I believe kids who excel at math is not due to them being smarter but due to the bypassing of the cultural conditioning. The bypassing can be due to the parents, teachers, some mentor, or their particular fa…

Imagine repeatedly telling those who truly struggle with mathematics that math is soothing, math is easy, even dumb people should be able to do math, etc.