[HN Gopher] Learning algebra in my 60s
___________________________________________________________________
Learning algebra in my 60s
Author : trocado
Score : 157 points
Date : 2022-08-07 12:28 UTC (1 days ago)
(HTM) web link (www.theguardian.com)
(TXT) w3m dump (www.theguardian.com)
| librarianscott wrote:
| Brain games do little for the older mind. You are much better off
| lowering blood pressure and walking. But, if you insist, read
| [1]. [1]: https://www.nia.nih.gov/health/cognitive-health-and-
| older-ad...
| bamboozled wrote:
| Not sure why but this just seems so useless and defeatist. Why
| is learning new things "brain games"?
| bitshiftfaced wrote:
| That link did mention the "Advanced Cognitive Training for
| Independent and Vital Elderly (ACTIVE)" study. Apparently it
| found that training can affect quality of life and reduced car
| accidents in driving seniors.
| fargle wrote:
| Probably not. Knowing more stuff doesn't make you "smarter". Nor
| would mastering kinds of math that many people find challenging
| prove you are more intelligent. Clearly many intelligent people
| are not great at math or don't know algebra.
|
| But exercising your curiosity and practicing the art of learning
| will always increase your mental fitness and the ability to
| _apply_ your intelligence better. It 's always good to keep in
| shape - mentally too.
|
| I don't think it matters much what the subject is. It doesn't
| have to be algebra, as long as it captures your interest. Algebra
| is very challenging because it requires thinking in abstract
| concepts, so it's a great subject. It will indirectly lead to
| deeper understanding in many other fields too, including computer
| science.
|
| Intelligent people are often curious and desire to learn new
| things, so No I don't think learning algebra makes you smarter -
| but I think the desire to is a possible symptom of this kind of
| curiousity.
| rongenre wrote:
| One of my retirement goals is to redo my math degree - turns out
| it's really easy to get hooked up with online tutors, and I
| figure it's totally worth paying $20-$40 an hour to work through
| a textbook.
| wrycoder wrote:
| If you have a math degree, you already know how to learn math.
| That hasn't changed. You're not going to try passing tests, so
| you don't really need a tutor. What you might want is a
| counselor to guide your "reading", as is done in places like
| Oxford. That can be worth the $40.
|
| All you need is updated texts, a plan of action, and maybe
| Wikipedia for an alternate take on sticking points. (Just using
| Wikipedia doesn't work well.)
|
| You could also look at the MIT or Harvard online
| curricula/syllabuses to get a feel for what the current
| subjects and texts are. Maybe take a couple of the free
| courses.
|
| Also, scan Kahn Academy first - that site is good for
| identifying holes in your background.
| waynesonfire wrote:
| interseting.. what are these counselors? what do they do?
| squeaky-clean wrote:
| Where do you find an online tutor? My first guess would be
| fiver but there's probably some sites with a more academic
| focus?
| miga wrote:
| Probably yes, since past research indicates that learning delays
| cognitive decay.
|
| Even more, there is research pointing that continuous learning
| will prolong your lifespan. Best indication is a study on how
| university professors tend to live longer.
| trocado wrote:
| Despite the title, I don't think the article is about algebra
| "making you smarter", but more whether you can learn it
| successfully and if by pursuing such an endeavor you can hamper
| age-induced cognitive decline.
| BirAdam wrote:
| I personally found that learning math was far easier as an adult.
| Like the author, I sucked in grade school. I learned multiple
| foreign languages. I did well in the sciences. History and
| geography were simple. Math? It was the one thing that escaped
| me. As an adult, after various misadventures, entering into
| college everything was different. My mind was more disciplined,
| my teacher was far better, and I was more humble. It was easy
| then.
| k_sze wrote:
| The title sounds like a non-question.
|
| I posit that learning anything (even fengshui or astrology), at
| any age, will make you smarter, as long as you bring a pair of
| critical eyes with you. Besides learning about the actual "thing"
| (which you can still fail at the end of the day), you can always
| introspect to figure out why/how you failed and what your limits
| are, and then you'll maybe find closure in saying "I've tried. It
| didn't work out. There was no 'wasted potential' - the potential
| was never there to begin with. And I'm a wiser person now."
| dangus wrote:
| The title is there to get you to click, there's not much need
| to analyze it. The article has little to do with it.
|
| This reminds me of On Cinema At the Cinema when Gregg
| Turkington, as a parody of the film critic persona, often bases
| his critical opinion of the films based on what he thinks of
| the title and runtime.
| photochemsyn wrote:
| Scientists can't agree to what extent human intelligence is
| inherited vs. acquired, and so asking if doing some activity x
| will result in an increase in personal smartness y will result in
| a wide variety of answers. (sort of a differential equation isn't
| it? Change in intelligence over time as a function of doing
| algebraic manipulations every day).
|
| There are some plausible benefits - it's likely to help with
| internal mental organization, for example. However, some thought
| should be given to how to set to work, in particular avoiding bad
| habits that eventually thwart future progress (this can happen in
| say, learning to play a musical instrument).
|
| Learning algebra (or any other field) conceptually is like
| constructing a connected map of various islands, although from
| reading this article it isn't clear whether that was
| accomplished. There's no mention of concepts like associativity,
| distribution, commutation, how this applies to the order of
| operations when you start mixing up + * - /, and the rather
| strange but also fundamental notion of identity. For example:
|
| https://byjus.com/maths/commutative-property/
|
| This is a bit sad because understanding these concepts opens the
| doors to fun higher math. Matrix multiplication is generally non-
| commutative, and this property makes it useful for quantum
| mechanics calculations. Symmetry operations - rotations and
| reflections and so on - are not generally commutative, but are
| associative. This is all very important in things like protein
| crystallography. A solid grasp of these ideas also allows for the
| introduction of the concept of groups as a way of looking at
| algebra. Here's a great series on that whole business:
|
| https://www.socratica.com/lesson/groups-motivation-for-defin...
|
| A good rule is to spend at least as much time understanding
| fundamental concepts and abstractions as on working out the
| results of specific explicit examples.
| wawjgreen wrote:
| My litmus test of someone being intelligent or not--among other
| such tests--is what they say about Shakespeare. This might anger
| the Brits, but Shakespeare is garbage--and the guy says: "I read
| .... most of Shakespeare."
|
| One famous guy lambasting Shakespeare (Tolstoy) was only
| marginally smarter, because according to his "Confessions", he
| read it many times over (incl. in English) and STILL could not
| find any artistic value in it. I could not have put up with that
| --for me, it was the first few pages that I knew this was a waste
| of time. You don't need to drink the whole festering junk of a
| decayed meal to know it is not edible.
|
| He also claims he wants to test the limits of his intelligence by
| tackling Mathematics once again--implying wrongly, as it were,
| that math education is fine and it is our brains that are
| incapable of learning. Not so fast: how am I less intelligent if
| the bozo teaching me does not know how to teach maths?
|
| Citing Carl Jung does not help either, though he namesdrops to
| mitigate the shame of not being able to do maths...Jung was not
| as bright as is commonly believed as well. He once claimed that
| UFO's were all imaginary-- a sweeping generalization by someone
| claiming to be scientist-- whereas I for sure know that UFO's are
| real for I have seen one and to prove that I wasn't seeing
| things, I had a camcorder ready which recorded the space-ship
| going vertically up very slowly. I have lost the video though but
| it certainly happened 20 years ago and I still remember it.
|
| EDIT: I see I have touched quite a few nerves.
| trebbble wrote:
| > EDIT: I see I have touched quite a few nerves.
|
| It reads like a post you made for yourself, not for the benefit
| of any reader.
| ziddoap wrote:
| > _Jung was not as bright as is commonly believed as well. He
| once claimed that UFO 's were all imaginary-- a sweeping
| generalization by someone claiming to be scientist-- whereas I
| for sure know that UFO's are real for I have seen one and to
| prove that I wasn't seeing things, I had a camcorder ready
| which recorded the space-ship going vertically up very slowly.
| I have lost the video though but it certainly happened 20 years
| ago and I still remember it._
|
| Your evidence that Jung was not as smart as believed is that
| you recorded a UFO 20 years ago (but can't prove it) and he
| said UFOs don't exist? Forgive me, but that's the opposite of
| convincing.
| JKCalhoun wrote:
| "Sometimes I dreamed that there were numbers falling from the sky
| into chasms I couldn't see the bottom of."
|
| Reminds me of the night terrors I had when I was younger. I
| remember often having dreams that involve the unknowable,
| unseeable or in some other way were impossible for my finite mind
| to comprehend. In the worst of those nightmares I was left shaken
| with fear -- feeling as though I had peered from just over the
| precipice of sanity itself.
| SketchySeaBeast wrote:
| As a child I experienced sleep paralysis where I would
| hallucinate shapes of impossible dimensions, both
| simultaneously incredibly large and incredibly small, hanging
| over my bed and leaving me to question the nature of
| everything. Looking back on it I can't even begin to understand
| why I dreamed those dreams.
| Wistar wrote:
| Despite that -- or, maybe, because -- my father was a math
| academic, I did poorly in school with math beyond basic
| arithmetic. When I was about 25 years old, a friend gave me three
| algebra textbooks written by an ex-military pilot, John Saxon. In
| a marathon session lasting about two weeks, I went through all
| three books from end-to-end and really learned algebra. For
| whatever reasons, the Saxon books worked really well for me --
| better than any other learning I have ever gotten from a
| textbook. Yes, I was really motivated but I attribute a lot of my
| success learning algebra to those books. Learning that opened a
| lot of doors since then, mostly because my confidence in taking
| on matters technical was so much improved.
| racl101 wrote:
| My grandpa used to enjoy doing Euclid's Elements compass and
| straightedge constructions and understanding proofs.
|
| He also enjoyed reproducing geometric proofs of certain
| equations.
|
| What's funny is he hated math as a student but for some reason or
| other got turned onto Euclid in old age.
|
| He then tried to get my dad to read Euclid but that didn't take.
|
| Then I came across the book as teen and took an interest so
| Grandpa and I bonded over that.
| 363849473754 wrote:
| The author also wrote this excellent article:
|
| https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
| mattwilsonn888 wrote:
| It's a shame he went to all the trouble of avoiding conventional
| education (for the wrong reasons), but then still ended up going
| down an _execution_ rather than _concept_ focused route. As soon
| as you start memorizing operations and treating math more like a
| narrow grind only approached through arbitrary problems solved
| primarily through computation or use of rote application of
| poorly understood technique, you lose the ability to understand
| math as anything other than a human attempting to be a computer.
|
| The author expresses his misintuition of variables - there are
| plenty of simple examples and thought exercises which can inform
| the intuition. Later he expresses getting mixed up dealing with
| fractions - this is a clear demonstration that a fundamental
| understanding of fractions was skipped in order to grind out
| solutions. Modern math education is spectacularly guilty of going
| too far too fast.
|
| Fundamental understanding gives exponential results yet most
| courses try and get the axioms and basics out of the way as fast
| as possible, working through basic proofs or derivations is cute
| sideshow, if anything. If the fundamentals are well understood,
| the applications feel trivial, intuitive, beautiful, clever -
| when the fundamentals are taken for granted, one word describes
| the relationship to the rest of the material: arbitrary.
| 2OEH8eoCRo0 wrote:
| > "Here is some advice," she said firmly. "I get it that you
| try to put things into a framework that you can understand.
| That's fine, but at first, until you become comfortable with
| the formal manipulation, you have to be like a child."
| sfpotter wrote:
| Many other comments here don't seem to have spent much time
| reflecting on why his niece said this.
|
| Speaking as an actual math teacher, this is a very important
| thing for students to try to come to grips with. Memorization
| and "learning without understanding" have a bad rep, but
| memorization is a tremendously valuable skill. Instead of
| thinking of "learning by memorization" as doing a disservice
| to learning math, consider that learning math might increase
| your capacity for memorization as a byproduct. This skill
| carries over to so many others, but I think its importance
| gets glossed over as a result of it being simple to look
| things up using the internet.
|
| Another thing to think about: if you have something memorized
| (e.g., an identity, a theorem, a formula...), then you can
| think about it when you're walking around. When you have many
| of them memorized, you'll be able to think about how they
| relate to each other. The more you have memorized, the fewer
| stumbling blocks there will be to trip over when you're
| mulling over a problem or trying to figure something out.
|
| There's a lot of apocrypha out there about how if you're
| learning math, you _shouldn 't_ try to memorize things. You
| should instead try to "pick it up". This elides two important
| details: 1) there's selection bias at work---many people who
| are extremely good at math have minds like steel traps,
| remembering many things after seeing them only a couple
| times, 2) the expectation is that you spend a significant
| amount of time drilling and working, to give yourself an
| opportunity to pick it up.
|
| Now, why did his niece say this? When you're learning the
| basics like this, becoming "comfortable with formal
| manipulation" is really a matter of running a large number of
| experiments. You try this and that manipulation and get
| comfortable with them over time. You apply a formula in error
| and observe the results. You puzzle over what happened. Over
| time, you discover the logical framework which holds
| everything together.
|
| Asking "why" at this point (when you're learning very basic
| math!) is like putting the cart before the horse. You would
| receive an answer that you would be in no position to
| understand. Hence, "be like a child": play around, make some
| mistakes, try to understand what's going on. As you grow, you
| will become more sophisticated in your approach and better
| able to shift the balance so that you can attempt to
| simultaneously understand more of what you're learning.
| bslorence wrote:
| Maybe "asking 'why'" isn't the right way to express it.
| Perhaps what the OP was really after here was a concrete
| illustration of the formula, rather than a rigorous
| derivation of the formula from first principles.
|
| There are many concrete geometrical illustrations of
| fundamental concepts in algebra, but most people aren't
| even aware that they exist.
|
| The niece's comment, oddly, both points to this and glosses
| over it. You don't teach a child multiplication by _only_
| forcing them to memorize times tables. You also show them
| stacks of coins, etc. But we _do_ teach children algebra by
| only forcing them to memorize symbolic procedures.
| Meanwhile Book 2 of Euclid is free online, or very cheap
| from Dover.
| hgomersall wrote:
| I can distinctly remember in my second or third year of
| engineering undergrad doing some stuff with the Radon
| Transform. At the time I had nothing to map it too. The
| maths to me was completely abstract. I just sat there,
| thought hard, and worked through problem after problem
| until it became intuitive. Now I have difficulty
| understanding what I found hard, because it all fits
| together with loads of other things into a coherent mass of
| concepts.
|
| It was interesting to me because it was one of a few times
| as an adult where I realised the framework was completely
| missing and I would just have to start from scratch, which
| meant lots of rote effort and thinking hard. There was no
| shortcut.
| photochemsyn wrote:
| I think an introductory course would ideally hop back and forth
| between the explicit execution of specific examples and the
| understanding of abstract concepts. As you say the concepts are
| very important - for example, the author never mentioned the
| associative, distributive, or commutative rules which apply to
| all cases of addition and multiplication (but not necessarily
| to subtraction and division). Those are universally applicable,
| and indeed understanding them is important for moving on to
| higher maths (group theory etc.)
|
| However, without doing a fair amount of ditch-digging, abstract
| general theories of ditches might be hard to grasp. For a
| programming example, consider data structures - lists, trees,
| graphs, and so on. The abstract concept is clearly what allows
| larger constructs to be made, i.e. entire programs, but if
| you've never actually implemented a linked list in any
| particular programming language, and try to, the result is
| almost certainly going to be a buggy disaster.
|
| The optimal math education might be one where for every hour
| the instructor spends on the conceptual approach, the student
| then spends about three hours on the execution approach as
| applied to specific examples of those concepts (ideally with a
| teaching assistant around to help if the student gets stuck).
| bowsamic wrote:
| I disagree. My favourite lecturer for general relativity said
| the important thing is fluency, which comes from repetition.
| It's great to know everything, but you should be able to do the
| basics, such as calculus or algebra, very quickly and without
| thinking. You will need that brain power for the new stuff. The
| best way to gain mathematical fluency is by lots of practise.
| Viliam1234 wrote:
| You can understand something first, and practice for the
| speed later. Understanding does not prevent practice.
|
| I know the result of 8x8 immediately, and I don't think I
| would ever forget it. But, hypothetically speaking, if I ever
| made a mistake here, I have the option to slow down and
| verify the result. If I wouldn't have the option, there is a
| risk that I would make a mistake and then keep making the
| same mistake forever, because once the lessons are over,
| there would be nothing to correct me.
| [deleted]
| jjice wrote:
| > ...still ended up going down an execution rather than concept
| focused route.
|
| This is something that made me really disappointed when I
| realized it in school. It was always a step by step process
| taught to me, and I loved math. I took my AP Calculus class in
| my senior year of high school and my teacher (fantastic man)
| showed us how the formula for a derivative is derived. I was
| blown away. We had just learned and it were told to memorize it
| the previous year and I had just accepted it.
|
| After that, math changed for me. Everything had a reason that
| was connected to everything else. If you understood where it
| came from, you could do even more than if you were just given a
| formula to follow. How beautiful.
|
| I realized that I had 5 years (assuming we only cared to do so
| after algebra) of match education that could have been better
| taught if done conceptually. I don't blame teachers. You need
| to get everyone to pass your class for state requirements, so
| you try to streamline it to get this year done, and not prepare
| for the future. Also, most math teachers I had did it as a job,
| not because they loved it, and there's nothing wrong with that.
| When I had that calculus class during my senior year though, I
| realized that that man loved math. Hard but fair. I learned so
| much and that teacher taught me how to develop a work ethic.
|
| I wish meta-learning and a more concept based approach could be
| applied to high school courses (at least looking back). I
| understand why they aren't, but man would I love to see how it
| would play out.
| ravi-delia wrote:
| When I was learning math in public school, my family couldn't
| make heads or tails of my homework. Even in Kindergarden we
| learned that '2 = 1 + 1' was the same as '1 + 1 = 2', and we
| got started with '1 + ? = 3' even in first grade. A focus was
| put on mental math, teaching techniques to break two digit
| numbers up, before we did pencil-and-paper work. Whenever
| possible there would be more than one technique, and the
| class would discuss which one we liked the most. Once we all
| got to high school, our teachers kept remarking on how much
| quicker our year picked up the new subjects than previous
| years (we were one of the first crops of kids taught in this
| way).
|
| My little cousin goes to the same elementary school as I did.
| When I looked over his homework, I saw none of what I was
| taught. A glance at his textbook showed they had switched
| back to the old method of teaching by rote. Apparently school
| board members who had run in part based on their skepticism
| of the new techniques had voted in a change in curriculum.
| One funeral at a time indeed.
| deelowe wrote:
| Common core. It's become a taboo term in certain circles.
| dhosek wrote:
| Arrrgggghhh! I was getting my math teaching credential
| when common core was getting its start and what I want to
| shout from the rooftops is:
|
| COMMON CORE DOES NOT SPECIFY PEDAGOGICAL METHODS!
|
| It is, essentially. a list of topics that should be
| mastered at each grade level.
| jsmith45 wrote:
| Seeing what people complaining about "common core"
| mathematics are complaining about, and it is
| understandable that they are complaining.
|
| At some point the idea came about that for things like
| addition, subtraction or multiplication, we should teach
| multiple different ways of viewing the same concept (like
| addition) in the hopes that even if students don't really
| understand the classic approach one of the other
| approaches makes sense. So perhaps the classic
| explanation of borrowing in subtraction does not make
| sense to some students, but one of a few other equivalent
| ways of handling it does make sense.
|
| But then it got turned into a system where all students
| need to learn all these different different methods, and
| apply them in both homework and on tests. Which totally
| defeats the point. The who idea was that some students
| may find some methods useful and intuitive, and others
| find different methods useful and intuitive, and as long
| as every student finds some method can can work with we
| are better off than only teaching the classic method of
| the concept.
| gpspake wrote:
| I kind of had a borderline religious experience in college
| calculus too. We had a handful of problems back to back: how
| long has a murder victim been dead? How hot does a fast food
| chain have to make their coffee so it burns their customers
| taste buds so they can't taste it? How long does it take the
| moon to orbit the earth? And we solved all these problems the
| same way. There was finally a grand order and purpose to all
| of it that was everywhere all the time.
| jonathanlydall wrote:
| When learning trigonometry in high school (in South Africa,
| '98), we were taught how to use the correct operation for a
| particular situation and then use a calculator to get the
| correct answer.
|
| I asked the teacher where the numbers which were spat out by
| the calculator came from and was only told "in my day we
| didn't have calculators and had to lookup the answers from a
| table on a book!".
|
| Which was such a thoroughly disappointing response and I
| hated trigonometry since it was mostly about memorisation
| rather than applying logic from understanding.
|
| About 10 years later I did a math course through the UK's
| Open University and the text book taught trigonometry through
| explaining the unit circle.
|
| It made me so happy to finally understand and see the actual
| logic as opposed to punching numbers into a calculator, but I
| was also sad and a bit upset the unit circle was not taught
| in high school.
| Kon-Peki wrote:
| > I did a math course through the UK's Open University and
| the text book taught trigonometry through explaining the
| unit circle. It made me so happy to finally understand
|
| It also has a downside... The British Empire depended on
| robust trigonometrical education. In those days, you
| couldn't efficiently navigate the world's oceans without it
| and they needed a steady supply of ship's masters for the
| thousands of ships that made the empire work. Earth is the
| unit circle.
| joshuaissac wrote:
| > It also has a downside
|
| What exactly is the downside and for whom?
|
| > The British Empire depended on robust trigonometrical
| education [...] and they needed a steady supply of ship's
| masters
|
| The British Empire was already collapsing rapidly when
| the Open University was founded in 1969. I doubt that
| ensuring a supply of ship masters was a motivation in
| choosing to include trigonometry in the syllabus.
| Kon-Peki wrote:
| Sure, if you want to believe that Open University
| invented a brand new pedagogy rather than continuing a
| traditional style of teaching trigonometry that had been
| honed over the course of 150 years of naval power
| projection.
| mkl95 wrote:
| > but I was also sad and a bit upset the unit circle was
| not taught in high school
|
| I'm not a mathematician, but that sounds like poor teaching
| to me. Even if it's not part of the programme, learning the
| unit circle takes literally a few minutes and is an
| invaluable tool afterward. I don't know if it's possible to
| develop an intuition for trigonometry without learning the
| unit circle.
| hgomersall wrote:
| The problem is, these tools always feel like they should
| be "the way" it's taught. I feel complex numbers should
| be taught as a special case of geometric algebra, though
| I suspect that would make things much more difficult. My
| understanding is that lots of university level maths _is_
| taught like this: start from the most general case.
| dhosek wrote:
| There's a nice diagram I created an EPS of (which I can't
| find now) that shows the unit circle derivations for
| tan/cot and sec/csc.
|
| Here's how to create this. Draw a unit circle centered on
| the origin. Draw a ray from the origin where th will be the
| angle measured from the positive _x_ axis. You know sin and
| cos already: you draw a perpendicular from the _x_ axis to
| where the ray intersects the circle. sinth is the distance
| from the origin to where your perpendicular hits the _x_
| axis and costh is the distance from the _x_ axis to where
| the perpendicular hits the circle.
|
| Now, draw another line perpendicular from the _x_ axis at
| (1,0) to your ray (tangent to the circle). Let's call the
| point where this line hits your ray _T_. The distance from
| _T_ to (1,0) is tanth and the distance from the origin
| (0,0) to _T_ is secth. (As an added bonus, you can easily
| see that 1+tan2th = sec2th). We can do the same process,
| drawing a tangent line from (0,1) instead of (1,0) to get
| cotth and cscth. You can do some simple math with similar
| triangles to get familiar formulae like tanth=sinth /costh
| etc.
|
| Explaining signs of the functions outside the first
| quadrant is left as an exercise to the reader.
| scythe wrote:
| >I asked the teacher where the numbers which were spat out
| by the calculator came from
|
| This is a bit of simple knowledge which is sadly
| unbeknownst to even most math majors and educators. Often,
| students are taught in calculus that cosines (and hence
| other trig functions) are computed with Taylor series,
| which is not really correct. In fact they use CORDIC, a
| highly optimized algorithm.
|
| But CORDIC is based on repeated use of the sum formula, and
| a simple version of this can be taught without calculus.
| Just notice that:
|
| cos(2x) = 2 cos(x)^2 - 1
|
| Then for small enough x, you have (can be shown by drawing,
| but kind of annoying):
|
| cos(x) [?] 1 - x^2 / 2
|
| Or rigorously: 1 - x^2 < cos(x)^2 < 1 / (x^2 + 1)
|
| So divide x by 2 until you get a small number (x < 0.1 is
| usually good enough), use the quadratic approximation, and
| repeatedly apply the doubling formula. CORDIC uses a
| similar iteration based on the sum formula. There is no
| need to wait for Taylor series, limits, convergence tests,
| etc. You can even bound the error term!
|
| I'm not entirely sure this would work (i.e. improve
| understanding), though -- I have not taught high schoolers
| before. And there is a decent bit of work required.
| jonathanlydall wrote:
| I can't remember the details now, but my recollection was
| that the unit circle enlightened me that they were
| ratios. It was even possible to reason about "round"
| angles (like 90deg and 45deg), in your head without the
| need of a calculator.
|
| I never tried a random angle like 13deg in my head, but I
| figured it was kind of on a scale between the "round"
| numbers which I knew, which would give a good sense of
| the actual number, even if not accurate enough.
|
| The point is, the unit circle explained the figures to me
| instead of them being just some magic numbers.
|
| Likewise, there was an actual explanation of concepts
| like cos and sin in that they are the names for
| relationships between angles/sides (again, I can't
| remember now, but it made sense), rather than them just
| being tools you're told to use when.
| dhosek wrote:
| Wasn't there at least right triangle math for sin =
| adjacent/hypotenuse and cos = opposite/hypotenuse? This
| sounds like serious educational malpractice.1
|
| [?]
|
| 1. Although when I think about educational malpractice, I
| remember a program I was teaching in where the director
| asked me about why the numerator and denominator were
| called what they were and I explained the denominator
| identified what _kind_ of fraction (and made the analogy
| to denominations of currency) and the numerator was how
| many (the "number") of that kind of fraction. She then,
| that same day, gave a talk to the assembled teachers
| saying that she'd asked me and I'd said that's just what
| they were called. I nearly stood up and screamed in
| anger. If I hadn't desperately needed the money, I would
| have quit on the spot.
| shard wrote:
| SOHCAHTOA (https://mathworld.wolfram.com/SOHCAHTOA.html) is
| how I learned about the trig functions back in school. It's
| stuck with me all these years, along with the quadratic
| formula (https://en.wikipedia.org/wiki/Quadratic_formula)
| from algebra class.
| mlyle wrote:
| I think those of us who are good at math greatly
| underestimate the amount of rote needed to reach competence--
| first, how much we actually needed, and second, how much
| _more of it_ most people need.
|
| A good math education in arithmetic, algebra, and calculus is
| a combination of concepts and drill. This was a concept that
| I rejected back in school. Now I'm teaching-- not
| mathematics, but I have tutored and helped students catch up
| who have had problems.
|
| There's a fair number of students struggling in pre-calc who
| have all the concepts just fine, from the bottom to the top.
| But when they're dealing with lots of terms and keeping a
| higher level goal in mind, their performance on a few simpler
| things, like fractions and quotient properties, falls apart.
| Maybe they missed a week or two in 6th grade when this was
| really solidified and practiced.
|
| The _terrible_ thing that tends to happen, once you stumble
| in math: the amount of concept content you have falls. The
| focus moves even more to rote, but focused on the "more
| difficult" stuff--- leaving whatever core deficit there is
| intact. There's solid reasons for this, but the outcomes are
| not great.
|
| I also have a kid who is going into AP Calculus pretty young.
| I'm kinda nervous that he just has not had enough reps of
| practice, even though he scores very high on placement tests.
| ravi-delia wrote:
| I think there are smarter drills to do. In another comment
| I lauded the way I was taught math, and how it built
| intuition, but it certainly had lots of drills too. I think
| what worked was that we were taught more than one way to do
| each thing, usually one more theory-heavy and another more
| technique-heavy, and we had lots of drills focused just on
| the primitive operations. There was lots of focus on mental
| math, which is just super handy for understanding more
| complicated problems since you don't even have to think
| about the simpler steps. While they did go over how
| fractions were just division in disguise (with pictures,
| which I found very helpful), knowing how to find a least
| common denominator is much faster than improvising every
| time, so even if I hated it I'm glad we did lots of
| drilling with those too.
| mlyle wrote:
| A lot of what you describe has become the standard way to
| teach elementary math-- from the "new math" onwards to
| Common Core pushing aspects of looking at problems the
| same way
|
| Our school does the much-lauded Singapore Math in
| elementary, which definitely tries to build intuition and
| looks at many approaches, and supplements with drills.
|
| And I teach a competitive math class which definitely is
| all about finding different ways around problems and
| comparing and contrasting.
| ravi-delia wrote:
| I'm thrilled to hear it's popular! In my hometown it was
| killed by doubters, but perhaps with expanding evidence
| they'll reconsider. I think perhaps it was lumped in with
| disastrous testing efforts, but the math at least was
| pretty great.
| thaumasiotes wrote:
| > I took my AP Calculus class in my senior year of high
| school and my teacher (fantastic man) showed us how the
| formula for a derivative is derived.
|
| This is pretty easy to do for a polynomial term, but much
| harder for x raised to a non-integer power[1], or sin(x).
| There is no one way to "derive the formula for a derivative";
| different functions have to be analyzed differently.
|
| [1] Wikipedia suggests that the simplest way to derive this
| formula is to begin by establishing that exp(x) is its own
| derivative. But that's not an approach I'd be likely to take
| with students new to calculus.
| dkarl wrote:
| > this is a clear demonstration that a fundamental
| understanding of fractions was skipped in order to grind out
| solutions
|
| Leaving aside that you concluded this from a one-sentence story
| about solving a single problem, it's funny that the idea that a
| "clear understanding of fractions" can make the skills easy to
| learn is something that virtually every trained teacher has
| believed for the last (I'm not sure how many) decades and yet
| nobody alive today is reporting that it happened that way for
| them.
|
| It's possible that every future teacher getting certified to
| teach politely pretends to believe these things, but secretly
| wishes to spend their career torturing children, so they
| abandon the teaching of concepts as soon as they step into a
| classroom. Or maybe they go through the motions of teaching
| kids to understand the meaning behind fractions, but they
| intentionally do it wrong and laugh with each other about it in
| the teacher's lounge. That's one explanation.
|
| Another explanation is that this opposition of "understanding"
| to "execution" and "concepts" to "skills" is an incomplete way
| of looking at things that has limited application to how people
| learn but isn't a panacea, and people still struggle with
| skills no matter how much you try to prepare them with
| concepts.
|
| I think it does a vast disservice to anyone when we assume that
| their job performance would be radically improved by the first
| cliche we ever learned about their field.
|
| If the idea is so magical, and we know that it has permeated
| the field of education for generations, where are the
| generations of students who remember math as being "trivial,
| intuitive, beautiful, clever?" Different educational
| philosophies have come and gone, emphasis has shifted, so
| where's the generation of parents saying math was much easier
| for them than for their kids? Where's the PTA meeting where all
| the parents say, "We think there's something wrong with the way
| math is taught here, because we all remember fractions being
| trivial and intuitive, but our kids are finding them hard?"
|
| Maybe teachers aren't such idiots that they consistently
| overlook the value of the most cherished and shopworn truism
| their field has produced in its history?
| thesuitonym wrote:
| I'll start my reply with this: I'm not a teacher, nor am I an
| understander of maths, but whenever I talk to teachers at any
| level, and especially mathematics teachers, they lament that
| they want to teach differently, but due to time constraints,
| or predetermined lesson plans, or any number of reasons, they
| simply can't. They have to teach in a way that gets students
| enough knowledge to pass the test, but only those that seem
| to have a natural aptitude end up really learning anything.
| The rest discard the information as soon as it's no longer
| necessary.
|
| I don't think it's a disservice to teachers to say it's a
| better way of teaching, because they say it themselves. It's
| just a model that only really works one on one. It can't be
| done in a room full of 30 bored children, and especially in a
| room of 400 university sophomores who are ready to get out of
| class to go to a party.
| dkarl wrote:
| You are saying some things that are guaranteed to generate
| skepticism. First, that the great power of this teaching
| method only manifests under circumstances that very few
| people get to experience. Second, that concepts make skills
| "trivial" and "intuitive" but testing skills prevents
| teachers from teaching concepts.
|
| Both of those statements call into question the
| effectiveness of what you're promoting.
| Viliam1234 wrote:
| It is okay to be suspicious when it sounds like someone
| just made this up, but actually in educational research
| it is one of the few well known things.
|
| See Wikipedia:
| https://en.wikipedia.org/wiki/Bloom%27s_2_sigma_problem
|
| Anyone who tried both teaching and tutoring knows that
| the difference is just incredibly large. A part of it is
| that 1:1 you can pay more attention individually; a
| similar argument can be used in favor of smaller classes.
| But the other part is that as a tutor, you are free to
| actually use your best judgment, while in school it is
| more of "yeah, I know that I should do X, but the rules
| say that I have to do Y instead".
|
| The school is just insanely ineffective, for various
| reasons. You need to follow a predetermined schedule,
| whether it makes sense for the given classroom or not.
| Your students are expected to already have some knowledge
| from their previous grade, and if they don't (which
| happens quite often) you don't get any extra time to
| catch up. There are all kinds of disruptions, like
| students who never pay attention and interrupt you and
| their classmates during lessons, but you must proceed at
| the speed that was designed for a hypothetical classroom
| without disruptions. The school inspection randomly
| checks whether you follow the latest fad, usually based
| on some pseudoscience, like whether your lessons are okay
| for both visual and kinesthetic learners, or whether your
| math lessons are sufficiently decolonialized.
|
| So the same teacher who fails to teach her class
| fractions at school, may be a successful tutor during the
| afternoon and explain the fractions properly.
| jrmg wrote:
| This is addressed in the article - just after the '(7/2) / 2 =
| 7' misunderstanding:
|
| _[I called] Deane Yang, my friend who is a mathematics
| professor at NYU._
|
| _"The way you remember procedures is you remember why," he
| said._
|
| _"Because?"_
|
| _"Because people learn math as a collection of procedures," he
| said. "When things get difficult, they're lost, and math
| becomes religion class. The teacher says what's right and
| wrong, and for all you know math came out of the sky, and some
| prophet told you how to do it, and it's just blind belief
| then._
|
| Interesting that he wants to know _why_ knowing why helps too!
|
| I do wonder if this is covered more in-depth in his book.
| shon wrote:
| This is insightful and rings true for me as well. A good blend
| of theory and practical knowledge is important to get the most
| out of any education. Do you have any recommendations for
| learning math this way?
| zozbot234 wrote:
| There's nothing wrong with practicing formal manipulations:
| these are just "the rules of the game" and are 100% rigorous.
| Now, if you want to know "why these rules and not others?",
| they just happen to follow from basic properties like
| associativity, commutativity, distributivity etc. There's not
| much else to it.
|
| The author's "misintuition" of variables is not critically
| important; variables can literally just be arbitrary symbols,
| and substitution understood separately. But the other "rules"
| are far more relevant.
| tankenmate wrote:
| The one thing that gave me deep insights into algebra was
| discovering Peano's axioms.[0] I had been taught the
| associative, transitive, and distributive laws in 6th grade
| at school, but discovering the axioms really opened my eyes
| in high school; especially under what circumstances they
| could and couldn't be applied. The last penny to drop was to
| realise that where you have numbers (e.g. x) you could also
| substitute formulas, provided you adhered to the rules of
| when the axioms could be applied. Obviously for similar
| reasons this also applies to matrices (e.g. when can matrices
| be inverted (additive and multiplicative), and hence why
| matrix addition is commutative, but multiplication is not).
|
| Don't get me started on subtractive anti-commutativity vs
| divisive anti-commutativity.
|
| Another thing that bothers me is when helping the my own kids
| with maths I came to realise that their teachers don't know
| the Peano axioms, so how the heck can my kids even stand a
| chance at properly learning algebra.
|
| [0] https://en.wikipedia.org/wiki/Peano_axioms
| erehweb wrote:
| People understood algebra just fine before Peano came up
| with this axioms.
| tankenmate wrote:
| The point of the Peano axioms is that it codified all the
| basic rules of Algebra in 8 (now 9) rules.
| greymalik wrote:
| What's the alternative? How does a neophyte go about
| cultivating the mindset and understanding you describe?
| rawgabbit wrote:
| Unfortunately, I don't have an easy answer. My son recently
| graduated from my high school and I noticed that while his
| school used Pearson's website to drill him on the mechanics.
| My son struggled with word problems precisely because he did
| not have the fundamental concepts.
|
| For example, I argued that division is more intuitive when we
| multiply using fractions. - Whole numbers have an implicit
| denominator of one. E.g., 7 is equal to 7/1. - It is more
| intuitive to think of division as multiplying by fractions.
| E.g., 7/2 is equal to (7/1) x (1/2). We multiply the
| numerators together. And multiply the denominators together.
| - Where this is helpful is when we deal with complex
| arithmetic like (7/2)/2. We have 7 pizzas. Divide by 2. And
| divide by 2 again. If you break it into fractions, this is
| equal to (7/1) x (1/2) x (1/2). The numerator is 7x1x1. While
| the denominator is 1x2x2. Putting it together, it is (7x1x1)
| / (1x2x2) which is equal to 7/4.
|
| This discussion usually leads to the question of what is
| multiplication (again)? I argued that multiplication can be
| thought of as a shortcut for addition or can be thought of as
| set theory. I have two bags of marbles and each bag contains
| 10 marbles each. How many marbles do I have? You can either
| write 10+10 (multiplication as a shortcut for addition) or
| you can write 2x10 (set theory). Taking it back to division.
| I have two bags of marbles and each bag contains 10 marbles
| and I have to share my marbles with my sibling (divide by
| two), how many marbles do I have?. You can write either
| (10+10)/2 or you can write 2 x 10 x (1/2).
| rawgabbit wrote:
| I agree. He tried to memorize his way through Algebra. It is
| hard to understand his mindset because of the stream of
| consciousness writing. But he appears to be confused with
| definitions versus equality. E.g., By definition (a+b)^2 is
| (a+b)(a+b). It is not just "equal".
| jorvi wrote:
| Yes. The clearest example of this for me was the fact that many
| things I was taught in high school were taught to me as an
| arbitrary concept first, without the accompanying visual
| relation.
|
| Being taught a2 + b2 = c2 with the triangle and accompanying
| squares drawn out much much later is the best example I can
| give.
| crispyambulance wrote:
| > ...still ended up going down an execution rather than concept
| focused route.
|
| Everyone will say, of course, that "concepts matter". But the
| reality is that there ARE points in a student's academic path
| where they HAVE TO memorize stuff and do rote operations like
| the multiplication tables.
|
| One can't move on to new concepts in math until the previous
| dependent concepts have been mastered. Mastery means practice,
| practice means drills, and all of that is boring but necessary.
| The best we can do is to make the mathematics more meaningful
| by tying it to problem solving. At some point many students
| will experience an "ah ha" realization and be further
| stimulated, while for some others math will always be
| meaningless drudgery. For most it will be something in-between.
| bslorence wrote:
| > _there ARE points in a student 's academic path where they
| HAVE TO memorize stuff and do rote operations like the
| multiplication tables._
|
| Sure, but imagine learning multiplication tables without
| having _any idea_ what it means to "multiply"; literally
| just memorizing sequences of symbols, without ever looking at
| piles of coins or whatever.
|
| Multiplication is so basic that this is hard to imagine, but
| I'm sure that, say, difference-of-squares rules feel like
| this to most beginning algebra students -- and for most
| people that probably never changes.
|
| I first encountered difference-of-squares in late middle
| school, memorized the procedure and used it handily through
| twelfth grade, and had _no idea_ that there was a
| visualizable geometric basis to it until I read Book 2 of
| Euclid 's Elements in college.
| BeetleB wrote:
| > I first encountered difference-of-squares in late middle
| school, memorized the procedure and used it handily through
| twelfth grade, and had no idea that there was a
| visualizable geometric basis to it until I read Book 2 of
| Euclid's Elements in college.
|
| Many mathematicians would disagree with your
| characterization. For them, the difference of squares _is_
| an abstract concept in algebra, and the geometric
| interpretation is merely a manifestation of it that just
| happens to work in some domain.
|
| As an example, the formula is equally valid for complex
| numbers, but I doubt you'd get there from Euclid's. No
| doubt _some_ geometric interpretation can be found for that
| as well, but then I 'd pick some other algebraic field
| where it's true and you'd have to search yet again for a
| geometric interpretation.
| bslorence wrote:
| Fair enough. Maybe I should have just said that it's
| possible to give concrete illustrations of basic
| algebraic concepts, and that doing this would probably
| help some students learn algebra, and might help others
| retain it.
|
| But for whatever reason this is generally skipped in
| middle/high-school algebra.
| BeetleB wrote:
| I'm torn. I think it's always good to show these - it
| certainly makes the subject more interesting!
|
| At the same time, if one is to use algebra for future
| studies/work, one _really needs_ to be able to manipulate
| those symbols in the abstract, without feeling a need for
| some deeper understanding. I can see teachers not wanting
| to deal with "But what does that really mean?" for every
| detail in algebra.
| bslorence wrote:
| This goes to the question of why anything other than
| basic arithmetic is compulsory. The famous 10th-grader's
| whine "what are we going to _use_ this for ", which
| infuriated my own 10th-grade Algebra 2 teacher, and even
| made me roll my eyes at the time, is actually a fair
| question _when algebra is taught as abstractly as it
| typically is_.
|
| I think you've explained exactly why here -- because the
| emphasis on abstract manipulation presupposes that this
| is useful for something that we need to get on to. But
| that's just false for almost all students. And yet
| they're required to take the class to get a diploma.
|
| My vote would be to treat any math beyond basic
| arithmetic as a liberal art, and do a lot less of it in
| compulsory curricula, but spend a lot more time on deep
| understanding. This would benefit everyone. The current
| approach pretends that everyone in the class is going to
| be a certain kind of engineer or scientist some day.
| User23 wrote:
| > you lose the ability to understand math as anything other
| than a human attempting to be a computer
|
| I get what you're saying and it's important, but I also
| disagree with how you're framing it. It's desirable to get to
| the point where you can automatically do purely mechanical
| symbol transformations as you work through a problem, thus
| saving your higher reasoning for the conceptual problem of
| searching for which transformations will lead to a solution.
|
| Frankly, I find the continuing concern with "intuition" with
| respect to purely mechanical transformations to be rather
| medieval. It is slowly changing though as the influence of
| computing science osmoses back into more traditional
| mathematics. Using Mathematica or similar tools makes it
| abundantly clear how distinct the mechanical and conceptual
| challenges are.
| WalterBright wrote:
| When I remarked yesterday that engineering curriculums that
| emphasized memorization of engineering formulas rather than
| understanding how to derive them were inferior, I was dismissed
| as arrogant and egotistical :-)
| mmmpop wrote:
| By whom? An instructor/professor or a student/engineering
| graduate?
| WalterBright wrote:
| It's yesterday in my comment history, you shouldn't have
| any trouble finding it. I prefer people interpret it for
| themselves as I said my piece in it.
| BeetleB wrote:
| Can't speak for the others, but you may come across as
| too one sided. I had your mindset, and one of the reasons
| I started getting left behind in grad school whereas
| others didn't was my stubborn refusal to memorize
| anything and insisting only on understanding and deriving
| as needed.
|
| Once you go deep enough, you'll often find yourself
| relying on N random theorems you learned some courses ago
| to solve a problem, and those who had memorized them were
| more likely to solve the problem than someone like me who
| happened to forget the theorem existed in some book and
| is trying to rederive everything from scratch.
|
| Of course, those who _mostly_ or _only_ memorize perform
| the worst.
|
| The other issue is in my other comment:
| https://news.ycombinator.com/item?id=32388595
|
| Basically, most undergrad engineering curricula will not
| teach you the math needed to properly understand the
| Laplace transform. So some level of "take it on faith and
| memorize a few items" are needed. There are other
| examples of this.
|
| Also, having studied under many top class physicists, I
| can tell you that a significant number of them cannot
| derive much of the mathematics they use, despite being
| wizards in applying the techniques.
| WalterBright wrote:
| 1. none of the exams at Caltech required memorization.
| They were open book and open note. Memorizing simply
| wouldn't have helped. For example, one physics exam
| question was: "Assume magnetic monopoles exist. Derive
| how Maxwell's Equations would then look." If you didn't
| _understand_ the ME derivation, you 'd be completely
| lost. The same for FFTs, where the exam question was
| derive the hyberbolic transforms.
|
| 2. one winds up inadvertently memorizing things used
| often, like I knew all the trig identities from excessive
| use. I never attempted to memorize any of them. Just like
| I know a lot of the hex opcodes for the x86 :-)
|
| 3. we were not expected to use the Laplace transform
| until its derivation was demonstrated. This applied to
| all the formulas used. FFTs too.
|
| 4. I've forgotten an awful lot in the 40 years since. But
| I took an online MIT course and was pleasantly surprised
| that it was still there, it just needed a bit of oiling.
|
| > most undergrad engineering curricula will not teach you
| the math needed to properly understand the Laplace
| transform
|
| Which is why some engineering schools are better than
| others. I applied to the USN&WR list of "top 10
| engineering schools in the US". Caltech was #2.
|
| I'm sorry you were left behind because of your insistence
| on learning it thoroughly. You were doing it right. You
| just were in a school that didn't value doing it right.
|
| BTW, I did not mean deriving a formula from scratch every
| time you used it. That would be silly. Just that at some
| point you did, and thereby understood where it came from,
| hence understood its limitations, and knew how to adapt
| it to a situation not in the book.
| BeetleB wrote:
| Much of your experience was Caltech specific, and will
| simply not translate to other schools. Caltech is famous
| for this - more so than schools like MIT, etc.
|
| 1. I learned the hard way, as did others, that one should
| still memorize with open book exams (or at the least make
| a 1-2 page cheat sheet). Why? Because there was a time
| limit and most professors would not alot enough time for
| people to even look up everything they needed in the
| textbook. Sure, if they increased the time by 50-100%,
| you'd do just fine with memorizing.
|
| I'm not even hypothesizing there. After one open book
| exam I went and asked everyone who got a good score -
| most had incorporated some level of memorizing. And
| _clearly_ most who did not memorize at all got a poor
| score (which is less surprising than it should be,
| because most students will do poorly regardless ;-)
| Still, I was the clear exemplar of one who improved from
| "below average" to "one of the top students" within the
| duration of one semester when I finally embraced that
| some level of memorization would be needed.
|
| 2. True, and I can relate to trig identities - most of
| which I remember 25 years later - and even after over a
| decade of not needing them. However, when I got to more
| advanced topics in math, the frequency with which I would
| _need_ to use them dropped significantly, and the
| approach of "Just do lots of problem sets and you'll
| passively memorize" failed me in grad school.
|
| > we were not expected to use the Laplace transform until
| its derivation was demonstrated. This applied to all the
| formulas used. FFTs too.
|
| Did they teach you measure theory before those
| transforms? Did they teach you measure theory before
| probability? Did they teach you the theory of
| distributions before the Dirac-Delta function?
|
| > Which is why some engineering schools are better than
| others. I applied to the USN&WR list of "top 10
| engineering schools in the US". Caltech was #2.
|
| I went to a top 5 engineering school. Can assure you
| Caltech's approach is not the norm.
| WalterBright wrote:
| > Much of your experience was Caltech specific
|
| I am sadly well aware of that. Caltech was known as
| unique at the time, I wonder what it is like 40 years
| later.
|
| 1. It was rarely necessary to look anything up. What was
| on the exam was reliably in the notes or in the assigned
| textbook. I don't recall ever memorizing things, and
| managed an A- average. My innate abilities were
| completely average there, and we all knew who the really
| smart ones were, like Hal Finney. What I did do to
| prepare, however, was ensure I attended every lecture and
| took comprehensive notes, make sure I could solve every
| homework problem, and every midterm problem (in trolling
| for the final). I did not look at prior year's stuff.
|
| 2. I didn't attend grad school, so can't comment there.
| But I did match wits with Masters engineers at Boeing,
| and would wind up fixing their work, too, though far more
| rarely. That group eventually offered me a position,
| though they had a Masters as a requirement.
|
| Measure theory wasn't taught, at least in the undergrad
| courses I took. Neither was the theory of distributions.
|
| > Can assure you Caltech's approach is not the norm.
|
| So I found out later :-(
| roenxi wrote:
| > I was dismissed as arrogant and egotistical :-)
|
| Was that related or unrelated to the comment? I can manage
| arrogant and egotistical without even remarking.
| #engineering.
|
| But more seriously (although that is normally how engineering
| formulas work) if an engineer is actually using the formula
| for anything they tend to pick up the intuition quickly. The
| memorisation in formal teaching is more limbering up the
| mental muscles so that it is easier to learn when & if the
| time comes.
| gamerDude wrote:
| I work with K12 schools in the U.S. and one of our really sad
| stories is how a calculus students failed to solve a "A
| t-shirt that costs $15 is 20% off today. How much does it
| cost after the 20% discount (don't include tax)?" <-- not the
| exact wording, but you get it.
|
| They responded, "I don't remember the formula for a sale".
| Only knowing formula's is terrible, it makes knowledge super
| fragile. So, I for one support your idea of emphasizing
| understanding over formula memorization!
| WalterBright wrote:
| I feel that way about the % key on calculators that also
| have a / key. If you need a % key, you have no business
| using one :-/
| ravi-delia wrote:
| Eh, the closer you can get to entering a formula without
| the mental effort of backtracking and lookahead the
| better. It's like fraction buttons; obviously you can
| just think ahead and use parenthesis with division, but a
| smart fraction button will save a lot of time
| WalterBright wrote:
| What's 20% of $59.22? .2 * 59.22
|
| What price is $2.99 milk with 9.2% inflation?
| 2.99 * 1.092
|
| What did $5.00 gas cost last year? 5.00
| / 1.092
|
| I'm not seeing backtracking and lookahead.
| ravi-delia wrote:
| You'll note that English is written left to right. In the
| string "20%" there is a "2", a "0", and a "%" arranged
| from left to right. To type that string into a
| calculator, one could press "2", "0", and "%" in that
| order, or "0", ".", "2", in that order. To know to lead
| with "0." rather than "20", you have to look ahead.
|
| For the others you'd have to do the mental work to append
| a 1 regardless, might as well stick with decimals
| As_You_Wish wrote:
| Right.
|
| >Fundamental understanding gives exponential results yet most
| courses try and get the axioms and basics out of the way as
| fast as possible, working through basic proofs or derivations
| is cute sideshow, if anything.
|
| This is so true. I was exactly like the author. I could not
| understand any math beyond algebra. Trigonometry was way beyond
| my ability.
|
| But one day, I found an old trig book lying around when I went
| to visit my parents - it was one of my sibling's books when she
| took it.
|
| I picked it up and had a thought - all those people who DID
| understand trig and higher math were not _that_ much smarter
| than me. I mean, yeah, the super smart ones were, but I 'm just
| talking about the regular students. So I got a bug to show
| myself that if _they_ could do it, then _I_ could do it.
|
| My approach that I would do is to read the first section of the
| first chapter until I understood it _completely_ and not move
| on until I did. I would not even move on from a sentence, or
| indeed a word or term, until I understood exactly what was
| happening. I would look all over the internet, watch youtube
| videos on that one single concept, or that one single word,
| until I understood, and I was absolutely adament about not
| moving on until I understood.
|
| Word by word, sentence by sentence, section by section, I
| rolled on. I wanted to completely completely completely
| understand the very very basics of trig. It was slow going, as
| you might imagine. Super slow.
|
| Well........all of a sudden, about 4 or 5 chapters into the
| trig text, my mind exploded. It was blown apart by a blinding
| light. In one single instant, I swear, one instant, I knew
| _all_ of trigonometry. I got it. All of trig. One single
| instant. It was probably one of the most singular moments in my
| life.
|
| So I then started paging through the entire rest of the trig
| book, and I knew it all. I swear. I would just glance over the
| pages and know what everything was about.
|
| Now, I was NOT an expert, by any stretch of the imagination,
| I'm not claiming I became a savant. No. I'm just saying I "got"
| it.
|
| As you said, I has the exponential results. So exponential, it
| was instantaneous.
|
| >If the fundamentals are well understood, the applications feel
| trivial, intuitive, beautiful, clever - when the fundamentals
| are taken for granted, one word describes the relationship to
| the rest of the material: arbitrary.
|
| This is it exactly. You describe it perfectly.
|
| I did not continue on, because I actually had other things in
| my life that I had to move on to. But it was great because I
| knew then that math was easy. No doubt I could do the exact
| same with any math class, no doubt in my mind whatsoever.
|
| For whomever wants to do great in almost any area of study,
| study HARD on those first 3-5 chapters. They set the tone for
| the rest of the course. If you study hard on them, they
| remainder of the course is trivial, with some things here and
| there that you need help with, but for the most part it is
| simple.
| WalterBright wrote:
| I've had a couple moments like that in my education. I've
| described it as stumbling about in a dark room, and suddenly
| the lights came on. One was when I struggling to learn how a
| computer worked. It was just a collection of arbitrary facts
| to me, more and more, until suddenly -- shazaam! And the rest
| was obvious and trivial.
|
| Thanks for your version. I enjoyed reading it!
| lake_vincent wrote:
| Very well said! It is true that once you grok this technique,
| you are basically unstoppable. It does make for slow progress
| at first, but it's worth it.
|
| Edit: I went to grad school for math, and I got there by
| pretty much doing what you described!
| DennisP wrote:
| Ok you two have inspired me. I've been wanting to learn
| math (and relearn college math from decades ago) and I
| always get bogged down and quit. I'm gonna try it your way.
| Viliam1234 wrote:
| The problem with using the right techniques is that there
| are essentially three kinds of techniques:
|
| a) fast at the beginning, then gradually slow down, e.g.
| rote memorization;
|
| b) slow at the beginning, then gradually speed up, e.g.
| deep understanding;
|
| c) slow at the beginning, then remain slow forever, e.g.
| doing something chaotic and stupid.
|
| The problem is that sometimes you have incompetent people
| who use the third kind of method. Then, when things blow
| up, people start paying close attention to the speed, and
| reject any method that is slow at the beginning, because
| they suspect it would be the same story.
|
| (In other words: premature optimization, technical debt.)
| stakkur wrote:
| I agree, but the fundamental problem is how most western math
| education is structured top-down to be a goals-driven
| curriculum. In other words, teaching to the test.
|
| In the US, it's a sprint towards meeting goals tied to
| government funding ("core curriculum ", etc.) And so many get
| left behind in that sprint, often feeling they're 'not smart
| enough' or 'not a math person'.
| sirsinsalot wrote:
| I left school at 15, never to return to education, in part
| because of how maths was taught.
|
| There was never an explanation of the _why_ of things. It was
| very frustrating, and the teaching was very poor.
|
| I ended up a programmer, in part, because I figured if teachers
| weren't teaching, i'd teach myself. This has been a good
| strategy for me, however there's serious holes in my pure maths
| knowledge.
| anthk wrote:
| Ditto. I've got more understanding of Math under
| mathisfun.com over months than in 4 years of edu in Spain
| akin to up to 10th grade in the US.
|
| Contextless Algebra is not intuitive.
| jerf wrote:
| One of the thoughts I've been developing over the past 10
| years or so is that you can't teach someone the solution to a
| problem they don't have. This encompasses the "why" question
| but even goes beyond it, because even the answer to "why" is
| often just another level of "why" and/or "who cares", quite
| reasonably. To learn something, you need a problem, you need
| to grapple with the problem for a bit, and then you can be
| presented with a solution. Then, the answer to why is clear:
| Because it will solve this problem.
|
| Education would still have to artificially give students
| problems; waiting for students to naturally have a problem
| for which taking the derivative of a tan function and then
| leaping in to discuss that just doesn't scale in all sorts of
| ways. But if we gave them problems first, and let them chew
| on them for a bit, I think it would work so much better.
|
| But that's an anathema to the current system. It would
| require not moving students through in cohorts because you
| need to give students enough time to chew on things and
| there's no way that will be standardized. And of course it
| requires admitting that Very Smart People Are Totally Wrong
| About Education, and I might as well ask for a pony while I'm
| at it. The Curriculum Must Not Be Changed. The Curriculum
| Must Not Be Questioned.
|
| I wish someone would pay me for a few years to try to develop
| a computer-based math curriculum based on this concept. I'm
| still waiting for education to move to the phase where
| computers are used as something other than "The Old
| Curriculum, But On A Computer!" I thought we'd be farther
| along on that path by now. I seem to have underestimated the
| inertia of the Holy Curriculum, Hallowed be its Name and
| Hallowed be The Heroes Who Practice It in its infinite glory,
| yea verily.
| gmays wrote:
| Re: computer based math curriculum, this is what Jason
| Roberts is doing with Math Academy here:
| https://www.mathacademy.us It's mainly targeted to kids,
| but has adult users as well. It is by far the best self-
| paced math program I've come across with the widest breadth
| (from the basics up to graduate level).
|
| He's been working on it for years and talks about it a lot
| on his podcast: https://techzinglive.com
|
| Edit: The link to use the beta software is here with more
| details on how the system works:
| https://www.mathacademy.us/beta-test-information
| agentultra wrote:
| There is plenty of pure mathematics that is beautiful and
| worth learning on its own without any practical
| application. A great deal of joy I get from mathematics is
| the delight in seeing a novel structure I hadn't before
| where the proofs fall effortlessly out of the definitions.
| jerf wrote:
| To be honest, _even in pure math this approach ought to
| be taken_. I love math too. But it 's not really a very
| good pedagogical approach even in pure math to start out
| with a full week of unmotivated definitions.
|
| I have no issue with the problems being posed being very
| abstract at a suitable level for the student. By the time
| you hit college, I have no issues with a professor
| introducing group theory with "Hey, look at this aspect
| of graph theory, and this aspect of topology, and this
| aspect of algebra... what commonalities do you think we
| could abstract from them?" But that's a way better
| introduction even at that level than "Let's spend 90
| minutes giving unmotivated definitions and hoping you
| pick up the pieces later."
|
| In a conventional school setting I expect the problems to
| be more concrete, by their nature. I can give another
| example myself: Taylor polynomials. In my opinion,
| they're one of the more important things to learn at that
| level. You can give the students a simple problem:
| "Having learned sin, cos, and tan, and by this point
| memorized some of the common values, please develop a
| procedure for taking an arbitrary sin/cos/tan of an
| angle." Give them some time to chew on it. They may even
| come up with some modestly clever things, maybe cover
| some more special cases or something. But then you can go
| into how we only "really" know how to add, subtract,
| multiply, and divide, and here's a tool that allows you
| to take a wide variety of functions that up to this point
| only existed in calculus and as magic buttons on your
| calculator, and turns them into problems we can do with
| real pencils on real paper using real human brains that
| do not come with a "sin" button. (And then, heh, be
| grateful you live in the 21st century and you don't
| actually have to.)
|
| That's now how I learned them. I learned them as just
| "Here's some Taylor polynomials. Do these homework
| problems." And I did. I learned them, and could do the
| math. It wasn't until years later in my computer hardware
| class that I realized this is what was motivating them.
| (Not the literal hardware, because of course Taylor
| polynomials greatly predate that, but the need to be able
| to calculate these things prior to computers.) And I'm
| not saying "oh, that's what they are"; math very often
| has the characteristic that something is discovered for
| reason X but then has both mathematical and practical
| applications well beyond it. My point here is that my
| understanding of Taylor polynomials is now much richer
| than what I got in the class I learned them in... but
| there was no reason for that insight to be delayed and
| almost coincidentally obtained. It could easily have been
| conveyed via a different teaching method.
| agentultra wrote:
| I think you have the right of it: it's hard to teach
| something like maths to someone who isn't curious or
| interested. And it is definitely difficult to hook
| someone's attention.
|
| When my children were still babies and quite young I was
| reading Zvonkin's book, _Math from Three to Seven_. And
| when they reached that age I started playing games with
| them myself to try and introduce these ideas to them.
| Like Zvonkin I found that one of my kids was more keen
| than the other... but the only way to keep them hooked
| was to avoid the "M" word: maths.
|
| What I think helped was to remind ourselves that we were
| _playing_ games. Any time I went into an area that
| required calculation: determining some value -- they
| would catch on to that and shut down. However if we stuck
| to exploration and fitting things together and exploring
| games together I could keep them interested for an hour
| some days.
|
| And as an adult that's what has kept me interested:
| Martin Gardners' articles in Scientific American and
| books; John Conway's playfulness (ONAG, the bloody game
| of life, etc) -- the stuff that wasn't simply rote
| calculation which I find many attempts at _practical_
| applications seem to focus on.
|
| I can appreciate definitions and proofs now because I've
| learned the language well enough to piece things
| together. However it was the fun, the absurd, and the
| playfulness of the completely impractical that kept me
| going. Games, thought experiments, what-ifs. That sort of
| stuff.
| scott_s wrote:
| Agreed with your overall point, and your specific
| example. Myself and one of my good friends I met in my
| physics classes in college both felt the importance of
| Taylor series had been massively undersold in our
| calculus courses, because it just kept coming up in our
| various physics courses. I learned it just as a thing
| that existed, but we kept relying on them when deriving
| things in courses like thermal dynamics or mechanics. We
| would joke that calculus professors should stop the class
| and just emphasize, "This is really important!" But of
| course, that wouldn't make the material land any better,
| for the reasons you've explained.
| ravi-delia wrote:
| Most of pure math does have practical applications - to
| pure maths! My favorite professors never launched into a
| subject without a motivating example, even if that
| motivation was often "Look at x, y, and z. Aren't they
| awfully similar?". My first exposure to Abstract Algebra
| started with a little number theory, moved on to rings,
| then to ideals, and only then to groups. Many people I've
| talked to about it are surprised we took axioms away
| rather than adding them, but the way we learned motivated
| each step. Indeed, groups themselves were introduced with
| permutations. Similarly, I found measure theory was best
| introduced by showing how handy cardinality was for
| finite sets. A "practical" application would have been
| probability, so perhaps this wasn't exactly application
| focused, but we certainly didn't start from the
| definition and work our way out.
| User23 wrote:
| I feel the same joy when seeing a derivation of a novel
| algorithm where the effective procedure falls
| effortlessly out of the definitions. A good example is
| Dijkstra's derivation of Smoothsort[1]. It's worth noting
| that he was educated as a professional mathematician, not
| as a computing scientist.
|
| [1] https://www.cs.utexas.edu/~EWD/transcriptions/EWD07xx
| /EWD796...
| politician wrote:
| I'd subscribe to your math app. Khan Academy Kids has
| greatly accelerated my kids' language and reading skills to
| the point where they were reading chapter books to us
| before Kindergarten.
|
| Brilliant and KA seem to be the leaders in self-directed
| learning, but I'm still waiting for the device described in
| Neil Stephenson's Diamond Age.
| gavinray wrote:
| Lol, I've been kicked out of class for asking the teacher to
| explain why something works in math here in the US
|
| I think it was long division and lattice multiplication in
| elementary school
|
| Doing math by drawing numbers in predefined shapes so that it
| magically worked out was the most ludicrous thing I'd ever
| seen
|
| "Because that's how it works." Wasn't really a satisfactory
| answer lol.
| BeetleB wrote:
| It's hard to explain to young kids what "out of scope for
| this class" means. And that the number of people who _need_
| these skills vastly the number of people who need to
| understand the derivation.
|
| Lattice multiplication will probably take algebra to
| explain. Long division definitely will require algebra. In
| my school, there's a gap of 5 years between teaching the
| two. A lot more people in the world need arithmetic than
| they need algebra (easily over a factor of 10). We can't
| put off teaching arithmetic till they learn algebra.
|
| A lot of people don't realize that this problem goes all
| the way to undergrad and grad education in engineering or
| science. Laplace transforms are very useful, but they
| require complex analysis to _begin_ to understand. If you
| blindly apply the integration that is normally taught, the
| Fourier transforms of several simple functions have
| integrals that simply, _clearly_ do not converge. Yet we
| 're taught tricks to indirectly calculate them. How is that
| possible? How do we get a result from something that
| clearly diverges?
|
| And don't even get me started on the Dirac Delta function.
|
| Recently I picked up an introductory analysis book - it
| starts from Peano axioms and builds up natural numbers,
| sets, integers, rationals, and then finally reals. It
| requires a fair amount of mathematical maturity to explain
| simple concepts, like how multiplying a positive with a
| negative could result in a negative, or how multiplying two
| positive numbers can result in an even smaller number
| (something that I did get upset about in my school days).
|
| While yes, it is convenient to cherry pick examples where
| it was taught poorly without intuition, the reality is that
| if you want to prepare someone to go into, say,
| engineering, there is a _lot_ of math one needs to cover,
| and teachers just can 't afford to spend time explaining
| things that are way out of scope.
| WalterBright wrote:
| Much of the math part was taught in math classes.
| Engineering classes worked with the math classes to
| ensure the students were ready for the math there.
|
| Chemistry, physics, thermo, dynamics, electronics, fluid
| mechanics, electronics, etc., were all math classes, in
| addition to a solid slate of required math classes.
|
| If one didn't care for math, Caltech was a very very
| wrong place to attend :-) You either got good at it, or
| you left. I definitely felt that 4 years of that rewired
| my brain.
| 2OEH8eoCRo0 wrote:
| One of my friends in HS was against calculators, and also
| very stubborn, and got in trouble for asking how to do sin,
| cos, tan functions without a calculator.
| nobodywasishere wrote:
| Good old taylor series, of course!
| WalterBright wrote:
| That's certainly the hard way!
|
| There are shortcut formulas, and before calculators a lot
| of calculations were done using drafting equipment.
| reacweb wrote:
| Sine Tables Charts is easier if you can bring them to
| exam.
| hgomersall wrote:
| Just read off the axis on a unit circle. If you need more
| precision, draw a bigger circle.
| auggierose wrote:
| Ever heard of new math?
|
| https://www.americanheritage.com/whatever-happened-new-math-...
|
| I agree with you, especially if you are in your 60s and have a
| math professor as your teacher, one would think a new math
| approach works best!
| [deleted]
| Viliam1234 wrote:
| If you know the history of education, the "new math" thing is
| a sad story of a mind-blowing stupidity that keeps hurting
| math students for decades, because it divided most people
| into two camps that keep promoting two different wrong ideas.
|
| Theories of education are often based on some psychological
| theory -- you have a theory how people think in general, and
| you use it to design a process to teach people.
|
| If we skip the medieval theories, one of the relatively
| modern ones was called "associationism". The theory was that
| human mind is basically a set of associations. We are born
| with zero associations; as we observe the world, we learn to
| associate this with that; and after many years we have
| learned to associate things properly and now we are smart
| adults. For extra nuance, some of us form new associations
| faster than others, probably for biological reasons; that is
| what intelligence is. Anyway, associations are all there is.
|
| Building an education theory on associanism is quite easy.
| You need a teacher who understands the subject (has the
| correct associations, a lot of them). Then the teacher stands
| in front of the classroom and keeps talking. The more he
| talks, the more associations the students can make. That's
| all there is. -- The order of lessons is not relevant;
| ultimately, after the teacher mentions everything, all
| associations will be properly connected into one large
| network; until then, you have to memorize. You don't wait
| until the students "understand", that would be a waste of
| time (there is no such thing as "understanding", you either
| have the right associations or you don't); the more you keep
| talking, the more associations the students can make. Of
| course you can (and should) repeat the facts, that's how the
| associations are deepened. But after a while, if some
| students don't get it, they are just hopeless: they had the
| opportunity to make the right associations, and yet they
| failed. It is their fate to remain farmers.
|
| This is a bit of a strawman, and yet many teachers follow
| this method intuitively, even without knowing the underlying
| theory. And their students complain that they don't get it.
| And the teachers reply that yeah, some students are just
| talented and some are not, "the camel has two humps", et
| cetera.
|
| In psychology, the next step after associationism was
| Piaget's "genetic epistemology". Where "genetic" is an
| adjective for "genesis", not the DNA. In modern language, we
| would probably call it "developmental epistemology", i.e. the
| study of the ontogenetic origins of understanding. The
| revolutionary approach was to watch how kids actually learn,
| rather than trying to shoehorn everything into a simplistic
| framework. One of the interesting findings was that kids
| actually do not make linear progress from "zero associations"
| to "correct understanding", but the process often takes a
| detour through a phase of magical thinking or some other kind
| of wrong understanding. Instead of "no opinion -> correct
| opinion", it is often "no opinion -> wrong opinion -> correct
| opinion". There are specific examples, not important now.
| Also, Piaget got some things wrong; this was later improved
| by Vygotsky. The important thing is the idea that the child
| is making mental models of the world. It is not just
| associations floating in a vacuum; the child has a paradigm,
| and tries to fit the new knowledge in that paradigm, and
| sometimes it doesn't work and the paradigm changes into a
| better one.
|
| An educational theory built on this, originally called
| "constructivism", says that the teacher should not just keep
| saying random true facts, but also check that the students
| have the right models. This is achieved on one hand by making
| the models explicit, saying the facts in proper order,
| putting them in the right context... and on the other hand by
| checking the students' models, finding the problems and
| fixing them. If you find out that many students keep making
| the same mistake, you should adjust your way of teaching
| accordingly: make it obvious at the beginning that it is X
| not Y, maybe change the order of lessons so that making the
| correct model becomes easier. Keep checking the students'
| models regularly, because the sooner you find the mistake,
| the easier it is to fix it. Etc.
|
| This is what many good tutors do intuitively, because if you
| teach 1:1, there is more interaction, and it is easier to
| catch the mistakes right when they happen and ask "why did
| you do this?" You do not wait until the student makes the
| same mistake hundred times to declare him a failure without
| talent; you notice when the mistake happens for the first
| time and keep "debugging" until the mistake is fixed. This is
| easy to do when tutoring; more difficult to make it scale to
| a classroom full of kids.
|
| And then... there is another thing, I do not know if it has a
| proper name in psychology, but "postmodernism" is what some
| people use (and other people object to this usage)... the
| "edgy" idea that knowledge transfer is actually impossible,
| everyone lives in their own different reality, trying to
| teach something is an oppression, if only we left the kids
| alone they would reinvent the civilization and make it much
| better (Rousseau's "Emile"). -- For stupid political reasons
| ("there are exactly two sides of the story, not more, not
| less", "the enemy of my enemy is my friend"), these people
| are typically associated with the constructivists, because
| they both oppose rote memorization. But although the opponent
| may be the same, the proposed solutions are quite different
| ("teaching better" vs "not teaching at all").
|
| To increase the confusion, the educational theory build on
| this was called "radical constructivism", misleadingly
| suggesting that this might be "something like the famous
| Piaget, only much more so", when if fact it is something
| completely different. The kids taught using this philosophy
| are left alone to reinvent the math... and fail predictably!
| Or sometimes they are taught dozen different methods how to
| do addition (bonus point if the method was used by some
| indigenous population, because, you know, "noble savage",
| doesn't matter if the specific method only works for adding
| 7+8 and 8+9), hoping that this will kickstart their math
| thinking so now they will develop the rest of the math
| independently. Predictably, that also never happens.
|
| So, how is this related to the "new math"? The curriculum of
| the "new math" was based exactly on this "radical
| constructivist" thought, except the authors did not emphasise
| the "radical" part enough and often just called it
| "constructivism". So you had the math curriculum that didn't
| work at all, and was a complete disaster. And when finally
| people got angry and returned to the traditional math
| education, the lesson everyone remembered was that
| "constructivism has been debunked".
|
| So now, whenever someone proposes to teach math in a way that
| emphasizes understanding over memorization, the kneejerk
| reaction is "haha, that sounds like constructivism... yeah,
| we tried that but that didn't work at all", plus a link to
| some web page that criticizes "new math". And if you try to
| explain how this is completely unrelated to Piaget, you are
| dismissed with "yeah right, the true constructivism has never
| been tried, comrades, hahaha".
|
| And then, ironically, people keep quoting the Feynman's story
| about how actually understanding physics is better than mere
| memorization that "light is waves". And the helpless
| (Piagetian) construstivist is like "yeah guys, that's exactly
| what I was trying to tell you all the time", but no one
| cares, and when it comes back to adding some understanding to
| the lessons, someone inevitably comes with the condescending
| "haha, but constructivism has been debunked" and a link to
| Ten Facts Why New Math Sucks.
|
| Possible solution: perhaps the brand of "constructivism" has
| been thoroughly poisoned, and we need to reinvent it and call
| it "Feynmanism". Then you can go and say "nope, I am totally
| not proposing a constructivist curriculum, that has already
| been tried and debunked, haha, what I am proposing instead is
| the Feynmanist curriculum", and then people will go online
| and say "wow, I hated math at school, but then we got a new
| teacher who used this new Feynmanist method, and the math
| finally started make sense and now I love it".
| bryanrasmussen wrote:
| sure https://www.youtube.com/watch?v=UIKGV2cTgqA I used to
| listen to that several times a day when I was a kid.
| Benobba wrote:
| I was disappointed by this. It's just a caricature of only
| one side - It's the easiest form of come. Almost things
| that are innovative will not be best on the first
| iteration. Why not offer (funny) ideas on how to improve
| it?
| auggierose wrote:
| You could read the article I linked, if you want to know
| why new math is problematic. It's not that easy to
| improve things. And who said that comedy needs to be
| balanced?
| auggierose wrote:
| Wow, great animations! I only read about the song in the
| article, I never actually listened to it until now, thanks.
| ikrenji wrote:
| it's way easier to understand the concept if you see it
| repeated many times in action though...
| inetsee wrote:
| In my opinion, nothing can make you smarter. Learning rationalist
| techniques may make you better at thinking through problems,
| learning memory techniques like Memory Palace may improve your
| memory, using learning techniques like Spaced Repetition may help
| you learn and retain large amounts of information with the
| minimum necessary effort, but nothing can make you smarter.
| haskellandchill wrote:
| I agree you are about as smart as you will ever be. However
| that does not apply to me thank you very much.
| water8 wrote:
| What about adderall? /s
| inetsee wrote:
| Adderall doesn't make you smarter; it just makes you more
| alert. It helps you focus longer on whatever intellectual
| task you're working on, be it problem solving or studying.
| (Not /s)
| jonathanstrange wrote:
| I'm pretty sure this is wrong. The brain adapts to whatever
| activity you do and, as far as I know, is way more plastic than
| researchers thought decades ago. If you're doing dumb things
| all the time you're making yourself dumber over time and when
| you're performing difficult tasks repeatedly and daily your
| brain adapts to them to be able to solve them easier. I
| realized that a long time ago in a completely different area,
| in martial arts. When you start it's extremely hard to follow
| complex movements shown by other people. Beginners struggle
| with this a lot. Over the years it's getting easier and easier
| and you can often grasp complicated movements after seeing them
| once. I suppose it's the same in dancing and ballet.
|
| The brain adapts to sports, painting, writing poems, or solving
| complex math puzzles. It just happens that some of these
| activities are considered smart and also (partly) measured with
| traditional IQ tests. You can even train to become better at IQ
| tests if you desire to acquire a completely useless skill set.
| magpi3 wrote:
| I think you have to define what "smart" means. I mean if you
| dropped me into some of my undergrad courses today, especially
| my CS courses, I would dominate like Wilt Chamberlain and
| people would think I am a genius. But what does it even mean to
| be smart anyway? I think in school a lot of the people we deem
| to be smart simply have more prior experience in the subject or
| just work very hard.
| synu wrote:
| This quote reminded me of learning chess as an adult as well:
|
| > In the paper Acquiring Skill at Mental Calculation in
| Adulthood, Neil Charness and Jamie Campbell say that middle-aged
| people perform as older ones do, but if they practise, they
| perform more as younger people do. If speed is valued more than
| accuracy, the decline in ability is obvious. If accuracy is
| valued more than speed, the decline is less obvious and maybe not
| even very pronounced.
| zxcvbn4038 wrote:
| I have to sit on my hands and not help my son with his math
| homework. The way they have taught him to do math is great - he
| can solve just about anything in his head - but I can't help him
| without introducing him to the old slow way I learned of doing
| everything.
| stinkytaco wrote:
| I really liked the Common Core early math curriculum. My
| daughter was taught to do calculations the way I had to come to
| them myself later on: break the problem apart into easier
| pieces. A simple example: If you are adding 73 and 29 break it
| to 70+20 and 3+9. Quick calculations takes the burden off on
| harder problems later on. Also, she learned to round and
| estimate and they did lots of practical math applications --
| especially in probability -- so she's learned to think of math
| in a "common sense" way, for lack of a better word. It's easier
| for her to look at an answer and say "wait, that doesn't make
| any sense considering the givens".
| victor106 wrote:
| My friends dad in his late 60's wanted to learn math and used the
| No Bullshit math and physics from https://minireference.com/ and
| loved it. Their Linear Algebra is quite good too.
| guidorice wrote:
| I highly recommend the Ivan Savof books as well. A Statistics
| title is upcoming.
| newbie2020 wrote:
| How did he get into college without knowing algebra?? The 70s
| were a wild time haha
| collimator wrote:
| No, but once you have learnt some algebra you'll be in a position
| to learn other things - electrical physics for example, or audio
| engineering. These subjects, and many others, use the language of
| algebra, because algebraic explanations are very compact and easy
| to follow.
| [deleted]
| dhosek wrote:
| I've been tempted to pick up an adjunct section of algebra at a
| local community college and invert the usual style of teaching
| and _start_ with word problems. A lot of people have an intuitive
| sense of how to figure out, say, how to scale a recipe but when
| it gets turned into symbolic math it becomes a challenge for
| them. I'd kind of like to take advantage of that and start from
| the word problem and then move to how to turn that into symbols
| so that instead of thinking _y_ = 1.4 _x_ -2.8 we'd say think in
| terms of you pay $1.40 for each donut but the first two are free.
| If all the _x_ 2 and _x_ 3 have particular meanings, you're less
| like to think that _x_ 2+ _x_ 3 = _x_ 5 but recognize that you
| can't simplify the expression until you know what _x_ is.
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