[HN Gopher] Children's arithmetic skills do not transfer between...
___________________________________________________________________
Children's arithmetic skills do not transfer between applied and
academic math
Author : rbanffy
Score : 84 points
Date : 2025-02-07 14:18 UTC (8 hours ago)
(HTM) web link (www.nature.com)
(TXT) w3m dump (www.nature.com)
| peterprescott wrote:
| This is really interesting, but I don't agree with this
| conclusion: "These findings highlight the importance of
| educational curricula that bridge the gap between intuitive and
| formal maths." (My own opinion is that educational curricula are
| generally not very important at all; that people are learning
| machines that learn what they need to in the contexts they find
| themselves; and that people -- as shown by this study -- struggle
| to effectively apply what they've learnt in one context into a
| different context.)
| csours wrote:
| > My own opinion is that educational curricula are generally
| not very important at all
|
| We spend an incredible amount of time and effort on educational
| curricula, so it's worth thinking about.
|
| _My opinion_ is everything you learn before you start actually
| using knowledge is "just" familiarization. In my opinion,
| pedagogical instruction should do a much better job of
| explaining this and incorporating this realization. I do think
| individual teachers understand this.
| delichon wrote:
| I'm afraid that this is because academic math is often taught and
| tested in a way that rewards memorization rather than
| understanding. Here's Richard Feynman's take on the problem:
|
| https://v.cx/2010/04/feynman-brazil-education
| 1970-01-01 wrote:
| Feynman was correct for science in school, however arithmetic
| is fundamental and maybe one level above the root of all
| mathematics. Children should be able to do most of it via
| mental lookup tables and apply that knowledge on paper. For
| some reason, they can't.
| freeone3000 wrote:
| Why should they? We have the tables in our pockets at
| literally all times; doing arithmetic without it might be
| useful, or a bit faster sometimes, but is hardly an essential
| skill.
| 1970-01-01 wrote:
| "Will you have enough fuel to make it over the mountain
| range?"
|
| "My phone is just rebooting" the pilot replied.
| freeone3000 wrote:
| The plane has an onboard calculator for this :)
| BeFlatXIII wrote:
| For catching when you keyed something in incorrectly.
| somenameforme wrote:
| It builds a numeric intuition. When you repeat something
| enough, it begins to do itself - you gain a subconscious
| mastery. Think about yourself as you read these words.
| Imagine if you were looking at the letters and actually
| trying to sound out each word, consciously thinking about
| each words meaning, and then finally trying to piece
| together the meaning. You'd spend 5 minutes reading a
| sentence or two, and oh God help yo if tere ws a tpyo.
| Instead it all just flows without you even thinking about
| it, even when completely butchered.
|
| And that sort of flow is, I think, obtainable for most of
| anything. But 100% for certain for numbers. Somebody who
| doesn't gain an intuitive understanding of basic arithmetic
| will have an extremely uncomfortable relationship with any
| sort of math, which mostly just means they'll avoid it at
| all costs, but you can't really. I don't even mean STEM
| careers, but everything from cooking (especially baking) to
| construction and generally an overwhelming majority of
| careers make heavy use of mathematical intuition in ways
| you might not consider, especially if you're already on
| good terms with numbers.
| em-bee wrote:
| that's why montessori math is so impressive. it starts
| with counting out beads one at a time by the hundreds
| until they have internalized that. then they get beads on
| a stiff wire 10 at a time, and repeat the process
| counting them out up to a 1000. and so forth until
| eventually they hold in their hands blocks of 1000 beads
| glued together in a cube, and only after they have
| internalized that the beads get replaced with more
| abstract woodblocks and sticks. and all that happens in
| the first year or so at the age of 3.
| em-bee wrote:
| so every time i go shopping i have to type all the prices
| of what i buy into my phone and also have the calculator
| connect to my bank account and not only make sure i have
| enough savings, but also tell me that i am not spending
| more than my average for weekly groceries? and when doing
| that i need to make sure to not make any typos because my
| lack of numeric intuition won't allow me to recognize where
| i made a mistake. and i also won't be able to tell if an
| item is overpriced. nor will i recognize a bargain unless
| it is marked with a big colorful sticker.
| cratermoon wrote:
| I suggest learning to use a slide rule and abacus.
| throwway120385 wrote:
| No, it goes beyond that. There's "arithmetic," the applied
| usage of addition, multiplication, subtraction, and division
| to permute numbers, and then there's Arithmetic, the set of
| theorems and axioms that give rise to that system of applied
| arithmetic. Memorization only works for the applied part, and
| children aren't usually taught that there is a system of
| reasoning behind those rules. Without that, no amount of
| mathematical dexterity in pushing symbols across a page will
| help them understand anything past the 100 level, and
| sometimes not even that.
|
| I also think there's a huge undercurrent of resistance from
| adults to having children learn that system of reasoning
| because adults don't understand why it's useful, and in my
| experience when people don't understand something they
| dismiss it.
|
| Edit: A nice example of another axiomatic system that might
| be more approachable is Euclid's Elements, in which five
| postulates are used to develop a system of geometry using an
| unmarked straightedge and a collapsible compass that you
| could, if you were careful, use to build bridges and other
| large buildings.
| 1970-01-01 wrote:
| The study was limited to cities in India. We shouldn't put
| much weight into this applying globally.
| throwway120385 wrote:
| I mean I remember seeing this first-hand student teaching
| Mathematics 13 years ago in the US. They got to me having
| never seen any of that stuff, and the curriculum
| attempted to provide a good education in mathematics. But
| the staff and the way the whole system is structured is
| to skip all of that and memorize the single rule you need
| to know to get through the test. So it's all done by rote
| and the only time you find out how you've been cheated is
| when you try to go through Calculus.
|
| And I remember that was how we learned everything when I
| was a kid, and the teachers chose not to do anything
| else. I also remember from my math ed curriculum one of
| the professors joking about the elementary education
| students complaining about having to learn middle school
| math from the college perspective. So I think portions of
| this apply here.
|
| I've also seen carpenters apply trigonometry very
| effectively to do things like cuts for roofs and stair
| jacks, so there's certainly a lot of truth to people
| learning maths by occupation and not in a formal setting,
| and I think part of it is the formal setting.
| sumtechguy wrote:
| Once I got to calc2 and 3. I was so mad. I realized I had
| spent nearly a decade memorizing things. When I could use
| calculus to have a factory that made formulas and the rules
| were on a whole simpler to remember and apply.
| Izkata wrote:
| Similar here: There were all sorts of volume and area
| equations I could never remember, then one slow day at
| work I decided to try and derive the volume of a spehere
| using what I'd just learned in calculus. After doing so
| each part of the equation made sense instead of appearing
| random, and two decades later I still remember it without
| having to derive it again.
| rcxdude wrote:
| Did you not get to doing lots of integrals? A whole new
| field of patterns to memorise, which I hated.
| epicureanideal wrote:
| Good point, but there are probably plenty of formulas
| that can be derived with simple integrals, as a backup
| for if those formulas are forgotten.
| Almondsetat wrote:
| Unfortunately, integration, as opposed to derivation, is
| made of pattern recognition. It's simply that way, like
| differential equations.
| Nihilartikel wrote:
| I had a lucky experience taking HS calculus the semester
| before as physics. I saw other students torturing
| themselves memorizing the formulae from the physics text
| and even then struggling to apply it to novel problems.
|
| For the most part, knowing basic calc, it was possible to
| just draw a free body diagram and either integrate or
| take a derivative to get the answer. Didn't memorize much
| beyond f=ma and v=IR, for better or worse.
|
| I still firmly believe that physics and calculus should
| be introduced together to provide a tangible and
| practical base to understand the mathematical theory.
| derbOac wrote:
| I had a similar reaction. I had a lot of reactions, and found
| the paper interesting.
|
| First, it reminded me of something a stats professor said in
| grad school: "there are two kinds of mathematicians, those who
| are good at arithmetic, and those who are not." He was speaking
| as someone who identified with the latter.
|
| I can't tell if this is something related to this domain of
| math in particular or something broader. My guess is it's
| something broader.
|
| I have colleagues (speaking as a professor) who have complained
| about admitted students who come in with very high grades and
| test scores, but who can't actually reason independently very
| well and despair when they are not "told exactly how to
| respond" on tests and whatnot. You have to be careful because
| sometimes these complaints hide bad teaching, but I think this
| is a common sentiment, and I've seen articles written about
| similar sentiments at other places.
|
| The paper touches on a lot of issues, like applied versus
| abstract concepts, generalizability of learning, "being a good
| student" versus actual cognitive ability, learning how to take
| tests versus learning concepts, the difficulty of measuring
| cognitive and academic ability, and the fallibility of
| measuring complex human attributes in general.
| zdragnar wrote:
| Even in lower education, as a student I hated word problems.
| Partly, I just wanted to be told what equation to solve. In
| retrospect, though, I think a lot of it was the framing.
|
| It was always presented as some variation of short exposition
| followed by a question. The question was usually framed as an
| outside observer asking for some fact about the story.
|
| Think of the classic "A train leaves station A headed west at
| 6:30 traveling at 30 miles an hour. A second train leaves
| another station at 7 traveling 50 miles an hour. When do they
| pass each other?". There's no problem here to solve. Who
| cares when they pass each other? Why do we care?
|
| Sure, a little exposition helps build up analysis and
| application skills, but it doesn't actually offer much in the
| way of engagement.
| almostgotcaught wrote:
| i hate when people quote random celebrities as authoritative on
| any topic, let alone as a counterpoint to actual authorities
| (google the authors of this study).
|
| Edit: hn is just as anti-intellectual as any other place these
| days but y'all style yourselves as intelligentsia because your
| celebrities are _special_.
|
| I'll repeat: check out the qualifications of the authors of
| this study and compare them to Feynman's _on this subject_. Any
| reasonable person would conclude that comparing them is exactly
| like comparing Kim Kardashian and Feynman 's on QED.
| gjstein wrote:
| Feynman is no random celebrity. In addition to be a renowned
| physicist, his famous "Feynman Lectures" and his thoughts on
| pedagogy are similarly legendary.
| BeetleB wrote:
| The Feynman Lectures are great at giving you an intuitive
| understanding, but is no substitute for the regular
| curriculum. You don't find many people who read only the
| Feynman Lectures who can then go on to solve physics
| problems well. You _do_ find many who read the regular
| textbooks and who can.
| wizzwizz4 wrote:
| In this case, Richard Feynman is just writing about his
| personal experiences of a well-known phenomenon.
| https://profkeithdevlin.org/wp-
| content/uploads/2023/09/lockh... ("Lockhart's Lament") would
| perhaps be a better reference, but nearly anyone who's been
| through the education system would be able to tell you this.
| almostgotcaught wrote:
| > Richard Feynman is just writing about his personal
| experiences
|
| Let's see
|
| 1. The personal experiences of a guy with no formal
| training in pedagogy or education
|
| 2. A research paper in nature written by expert education
| economists
|
| Hmmmmmmmm
| erikerikson wrote:
| I wanted to upvote your other comment because it caught a
| detail of "how much" that may have slipped past the other
| commenter's or other reader's minds but...
|
| 0. The Kardashians
|
| The distance between 0 and 1 is vast compared to the
| distance between 1 and 2. Feynman was a professor and
| also beloved for his ability to bridge across the
| academic to pragmatic divide that is the subject of this
| paper.
| almostgotcaught wrote:
| What is the relevance of this point? No one has linked a
| Kardashian's take on anything? So who cares if the
| distance between 0 and 1 is larger than the distance
| between 1 and 2 - we are only discussing the distance
| between 1 and 2.
| erikerikson wrote:
| The original comment you responded to made no comparative
| claims. It simply offered another person's attempt to
| describe. Feynman is fairly famous but nonetheless an
| authoritative source relative to most of the population
| (probably more so than both of us, though I don't know
| you do have little basis beyond priors [sorry if you have
| greater credibility than Feynman, I didn't know]).
| Feynman is less authoritative on the subject than the
| authors of the article but still... Being well known
| doesn't remove the authority level that Feynman does have
| on the topic.
|
| I should, perhaps, have used:
|
| 0. Average person
| wizzwizz4 wrote:
| Richard Feynman is famous _for being an educator_ , and
| he's clearly quite good at it. Who cares if he has no
| formal training? I reckon he deserves at least a 1.2 on
| this scale.
| almostgotcaught wrote:
| > Richard Feynman is famous for being an educator
|
| It's amazing how deep the celebrity worship goes. No he's
| famous for being a mathematical physicist (his Nobel is
| in physics not education). He was actually a very
| mediocre educator - you can read his own assessments of
| his success/failure in teaching the "famous" intro
| courses.
|
| Or you can ask literally any physics major that's
| actually had to use those books (they are horrible for
| actually learning from).
| wizzwizz4 wrote:
| Most Nobel winners are not famous. I never said he was a
| stellar educator: I said he is _famous_ for it, and that
| he is _quite good_ at it.
| zyklu5 wrote:
| And why should I simply assume that "Education
| Economists"* really know the subject they purport to talk
| about? Because they are credentialed members of
| university departments with some label? Because a few of
| them won some Bank of Sweden award?
|
| Just because a particular department or field of study
| exists in academia does not magically give them the
| imprimatur you think it does.
|
| * Btw, I know for a fact that a few of them are not
| "education economists"
| almostgotcaught wrote:
| > And why should I simply assume that "Education
| Economists"* really know the subject they purport to talk
| about?
|
| I'm glad we've arrived at Fox News level takes. At least
| we can all admit what we are here.
| kbelder wrote:
| Wow, it's really a no-brainer when you phrased it that
| way.
|
| Unfortunately it destroyed your argument.
| deadbabe wrote:
| You delievered this comment as if he had just quoted kim
| kardashian.
| throwway120385 wrote:
| More to the point, if Kim Kardashian delivered an essay
| with similar arguments I don't think we should care that it
| was written by a Kardashian.
| PaulRobinson wrote:
| Did you just call Richard Feynman a "random celebrity", who
| isn't authoritative on the subject of science?
|
| Hint: I might need to google the authors of the study, I
| don't need to google Richard Feynman...
| almostgotcaught wrote:
| > don't need to google Richard Feynman...
|
| My friend that is the textbook definition of celebrity
| status.
| inglor_cz wrote:
| "Celebrity" in colloquial use implies someone "famous for
| being famous", like Kim Kardashian.
|
| To use such word with regard to one of the most talented
| and innovative physicists of the 20th century debases the
| entire conversation.
| almostgotcaught wrote:
| > debases the entire conversation.
|
| If my eyes rolled back any further in my head I could
| look into the past.
|
| But you've proven your own point: you know him because
| he's known but because you're actually familiar with his
| work.
|
| So again: textbook celebrity worship masquerading as
| intellectualism.
| jcranmer wrote:
| Linus Pauling's authority in chemistry doesn't make his
| cuckoo theories about Vitamin C any less cuckoo. Feynman
| may have been an important physicist, but that doesn't make
| him knowledgeable about education!
|
| And, to be honest, there's a reason why there are memes
| about physicists' competence in other fields, like
| https://xkcd.com/793/.
| PaulRobinson wrote:
| Feynman was extremely knowledgeable about pedagogy, and
| his lectures are considered some of the finest works in
| the field.
|
| I'm very aware of the "spherical cow" joke about
| physicists, but this isn't a "random celebrity".
| somenameforme wrote:
| As well as the texts based off his lectures. [1] His
| ability to teach was completely unreal. Those 'books'
| dramatically deepened my _understanding_ of physics.
|
| [1] - https://www.feynmanlectures.caltech.edu/
| fragmede wrote:
| He taught a two-year introductory physics course at
| Caltech from 1961 to 1964, which gives him _some_
| experience with the matter though. He was known as "The
| Great Explainer", due to his ability to help people
| understand and more importantly, be inspired by science
| and the world around them*. His materials from those
| lectures were converted into "The Feynman Lectures on
| Physics", a highly regarded physics textbook. so I
| wouldn't have chosen education as my example.
|
| In support of 793 however, he didn't do well with
| bureaucracy so I'd not listen to his advice on how to run
| something that favored rigorous rule following even when
| the rules don't make sense**.
|
| * https://www.commoncraft.com/honor-richard-feynman-
| great-expl...
|
| ** https://laurakornish.com/2019/04/no-joking-matter-
| feynman-on...
| watwut wrote:
| He is celebrity. He is not authoritative about subject of
| teaching math to kids. He is authoritative about his area
| of physics. He also is authoritative about writing popular
| books for physics that demystify physics to adults. But
| again, not about kids and math.
| dambi0 wrote:
| Regardless of the people involved, being asked to consider
| someone's opinion on a matter is a world apart from claiming
| they are an authority on the topic.
| timerol wrote:
| My parent comment is especially jarring because Feynman's
| findings agree with, and propose a mechanism for, the
| findings of the study. The comment seems to imply that
| there's some great tension between "arithmetic skills do not
| transfer between applied and academic mathematics" and "the
| students had memorized everything, but they didn't know what
| anything meant". Or between "These findings highlight the
| importance of educational curricula that bridge the gap
| between intuitive and formal maths" and the less academically
| worded "There, have you got science? No! You have only told
| what a word means in terms of other words. You haven't told
| anything about nature."
|
| Nobody's quoting Feynman "as a counterpoint to actual
| authorities". Feynman's excerpt provides first-hand testimony
| from a teacher on the front lines that fully validates what
| the study found.
| yesfitz wrote:
| It's not a counterpoint. The Feynman excerpt and the paper
| support each other.
|
| The paper's abstract ends, "These findings highlight the
| importance of educational curricula that bridge the gap
| between intuitive and formal maths." The Feynman excerpt is
| about the issues caused by a lack of practica in education
| and how they should be resolved.
|
| The paper's authors wrote, "These findings call for a maths
| pedagogy that explicitly addresses these translational
| challenges through curricula that connect abstract maths
| symbols and concepts to intuitively meaningful contexts and
| problems." And provide 2 examples of Randomized Control
| Trials of math courses in Brazil and India respectively that
| address the challenges successfully.
|
| Even if you remove Feynman's name, it's still interesting
| that a Theoretical Physics professor and educator wrote
| clearly about a very similar issue they encountered over 60
| years before the paper in question was published.
| mvieira38 wrote:
| This tracks nearly 1 to 1 with my experience growing up in
| Brazil. Nice to see someone so accomplished pointing it out,
| thanks for sharing
| blackeyeblitzar wrote:
| I wonder if this is true for all cohorts. There are a lot of
| children who are just fundamentally not intelligent, and deal
| with math classes by basically memorizing things and repeating
| them without real understanding. But for children who are
| understanding what they're learning, I would expect academic
| learning to translate to other things.
| greentxt wrote:
| "ask whether, in the urban Indian context, the arithmetic skills
| that are used in market transactions transfer to the more
| abstract maths skills taught in school."
|
| I have a hard time with this notion of 2 different maths. I
| wonder if it is specific to the "urban Indian context" as the
| authors seem to suggest in their literature review -- I didn't
| pursue their references. Intuitive math that is not associated
| with memorization sounds like g.
| dkarl wrote:
| My father, who had a PhD in history, had exactly the kind of
| math skills the study describes in Indian children. With
| ordinary arithmetic, he was fast and accurate. Money and
| baseball statistics were no problem. Algebra? No fucking way.
| As soon as x and y were involved, he struggled. Strike that: he
| wasn't able to struggle. He wasn't able to engage the gears
| that would allow him to apply effort.
|
| One time, when I was in high school and already contemplating
| majoring in math in college, he told me that a math professor
| had told him that in modern mathematics you didn't have to know
| what you were talking about. All their theories could apply to
| anything. Like you could pick up a paper, and they're talking
| about X, and you could decide X was Donald Duck! He told me
| this like it was an exotic glimpse into another culture -- he
| knew that it looked ridiculous to him, but he also knew that it
| probably looked ridiculous because he didn't understand what he
| was looking at. You could tell that one part of his brain felt
| like it was a gotcha moment for the mathematicians, but another
| part of his brain could see that they weren't embarrassed about
| it, and he taught WWII every year so he knew that Donald Duck
| could also be artillery shells or atoms. He had that last
| defense of common sense that stops people from embracing crank
| theories about other fields of study.
|
| This was a guy who taught recent history and accepted the
| abstract ideological struggles of the 20th century without
| blinking, but when you told him that someone could write an
| entire doctoral thesis about X without knowing _concretely_
| what X was, it was such an alien idea that it was out of range
| of his curiosity.
|
| From this I would be confused about how he got through high
| school math, except that my sister, who also has a PhD in
| history, explained how she got through calculus: she studied
| all the homework problems and all the solutions over and over
| until she had memorized them, and she reproduced the patterns
| on the tests. At our high school, that was good enough for As.
| In college, it was good enough for Bs.
|
| (It makes me feel a tiny bit more empathy for the condescending
| mathphobes who denigrate virtually all school mathematics work
| as pointless, deadening rote learning. For many of them it
| might be a sincere belief. They might have actually experienced
| it that way and never experienced any of the worthwhile aspects
| of it. But, on the other hand, they should have the grace my
| dad did to stop short of declaring it worthless just because
| they didn't get it.)
| greentxt wrote:
| It is very sad to me that so many people can't enjoy that
| aspect of math. I was lucky, pbs used to show math stuff to
| kids, so it was fun and interesting before it was a school
| thing. Of course a huge part of math learning is just hatd
| work for most if us. But kids should taste the delight first,
| it motivates them to do the less delightful practice.
| dkarl wrote:
| Unfortunately, the pleasure or displeasure of doing math
| compounds quickly. A kid who doesn't enjoy it is going to
| do the minimum required, and if that isn't enough, it will
| become ever harder and less enjoyable in the future. You
| need practice, and practice is a dirty word in pedagogy.
|
| Someday education is going to catch up with music and sport
| in its attitude towards practice. As adults we all
| understand that in music or sports, the most elite of the
| elite, the top 0.01% of human beings in attainment, are
| never too good to need more practice and polish of basic
| skills. But when we look at children learning math,
| repetition becomes anathema. My question is, how could
| anybody ever enjoy math without repetition? You need to
| make the boring stuff easy and then keep it easy. How does
| that happen without repetition? If you don't practice, the
| boring stuff becomes hard again, and you don't have brains
| to spare for the interesting stuff.
| greentxt wrote:
| I agree. Kids need to eat their vegetables, but we can
| make vegetables quite delicious if we try. Also, the
| music metaphor is imperfect because some musicians are
| performers others composers. I prefer sport as the
| metaphor: a mix of short laregely repituous training and
| longer term strategy, different styles for different
| athletes, influence of genetics and talent heterogeneity
| acknowledged at the elite end of the spectrum.
| brnaftr361 wrote:
| Anecdotally, this is true even as an adult. As a non-trad student
| the application side of things as they are taught are fairly
| effete, we learn how to translate graphs in algebra, quadratics,
| polynomials... I don't recall much in the way of meaningful
| application in either trig or algebra and what was there was
| remembered solely in the context of future examination.
|
| In one hand I would argue there is virtually no incentive for
| play, or discovery, or superfluous activity with the math due to
| grading, and in fact it's disincentivized as it is one factor of
| a multivariate optimization problem. On the other hand I would
| argue that it's taught too fast as an effect of the former
| condition - as someone who isn't in a highly mathematical branch
| of STEM the use of mathematics is comparably infrequent when
| considering the TEM, as such atrophy sets rapidly after
| examination. And this could be said more generally with the S as
| well, though there is some degree of reinforcement there.
|
| As things are, I feel that the timeline is askew, the won't if
| these institutions to produce biologists along the same timeline
| as they did a few decades ago is a little ridiculous considering
| the ballooning of quantitative discovery that has occured since,
| for instance, it wasn't so long ago that DNA was a conceptual
| exercise.
|
| Moreover, the failure of education to keep with the times and
| adapt a realistic curricula for the modern era is also
| inhibitory. Indeed I would argue that the current academic
| zeitgeist is working against itself. At once being a trade
| program and while also trying to facilitate the development of
| "academia" itself are forces acting against one another. The
| number of premed students running the gauntlet in my program far
| outweigh the number of people with [let's say] legitimate
| interest in learning about the concepts in the program, which are
| also made to compete with the premeds in the limited slots
| available for lab internships. In my experience this leads to a
| chilling effect. Fortuitously, once in a lab things tend to be a
| little more facorable in terms of rapport.
| robertclaus wrote:
| On the surface it's pretty obvious that classroom learning
| doesn't immediately translate to real world experience, but the
| paper's finding seems to be more about the extreme degree of
| discrepancy between the two cohorts. It almost feels like a
| comment on social class distinctions - there are children who get
| classroom educations that don't have to use it, while others have
| to use the skills but don't get the related education.
| cjs_ac wrote:
| Jean Piaget was one of the foundational researchers in cognitive
| development, particularly in 'constructivist' circles.
| Constructivism is the theory that learning is an active
| psychological process, and is contrasted with behaviourism (often
| associated with B.F. Skinner), in which learning is a passive
| process. Constructivism is popular amongst school teachers and
| behaviourism is unpopular.
|
| Piaget's theories are very much rooted in learning about the
| physical world, and are thus more popular amongst teachers in the
| 'STEM' disciplines. The other foundational researcher in
| constructivism was Lev Vygostky, who worked in the Soviet Union
| under Stalin, and his theories reflect the political pressures of
| that environment; his earlier work remains influential especially
| amongst humanities teachers, whereas his later work is
| excessively ideological.
|
| Piaget proposed a stage-based model of development, in which
| children progress through four different stages of cognition. The
| last two stages he called the _concrete operational stage_ and
| the _formal operational stage_. In the concrete operational
| stage, children become capable of _one_ layer of abstraction or
| symbolic indirection; it is not until they reach the formal
| operational stage that they are capable of multiple layers of
| abstraction or symbolic indirection.
|
| It is interesting to note that Piaget suggested that the
| transition between these two stages usually occurs around the age
| of eleven, which is when British children transition from primary
| to secondary school.
|
| So, to contextualise the linked article, the studied children
| developed an applied understanding of mathematics that fit their
| concrete operational cognition: the mental operations were
| readily made concrete by manipulating coins and banknotes.
|
| However, when children study mathematics in school, it is very
| much done as formal operations: the numbers are not intended to
| represent anything physical at all. Consequently, many children
| learn mathematics as a set of arcane rituals for manipulating
| symbols on paper, because they can't yet understand the abstract
| meanings of those symbols.
|
| Piaget's theory on Wikipedia:
| https://en.wikipedia.org/wiki/Jean_Piaget#Theory
|
| Paul Lockhart writes something parallel to this argument in _A
| Mathematician 's Lament_:
| https://archive.org/details/AMathematiciansLament/mode/2up
| greentxt wrote:
| "many children learn mathematics as a set of arcane rituals for
| manipulating symbols on paper, because they can't yet
| understand the abstract meanings of those symbols."
|
| In the US?
|
| I had pretty typical US public school education. Word problems
| and application were ubiquituous. Perhaps my experience is non-
| representative, or you like the study authors are speaking of
| other educational contexts (Asia).
| cjs_ac wrote:
| I grew up in Australia and taught in Australia and the UK;
| and yes, the word problems are ubiquitous in those countries
| too. But those word problems are always deeply inauthentic,
| and extracting the relevant information from them becomes
| just another arcane activity.
|
| One that comes to mind involves a farmer with a given length
| of fencing, and the student has to find the area of the
| largest rectangular field the farmer can surround with that
| fencing. It's a good mathematical puzzle, but the actual
| real-world problem is how much fencing the farmer needs to
| surround a given field.
|
| Coming up with genuine, real-world applications for every
| mathematics lesson is extremely time-consuming, and maths
| teachers simply don't have the time.
| greentxt wrote:
| Maybe there's an element of both time and skill? How many
| folks that pursue math can point to a great early teacher
| as being influential. Not all, but I _want_ to believe it
| 's common and maybe true for most.
| anon291 wrote:
| With ChatGPT et al... it should start to be pretty easy. I
| envision a question answer game between humans and the AI.
| The AI would set up the scenario and the kids would have to
| ask the right questions. Teachers could supervise and
| evaluate
| aidenn0 wrote:
| Some children read word problems, understand the question and
| apply math. Most children look for key words, use those key
| words to guess what operation to apply, then apply it to the
| numbers in the question (e.g. there were 3 numbers and the
| word "total" so I'll sum the numbers).
| anon291 wrote:
| In my experience, this is as much an issue with reading
| comprehension as it is with math
| hyeonwho4 wrote:
| I'm surprised this was published now, given that I saw a talk on
| this at a math conference in either 2008 or 2009. The memorable
| anecdote was that they filmed children who worked with cash in
| the market, brought them to the classroom, and given pen and
| paper in the classroom they would be unable to duplicate the
| calculations they had already done in the market. The speaker was
| promoting VR to simulate the market context in the classroom.
|
| I guess what this paper adds is a higher N and the reverse case,
| that classroom skills don't transfer to the market.
| tombert wrote:
| When I was an adjunct a couple years ago, I would use money for
| the more introductory stuff in Python.
|
| I figured that for better or worse, every single person in that
| classroom will have to deal with some amount of "money math" in
| their life, and "money math" is still "real" math, and programs
| involving money are still "real" programs. If nothing else, I
| couldn't really get the "when will I ever use this????" kinds
| of questions.
|
| A lot of people seem to have almost a "phobia" of mathematics;
| they are perfectly fine doing the relevant calculations in
| regards to stuff that's directly used, like money, but seem to
| shut down when mathematical notation is used.
| WillAdams wrote:
| My father's comment after we finally nagged him into buying
| the family a copy of the boardgame _Monopoly_ and playing a
| game:
|
| >If I'd known you kids would get so much math practice from
| this I would have bought it a long time ago.
| tombert wrote:
| Yeah, similarly, a friend of mine's kid got really into
| Kerbal Space Program. That friend didn't mind his kid
| playing that one for long periods of time, because there's
| a ton of real math and physics being used, but the game is
| relatively fun.
| Joel_Mckay wrote:
| The development challenges posed by wealth inequality was
| quantified in numerous studies: "Outliers: The Story of Success"
| (Malcolm Gladwell, 2011)
|
| https://www.amazon.com/Outliers-Story-Success-Malcolm-Gladwe...
|
| And the psychological impact on vulnerable children has also been
| studied: "Inducing learned helplessness: video fragment"
|
| https://www.youtube.com/watch?v=MTqBP-x3yR0
|
| Notably, the advanced mathematics guidance programs from the
| institutional outreach for children were usually by invitation
| only. Thus, unless you were identified as "gifted" early on, the
| academic content for your development is very different.
|
| Personally, I think grade-school kids in grade 7 should be
| introduced to physics with calculus, discreet mathematics, and
| linear algebra. However, the content should not be part of a
| graded curriculum, but rather a set of trivial daily puzzles to
| solve.
|
| The current academic institutions are often engaged in all sorts
| of policies that have nothing to do with science, or students
| cognitive development.
|
| YMMV, and hold out hope for tenure... lol =3
| taeric wrote:
| I'm curious on this. My gut would be that NOTHING transfers
| between contexts by default. Instead, learning to transfer
| something between contexts is itself a skill that needs effort.
|
| As an easy example, just counting beats is clearly just counting.
| Yet counting beats and aligning transitions/changes on or off a
| beat is a skill you have to work on.
|
| Indeed, the same counting can help people that are running to
| start to even breaths out. Simple meditations often have you
| count a breath in, and then count it out. Just because you can
| count doesn't mean you will automatically be good at any of these
| things.
|
| There is also the question of what it means to count? Do you
| literally hear a voice in your head speaking the numbers away? Do
| you see a ticker? Do you have some other mental tally system?
| milesvp wrote:
| What you describe is one of the fundamental "problems" of
| associative memories. Which is doing or recalling a thing in
| one context does not mean you are capable of doing or recalling
| the exact same thing in another context. Neurons light up based
| on all the current inputs, and if none of the current inputs
| light up the neurons that can trigger a skill, good luck doing
| that skill. This is why practicing in a wide variety of
| contexts is really important for mastery, you're essentially
| increasing the odds that different inputs have a chance to
| trigger the knowledge that's locked away in the structure.
| taeric wrote:
| Is that, essentially, what the article is looking at?
|
| My curiosity is that this is my prior. The article is clearly
| framed in the opposite direction. Am I just putting too much
| emphasis on the headline?
| Someone wrote:
| > My gut would be that NOTHING transfers between contexts by
| default.
|
| Once you've learned to write, you can write with a pen between
| your toes (crudely because of a lack of fine motor control),
| with a chisel in wood, with a spray can, men can write while
| peeing into snow, etc.
|
| > As an easy example, just counting beats is clearly just
| counting.
|
| Agreed.
|
| > Yet counting beats and aligning transitions/changes on or off
| a beat is a skill you have to work on.
|
| But that's not _just counting_ , is it? It's counting _and_
| aligning transitions/changes on or off a beat, so it requires
| detecting transitions/changes while counting.
| taeric wrote:
| I'm not clear that writing with different tools is really the
| same as writing in different contexts? Writing in a different
| context would be that you have learned to write your letters,
| but now you are learning to write poetry. Different kinds of
| poetry, even.
|
| My point for aligning changes on beats is that you are
| learning to count a mixed radix, effectively. We don't teach
| it that way, anymore, as positional numbers have grown to be
| what many of us think of as numbers. But mixed radix counting
| is the norm in the world in ways that people just don't
| realize anymore.
| layman51 wrote:
| The idea of transferring knowledge being a separate skill
| reminds me of the Wason selection task[1]. I first learned
| about this in a course on education and it felt pretty
| shocking to see so many classmates struggling with the logic
| puzzle version of the question. But then if you set up the
| same task with a story about being a bartender then it
| becomes more straightforward to solve.
|
| [1]: https://en.wikipedia.org/wiki/Wason_selection_task
| timerol wrote:
| > Once you've learned to write, you can write with a pen
| between your toes (crudely because of a lack of fine motor
| control), with a chisel in wood, with a spray can, men can
| write while peeing into snow, etc.
|
| This is not accurate. Find a child who has just learned how
| to write an A, and ask them to write an A with their feet, or
| even their non-dominant hand. It will be just as hard as
| getting them to write a B. The connection between shape and
| motion is a relatively simple one, but your first attempt at
| writing a word with piss in snow is gonna look awful.
| Penmanship needs to be learned in the new context.
|
| A fun example I like to bring up involves yelling in foreign
| languages. Even if you have an impeccable accent in your
| second language, if you've never practiced talking loudly,
| the first time you need to order lunch over the noise of a
| passing subway train, your accent will entirely fall apart as
| you try to say the same thing, but louder. (Yes, this is a
| personal anecdote, with a passable accent in my second
| language, as opposed to impeccable.)
| jrm4 wrote:
| Completely unsurprising. (Former?) Math nerd and common core
| hater here; having watched myself and other kids go through all
| of this; beyond EARLY algebra, maybe earlier we literally should
| not teach any math that doesn't have an immediate and obvious use
| to children.
|
| (My biggest pet peeve is how Common Core teaches fun -- but
| unnecessary for learning -- math nerd tricks. I LOVE math nerd
| tricks, but they should be discovered independently and entirely
| optional)
|
| Now, the silver lining here is; we have a thing that does hit a
| lot of advanced high-school math easily. Just let them kids learn
| video game programming and be done with it.
| wordpad25 wrote:
| Advances math helps with developing logical thinking and in
| general makes you smarter.
|
| But I agree we could achieve same results by teaching more
| practical skills. Maybe, programming (data structures and
| algorithms) over calculus?
| SamoyedFurFluff wrote:
| I would say data structure and algorithms are not more
| practical to actual literal children. Better off letting them
| cook/bake, where they have to work out ratios, temperatures,
| giving one third more or quartering other portions, etc etc.
| let them do scoring, figure out how to split 300g of
| chocolate between 5 friends and the like.
| MajimasEyepatch wrote:
| Calculus is so important for understanding the natural world.
| Being able to reason about rates of change is a valuable
| skill.
|
| That said, spending a bunch of time memorizing stupid
| identities to compute integrals is probably not the best way
| to teach introductory calculus.
| aidenn0 wrote:
| I can't find it now, but there was a blog article from maybe
| 15-20 years ago where the author, a math educator, lamented how
| students seem to put math in a "math box" and completely divorce
| it from reality.
|
| One example given was a word-problem in which a round-trip was
| involved and the student complained that there was no way to know
| they had to double the time, but then demonstrated obvious
| knowledge of how this worked when asked a real-world question.
| timerol wrote:
| Are you thinking of Lockhart's Lament?
| https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician'...
| aidenn0 wrote:
| That is not it, but was a good read, thanks!
| aidenn0 wrote:
| My second daughter (16 years old) can bake (including e.g. making
| a 50% larger amount than the recipe calls for). She can beat me
| at games where reasoning about probabilities and numbers is
| involved. She can relate exactly none of that to what she is
| learning in her algebra classes.
| empath75 wrote:
| This really has nothing to do with pedagogy and everything to do
| with the difference between calculations and mathematics. It's
| similar to the split between science and engineering.
|
| In engineering, you're applying rules of thumb to get some
| desired outcome. Those rules of thumb may be based on math or
| science, but they could also be practices that have worked
| before. You don't have to understand why the rules of thumb work
| to apply them, although it is helpful, if you do.
|
| What those kids are doing is something similar in the world of
| math. They have a desired outcome -- getting the correct answer,
| and they have a memorized list of "tricks" that get them to the
| correct answer quickly. They may or may not know why they work,
| only that they do.
|
| None of that is really helpful for learning math in a school
| setting, which is not just about getting the correct answer, but
| about understanding why the answer is correct, and understanding
| why those various rules work or don't work. They don't usually
| even teach mental math tricks in schools because the point isn't
| being able to calculate quickly. Probably having a catalog of
| addition tricks makes it even harder to learn basic school
| arithmetic because you have to ignore what you already know to
| work back up from first principles.
| inglor_cz wrote:
| Even as a maths professional (20 years ago), I had trouble
| programming the same abstract algorithms to be fast and generally
| usable. It is ... non-intuitive for me.
|
| That is why I always admired work of people like Peter Montgomery
| or Donald Knuth.
| anon291 wrote:
| Arithmetic in particular is from India. Indians have a long
| cultural history of using arithmetic. Even today, my mom who is
| categorically not a technical person, will randomly use a
| computational technique I'd never seen before to do her sewing,
| cooking or measuring. My grandmother, barely educated, did the
| same thing. They were extremely good at daily calculations. To
| the point where whatever I picked up off of them helped me win
| several 'fast math' competitions as a kid in LA. I'm not
| particularly good at arithmetic.
|
| My guess is that even a not really educated / barely literate
| Indian kid will be able to do arithmetic, simply due to cultural
| knowledge.
|
| The schoolchildren do well because they've been exposed to logic
| as well as arithmetic, and also given a systematic overview of
| the whole thing, whereas the market kid is just applying whatever
| was passed down culturally.
|
| However, I don't think we can necessarily transfer this study
| outside of India to assume that kids from other areas who are
| forced to work in markets will be good at arithmetic either. In
| my opinion, this is a unique cultural condition in India which
| reflects arithmetic's development there. Indian numerals and
| arithmetical algorithms were first introduced to Europe in the
| 13th century. The first records of their use in India go back to
| the first century AD. Thus, it's had a lot more time to 'soak in'
| to the common culture. I think it would be a good exercise in
| 'ethnomathematics' (is that a thing) to document all the
| algorithms used by the various laborers / uneducated productive
| workers in India. There'd probably be a few gems there.
|
| My two cents; maybe a controversial take.
| locallost wrote:
| It means almost everything is a skill that needs to be learned.
| Thus being successful in a school setting could mean one has
| successfully learned how to succeed at exams, but not really
| learning anything useful. I recently needed to take a test for a
| license of some kind, and while preparing with a set of questions
| noticed I can pick the correct answer without even reading. The
| correct answer was always much longer than the wrong answers and
| much more formal which was obvious by even just skimming.
|
| It also means the differences between people and their potential
| are in general for the vast majority of people small. Truly
| gifted or special is rare, as is truly giftless.
| Workaccount2 wrote:
| My pet theory is that mathematics is largely taught by people who
| enjoy math, and who see math as a puzzle game with rules you
| follow to solve puzzles. Kind of like people who are addicted to
| sudoku or wordle.
|
| Therefore it is taught from this totally abstract perspective and
| just hooks others who like the game of math. Whereas I think math
| would have a much greater impact if it was taught from an
| engineering or science perspective, where math is a tool used to
| explore the world rather than as something that would be a game
| you find on the back page of the newspaper.
| chilmers wrote:
| Here's the thing: I bet if you were to take the children who
| performed well at complex abstract mathematical problems and
| placed them in a job that required complex applied mathematical
| skill, the ones who _eventually_ performed best would be the
| same. And vice-versa for the children who excelled at complex
| applied mathematics. So mathematical skill might not be
| immediately transferable, but would be a useful indicator of
| innate ability.
|
| This relates to the (somewhat controversial) economic theory that
| much of higher education is _not_ about training or creating
| transferable skills, and instead is about "signaling".
| Essentially, we place children in simulated quasi-work settings
| and use it to determine who has the most of certain economically
| valuable abilities. By performing well at abstract mathematics, a
| child signals that they have the requisite mental ability to
| tackle similarly complex "real world" problems. But also, they
| signal that they are capable of obeying authority, working with
| diligence, delaying gratification, etc.
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