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Many elementary teachers don’t understand math, and it makes them anxious

• _bxg1 7y ago
  I still don't know why long-division works. I don't think I even remember how to do it as an adult. I've always had a very intuitive grasp of math, which meant I excelled at solving things in my head but struggled with (and resented) having to show my work in the arbitrary algorithms we were taught. I understand division despite my education. They really should try to come up with algorithms that are more intuitively linked to a spatial comprehension of the numbers. That underlying grasp is really all that sticks with you into the real world (and it's also what makes math engaging). If students aren't getting that, they're wasting their time in my opinion.
  • novok 7y ago
    Division is a repeated subtraction operation, and the long division algorithm is just a nice way to lay it out. Multiplication = repeated addition Addition = a form of counting
    • threatofrain 7y ago
      I prefer to think of subtraction as the inverse of addition, and division as the inverse of multiplication. You only need addition and multiplication if you're dealing with rationals.
    • _bxg1 7y ago
      That makes sense. And actually I think I could re-derive the algorithm from that explanation. Maybe they need to explain it in those terms. Or use pictures.
    • dboreham 7y ago
      I've never thought of division as repeated subtraction: it's the inverse of multiplication, or literal division as in I have a pile of beans that I divide equally between N piles. I don't think you're wrong, just seems an odd way to think about it, and fails to account for any remainder.
      • pjscott 7y ago
        The remainder is what's left after you've repeatedly subtracted, and can't subtract any more before going negative. (I wish that one of my teachers had said this to me, but they were too busy getting everybody to memorize the steps.)
  • alfiedotwtf 7y ago
    It's weird because I was taught long division in primary school that was different to everyone else in class (different maths class). Everyone seems to do some weird subtraction for each digit, whereas I was taught to carry right (i.e almost like how carries work with addition). Way faster, and it works the same all the way to decimal places, no weird remainders. Not sure why that wasn't the standard approach.
    • alfiedotwtf 7y ago
      Copied from comment below... Oh :) I just worked it out - I was taught to fold the subtraction and carry right in one step. I'm an idiot!
      • overlordalex 7y ago
        From elsewhere in this thread I learned that this is actually called "short division"! I can distinctly remember learning long division as a kid, and I wonder why this isn't immediately taught afterwards as the logical progression
    • deleted 7y ago
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  • pjscott 7y ago
    My parents invented a simplified division algorithm so they could teach me the four basic arithmetic operations by third grade, and it was so much easier than long division that I could never remember how to do things the conventional way. I just pretended to follow the teachers' instructions for the next few years. (All these methods had the same asymptotic complexity. Mine was slower by a small constant factor; on the other hand it actually made sense. It was just repeated subtraction!)
    • ravenstine 7y ago
      My mom got in trouble with the teacher for doing that kind of thing with me.
      • paleotrope 7y ago
        Your mom didn't get in trouble she just exposed something that the people with PhDs in Education don't want exposed.
      • pjscott 7y ago
        Teachers are a very mixed bag. Some of them are delighted to see a student learning, in whatever way that learning may happen -- and some are just annoyed that the classroom's reality fails to match up perfectly with their lesson plans. If you're a nail that sticks up, some will try to uplift you further and others will angrily try to nail you down. (And most are somewhere in between.)
    • _bxg1 7y ago
      I repeatedly got in trouble for not showing my work in the format they wanted, whether because of intuitive leaps or because of tricks I came up with myself.
      • anongraddebt 7y ago
        This worked for me up until Algebra 2. Up through Algebra 1 I'd just do everything in my head, never show my work, and thus solved a lot of problems using non-standard/creative methods on the fly. Ended up hurting myself a lot. I didn't have any muscle memory for solving the trivial parts of problems as I progressed from Algebra 2 through BC Calc. This led to me choosing a non-mathy degree in undergrad. I could breeze through higher-level logic courses, but was essentially remedial for higher-level math.
  • geomark 7y ago
    I don't understand why long division would be a mystery. It's just repeated subtraction - you are finding how many times you can subtract the divisor from the dividend, in a systematic way.
    • pjscott 7y ago
      It's almost never explained with such lucidity. If you find a hundred people who can perform long division, I would be surprised if twenty of them have any idea how it works.
      • dboreham 7y ago
        You're never going to find 100 people who can perform long division.
        • pjscott 7y ago
          Sure you can! Just look at a typical fifth- or sixth-grade classroom. Most of them can perform long division, briefly, before the test. They may even remember it for several days after the test.
          • mavhc 7y ago
            What none of them can tell you is why they've been taught long division, when short division is faster. And it's because you need long division to do division in algebra
            • abakker 7y ago
              What is short division? I only remember being taught long division and having a calculator. Is there another method?
              • Tomte 7y ago
                Fascinating. I didn't know, either. After looking at the English Wikipedia I found that a) what I learned in school is long division b) the notation with the dividend and the divisor separated by a parenthesis and the result on top of a horizontal bar is totally alien to me c) I've never seen short division before d) the German Wikipedia doesn't even have an article about short division
  • deleted 7y ago
    [deleted]
  • keenmaster 7y ago
    That’s a good point. You know what, it’s way easier to teach how to solve for x when y + x = 3x + 3 and y = 4 than it is to teach long division. As far as I could tell, that basic algebra is also much more important than long division, so why not teach it at an earlier age? The symbology of algebra is probably great for developing minds. Just show long division for “proofs” while introducing the concept of division, and let kids use a calculator for division after that (supplemented with calculator-free drills). That way there will be more lesson time for advanced topics. Same thing with long multiplication. Show the proofs using long multiplication, then have kids do progressively harder multiplication drills where all the math is done in their head. In fact, kids might grok first principles better that way.
    • LegitGandalf 7y ago
      Some guys in Norway made an app that teaches algebra concepts to 5 year olds using pictures, as the game levels up it slides variables and numbers in.
      • keenmaster 7y ago
        Are you referring to Dragon Box? This looks awesome. https://apps.apple.com/us/app/dragonbox-algebra-12/id634444186 https://apps.apple.com/us/app/dragonbox-algebra-12/id6344441...
        • LegitGandalf 7y ago
          Yeah,it's amazing, I've hooked multiple friends on this app. It turned my friend's girls math fear around, which is really important given that girls are learning to fear math from teachers who are themselves afraid of math. https://news.uchicago.edu/story/female-teachers-can-transfer-fear-math-and-undermine-girls-math-performance https://news.uchicago.edu/story/female-teachers-can-transfer...
        • copperx 7y ago
          You linked to the iOS version. Do you have a link to the app webpage, which will surely have both Play Store and App Store links?
  • buzzkillington 7y ago
    You should probably read about it as an adult: https://en.wikipedia.org/wiki/Long_division https://en.wikipedia.org/wiki/Long_division Children are mostly idiots. Education is the process of painfully breaking them out of their idiocy. Bonus points if you implement a long division algorithm that works on strings in your favorite language.
    • _bxg1 7y ago
      Thing is, even now I find it useless. I can do nearly any division in my head whose digits themselves fit in my head (Miller's Law). For anything bigger, I have a pocket supercomputer with me at all times. You're right that it could be a fun programming exercise, though
      • buzzkillington 7y ago
        If you do code in any capacity not being able to understand long division is a huge red flag. In terms of difficulty it's somewhere between regex and being able to write a recursive function.
        • _bxg1 7y ago
          I didn't understand why it worked (a comment here pointed out that it's just division-by-repeated-subtraction which does make sense now). But I was still able to perform it, during the time when it was relevant to my life.
          • copperx 7y ago
            What's a red flag is being frustrated by long division and at the same time not being curious or resourceful enough to resolve your frustration. You blame your lack of understanding on other people. I can understand if most people shared your frustration, but not a PhD holder. Your degree basically says that you're capable of doing long, intensive research on arcane topics. What am I missing here?
  • jknz 7y ago
    Dividend is 123456 and divisor is 37. 37>1, 37>12 but 37*3<=123. Write down 3 (meant for 3000) and start again with 123456-37x3000 as the new dividend. I wish the algorithm was taught by writing the full 3000 instead of only the digit 3; this appears to be a source of major confusion and to hide the underlying additions/multiplications.
  • DanBC 7y ago
    > They really should try to come up with algorithms that are more intuitively linked to a spatial comprehension of the numbers. Have you heard of "chunking"? This video shows old style long division (and you're right, the method is confusing). It also shows an easier method that is based on understanding numbers rather than just applying an algorithm by rote. https://www.youtube.com/watch?v=K2rBaSj2L74 https://www.youtube.com/watch?v=K2rBaSj2L74 Here's another explanation: https://www.theschoolrun.com/what-is-chunking https://www.theschoolrun.com/what-is-chunking
  • cjfd 7y ago
    Not really something to be proud of. You basically have just admitted on a forum for programmers that you don't have very much insight in algorithms.
  • layoutIfNeeded 7y ago
    Long division is just polynomial division with x = 10.
    • nathanwh 7y ago
      This comment is hilariously on the nose for how a lot of math texts read (at least in my experience) $SIMPLE_CONCEPT that you’re trying your best to understand is easy if you think of it as a specific example of $MORE_COMPLICATED_CONCEPT that you surely can’t understand without understanding $SIMPLE_CONCEPT
      • jiggawatts 7y ago
        Wait till you get into Theoretical Physics, where the same thing is being explained by people who don't even have a firm grasp of $MORE_COMPLICATED_CONCEPT, but know just enough to be dangerous with it. E.g.: I distinctly remember that many of the original articles about the Amplituhedron spent like a page explaining what a triangle was, then jumped straight into functions over higher-dimensional analogues without explaining any of the intermediate steps. Something-something-logarithm-of-a-simplex. I can grok what a simplex is, but they literally just hand-waved over the bit where they take a logarithm of it in some way. The page of explanation of what a triangle is was of zero assistance.
      • jeltz 7y ago
        That was quite an eye opener though when we were taught polynomial division in high school. Not necessarily for how long division works because I already understood that, but because it was an example where short division could not be used.
  • username90 7y ago
    This is the problem with rewarding conscientious kids, they just follow instructions without ever asking themselves why. If you were more rebellious and refused to follow steps you didn't understand then you'd get worse grades but you'd learn more.
  • cptskippy 7y ago
    Then you never knew it? Perhaps what you were taught wasn't long division. My kids were not taught the same math that I was taught in the 80s. My 17 year old was struggling with the division algorithm they were teaching in Elementary school at the time (late 2000s). I can't remember the details but it was something I had never seen before that was a step up from throwing poop at the wall. I showed her how to do long division and she still uses it to this day. It's simple and there is a history of work to easily identify errors. The homework scene from The Incredibles 2 really hit home for me. I feel like book publishers development new methods or ways to teach the methods as a means to sell new editions of books, not because the new ways are better. This causes a rift in society because people no longer share a common understanding of a subject.
    • kasey_junk 7y ago
      Conversely I have been immediately impressed with the way my first grader is being taught math. It’s completely different than the way I was taught so I have to invest in figuring it out but it’s led them extremely quickly through multiplication & division and into algebra. I’d characterize it as “multi-algorithmic”. They come at it from a variety of algorithms, assumedly so that each student can find one that works for them. As someone who had to teach themselves a different algorithm than I was taught many painful years later this resonates. Perhaps the rift was always there and you ended up in the lucky side of it?
      • thawaway1837 7y ago
        I remember when the common core debates were taking place, one of the memes running in parenting circles was a comparison of long division vs how common core taught division, and how much more “complicated” the common core version was. But in reality the common core way explained the actual algorithm and the reason it worked so much better, and further, the only reason it appeared more complicated was because the people who were targeted by the meme already knew long division. If you took someone completely new to both methods, they’d find the more “complicated” algorithm far easier, because a large part of its complexity was explaining why the division worked, whereas long division was about memorizing a process.
  • kasey_junk 7y ago
    I was taught math via spatial properties and it wasn’t until I came back to it from a logical perspective that I was able to understand it. Just another anecdotal data point but my experience was completely opposite yours.
    • _bxg1 7y ago
      True; part of the problem is that everyone is different.
  • dannypgh 7y ago
    > I still don't know why long-division works I was taught long division in the 4th grade (age 9) but wasn't taught a proof of why it works until a college algebra course (age 19). For whatever it's worth a different type of long division is taught now under common core. The common core math stuff that I've seen is impressive - there is more focus on teaching things in a way where comprehension of deeper concepts should be possible. It's a shame there's so much fear of it from parents who are upset kids are learning differently than they did.
    • brianwawok 7y ago
      Yes I love the Facebook memes by parents talking about how math is so complicated now. Most American adults struggle to solve a basic algebraic equation, because it’s not used in day to day life. If a new teaching style gives us a better way to teach concepts, seems worth a try..
  • tasty_freeze 7y ago
    I think one reason why it doesn't make sense to a lot of people is that when doing long division, it is taught with notational shortcuts that obscure what the student is really doing. Eg, say you are dividing 76543 by 5. The normal procedure is to say something like: - 5 goes into 7 once. The first quotient digit is 1. - 1 times 5 is 5, write that under the 7, and draw a short line under the five - 7 minus 5 is 2; write that under the line you just drew - the next digit is 6, so write that next to the 2 you just drew, ie, 26 - 5 goes into 26 five times; put that up in the quotient, etc That minimizing the writing, but it hides what is really happening. Really what is going on is this. How many times can you subtract 5 from 76543? One procedure would be to subtract five over and over until the remainder is less than five. The number of times you subtracted is the quotient. But in this example you'd have to subtract more than 15000 times. Instead of subtracting 5 at a time and incrementing a count, it would be faster to subtract 50 at a time and increment your quotient by 10 until the result is less than 50, and then switch to subtracting by 5s and incrementing by 1 as before. If you can see why that works, then you realize it would be even faster to subtract by 500 at a time and increment by 100, then switch to subtracting 50 at a time and increment by 10, then subtract 5 at a time and increment by 1. Etc. So rather than minimizing the amount of digit copying, it would be better at first to write everything out. In this particular case: - write the dividend: 76543 - remove groups of 50000. we can see we can subtract 50000 from this once; the leading digit of the quotient is 1 - write 50000 (150000) under the 76543 - subtract, writing 26543 (this is the current remainder) - remove groups of 5000. now we see we can subtract 25000 from this, so the next digit of the quotient is 5. - write 25000 (55000) under the 26543 - subtract, writing 1543 (current remainder) - remove groups of 500. 500 goes into that 3 times, so the next digit of the quotient is 3 - write 1500 (3500) under 1543 - subtract, writing 43 (current remainder) - remove groups of 50. it is larger than the current remainder, so the next digit of the quotient is 0. - remove groups of 5. 5 goes into 43 eight times, so the next quotient digit is 8 - write 40 (85) under the 43 - subtract. the final remainder is 3. The sequence of quotient digits was 15308. The standard way long division is taught, the child skips writing down all the digits of the remainder and it is confusing as to why this all works.
• alfiedotwtf 7y ago
  On the flipside, I was in a Calculus lecture at uni and turned to the girl next to me who was crying. I asked her if she was ok, and she said she didn't understand why she had to sit in the lecture learning Laplace transforms when she "only wanted to be a primary school teacher". From the sounds of it, it was a prerequisite? Edit: Oh :) I just worked it out - I was taught to fold the subtraction and carry right in one step. I'm an idiot!
  • analog31 7y ago
    Seems unlikely, though it may vary from one country to another. Here in the states, you would probably need to take a year of math, but there are levels of math "below" calculus that you can take to get a K-12 general education certificate. Also in the states, engineering students ask the same question about Laplace transforms. ;-)
    • buzzkillington 7y ago
      >Also in the states, engineering students ask the same question about Laplace transforms. ;-) Because I want you to build things that don't fall apart because of resonance?
      • analog31 7y ago
        I'm playing devils advocate here, but the argument you'll get is that there's software for that. Also, in a typical department, there may be one or two engineers who handle the quantitative problems for the team. Many engineering students go through their education, having heard from someone, that they won't need their math after they graduate. And things do fall apart.
        • buzzkillington 7y ago
          >I won't need math, so I'll blindly trust this 20 year old 32 bit half precision floating point thing someone coded up in an afternoon before their holiday break. Don't get me started on engineers who think floating point numbers can be treated like real numbers for equality testing.
• acollins1331 7y ago
  I feel like anyone should be anxious if they don't understand math. How does one get through life without understanding math? Probably by getting had a lot.
  • analog31 7y ago
    Math is taught so badly, that it may seem more complicated than it really is. For instance, many students come away from their math classes, convinced that math is subjective. They believe that, like in English class, the correct answer is the one that the teacher wants to see, and that there is a "trick" for producing this answer. I saw this, teaching college math for one semester long ago, and also in talking with adults about their experience in high school and college math. As a consequence, a person may be completely functional at basic math, such as balancing their checkbook, while simultaneously believing that they don't understand math.
• optimiz3 7y ago
  Routinely we have to explain to users in a crypto related business that 0.00835 Satoshis is the same as 0.008350 Satoshis. Adult humans don't handle decimals.
  • tmh88j 7y ago
    That reminds me of A&W killing off the 1/3 lb burger from their menu because people didn't realize it's more than a 1/4 pound and thought they were getting ripped off. https://www.mentalfloss.com/article/76144/why-no-one-wanted-aws-third-pound-burger https://www.mentalfloss.com/article/76144/why-no-one-wanted-...
  • xiphias2 7y ago
    Why does it even matter? Sub-Satoshis are used only as transaction fees in lightning network, and even there 1 milli Satoshi is the smallest amount that's allowed for now AFAIK.
    • optimiz3 7y ago
      It's a hypothetical example based on actual customer cases. Change the units to BTC/LTC/whatever if you'd like to make it more precisely simulate the user interaction.
• dnprock 7y ago
  The author of this article does not understand math :). Learning math is difficult. There're two things you need to learn: fundamental concepts and skills. These two things are related and commingled. Mastering both requires extraordinary effort. I come from a skill-focused math education. I was pretty good with skills. But I didn't understand fundamental concepts. I still remember my struggle with prime numbers. Those problems were the hardest for me. I didn't understand why prime numbers are popular. In college, I discovered the meaning of prime numbers and their applications. Then, I was reading a lot about prime numbers just for fun. My kids are now learning in the American education system. I feel like there is a shift to skill-focused math education. I suppose American educators feel like we need to "catch up" with other countries in terms of test scores. I find those objectives silly. Comparing test score is very easy. Gauging concept understanding is much harder. I myself will encourage my kids to understand the fundamental concepts.
  • barry-cotter 7y ago
    Being able to mechanically perform an algorithm is the first step on the road to in some sense actually understanding it. If you think you understand the fundamental concept but you can’t solve a problem using the algorithm you probably don’t. > Young man, in mathematics you don't understand things. You just get used to them. - John von Neumann
    • buckminster 7y ago
      And the best way to test you really understand all the details is to teach a computer the algorithm.
• elchief 7y ago
  My ex taught grade 6 geography. Is a principal now. Thought the seasons were due to earth tilting back and forth. Didn't get that the tilt was the same, but the position around the sun was different
  • BubRoss 7y ago
    That doesn't seem so bad in the grand scheme of things.
    • tbihl 7y ago
      The particular non-fact is unremarkable, but the lack of inquisitiveness is off-putting. It falls down after not so many "why"s. You'd like to imagine a greater love of discovery in the people who are (hopefully) shepherding dozens of kids through their own discoveries, day in and day out.
  • dboreham 7y ago
    Just a reference frame change right?
    • 6gvONxR4sf7o 7y ago
      In the same way that "the earth doesn't orbit the sun" is true, yeah it's just a reference frame change.
      • dboreham 7y ago
        Invariant under Lorentz transform.
  • Gunax 7y ago
    While I am also appalled, I imagine everyone has some fundamental misunderstanding about something that is 'obvious' to everyone else. Something akin to 'today's 10,000"
  • cultofmetatron 7y ago
• dboreham 7y ago
  In my kids elementary school time. I had the interesting situation that while the teachers weren't strong in math, their boss the principal was an actual mathematician (or at least he had a real degree in Mathematics). Generally principals seem to have a psych background but not this one. Good times!
• LegitGandalf 7y ago
  The worst part is that female students are more likely to notice the teacher's fearful body language, so you get this never ending cycle of girls developing fear of math, going into teaching and passing the fear on to the next generation.
  • _bxg1 7y ago
    That seems like a stretch.
    • LegitGandalf 7y ago
      University of Chicago study for you to peruse https://news.uchicago.edu/story/female-teachers-can-transfer-fear-math-and-undermine-girls-math-performance https://news.uchicago.edu/story/female-teachers-can-transfer...
      • wnoise 7y ago
        Even assuming the study holds up, that doesn't mean that only the girls picked up the fear. It means only the girls were affected by the fear, by overly generalizing from this one example.
        • LegitGandalf 7y ago
          Yeah it could be that the boys noticed and ignored the fear. The impact on the girls is troubling though.
• symplee 7y ago
  A couple examples from the article: >> A colleague told me of vainly trying to persuade a college student that .015 was less than .05; the student insisted “but 15 is more than five.” >> To [students], [the equals sign] doesn’t signify equality, but instead means “put the answer here.” Imagine their confusion when, in algebra, they first encounter problems with numbers on both sides of the equal sign.
  • hanoz 7y ago
    >> To [students], [the equals sign] doesn’t signify equality, but instead means “put the answer here.” Imagine their confusion when, in algebra, they first encounter problems with numbers on both sides of the equal sign. Is this really true? I'm sure I and my children (in the UK) were presented with fill in the box questions like: 7 + [ ] = 10 at the very earliest stages of arithmetic.
  • pbhjpbhj 7y ago
    Sounds like you can just say (as happens) we have to get the answer we're looking for on the "put the answer here" side. 7 + x = 12 "The 'x' represents the answer, so we have to get it on the 'put the answer here side'; lets write 12 = x + 7 ; now we just need to take 7 from the 'put the answer here side' and we're done. But whatever we do on one side we have to do on the other. Take 7 off. 12 - 7 = x + 7 - 7 => 5 = x ... bingo".
• aaron695 7y ago
  This is because of a lack of men in elementary schools, interesting they skipped the reason. But this is not solvable in the West I guess, it will continue to decrease. Need to offshore it. IE. Produce good external lesson plans that are free to use.
  • _bxg1 7y ago
    ...yikes.
    • LegitGandalf 7y ago
      Ha,I see why you flipped an exception on my comment now. Let me assure you that I'm not attributing the decline of Western Civilization to women teaching math! Did you read that Chicago study? I'm interested to see if your view shifted any.
• blt 7y ago
  If we are serious about trying to "break the cycle", one obvious solution is to pay teachers more. Imagine if teaching paid as well as software development.
  • zamfi 7y ago
    > Imagine if teaching paid as well as software development. I’d love this personally. But the organizations that employ teachers do not make the same amount of money, per employee, as organizations that employ software engineers. Depending on your state, your property taxes would need to double or triple to make the math work out. To be clear: I’m not making a comparison of the value provided, just the accounted-for profit. Society as a whole would obviously benefit from exceptional math teachers at every school, but we would need to get creative to pay for it. Perhaps we should be spending less on wars.
    • rocqua 7y ago
      Its weird that education is payed for completely by property taxes in the US. The benefits of education go to more than just the (parents of) the recipients of that education. You also get weird incentives for people to reduce these taxes if they send their children to private schools.
      • war1025 7y ago
        I believe the idea of schools being funded by property taxes is that you then spread the cost load out over people of all age groups, in a way to acknowledge that having educated children is a benefit to the community.
      • barry-cotter 7y ago
        > Its weird that education is payed for completely by property taxes in the US. This isn’t true. From the CPC Popular Front funded Brookings Institution. https://www.brookings.edu/research/how-progressive-is-school-funding-in-the-united-states/ https://www.brookings.edu/research/how-progressive-is-school... > School funding systems vary by state, but in the typical state, the financing of K-12 education is shared roughly equally by the state and local governments (with the federal government covering less than 10 percent of the cost). Most states have provisions designed to target additional funding to districts attended by low-income students, with the goal of making up for these districts’ limited capacity to raise revenue through property taxes and other local sources. > Nationwide, per-student K-12 education funding from all sources (local, state, and federal) is similar, on average, at the districts attended by poor students ($12,961) and non-poor students ($12,640), a difference of 2.5 percent in favor of poor students.
• rahuldottech 7y ago
  Huh. I'm from India, and from what I can tell, maths is taught much better here than in the US, at least till 10th grade, after which it's no longer compulsory. It's actually taught in a very intuitive manner, and most students "get" it quite well. Edit: the textbooks are freely available on the web, if anyone wishes to take a look. Look up "NCERT class 10 maths textbook".
  • Gunax 7y ago
    Im my experience there is so much variation between countries, between different regions within the same country, between schools within the same region, between teachers within the same school and even students' subjective experiences within the same classroom that I am generally skeptical of broad conclusions drawn between countries like that.
  • kkarakk 7y ago
    My experience has been the opposite. Indian kids can solve math problems in the "curriculum" quite well but ask them to do anything WITH the math and they're lost. Ask them how the theorem relates to the natural world and they don't know. Ask them what applications it has and they don't know. Ask them to extrapolate their current knowledge base to a theoretical future and they are completely lost on what a "next step" would look like. Information is laser focused on getting into an "Engineering college" and not actually on "knowing" anything. It's like teaching someone about spoons and forks but never giving them any food to actually interact with but vaguely mention "food will be relevant when you get a job, not right now" Better than american kids sure but i'm not sure anything more of value is earned by making these kids essentially memorize theorems and algorithms
    • y4mi 7y ago
      Try asking the average western student the same question. Most people -be they young or old- aren't able to do that. Nor is that a skill you actually need in real life. You should also think about if it's more important (to the people you're telling that message) to get a good wage or if they're fine taking a few years to catch up. "Understanding" concepts is great and all, but unless they're doing that additionally to training the simple application, they'll most likely start having issues once the tests are up. Having good grades (and being able to prove their ability on the spot) is definitely correlated with good starting wages. Not so much later on, but it's a massive difference at the start.
• Gunax 7y ago
  This is a consequence of refusing to acknowledge math as a required course. Teachers and schools loathe to fail a student--especially one who is actually displaying effort and is otherwise a good student. My high school had an elaborate system of ways for people to avoid doing math. Even though Algebra was technically required to graduate, there were enough loopholes and alternative classes that students could effectively skip learning actual math. Fail an exam? The teacher would give extra credit: presenting about a famous mathematician--certainly interesting work, but does not really show that the student actually understands the core math material. This is how you get incoming first-year undergrads who don't know how to reduce fractions even though their curriculum indicates they should have learned it multiple times over. They have managed to avoid jumping through the hoop so many times because we gave them a rope to skip instead.
• Gunax 7y ago
  > If math knowledge is limited to “when I see this sort of problem, I do that” it’s inflexible; if problems are phrased just a little differently, the student is often stumped. And once students hit algebra, math increasingly demands thinking through a sequence of steps and picking the right mathematical tool for each. That’s hard if you don’t understand what the tools actually do. That's easy to say, but hard to do. I find in my math experiences that I often don't truly master a concept until I am a level or two beyond it. I learned fractions when I was in grade 5 or 6 and could demonstrate adding, multiplying, reducing, etc. I think if you asked me 'why can you multiply across when multiplying fractions' I may have said I knew, but I don't think I actually mastered that until high school--when I had the skills to derive the fraction operation rules. Ditto with things like FOIL, completing the square, etc which I don't think I mastered until I was out of high school. And of course, everything seems easy and obvious in hindsight. I suppose what I mean is: I don't think there is anything new about what the author lauds as a goal. It's what everyone has been trying to do for 50 years. It just isn't easy, and so far no one has found a magic bullet aside for lots of hardwork and practice.
• BrandoElFollito 7y ago
  I have a hard time explaining to my children how addition of fractions work. Not how to do that, but how to feel the intuitive way. And percentages, to some extend. The sad part is that I have a PhD in physics and an engineering degree in CS. I used to teach physics at the uni and loved it. My children like me to explain them physics because I love it so much and they appreciate the analogies, their limits etc. It is just these freaking fractions and their convoluted addition I do not know how to properly convey.
  • ezequiel-garzon 7y ago
    To me what is utterly crucial is to use pie pictures to convince children (and adults...) that 1/2 + 1/3 cannot possibly equal 2/5. If they actually buy this that’s a pretty good inflection point in their education on fractions.
    • BrandoElFollito 7y ago
      After going though the education of two children who like math very much I realize that fractions on their own are very natural. It is their addition which is not, as it does not have any obvious counterpart in simple nature. I had a math teacher at the university who told us once "there is a neat trick which is normally taught next year but is useless by then and super useful today. Just use it without too much thinking." This is my reasoning too, I showed them the always working way to add fractions (cross multiplication) and hope that someday they will appreciate the beauty of the thing.
      • sethammons 7y ago
        As a former high school math teacher, chances are you did a disservice to them. Many kids never "get" fractions, but do remember cross multiplication! And so they use it for anything with a fraction. 1/3 + 4/5=? CrossMiltipy! 1/3 x 4/5=? CrossMiltipy! 1/3 - 4/5=? CrossMiltipy! 1/3 ÷ 4/5=? CrossMiltipy! 1/3 + ? = 4/5 ... CrossMiltipy? Lots of kids complain about math because there is so much to remember. That's a bad sign because there is so little to remember if you understand the concepts as you can always re-discover forgotten tricks. In similar fashion, as a little kid, I was not allowed Velcro shoes when learning to tie my shoes. After I showed I could master tying them, then I had the option to go back to the easier Velcro option.
• projektir 7y ago
  Wonder why math is such a problem. Growing up in the Ukrainian school system I was doing very well in math for 3 years and was doing double exams in the span of one. Moved to the US, math was easier and it kinda fell off. Was back to Ukraine for some time, but it was never the same afterwards, and now I have the same math anxiety many other people do.
• jimhefferon 7y ago
  I am a math professor. I teach in a SLAC, St Michael's College in VT. In our state education majors must also major in another subject (on an approved list). Every year we graduate one or two future Elementary Educators who are math majors. They take the same courses as any other math majors, and do just as well. One thing it helps is that they are quite employable since their resume stands out.
• nopinsight 7y ago
  For over a decade, I have run a successful tutoring center focused on math and science for elementary school students in a developing country. Our students won numerous awards both at the national and international levels. We usually employ teachers with degrees in math or science, or sometimes engineering. Occasionally for early elementary levels (grades 1-3), some teachers have a degree in another discipline. Regardless of major, we always test them for math aptitude. The test includes competitive exam questions focused on mathematical understanding and problem solving skills, rather than advanced math knowledge. (The other part of teacher selection is interviewing for teaching skills and trial teaching with student and expert evaluation.) Somewhat surprisingly, even some PhD candidates in engineering may fail these tests, while a few liberal arts degree grads passed the lower levels of these tests. So math aptitude is not limited to those who majored in math, science, or engineering. (Note that many of our elementary school students who have studied with us for a couple of years also pass them at a high level.) It seems that countries with successful math programs, like Singapore, also utilize a rigorous teacher selection process that favors subject matter specialists even for teaching at an elementary level. This is crucial for math, which is particularly hard to catch up later on with shaky foundation. (In fact, kids are great at absorbing their teacher’s attitude toward the subject. It is no surprise many kids may sense “math fear” from their teacher. Having a teacher with positive attitude toward the subject they teach is essential!)
  • leetcrew 7y ago
    I always found it amusing that most of my math and cs professors in college seemed to have terrible arithmetic skills. I'm sure part of it is just that it's hard to concentrate on the work when you're also narrating the process to the class, but I suspect it's also that basic arithmetic is a relatively unimportant skill in most advanced research. hiring this sort of person to teach/tutor grade school math might be like hiring a porsche engineer to repair your car. they'll get the job done, but maybe not as efficiently as your local mechanic.
    • scruple 7y ago
      "My degree is in Mathematics, not arithmetic." was a common one that I'd hear from friends with Math degrees or in the Math department.
• roenxi 7y ago
  > But many elementary students don’t understand the meaning of the equal sign. Some people might not have made the connection, but by and large programming languages make this mistake too. All the major ones are using the = glyph for assignment.
  • purple-again 7y ago
    What mistake is that? The left hand variable equals the right hand value assigned to it. Foo = bar and 1 = 1 are equivalent in my mind.
    • pbhjpbhj 7y ago
      So 2 = 1? How do you square that circle, "fine" for assignment; wrong for equality.
    • ken 7y ago
      The mistake is assuming "=" is a statement of fact (as in mathematics), rather than a request for assignment (as in programming). Depending on which programming language you're using, "foo = bar" can mean either "make foo now equal to what bar is" or "is foo equal to bar now?". Neither one is the same as in mathematics, which is "foo is categorically equal to bar". Consider: foo = 1 bar = 2 foo = bar In mathematics, this would be a logical contradiction, and either you'd realize you made a mistake, or you'd jump back to the step where you said "now assume ..." and say "now we've proven this can't be true". In programming, this is perfectly valid, and means "ok, ignore what I wrote just 2 lines ago, and now make foo equal to 2, also". You say you think it means "the LH var equals the RH value assigned to it", but you don't say what you think it means to assign to the literal 1.
      • purple-again 7y ago
        Thank you for your well thought out explanation! It is clear now and I was committing the cardinal sin of speaking generally while assuming context that was explicit only to my understanding of the situation. Have a great day!
  • 7y ago
• olooney 7y ago
  Perhaps base 10 is the problem. In base 2, you can learn the "multiplication table" in 10 seconds (0 X n=0, 1 X n=n, we're done.) The long multiplication algorithm for n and m is just multiply by n by the least significant bit of m, add to the accumulator, left shift, repeat. You can do subtraction by adding the two's complement (just flip every 1 to a zero and every zero to a one - trivial for a child to learn) instead of using the "borrowing" algorithm. You never have to "guess" when doing long division in base 2. All of these operations can be justified in terms of the Peano axioms with very short arguments: prove the associative and distributive properties, and the multiplication algorithm is obvious, etc. Arithmetic in base 2 is so easy, you can literally teach your pet rock to do it (assuming your pet rock is a reasonably pure silicon wafer with scattered boron and phosphorous impurities shot through with thin veins of copper.) And no, I'm not advocating teaching kids to convert between different bases, which they almost universally find confusing - I'm saying we should teach kids arithmetic purely in base 2, and only expose them to base 10 once they understand the fundamental concepts. Am I being serious? Well, it's certainly a lot easier to dismiss this approach out of hand then to pinpoint what - if anything - is actually wrong with it.
  • koonsolo 7y ago
    The real world depends on base 10, and so it's way more practical to learn calculations in base 10. I said calculations, not math. I think there is a big confusion between math and calculations. Kids learn calculations, not math. Learning the multiplication table is just a way of how to do quick calculations inside your head. I need to pay 35.15 euros, so how much do I get back? That is calculation not math. In my opinion the most practical thing that kids need to learn first. It's more about remembering than logic. When the get older, math comes into the picture. They can leave calculations up to calculators or computers, and learn the language of math. So in my opinion, there needs to be a more clear distinction between calculations and math, because they are definitely not the same.
    • pharke 7y ago
      Wasn't the premise of the article that the reason U.S. students perform so poorly in math in higher grades is because they are only being taught to memorize arithmetical facts and algorithms without properly learning the reasoning behind them? It seems you're prescribing the cause of the problem.
• spodek 7y ago
  Or: why do so few people who understand math teach elementary school? Once someone learns math, what incentives do they have to choose teaching elementary school over alternatives? I could imagine that the skills to learn math don't overlap with the skills to teach elementary school and that time spent one detracts from the other. If mastery in each takes a decade or so and few people dedicate themselves to double time in school for a lifetime in communities that misunderstand them, I could see teaching k-12 math as a career and life disaster.
  • dannypgh 7y ago
    I would love to teach math to elementary students but I make a base-10 order of magnitude more as a software engineer for a FAANG. I'd take a pay cut of a factor of 4 or 5 to do it in a heartbeat... but having to go back to school to get a teacher's certificate (and likely a master's degree) to get paid 1/10th as much is challenging. I may end up doing it in a couple of years when I'm even more solidly financially independent, but we as a society should probably just be paying teachers a lot more across the board. I guess there's more money in advertising than educating the next generation.
    • galangalalgol 7y ago
      Also consider FAANG pay 2-3 times more than other software development jobs. When the next bust comes some of these problems will be alleviated.
  • 6gvONxR4sf7o 7y ago
    The fewer people who are capable at something, the more lucrative it is to know that thing. The more lucrative it is to know something, the more of a pay cut it takes to teach it, assuming teacher pay at this level isn't subject dependent. The more of a pay cut it takes to teach, the fewer the skilled people willing to take a pay cut, and the fewer teachers there are with those skills. The fewer teachers there are with this skill, the worse the students turn out at it, leading to fewer people who are capable at it. It's a vicious cycle. I'm a data scientist in SF and I'd make 5-10x less if I taught grade school.
• scscsc 7y ago
  The author unfortunately has a misunderstanding of equality. Quoting him, "The equal sign is another mathematical concept that’s often misunderstood. It means, of course, that whatever is on either side of the equal sign is equivalent." This is actually wrong. The equals sign (=) is a shorthand for stating not that the two sides are equivalent, but that they are the same (i.e., they are equal). If they were just equivalent, we would use another sign, like ≡ (unicode U+2261).
  • adriand 7y ago
    (2 + 3) and 5 are not “the same”. If we’re using ordinary English language to discuss this issue I think “equivalent” is a sensible choice.
    • scscsc 7y ago
      As expressions, sure, they are not the same. But "2 + 3 = 5" is a statement about numbers, not about expressions. When you talk about numbers (the object of study), you do not (usually) talk about arithmetic expressions. Arithmetic expressions are part of the language used to talk about numbers. When talking about arithmetic expressions themselves, you would use a meta-language (e.g., English). The arithmetic expressions would then be called the object-language (the language being studied). All of this should be pretty clear after taking a serious course in Logic. I am not saying the choice of words is not sensible, I'm saying it it factually wrong.
  • deleted 7y ago
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• ausjke 7y ago
  Absolutely correct, but it's worse, since they don't truly understand math, they taught in such a way that confused/misled students, no wonder USA had to import so many STEM engineers(and graduate students to fill college classrooms) from other countries, and it's getting worse. The fix is to have specific teacher undergraduate or graduate degree, or education departments or state-funded full-blown normal universities, pay them well, make teacher profession a honorable career, only then the best will seek a position in k-12 schools and our next generation can keep getting better.
• dclowd9901 7y ago
  > If you know that to divide one fraction into another you should “flip the numerator and denominator of one, then multiply them,” is it important to be able to explain why it works? This is, of course, a well known and reliable way of solving fractional division problems, but it seems to be the _only_ way I’ve seen people solve fractional division problems (by using the reciprocal rule). What would be the approach that doesn’t involve using multiplication?
• esmi 7y ago
  I suppose this is what happens in a teacher shortage. https://www.epi.org/publication/the-teacher-shortage-is-real-large-and-growing-and-worse-than-we-thought-the-first-report-in-the-perfect-storm-in-the-teacher-labor-market-series/ https://www.epi.org/publication/the-teacher-shortage-is-real...
• deleted 7y ago
  [deleted]
• epx 7y ago
  Was evicted once from the class by a teacher that said 5^0=0 and I proved on board it was =1.
• joker3 7y ago
  The author of this article is a researcher in cognitive psychology specializing in education. He wrote a pop-sci book detailing how cognitive science explains what's wrong with school, and it's very much worth a read if you're at all interested in the topic. Link is https://www.amazon.com/Why-Dont-Students-Like-School/dp/047059196X https://www.amazon.com/Why-Dont-Students-Like-School/dp/0470....
• GordonS 7y ago
  I struggled with maths as a kid, but gained a much better grasp as an adult, when I had to use it for practical applications. I really think the abstract nature of the way maths is taught (at least in the UK) is a big problem that holds people back - kids don't understand the point of the more complex stuff. I recall asking my secondary school maths teacher what the point of learning about some concept was (logarithmic equations, I think), what practical applications it had - he couldn't answer that. If I'd understood how such concepts could be used in the real world for interesting things, I'm certain I and others in the class would have been better able to "get it", and would certainly been more enthused.
  • BeetleB 7y ago
    It's a double edged sword. I loved math because I liked physics, and calculus text books had plenty of physics examples. Then I tutored people doing "business math" - calculus but using finance for all its applications. Most people struggled not because of the math but because of the difficulty in understanding the finance part. And it was hard for me to give help because I had to learn the financial aspects to help them with their HW. If the application doesn't click with you, then you now have two problems.
    • r00fus 7y ago
      Exactly - the applicability has to mean something for the person. It could require several regressions before you get to a core interest of any specific student.
  • orbifold 7y ago
    I never understood people that questioned the practical applicability of math. Surely they must realize that these things are so elementary that there is no way there was no practical use for them. Logarithms specifically allow you to multiply values quickly by looking up their logarithm in a table, add the values and do a reverse lookup in the same table. This is the foundation of slide rules, which is the way engineers did calculations before computers were common.
    • deleted 7y ago
• gok 7y ago
  This could be written about essentially any school subject. Many elementary school teachers don't understand history, science, or writing either.
• m3kw9 7y ago
  Or they think there is actually more to elementary math than meets the eye
• 6gvONxR4sf7o 7y ago
  This is such a vicious cycle. The worse we teach math, the worse our math labor force, including teachers. The worse our labor force, the more of a pay cut it takes to teach math for the rarer and rarer people who really get it. To cut this cycle, I'm tempted to say we just need to throw money at it for a while. Pay people who really get math to teach it. Pay them enough to draw them away from whatever industry would pay them to do. Pay for them to be trained to teach well. Or maybe require more stringent testing to become an elementary school teacher in the first place. If this means too few eligible candidates, pay more. It seems like a no brainer use of tax dollars. I'm appalled that it's so acceptable to let people who don't understand a thing teach it. Maybe it just comes down to "pay teachers more."
• nyxtom 7y ago
  Leave to journalists to constantly dictate this so often used phrase: “X... and that should scare you” I would really wish the emotional appeal journalism would make its way to the door but it doesn’t look like that will ever go away.
• renaudg 7y ago
  Research also shows that, compared with other college students, future elementary teachers are especially prone to math anxiety —apprehension about doing math that’s so severe it interferes with actually doing it. That anxiety remains once they are in classrooms, and studies show that students learn less math from a math-anxious teacher. Reading this paragraph gives me a strong hunch that it could be gender related. I'm wondering what the gender skew is like in elementary school teaching, similar to nursing ? While research hasn't shown any difference in STEM ability between genders, it most definitely has when it comes to intrinsic interest (and I know this is controversial, but contrary to what is commonly said it's not because of some evil patriarchal plan to keep girls out : gender-related "things vs people" affinity has been found in baby monkeys too, and seems correlated to pre-natal testosterone exposure !) One can easily imagine a bunch of people-orientated young female students going into this field for the kids and people interaction not being too thrilled with the topic.
• Falkon1313 7y ago
  When I was in school, the problem was the way it was taught. There was no real explanation of anything or questioning that would lead to learning the fundamentals until the higher grades, when kids were already expected to understand things that they'd never been taught. They could have taught us about arithmetic, but instead it was "Just stare at this table until your attention wanders off, then daydream until the bell rings and hope by chance some bit of it sticks in your memory." No phonetics or linguistics to explain how things were spelled, instead just "Write this word over and over until your hand cramps, then do the same with the next one." As if by getting enough hand cramps you would magically learn how language works. Just recite "In 1492 Columbus sailed the ocean blue." Never any question into why the Portuguese, with their lead on exploration and colonization, did not end up with a bigger overseas empire than the British. As if that should be obvious as long as you can recite that rhyme and get the date right. If the teachers are around my age, most of them were probably taught the same way, which explains why they wouldn't understand math.