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Chunking & Gridding - Why parents can't do math today

123 points by KoZeN 16y ago

RiderOfGiraffes 16y ago
I've spoken with many people about these systems and methods. The interesting thing that I've found is that lots of adults are dead set against them, even though some research shows that they are far more effective than previous methods. (No, sorry, I can't cite the research off-hand, it's been shown to me by others and I don't have the references) Adults don't learn in the same way children learn. Much of the criticism levelled by adults against these systems (and others) is based on their own ability (or more accurately, inability) to use or understand them. Such a basis for criticism is fundamentally broken. We need to know how children learn, and how they can learn in a way that enables understanding, and enables future progress. "Chunking" in particular seems to work especially well as a basis for moving on and developing a deeper understanding of division and related processes. We need fewer fads and more evidence-based child-specific learning systems, especially those that take into account and control for effects such as teacher enthusiasm/understanding and the Hawthorne effect. (Note: we also need to understand that there is a huge variation even amongst children, but that's another rant.)
- KoZeN 16y ago
  We need fewer fads and more evidence-based child-specific learning systems Couldn't agree more. My partner is a science teacher and this is one of her biggest complaints. Every week, teachers are faced with new research indicating that what their doing is either wrong or can be dramatially improved through dramatic change. Less criticism, more evidence.
  - tonygonz 16y ago
    Please allow me to cast a bit of skepticism on this topic. I have an MA in Math Education, and am currently working on a PhD in Education, so I have had a lot of first-hand experience with "how the sausage is made" with regards to education research. Teaching fads are all coming from research-oriented schools of education like those that I am getting my degree from. While the new methods that are being introduced are "research based", I find much of the research being conducted as highly questionable. Since around the 1970s there has been a strong shift away from quantitative research towards qualitative research in education, which IMHO allows researchers to pretty much "prove" anything that they want. I think that when most people hear the phrase "better teaching methods" they tend to assume that this means "better ways of getting students to learn the target curriculum", but in my experience actual mastery of, say, math skills now takes a back seat to, for example, higher self esteem or even just hating math less. The result is that over the past several decades students in the US, at least, where such trends towards new teaching methods seem to be the strongest, are performing worse and worse on international comparisons of math and science skills (see, for example, the TIMMS: http://nces.ed.gov/timss/ http://nces.ed.gov/timss/). Meanwhile the countries that now have the best performing students in terms of mathematics ability are largely from Asian countries that predominantly use methods now considered passe here. The elephant in the room is that the biggest reasons for poor math skills are not related to teaching methods, but rather are cultural issues---starting especially in the middle school years, most students just don't want to be seen as being good at math, as that can be a social stigma for many students of that age (I'm sure that many readers of this site can relate...). A student who wants to learn will do so, no matter how bad the teacher or the teacher's methods. A student who doesn't want to learn will not, no matter how good the teacher or the methods.
    - RiderOfGiraffes 16y ago
      How much research is done on "Drill (and kill)" methods versus the more "modern" alternatives? Probably none. The dogma is that "drill and kill" is really bad, so no one actually does research on whether it works. It seems to me (not in the math education research field, but not that far away) that the "research" being done is often bedevilled and ham-strung by these preconceived dogma. (Primarily I'm agreeing with you) I raise a cautious note about one of your points though: > A student who wants to learn will do so, no matter > how bad the teacher or the teacher's methods I used to believe this, and I want to believe it, but I've now seen several instances of otherwise bright children with bad teachers making no progress. Then a brief intervention by a gifted tutor has led to the child catching up and surpassing the other in their class. You might claim that this is a gifted tutor by-passing the cultural issues by providing enough interest to make the child engaged despite the peer pressure. Perhaps. But I now do believe that a poor to middling teacher in math will actively prevent many otherwise perfectly capable students from achieving their potential. That's why I'm heavily involved in math enrichment programs.
    - deleted 16y ago
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- tel 16y ago
  I'm not sure that adults learning differently from children is vital here. I think it just comes into play that adults already learned these things and have an expectation to know them; children are looking fresh and are allowed to not know things. I think many adults would benefit from relearning math to use grids and chunks. To this day I think about arithmetic using my own analogues to the grid and chunk methods. They're far closer to the intuitive meanings of the operators and are more useful when you start generalizing. They get at the heart of the divide between meaning and representation. (And for that, see GEB) Algebra is representation of abstract objects via formal computational procedures (sort of). The formal procedures are the mechanism for proof, but really have no weight of meaning. I to this day cannot long divide (without regenerating the rules anyway) and get weird stares when I tell people that. But then I stare back: how many ways can you describe what division means?
- wiredfool 16y ago
  At least a few of the people dead set against these methods are college professors seeing new students come in with no math skill whatsoever. See: http://cliffmass.blogspot.com/2010/01/how-good-are-uw-students-in-math.html http://cliffmass.blogspot.com/2010/01/how-good-are-uw-studen... The 'classic' methods are algorithims. They work. They aren't any more complicated than the various other methods, but at least they work in all situations. The other methods, while they may be useful for fermi problems, don't always have a clear way to progress when the problems become harder. And the kicker is, they're no simpler when you account for all the 'non-easy' numbers. See: http://www.youtube.com/watch?v=Tr1qee-bTZI http://www.youtube.com/watch?v=Tr1qee-bTZI Whatever the reason, there's a dramatic decline in the understanding of basic math by students at reasonably good universities over the last 20ish years. I've done the math, and I think I was on the tail end of a conservative swing. There's been one new math swing in that time, and I think we're heading back again. (This as the parent of small kids who's done way too much reading about the various curricula that are available, and how even the best ones used in classrooms pretty much suck. Jump and Singapore are the two that I know of that are pretty good, but they're not not USAian, so they don't get used in the schools. It is kinda funny going over problems in Jump and seeing the canadian cultural content.)
  - seltzered 16y ago
    Came in here to post that exact video, it made perfect sense to me. I learned with the classic algorithms, and have kindof self-realized some of the newer methods after college when needing to solve some basic math in my head to "eyeball" whether something will work. It's convenient, but not exact. You can't use it in areas where exact answers are required to make sure you're right, and the classical equations serve as "debug/trace information" to see where you went wrong.
    - Someone 16y ago
      I disagree with that video. The alternative methods for multiplication and long division are fine. For long division, the classical one doesn't even work on harder problems. Listen carefulley, and you'll hear several "because we know two times six equals twelve"; for most, that will not work when dividing by say 4367. Also, I doubt that price of books is part of this problem. There is one argument that I think is valid: kids get insufficient exercise to master these. Even that need not be a bad thing. I cannot recite the bible, but the time I gained by not even attempting to master that (I think) gave me time to read a zillion other books.
- godDLL 16y ago
  There is a part in the article where it says that some children feel long division is a pointless chore. I still feel like that. When I had to do this long division stuff in school, I'd split up the numbers into chunks in my head, and calculate the result with these small chunks (27x43 = (27x40) + (27x3) = 1080 + 81 = 1161); just to be sure. My head could not get around long division back then. I can now, but it would feel like a pointless chore – why do that when I can just chunk it and sum it up in my head?
jgrahamc 16y ago
I bet half the problem is calling repeated subtraction 'chunking' and breaking a multiplication down into multiple simple multiplications 'gridding'. Those names sound intimidating. Add to that the confusion of mathematics and arithmetic and the fact that many parts are convinced they 'can't do maths' and you have a recipe for disaster.
- RiderOfGiraffes 16y ago
  FWIW I agree, but to add a counter point ... Jargon emerges and evolves because it serves a real need. Short names make things easy to refer to, and hence easy to discuss. Once the definitions are made clear, often with just a simple example or two, the terminology makes things easier, not harder. Until you do that, it makes things worse, which is your point.
  - pohl 16y ago
    I was unaware of this new jargon until today. I welcome one of them in particular, the "number bond", because it draws specific attention to something valuable that I was never able to transmit to my teenagers. They have always believed that understanding the concepts is the sole goal of education, but I have always maintained that building a large base of instantaneous recall, through practice, also has enormous value. "Number bond" explicitly addresses that.
lmkg 16y ago
The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed, rather than as a common-sense way of systematizing simple common concepts. Part of this is that teachers tend to have a poor quantitative background themselves, so they can do math but they don't grok it. I've seen transcripts of grade school teacher math lessons, and it's clear that they perceive math in a legalistic sense, using phrases like "those are the rules, right?" The result is that the students only see a set of rules, and not the reasoning, so the set of rules looks basically arbitrary. If you explain well enough the concepts of place-value and multiplication, most children can independently invent an algorithm for long multiplication. They tend to arrive at similar solutions which aren't as well-organized or space-optimized as the traditional long multiplication algorithm, but they made it themselves so they understand not just how to do it but why it works. The exact algorithms being taught for math education may make a difference in exposing the concepts, but I think the real determiner of math education is not the algorithms but how the material is presented. Addition and multiplication make sense, if they're taught as concepts, but the focus on test scores emphasizes ability to plug-and-chug over ability to understand what's going on under the hood. This is also why everybody hates word problems, and why I think math testing should use more word problems. Math isn't about your ability to perform rote tasks, it's about your ability to apply general concepts to specific situations, and that in itself is as important as the rest of schooling combined (indeed, it makes the rest of schooling useful). Relevant parting anecdote: some school district knew that the Pythagorean Theorem would be on their standardized algebra tests that year, so they drilled all their students on it. The students could use the formula flawlessly. The test rolled around, and surprise! the Pythagorean Theorem question was a word problem, and most of the students bombed the question.
- btilly 16y ago
  The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed, rather than as a common-sense way of systematizing simple common concepts. http://randomgraphs.blogspot.com/2009/06/feynman-on-education.html http://randomgraphs.blogspot.com/2009/06/feynman-on-educatio... is a good cautionary tale on what this tendency leads to. Part of this is that teachers tend to have a poor quantitative background themselves, This is not a new problem. In the 50s a survey of elementary school teachers found that most thought that 3/5 was larger than 2/3. Based on anecdotal experience and the impressions of people I've talked to, I would expect the same result if the same question was ask of today's teachers. Speaking of anecdotes, the first time I won an argument with a teacher was in grade 6. I hadn't done my homework, so I sat reading the textbook while the class went through homework. I understood the textbook's explanation, and realized that the teacher had been teaching us the wrong thing for the last 2 weeks, and that explained why the class had consistently found that the answers in the official answer key were "wrong". It took me 20 minutes of arguing to convince the teacher, and everyone had to relearn that 2 weeks of material. (We had been practicing doing arithmetic in bases other than 10, and he had been teaching us to just calculate the answer in base 10 then convert into the right base. What we needed to do was convert the number into base 10, calculate, then convert back into the right base.)
btilly 16y ago
Interestingly the complaints would disappear if they just got rid of homework. No, I'm not being facetious. If you read The Homework Myth you'll find out that homework in grade school is a net neutral from a learning perspective. The benefit is that you get more practice. The downfall is that without supervision it is as easy to practice the wrong thing as the right thing, and practicing the wrong thing sets you back. Studies have shown that, in practice, the net effect is that homework causes academic performance to become more strongly correlated with the parent's socioeconomic status, but overall across the whole population there is no net change in learning outcomes. Plus assigning homework is a source of stress for families that causes students to dislike school. The strength of these effects is roughly linear in the amount of homework assigned, all of the way down to no homework at all. (Instead students have to do in class exercises, under the supervision of teachers. Which they have to do anyways.) The reason for that is that homework causes enforcing correct practice to be the job of the parents, and so how well the parents understand the material determines whether homework benefits or hurts. While my family educationally benefits from the trend (me and my wife can assist correct practice), reduced stress and inequality makes no homework seem fair to me. And hence the problem underscored by this article. With homework, when we teach students different techniques, we need to think both about how easily the students will learn them, and how well the parents will understand them. Without homework the parents lose that responsibility, and we're free to just focus on what works for the kids.
- confuzatron 16y ago
  the net effect So homework does help some kids. But it's unfair to do anything that helps some kids but not others, so it should be got rid of. The ideal is perhaps for parents to be able to choose whether to send their kids to a school that has their kids best interests at heart, rather than 'fighting inequality' by deliberately teaching them less well.
  - btilly 16y ago
    So homework does help some kids. Homework helps some kids and hurts some other kids. And causes stress all around. The kids that it helps are exactly the ones whose families are most able to send them to private schools. The kids that it hurts are the ones whose families are least able to send them to private schools. The ideal is perhaps for parents to be able to choose whether to send their kids to a school that has their kids best interests at heart, rather than 'fighting inequality' by deliberately teaching them less well. You're using loaded language that shortcuts thinking. If we TRULY have the kid's best interests at heart then it seems to me we should not assign homework in public schools (avoid hurting the kids who get no other choices), and allow private schools to assign as much as they want (so kids who would be helped can be allowed that experience). This maximizes the chance of having kids encounter the homework strategy that will be most effective for them. That seems better to me than the current situation where half the kids encounter a homework policy that actively harms them. And have no alternatives.
pohl 16y ago
For reference, the four items mentioned in the second paragraph: http://en.wikipedia.org/wiki/Chunking_(division) http://en.wikipedia.org/wiki/Chunking_(division) http://en.wikipedia.org/wiki/Carroll_diagram http://en.wikipedia.org/wiki/Carroll_diagram http://en.wikipedia.org/wiki/Number_bond http://en.wikipedia.org/wiki/Number_bond http://en.wikipedia.org/wiki/Grid_method http://en.wikipedia.org/wiki/Grid_method
- eldenbishop 16y ago
  I have always been terrible at multiplication and when I am forced to do it always use a method identical to the grid method. Except I didn't know it had a name. I just cheated and did this because it seemed like a non-optimal but straight forward way to an answer that was self evidently correct.
og1 16y ago
I was taught with the older techniques, but I like the new chunking and gridding system better. I think it is a lot closer to how you would break the problem down if you couldn't use pen and paper and had to do everything in your head.
- sesqu 16y ago
  This is exactly how I do it in my head. I was never taught to, but found myself multiplying multifigure numbers in my head often enough to come up with the gridding system. It becomes laborious after five figures or so. I was never that fond of the long forms, and didn't use them after they stopped appearing in the exams. Incidentally, I didn't bother with a calculator until 12th grade. Since then, I've started making more mistakes, and now have to do any mental calculations several times over.
- microtherion 16y ago
  Yes, the advantage of gridding seems to be that it decomposes the multiplication into simpler steps. I would suspect, however, that if you habitually had to do long multiplication, the old method would be faster (precisely because you do a bit more in your head and less on paper). Multiplying two 5 digit numbers involves adding up 5 numbers instead of 25. But what has changed over the last 40 years is that NOBODY does habitual long multiplication anymore, because calculators are ubiquitous. Therefore, the technique is purely a building step for further understanding, and picking an algorithm optimized for casual users of multiplication is appropriate.
ryanricard 16y ago
I think this is my first time encountering the "Grid Method," maybe I've seen it before but I was definitely taught the "old way" in school. It's genius. I remember the thing I hated the most about long multiplication or division was having to go back and "do it all over again" if I got the answer wrong. The nice thing about the Grid, though, is if you screw up the answer in one of the cells, all the remaining cells might still be good. Then you only have to re-do the addition. Much more user-friendly. I don't have kids, but it makes me sad that any parent could see their kid come home with that respond with anger. Hell, once you've let your kid show you their way, maybe you can teach him the old way too. The far more important lesson, I think, is that there's more than one way to do it.
orangecat 16y ago
Interesting. It seems that gridding and chunking would better generalize when students start taking algebra and multiplying x+1 and 3x-5 instead of 11 and 25.
- patio11 16y ago
  Gridding is indeed very similar to concepts you'll learn in geometry or algebra, such as FOIL for evaluating the product of expressions. Here, I whipped up a geometric explanation for gridding. This would be taught after you've demonstrated with counting blocks that multiplication is fast addition of rows in a rectangle, i.e., that multiplication is the calculation of area. http://images1.bingocardcreator.com/blog-images/hn/multiplication-is-calculation-of-area.png http://images1.bingocardcreator.com/blog-images/hn/multiplic... You can teach this to third graders very easily. (Well, if you are a third grade teacher who understands how to calculate the area of a 5x6 rectangle. That is, sadly, not universal.)
  - michael_dorfman 16y ago
    Patrick: Have you ever considered creating additional products (e.g., teaching aids like this graphic) aimed at your primary BCC market? You obviously know how to reach this market, and what kinds of things they need.
    - patio11 16y ago
      I'm very good at getting to them when they're looking for bingo, but not very good at reaching them otherwise. Also, candidly speaking, after four years I'm sort of ready for an audience with new challenges and, ahem, a higher dollars-to-crisis ratio.
njharman 16y ago
Huh, I was never taught the chunck/grid system (39yr old US public school) but that is almost exactly how I taught myself in later years, (mid 20ies) to do multiplication in my head.
- blahedo 16y ago
  I had the same reaction: it's really remarkable how well those two methods map to my own mental arithmetic algorithms. The grid multiplication technique would also prepare the student pretty well for polynomial multiplication (which is how I originally derived it).
  - Natsu 16y ago
    Same here. I taught myself the same methods and several other similar things that I found uses for later, when I got my degree in mathematics.
- JunkDNA 16y ago
  Same reaction here. The grid method makes so much more intuitive sense to me than the way I was taught. I do it without even thinking about it sometimes.
- weavejester 16y ago
  I do pretty much the same thing. I was taught traditional long multiplication in school, but its a technique that is designed to be worked out with a pen and paper, and doesn't tell you a lot about why the technique works. The grid system seems more geared toward mental arithmetic and understanding, and allows one to easily approximate an answer, rather than working it out exactly. This is pretty much exactly what's required in the modern world, and that's probably why lots of people have discovered this technique independently.
- poutine 16y ago
  This is always how I've done multiplication in my head as well. Just seems natural and pretty easy to do up to 99x99, after which it's too annoying to track.
Hovertruck 16y ago
Hmmm. I'm pretty sure I learned all of those methods in school as a child (20 now). I stick to doing the traditional way typically.
etal 16y ago
This is neat for the explanation of griding and chunking alone. - Gridding is how I casually multiply things in my head already. I read left-to-right, so I usually start multiplying with the leftmost digits and track the zeros, then add things up until I've reached the desired accuracy. Really, it's just the traditional method in reverse -- this 10-slide explanation is much easier for teaching the basic concept of multiplication, though. - Chunking is division explained as "how many portions does this make" rather than the traditional problem of "how big would each of these equal-sized portions?" -- which is great for two reasons: (1) physically performing the alternative -- measuring out a fluid into N equal-sized portions -- is hard! (2) it's more like how computers work, and makes the modulo operation trivial to explain; this kind of quotient clearly ignores the issue of remainders in the initial problem, and then later it's clear what you can do with the remainder to make a compound fraction. But it's unfortunate that memorization has become taboo in Western education. Sometimes you actually need to memorize a table of facts and be able to recall them quickly, as with single-digit multiplication -- if you have to add up seven eights each time you encounter 7x8, you'll just be too slow to keep up. Know how to confirm what you've memorized, but also know that 7x8=56 as a basic fact. (It's not that hard if you take advantage of spaced-repetition learning, as in: http://www.mnemosyne-proj.org/ http://www.mnemosyne-proj.org/ )
- abecedarius 16y ago
  In grade school, I decided to learn the times table by just referring to one as needed while doing the exercises, and expecting it to soak in implicitly. Worked for me at least.
  - ElliotH 16y ago
    Exactly how I learned mine. My times tables were abysmal until I did A level maths, there's only so many times you can use your calculator for 6x7 before you just remember it as 42.
    - jleader 16y ago
      That's a funny coincidence that you'd mention that particular multiplication "fact". I never successfully memorized the whole 10x10 times table in elementary school. I memorized a subset initially, and did the others by factoring and re-grouping them into combinations of the ones I'd memorized. Gradually I memorized most of them simply from seeing the answer so many times, but 40 years later, 6x7 is about the only one I haven't yet memorized. I still work it out by factoring and re-grouping: 6x7 = 2 x 3x7 = 2 x 21 = 42. Though at this point I'm not sure if perhaps I've actually memorized it, and I recite that little sequence to myself out of habit rather than necessity. (I don't recommend this approach; long division was quite painful until I got quicker at multiplication)
ori_b 16y ago
It's interesting that this is exactly how I ended up doing these calculations. It just seemed intuitive to me that this is the way it should be done over time, so I eventually just dropped the method taught in schools. For once, I agree with the changes being made.
deleted 16y ago
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mcn 16y ago
The article reminded me of Tom Lehrer's 1960's song "New Math." http://www.youtube.com/watch?v=I8aW4YuFSiY&feature=channel http://www.youtube.com/watch?v=I8aW4YuFSiY&feature=chann... (3 minutes, song starts at 1:15 but it's all relevant.) So I guess the way I learned in school would have been considered "new math" in the 60's. If a new method really is better we should go for it.
cageface 16y ago
This is always how I've done arithmetic. It falls naturally out of the distributive property you learn in algebra, so it makes a lot of sense to teach kids this early, I think.
code_duck 16y ago
Mm, it doesn't sound like the parents "can't do math", it sounds like they aren't familiar with the algorithms and methods being taught these days. I haven't taken the time to understand the new methods, but just from the names I get the feeling it's more like the methods I "natively" used to do multiplication and division very quickly in my head when I was a child (under 12). My methods for both division and multiplication involved breaking the numbers into easy parts. For instance, if I had to multiply 16x7, I'd split it into (16x5) + (16x2). 16x5 I'd use a quick rule that it is the same as 1/2 of (the first number x 10), or just (half of the first number times 10). Multiplying something by 2, 3 or 4 is always easy, so even if it was like 12.3 x 47, it always works. For the latter, I'd so something like (12.3x50)-(12.3x3) since multiplying something by 50 (broken into (X5)10) is always easy, for instance. For division, I'd do something similar, finding the nearest easy number, and then iteratively working on the difference. For instance, to divide 412 by 17, I'd start by saying 17x10 is 170, and that times two is 340, so we have 20 plus (the difference between 412 and 340, divided by 17). Then, I'd take an easy block of that, like 17x2, and see that we have 17x2=34, so you have 4 times and a remainder of 4. So, my answer would be 24 and a remainder of 4. Of course, teachers didn't want to see this. The most boring and damaging teachers would insist that I use the standard 'long division' algorithm, though I could do division like this faster in my head than most kids could do it on paper. Of course, that method is valuable too, and necessary for more tricky problems, but the discouragement of my natural intuition didn't help at all.
- dgordon 16y ago
  In fact, in this case (412 divided by 17) what you did is exactly what the long division algorithm would have you do -- try to divide 41/17, which is really 410/17, and take the closest-without-going-over multiple of 10 -- 20 in this case, then recognize how much you actually "used up" -- 340 -- and then divide into what's left. So it's even worse than you imagined with your ignorant teachers -- you were doing long division after all and they didn't recognize it.
maxawaytoolong 16y ago
These are "speed arithmetic" techniques which have existed for decades in Western society (check out any of the half dozen $7 Dover books about the topic). It's also based on abacus arithmetic techniques which have existed for centuries in Japan and other countries. It's good that this is being taught to young kids, most adults have 12 years of bad habits to unlearn.
- jjcm 16y ago
  I wouldn't say that adults have 12 years of bad habits - how you do an equation isn't as important as understanding the equation. I've found that if you teach kids the "speed" methods right off the bat, they don't fully understand the methods behind it and instead see it as a set of rules and tricks to follow. Think of it like learning how to take the derivative of an equation in calculus - you'll never learn the fast tricks right away because you wont properly understand what you're actually doing if you do. However, after you have this foundation you'll often be taught much faster methods of arriving at a derivative.
bluemetal 16y ago
I've been using a slightly different version of that gridding method to multiply numbers since I was in grade 5. Some guy came into school and showed us some "novel math", and it just clicked for me where long multiplication never really had. These days I can do either, but I'm far quicker doing my grids once I learnt how to draw them fast. If my experience teaches me anything, it's that if these children progressed up through a higher education system similar to what I have, where the people marking your work seem to have some kind of outright fear of the new and different, they're going to take a beating on their grades. At least until it becomes more widely accepted. Would suck to go first.
happy4crazy 16y ago
I try to spend a half hour or so a day playing around with Bill Handley's Speed Mathematics book (http://amzn.to/cE5hDu http://amzn.to/cE5hDu) The techniques aren't quite as homespun as chunking/gridding, but they're definitely fun.
anigbrowl 16y ago
Several people have argued that multiplication by gridding is more natural and they came up with it themselves after having learnt the old way. So did I, but I'm inclined to think that I came up with because I had plenty of multiplication practice in the first place. Most people are familiar with the idea of coming up with a loose approximation of the answer, then working it out exactly, and for that you are basically doing the same thing. Gridding is like big-endian multiplication - you start with the most significant bits and refine towards a solution, whereas traditional multiplication is little-endian. I suspect that this became the norm not because older math teachers are sadists, but because if you get a traditional multiplication calculation badly wrong (in terms of not understanding the process), it'll be screamingly obvious, whereas if you make an error with the gridding method it will still look like a good approximation...those kind of errors turn to be expensive in the real world, so trapping them early is a good idea. Yes, we do need fewer fads and more evidence...and we also need to stop thinking that there is one eternally best method for doing things. If you use the traditional method to multiply two large numbers, your calculations will extend down the page. If you use the gridding method, your calculations will extend sideways (total = a +b +c +d +...). In both cases you are trying to reduce the problem to single-digit multiplication and use base-shifting. The traditional way is little-endian (least significant bit): you multiply, add/carry and shift, for big-endian you shift and multiply, but you have to perform more addition operations at the end. What we're doing with numbers and digits resembles what a CPU does with bytes and bits when dealing with floating point math. I can multiply 2 4 digit numbers in my head by either method, but if accuracy matters then I prefer the little-endian one because you increment your exponent based on the depth of your stack. So if you multiply two integers with m and n digits you end up with a stack of depth n. The numbers on the stack look more complex than with gridding, but none of them will have more than m + 1 new digits (zeros at end added by you) and you only need one register and one carry digit. If you go with the other method you end up with a final stack of length m * n - with ~half of them being zeros, which is your information penalty for postponing the addition and carrying operations (although only m * n are significant carries, the others are incremental ones when a carry + product goes over a decimal boundary - eg if we carried 7 and next multiply 6 x 9, 54 + 7 = 61 and we carry 6 instead of 5). Would you rather do m single-digit multiply-adds n times followed by m additions of n terms, or or m * n single-digit multiplications and base power additions, followed by adding up m * n terms? For small numbers it's not a big deal, but if you multiply two 4 digit numbers you're going to have add up a string of 16 numbers at the end. Indeed, you could do the little endian method by gridding too; the numbers on each line in long multiplication are the totals of the grid columns read from the bottom right. But multiplying any number by a single digit is easy. We add numbers with a little-endian method because otherwise we'd have to keep stepping backwards to update the leading digit. As a bonus we know from the first digit that the numbers are correctly aligned if we arrange them vertically, rather than adding 300 to 4000 to get 700? - oops). By multiplying and adding the carry as we go along we're just extending that technique, and we still get the benefit of the ordinal alignment instead of writing down all those zeros - we just have to remember to pad each new line with one extra zero. The grid method encourages people to start multiplying numbers on the left...which will then have to be added up by working from the right. If you made any errors with the large # of zeros you are going to be screwed; multiple digits will be corrupted and the total will be significantly farther away from the correct answer. Considering that multiplying 2 4-digit numbers will yield 42 zeroes spread over 16 products vs 6 over 4 for the traditional method, I'd say it multiplies the potential for a mistake.

123 points by KoZeN 16y ago

I've spoken with many people about these systems and methods. The interesting thing that I've found is that lots of adults are dead set against them, even though some research shows that they are far more effective than previous methods. (No, sorry, I can't cite the research off-hand, it's been shown to me by others and I don't have the references) Adults don't learn in the same way children learn. Much of the criticism levelled by adults against these systems (and others) is based on their own ability (or more accurately, inability) to use or understand them. Such a basis for criticism is fundamentally broken. We need to know how children learn, and how they can learn in a way that enables understanding, and enables future progress. "Chunking" in particular seems to work especially well as a basis for moving on and developing a deeper understanding of division and related processes. We need fewer fads and more evidence-based child-specific learning systems, especially those that take into account and control for effects such as teacher enthusiasm/understanding and the Hawthorne effect. (Note: we also need to understand that there is a huge variation even amongst children, but that's another rant.)

KoZeN 16y ago
We need fewer fads and more evidence-based child-specific learning systems Couldn't agree more. My partner is a science teacher and this is one of her biggest complaints. Every week, teachers are faced with new research indicating that what their doing is either wrong or can be dramatially improved through dramatic change. Less criticism, more evidence.
- tonygonz 16y ago
  Please allow me to cast a bit of skepticism on this topic. I have an MA in Math Education, and am currently working on a PhD in Education, so I have had a lot of first-hand experience with "how the sausage is made" with regards to education research. Teaching fads are all coming from research-oriented schools of education like those that I am getting my degree from. While the new methods that are being introduced are "research based", I find much of the research being conducted as highly questionable. Since around the 1970s there has been a strong shift away from quantitative research towards qualitative research in education, which IMHO allows researchers to pretty much "prove" anything that they want. I think that when most people hear the phrase "better teaching methods" they tend to assume that this means "better ways of getting students to learn the target curriculum", but in my experience actual mastery of, say, math skills now takes a back seat to, for example, higher self esteem or even just hating math less. The result is that over the past several decades students in the US, at least, where such trends towards new teaching methods seem to be the strongest, are performing worse and worse on international comparisons of math and science skills (see, for example, the TIMMS: http://nces.ed.gov/timss/ http://nces.ed.gov/timss/). Meanwhile the countries that now have the best performing students in terms of mathematics ability are largely from Asian countries that predominantly use methods now considered passe here. The elephant in the room is that the biggest reasons for poor math skills are not related to teaching methods, but rather are cultural issues---starting especially in the middle school years, most students just don't want to be seen as being good at math, as that can be a social stigma for many students of that age (I'm sure that many readers of this site can relate...). A student who wants to learn will do so, no matter how bad the teacher or the teacher's methods. A student who doesn't want to learn will not, no matter how good the teacher or the methods.
  - RiderOfGiraffes 16y ago
    How much research is done on "Drill (and kill)" methods versus the more "modern" alternatives? Probably none. The dogma is that "drill and kill" is really bad, so no one actually does research on whether it works. It seems to me (not in the math education research field, but not that far away) that the "research" being done is often bedevilled and ham-strung by these preconceived dogma. (Primarily I'm agreeing with you) I raise a cautious note about one of your points though: > A student who wants to learn will do so, no matter > how bad the teacher or the teacher's methods I used to believe this, and I want to believe it, but I've now seen several instances of otherwise bright children with bad teachers making no progress. Then a brief intervention by a gifted tutor has led to the child catching up and surpassing the other in their class. You might claim that this is a gifted tutor by-passing the cultural issues by providing enough interest to make the child engaged despite the peer pressure. Perhaps. But I now do believe that a poor to middling teacher in math will actively prevent many otherwise perfectly capable students from achieving their potential. That's why I'm heavily involved in math enrichment programs.
  - deleted 16y ago
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tel 16y ago
I'm not sure that adults learning differently from children is vital here. I think it just comes into play that adults already learned these things and have an expectation to know them; children are looking fresh and are allowed to not know things. I think many adults would benefit from relearning math to use grids and chunks. To this day I think about arithmetic using my own analogues to the grid and chunk methods. They're far closer to the intuitive meanings of the operators and are more useful when you start generalizing. They get at the heart of the divide between meaning and representation. (And for that, see GEB) Algebra is representation of abstract objects via formal computational procedures (sort of). The formal procedures are the mechanism for proof, but really have no weight of meaning. I to this day cannot long divide (without regenerating the rules anyway) and get weird stares when I tell people that. But then I stare back: how many ways can you describe what division means?
wiredfool 16y ago
At least a few of the people dead set against these methods are college professors seeing new students come in with no math skill whatsoever. See: http://cliffmass.blogspot.com/2010/01/how-good-are-uw-students-in-math.html http://cliffmass.blogspot.com/2010/01/how-good-are-uw-studen... The 'classic' methods are algorithims. They work. They aren't any more complicated than the various other methods, but at least they work in all situations. The other methods, while they may be useful for fermi problems, don't always have a clear way to progress when the problems become harder. And the kicker is, they're no simpler when you account for all the 'non-easy' numbers. See: http://www.youtube.com/watch?v=Tr1qee-bTZI http://www.youtube.com/watch?v=Tr1qee-bTZI Whatever the reason, there's a dramatic decline in the understanding of basic math by students at reasonably good universities over the last 20ish years. I've done the math, and I think I was on the tail end of a conservative swing. There's been one new math swing in that time, and I think we're heading back again. (This as the parent of small kids who's done way too much reading about the various curricula that are available, and how even the best ones used in classrooms pretty much suck. Jump and Singapore are the two that I know of that are pretty good, but they're not not USAian, so they don't get used in the schools. It is kinda funny going over problems in Jump and seeing the canadian cultural content.)
- seltzered 16y ago
  Came in here to post that exact video, it made perfect sense to me. I learned with the classic algorithms, and have kindof self-realized some of the newer methods after college when needing to solve some basic math in my head to "eyeball" whether something will work. It's convenient, but not exact. You can't use it in areas where exact answers are required to make sure you're right, and the classical equations serve as "debug/trace information" to see where you went wrong.
  - Someone 16y ago
    I disagree with that video. The alternative methods for multiplication and long division are fine. For long division, the classical one doesn't even work on harder problems. Listen carefulley, and you'll hear several "because we know two times six equals twelve"; for most, that will not work when dividing by say 4367. Also, I doubt that price of books is part of this problem. There is one argument that I think is valid: kids get insufficient exercise to master these. Even that need not be a bad thing. I cannot recite the bible, but the time I gained by not even attempting to master that (I think) gave me time to read a zillion other books.
godDLL 16y ago
There is a part in the article where it says that some children feel long division is a pointless chore. I still feel like that. When I had to do this long division stuff in school, I'd split up the numbers into chunks in my head, and calculate the result with these small chunks (27x43 = (27x40) + (27x3) = 1080 + 81 = 1161); just to be sure. My head could not get around long division back then. I can now, but it would feel like a pointless chore – why do that when I can just chunk it and sum it up in my head?

jgrahamc 16y ago

I bet half the problem is calling repeated subtraction 'chunking' and breaking a multiplication down into multiple simple multiplications 'gridding'. Those names sound intimidating. Add to that the confusion of mathematics and arithmetic and the fact that many parts are convinced they 'can't do maths' and you have a recipe for disaster.

RiderOfGiraffes 16y ago
FWIW I agree, but to add a counter point ... Jargon emerges and evolves because it serves a real need. Short names make things easy to refer to, and hence easy to discuss. Once the definitions are made clear, often with just a simple example or two, the terminology makes things easier, not harder. Until you do that, it makes things worse, which is your point.
- pohl 16y ago
  I was unaware of this new jargon until today. I welcome one of them in particular, the "number bond", because it draws specific attention to something valuable that I was never able to transmit to my teenagers. They have always believed that understanding the concepts is the sole goal of education, but I have always maintained that building a large base of instantaneous recall, through practice, also has enormous value. "Number bond" explicitly addresses that.

lmkg 16y ago

The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed, rather than as a common-sense way of systematizing simple common concepts. Part of this is that teachers tend to have a poor quantitative background themselves, so they can do math but they don't grok it. I've seen transcripts of grade school teacher math lessons, and it's clear that they perceive math in a legalistic sense, using phrases like "those are the rules, right?" The result is that the students only see a set of rules, and not the reasoning, so the set of rules looks basically arbitrary. If you explain well enough the concepts of place-value and multiplication, most children can independently invent an algorithm for long multiplication. They tend to arrive at similar solutions which aren't as well-organized or space-optimized as the traditional long multiplication algorithm, but they made it themselves so they understand not just how to do it but why it works. The exact algorithms being taught for math education may make a difference in exposing the concepts, but I think the real determiner of math education is not the algorithms but how the material is presented. Addition and multiplication make sense, if they're taught as concepts, but the focus on test scores emphasizes ability to plug-and-chug over ability to understand what's going on under the hood. This is also why everybody hates word problems, and why I think math testing should use more word problems. Math isn't about your ability to perform rote tasks, it's about your ability to apply general concepts to specific situations, and that in itself is as important as the rest of schooling combined (indeed, it makes the rest of schooling useful). Relevant parting anecdote: some school district knew that the Pythagorean Theorem would be on their standardized algebra tests that year, so they drilled all their students on it. The students could use the formula flawlessly. The test rolled around, and surprise! the Pythagorean Theorem question was a word problem, and most of the students bombed the question.

btilly 16y ago
The fundamental problem with math education in the United States (yeah, I know the article is about UK, but it's Friday and I'm in a ranting mood) is that math is taught as a set of rules that must be followed, rather than as a common-sense way of systematizing simple common concepts. http://randomgraphs.blogspot.com/2009/06/feynman-on-education.html http://randomgraphs.blogspot.com/2009/06/feynman-on-educatio... is a good cautionary tale on what this tendency leads to. Part of this is that teachers tend to have a poor quantitative background themselves, This is not a new problem. In the 50s a survey of elementary school teachers found that most thought that 3/5 was larger than 2/3. Based on anecdotal experience and the impressions of people I've talked to, I would expect the same result if the same question was ask of today's teachers. Speaking of anecdotes, the first time I won an argument with a teacher was in grade 6. I hadn't done my homework, so I sat reading the textbook while the class went through homework. I understood the textbook's explanation, and realized that the teacher had been teaching us the wrong thing for the last 2 weeks, and that explained why the class had consistently found that the answers in the official answer key were "wrong". It took me 20 minutes of arguing to convince the teacher, and everyone had to relearn that 2 weeks of material. (We had been practicing doing arithmetic in bases other than 10, and he had been teaching us to just calculate the answer in base 10 then convert into the right base. What we needed to do was convert the number into base 10, calculate, then convert back into the right base.)

btilly 16y ago

Interestingly the complaints would disappear if they just got rid of homework. No, I'm not being facetious. If you read The Homework Myth you'll find out that homework in grade school is a net neutral from a learning perspective. The benefit is that you get more practice. The downfall is that without supervision it is as easy to practice the wrong thing as the right thing, and practicing the wrong thing sets you back. Studies have shown that, in practice, the net effect is that homework causes academic performance to become more strongly correlated with the parent's socioeconomic status, but overall across the whole population there is no net change in learning outcomes. Plus assigning homework is a source of stress for families that causes students to dislike school. The strength of these effects is roughly linear in the amount of homework assigned, all of the way down to no homework at all. (Instead students have to do in class exercises, under the supervision of teachers. Which they have to do anyways.) The reason for that is that homework causes enforcing correct practice to be the job of the parents, and so how well the parents understand the material determines whether homework benefits or hurts. While my family educationally benefits from the trend (me and my wife can assist correct practice), reduced stress and inequality makes no homework seem fair to me. And hence the problem underscored by this article. With homework, when we teach students different techniques, we need to think both about how easily the students will learn them, and how well the parents will understand them. Without homework the parents lose that responsibility, and we're free to just focus on what works for the kids.

confuzatron 16y ago
the net effect So homework does help some kids. But it's unfair to do anything that helps some kids but not others, so it should be got rid of. The ideal is perhaps for parents to be able to choose whether to send their kids to a school that has their kids best interests at heart, rather than 'fighting inequality' by deliberately teaching them less well.
- btilly 16y ago
  So homework does help some kids. Homework helps some kids and hurts some other kids. And causes stress all around. The kids that it helps are exactly the ones whose families are most able to send them to private schools. The kids that it hurts are the ones whose families are least able to send them to private schools. The ideal is perhaps for parents to be able to choose whether to send their kids to a school that has their kids best interests at heart, rather than 'fighting inequality' by deliberately teaching them less well. You're using loaded language that shortcuts thinking. If we TRULY have the kid's best interests at heart then it seems to me we should not assign homework in public schools (avoid hurting the kids who get no other choices), and allow private schools to assign as much as they want (so kids who would be helped can be allowed that experience). This maximizes the chance of having kids encounter the homework strategy that will be most effective for them. That seems better to me than the current situation where half the kids encounter a homework policy that actively harms them. And have no alternatives.

pohl 16y ago

For reference, the four items mentioned in the second paragraph: http://en.wikipedia.org/wiki/Chunking_(division) http://en.wikipedia.org/wiki/Chunking_(division) http://en.wikipedia.org/wiki/Carroll_diagram http://en.wikipedia.org/wiki/Carroll_diagram http://en.wikipedia.org/wiki/Number_bond http://en.wikipedia.org/wiki/Number_bond http://en.wikipedia.org/wiki/Grid_method http://en.wikipedia.org/wiki/Grid_method

eldenbishop 16y ago
I have always been terrible at multiplication and when I am forced to do it always use a method identical to the grid method. Except I didn't know it had a name. I just cheated and did this because it seemed like a non-optimal but straight forward way to an answer that was self evidently correct.

og1 16y ago

I was taught with the older techniques, but I like the new chunking and gridding system better. I think it is a lot closer to how you would break the problem down if you couldn't use pen and paper and had to do everything in your head.

sesqu 16y ago
This is exactly how I do it in my head. I was never taught to, but found myself multiplying multifigure numbers in my head often enough to come up with the gridding system. It becomes laborious after five figures or so. I was never that fond of the long forms, and didn't use them after they stopped appearing in the exams. Incidentally, I didn't bother with a calculator until 12th grade. Since then, I've started making more mistakes, and now have to do any mental calculations several times over.
microtherion 16y ago
Yes, the advantage of gridding seems to be that it decomposes the multiplication into simpler steps. I would suspect, however, that if you habitually had to do long multiplication, the old method would be faster (precisely because you do a bit more in your head and less on paper). Multiplying two 5 digit numbers involves adding up 5 numbers instead of 25. But what has changed over the last 40 years is that NOBODY does habitual long multiplication anymore, because calculators are ubiquitous. Therefore, the technique is purely a building step for further understanding, and picking an algorithm optimized for casual users of multiplication is appropriate.

ryanricard 16y ago

I think this is my first time encountering the "Grid Method," maybe I've seen it before but I was definitely taught the "old way" in school. It's genius. I remember the thing I hated the most about long multiplication or division was having to go back and "do it all over again" if I got the answer wrong. The nice thing about the Grid, though, is if you screw up the answer in one of the cells, all the remaining cells might still be good. Then you only have to re-do the addition. Much more user-friendly. I don't have kids, but it makes me sad that any parent could see their kid come home with that respond with anger. Hell, once you've let your kid show you their way, maybe you can teach him the old way too. The far more important lesson, I think, is that there's more than one way to do it.

orangecat 16y ago

Interesting. It seems that gridding and chunking would better generalize when students start taking algebra and multiplying x+1 and 3x-5 instead of 11 and 25.

patio11 16y ago
Gridding is indeed very similar to concepts you'll learn in geometry or algebra, such as FOIL for evaluating the product of expressions. Here, I whipped up a geometric explanation for gridding. This would be taught after you've demonstrated with counting blocks that multiplication is fast addition of rows in a rectangle, i.e., that multiplication is the calculation of area. http://images1.bingocardcreator.com/blog-images/hn/multiplication-is-calculation-of-area.png http://images1.bingocardcreator.com/blog-images/hn/multiplic... You can teach this to third graders very easily. (Well, if you are a third grade teacher who understands how to calculate the area of a 5x6 rectangle. That is, sadly, not universal.)
- michael_dorfman 16y ago
  Patrick: Have you ever considered creating additional products (e.g., teaching aids like this graphic) aimed at your primary BCC market? You obviously know how to reach this market, and what kinds of things they need.
  - patio11 16y ago
    I'm very good at getting to them when they're looking for bingo, but not very good at reaching them otherwise. Also, candidly speaking, after four years I'm sort of ready for an audience with new challenges and, ahem, a higher dollars-to-crisis ratio.

njharman 16y ago

Huh, I was never taught the chunck/grid system (39yr old US public school) but that is almost exactly how I taught myself in later years, (mid 20ies) to do multiplication in my head.

blahedo 16y ago
I had the same reaction: it's really remarkable how well those two methods map to my own mental arithmetic algorithms. The grid multiplication technique would also prepare the student pretty well for polynomial multiplication (which is how I originally derived it).
- Natsu 16y ago
  Same here. I taught myself the same methods and several other similar things that I found uses for later, when I got my degree in mathematics.
JunkDNA 16y ago
Same reaction here. The grid method makes so much more intuitive sense to me than the way I was taught. I do it without even thinking about it sometimes.
weavejester 16y ago
I do pretty much the same thing. I was taught traditional long multiplication in school, but its a technique that is designed to be worked out with a pen and paper, and doesn't tell you a lot about why the technique works. The grid system seems more geared toward mental arithmetic and understanding, and allows one to easily approximate an answer, rather than working it out exactly. This is pretty much exactly what's required in the modern world, and that's probably why lots of people have discovered this technique independently.
poutine 16y ago
This is always how I've done multiplication in my head as well. Just seems natural and pretty easy to do up to 99x99, after which it's too annoying to track.

Hovertruck 16y ago

Hmmm. I'm pretty sure I learned all of those methods in school as a child (20 now). I stick to doing the traditional way typically.

etal 16y ago

This is neat for the explanation of griding and chunking alone. - Gridding is how I casually multiply things in my head already. I read left-to-right, so I usually start multiplying with the leftmost digits and track the zeros, then add things up until I've reached the desired accuracy. Really, it's just the traditional method in reverse -- this 10-slide explanation is much easier for teaching the basic concept of multiplication, though. - Chunking is division explained as "how many portions does this make" rather than the traditional problem of "how big would each of these equal-sized portions?" -- which is great for two reasons: (1) physically performing the alternative -- measuring out a fluid into N equal-sized portions -- is hard! (2) it's more like how computers work, and makes the modulo operation trivial to explain; this kind of quotient clearly ignores the issue of remainders in the initial problem, and then later it's clear what you can do with the remainder to make a compound fraction. But it's unfortunate that memorization has become taboo in Western education. Sometimes you actually need to memorize a table of facts and be able to recall them quickly, as with single-digit multiplication -- if you have to add up seven eights each time you encounter 7x8, you'll just be too slow to keep up. Know how to confirm what you've memorized, but also know that 7x8=56 as a basic fact. (It's not that hard if you take advantage of spaced-repetition learning, as in: http://www.mnemosyne-proj.org/ http://www.mnemosyne-proj.org/ )

abecedarius 16y ago
In grade school, I decided to learn the times table by just referring to one as needed while doing the exercises, and expecting it to soak in implicitly. Worked for me at least.
- ElliotH 16y ago
  Exactly how I learned mine. My times tables were abysmal until I did A level maths, there's only so many times you can use your calculator for 6x7 before you just remember it as 42.
  - jleader 16y ago
    That's a funny coincidence that you'd mention that particular multiplication "fact". I never successfully memorized the whole 10x10 times table in elementary school. I memorized a subset initially, and did the others by factoring and re-grouping them into combinations of the ones I'd memorized. Gradually I memorized most of them simply from seeing the answer so many times, but 40 years later, 6x7 is about the only one I haven't yet memorized. I still work it out by factoring and re-grouping: 6x7 = 2 x 3x7 = 2 x 21 = 42. Though at this point I'm not sure if perhaps I've actually memorized it, and I recite that little sequence to myself out of habit rather than necessity. (I don't recommend this approach; long division was quite painful until I got quicker at multiplication)

ori_b 16y ago

It's interesting that this is exactly how I ended up doing these calculations. It just seemed intuitive to me that this is the way it should be done over time, so I eventually just dropped the method taught in schools. For once, I agree with the changes being made.

deleted 16y ago

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mcn 16y ago

The article reminded me of Tom Lehrer's 1960's song "New Math." http://www.youtube.com/watch?v=I8aW4YuFSiY&feature=channel http://www.youtube.com/watch?v=I8aW4YuFSiY&feature=chann... (3 minutes, song starts at 1:15 but it's all relevant.) So I guess the way I learned in school would have been considered "new math" in the 60's. If a new method really is better we should go for it.

cageface 16y ago

This is always how I've done arithmetic. It falls naturally out of the distributive property you learn in algebra, so it makes a lot of sense to teach kids this early, I think.

code_duck 16y ago

Mm, it doesn't sound like the parents "can't do math", it sounds like they aren't familiar with the algorithms and methods being taught these days. I haven't taken the time to understand the new methods, but just from the names I get the feeling it's more like the methods I "natively" used to do multiplication and division very quickly in my head when I was a child (under 12). My methods for both division and multiplication involved breaking the numbers into easy parts. For instance, if I had to multiply 16x7, I'd split it into (16x5) + (16x2). 16x5 I'd use a quick rule that it is the same as 1/2 of (the first number x 10), or just (half of the first number times 10). Multiplying something by 2, 3 or 4 is always easy, so even if it was like 12.3 x 47, it always works. For the latter, I'd so something like (12.3x50)-(12.3x3) since multiplying something by 50 (broken into (X5)10) is always easy, for instance. For division, I'd do something similar, finding the nearest easy number, and then iteratively working on the difference. For instance, to divide 412 by 17, I'd start by saying 17x10 is 170, and that times two is 340, so we have 20 plus (the difference between 412 and 340, divided by 17). Then, I'd take an easy block of that, like 17x2, and see that we have 17x2=34, so you have 4 times and a remainder of 4. So, my answer would be 24 and a remainder of 4. Of course, teachers didn't want to see this. The most boring and damaging teachers would insist that I use the standard 'long division' algorithm, though I could do division like this faster in my head than most kids could do it on paper. Of course, that method is valuable too, and necessary for more tricky problems, but the discouragement of my natural intuition didn't help at all.

dgordon 16y ago
In fact, in this case (412 divided by 17) what you did is exactly what the long division algorithm would have you do -- try to divide 41/17, which is really 410/17, and take the closest-without-going-over multiple of 10 -- 20 in this case, then recognize how much you actually "used up" -- 340 -- and then divide into what's left. So it's even worse than you imagined with your ignorant teachers -- you were doing long division after all and they didn't recognize it.

maxawaytoolong 16y ago

These are "speed arithmetic" techniques which have existed for decades in Western society (check out any of the half dozen $7 Dover books about the topic). It's also based on abacus arithmetic techniques which have existed for centuries in Japan and other countries. It's good that this is being taught to young kids, most adults have 12 years of bad habits to unlearn.

jjcm 16y ago
I wouldn't say that adults have 12 years of bad habits - how you do an equation isn't as important as understanding the equation. I've found that if you teach kids the "speed" methods right off the bat, they don't fully understand the methods behind it and instead see it as a set of rules and tricks to follow. Think of it like learning how to take the derivative of an equation in calculus - you'll never learn the fast tricks right away because you wont properly understand what you're actually doing if you do. However, after you have this foundation you'll often be taught much faster methods of arriving at a derivative.

bluemetal 16y ago

I've been using a slightly different version of that gridding method to multiply numbers since I was in grade 5. Some guy came into school and showed us some "novel math", and it just clicked for me where long multiplication never really had. These days I can do either, but I'm far quicker doing my grids once I learnt how to draw them fast. If my experience teaches me anything, it's that if these children progressed up through a higher education system similar to what I have, where the people marking your work seem to have some kind of outright fear of the new and different, they're going to take a beating on their grades. At least until it becomes more widely accepted. Would suck to go first.

happy4crazy 16y ago

I try to spend a half hour or so a day playing around with Bill Handley's Speed Mathematics book (http://amzn.to/cE5hDu http://amzn.to/cE5hDu) The techniques aren't quite as homespun as chunking/gridding, but they're definitely fun.

anigbrowl 16y ago

Several people have argued that multiplication by gridding is more natural and they came up with it themselves after having learnt the old way. So did I, but I'm inclined to think that I came up with because I had plenty of multiplication practice in the first place. Most people are familiar with the idea of coming up with a loose approximation of the answer, then working it out exactly, and for that you are basically doing the same thing. Gridding is like big-endian multiplication - you start with the most significant bits and refine towards a solution, whereas traditional multiplication is little-endian. I suspect that this became the norm not because older math teachers are sadists, but because if you get a traditional multiplication calculation badly wrong (in terms of not understanding the process), it'll be screamingly obvious, whereas if you make an error with the gridding method it will still look like a good approximation...those kind of errors turn to be expensive in the real world, so trapping them early is a good idea. Yes, we do need fewer fads and more evidence...and we also need to stop thinking that there is one eternally best method for doing things. If you use the traditional method to multiply two large numbers, your calculations will extend down the page. If you use the gridding method, your calculations will extend sideways (total = a +b +c +d +...). In both cases you are trying to reduce the problem to single-digit multiplication and use base-shifting. The traditional way is little-endian (least significant bit): you multiply, add/carry and shift, for big-endian you shift and multiply, but you have to perform more addition operations at the end. What we're doing with numbers and digits resembles what a CPU does with bytes and bits when dealing with floating point math. I can multiply 2 4 digit numbers in my head by either method, but if accuracy matters then I prefer the little-endian one because you increment your exponent based on the depth of your stack. So if you multiply two integers with m and n digits you end up with a stack of depth n. The numbers on the stack look more complex than with gridding, but none of them will have more than m + 1 new digits (zeros at end added by you) and you only need one register and one carry digit. If you go with the other method you end up with a final stack of length m * n - with ~half of them being zeros, which is your information penalty for postponing the addition and carrying operations (although only m * n are significant carries, the others are incremental ones when a carry + product goes over a decimal boundary - eg if we carried 7 and next multiply 6 x 9, 54 + 7 = 61 and we carry 6 instead of 5). Would you rather do m single-digit multiply-adds n times followed by m additions of n terms, or or m * n single-digit multiplications and base power additions, followed by adding up m * n terms? For small numbers it's not a big deal, but if you multiply two 4 digit numbers you're going to have add up a string of 16 numbers at the end. Indeed, you could do the little endian method by gridding too; the numbers on each line in long multiplication are the totals of the grid columns read from the bottom right. But multiplying any number by a single digit is easy. We add numbers with a little-endian method because otherwise we'd have to keep stepping backwards to update the leading digit. As a bonus we know from the first digit that the numbers are correctly aligned if we arrange them vertically, rather than adding 300 to 4000 to get 700? - oops). By multiplying and adding the carry as we go along we're just extending that technique, and we still get the benefit of the ordinal alignment instead of writing down all those zeros - we just have to remember to pad each new line with one extra zero. The grid method encourages people to start multiplying numbers on the left...which will then have to be added up by working from the right. If you made any errors with the large # of zeros you are going to be screwed; multiple digits will be corrupted and the total will be significantly farther away from the correct answer. Considering that multiplying 2 4-digit numbers will yield 42 zeroes spread over 16 products vs 6 over 4 for the traditional method, I'd say it multiplies the potential for a mistake.