9 ms·

> But drill isn’t popular. It’s widely seen as boring. Worse, there’s a body of opinion, especially in education, that considers it an ineffective way to practi

> But drill isn’t popular. It’s widely seen as boring. Worse, there’s a body of opinion, especially in education, that considers it an ineffective way to practice. That’s a shame, because it’s the most effective way to build your skills. This resonated with me as I watch my kids go through school. Math used to be taught with more drill style. Now with common core every single problem is an epic quest of 10 frames and double pluses. It seems so ridiculous. I'd rather them crush a worksheet of 20 problems that practices a single skill then 2 problems that try to include everything from reading to drawing just for a simple subtraction problem.

kiba 5y ago
Math used to be taught with more drill style. Now with common core every single problem is an epic quest of 10 frames and double pluses. It seems so ridiculous. I'd rather them crush a worksheet of 20 problems that practices a single skill then 2 problems that try to include everything from reading to drawing just for a simple subtraction problem. The problem with these 20 problems of basically the same identical challenge is that it's actually less effective than intermixing of different kind of problems, at least according to learning science. You and I may prefer 20 problems that practice straight subtraction, but that's not what the science says is actually the most effective learning strategy. You want different kind of problems in a problem set. It shouldn't be straight subtraction, but also additions, word problems, and so forth. This creates a level of desirable difficulty, which embeds knowledge more deeply than something that is very easy to do by rote.
- ipaddr 5y ago
  You are combining too many things. Focusing on 20 simple with increasing difficult problems builds visual memory, pattern matching which is lacking with a few compounds problems. The science must be missing some inputs because the current theory is lacking.
  - wisty 5y ago
    The current theory isn't particularly lacking, the average teacher (let alone layperson's) understanding of it is lacking. Plus there's dozens of rubbish theories that are sorely lacking so you have to find the researchers that actually do solid research. IIRC (see the above, it applies to random comments on the internet) drill work is more effective (but feels less effective to both teachers and students) if it's mixed up with different topics or question types, kind of like how doing a kata is better than doing exactly the same punch 10x (obviously katas are not ideal either, at least not as the only tool). Really you need a bit of diversity, and IMO two of the big traps to fall into are overly homogenous drill work (which doesn't retain as well as mixed drills, but looks effective because anyone who doesn't eat their crayons can do it without thinking too hard) and one-off problems (do an assignment where you solve a heavily obfuscated problem once, then pretend that it's now something that students actually understand, when they've literally just answered one single question assuming they even did it themselves).
    - stenl 5y ago
      In first grade we were given a workbook for learning how to write the letters. There was one whole page of A, one page of B, etc. You had to write maybe 100 As in a row. The kids quickly figured out that the fastest way to do it was to first write the left slant 100 times, then the right slant 100 times and finally all the 100 crossbars. So yes, you do need a little variety to defeat such shortcuts.
      - varjag 5y ago
        What a brilliant way to teach kids economies of scale.
      - TrispusAttucks 5y ago
        According to the article this is actually a really good way for kids to learn and improve. The point being that just practicing the lines is hugely important.
        TeMPOraL 5y ago
  - diordiderot 5y ago
    "The science is wrong because I disagree" - ipaddr
    - dvfjsdhgfv 5y ago
      They are not the only one, I see the same. Kids without math drills have problems in storing crucial bits of information in long-term memory and consequently fare worse at solving simple arithmetic problems than myself when I was much younger. I'm not talking about complex things but basic arithmetic, like multiplying digits, adding fractions or, more importantly, dealing with ratios. Drills give you a considerable advantage here.
      - doix 5y ago
        I have another peice of anecdata with my parents/grandparents. They grew up in the USSR and went through school there, drilling (according to them) was extremely common. They can still remember some peoms verbatim over 70 years later (in my grandfathers case). And they still remember/understand pretty much all the math they were taught. When I was doing my Advanced Highers (final exams in Scotland) I was asking my parents for help and they could answer all the questions without looking things up. I looked up the exam paper[0] I sat, I'm pretty sure there's no way I'd get an A again if I sat it right now without studying for it. But I'm pretty sure my parents still woudl. [0] https://www.advancedhighermaths.co.uk/wp-content/uploads/2016/02/AH-Maths-2008.pdf https://www.advancedhighermaths.co.uk/wp-content/uploads/201...
        gotorazor 5y ago
        I had a bit of schooling in that kind of educational system (Asia) before continuing schooling in North America. I'd say that you there is no free lunch. You're always giving up something for something else. I had a job 10 years ago doing in-person training at a company trying to digitize their paper-based office for the first time. They were in a commodity distribution business, so while the math isn't hard, there is a lot of day-to-day arithmetic (conversion between unit of measure and price/unit vs total price) for all the employees from the warehouse guy to the sales staff. The system introduced a change in their workflow. Before in their old manual paper system, people just kind of put things on a truck and figure out later how much got shipped and how much to invoice a customer. The whole can be very hand-wavy. There was no live inventory system either. Digitization meant that sales have to write sales orders that had precise units to be sold. Based on inventory, they know how much they will actually ship and they know that down to the dollar. Everybody suddenly had to start being aware of the math involved in their work. It was kind of funny to see a bunch of blue-collar, ex-con, high school dropouts learning faster than all the college-education office workers. The college-educated guys were too drill-orientated and approached the work like the math worksheets that everybody is talking about. The ex-cons had a working relationship with the numbers on the screen and the things that are hanging off their forklifts. Many of the white-collar clerks had been getting by memorizing formulas. They had no idea what any of those formulas mean.
- interesting22 5y ago
  This is very interesting and I'd like to share info about this with my wife, as we're approaching this challenge right now. Do you have any articles or references that you'd personally recommend, in order to learn more?
  - SiVal 5y ago
    Do yourself a big favor and read the book "Why Don't Students Like School?" by Prof. Daniel T. Willingham. He's a prof of psych at the University of Virginia specializing in the application of cog sci and neuro sci to K-12 education. I don't know who chose the title, but it doesn't describe the book, which is really a collection of articles about the results of experiments comparing various learning & teaching techniques. Only one chapter is about why children, who like learning some things, don't like school. Willingham publishes in academic journals and in journals for educators, so you can find other writings online. He tries to persuade teachers that so much of what the elite grad schools of education teach is intellectual fashion out of touch with actual cog sci findings, but all he cares about are the science experiments.
    - TheOtherHobbes 5y ago
      Huge upvotes for this. The book isn't just about learning, it's about how our minds really work - as opposed to how we think they work. It's as useful for insights into practical intelligence as it is for theory-of-teaching.
- dataviz1000 5y ago
  Along the same lines, Alfred North Whitehead had a very similar approach to education and learning as a whole saying it is cyclic with one important stage being precision analogous to the idea of practice with a later stage of generalization analogous to the idea of performance. > Whitehead conceives of the student’s educational process of self-development as an organic and cyclic process in which each cycle consists of three stages: first the stage of romance, then the stage of precision, and finally, the stage of generalization. The first stage is all about “free exploration, initiated by wonder”, the second about the disciplined “acquirement of technique and detailed knowledge”, and the third about “the free application of what has been learned” [0] [0] https://plato.stanford.edu/entries/whitehead/#PhilEduc https://plato.stanford.edu/entries/whitehead/#PhilEduc
- Treegarden 5y ago
  In reality the problem is more complicated and the issue is that the current media of teaching via worksheets and teachers is lacking and insufficient, which renders this debate obsolete. What you have is different parts, learning new stuff, practicing it, but also spaced repetition. Those need to be in balance with each other but also rely on cognitive overload, tiredness and motivation (among others) of the learner. So what you really need for a solution is software that replaces those work sheets and does a good job (as opposed to many of the current cheap learning apps) of giving you the right task at the right time. Eg. it knows that you have been drilling stuff and gotten good at it and so now its time for some more mixed stuff that could be paired with srs (stuff that needs refreshment). I think apps like kahn academy are good but could be improved and personally tried to build my own language learning flashcards app[0] after becoming frustrated with duolingo where I have a 2000 day streak. 0 - https://ling-academy.com/ https://ling-academy.com/ It's a bit of a mess. I found that its pretty hard to build a language app.
  - mushishi 5y ago
    Cool idea, my first impression is that there should be some kind of mini-arcs or long phrases that make the content more digestible. What I mean is that now the speaker goes on and on, and the subtitles are abruptly changed to another set of words. As a learner, I would like to have better sense when the words will disappear. I noticed it's arbitrary Youtube content so it's hard.. A simple testable solution for arbitrary content: option to _automatically_ pause/slow down the video just a for second before changing to the next set of text. Another simple and stupid idea would be to show previous words below the the current context. In the long run usage of a particular user, maybe the app should not highlight words that it knows you have encountered in previous videos many times or if the system knows you have mastered that word -- in case the system is testing user, did not notice if it does. Btw. Not sure what is the difference between green and black colored words. (Don't know Spanish so hard to guess.) Also there's some typos at the landing page, at least these: "Activley", "wont be" Seems like a great concept, good luck :) (I don't find Duolingo that effective either.)
    - Treegarden 5y ago
- wirrbel 5y ago
  I am of course not an empirical scientist in that area but I tutored math middle school students when in High School. [Note: This was in Germany and not in the US so it wasn't 'common core'] You see, the students who failed at the interleaved problems initially, were rocking it when I had them work through like 3 of these 20-similar problems worksheets before moving on to the 'pedagogically designed' problems. And epistemically I think it makes a lot of sense to train basics and build upon that. I think Math education could benefit a lot if we split the subject in two courses, 3h per week on drill (Arithmetics), 2h per week on the beautiful math (can also expose the student to axioms there, functions, mappings, etc, more complex problems and solving that with math, potentially with CAS support). Best separated with different teachers. Fact is, most high school graduates will find it challenging in their lives to apply the 'rule of three'. During the covid pandemic we have seen that members of the executive branch have no understanding of exponential growth (bad during the pandemic, but I wonder how the fiscal policy is affected by that??). Maybe we need to rethink mathematical education once again.
  - stjohnswarts 5y ago
    I don't know this was how I was taught. Mind numbingly boring rote style in 1-6, them a mix of stuff like reading problems and assignments along with practice assignments, by high school it was fairly even mix of progressively harder problems mixed with "reading problems" and projects (geometry, trig, calculus, diffeq)
- kebman 5y ago
  > You and I may prefer 20 problems that practice straight subtraction, but that's not what the science says is actually the most effective learning strategy. Hi, I'm an pedagogue and a licensed teacher. Another way to phrase that, is that humans tend to find repetitive tasks overwhelming and boring. Got a load of dishes you have to do? I bet most people feel right at home in that gnawing urge to postpone that mundane and monotonous task. I mean how many times haven't you sat there with a really dull chore and started daydreaming until someone snapped you out of it? The fact is, humans need variety, but more importantly we need a sense of agency. You kinda lose that when you're forced to do something repetitive over and over, and so naturally it's not a very effective way to learn or teach. If you're faced with repeating something 20 times, even with slight variations; first off it's overwhelming, and second if you feel that it's forced on you, then you lose agency. In other words, you're no longer the owner of the task. In turn that means you're no longer in control, so why would you slave away for that "evil" tutor over there? This is why repetition isn't very effective pedagogically speaking, because worst case it can even create antipathy towards you or the task you're trying to teach. On the other hand, it's exactly repeating something over and over that makes you master it, though... But how can you master a thing when it's too bloody boring to learn in the first place? Enter motivational strategies! And tactics to heighten morale. This is explains why you may prefer solving 20 problems that practice straight subtraction, because you're already motivated for it, and then it's easy. But when you're dealing with an entire class of pupils, you have to make sure as many of them as possible feel the same way about those tasks, or they'll fall behind. And so, at the most basic level, teachers need to vary their approach to a topic in order to effectively teach it. This means finding new ways, new angles, to look at a problem, and make sure you get some variety in between, so the thing doesn't become boring. Meanwhile, if you already know that your pupils are very motivated, you can get away with more straight repetition.
  - kiba 5y ago
    My understanding is that it's not about motivation. It's not about whether it's boring or not when it comes to intermixing versus blocked practice. It's that intermixing require more thought and thinking rather than just following steps by rote.
  - 5y ago
nyanpasu64 5y ago
I was taught math with the drill style. Being forced on a daily basis into repeatedly solving problems you struggle with, trapped with no ability to escape, by people with power over you (eg. parents), having your worth judged based on your ability to solve problems forced upon you, and once you learn a skill your parents find a new weakness to torment you with, is traumatic.
- foxes 5y ago
  That is not a problem with the drill style per say but toxic teachers and parents. You can make someone drill without tormenting them for failing. I think doing drills is good if done in a supportive setting, but it is also and should not be the only way to teach.
  - nextaccountic 5y ago
    Do you know any kid that signed up for math drills? It's nearly always imposed by parents or teachers.
    - odshoifsdhfs 5y ago
      My kid would be super happy with it. His teacher as a 'reward' for doing his language-related stuff on time/quicker, lets him do math worksheets
    - throwyuno 5y ago
      I loved math drills. In first grade we would get a sheet of simple math problems and the teacher would give us 5 minted to complete as many as possible. I was good at it, and it was one of my first experiences of competition and being better at something than my peers. I don’t know if any of that is a good thing, and it definitely would have sucked if I had been slow at math.
    - Aunche 5y ago
      I was certainly more excited by math drills more than attempts to make math "fun."
dr_dshiv 5y ago
We made factflow.io as a side project to help parents drill with their kids. I found it useful with mine, anyway. It's free.
MisterBastahrd 5y ago
When I was in middle school, we used the Saxon method of learning math. Every lesson set contained new concepts AND forced students to answer questions to old concepts. By the end of the year, we were fluent in all the concepts regardless of the last time they were taught in class, because we were constantly forced to solve equations from all parts of the book. We all loved that approach because it kept us in practice.
danielheath 5y ago
There's a big difference between the kind of learning that's effective in going from zero to knowing basics VS the kind that's effective in going from basics to mastery. I don't think drill is an effective method for early-stage learning, because you don't have the mental framework to hang the new knowledge off yet.
grecy 5y ago
> But drill isn’t popular. It’s widely seen as boring. Worse, there’s a body of opinion, especially in education, that considers it an ineffective way to practice. That’s a shame, because it’s the most effective way to build your skills. When you sign up to become a snowboard instructor in Canada they basically say "You should already be able to snowboard. We're not going to teach you that. We're going to teach you how to teach". As you move up the levels, you spend more and more time on pedagogy (teaching other teachers). Virtually all of it is drills - breaking down a small skill into an exercise or challenge or "do this 100 times before you get to the bottom" - then you "put it all back together" and ride. I can teach someone in half a day what took me a month to teach myself when I learned with no instruction. Drills are an awesome way to teach & learn
- smw 5y ago
  Any chance you know of a decent video showing how to teach someone to snowboard? (or ski?) I'd love to understand the best way to do that.
  - grecy 5y ago
    I've never seen a video that actually lays it out. The course to be a level 1 instructor (the lowest level) is three days intensive. Level 2 is four days. Level 3 is a five day course, plus separate two day exams (on and off snow). Level 4 (highest level).. well, I'm not there yet, it takes most people three years to do all the training and pass all the tests. For virtually everyone you have to snowboard/ski full time (100+ day seasons) working on your technique and teaching at least every day for about ten years to pass level four. The teaching isn't something you can learn from a couple of hour video.
eloisius 5y ago
Resonates with me too. I'm studying Chinese full-time and the classroom format is extremely drill-heavy. Here's a sentence, now say it this other way, using this new grammar device that we just learned. Here's another, and another, and another. After that we _mostly_ rote memorize characters (mostly, because they do have fragments of meaning that you can reason about sometimes). We drill on reading and writing characters daily. There is little to no "creative" homework or classroom activities like writing a dialog and acting it out in front of the class, as we did in Spanish class when I was in high school in the States. At first I was afraid that this learning style would be ineffective. Foreigners here often malign the Taiwanese education system as full of rote memorization, drills, and testing. Yet, the average Taiwanese can put together functional English sentences. A great many of them can speak fluently, even if they've never gone to an English speaking country. That's a lot more than you can say for the Spanish-speaking abilities of non-Hispanic Americans. Drilling like this let's me build confidence, memory muscle, and trains you to quickly pattern match and respond without giving the logical part of your brain time to get in the way and start translating slowly. It actually works outside the classroom too, I'm repeatedly surprised how natural words or grammar I drilled in class feel when it comes up in daily life. I cannot imagine the American style of incorporating a bunch of other activities into the exercise would help at all.
- CorrectHorseBat 5y ago
  Yes, some kind of drilling is definitely necessary, the issue is that traditional drilling is not efficient and there is so much more to language learning. I don't think traditional drilling doesn't work at all, it's just super inefficient and boring. That said, I also kind of feel that dialogue with other students isn't that useful. I don't know about English in Taiwan, but if it's similar to Mainland China then then many of them started with learning English when they were 3 or so. The level of English you get from that is rather disappointing.
- tasogare 5y ago
  > Yet, the average Taiwanese can put together functional English sentences. A great many of them can speak fluently Not true at all in my experience. I met only a few people there speaking English or French and all where young and most studied in a language department at university. There's surely a big divide along age categories and probably a North/South divide as I sometimes read people on the internet claiming Taiwanese are somewhat good in English, while I haven't seen that at all where I went (mostly Southern part). That being said I agree with the rest of your post. Anything trying to make learning Chinese fun is actually a waste of time, and rote memorization is extremely effective. In fact, it's one of the most effective way to learn vocabulary (Nation, 2001). I find it very sad that bad methods like Remembering the Kanji are hugely popular when they are in fact a waste of time. The amount of bad content on the internet is staggering. I think the biggest issue is that most people lost the willingness to put efforts in learning, and want everything immediately. As for learning Chinese, it also helps speakers of Chinese are usually very keen on correcting mistakes and teaching things even when not asked.
Zababa 5y ago
When I was in school we usually had at first some drill exercises, and then at the end some bigger problems. I think it was a nice balance, and the problems acted a bit like a reward.
dalbasal 5y ago
>> Math used to be taught with more drill style. There are many ways to suck. We tend to think in folksy wisdom: simples rules, simplifications, generalities. Practice makes perfect. 10k hrs. Problem solving. Etc. Often, we bounce back and forward between one such slogan and another. Folksy wisdom requires folk to be wise. You can't just distil it into a statement and run with that. A great instructor might be extremely focused on drill X or exercise Y. In reality, X or Y outside the greater context is not the same. The 10k hrs "rule" is a good example. I reckon I'm closing in on 5k hrs of chess. I'm not very good. I could have probably improved more than I would be with just 500 hrs training on a team, with an instructor, game analysis, tactic training, competition, etc. 5k hrs of bullet while on the toilet is not that. Now... I'm not saying that the book does claim that playing 10k hrs of ultra-casual chess while on the toilet leads to mastery. You need more context. Drill & Scrimmage, in this article's terms. "Deliberate practice" in Anders Ericsson's. Competition in other's terms. No matter what though, I think that the actual formula is not expressible. There will be a way of sucking while still ostensibly following the formula. You need the subjective human element. A person, training themselves or others who is focused on the goal of improvement, with the methods used as tools. A lot of canonical examples like sports, art or whatnot us an "art & science" adjacent terminology. TLDR, you'll also find plenty of example of rampant suckage and plateaus using drill oriented methods of teaching.
porb121 5y ago
> That’s a shame, because it’s the most effective way to build your skills. this is just not true. blocked practice (i.e. practicing the same task repeatedly) is generally worse for long-term retention than mixed practice strategies where you vary the practice conditions or interleave different tasks in practice e.g. if you go through 100 problems of 2 digit multiplication, you will probably have worse retention than if you went through 10 of those problems, then 10 division problems, then 5 word problems, then 10 3 digit multiplication problems, and so on, equating practice time drilling _feels_ like a really effective way to learn because you do better at it and quickly develop muscle memory or mental shortcuts, but your performance on practice tasks is really not a good signal of your actual learning or retention. https://www.researchgate.net/publication/275355435_Learning_Versus_Performance_An_Integrative_Review https://www.researchgate.net/publication/275355435_Learning_...
analog31 5y ago
Math teaching has never been successful. Nobody knows why we do it. The parents all learned it the old way, meaning that they got good test scores but did not retain any useful facility with math beyond their obligatory high school and college courses. A few people who managed to carry math to the level of being an art, like painting, probably can't tell you how or why that happened. My kids did lots and lots of drill. But no proofs. I struggled with math until we started to do proofs. Then it came alive for me. I loved sets. My school also used a series of textbooks in which some of the problems had no answer, and you were supposed to write "no answer." Those problems were a special rare treat that motivated me to do all of the problems. For most kids and their parents, math functions as some sort of diligence / obedience / IQ training that they hope will get them into a better college and job before it is forgotten.
- the__alchemist 5y ago
  A Mathematician's Lament (A Mathematician’s Lament): https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... is an outstanding article on the topic.
  - analog31 5y ago
    I wonder if a problem with math is that there are (at least) two sides to it: Practical and artistic. And I wonder if the practical or useful side of math could be taught today via computation. Even the useful side of math -- the stuff supposedly taught in service to the science and engineering courses -- is taught with a computer nowhere in sight. Yet people who use math in those fields, including myself, always do it with the assistance of a computer, whether it's to help avoid dumb mistakes or to automate tasks involving large amounts of numbers. And the physics problems that can really be solved with purely symbolic math are few and far between, and rather contrived. It actually leads students to believe that the formulas are false when applied in the real world. I don't think teaching computation would harm the handful of students who develop an interest in math as a liberal art. If they want to explore, they will, either on their own or with guidance. There's no reason why they can't be introduced to abstract math, and proofs, along the way.
btkramer9 5y ago
I've noticed something similar in University that was very frustrating for me. Freshmen and Sophmore year all math classes had tens or hundreds of problems that get progressively harder and bring out every corner case e.g. take then derivative of x^2 then 2x^2 then x + 2x^2 then sqrt(x) + 2x^2 + 3x^3. Eventually you could derive the most archaic equations Junior and Senior year Engineering homework is like 2 Epic questions with 10 parts that feed into eachother. I would have learned more and been more confident if they gave like 10-20 starter problems and then one final epic one at the end. I feel like this is similar to the story of a pottery class having half the class make as many items as possible while the other half had to make just 1 perfect piece. The group that was targeting quantity actually produced better pieces then the group that was targeting 1 perfect piece edit: typo
todd8 5y ago
> Math used to be taught with more drill style. This comment has engendered a lot of responses. Some arguing a position on personal experience and others arguing the supposed science of learning. A different and perhaps useful way to look at this is to consider how outcomes differ between large groups of schools around the world following different strategies for teaching math. The Program for International Student Assessment does world wide testing of 15 year olds in mathematics (and other subjects as well). The evidence suggests that whatever the US is doing isn't working. The USA ranks 25th in 15 year olds' mathematical ability with overall scores far lower that many other countries. Unfortunately, I can't read Mandarine, Japanese, Korean or Estonian so most of the effective text books (based on the PISA student scores) are inscrutable to me. However, Singapore's students have always scored near the top in math (second place in the most recent tests) and their text books are in available in English. I used Singapore math books to help my daughter improve her math after her school switched to Everyday Mathematics, a typical US math program developed at the University of Chicago. It wasn't terrible, but it wasn't good. The Singapore Math books are quite small and printed in black in white. They introduce the concepts with diagrams, etc. and have a set of easy introductory exercises to make sure the students understand the concepts. The books then have harder problems using the concepts and include word problems. The focus is on learning and practicing the techniques of mathematics. In Singapore Math, there is rote learning going on; in Everyday Math there is a kind of impressionistic emphasis where alternative approaches are encouraged, unusual methods of computation are introduced, inefficient solutions are treated as just as valuable as better approaches, and calculators replace fluency with basic operations. I've read the research paper the Everyday Math web site uses to justify the principles underlying the program. It's garbage and not any sort of science of learning. It even cites Knuth's Art of Computer Programming: Searching and Sorting as justification for introducing multiple non-standard ways to perform subtraction, multiplication, and division to elementary school children. This is completely ridiculous.

kiba 5y ago

Math used to be taught with more drill style. Now with common core every single problem is an epic quest of 10 frames and double pluses. It seems so ridiculous. I'd rather them crush a worksheet of 20 problems that practices a single skill then 2 problems that try to include everything from reading to drawing just for a simple subtraction problem. The problem with these 20 problems of basically the same identical challenge is that it's actually less effective than intermixing of different kind of problems, at least according to learning science. You and I may prefer 20 problems that practice straight subtraction, but that's not what the science says is actually the most effective learning strategy. You want different kind of problems in a problem set. It shouldn't be straight subtraction, but also additions, word problems, and so forth. This creates a level of desirable difficulty, which embeds knowledge more deeply than something that is very easy to do by rote.

ipaddr 5y ago
You are combining too many things. Focusing on 20 simple with increasing difficult problems builds visual memory, pattern matching which is lacking with a few compounds problems. The science must be missing some inputs because the current theory is lacking.
- wisty 5y ago
  The current theory isn't particularly lacking, the average teacher (let alone layperson's) understanding of it is lacking. Plus there's dozens of rubbish theories that are sorely lacking so you have to find the researchers that actually do solid research. IIRC (see the above, it applies to random comments on the internet) drill work is more effective (but feels less effective to both teachers and students) if it's mixed up with different topics or question types, kind of like how doing a kata is better than doing exactly the same punch 10x (obviously katas are not ideal either, at least not as the only tool). Really you need a bit of diversity, and IMO two of the big traps to fall into are overly homogenous drill work (which doesn't retain as well as mixed drills, but looks effective because anyone who doesn't eat their crayons can do it without thinking too hard) and one-off problems (do an assignment where you solve a heavily obfuscated problem once, then pretend that it's now something that students actually understand, when they've literally just answered one single question assuming they even did it themselves).
  - stenl 5y ago
    In first grade we were given a workbook for learning how to write the letters. There was one whole page of A, one page of B, etc. You had to write maybe 100 As in a row. The kids quickly figured out that the fastest way to do it was to first write the left slant 100 times, then the right slant 100 times and finally all the 100 crossbars. So yes, you do need a little variety to defeat such shortcuts.
    - varjag 5y ago
      What a brilliant way to teach kids economies of scale.
    - TrispusAttucks 5y ago
      According to the article this is actually a really good way for kids to learn and improve. The point being that just practicing the lines is hugely important.
      - TeMPOraL 5y ago
- diordiderot 5y ago
  "The science is wrong because I disagree" - ipaddr
  - dvfjsdhgfv 5y ago
    They are not the only one, I see the same. Kids without math drills have problems in storing crucial bits of information in long-term memory and consequently fare worse at solving simple arithmetic problems than myself when I was much younger. I'm not talking about complex things but basic arithmetic, like multiplying digits, adding fractions or, more importantly, dealing with ratios. Drills give you a considerable advantage here.
    - doix 5y ago
      I have another peice of anecdata with my parents/grandparents. They grew up in the USSR and went through school there, drilling (according to them) was extremely common. They can still remember some peoms verbatim over 70 years later (in my grandfathers case). And they still remember/understand pretty much all the math they were taught. When I was doing my Advanced Highers (final exams in Scotland) I was asking my parents for help and they could answer all the questions without looking things up. I looked up the exam paper[0] I sat, I'm pretty sure there's no way I'd get an A again if I sat it right now without studying for it. But I'm pretty sure my parents still woudl. [0] https://www.advancedhighermaths.co.uk/wp-content/uploads/2016/02/AH-Maths-2008.pdf https://www.advancedhighermaths.co.uk/wp-content/uploads/201...
      - gotorazor 5y ago
        I had a bit of schooling in that kind of educational system (Asia) before continuing schooling in North America. I'd say that you there is no free lunch. You're always giving up something for something else. I had a job 10 years ago doing in-person training at a company trying to digitize their paper-based office for the first time. They were in a commodity distribution business, so while the math isn't hard, there is a lot of day-to-day arithmetic (conversion between unit of measure and price/unit vs total price) for all the employees from the warehouse guy to the sales staff. The system introduced a change in their workflow. Before in their old manual paper system, people just kind of put things on a truck and figure out later how much got shipped and how much to invoice a customer. The whole can be very hand-wavy. There was no live inventory system either. Digitization meant that sales have to write sales orders that had precise units to be sold. Based on inventory, they know how much they will actually ship and they know that down to the dollar. Everybody suddenly had to start being aware of the math involved in their work. It was kind of funny to see a bunch of blue-collar, ex-con, high school dropouts learning faster than all the college-education office workers. The college-educated guys were too drill-orientated and approached the work like the math worksheets that everybody is talking about. The ex-cons had a working relationship with the numbers on the screen and the things that are hanging off their forklifts. Many of the white-collar clerks had been getting by memorizing formulas. They had no idea what any of those formulas mean.
interesting22 5y ago
This is very interesting and I'd like to share info about this with my wife, as we're approaching this challenge right now. Do you have any articles or references that you'd personally recommend, in order to learn more?
- SiVal 5y ago
  Do yourself a big favor and read the book "Why Don't Students Like School?" by Prof. Daniel T. Willingham. He's a prof of psych at the University of Virginia specializing in the application of cog sci and neuro sci to K-12 education. I don't know who chose the title, but it doesn't describe the book, which is really a collection of articles about the results of experiments comparing various learning & teaching techniques. Only one chapter is about why children, who like learning some things, don't like school. Willingham publishes in academic journals and in journals for educators, so you can find other writings online. He tries to persuade teachers that so much of what the elite grad schools of education teach is intellectual fashion out of touch with actual cog sci findings, but all he cares about are the science experiments.
  - TheOtherHobbes 5y ago
    Huge upvotes for this. The book isn't just about learning, it's about how our minds really work - as opposed to how we think they work. It's as useful for insights into practical intelligence as it is for theory-of-teaching.
dataviz1000 5y ago
Along the same lines, Alfred North Whitehead had a very similar approach to education and learning as a whole saying it is cyclic with one important stage being precision analogous to the idea of practice with a later stage of generalization analogous to the idea of performance. > Whitehead conceives of the student’s educational process of self-development as an organic and cyclic process in which each cycle consists of three stages: first the stage of romance, then the stage of precision, and finally, the stage of generalization. The first stage is all about “free exploration, initiated by wonder”, the second about the disciplined “acquirement of technique and detailed knowledge”, and the third about “the free application of what has been learned” [0] [0] https://plato.stanford.edu/entries/whitehead/#PhilEduc https://plato.stanford.edu/entries/whitehead/#PhilEduc
Treegarden 5y ago
In reality the problem is more complicated and the issue is that the current media of teaching via worksheets and teachers is lacking and insufficient, which renders this debate obsolete. What you have is different parts, learning new stuff, practicing it, but also spaced repetition. Those need to be in balance with each other but also rely on cognitive overload, tiredness and motivation (among others) of the learner. So what you really need for a solution is software that replaces those work sheets and does a good job (as opposed to many of the current cheap learning apps) of giving you the right task at the right time. Eg. it knows that you have been drilling stuff and gotten good at it and so now its time for some more mixed stuff that could be paired with srs (stuff that needs refreshment). I think apps like kahn academy are good but could be improved and personally tried to build my own language learning flashcards app[0] after becoming frustrated with duolingo where I have a 2000 day streak. 0 - https://ling-academy.com/ https://ling-academy.com/ It's a bit of a mess. I found that its pretty hard to build a language app.
- mushishi 5y ago
  Cool idea, my first impression is that there should be some kind of mini-arcs or long phrases that make the content more digestible. What I mean is that now the speaker goes on and on, and the subtitles are abruptly changed to another set of words. As a learner, I would like to have better sense when the words will disappear. I noticed it's arbitrary Youtube content so it's hard.. A simple testable solution for arbitrary content: option to _automatically_ pause/slow down the video just a for second before changing to the next set of text. Another simple and stupid idea would be to show previous words below the the current context. In the long run usage of a particular user, maybe the app should not highlight words that it knows you have encountered in previous videos many times or if the system knows you have mastered that word -- in case the system is testing user, did not notice if it does. Btw. Not sure what is the difference between green and black colored words. (Don't know Spanish so hard to guess.) Also there's some typos at the landing page, at least these: "Activley", "wont be" Seems like a great concept, good luck :) (I don't find Duolingo that effective either.)
  - Treegarden 5y ago
wirrbel 5y ago
I am of course not an empirical scientist in that area but I tutored math middle school students when in High School. [Note: This was in Germany and not in the US so it wasn't 'common core'] You see, the students who failed at the interleaved problems initially, were rocking it when I had them work through like 3 of these 20-similar problems worksheets before moving on to the 'pedagogically designed' problems. And epistemically I think it makes a lot of sense to train basics and build upon that. I think Math education could benefit a lot if we split the subject in two courses, 3h per week on drill (Arithmetics), 2h per week on the beautiful math (can also expose the student to axioms there, functions, mappings, etc, more complex problems and solving that with math, potentially with CAS support). Best separated with different teachers. Fact is, most high school graduates will find it challenging in their lives to apply the 'rule of three'. During the covid pandemic we have seen that members of the executive branch have no understanding of exponential growth (bad during the pandemic, but I wonder how the fiscal policy is affected by that??). Maybe we need to rethink mathematical education once again.
- stjohnswarts 5y ago
  I don't know this was how I was taught. Mind numbingly boring rote style in 1-6, them a mix of stuff like reading problems and assignments along with practice assignments, by high school it was fairly even mix of progressively harder problems mixed with "reading problems" and projects (geometry, trig, calculus, diffeq)
kebman 5y ago
> You and I may prefer 20 problems that practice straight subtraction, but that's not what the science says is actually the most effective learning strategy. Hi, I'm an pedagogue and a licensed teacher. Another way to phrase that, is that humans tend to find repetitive tasks overwhelming and boring. Got a load of dishes you have to do? I bet most people feel right at home in that gnawing urge to postpone that mundane and monotonous task. I mean how many times haven't you sat there with a really dull chore and started daydreaming until someone snapped you out of it? The fact is, humans need variety, but more importantly we need a sense of agency. You kinda lose that when you're forced to do something repetitive over and over, and so naturally it's not a very effective way to learn or teach. If you're faced with repeating something 20 times, even with slight variations; first off it's overwhelming, and second if you feel that it's forced on you, then you lose agency. In other words, you're no longer the owner of the task. In turn that means you're no longer in control, so why would you slave away for that "evil" tutor over there? This is why repetition isn't very effective pedagogically speaking, because worst case it can even create antipathy towards you or the task you're trying to teach. On the other hand, it's exactly repeating something over and over that makes you master it, though... But how can you master a thing when it's too bloody boring to learn in the first place? Enter motivational strategies! And tactics to heighten morale. This is explains why you may prefer solving 20 problems that practice straight subtraction, because you're already motivated for it, and then it's easy. But when you're dealing with an entire class of pupils, you have to make sure as many of them as possible feel the same way about those tasks, or they'll fall behind. And so, at the most basic level, teachers need to vary their approach to a topic in order to effectively teach it. This means finding new ways, new angles, to look at a problem, and make sure you get some variety in between, so the thing doesn't become boring. Meanwhile, if you already know that your pupils are very motivated, you can get away with more straight repetition.
- kiba 5y ago
  My understanding is that it's not about motivation. It's not about whether it's boring or not when it comes to intermixing versus blocked practice. It's that intermixing require more thought and thinking rather than just following steps by rote.
- 5y ago

nyanpasu64 5y ago

I was taught math with the drill style. Being forced on a daily basis into repeatedly solving problems you struggle with, trapped with no ability to escape, by people with power over you (eg. parents), having your worth judged based on your ability to solve problems forced upon you, and once you learn a skill your parents find a new weakness to torment you with, is traumatic.

foxes 5y ago
That is not a problem with the drill style per say but toxic teachers and parents. You can make someone drill without tormenting them for failing. I think doing drills is good if done in a supportive setting, but it is also and should not be the only way to teach.
- nextaccountic 5y ago
  Do you know any kid that signed up for math drills? It's nearly always imposed by parents or teachers.
  - odshoifsdhfs 5y ago
    My kid would be super happy with it. His teacher as a 'reward' for doing his language-related stuff on time/quicker, lets him do math worksheets
  - throwyuno 5y ago
    I loved math drills. In first grade we would get a sheet of simple math problems and the teacher would give us 5 minted to complete as many as possible. I was good at it, and it was one of my first experiences of competition and being better at something than my peers. I don’t know if any of that is a good thing, and it definitely would have sucked if I had been slow at math.
  - Aunche 5y ago
    I was certainly more excited by math drills more than attempts to make math "fun."

dr_dshiv 5y ago

We made factflow.io as a side project to help parents drill with their kids. I found it useful with mine, anyway. It's free.

MisterBastahrd 5y ago

When I was in middle school, we used the Saxon method of learning math. Every lesson set contained new concepts AND forced students to answer questions to old concepts. By the end of the year, we were fluent in all the concepts regardless of the last time they were taught in class, because we were constantly forced to solve equations from all parts of the book. We all loved that approach because it kept us in practice.

danielheath 5y ago

There's a big difference between the kind of learning that's effective in going from zero to knowing basics VS the kind that's effective in going from basics to mastery. I don't think drill is an effective method for early-stage learning, because you don't have the mental framework to hang the new knowledge off yet.

grecy 5y ago

> But drill isn’t popular. It’s widely seen as boring. Worse, there’s a body of opinion, especially in education, that considers it an ineffective way to practice. That’s a shame, because it’s the most effective way to build your skills. When you sign up to become a snowboard instructor in Canada they basically say "You should already be able to snowboard. We're not going to teach you that. We're going to teach you how to teach". As you move up the levels, you spend more and more time on pedagogy (teaching other teachers). Virtually all of it is drills - breaking down a small skill into an exercise or challenge or "do this 100 times before you get to the bottom" - then you "put it all back together" and ride. I can teach someone in half a day what took me a month to teach myself when I learned with no instruction. Drills are an awesome way to teach & learn

smw 5y ago
Any chance you know of a decent video showing how to teach someone to snowboard? (or ski?) I'd love to understand the best way to do that.
- grecy 5y ago
  I've never seen a video that actually lays it out. The course to be a level 1 instructor (the lowest level) is three days intensive. Level 2 is four days. Level 3 is a five day course, plus separate two day exams (on and off snow). Level 4 (highest level).. well, I'm not there yet, it takes most people three years to do all the training and pass all the tests. For virtually everyone you have to snowboard/ski full time (100+ day seasons) working on your technique and teaching at least every day for about ten years to pass level four. The teaching isn't something you can learn from a couple of hour video.

eloisius 5y ago

Resonates with me too. I'm studying Chinese full-time and the classroom format is extremely drill-heavy. Here's a sentence, now say it this other way, using this new grammar device that we just learned. Here's another, and another, and another. After that we _mostly_ rote memorize characters (mostly, because they do have fragments of meaning that you can reason about sometimes). We drill on reading and writing characters daily. There is little to no "creative" homework or classroom activities like writing a dialog and acting it out in front of the class, as we did in Spanish class when I was in high school in the States. At first I was afraid that this learning style would be ineffective. Foreigners here often malign the Taiwanese education system as full of rote memorization, drills, and testing. Yet, the average Taiwanese can put together functional English sentences. A great many of them can speak fluently, even if they've never gone to an English speaking country. That's a lot more than you can say for the Spanish-speaking abilities of non-Hispanic Americans. Drilling like this let's me build confidence, memory muscle, and trains you to quickly pattern match and respond without giving the logical part of your brain time to get in the way and start translating slowly. It actually works outside the classroom too, I'm repeatedly surprised how natural words or grammar I drilled in class feel when it comes up in daily life. I cannot imagine the American style of incorporating a bunch of other activities into the exercise would help at all.

CorrectHorseBat 5y ago
Yes, some kind of drilling is definitely necessary, the issue is that traditional drilling is not efficient and there is so much more to language learning. I don't think traditional drilling doesn't work at all, it's just super inefficient and boring. That said, I also kind of feel that dialogue with other students isn't that useful. I don't know about English in Taiwan, but if it's similar to Mainland China then then many of them started with learning English when they were 3 or so. The level of English you get from that is rather disappointing.
tasogare 5y ago
> Yet, the average Taiwanese can put together functional English sentences. A great many of them can speak fluently Not true at all in my experience. I met only a few people there speaking English or French and all where young and most studied in a language department at university. There's surely a big divide along age categories and probably a North/South divide as I sometimes read people on the internet claiming Taiwanese are somewhat good in English, while I haven't seen that at all where I went (mostly Southern part). That being said I agree with the rest of your post. Anything trying to make learning Chinese fun is actually a waste of time, and rote memorization is extremely effective. In fact, it's one of the most effective way to learn vocabulary (Nation, 2001). I find it very sad that bad methods like Remembering the Kanji are hugely popular when they are in fact a waste of time. The amount of bad content on the internet is staggering. I think the biggest issue is that most people lost the willingness to put efforts in learning, and want everything immediately. As for learning Chinese, it also helps speakers of Chinese are usually very keen on correcting mistakes and teaching things even when not asked.

Zababa 5y ago

When I was in school we usually had at first some drill exercises, and then at the end some bigger problems. I think it was a nice balance, and the problems acted a bit like a reward.

dalbasal 5y ago

>> Math used to be taught with more drill style. There are many ways to suck. We tend to think in folksy wisdom: simples rules, simplifications, generalities. Practice makes perfect. 10k hrs. Problem solving. Etc. Often, we bounce back and forward between one such slogan and another. Folksy wisdom requires folk to be wise. You can't just distil it into a statement and run with that. A great instructor might be extremely focused on drill X or exercise Y. In reality, X or Y outside the greater context is not the same. The 10k hrs "rule" is a good example. I reckon I'm closing in on 5k hrs of chess. I'm not very good. I could have probably improved more than I would be with just 500 hrs training on a team, with an instructor, game analysis, tactic training, competition, etc. 5k hrs of bullet while on the toilet is not that. Now... I'm not saying that the book does claim that playing 10k hrs of ultra-casual chess while on the toilet leads to mastery. You need more context. Drill & Scrimmage, in this article's terms. "Deliberate practice" in Anders Ericsson's. Competition in other's terms. No matter what though, I think that the actual formula is not expressible. There will be a way of sucking while still ostensibly following the formula. You need the subjective human element. A person, training themselves or others who is focused on the goal of improvement, with the methods used as tools. A lot of canonical examples like sports, art or whatnot us an "art & science" adjacent terminology. TLDR, you'll also find plenty of example of rampant suckage and plateaus using drill oriented methods of teaching.

porb121 5y ago

> That’s a shame, because it’s the most effective way to build your skills. this is just not true. blocked practice (i.e. practicing the same task repeatedly) is generally worse for long-term retention than mixed practice strategies where you vary the practice conditions or interleave different tasks in practice e.g. if you go through 100 problems of 2 digit multiplication, you will probably have worse retention than if you went through 10 of those problems, then 10 division problems, then 5 word problems, then 10 3 digit multiplication problems, and so on, equating practice time drilling _feels_ like a really effective way to learn because you do better at it and quickly develop muscle memory or mental shortcuts, but your performance on practice tasks is really not a good signal of your actual learning or retention. https://www.researchgate.net/publication/275355435_Learning_Versus_Performance_An_Integrative_Review https://www.researchgate.net/publication/275355435_Learning_...

analog31 5y ago

Math teaching has never been successful. Nobody knows why we do it. The parents all learned it the old way, meaning that they got good test scores but did not retain any useful facility with math beyond their obligatory high school and college courses. A few people who managed to carry math to the level of being an art, like painting, probably can't tell you how or why that happened. My kids did lots and lots of drill. But no proofs. I struggled with math until we started to do proofs. Then it came alive for me. I loved sets. My school also used a series of textbooks in which some of the problems had no answer, and you were supposed to write "no answer." Those problems were a special rare treat that motivated me to do all of the problems. For most kids and their parents, math functions as some sort of diligence / obedience / IQ training that they hope will get them into a better college and job before it is forgotten.

the__alchemist 5y ago
A Mathematician's Lament (A Mathematician’s Lament): https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... is an outstanding article on the topic.
- analog31 5y ago
  I wonder if a problem with math is that there are (at least) two sides to it: Practical and artistic. And I wonder if the practical or useful side of math could be taught today via computation. Even the useful side of math -- the stuff supposedly taught in service to the science and engineering courses -- is taught with a computer nowhere in sight. Yet people who use math in those fields, including myself, always do it with the assistance of a computer, whether it's to help avoid dumb mistakes or to automate tasks involving large amounts of numbers. And the physics problems that can really be solved with purely symbolic math are few and far between, and rather contrived. It actually leads students to believe that the formulas are false when applied in the real world. I don't think teaching computation would harm the handful of students who develop an interest in math as a liberal art. If they want to explore, they will, either on their own or with guidance. There's no reason why they can't be introduced to abstract math, and proofs, along the way.

btkramer9 5y ago

I've noticed something similar in University that was very frustrating for me. Freshmen and Sophmore year all math classes had tens or hundreds of problems that get progressively harder and bring out every corner case e.g. take then derivative of x^2 then 2x^2 then x + 2x^2 then sqrt(x) + 2x^2 + 3x^3. Eventually you could derive the most archaic equations Junior and Senior year Engineering homework is like 2 Epic questions with 10 parts that feed into eachother. I would have learned more and been more confident if they gave like 10-20 starter problems and then one final epic one at the end. I feel like this is similar to the story of a pottery class having half the class make as many items as possible while the other half had to make just 1 perfect piece. The group that was targeting quantity actually produced better pieces then the group that was targeting 1 perfect piece edit: typo

todd8 5y ago

> Math used to be taught with more drill style. This comment has engendered a lot of responses. Some arguing a position on personal experience and others arguing the supposed science of learning. A different and perhaps useful way to look at this is to consider how outcomes differ between large groups of schools around the world following different strategies for teaching math. The Program for International Student Assessment does world wide testing of 15 year olds in mathematics (and other subjects as well). The evidence suggests that whatever the US is doing isn't working. The USA ranks 25th in 15 year olds' mathematical ability with overall scores far lower that many other countries. Unfortunately, I can't read Mandarine, Japanese, Korean or Estonian so most of the effective text books (based on the PISA student scores) are inscrutable to me. However, Singapore's students have always scored near the top in math (second place in the most recent tests) and their text books are in available in English. I used Singapore math books to help my daughter improve her math after her school switched to Everyday Mathematics, a typical US math program developed at the University of Chicago. It wasn't terrible, but it wasn't good. The Singapore Math books are quite small and printed in black in white. They introduce the concepts with diagrams, etc. and have a set of easy introductory exercises to make sure the students understand the concepts. The books then have harder problems using the concepts and include word problems. The focus is on learning and practicing the techniques of mathematics. In Singapore Math, there is rote learning going on; in Everyday Math there is a kind of impressionistic emphasis where alternative approaches are encouraged, unusual methods of computation are introduced, inefficient solutions are treated as just as valuable as better approaches, and calculators replace fluency with basic operations. I've read the research paper the Everyday Math web site uses to justify the principles underlying the program. It's garbage and not any sort of science of learning. It even cites Knuth's Art of Computer Programming: Searching and Sorting as justification for introducing multiple non-standard ways to perform subtraction, multiplication, and division to elementary school children. This is completely ridiculous.