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.
0 - https://ling-academy.com/ It's a bit of a mess. I found that its pretty hard to build a language app.
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.)
> 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]
The science must be missing some inputs because the current theory is lacking.
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).
I don't have an opinion on the issue at hand. "Because the science says" With nothing in support makes me really suspicious. It really starts looking like "Because $authority says so you may not question" Which is the opposite of what scientific inquiry is meant to be.
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/201...
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.
I am not telling you to do multiplying digit only 1 times. That would be silly. I would be telling you should mix up multiplying digits with other previously learned concepts, say 10 addition and 10 subtraction questions, and the rest can be 80 multiplying digit problems. I don't know the optimal intermixing ratio here, but it shouldn't be a straight 100 multiplying digit problems which all use the same algorithm to solve it.
Drilling and repetition is good, but there's the danger of having illusory mastery because it's already there in short term memory. Your goal is to encode those skills into long term memory.
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.
One reason this topic is hard to talk about, I think, is that, "what is best for my kid" and "what is the best strategy for a roomful kids from various backgrounds" are often not the same thing; in fact, they can be in direct opposition to one another. Educators think about the latter, naturally, but parents think primarily about the former.
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.
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.
Do you have any articles or references that you'd personally recommend, in order to learn more?
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.
It's as useful for insights into practical intelligence as it is for theory-of-teaching.
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.
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.
Memorizing solutions isn't useful anymore. We have google to list out formulas. A deep understanding of problem solving is far more important and something you cant trivially search.
Same with understanding anything else in life - you don't really understand anything until you get to the point you can, in your head, predict a specific outcome, test it, and be proven correct, repeatedly.
(At some point you may learn to gain robust understanding purely from simulating things in your head, deriving insight from lower-level principles. But this itself is a skill, a hard one, which few people master. It's not something a random kid, or adult, can do.)
At a low level, you can point out all the patterns that are in multiplication tables, but they won’t be remembered until the student has internalized the facts.
At a higher level, teaching epsilon-delta proofs isn’t a good way to learn calculus. Memorizing the building blocks and the chain rule is.
Downplaying it to 'dumb ones' is not helpful when they can solve polynomials without much trouble but are getting burned on concepts that should have been drilled down properly in middle school.
They were saying that back in the day as well: https://en.wikipedia.org/wiki/Why_Johnny_Can%27t_Add
If a person's approach to educating their kids is coercive ("jump through our hoops or you'll be working at McDonalds your entire life") or downright abusive ("you're a worthless child to us if you don't meet this grade") then the results can be catastrophic. For every success story, this kind of maltreatment will produce many people who give up on learning altogether or drive themselves headfirst into mental illness. I definitely think history will judge this period as a bit of a dark age in education, the fact that people who've long retired still report exam nightmares says a lot about the completely arbitrary and needless pressure we put our children under.
In my experience being "well-spoken" (ie having an accent that's fairly close to RP) and being quick at picking things up has served me far better than any qualifications I have, both in the tech industry and out of it.
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_...
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.
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.
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.
Are you talking about this book? https://www.cambridge.org/core/books/learning-vocabulary-in-...
I haven't read the whole book, but quick skimming got me to this part:
> The highest vocabulary test scores were from the small number of learners reporting mnemonic techniques. The most commonly used strategies were effective but not as effective as the lesser used visualisation, mnemonic, oral rote rehearsal and retrieval strategies. Clearly, strategy training in memory-enhancing techniques could have useful effects.
I can't imagine learning Chinese characters without any form of mnemonics.
Username checks out :). Have you learned Chinese? I'm not convinced that my method is the best by a long shot, but I don't use mnemonics. I've tried memorizing other things with tricks like the memory palace, and either I don't know how to do it or my brain is busted, but I've never been able to create a "palace" much less store information I want to remember within it. I recently read about the mnemonic peg system and thought it might be useful if I could use radicals as "pegs." So far I just haven't been able to employ any of these tricks to my advantage.
My memorization routine (about 10 new words per day) is to write down a whole list ~30 words, writing the character and it's pronunciation (I use zhuyin). Sometimes I do this in class while we're going over the chapter's vocab. I set a timer for 5-8 minutes depending on how many new words there are, how "hard" some of them look at first glance, how many have unusual radicals I haven't used frequently, etc. Looking at just the pinyin or zhuyin I try to write them all down before the timer goes off. If I can't write a whole character, I at least try to write down some of the radicals, if nothing at all I just skip it and move on. Afterwards I grade myself and looking at the characters in my textbook again, I write a new list of just the words I couldn't write. I practice writing them several times until I feel like I have the hang of it. Then I test again. Repeat until I can write the whole list. After I can write them all, ideally, I would retest again after some time, sometimes I do. Unfortunately, I usually don't have enough time and I have just barely gotten them memorized before it's time for a chapter test and then a new load of vocab. Fortunately old words get rolled into new chapters and having to write essays and stuff gives me more practice after they've had a while to stew.
If there is some One Easy Trick that I'm missing that would make my routine 10x more efficient, I hope someone can share it, because this routine is very tiresome and time consuming. I have played a bit with visualization, where I would close my eyes and imagine every stroke of a character without actually writing it. It works, but I need a really quiet environment for that.
Yes, and still am. Not really active anymore since I'm busy with other things. I don't read much, but I can read a book, albeit slow and with much difficulty. Writing I do even less, and handwriting I never really did apart from tests and filling in forms. Speaking and listening I do every day. After a while I just decided to not focus really much on handwriting because it easily takes the most effort for something I actually barely use.
I've played a bit with memory palaces for fun and found them to be effective, but really not for Chinese Characters. I mostly used my own flashcards with Anki. I used pictures and tone colored characters to help. That worked fine, but making the flashcards also took a lot of time so it wasn't really perfect either. Having many synonyms also complicates the whole ordeal. But overall spaced repetition really, really helps with managing lots of vocabulary. You can have a deck of several thousands of words without too many issues. If you find words hard to remember it's also OK to just delete them (Anki also helps with this by automatically labeling cards you often fail as leech).
The mnemonics I use are the "build in" ones in the characters, maybe you already do this by yourself and actively learning them won't change too much. Nearly all characters have some meaning and/or pronunciation component hidden in them which can help you remembering them, or guessing their meaning or sound when you first encounter them.
I don't really think about radicals since those are just arbitrarily chosen components used only for paper dictionaries (and who wants to use those in the age of smartphones...). Some random examples of interesting components: 疒, is a character that is not used anymore (I think) but it means disease and if any character has this component you can be very sure it's some kind of disease.
月 is a tricky one. It's moon and it often really doesn't make sense in words until you know that 肉 is often corrupted into 月. The 月 in 脑 doesn't mean moon but meat/flesh.
Some characters change depending on where they appear as components: 水/氵, 人/亻.
I like to use the Outlier Linguistic Dictionary on Pleco for looking up how characters are build up and see if it can help me remembering the character. I find little stories help me to remember words/characters. A character I forgot how to write and just looked up: 虹 (rainbow). It' made of the component snake/insect/worm because the ancient Chinese thought it looked like a snake in the sky. 工 is there for sound because gong/hong sound similar. Or 取 (take/get/fetch), it's literally a hand taking an ear.
I don't think mnemonics is the One Easy Trick that will make your routine 10x more efficient, but depending on how you learn now it could really boost your efficiency. If you're not using any form of spaced repetition yet I think that really could be it.
I use Pleco flashcards to do spaced repetition. Not sure how it compares to Anki, but it takes almost no effort to create new cards, and that’s good for me because I can easily get sucked into fiddling with tools instead of using them.
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.
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
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
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.
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.
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.
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.