Chunking & Gridding - Why parents can't do math today
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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.)
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.
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/). 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.
(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
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.
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?
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
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.)
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.
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.
And the lattice method et al are really the same as long division.
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?
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.
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.)
> If you explain well enough the concepts of
> place-value and multiplication, most children
> can independently invent an algorithm for long
> multiplication.
But this is not the norm.
Evidence suggests - and my experience agrees - that correctly teaching process creates the ground-work for moving to understanding. Correct methods, practised over time and in different context, result in an internalisation that leads to an ability to generalise and apply in wider contexts.
This is my point about research and evidence-based teaching systems. The existing fads are borne of insufficiently researched and poorly developed methods, badly applied. They are often driven by dogma, and then delivered by teachers who don't properly understand what they're doing.
There has to be a better way, and unfortunately, nearly everyone has an opinion, and none of them are supported by anything like enough evidence. And so we continue to damage our children.
In parting, much of what you say I agree with, but not all. This isn't really the place to have a lengthy debate, but our positions aren't as far apart as my comment might have you think at first sight.
To the contrary, this is Hacker News!
Unfortunately, the situation that I'm referring to with children inventing their own long-multiplication algorithm was presented at a talk, so I can't point to the specific results, if they're even published yet, but my contention is based on an academic research study. The punchline of the experiment was that a full class of approximately-average grade-schoolers were taught math in a novel method, and developed their own multiplication algorithms, and went on to have above-average improvements in math test scores. The development of those algorithms were probably guided to a certain extent, but they looked to me like someone who was working the problem out from the first principles based on the concept of place value. That said, I'm definitely willing to accept the evidence you claim to have (especially since it leaves open the meaning of "correct").
I definitely agree that education should be based on science and not on dogma. Just because we think something ought to work doesn't mean that it works (it usually doesn't), and education is something you don't want to fuck up because the stakes are high and you only get one chance. But, I'm not sure we're even at the point where we can do research yet. Before you can measure results, you need to decide what results you're looking for and how you're measuring them.
I'm a mathematician, and I'll be the first to admit that being able to perform long-multiplication isn't actually that useful of a skill, and yet that's mostly what we measure on standardized tests. Understanding what multiplication means and being generally numerate is far more useful and not measured very often (the best we have is word problems). The more desirable side-effects, such as the ability to abstract concepts, to generalize processes, and to apply knowledge from one domain to another, aren't being measured at all, and those are really the only things from my math education that I use in my day-to-day life. Similar things happen across other subjects--knowing what year the War of 1812 happened in isn't useful, while understanding the social, economic, and political conditions that led to that war is useful for the average citizen in a democracy (but doesn't fit well in a scantron).
I think we as a society need to first have a big sit-down and figure out what the point of education is. We know it's important, but we've never coherently stated particularly why, and that lack of explicit direction is what lets politics, dogma, and fads more easily take control of the process. Once we've done that, we can let science guide the structure of education. Or rather, we should. There's a bit of a chicken-and-egg problem that without solid backgrounds in quantitative and logical thinking, most people have difficulty making data-driven decisions, and people in general won't get better at that until the education system improves.
I've read that before too, and I'm guessing that there is a cite for this in Alfie Kohn's book The Schools Our Children Deserve. However, it's been a few years since I've read it now so I can't remember if it's actually in that book or if it was Eleanor Duckworth's book The Having of Wonderful Ideas.
On the other hand, going through 100 lessons on reading, spelled out to the word what I was to say and do, and 9 months after starting he's reading 100 pages of books like The Way Things Work when he's supposed to be sleeping.
Sometimes rote methods work _really_ well.
Considering you are talking about evidence, where is yours?
I agree that a lot of people make unsupported generalizations about "what works" and "what doesn't work". I'm simply doubtful you can really propose a practical alternative, not even hypothetically practical alternative.
I would assert that part of this how extremely difficult it to really determine better teaching methods scientifically, controlling for expectation, controlling for multiple socio-economic factors, controlling for each child's background, etc, etc.
The rather rough experiments that anyone can point to show at best that "X works reasonably with teacher of X ability with student of X socio-economic position".
Inherently, just about every teaching method is "insufficiently researched". Just about every way that we deal with human beings is "ready, fire, aim" because we simply don't have the resources to fully, scientifically tease out what's the best dealing with the many, many dimensions of human behavior.
I'm not against social science because it's better to try to be a systematically as possible. I am against deluding oneself into thinking social science will work like natural, that it will come up with generalized laws that one can follow without constant adjustment.
> Considering you are talking about evidence,
> where is yours?
> 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
I've seen a lot of papers written and witnessed a lot of different techniques. I've seen first hand several fads, and most things work for some children with the right teachers. But I am concerned with the way the whole thing is approached, in part because I have a PhD in math and I know that "research" when in an area that involves people is really, really hard.
Especially with children.
I know that things have to be done differently in education research, and people who do the work are doing their best. I observe that the situation is still appalling - there's more that needs to be done.
I have my personal opinions about what might work, but they are unsupported by anything like scientific evidence, and I'm working closely with both teachers and education researchers to see if my ideas can be tested, and if so, whether they can be made to work. But I won't air them here.
ADDED IN EDIT: Wow - a down-vote. Wonder what I said that made someone feel that this comment is of negative value. Interesting.
It's really not worth making an issue of a single action of an random person on the internet.
Yes, but this discussion can't really be understood without understanding what it is that the US education system is trying to optimize itself for. In the US, teaching methods are NOT optimized for mathematical proficiency. They are designed to implement a "progressive" social agenda, "social justice," that puts "closing the achievement gap" ahead of all other considerations. The more rigorous the math curriculum, the more obvious our achievement gap, so the progressive ed schools have decided to de-emphasize objectively measurable math performance and emphasize other things that reveal less of an achievement gap.
Instead of reasoning through geometry problems and "finding the measure of angle A," which can be easily graded for correctness, we have them increase the amount of time spent "engaging in mathematical discourse." ("Who can name a job that uses math?") Of course all answers are correct so, voila, no more achievement gap.
We de-emphasize calculational proficiency--too likely to produce an achievement gap--and emphasize "conceptual understanding," which sounds great and can mean whatever you need it to mean for political purposes. "Do we all understand that there are many ways to multiply?" Sure we do!
Instead of "find the answers to these problems," our kids are assigned to "find pictures in magazines of math being used in real life, cut them out, and bring them to class." This is called "engagement." Who can't cut pictures out of magazines? Bye-bye achievement gap, hello social justice.
While the East Asians train their primary math teachers in techniques that have been found to maximize math proficiency, our teachers get a very different sort of preparation for teaching out kids:
http://www.city-journal.org/html/8_2_a1.html
Read the above article, consider what the proficiency-focused Asian schools are doing instead, and imagine the implications for our future.
http://frwebgate1.access.gpo.gov/cgi-bin/PDFgate.cgi?WAISdoc...
While American 5th graders (to use your example) get one homework assignment after another requiring them to cut, paste, draw, and color, their equivalents in Japan, Korea, Singapore, etc. are grinding away at difficult, real math. The Asian American students you describe are not exempt from this nonsense if they're in the public schools.
You can find the reason for all this described in the article I link to in my post above. If you don't believe this is going on, here's an example article from an academic journal describing the difficulty the education schools are having "breaking down the resistance" of math teachers to the social justice agenda:
http://www.informaworld.com/smpp/1045476611-60265133/content...
Here's how the abstract begins. It sounds like something from a dystopian future science fiction story:
"This paper reports on an action research project designed to explore the complexities of pre-service mathematics teacher resistance to social justice issues. Research on equity and mathematics education has indicated that such resistance seems particularly strong for mathematics teachers."
Maybe you need to read fewer right-wing rants about what educators say they're trying to get teachers to do, and pay more attention to what teachers are actually doing in classrooms.
Also, I believe most of your citations pre-date "No Child Left Behind", which from what I understand greatly increased the emphasis on standardized test scores.
The difference is stark. In Asia, they teach people math the way you would expect to teach someone if you were trying to maximize their proficiency in almost any skill: sports, chess, whatever. In the US, the education industry is pushing a very different approach, one whose primary objective is NOT maximizing the proficiency of each student.
But, you don't see it, and the US education industry (meaning the ed schools and their progeny, not individual teachers) would prefer that people like you go on ascribing any observation of their inferior results to "right-wing ranting" so you'll stay out of their way or, even better, go out of your way to try to discredit their critics.
Of course many teachers, even whole school districts, resist this pressure, and the ed schools describe that resistance in peer-reviewed journal articles such as the one I cited. If what I'm describing were merely the imaginings of those kooky right-wingers, like the leader of the Democratic Party in the Senate I cited above, then what is all this talk by the ed schools of "resistance"? Resistance to what, something that would increase math proficiency? Why would math teachers be "especially resistant" to maximizing math proficiency, or are they resisting pressure to make changes that would reduce math proficiency?
If your son had been in East Asia, he would have done the half hour of drills every day BEFORE taking that test and probably would have avoided the problem he apparently had. How do you think he, or his classmates, would do now on a test for Asian 5th graders? (Ex: 1/6 of A is equal to 5/9 of B. 2/5 of A is 270 more than 1/3 of B. What is the ratio of A:B in simplest form? Source: misskoh.com)
My kids are learning from the materials I brought back from Asia, and I'm trying to teach using the methods I saw (though I feel so amateurish compared to those amazing Asian teachers) and the results are, well, Asian-level. These are materials and teaching methods optimized for math proficiency and nothing else. Their friends think they are geniuses, and I have to keep reminding them that in Asia, their whole class (though not each individual student) would be operating at their level.
If I could, I would switch the whole country to these materials and methods, but guys like you can be counted on to say, "I don't see the problem, it's just more uninformed nonsense from someone who reads too much right-wing ranting". With your help, our public education industry can continue unimpeded by guys like me. My own kids will face less local competition as a result, but I fear the consequences for my country as a whole will be dire, so I would change it if I could.
Well that's actually how most of the stuff in school is taught. Whenever a kid goes "why am I learning this I'm never going to use this" that's a sign they don't know what they're learning.
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.
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.
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.
Adults today don't like doing their equivalent homework either. Most adults dislike bringing extra work home with them, taking phone calls outside of work hours and talk on and on about work/life balance and so on.
Also for kids who don't have environments conductive to studying at home would do better too. If they don't have a computer at home, they don't have stable, quiet environment to study in, or they're parents are clueless, they would all benefit staying at school to do their 'home work'. Often schools cannot let the kids stay for an extra amount of time, since computer labs would close down, staff often lock the school and do not have the time to supervise the kids.
This is a disingenuous straw man of a summary. If it were the case that it "helps some kids" but the others felt no effect, and they kiboshed the program for that reason, you'd have a point. But the problem here is that it "helps some kids", AND HURTS OTHERS. That's a big problem. The fact that it correlates with SES makes it even worse, because it means that the ones it's hurting most are the ones least capable of defending themselves, but the basic logic would be sensible even if the help/hurt spectrum were uncorrelated.
"Eliminating homework" does not mean outlawing homework and sending the police after anyone that attempts to do any homework.
Worse than weeknight homework: weekend homework. I absolutely reject this idea; I plan my family time, NOT the lazy teacher.
I don't understand your slight against teachers. Preparing homework takes time and effort if it's to be a proper part of the curriculum of learning to be set. Even if it's an afterthought it still has to be marked and assessed and recorded (in the UK it does).
Surely the lazy teacher would just say "for homework tonight watch TV" or some-such.
Naively, it seems reasonable to me that there is going to be a fraction of the necessary practice which must be done under the supervision of the teacher, and a fraction which can be done alone. Now, as people move up the academic ladder and, presumably, become better at teaching themselves, the amount of class time decreases and the amount of self-directed time increases.
At the 3rd grade level, where they spent 6-8 hours in class each day, it seems plausible that the correct amount of homework is roughly 30-60 minutes per night. It also seem plausible that the homework teaches a student not only the material, but also how to learn on their own.
From Wikipedia: "In England, the year of school that is the same age as 3rd Grade is Year 4, which is the fifth year of school."
Classic.
Personally I don't think homework should be mandated for young children (< c.11 yo).
The second type of homework doesn't apply to all classes, but a good candidate for this is math (spelling/grammar/reading is another). I'm a believer in children needing a good foundation in basic concepts before advancing to higher ones. If you can't do 14 - 9 in your head, instantly, then it's pointless to try multiplication or division. Directed practice can make children perform mathematics quickly and accurately. They don't come like that out of the box.
From this I would guess that if you want your child to really do well, you need to make sure they practice the right things. So make your 3rd grader to the homework and help them so that you know they're practicing the right thing. The studies cited by the grand-parent comment never say that practice of the right thing is bad.
However in households with lower socioeconomic status the problem isn't lack of attention from the parents. The parents TRY to help. But they are less likely to succeed, because they lack the academic skills needed.
This is the opposite of the educators in my family, my own experience as an educator, my book learning on pedagogy, and my experience as a student. It also runs screamingly against the difficulty that, in the United States, children with lower socioeconomic status are disproportionately unlikely to have parents.
Some school districts (including Arcadia, California, where I live) set standards specifying how much time different grades should spend on homework "on average". For 5th grade, the standard here is 50 minutes. So the teacher "feels the need" because that's what his/her bosses say to do.
From what I've heard, those standards are driven at least in part by the expectations of the parents.
However, while studying math in university I did lots of homework and it actually helps with understanding. Also in mathematics you know when you are cheating yourself.
I find that comment interesting and befuddling all at the same time. Which bit of your course was homework. I did a modular course in the UK including modules of Maths at honours level - we had lectures and tutorials on subject fields, did library study, computer based exercises and "labs", example problems, proofs and the like.
Which bits are what you consider "homework"? Did people really get degrees without studying outside of lectures?
When your entire life is self-motivated study I'm not sure how you draw a line and say "this bit is homework"?
Germany has a history of very self-directed university study.
Of course you were free to do library study and the like. It was often a good idea to work on the homework in teams.
I found it quite easy to get through the exams with just attending lectures, tutorial and doing the assigned homework. But most people studied a lot more. (I got decent but not perfect grades.)
http://en.wikipedia.org/wiki/Chunking_(division)
http://en.wikipedia.org/wiki/Carroll_diagram
- 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/ )
(I don't recommend this approach; long division was quite painful until I got quicker at multiplication)
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.
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.
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/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.)
You obviously know how to reach this market, and what kinds of things they need.
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.
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.
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.
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.
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.
The techniques aren't quite as homespun as chunking/gridding, but they're definitely fun.
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.
For once, I agree with the changes being made.
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.