
Chunking & Gridding - Why parents can't do math today - KoZeN
http://www.bbc.co.uk/news/magazine-11258175
======
RiderOfGiraffes
I've spoken with many people about these systems and methods. The interesting
thing that I've found is that lots of adults are dead set against them, even
though some research shows that they are far more effective than previous
methods. (No, sorry, I can't cite the research off-hand, it's been shown to me
by others and I don't have the references)

 _Adults don't learn in the same way children learn._

Much of the criticism levelled by adults against these systems (and others) is
based on their own ability (or more accurately, _inability_ ) to use or
understand them. Such a basis for criticism is fundamentally broken.

We need to know how children learn, and how they can learn in a way that
enables understanding, and enables future progress. "Chunking" in particular
seems to work especially well as a basis for moving on and developing a deeper
understanding of division and related processes.

We need fewer fads and more evidence-based child-specific learning systems,
especially those that take into account and control for effects such as
teacher enthusiasm/understanding and the Hawthorne effect.

(Note: we also need to understand that there is a huge variation even amongst
children, but that's another rant.)

~~~
wiredfool
At least a few of the people dead set against these methods are college
professors seeing new students come in with no math skill whatsoever. See:
[http://cliffmass.blogspot.com/2010/01/how-good-are-uw-
studen...](http://cliffmass.blogspot.com/2010/01/how-good-are-uw-students-in-
math.html)

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.)

~~~
seltzered
Came in here to post that exact video, it made perfect sense to me.

I learned with the classic algorithms, and have kindof self-realized some of
the newer methods after college when needing to solve some basic math in my
head to "eyeball" whether something will work. It's convenient, but not exact.

You can't use it in areas where exact answers are required to make sure you're
right, and the classical equations serve as "debug/trace information" to see
where you went wrong.

~~~
Someone
I disagree with that video. The alternative methods for multiplication and
long division are fine. For long division, the classical one doesn't even work
on harder problems. Listen carefulley, and you'll hear several "because we
know two times six equals twelve"; for most, that will not work when dividing
by say 4367.

Also, I doubt that price of books is part of this problem. There is one
argument that I think is valid: kids get insufficient exercise to master
these. Even that need not be a bad thing. I cannot recite the bible, but the
time I gained by not even attempting to master that (I think) gave me time to
read a zillion other books.

------
lmkg
The fundamental problem with math education in the United States (yeah, I know
the article is about UK, but it's Friday and I'm in a ranting mood) is that
math is taught as a set of rules that must be followed, rather than as a
common-sense way of systematizing simple common concepts. Part of this is that
teachers tend to have a poor quantitative background themselves, so they can
do math but they don't grok it. I've seen transcripts of grade school teacher
math lessons, and it's clear that they perceive math in a legalistic sense,
using phrases like "those are the rules, right?" The result is that the
students only see a set of rules, and not the reasoning, so the set of rules
looks basically arbitrary.

If you explain well enough the concepts of place-value and multiplication,
most children can independently invent an algorithm for long multiplication.
They tend to arrive at similar solutions which aren't as well-organized or
space-optimized as the traditional long multiplication algorithm, but they
made it themselves so they understand not just how to do it but why it works.

The exact algorithms being taught for math education may make a difference in
exposing the concepts, but I think the real determiner of math education is
not the algorithms but how the material is presented. Addition and
multiplication make sense, if they're taught as concepts, but the focus on
test scores emphasizes ability to plug-and-chug over ability to understand
what's going on under the hood. This is also why everybody hates word
problems, and why I think math testing should use more word problems. Math
isn't about your ability to perform rote tasks, it's about your ability to
apply general concepts to specific situations, and that in itself is as
important as the rest of schooling combined (indeed, it makes the rest of
schooling useful).

Relevant parting anecdote: some school district knew that the Pythagorean
Theorem would be on their standardized algebra tests that year, so they
drilled all their students on it. The students could use the formula
flawlessly. The test rolled around, and surprise! the Pythagorean Theorem
question was a word problem, and most of the students bombed the question.

~~~
RiderOfGiraffes

      > If you explain well enough the concepts of
      > place-value and multiplication, most children
      > can independently invent an algorithm for long
      > multiplication.
    

You're going to stare at your screen in disbelief, my experience suggests
otherwise. A small number of gifted children have, in my experience, been able
to work out their own methods, and they've become more efficient over time,
and eventually converged to something isomorphic to 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.

~~~
wiredfool
Evidence suggests that if my oldest were to come up with a method for
multiplication it would involve: scissors, a stapler, and a stick, 5
complicated steps, 2 requests for materials from one of the parents, and it
would be abandoned halfway through for some method of making a crane that
involves an even bigger stick and rope.

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.

~~~
WalterBright
We have a 1000 years of history of what works in education, but people keep
ignoring that.

------
btilly
Interestingly the complaints would disappear if they just got rid of homework.

No, I'm not being facetious. If you read _The Homework Myth_ you'll find out
that homework in grade school is a net neutral from a learning perspective.
The benefit is that you get more practice. The downfall is that without
supervision it is as easy to practice the wrong thing as the right thing, and
practicing the wrong thing sets you back.

Studies have shown that, in practice, the net effect is that homework causes
academic performance to become more strongly correlated with the parent's
socioeconomic status, but overall across the whole population there is no net
change in learning outcomes. Plus assigning homework is a source of stress for
families that causes students to dislike school. The strength of these effects
is roughly linear in the amount of homework assigned, all of the way down to
no homework at all. (Instead students have to do in class exercises, under the
supervision of teachers. Which they have to do anyways.)

The reason for that is that homework causes enforcing correct practice to be
the job of the parents, and so how well the parents understand the material
determines whether homework benefits or hurts. While my family educationally
benefits from the trend (me and my wife can assist correct practice), reduced
stress and inequality makes no homework seem fair to me.

And hence the problem underscored by this article. With homework, when we
teach students different techniques, we need to think both about how easily
the students will learn them, and how well the parents will understand them.
Without homework the parents lose that responsibility, and we're free to just
focus on what works for the kids.

~~~
MattGrommes
I just had this discussion with my 3rd grader's teacher and have talked about
it with all her previous teachers after reading that book. Her point is that
the homework is just practice and only takes a few minutes which makes me
dislike it just on the principle of wasting "just a few minutes" of my
family's time for no benefit. I try to give teachers the benefit of the doubt
on this and not argue too much but in the past I've sent a note to school
saying she just won't be doing the homework and it's not her fault, it's my
decision.

~~~
jessriedel
This is an honest question from a childless person: how much time does your
3rd grader spend doing math homework per night?

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.

~~~
wikyd
I've seen the rule of thumb of 10 minutes of homework per grade in school. So,
1st graders would have 10 minutes of homework per night and 3rd graders would
have 30m of homework per night.

~~~
moultano
In fourth grade I had 5 hours.

~~~
eru
Wow. And there was no way out?

~~~
pstuart
Fifth grade.

------
pohl
For reference, the four items mentioned in the second paragraph:

<http://en.wikipedia.org/wiki/Chunking_(division)>

<http://en.wikipedia.org/wiki/Carroll_diagram>

<http://en.wikipedia.org/wiki/Number_bond>

<http://en.wikipedia.org/wiki/Grid_method>

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

------
etal
This is neat for the explanation of griding and chunking alone.

\- Gridding is how I casually multiply things in my head already. I read left-
to-right, so I usually start multiplying with the leftmost digits and track
the zeros, then add things up until I've reached the desired accuracy. Really,
it's just the traditional method in reverse -- this 10-slide explanation is
_much_ easier for teaching the basic concept of multiplication, though.

\- Chunking is division explained as "how many portions does this make" rather
than the traditional problem of "how big would each of these equal-sized
portions?" -- which is great for two reasons: (1) physically performing the
alternative -- measuring out a fluid into N equal-sized portions -- is hard!
(2) it's more like how computers work, and makes the modulo operation trivial
to explain; this kind of quotient clearly ignores the issue of remainders in
the initial problem, and then later it's clear what you can do with the
remainder to make a compound fraction.

But it's unfortunate that memorization has become taboo in Western education.
Sometimes you actually _need_ to memorize a table of facts and be able to
recall them quickly, as with single-digit multiplication -- if you have to add
up seven eights each time you encounter 7x8, you'll just be too slow to keep
up. Know how to confirm what you've memorized, but also know that 7x8=56 as a
basic fact.

(It's not that hard if you take advantage of spaced-repetition learning, as
in: <http://www.mnemosyne-proj.org/> )

~~~
abecedarius
In grade school, I decided to learn the times table by just referring to one
as needed while doing the exercises, and expecting it to soak in implicitly.
Worked for me at least.

~~~
ElliotH
Exactly how I learned mine. My times tables were abysmal until I did A level
maths, there's only so many times you can use your calculator for 6x7 before
you just remember it as 42.

~~~
jleader
That's a funny coincidence that you'd mention that particular multiplication
"fact". I never successfully memorized the whole 10x10 times table in
elementary school. I memorized a subset initially, and did the others by
factoring and re-grouping them into combinations of the ones I'd memorized.
Gradually I memorized most of them simply from seeing the answer so many
times, but 40 years later, 6x7 is about the only one I haven't yet memorized.
I still work it out by factoring and re-grouping: 6x7 = 2 x 3x7 = 2 x 21 = 42.
Though at this point I'm not sure if perhaps I've actually memorized it, and I
recite that little sequence to myself out of habit rather than necessity.

(I don't recommend this approach; long division was quite painful until I got
quicker at multiplication)

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

~~~
blahedo
I had the same reaction: it's really remarkable how well those two methods map
to my own mental arithmetic algorithms. The grid multiplication technique
would also prepare the student pretty well for polynomial multiplication
(which is how I originally derived it).

~~~
Natsu
Same here. I taught myself the same methods and several other similar things
that I found uses for later, when I got my degree in mathematics.

------
ryanricard
I think this is my first time encountering the "Grid Method," maybe I've seen
it before but I was definitely taught the "old way" in school.

It's _genius_. I remember the thing I hated the most about long multiplication
or division was having to go back and "do it all over again" if I got the
answer wrong. The nice thing about the Grid, though, is if you screw up the
answer in one of the cells, all the remaining cells might still be good. Then
you only have to re-do the addition. Much more user-friendly.

I don't have kids, but it makes me sad that any parent could see their kid
come home with that respond with anger. Hell, once you've let your kid show
you their way, maybe you can teach him the old way too. The far more important
lesson, I think, is that there's more than one way to do it.

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

~~~
patio11
Gridding is indeed very similar to concepts you'll learn in geometry or
algebra, such as FOIL for evaluating the product of expressions.

Here, I whipped up a geometric explanation for gridding. This would be taught
after you've demonstrated with counting blocks that multiplication is fast
addition of rows in a rectangle, i.e., that multiplication is the calculation
of area.

[http://images1.bingocardcreator.com/blog-
images/hn/multiplic...](http://images1.bingocardcreator.com/blog-
images/hn/multiplication-is-calculation-of-area.png)

You can teach this to third graders very easily. (Well, if you are a third
grade teacher who understands how to calculate the area of a 5x6 rectangle.
That is, sadly, not universal.)

~~~
michael_dorfman
Patrick: Have you ever considered creating additional products (e.g., teaching
aids like this graphic) aimed at your primary BCC market?

You obviously know how to reach this market, and what kinds of things they
need.

~~~
patio11
I'm very good at getting to them when they're looking for bingo, but not very
good at reaching them otherwise. Also, candidly speaking, after four years I'm
sort of ready for an audience with new challenges and, ahem, a higher dollars-
to-crisis ratio.

------
code_duck
Mm, it doesn't sound like the parents "can't do math", it sounds like they
aren't familiar with the algorithms and methods being taught these days.

I haven't taken the time to understand the new methods, but just from the
names I get the feeling it's more like the methods I "natively" used to do
multiplication and division very quickly in my head when I was a child (under
12).

My methods for both division and multiplication involved breaking the numbers
into easy parts. For instance, if I had to multiply 16x7, I'd split it into
(16x5) + (16x2). 16x5 I'd use a quick rule that it is the same as 1/2 of (the
first number x 10), or just (half of the first number times 10). Multiplying
something by 2, 3 or 4 is always easy, so even if it was like 12.3 x 47, it
always works. For the latter, I'd so something like (12.3x50)-(12.3x3) since
multiplying something by 50 (broken into (X _5)_ 10) is always easy, for
instance.

For division, I'd do something similar, finding the nearest easy number, and
then iteratively working on the difference. For instance, to divide 412 by 17,
I'd start by saying 17x10 is 170, and that times two is 340, so we have 20
plus (the difference between 412 and 340, divided by 17). Then, I'd take an
easy block of that, like 17x2, and see that we have 17x2=34, so you have 4
times and a remainder of 4. So, my answer would be 24 and a remainder of 4.

Of course, teachers didn't want to see this. The most boring and damaging
teachers would insist that I use the standard 'long division' algorithm,
though I could do division like this faster in my head than most kids could do
it on paper. Of course, that method is valuable too, and necessary for more
tricky problems, but the discouragement of my natural intuition didn't help at
all.

~~~
dgordon
In fact, in this case (412 divided by 17) what you did is _exactly_ what the
long division algorithm would have you do -- try to divide 41/17, which is
really 410/17, and take the closest-without-going-over multiple of 10 -- 20 in
this case, then recognize how much you actually "used up" -- 340 -- and then
divide into what's left.

So it's even worse than you imagined with your ignorant teachers -- you were
doing long division after all and they didn't recognize it.

------
og1
I was taught with the older techniques, but I like the new chunking and
gridding system better. I think it is a lot closer to how you would break the
problem down if you couldn't use pen and paper and had to do everything in
your head.

~~~
sesqu
This is exactly how I do it in my head. I was never taught to, but found
myself multiplying multifigure numbers in my head often enough to come up with
the gridding system. It becomes laborious after five figures or so. I was
never that fond of the long forms, and didn't use them after they stopped
appearing in the exams.

Incidentally, I didn't bother with a calculator until 12th grade. Since then,
I've started making more mistakes, and now have to do any mental calculations
several times over.

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

~~~
RiderOfGiraffes
FWIW I agree, but to add a counter point ...

Jargon emerges and evolves because it serves a real need. Short names make
things easy to refer to, and hence easy to discuss. Once the definitions are
made clear, often with just a simple example or two, the terminology makes
things easier, not harder.

Until you do that, it makes things worse, which is your point.

~~~
pohl
I was unaware of this new jargon until today. I welcome one of them in
particular, the "number bond", because it draws specific attention to
something valuable that I was never able to transmit to my teenagers. They
have always believed that understanding the concepts is the sole goal of
education, but I have always maintained that building a large base of
instantaneous recall, through practice, also has enormous value. "Number bond"
explicitly addresses that.

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

------
happy4crazy
I try to spend a half hour or so a day playing around with Bill Handley's
Speed Mathematics book (<http://amzn.to/cE5hDu>)

The techniques aren't quite as homespun as chunking/gridding, but they're
definitely fun.

------
bluemetal
I've been using a slightly different version of that gridding method to
multiply numbers since I was in grade 5. Some guy came into school and showed
us some "novel math", and it just clicked for me where long multiplication
never really had. These days I can do either, but I'm far quicker doing my
grids once I learnt how to draw them fast.

If my experience teaches me anything, it's that if these children progressed
up through a higher education system similar to what I have, where the people
marking your work seem to have some kind of outright fear of the new and
different, they're going to take a beating on their grades. At least until it
becomes more widely accepted. Would suck to go first.

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

For once, I agree with the changes being made.

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

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

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

------
anigbrowl
Several people have argued that multiplication by gridding is more natural and
they came up with it themselves after having learnt the old way. So did I, but
I'm inclined to think that I came up with because I had plenty of
multiplication practice in the first place.

Most people are familiar with the idea of coming up with a loose approximation
of the answer, then working it out exactly, and for that you are basically
doing the same thing. Gridding is like big-endian multiplication - you start
with the most significant bits and refine towards a solution, whereas
traditional multiplication is little-endian. I suspect that this became the
norm not because older math teachers are sadists, but because if you get a
traditional multiplication calculation badly wrong (in terms of not
understanding the process), it'll be screamingly obvious, whereas if you make
an error with the gridding method it will still look like a good
approximation...those kind of errors turn to be expensive in the real world,
so trapping them early is a good idea.

Yes, we do need fewer fads and more evidence...and we also need to stop
thinking that there is one eternally best method for doing things. If you use
the traditional method to multiply two large numbers, your calculations will
extend down the page. If you use the gridding method, your calculations will
extend sideways (total = a +b +c +d +...). In both cases you are trying to
reduce the problem to single-digit multiplication and use base-shifting. The
traditional way is little-endian (least significant bit): you multiply,
add/carry and shift, for big-endian you shift and multiply, but you have to
perform more addition operations at the end. What we're doing with numbers and
digits resembles what a CPU does with bytes and bits when dealing with
floating point math.

I can multiply 2 4 digit numbers in my head by either method, but if accuracy
matters then I prefer the little-endian one because you increment your
exponent based on the depth of your stack. So if you multiply two integers
with m and n digits you end up with a stack of depth n. The numbers on the
stack look more complex than with gridding, but none of them will have more
than m + 1 new digits (zeros at end added by you) and you only need one
register and one carry digit. If you go with the other method you end up with
a final stack of length m * n - with ~half of them being zeros, which is your
information penalty for postponing the addition and carrying operations
(although only m * n are significant carries, the others are incremental ones
when a carry + product goes over a decimal boundary - eg if we carried 7 and
next multiply 6 x 9, 54 + 7 = 61 and we carry 6 instead of 5).

Would you rather do m single-digit multiply-adds n times followed by m
additions of n terms, or or m * n single-digit multiplications and base power
additions, followed by adding up m * n terms? For small numbers it's not a big
deal, but if you multiply two 4 digit numbers you're going to have add up a
string of 16 numbers at the end.

Indeed, you could do the little endian method by gridding too; the numbers on
each line in long multiplication are the totals of the grid columns read from
the bottom right. But multiplying any number by a single digit is easy. We add
numbers with a little-endian method because otherwise we'd have to keep
stepping backwards to update the leading digit. As a bonus we know from the
first digit that the numbers are correctly aligned if we arrange them
vertically, rather than adding 300 to 4000 to get 700? - oops). By multiplying
and adding the carry as we go along we're just extending that technique, and
we still get the benefit of the ordinal alignment instead of writing down all
those zeros - we just have to remember to pad each new line with one extra
zero.

The grid method encourages people to start multiplying numbers on the
left...which will then have to be added up by working from the right. If you
made any errors with the large # of zeros you are going to be screwed;
multiple digits will be corrupted and the total will be significantly farther
away from the correct answer. Considering that multiplying 2 4-digit numbers
will yield 42 zeroes spread over 16 products vs 6 over 4 for the traditional
method, I'd say it multiplies the potential for a mistake.

