
Do you understand this first-grade child's homework? - shrikant
http://www.boingboing.net/2009/11/12/do-you-understand-my.html
======
RiderOfGiraffes
Firstly, I agree with other contributors that memorization is under-rated, and
agree that lots of memorization gives a basis of "instant knowledge" for when
you are working on more difficult problems.

However, this is a pretty good way of learning a general principle. Take the
first example, "8+3". Over on the right you have boxes in groups of 10, and
the first box has eight "counters" filled in. Now you add three by filling in
another three boxes. This gives you ten full boxes, and another one left over.
Thus 8+3 goes to 10+1.

ADDED IN EDIT: Now that I think of it, how many of you simply ignored the
instructions to fill counters in the boxes to help you, and tried to see what
was going on just by looking at the numbers?

Of the three that you are adding, two are used to make the eight up to ten, so
you only have one left. In a sense you've moved 2 from the 3 into the 8,
making the sum easier. Moving stuff around in an addition is very similar to
moving stuff around (and then making a correction) in multiplication. This is
how Art Benjamin squares two and three digit numbers faster than you can punch
them into a calculator.

53 squared is 53 times 53. Move three from one into the other (remember the 3
you've moved around) so you get 50 times 56. Use the same trick but in a
different way: move a _factor_ of two from the 56 into the 50 so you get 100
times 28, or 2800. Now add on the square of the three you moved earlier,
giving 2809.

ADDED IN EDIT: How this works can be demonstrated clearly with an appropriate
diagram.

If you're multiplying, moving _factors_ is the same sort of operation (both
multiplications) so you don't need a correction. If you move by
addition/subtraction it's a different operation, so you will need a
correction.

However, all of this fits into a larger framework. Getting only a small part
lets you do those bits, but as it fits together you end up with more than just
the parts, you get a much larger framework.

This is actually, tangentially, related to the item I submitted 10 hours ago
and which sank without a trace:

<http://news.ycombinator.com/item?id=951250>

The same trick is used twice in two different context to give the Infinite
Ramsey Theorem: Every infinite graph contains an infinite complete subgraph or
an infinite null graph.

Very similar.

~~~
viraptor
> ADDED IN EDIT: Now that I think of it, how many of you simply ignored the
> instructions to fill counters in the boxes to help you, and tried to see
> what was going on just by looking at the numbers?

I didn't ignore it - yet it didn't help. The instructions are ambiguous. How
are you supposed to fill them in? What is the connection between the 8 in the
first example and 8 counters? Are they the same by accident? Why should both
results be the same (8+3 and 10+X) - I thought about adding 10 new counters,
so that's 8+3=11, 10+8=18, 8+10=18 (although it doesn't make much sense
either). Then thought about adding 3 to the lower box (not good). There are
simply not enough information to "solve" this. Even if most people can guess
what the right solution is, I don't think that's good enough for homework.

~~~
RiderOfGiraffes
As has been said elsewhere, most likely examples were done in class, and this
is extra work to act as spaced repetition. Without that lesson it might not be
obvious to some. To me, it was, but only after I'd continued filling in boxes
in the grid.

In general, homework is not intended to contain the entire lesson all over
again. That means parents often don't know what to do, or how to help.

Further, lessons and homework are often not intended to be done by creative
people without instruction. Creative people find unexpected ways to follow
instructions the teachers (or lesson planners) thought were clear, obvious and
unambiguous.

I see a lot of criticisms of lessons and homework like this, but I've also
seen the frustration from teachers trying to create lessons that are EXCITING
and ENGAGING and ENTERTAINING and THRILLING and WONDERFUL, because now they're
not expected just to be good at teaching and experts in their subjects, but
also entertainers.

This lesson/homework isn't perfect, but I think many of the criticisms aimed
at it are misplaced and ignoring the context. This homework is being assessed
as if it were a lesson, and that's unfair.

Finally, homework that my colleague's child is set is accompanied by a sheet
explaining the lesson. My colleague hos no trouble helping with the homework,
and constantly praises the variety of approaches and integrity of the work.

~~~
DanielStraight
Right. I think this really sums it up.

My answer to the original question is, "No, but I would if I'd seen it
before." The children, presumably, have seen it before.

------
albertsun
In a lot of teaching (particularly math teaching) it seems that when a student
doesn't understand the concept, the immediate conclusion is that there must be
something wrong with the way it's being explained. Thus we have convoluted
methods of teaching addition like this that in some way do make sense, but are
mostly just another layer of confusion.

The proper way, I believe, is to just have the student learn the concept by
rote. If you can't grasp conceptually why 8+3=11 then you should just memorize
it.

After repeating enough times, eventually it will click and you'll understand
it.

Additionally, there's no point in trying to teach the concepts of arithmetic.
You'll be crippled trying to learn things like decimals, fractions, exponents
or god forbid algebra or calculus if whenever you have to add you have to be
"making ten" in your head. It needs to be memorized.

I believe this also holds true up to even much more complicated concepts than
basic arithmetic.

For example, I find if I don't understand the proof of some theorem in one of
my math classes, if I just copy it out of the book by rote, (translating the
notation line by line to something I'm more comfortable with) then I sometimes
will suddenly understand it.

\--

Addendum: I believe Kumon has a good method for teaching math. They put a very
strong focus on a rock solid understanding of basic concepts. If things
haven't changed, they give kids timed problem sets of basic problems to solve
(memorize) with the expectation of 100% accuracy.

~~~
rdtsc
> You'll be crippled trying to learn things like decimals, fractions,
> exponents or god forbid algebra or calculus if whenever you have to add you
> have to be "making ten" in your head. It needs to be memorized.

/joke: they should just start with category theory -- it will produce a fine
new generation of HN readers ;-)

> Thus we have convoluted methods of teaching addition like this that in some
> way do make sense, but are mostly just another layer of confusion.

I think everyone conceptualizes and encodes basic arithmetic stuff in their
own way. Some memorize it symbolically, some visualize, some think of number
lines, graphs and so on.

Trying to "help" by providing tables, coins and all kind of "aides" just
confuses some students who started to encode it differently.

~~~
RevRal
_I think everyone conceptualizes and encodes basic arithmetic stuff in their
own way. Some memorize it symbolically, some visualize, some think of number
lines, graphs and so on._

I've always had a tetris thing going on in my head.

------
zephjc
We can guess they want different groups of numbers that lead to the same sum
(e.g. 7+4=12; 10+2=12) but this lesson clearly has context outside of the
paper itself - that is, the teacher explained what to do, and these are
exercises based around that. Without that context, we're just guessing what
the goal is.

~~~
RevRal
That's a good point, but that doesn't make this assignment any less poorly
crafted. It is common for parents to help their children with homework, and I
would expect that to be taken into consideration when creating a worksheet
that is less straight forward.

Even if the child knew what they were supposed to do, they'd have the explain
the assignment to the parent... which they might explain incorrectly. Just
another level of confusion.

~~~
bartl
On similar homework from my kids, there is always a presolved similar problem
at the top of the page.

I can guess the intent of the exercise, but it would have been a lot easier if
there had been a solved example.

~~~
DougBTX
That was probably on the page before, no?

------
gcheong
I think I understand it, but for a second there I'm sure I felt a cold chill
go up my spine from some distant, forgotten, childhood trauma.

------
Tichy
I don't like this kind of "illustratory" maths. In my opinion it will only
distract the brain and make maths look more complicated than it is. My
neighbour's kid once had to draw squares on paper to do multiplication. If I
had been forced to draw 100 squares just to multiply 10*10, I would probably
have refused to do any more maths for the rest of my life.

~~~
RiderOfGiraffes
You're being up-modded for that comment, so people obviously agree with you. I
think it's important to realise that not everyone does, because not everyone
thinks the same way.

I've helped out in primary classes, and sometimes when I've done something
like this kids have been so excited they've jumped up and run around, unable
to contain themselves. They've suddenly seen _why_ something works, not just
been told to memorize it.

It's the _range_ of ways of thinking about things that matters. Here's one
that doesn't work for you. Fine. Find another. Use both. See how they're the
same thing, but from different points of view.

Sorry, I'll go away and stop ranting now - I'm just getting angry and will say
something I regret.

~~~
Tichy
Sure, if it works for others, I have nothing against them using it. But don't
force it on kids and make them hate maths.

Another silly thing I read about is associating numbers with animals. So "1"
is the crocodile and "2" is the elephant or whatever. I can only say - wtf?
Just another example - numbers are numbers, not elephants...

I just think to understand maths is to REDUCE the number of concepts (that's
the beauty of maths). A lot of teaching concepts seem to increase the number
of concepts instead.

I am not a teacher, though.

------
manvsmachine
To play the role of devil's advocate, I can see what this assignment is trying
to do. It's trying to teach that numbers (particularly discrete ones) are
representations of a _quantity_ , not just abstract symbols that follow some
arbitrary set of rules. I wouldn't be suprised if the kids who successfully
learned this way would have an advantage over the rote memorization kids if
you were to stick them in a Discrete class 12 years later. As soon as I saw
the grids, my mind was screaming "pigeonhole principle!" But that might just
be because I have a combinatorics exam in 6 hours.

------
wisty
In modern educational parlance, this is "present, practice, produce". But it's
missing the bloody present part. And the production.

A single worked example would save the student a lot of time.

Some educators like to say that students should be able to figure it all out
for themselves. That students are better off not being taught. I guess it
depends on the teacher - some teachers can add value.

------
RevRal
There seems to be a lot of clues but they all lead nowhere.

I don't think the goal has to do with adding to ten....

e: I think I get it, but I don't know what you're supposed to do with the
second grid.

e2: Oooohhh. The grids are there to _help_ you get to your answer.

------
billswift
From one of the comments "Whoever wrote this damned workbook is a mountebank
and a fool." That would be an ed major, maybe even an EdD.

------
vegashacker
It's funny, I just realized the way I do certain simple arithmetic is by
actually counting in my head. 8 + 3 is 8... 9, 10, 11. But something like 6 +
5 I just have memorized. Maybe cause my brain has decided it takes too long to
go 6...7, 8, 9, 10, 11. Or that I'd forget where I was without using my
fingers or something. And now that I think about it, for 8 + 7 I use a third
technique: I double 8, and then subtract one. Even though I'm pretty sure I
have that one memorized correctly, I still seem to always do that check in my
head. Ok, I think it's bedtime. Probably shouldn't admit any of this in a
public forum. :)

~~~
gb
I'm the same actually, although I am absolutely pathetic at arithmetic. 8+7
would probably involve counting for me.

------
RevRal
As far as teaching children the facts of numbers, I've always liked Eliezer
Yudkowsky's The Simple Truth:

<http://yudkowsky.net/rational/the-simple-truth>

Especially how it initially presents the concept as _magic._ Actually, maybe
not a good way of explaining numbers to children. But I like the story
nonetheless.

------
Luc
I don't get the strong reactions to this homework. As far as I understand the
assignment, it pretty much describes how I do arithmetic. It seems like a
natural way of working to me, since nobody taught me to do it this way (if my
memory doesn't fail me).

------
dazzawazza
I don't know if any of you have kids but this seems like a pretty normal work
sheet to me.

Don't condemn or applaud it, it's just one of many ways the children will be
taught to add. Some kids get it straight away and some don't.

It's just practice and practice is good.

------
dfarm
So simple that only a child could do it.

<http://www.youtube.com/watch?v=RA-dMSDPFhA>

------
mariorz
just change the arrows into equal signs and it looks pretty clear, though i'm
not sure if a first-grade child is more familiar with that notation. anyway
this homework is part of a larger lesson in which almost identical exercises
were solved ad nauseam. unclear textbooks isn't really one of the problems
with education at any level. also: flagged.

~~~
RevRal
I think the grids are throwing people off.

This is what the child is supposed to do: Grid (Draw in counters for second
number, then count total of both grids) -> Fill in both bottom blanks with
counted total -> Ten plus what equals bottom blank?

This is what we're doing: Add numbers -> Why the hell is there only one grid?
-> Then we post our child's homework onto the internet

~~~
sheena
Agreed. The visual placement of the grid is a large part of the problem. You'd
be able to discern the flow (and logic) of the assignment far better if the
grid were placed in between the two sums, for example.

------
gojomo
Before giving this out, several identical questions would have been worked out
in front of the class. So it's not a "figure this out from the worksheet"
matter; it's a "recognize it's like the problems we did earlier".

It that context, it's a plausibly useful exercise; it's analogous to how I
(still) total and multiply numbers in my head: determine the similar problem
that gets to either a nice round number, or a memorized result -- then adjust
for the remainder.

