
Can 1/3 and 1/3 = 2/6? It seemed so - ColinWright
http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-so/
======
forbiddenvoid
The units are missing, and I think that's a key factor here.

Both of the equations up on the board at the end are correct because they are
counting different things. This is a huge miss if you use a numeric only
approach to fractions.

The student came up and wrote 1/3 + 1/3 = 2/6.

What they meant by that is 1/3 (of the students at a table) + 1/3 (of the
students at a different table) = 2/6 (of the students at those tables).

The teacher then demonstrates an entirely different formula: 1/3 (of the
students at a table + 1/3 (of the students at a table) = 2/3 (of the students
at a table).

The confusion comes because no one calls out that they're talking about
fractions of different things.

Edit: There are a whole range of exploratory questions you can follow on from
here as well.

Imagine if the tables have different numbers of students or if there are more
than two tables. Helping students navigate these types of ratio
transformations is why keeping track of units is so important. Otherwise,
things can get hairy for the students very quickly.

~~~
Wowfunhappy
You're right, but the tricky part is, how do you explain that to children who
are _just_ being introduced to the concept of adding fractions, without
leading them off track? That's _hard!_

> The confusion comes because no one calls out that they're talking about
> fractions of different things.

But she did: "When thinking about fractions, it’s important to keep your
attention on what the whole is. [...] you’re thinking about the two tables
together."

Now, I think something closer to what you're suggesting is, the teacher could
have written the following two equations on the board:

 _" 1/3 of the students at the first table + 1/3 of the students at a second
table = 1/3 of the students at_ both _tables "_

 _" 1/3 of the students at the first table + 1/3 of the students at the first
table = 2/3 of the students at the_ first _table "_

Accompanied by some drawings, maybe that would have worked. But I think it
could just as easily end up confusing everyone—you've made the concept of
addition _much_ more complicated! And sure, the real world is more complicated
too, but you've got to learn the basics first.

\---

The more I think about it, the more I think the best response might have been:
"No, you can't do that, because those kids are at a different table. If we
added another third of the kids at the _same_ table...", and move on. Ignore
the confusing example and refocus on the simple one.

~~~
DigitallyFidget
The issue is that it's ratios, not fractions. 1:3. You take 1:3 and 1:3 and
it's still 1:3 or 2:6. You haven't changed the ratio for this at all by
showing it as fractions, it's simply being represented and presented without
correct context.

~~~
TeMPOraL
Isn't this why you end up seeing these "silly" units like kg/kg in chemistry?
So that, while the value is technically dimensionless, it doesn't get added to
another dimensionless value (e.g. l/l) that's a ratio of values of a different
dimension?

~~~
romwell
This hits the nail on the head:

1 (person at table A) / 3 (people at table A) can't be added to 1 (person at
table B) / 3 (people at table B) without conversion of units.

~~~
dwild
And then it's easy to make them see how her answer was right in its own way by
adding another unit, (person at table a+b and people at table a+b) and show
why it may be harder to works like that for now.

I heard so many people complains that each year in maths they would
essentially learn that everything they learned the year before was wrong...
can we fix that please?!

------
wcarey
This is a lovely (edit: having been in similar shoes, also terrifying-in-the-
moment) example of a broader problem in teaching mathematics: the language we
use to describe mathematical reasoning is a natural language, like English or
Latin, and therefore full of the sorts of bizarre irregularities you'd find in
a natural language. Mathematics is also a language about rigorously and
precisely defined objects. The conceptual shear between those two things is
murder for lots of students.

Just like Tacitus omits his verbs (!), when we describe fractions we often
omit the implicit definition of the whole. Turns out that's a problem for many
students.

It's a bit like trying to learn a context-dependent programming grammar with
an inconsistent API, but worse, because it's your first "mathematical"
language so you're also trying to learn what the abstract objects the language
manipulates _are_.

Some other lovely examples:

3(5) means three times five. 3(x) means three times x. 35 means three times
ten plus five. 3x means three times x. x(3) means that x is the name of a
function taking, in this instance, 3 as its input.

x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.

\sin{30}. Radians or degrees? Probably the writer means degrees, but there's
no way to tell.

There are many more.

~~~
twic
> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.

That notation for inverse functions is truly appalling. I don't know how the
first mathematician to think of that didn't immediately discard it as
nonsensical and misleading.

~~~
rocqua
It's not because 'function powers' make sense, yhey are just iterated function
application. That's how they work for the natural numbers and when you extend
that to the integers, you immediately get f^-1 for the inverse.

Notation in higher level maths is almost always very ambiguous. Because many
concepts are analogues of each other and to reflect that notation is just
taken from the analogue. Within a single domain, (like high-school arithmetic)
you will usually not have this ambiguous problem. But once you move past that,
it is something to get used to.

~~~
JadeNB
> It's not because 'function powers' make sense, yhey are just iterated
> function application.

To be clear, I think you aren't saying "It's not because 'function powers'
make sense" (which seems to apply that's not the reason it's done, and
possibly that the reason isn't correct), but rather "It's not [an appalling
notation], because 'function powers' make sense"—to me, that extra comma
changes the meaning!

~~~
rocqua
Indeed!

------
adrianmonk
The difficulty isn't with fractions. It's about understanding what "+" means.

Words have multiple meanings/senses. Addison knows the word "plus" already and
knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean
combining things in other ways ("tonight, we'll eat pizza plus see a movie").

His teacher has introduced "+", and pronounced it "plus", so it's reasonable
for him to apply what he knows about the word "plus" to the symbol. People
even use "+" (rather than "plus") to mean combining things. Maybe Addison saw
"Nature's Path Pumpkin Seed + Flax Granola" at the grocery store. So why
shouldn't he try using it that way?

Somebody needs to communicate to Addison that, in math class, "+" always means
something specific. He doesn't know that yet, but he has been asked to use the
symbol anyway.

Math uses a whole lot of lingo. If it's not covered well enough, stumbling
over the terminology can be an impediment to learning. This includes both new
words ("quotient", "integer") and words that are used in everyday language but
differently in math ("where" meaning condition or definition instead of place,
"of" meaning multiplication, "real" numbers).

~~~
Phlogistique
That's my favourite explanation so far. That's how I would put it to the
student:

Yes, that's a perfectly correct operation. When you take 1/3 of the first
table and you put them toghether with 1/3 of the second table, you get 2/6 of
both tables. However, that's not what mathematicians mean when they use the
sign "+". Let's explain the difference with examples:

* At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table.

* Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables.

The first operation is what mathematicians call "+". They write "1/3 + 1/3 =
2/3"

The mathematicians do not have a good name for the second one, so let's invent
one: "1/3 1/3 = 2/6"

What's better with this explanation, compared to the "ratios vs. fractions"
thing, or the units thing, is that you do not have to introduce a separate
category of numbers that sound very similar but act differently.

~~~
wcarey
We explicitly encourage the "let's invent name for this similar but unnamed
thing" with our students, and use whatever name they come up with for the
thing for the rest of their time with us.

Sometimes those student created names become legendary. They really enjoy the
idea that they can have "Jane's relation" be used by students after they
graduate. (Of course, sometimes Jane's relation is really Euler's totient
function, and we have to encourage them that even though they found something
a very good mathematician also found, we're going to call it by the common
name.)

------
pdkl95
As others have already pointed out, the question is ambiguous without more
information. _Always_ include the unit; bare magnitudes could mean anything!
The student was describing the _mediant_ [1], which was the correct solution
when the problem is interpreted with forbiddenvoid'a units[2].

"Adding" fractions with the mediant leads to fun things like the Farey
sequence[3] (related to Ford circles[4]) and the very interesting Stern–Brocot
tree[5]. (Numberphile has a nice introduction[6] to the fun properties of the
Farey sequence)

[1]
[https://en.wikipedia.org/wiki/Mediant_%28mathematics%29](https://en.wikipedia.org/wiki/Mediant_%28mathematics%29)

[2] "1/3 (of the students at a table) + 1/3 (of the students at a table) = 2/6
(of the students in the room)"

[3]
[https://en.wikipedia.org/wiki/Farey_sequence](https://en.wikipedia.org/wiki/Farey_sequence)

[4]
[https://en.wikipedia.org/wiki/Ford_circle](https://en.wikipedia.org/wiki/Ford_circle)

[5]
[https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree](https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree)

[6]
[https://www.youtube.com/watch?v=0hlvhQZIOQw](https://www.youtube.com/watch?v=0hlvhQZIOQw)

------
CydeWeys
I guess you'd have to come up with a way to explain that adding numbers (which
is what you're doing with 1/3 + 1/3) is not the same as combining/averaging
fractions, i.e. when you're totaling subgroups into a larger group. It's
almost like we need a different "combining" operator for the latter that means
to add both the numerator and denominator, because + isn't right for this. Now
that I think about it, I'm surprised there is no such operator for averaging.

It's not as bad of a nightmare as trying to explain the answer to the missing
dollar riddle:
[https://en.m.wikipedia.org/wiki/Missing_dollar_riddle](https://en.m.wikipedia.org/wiki/Missing_dollar_riddle).
That's an absolute nightmare.

~~~
didgeoridoo
Someone in the comments makes a good point that the best thing to do here may
be to introduce ratio notation for proportions (e.g. 2:4) which CAN be
added/combined according to the kids’ intuitions — 1:2 combined with 1:2 does
indeed equal 2:4, which reduces back to 1:2.

You could then teach how to go from ratios to fractions by adding the ratio
sides together and putting that in the denominator for each side... poof,
you’ve invented averages!

No way someone would have come up with that approach on the fly, though.

~~~
CydeWeys
Yeah that's a good approach. The problem remains though that if you use the +
operator on ratios you're still overloading it to mean something different in
a way that doesn't retain its meaning when you start expressing things as
fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you
still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just
spitballing here. There's definitely enough options in Unicode that an
existing operator should be suitable for this purpose:
[https://en.wikipedia.org/wiki/Mathematical_operators_and_sym...](https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode)

~~~
wcarey
And + is overloaded in a bunch of ways students encounter in high school, and
much time is spent talking about when you're allowed to add and when you're
not and which rules apply when. Examples:

1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set
of rules.

1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe
modular.

15% + 20% - who knows? 15% of what? 20% of what?

(1,2) + (2,4) - can't be done.

a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.

It would be lovely if mathematics were taught as a strongly typed language
without overloaded operators, alas all our corpus is in the language it's in.

~~~
uryga
> (1,2) + (2,4) - can't be done.

are you talking about points or open intervals? the notation is ambiguous ;)
but addition of points is fine if you look at them as vectors

> a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.

what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of
Pythagoras theorem which is widely considered "fine" afaik ;)

~~~
wcarey
Indeed. And if they're cartesian vectors, you're good. But if that second
number is an angle measured in radians, you use yet a different set of rules
for the addition.

The conversation with 16 year olds when you explain that their previous
teacher who told them that you couldn't add points wasn't lying, but was,
perhaps oversimplifying things to make their life easier, is a fun and fraught
one.

I've had to reason kids through the fact that a^2 + b^2 is not equal to (ab)^2
or even (a+b)^2 more times than I can count. What's particularly difficult is
that, confronted with the fact that 25 and 49 are manifestly different
numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the
"law of distribution", which they haven't learned as the "law of distribution
of multiplication of monomials over addition, and only that".

~~~
CydeWeys
How did I go all of my life so far without hearing the super useful word
"monomial"? It's such an obvious concept to have a meaningful name, and yet I
don't recall anyone ever having said that word.

~~~
wcarey
I also didn't encounter it until I was relatively older. Our school has been
pushing to introduce more rigorous language and definition in our 7-12th math
program. For some students, it really seems to help. For others, it's really
hard.

------
JBorrow
A great opportunity to introduce multiplications of fractions!

1/3 of the students at table A are girls. 1/3 of the students at table B are
girls. What fraction of the tables does table A represent? A is one out of two
tables, so A is 1/2 of the tables. Likewise, B is 1/2 of the tables, too.

When we want to consider the whole here, we need to take into account what
fraction of the whole each proportion represents.

The question that we want to answer is 'what proportion of _all students_ at
_all of the tables_ are girls?'. This is a combination of the question 'what
proportion of students at table A are girls, and what proportion of students
at table B are girls', and 'what proportion of all of the tables does each
table represent'? That second question might seem quite convoluted but it is
important!

To do this, we need to multiply the fractions together like so:

(Fraction of tables that A represents) * (Fraction of students at table A that
are girls) + (Fraction of tables that B represents) * (Fraction of students at
table B that are girls) = (Fraction of students at tables A AND B that are
girls).

So in this case we would have:

(1/2) * (1/3) + (1/2) * (1/3) = (1/6) + (1/6) = (1/3).

This is even clearer when we consider the case where there are two girls at
table B. There, we can do the same thing:

(1/2) * (1/3) + (1/2) * (2/3) = (1/6) + (2/6) = (3/6) = (1/2).

~~~
quietbritishjim
I think this one of those situations where you do a bunch of working out and
get to the end and see that, mathematically, the problem is fixed, but in your
heart it still _feels_ like the original problem is still there. (1/2) * (1/3)
+ (1/2) * (2/3) might seem like a small calculation to us, but if you've just
encountered fractions for the first time I think that is a huge amount of
abstract notation.

Instead, I think it's better to follow CydeWeys's suggestion of saying that
both are correct results of combining 1/3 with 1/3, but they're two different
ways of combining them. Say that when you combine two fractions within the
same group we call it "addition" and use a plus, but when we combine two
fractions from different groups we call it "averaging" (and maybe make up your
own symbol for it).

Once you've talking about averaging a bit you can move on to multiplication,
which in some ways is a more basic concept but, for fractions, is actually a
bit less intuitive.

------
wondringaloud
I think this is a perfect example of why people find math hard. Not because
there's something inherently difficult, but because we have teachers like this
one in charge of open and curious minds at a young age. You get a teacher who
doesn't quite understand what it is she's trying to teach, and this confusion
is multiplied over and over again for each batch of students that pass
through. Not too long afterwards you get kids who think they don't understand
math, when in fact it was a failing of the teachers who couldn't correctly
explain it in the first place.

------
Wowfunhappy
Something I missed the first time around—in the comments below the article,
the teacher expands on what she actually did. This was originally one long
paragraph, but I've added paragraph breaks for readability:

\---

> At the time, I let it go so I could think and regroup myself. There was more
> than fractions to think about: Is it possible for both equations to be true:
> 1/3 + 1/3 = 2/6 and 1/6 + 1/6 = 2/6\. This was a long discussion that kept
> their interest, in which I learned that many (most?) of the students didn’t
> think of fractions as numbers. That was another hurdle. I kept going back to
> what we know about adding whole numbers.

> Then I took another approach, once I was sure that they understood that 1/3
> and 2/6 are equivalent, so how can I add a number to itself and wind up with
> a sum that’s the same as one of them? In all of these discussions, students
> would change their thinking. We looked at adding on a number line. I used
> pattern blocks to explore the same problem. I kept talking about keeping our
> attention on what was 1 whole.

> My, this all takes time, and the time is important for students to develop,
> cement, and extend their understanding. What I didn’t do, that I’ve been
> thinking about now, is to make it part of a writing workshop on persuasive
> writing and have them choose a conjecture and write. I think I could spend
> most of the year on this with students.

------
Spivak
I can't imagine doing this to a class of fourth graders but I don't think her
thinking is wrong. I think the correct way to views maths notation is that
it's a language and we should treat people using it "incorrectly" as a grammar
mistake and try to understand the idea they're trying to express.

The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get
across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled."

It's hard when the language required to express the ideas they're having is
just a little to advanced. And nothing about this stops when you get older.
The "just a little outside your knowledge" keeps stretching on forever.

~~~
macspoofing
>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to
get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled."

I disagree. The fourth-grader is perfectly correct within the context of the
analogy that the teacher used. The problem is the analogy is wrong which is a
general problem of relying on metaphors and analogies to explain rigorous
technical concepts. Fractions are not like tables of girls and boys. There are
rules for how you add fractions that flow from the underlying axioms. Those
rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of
any intuitive reason, but because it's disallowed by the 'rules' of fraction
addition - that's it.

~~~
tsimionescu
> Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not
> because of any intuitive reason, but because it's disallowed by the 'rules'
> of fraction addition - that's it.

That is absolutely incorrect. There is an intuitive reason why 1/3 + 1/3 !=
2/6\. That reason is that 1/3 + 1/3 = 2/3, and 2/3 != 2/6.

The important thing here is to help students build the intuition that
mathematical notation shouldn't be treated mechanically, you should think
about what the notation represents. The temptation to say 1/3 + 1/3 = 2/6 only
comes when you're blindly applying operators to notation.

Now, the example from the classroom is more subtle, because it deals with an
improper translation of the English into mathematical notation. If I say 'one
third of the students at table 1 are girls', that should be translated to 1/3
* 3, not 1/3\. Applying this rule gives 1/3 * 3 + 1/3 * 3 = 2/6 * 6,which is
perfectly correct. Similarly, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 is obviously
correct.

~~~
macspoofing
My only point is that analogies are flawed. The abstraction that they provide
hides complexity that at some point will leak out.

>There is an intuitive reason why 1/3 + 1/3 != 2/6.

You and I have different ideas of what 'intuitive' means. It's 'intuitive'
once you understand the rules of fractions and what they mean. It's not so
easy to derive this rule if you're working in the space of real world things.

And sure, I agree you can ad hoc extend the analogy of tables to bring it in
line with the underlying mathematical rules, but then your analogy is no
longer as simple as it was. The complexity is leaking out of the abstraction
you had it under.

>The important thing here is to help students build the intuition that
mathematical notation shouldn't be treated mechanically, you should think
about what the notation represents

Sure, using abstractions and analogies is a powerful way of teaching. All I
did was point out that analogies have limits and at some point they can become
detrimental to understanding the fundamental concepts.

This is a common complaint by Physicists when doing public lectures on Quantum
Mechanics and then having people extrapolate from the metaphors to derive
incorrect physical rules (e.g. faster-than-light communication from a shallow
understanding of quantum entanglement).

>because it deals with an improper translation of the English into
mathematical notation.

It isn't just about the improper translation to English. It is also about the
improper mapping of fractions to real-world things. 2 girls out of 6 kids in a
table maps nicely to the fraction 2/6\. But even though 2/6 is equivalent to
80/240, the latter is a little harder to map to a table of 6 kids and 2 girls
- don't you think?

>The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying
operators to notation.

I disagree with that in context of learning how fraction operators work. The
fourth-grader logically extended the analogy that they were given because
conceivably, there could have been an operator defined that matched their
intuition, for example, let's call it '@' and define it (not rigorously) as
"a/b @ c/d = (a+c)/(b+d)". This operator, if existed, would work very well for
combining tables of boys and girls and getting the fraction of girls to match
the fourth grader's intuition. The fourth-grader is learning fractions for the
first time, and that operation could have conceivably existed - so the only
reason they were wrong is that they haven't been told what the rules of
fraction addition are and NOT that they misunderstood the analogy. The problem
is that the "+" operator does not work that way because it isn't defined this
way as per axioms for fractions.

~~~
tsimionescu
I still don't agree. Sure, there are limits to particular intuitions, but all
of the rules make perfect sense with real world quantities.

For example, 1/3 of an orange + 1/3 of an orange is actually 2/3 of an orange,
not 2/6 of one. And 1/3 of 1 kg of flour is exactly 2/6 or 300/900 of that kg
of flour. Sure, it's hard to talk about 1 Graham's number / 3 graham's number
of 1kg of flour, so it does break down at some point, but unless you go
overboard with quantities, all of the rules for fractions are in fact
intuitive, and important for day to day things like cooking and money
management. In fact, fractions and their operations are probably older than
the idea of abstract rules, because they are fundamentally useful things.

The child in this example wasn't even making the mistake of thinking the rule
for + is the rule for your @ operation. They were confused because they were
trying to apply the intuition they had built up for how to translate real-
world problems into fractions in the wrong way. Their result was in fact
physically true: it was true that 1/3 of the children at one table + 1/3 of
the children at the other table was equal to 2/6 of the children at both
tables. This was confusing them because it suggested a different way of
manipulating the numbers than they had just been shown.

The right solution, again, was to teach them how to translate 'a fraction of
something' to rational numbers - that is, to multiply the fraction by the
something, with only a special notational case when that something is 1. If
they had known to do this, their intuition would have translated directly into
the correct algebraic formula. No need to learn the abstract rules yet.

------
jnbiche
In the comments, an educator references an excellent article related to this
confusion, "When Can you Meaningfully Add Rates, Ratios, and Fractions" that
implicitly suggests some pedagogical approaches: [https://flm-
journal.org/Articles/11019C10CF34E90DC5866E53E90...](https://flm-
journal.org/Articles/11019C10CF34E90DC5866E53E905E8.pdf)

------
xigency
When you want to add fractions of a whole together.

    
    
         1     1     2
        --- + --- = ---
         3     3     3
    

When you want to add wholes together and make a new fraction.

    
    
         1  +  1     2
        --------- = ---
         3  +  3     6

------
knappa
From a 'not currently in front of the class' perspective, it's pretty clear
what the student meant by 1/3 + 1/3 = 2/6\. They were taking + to mean
something like a general 'and' or combining action, not in the strict sense of
standard fraction addition. It even has a name in mathematics, 'Farey
addition'.

The kids clearly want to write it in shorthand, so maybe the thing to do is to
come up with another symbol for this similar but distinct operation. For
example, ⊕.

------
chx
I happen to have a math teacher masters though I myself do not teach (but I do
help with the program of a tiny, tiny reform school). This teacher here had
painted themselves into a corner and it's hard to get out of it. Do not
explain fractions with things you can't change the denominator of.

Rather tell it with money. Say, 1/3 means if the table has 3 coins , one kid
gets 1 coins. If there are 6, 9, 12, 15, one kid gets how many? If the other
table has five kids and 5 coins then 1/5 means when splitting five coins a kid
gets one. Figure out together what happens with splitting 10, 15, 20 coins.
Now putting together the two tables we want to calculate 1/3+1/5, how many
coins can we do that with? Step through it, we practice 1/3 with 3, 6, 9, 12
... but can you tell what the fifth of nine coins are? You can't ... Then find
15 and then show them 1/3+1/5 means 8/15\. This is all play and very smooth.

To answer the question posed in the blog post: I would plan the class
carefully to avoid the entire situation. But if I must, I'd point out 1/3 is a
mere shorthand for division, 1:3 and writing "1:3 + 1:3" is adding two
operations together and it does not even make sense. We can restore sanity but
that has its own rules.

------
ash
One of the commenters gave a great suggestion:

> I would use bar models to show how the whole changes. First, draw one bar
> (table) with 3 students inside; label the bar one whole, star one student
> and label that student as one third of that whole. Do the same thing beside
> the first bar model, again showing the bar as one whole, starring one
> student and labelling that student as one third. Then, push the two bar
> models together (I’d use a Smart Board) to show a new whole: the two bars
> together with 6 students in the one bar is labeled as the new whole. Point
> out that this is a NEW whole, with a different # of pieces. Now they would
> see the 2 starred students in the one new bar made up of 6 total students,
> so the “old” 1/3 student becomes the “new” 1/6 student once the whole
> changes. This is like a name change, once the size of the whole changes. You
> can also show them that by cutting each “student” in half, you would get an
> equivalent fraction of 1/3 = 2/6 of the first one whole bar. So 1/3 = 2/6,
> not two 1/3’s = 2/6.

[http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-
seeme...](http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-
so/#comment-27347)

------
jessermeyer
Proportions are tricky to introduce since they are the first obvious move away
from absolute quantities. We're taught that division is just fancy
subtraction, but it's actually the more subtle idea of proportionality.
Similar with multiplication as dimensionality.

From here, it feels like the natural setup to show that you can't just
'combine' proportionalities without accounting for what portion these
proportions contribute to the new whole.

~~~
wcarey
Perhaps an argument for arithmetic followed by geometry using Nicomachus and
Euclid. We sit on that until 9th grade, but I wonder how young you could go
with it?

~~~
jessermeyer
Introducing line segments as alternative representations of numbers at this
point feels very natural, and is already implied by most circulum with the
standard 'number line'. As you say, we don't do anything with that until much
later.

Let's leverage that early on!

------
gilbetron
"If we reduce 2/6, we get 1/3, so 1/3 + 1/3 = 1/3, that seems weird, what did
we do wrong?"

------
logfromblammo
This is why I taught my kids to label units.

( 1 girl / 3 students at table L + 1 girl / 3 students at table R ) can't be
added, because the denominators are different.

To fix that, you can multiply by the factors ( 3 students at table L / 6
students at both tables ) and ( 3 students at table R / 6 students at both
tables ) so you get ( 1 girl / 6 students at both tables + 1 girl / 6 students
at both tables ) = ( 2 girls / 6 students at both tables ).

To hammer it home, you point to the table that has 2 or 4 students at it, and
ask someone how to add the proportion of girls at that table to the two
already under consideration.

------
matvore
I think the problem is the teacher moved from one type of "fraction problem"
to another kind.

The example of drinking 4 bottles of water to only have 2/6 left of the
original pack is doing an integral problem and wrapping up the answer as a
fraction. Same deal with the pack of 12 pencils.

The students can grasp those problems as e.g. (1+1)/6 rather than (1/6 + 1/6).
In other words, there is only one "whole" in the problem.

When you're adding the fractions of desks filled with students, the fraction
is counter-intuitive because both desks have to have the same number of
students for it to make sense. (1/3 of a 5000-student round table is different
from a 3-student table). And to say "the units are wrong" is kind of a limited
way of explaining this. The units _also_ need to have the table capacity as
part of the unit identity (i.e. the units would have to be 3-student-table).
That's a pretty sophisticated way of thinking about units.

I think that after the pencil/water examples, transitioning from the
pencil/water pack examples to a more "pure" fraction example would be better.
e.g. one group of students eats 2/3 of a pizza, and another group eats 2/3 of
a pizza. Now you can throw away one of the original pizza boxes and put the
two remaining 1/3 in a single box which is 2/3 full. Now the "whole" for each
fraction is no longer arbitrary (like 6 bottles of water or 12 pencils).

------
biddlesby
I would say it was a mistake to introduce the concept of adding fractions from
two different "wholes". Instead, teach that in order to add fractions, you
first have to get everything into the same "whole".

Like, you can add Jack / 3 to Ben / 3 because they were at the same table. But
adding the boys from one table to another is quite a different thing.

Instead, you should teach that you should first make the fractions with both
tables as the whole. Only then are you allowed to add. This could come as a
later concept

------
karmakaze
Use pie graphs to show the original tables split up into thirds of one girl
two boys.

Then show two ways of combining two tables.

(>-) combined with (>-) is a table of six: 2 girls, 4 boys show all boys and
girls moving from tables with 3 seats to the same table with 6 seats.

Now show a table with three seats, one occupied by a girl. Show another table
with same configuration. One girl moves from one table to join the first
table. 1/3 full + 1/3 full (at the same table) is 2/3 full.

------
ValorieO13
I solved this by then introducing the kids to slices in a pie. Visually, the
kids can see that 1/3 of a pie plus 1/3 of a pie is not the same as 2/6\.
Fractions are often introduced as "parts of a set" which is great for the
first day. It's easy for kids to see and understand, but kids are more
familiar with trying to slice the birthday cake for all the kids at the party.
After the kids realize there are _two_ kinds of fractions, we spend time at
each problem thinking through is this a set (which we'll return to in ratios)
or a whole (which can't be expanded) and visually representing fractions both
ways to check their answers. Adding units is absolutely important, but when I
teach it, that's what I add in fourth or fifth grade when I'm teaching them to
reduce.

------
08-15
That's what happens when you try to "build an intuition" for a concept instead
of defining it.

Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack
is three bottles. I can even add them: one third (of a sixpack) plus one half
(of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a
sixpack? _That don 't make no sense!!_

The whole point of common fractions is to close integer arithmetic over
division by introducing a new kind of numbers, precisely those fractions that
_don 't_ correspond to integers. The question isn't so much "What _is_ half a
bottle of water?", it's " _How_ do you calculate with one half (a bottle or
whatever)."

Math is beautiful, but only if built up from simple rules. It's amazing how
much you can make from counting and a handful of convenient notations. Even
children can recognize that beauty. But high school instead teaches math as a
jumbled mess of things you have to memorize, without structure, without rhyme
and reason. Needless to say, I hated it. (School, not math.)

~~~
wcarey
As someone who has pushed pretty hard towards precise definition and rigor as
a teacher, you're in the (happy!) minority being able to approach mathematics
that way. For lots of students, the process of moving from definitions and
simple rules to other statements is extraordinarily difficult.

I teach the properties of exponents to 14 year olds as a unit on logical
necessity - the properties all flow necessarily from the definition of the
exponent. About a third of the students say, "cool" and never miss anything on
any assessment again because the answers flow necessarily from the givens.

About a third work their way through it fine.

About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite
working many particular examples where that is manifestly false.

~~~
08-15
> About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2

...and I bet, their justification is "It looks right!" or "Why not?" I've seen
this, too.

A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b.
Why? Because it looks right, and why not? And it isn't so different from (a
_c) /(b_c) = a/b, is it? Of course, she never asked _why_ a certain
manipulation is legal. Arithmetic manipulation is legal because the teacher
said so.

I suspect, this isn't a property of their personality; rather it's a failure
of their math teachers to convey the idea, that everything has a reason, and
if you don't know the reason why some manipulation would be allowed, it very
likely isn't. By the time you tried to teach exponentiation to these students,
the damage was long done.

------
hathawsh
This is fantastic material. I'm going to present this quandary to my kids.
Some of my kids struggle at math, so it will help them understand that math is
a language and, like any language, it has abbreviations that sometimes lead
you down a confusing path until you spell out more. Some of my kids are good
at math and it will help them relate with others.

------
benlivengood
I love this discussion and want to throw in my two cents. Why not teach
beginning students group theory as the foundation to addition and subtraction?

While it wouldn't solve this particular problem since fractions require
fields, it would give teachers the tools to explain precisely why addition of
two tables is not a group or field operation; they are from different sets and
require _measurement_ with _units_ to define a new inclusive set.

My own understanding of arithmetic and algebra was enhanced immensely by a
book outlining the derivation from Peano axioms.

Ultimately the concepts are very simple and kids intuitively do the steps
without having the names for the mental processes that they're doing. The
educational hurdles to me seem to be 1) identifying the memorized names of the
sequence of natural numbers with a procedure for generating them; 2)
distinguishing between procedures, objects, numerals, and numbers; 3) basics
of set theory; 4) natural induction.

~~~
GuB-42
Bad idea!

They tried exactly that in France in the 60s and 70s. It was called Maths
modernes (New maths). I think it was the same in most of the western world. It
was a disaster and they abandoned it in the 80s. The problem was that while it
may have helped the best students understand advanced concepts later on, it
produced a generation of people who lacked practical counting skills.

EDIT: By "bad", I don't mean "stupid". The people who came up with new maths
were experts in their field, with the goal of having a more scientifically
literate population. No doubt very smart people, but maybe a little too smart
for their own good. Turned out these concepts were too much for most young
kids, and too remote from the way math is used on a day to day basis.

------
EmilioMartinez
My take: Adding proportions is kinda "semantically incorrect". It's well
defined as a value, it just doesn't mean what you think it means, akin to
thinking that union of two sets is written A + B (which would be the set of
pair-wise additions of elements). Now, dovetailing the equation with units
that signify concrete quantities (as suggested by others) circumvents this by
making it meaningful, at the cost of no longer working with the abstract
concept of proportion.

An example of semantically meaningful manipulation of proportions is the
weighted combination of proportions. This here's a more verbose version of
what the teacher wrote: (3/6)* (1/3) + (3/6)*(1/3) = 1/3 (that is, the
weighted average of the proportions give the final proportion)

------
luord
> What would you do now???

I would probably start crying, to be honest.

But in all seriousness, this is the kind of thing that makes me dread
teaching. I mean, I know what is wrong, I just can't wrap my head around how
to explain it, specially without going into stuff that would be too advanced
for these particular students.

------
devit
It's easy to see that this cannot be a valid definition for fraction addition
because applying it along with the rule that a/b = (ak)/(bk) gives a
contradiction:

3/7 = 2/4 (+) 1/3 = 1/2 (+) 1/3 = 2/5, absurd because 2/5 != 3/7

Once this is realized, it's easy to see that the correct formulation for the
"table joining" operation used here is a weighted arithmetic mean, i.e. a(/)b
(+) c(/)d = weighted arithmetic mean of a/b and c/d with relative weights b
and d = ((a/b) * b + (c/d) * d) / (b + d) = (a + c)/(b + d).

On the other hand, fraction addition is clearly determined as a/b + c/d =
ad/bd + bc/bd = (ad + bc)/bd given the a/b = (ak)/(bk) and (a/b + c/b) = (a +
c)/b axioms.

~~~
karatestomp
2/4 + 1/3 would _clearly_ be 5/12.

You halve each to put them in terms of the two wholes you're combining (which
you must be doing, since you're adding them—else where do those go?) then
combine those. Same process that gives you 1/3 + 1/3 = 1/3.

Worked out (conventional meaning of + for clarity; we'd need a new operator
otherwise):

1/4 + 1/6 = 6/24 + 4/24 = 10/24 = 5/12

------
zuminator
1/3 + 1/3 does equal 2/6.

1/3 of one table + 1/3 of one table = 2/6 of two tables.

------
wwarner
I would express this in a diagram with a dotted line around the set under
consideration. When dealing with thirds, make a diagram containing three
objects with their respective properties, and end by drawing a dotted line
around the set. When switching to sixths, add three more objects, _erase_ the
line around the first three, and draw a new dotted line around the whole set
of six. Then go on to sevenths by adding one more object, erasing the dotted
line for the six and drawing it around all seven.

If I do it this way, my 4th grader can add fractions just like numbers without
resorting to common denominators.

------
djmips
As the concept of fairness is innate in primates including humans in the 4th
grade I wouldn't stop and correct the students but try and re-frame the
problem in terms of a problem where fairness was involved and then immediately
the ratios and the units would come into more clear focus. As an example I
might say that if I promised to give you 1/3 of the pizza as a reward for
cleaning your room. And then how much should you get if I had 1 pizza. How
about 2 pizzas? Things like that.

------
boomlinde
Of course 1/3 + 1/3 = 2/6, and in this case you can make an error in reasoning
that is probably common among children who are just learning fractions that
will still arrive at the same answer.

In this case I would have demonstrated a case where the error in reasoning
that was possibly made does not lead to the correct answer, for example 1/8 +
1/4\. The erroneous intuition of simply adding the groups and their subsets
together breaks down when you can show that 1/8 + 1/4 ≠ 2/12 but 3/8

~~~
boomlinde
And now I feel stupid for obvious reasons

------
kingkawn
It’s 1/3rd of something. what that something is matters. Fractions express
proportions of a whole, so you can’t add and subtract them without taking into
account what whole they refer to.

------
prvc
Her first example, "proportion of a six-pack of water" is actually very
complex, with a lot of potentially distracting details. Nowhere does she
mention that's what her fractions are measuring, either! The whole lecture was
simply a mass of detail with no reference to the underlying concept it was
putatively intended to explain. No wonder the children were confused. Why not
start with cutting a pie? Guess that doesn't lead to nearly as much busy-work.

------
Geee
I would have said that a whole table is 3/3 and that both tables together are
6/3 and not 6/6 (maybe they'll get it that 6/3 equals to 2). Probably a good
idea to explain that the "whole" isn't all pens and bottles but the group of 3
or 6 or 12 or whatever.

Maybe start with explaining from the start what happens if there are 7
bottles, when bottles are counted in the groups of six-packs. How many six-
packs can you make from 15 bottles?

------
0xfaded
Not that I would have thought of this on the spot, but ...

The numerator and denominator represent different things. The numerator is
"how many things", and the bottom is "of how many spaces". Asking what
fraction of the table is girls is dangerous, because the answer is (1 girl) /
(1 girl + 2 boys). Still a stretch, rephrasing as "what fraction of seats are
filled by girls" might have helped show the idea that the number of spaces is
fixed.

------
imetatroll
Have the kids stand up and count out numbers.

Now they know how many kids are in the room.

Then have them go and stand in two different corners. Count one corner.

Then have them stand in three different corners. Count one corner.

Finally the four corners. Count one corner.

Then have the two tables distribute their students to the corners and figure
out these elusive fractions.

Have each corner be a Pokemon or something. That should do the trick.

------
Jabbles
Don't use tables of the same size:

1/3 + 1/4 = 2/X

There would be no confusion as to what X should be.

And as someone else said, use units:

1/3 of table A + 1/4 of table B = 2/7 of both tables.

------
carapace
[http://iconicmath.com](http://iconicmath.com)

We probably haven't come up with the best visual abstractions yet, eh?

------
contextfree
Wow, I had no idea this blog existed - I loved Marilyn Burns's children's math
books as a kid, awesome to see her still in action.

------
krm01
That’s the same as saying: if you put 1/3rd pizza slice on top of an other
1/3rd pizza slice, you get 2/6 pizza. Layered math.

------
lucio
When you put two tables together, each table has 1/2 of the total number of
kids. By having 2 tables, now you have 6 kids, so:

the original 1/3 in the first table becomes 1/6

the original 1/3 in the second table becomes 1/6

so adding the two tables means 2/6 are girls

She needs to explain 1/3 of 1/2 (fraction of a fraction) before using an
example where she joins two units (two tables)

------
joshocar
This reminds me of a homework problem I did in numerical methods where I made
a small mistake in the program but still got an answer that was super close
the the actual answer out of pure luck. The professor spent a while looking at
it knowing it had to be wrong before he ran out the decimals and showed that
it didn't converge.

------
mynegation
I would try to explain semantics of addition (without the word “semantics” of
course). I would say, in the first group 1/3 is the girl so if we take out one
boy and replace it with another 1/3 (another girl), the resulting proportion
is 2/3\. Once we mix both groups, that is not what “plus” is.

------
alakadfs
I would explain that the "+" sign is not in the right place in the equation
and write

(1+1)/(3+3) = 2/6

instead.

------
bjourne
Many students at beginner university level math courses do the same kind of
mistakes, albeit with larger numbers: 10/13 + 7/21 = 17/34\. Decimals are much
easier to deal with imo, but mathematicians does not like them because they
are inexact.

------
pdevr
The moment you add three more students, the first girl is no longer one girl
out of a group of three students, she is one girl (out of the two girls) out
of a group of six students.

Same with the second girl.

So, once you add 3 more students, 1/3 becomes 1/6.

1/6 + /6 = 1/3.

------
rmtech
"+" is the wrong mathematical operation for combining fractions of distinct
sets.

The right operation is a weighted sum:

1/3×1/2 + 1/3×1/2 = 1/3

The "1/2" is there because each of the two sets that we are combining is the
same size.

------
sova
Really great anecdote! That first plus sign that Addison wrote is actually
more of an "And" instead of an arithmetic "plus"

------
jfkienennd
My favourite: I got 1/3 on the first test and 1/3 on the second test. So in
total I got 2/6\. So 1/3 + 1/3 = 2/6.

------
fallingfrog
I think that one key issue is that the students are confusing addition with
the more general concept of considering two things together.

------
macspoofing
This is a great example of the downside of over-relying on metaphors and
analogies to teach mathematical concepts. The reality is that fractions are
not like groups of pencils or tables of girls and boys. They are rigorously
defined mathematical constructs that occasionally can be mapped to real-world
things (and usually with severe constraints). 1/3 + 1/3 isn't 2/6 because it
doesn't follow from the underlying axioms that define rational numbers - and
not because of anything else.

~~~
bsaul
Except mathematical concepts emerged from real-world observation and problems.
We didn't invent fraction out of nowhere.

The moment when you have to stop relying on intuition is a pretty delicate
matter, but it's still interesting to try to rely on metampho, and then
understand when and why a particular metaphor stops working.

Much better (imho) than teaching math as a purely formal and transcendent
topic that happens to apply to real-world problems, and start with axiomatic
definitions (which is the way maths are often taught).

Note : i'm sure you're not advocating for that as well, and i don't mean to
contradict you. Just that i think it's better to start with a partially broken
metaphor, then fix it using formal definition, than not try at all.

~~~
macspoofing
>We didn't invent fraction out of nowhere.

But we did. When mathematicians provided a rigorous definition of fractions
(rational numbers) they separated them from the real world. Rational numbers
do not exist in the real world. Real-world does not have infinities. It does
not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way
that the fourth-grader applied the analogy.

>The moment where you have to stop relying on intuition is a pretty delicate
matter

I didn't argue that metaphors and analogies shouldn't be used. I argued that
analogies and metaphors are intrinsically flawed and this article provides a
great example. At some point, you have to give up on the analogy and fallback
on the underlying axioms. You can't do "1/3+1/3=2/6" not because it doesn't
make sense for tables of boys and girls (because it does) but because it's
against the rules for adding fractions.

~~~
jancsika
> You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of
> boys and girls (because it does) but because it's against the rules for
> adding fractions.

It is pedagogically superior to choose the route implied by the comments about
this being a type error.

That is, if you teach the students to "type" all those fractions (e.g., 1/3 of
this blue table, etc.), you gift them a tool they can use to map between the
real world and basic unitless mathematical notation. (I'd even add explicit
operator definition to that.)

For example-- such an educated student could hear your ascetic declaration
that "it's against the rules" and quickly grasp something like the following:

1\. "1/3+1/3=2/6" doesn't have any units, but it must _somehow_ map to
operations with units.

2\. _If_ unitless math can be applied regardless of units, then perhaps
"1/3+1/3" may mean "1/3 blue table + 1/3 of the red table, where + means
joining the two tables." That would equal 2/6 of the joined tables. But "1/3
blue table + 1/3 (same) blue table" would give 2/3 of that blue table, with +
mapping to adding those two fractions of the same table.

3\. 2/3 does not equal 2/6, so unitless math can't map to both operations.

4\. macspoofing said that 2/6 is wrong.

5\. Therefore, unitless fraction addition _implies_ addition of things of the
same units, and not joining two different things together and finding the new
fraction of the new joined unit thingy.

If on the other hand a student of your apparent method of declaring rules for
unitless math came to a class that had practiced explicitly mapping unitless
<-> unit math, they wouldn't have any tools to understand the mapping. (Well,
at least if the teacher made a similarly ascetic declaration regarding
mapping.)

I offer into evidence this very article to show what happens when a student of
your apparent method becomes the teacher and encounters the most trivial of
unit -> unitless mapping errors.

Edit: clarification

~~~
macspoofing
>That is, if you teach the students to "type" all those fractions (e.g., 1/3
of this blue table, etc.),

Sure. You can certainly ad hoc extend the analogy to bring it in line with the
mathematical rules. But at this point, you do hit a higher level of
complexity. Your simple analogy is gone and you're slogging through the weeds.
That was my point. Analogies are flawed. The teacher started with a very
simple rule that worked well and communicated the ideas under certain
constraints and then those rules were extended in a logical but incorrect
manner by a fourth-grader ... and now the complexity that was hidden in the
abstraction is leaking out.

Your explanation is more confusing to me and wouldn't be grasped by the vast
majority of fourth-graders. At some point, simply stating that fractions have
different rules is the most simple (and correct) explanation.

------
poleguy
1+1=50.8 My forth grader understood that this equation makes the same mistake.
Hint: units of inches and mm must be made explicit.

------
nartz
If you make it extreme, then its easier to see the folly / error -

500/1500 students + 1/3 students = ?

------
tosh
related: notation as a tool of thought

[https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=tr...](https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=true)

1/3 relates to one of the tables

2/6 relates to both tables

------
Gravityloss
You're taking a weighted average, not adding the fractions.

It takes a while to learn what that is used for.

------
JoeAltmaier
No cat has nine tails. Every cat has one more tail than no cat. Ergo, every
cat has 10 tails.

~~~
twic
Half a loaf is greater than nothing. Nothing is greater than God. Therefore,
half a loaf is greater than God.

------
mintyc
The and operation could be logical so the result of anding with one self is a
mop

------
pfortuny
> It’s hard to think and teach at the same time!

So true...

You do not see anything when you are at the blackboard.

------
inimino
This kind of thing fascinates me. I've always felt that you don't really
understand a thing, even a simple thing like fractions, until you can find and
resolve all the paradoxes that arise in using it naively.

My take when reading the 1/3 + 1/3 = 2/6 equation is that Addison was right
but the "+" here is a different operation than the "+" of 1/3 + 1/3 = 2/3\. In
one case we are adding disjoint parts of the same whole and counting how much
of the parts of the whole are now included, and get 2/3, and in the other case
we are adding two wholes and counting how much of the new whole is included.

People who pass primary education without internalizing this distinction will
have to do so later if they learn statistics or probability theory.

The answer "what would you do now?" is the same one I would try to ask the
students. First I would stop, confused, and write 1/3 + 1/3 = ? elsewhere on
the board, and then wait for the students to see why I'm confused. They
already understand that we want mathematics to be consistent, and getting
different answers means we were wrong at least once, but they will have a hard
time to find out or explain why 2/6 and 2/3 both are intuitively and obviously
right. You have to compare the intuitions directly to each other to see that
in one case you're adding parts of the same whole (set or group or table of
kids) and in another case adding the wholes.

From what I know of elementary school, we focus a lot on fractions and the
formal manipulation of them as unitless values, and drill into kids that 2/4,
1/2, and 0.5 are all "the same thing". Bullshit! It's no wonder they have
problems with word problems! Two-out-of-four apples and one-out-of-two apples
and a half apple are very different things, as every elementary student knows
well. How can we expect them to believe something that's obviously false
unless we've introduced the narrow context in which it's true? When used as
unitless "coefficients" or bare numbers that exist only to scale another value
that represents a real quantity _of_ something, _then_ 1/2 and 2/4 are the
same. _This_ is what we should be teaching when we teach fractions, not some
mindless rote symbol manipulation like cross-multiplying or what have you
without any ability to intuitively derive those rules yourself.

I'd spend the rest of the class time letting them work out all the different
ways of looking at it, and the connections to everything else fractions and
ratios and addition are good for.

Of course we want to be able to have "+" be well-defined, especially in
elementary school, and the students understand that! We've now found a
perfectly intuitive meaning of addition for which our standard "+" definition
does not fit! This gives an opportunity to explore the connections between
mathematics and intuition and the symbols and the real world operations they
stand for. It gives us an excellent chance to, as Polya puts it, "introduce a
suitable notation" for the setwise addition of "marked" sets that we have just
discovered. Then we can explore the properties of this operation and the ones
between it and the familiar addition operation. We can show that the plus sign
and the addition operation we use it to represent are distinct, and our use of
one for the other is a choice, and one we could make differently.

I saw some mentions of units in the comments, and yet 1/3 + 1/3 = ? is still
ill-defined if we know only that "1/3" means "one out of three students". What
we need to know is which students they are! So if one out of Alice, Bob, and
Charlie has red hair, and one out of Charlie, Doris, and Evan has red hair,
how many of Alice, Bob, Charlie, Doris, and Evan have red hair? Well, we made
the question challenging even for adults if we ask it that way! And now you
can introduce uncertainty and degrees of belief. (What if three people tell
you they have one red-headed friend, in a class of 23 students? Do they all
have the same friend, or is there more than one redhead?)

Of course what most of us will actually do as the teacher in this situation is
freeze and think "oh no, I've made a mistake and the teacher (in your mind) is
going to point it out and the class is going to laugh at me!". Because that's
what we've learned happens in these situations. If we interpret confusion and
being wrong as painful and embarrassing, little wonder that mathematics seems
a long and painful road that we avoid whenever we can.

------
andred14
Apologies but I don't see the mystery here?

A fraction describes a proportion ie. the amount of something(s) in relation
to another amount of something(s) thus, if we double all of the things in a
fraction the proportion and hence the fraction is equivalent.

------
dgentile
1/3 and 1/3 is 1/6 ;)

------
nurettin
This is pretty simple. Put two tables together, 0/6 girls. Now put only one
girl: 1/6 put the other girl: 2/6

------
tyingq
One third of one half, plus one third of the other half, is one third of the
whole?

~~~
Dumblydorr
Yeah the writer didn't have the words "of one half" or "of the other half",
therefore it seemed a lot more confusing than it should have.

------
jmmcd
This is exactly why third-level teaching is so much easier.

------
ValleZ
I don't understand why this simple 2nd grade math is featured on hacker news.
I guess the next topic will be a long discussion what would be 1/2 divided by
2/3.

~~~
CydeWeys
A lot of people are at home right now trying to teach their kids who are also
at home.

~~~
jpindar
I'm surprised I haven't been seeing more discussions about education recently
on HN and everywhere else.

~~~
CydeWeys
I suspect we skew young enough on here that most of us don't (yet?) have kids.

~~~
LyndsySimon
I very strongly suspect that’s not true.

~~~
CydeWeys
It would be interesting to see a demographic survey of HN
commenters/contributors. Wonder if one has ever been done?

------
dr0nkprogrammer
Javascript:

console.log((0.1 + 0.2) === 0.3) // => false

C#:

(0.1m + 0.2m) == 0.3m // => true

see here: [https://dotnetfiddle.net/zVnrNQ](https://dotnetfiddle.net/zVnrNQ)

Javascript is such a poor language, there is no even a good way to work with
currencies or exact numbers and you have to multiply numbers with a factor to
avoid such problems. The worst language I ever used.

