> 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.

She's already introduced bottles of water and pencils. So one interesting question is, "If you take three bottles of water, and add three pencils, do you now have all of the bottles of water?"

In grade school where these kids are, they won’t understand the concept of an “expression”, a “variable”, and barely the importance of “units”.

So yes, telling kids that there are more complications when approaching certain problems that you can explain later might be a good approach.

I have a PhD in physics and the more I think about that, the less obvious it becomes (or at least makes you seriously think about it).

The average person is probably in the easiest situation because they learned how to add fractions without further philosophy and they can live happily after.

1/3 + 1/3 is not a correct mathematical description of the problem.

1/3 [students at table A] + 1/3 [students at table B] is.

Let's shorten this to: 1/3 sTA + 1/3 sTB. Then, you factor out the 1/3, to get: 1/3 * (sTA + sTB). And now it's it's impossible to give the wrong answer "2/3".

Or, in other words, it's best to "keep your attention on what the whole is" by keeping it explicitly written out in the equation.

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.

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?!

  : = Out Of
  A = Table A
  B = Table B
  E = Every Table Added (A + B)
  
  1:3 * A + 1:3 * B = (1:3 + 1:3) * E
  2 out of 6 among Every Table Added

Glossing over critical mental models while explaining new concepts is what creates the confusion. Absolutely using drawings would have made things easier. That's a good way to establish those correct mental models.

"No you can't do that" is probably not the right approach.

The kid's instinct was spot on. The calculation they did was correct, just out of context. Rather than telling them they did it wrong and disrupt their correct mental model, I think it makes way more sense to talk about how what they did is different.

Ignoring it does feel wrong, but I don't know that it is. There's only so much time in a class period, etc.

That said, I really liked where mnsc went with it: https://news.ycombinator.com/item?id=23312266

- Fractions

- Factoring

So yeah, I'd say great care is warranted in teaching kids fractions. Screw that up and there's a good chance you've lost them for life.

Except that the kids were all agreeing with her. Whether she did made that statement or not, they still believed that made sense.

I agree with you about the solution from mnsc, that was a good way to skip that without confusion.

"No, you can't do that, because those kids are at a different table. If you want to add Jack and Brad you have to go back and see their fraction _of both tables_. They are each 1/6 so together they are 1/6 + 1/6 = 2/6"

Then you could talk about if the ratios are same at both tables 1/3. Adding both tables together keeps the ratio. 1/3 + 1/3 = 1/3.

This is closer to what the teacher actually did—but then she became focused on explaining why the original supposition was wrong, as opposed to moving on to clearer examples.

1/3 of two tables = 2 students.

1/3 of one table + 1/3 of one table = 2/3 of one table.

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

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

--

It would take a lot longer to explain than to write, but that's how I'd be tempted to proceed.

As far as learning the basics, it sounds like this class was just getting them introduced to the concept of fractions, while the arithmetic rules would come later...

You add slices of pie thus /, and win a vote by ratios thus :.

Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do.

Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing anomalies is the best habit you want the next generation of engineers or statespeople to learn as a reflex from an early age?

this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves.

there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning experience in general.

now, if you are homeschooling a child or somehow in a 1 on 1 (or at least, working with a very small teacher:student ratio), then perhaps a more freeform exploration of fractional arithmetic might be a wonderful thing.

doing so with a general class of kids? i strongly suspect you're wrong. they won't even understand that there is an anomaly, because they don't even understand the basic concepts that make it an anomaly. why shouldn't adding two fractions (never seen those before!) result in different answers? in fact, why is 2/3 different from 2/6 anyway? and so on.

this is not about shutting down curiosity IMO. it's about nurturing the basic concepts so that curiosity can grow amidst them in the (near) future.

> why is 2/3 different from 2/6 anyway

Because 2 pieces of a pie that you cut into 3 pieces is more than 2 pieces of a pie that you cut into 6 pieces. She understands that 4/8 == 2/4 == 1/2. Sometimes she needs to think about it a bit, but she does "get it".

My daughter's now 25 and has been quite the nerd herself through the years (eventually landing in linguistics and speech pathology), but I'd never assume that the fascinations she had instrinsically and that I helped foster as a parent were really typical. I wish they were - and hey, here I am reading HN too :)

That homework expects a reasonable understanding of fractions. Enough that the child doing them can understand the difference between the number of slices in a pie being representative of the bottom number of a fraction.

Sure, I do math exercises with my daughter that aren't representative of what they teach in school (square roots, the fact that parallel lines _do_ meet at the vanishing point in the real world, etc). But those things aren't what I'm basing my assumptions are; the expectations the school has of her are.

They are not stumbling into it by themselves! They have a guide! The guide's job is to point out the works in the museum, and to make sure the kids don't get lost. It's not to stand in front of the art so the kids can't even see it and lecture about it!

There's nothing complex or sophisticated about recognizing that "+" has a meaning that we chose, and we could have chosen others, and in some cases other choices would be more natural.

You don't think students learning fractions know that there is one right answer, or that 2/3 and 1/3 can't both be right? Their short little lives have already been filled with enough test-taking to teach them that, at least.

And if they don't understand that 1/3 and 2/3 represent something different about the real world, and that 2/6 and 1/3 are different in a different way from 1/3 and 2/3, then why are we going on to teach them even more complex sophisticated concepts before they have grasped these most basic of basics?

There's never been a pedagogical program to shut down the curiosity of primary students. We do it by accident, by "nurturing the basic concepts" (the ones you learned at that age) so that curiosity can grow amidst them "in the future," which means, maybe, after they finish calculus, if they are lucky.

However, your conception of how this could work starts from the supposition that the kids are actually interested enough to wonder. I don't doubt that there's something that will get every child wondering (and likely, more than one thing). But if we're going to actually require the teaching of fractional arithmetic (implying that we're requiring the learning of it, really) then we need to accept that we'll be teaching it a great many children who are not interested in the conundrum, and who even in the presence of a great teacher will remain not interested in it.

Self-led discovery learning is without doubt the best kind, but its not compatible with the current goals of education in a (post)industrial highly structured society, because it will naturally lead to people who for their own reasons chose never to learn things that we consider vital. I might be entirely willing to agree that they are not vital, and even that a (post)industrial highly structured society may also be a bit of an issue, but pretending that every child will just be naturally curious about 1/3+1/3=2/6 vs 1/3+1/3=2/3 is, IMO, not ground in reality.

In the original article, the guide isn't "standing front of the art". She's just taken them over to a corner that has a piece called "no mammals lay eggs", and then noticed that right next door to it, there's a platypus. She's wondering what to say next, or whether to say anything.

Does every student learn to understand fractions under the current regime? Not in my experience.

Edit: And yes, you should say something about the platypus, because understanding that models are simplifications and incomplete can be enough of an escape hatch for the smart kids (the ones that usually hate math class) to notice that even the teacher knows that there's always more, and that can be enough to keep them from throwing it all away in disgust as a useless mishmash of arbitrary, conflicting, and incorrect rules.

Yea, the typing (or w/e) is the problem here. I like another posters examples of apples + oranges = fruits.

Otherwise you could further blow their minds. Where 1/3 + 1/3 == 2/6 and 2/6 is equal to 1/3 so 1/3 + 1/3 == 1/3. Hah. Understanding the incompatibility, or at least difference between the first, second and resulting fractions seems essential to understand.

I think it's the opposite: if you are taught the procedure for adding fractions without understanding what's going wrong here, then you just know a procedure, not a concept. Taking the detour into the detailed discussion of what 'the whole' is takes longer, but cements that crucial idea, and so lays the foundation for understanding addition of fractions more generally.

"1/3 of the students at the first table + 1/3 of the students at a second table =

1/3 of 1/2 of all the students at both tables [ie. those students at the first table] + 1/3 of another 1/2 of all the students at both tables [ie. those at a second table] =

1/3 of the students at both tables"

In US math education I noticed there is a lot emphasis on specific patterns/formulas/etc., kind of "deus ex machina", and a lot of excercises to mechanize their application, yet too little is spend on where it comes from, manual derivation/proval of formulas, establishing logical connections, etc.

(1 girl at table A)/(3 students at table A) + (1 girl at table B)/(3 students at table B) = (2 girls at tables A & B) / (6 students at tables A & B)

vs

(1 boy at table A)/(3 students at table A) + (1 boy at table A)/(3 students at table A) = (2 boys at table A) / (3 students at table A)

"One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.'

Their written representation of the mental model was incorrect because their instruction was focused only on abstract numbers instead of concrete labels. If those fractions (or ratios or whatever) are labeled properly the equation is completely correct.

The student was taking the model for ratios and applying it to fractions. If I need to add 2 + 2, and I multiply instead, I did the wrong calculation. It does not matter whether I multiplied correctly, nor does it matter that, in this case, both operations equal four.

The confusion of what is wrong with what is correct but non-standard is in fact quite standard and quite wrong.

The confusion of the mental model with the notation is also a standard mistake.

The only "whole" in this case is the entire set of students (or seats or whatever it is that is being talked about).

This is another instance of situations where teaching compsci or at least programming could help teaching maths rather than the other way around as it is traditionally thought.

Note that I'm using ratio notation because the answer for the above is not the same as adding 5:10 apples and 1:2 oranges; in other words, the exact numerator and denominator both matter, so it's not really a simple fraction; a simple fraction can be reduced to its lowest terms (e.g. 5/15 becomes 1/3), but you can't do that here and still support the mediant operation. https://en.wikipedia.org/wiki/Mediant_(mathematics)

You can convert apples to fruit and oranges to fruit and then do operations on fruit, but you can't go back from the result to apples and/or oranges. For example, 1/7 of (3 apples + 4 oranges) is 1 fruit, but we can't tell if it's one orange, or 1 apple, or 1/3 apple and 2/3 orange.

The lesson mixes two of the physical world concepts that are often modeled by fractions.

One is an incomplete whole (the six pack) and the other is the proportions of a mixture (table demographics).

Fractions always have the same rules, but the rules map differently to the different territory being modeled.

    abstract class Fruit:
      prop name: string
      prop rotten: bool

    class Apple (Fruit):
      prop name: string = "apple"
      prop rotten: bool = False

    class Orange (Fruit):
      prop name: string = "orange"
      prop rotten: bool = False

    func add (a: List<Fruit>, b: List<Fruit>) -> List<Fruit>:
      return foldl(lambda l, f: l.append(f), a, b)

    func filter (p: Func<[Fruit], bool>, l: List<Fruit>) -> List<Fruit>:
      return [f for f in l if p(f)]

    func is_rotten (f: Fruit) -> bool:
      return f.rotten

    a1 = Apple()
    a2 = Apple()
    a3 = Apple()
    a3.rotten = True
    apples = [a1, a2, a3]

    o1 = Orange()
    o2 = Orange()
    o3 = Orange()
    o3.rotten = True
    oranges = [o1, o2, o3]

    fruits = add(apples, oranges)

    print(len(filter(is_rotten, apples)), "/", len(apples))
    > "1/3"

    print(len(filter(is_rotten, oranges)), "/", len(oranges))
    > "1/3"

    print(len(filter(is_rotten, fruits)), "/", len(fruits))
    > "2/6"

And of course this only works if 1 apple = 1 orange = 1 fruit, which means that they are the same unit of measure, or, equivalently they are of substitutable types. It's even debatable of it's correct to say that 'if I have 1/3 of an apple and 1/3 of an orange, I have 2/3 of a fruit', so depending on what you want to do with your apples, oranges and fruit, your hierarchy may in fact break Liskov substitution. For example, if 1 apple + 1 orange = 2 fruit, so 2 fruit - 1 apple = 1 orange, so 1 apple = 1 orange, which is not true.

> In normal mathematical notation, 1/3 is interpreted as 1/3 of 1

Why would you deny the possibility of 1 being 1 list of fruits rather than 1 fruit?

In the original example, it was 1 table of students, not 1 student.

Everything you say after that is based on this broken assumption.

And then by way of analogy of course, (e apples)^(pi oranges) + 1 apple = 0.

For example, 1/3 + 1/3 = 1 is true, if I mean 1/3 of 3 + 1/3 of 0. 1/3 + 1/3 = 2/6 can only be true if we add some quantities, we can't make it true by choosing the right types (we could fix it by choosing a different definition of +, or =, though).

  [<Measure>] type lbFeather 
  [<Measure>] type lbGold
  [<Measure>] type lbLead

  let convertGoldToLead ( weight : float<lbGold> ) = 1.0<lbLead> / 1.0<lbGold> * weight

  // works
  let barOfGold = 1.0<lbGold>
  let barOfLead = convertGoldToLead barOfGold
  // doesn't work
  let bagOfFeathers = 1.0<lbFeather>
  let leadDuck = convertGoldToLead bagOfFeathers
  // Error reported: mismatched units.

"Janelle Schorg says:

"This is why students are confused and have misconceptions about ratios in middle school. When we teach fractions it is part(s) of a whole (Water bottles and pencils context) and when we teach ratios they are sets (boys and girls). It is actually okay to add ratios (as fractions) by combining the numerators and denominators, no common denominators needed. In my opinion, ratios should not be written like fractions until later after students have conceptual understanding and fractions should never be taught with sets in the 3-5 work. Many teachers are not even aware of this difference and misconception we are creating in student understanding."

You can't add 1/3 and 1/3 to get 2/6 if the first 1/3 was reduced from 3 of 9 and th second was 1/3. Well, you can, but that only works in the degenerate case where the items you add (actually, average) are equal and there's no point in adding in the first place.

1+1=1

One shoe plus one shoe equals one pair of shoes.

Once they grasp that explaining the fraction issue should be easier.

Like, "students at table A" and "students at table B" are both of the same dimension as "students", but are different units. Like meters and miles. You can add them together, but only if you know the factors needed to convert them to a common unit. In this case, the conversion factors are, how many students are at table A and how many at table B.

Wonder if the kids can grasp that concept though.

> 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).

That’s not what ‘+’ means. Addition doesn’t mean “I have this thing and the other thing; please describe the result”; addition means a specific operation on numbers (or on elements of an additive group, or on numbers with units, etc). But you cannot fully describe 1 student at table of three people as 1/3. Sure, 1/3 of the students at that table are that one student, but if you want to add across tables, you need more information and a better description.

Explaining this in a classroom setting may be quite challenging indeed.

You say "you cannot fully describe 1 student at table of three people as 1/3", which is true: What's missing is the unit (or dimension, I'm ignoring the difference here).

You can only add two things if they have the same units, as per "dimensional analysis" [1].

So this is an entirely meaningless statement:

[students]/[seats at table 1] + [students]/[seats at table 2]

But you can fix the units with some multiplication (because dimensions do form an Abelian group under multiplication):

([students]/[seats at table 1]) * ([seats at table 1]/[total seats]) + ([students]/[seats at table 2]) * ([seats at table 2]/[total seats])

Which simplifies to:

[students]/[total seats] + [students]/[total seats]

Now that's a statement with meaning!

Since I know that

[seats at table 1]/[total seats] = 1/2

[seats at table 2]/[total seats] = 1/2

I've just derived the calculation that I really wanted to do:

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

[1] https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...

If you add up a sixth of a six-pack and a sixth of another six-pack you get a sixth of two six-packs -- two twelveths.

The student's misunderstanding comes from being taught fraction addition in terms of items in a collection -- which only holds if you keep to the same set (what you called scale).

This is a common choice -- "students already know how to add integers, so let's start from there", but as it did in this case, it doesn't always work as intended.

This is a great example of taking an analogy so far that the student didn't learn anything new at all. Everyone feels happy -- teacher's teaching, student's learning -- until you test what your knowledge on outside the domain of the analogy.

Fraction and integer addition are one and the same, yes -- but from the point of view of fractions, wherefrom integer addition is a special case. It remains challenging to teach and understand from the point of view of the integers, which is where the student stands.

The two formulas on the board are essentially:

1/3x + 1/3y = 2/6z

1/3x + 1/3x = 2/3x

Both are correct, but without units labeled you wouldn't know that.

It is a problem of implicit refernces. The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other).

The correct equation would have been 1/3 * 3 + 1/3 * 3 = 2/6 * 6. Note that 3, 3 and 6 have the same unit. If they didn't, then this would be meaningless. 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals) ...).

That is, precisely, the problem of units. Each object is its own unit here.

> The correct equation would have been 1/3 3 + 1/3 * 3 = 2/6 * 6.*

That's not how you add fractions. The correct equation would have been, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 (or, 1+1=2), if these 3 all truly had the same units. But they don't, so you can't add like that.

> 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals)

That's the point (but it's 1/3, not 2/6). Also, 1/3 of a meter, + 1/3 foot = 1/3 (1 meter + 1 foot). Different units, but same dimension, so if you know the conversion factor (here, 1 meter = 3.3 feet), you can change it into (1/3 meter * 3.3 feet/meter) + 1/3 foot = 1.1 foot + 1/3 foot = 33/30 feet + 10/30 feet = 43/30 feet = 1.43(3) feet.

You can do the same math with students at tables.

The way you can write this observation mathematically is as I did: ((1/3) × 3) + ((1/3) × 3) = ((2/6) × 6), or 1 + 1 = 2, after computing the fractions. The student's observation was perfectly correct, but he was missing the proper explanation, as it is not about the addition of fractions (it is almost a coincidence that the fractions used on one side of the equation happen to have the sum of their numerator and the sum of their denominators equal to the numerator and denominator of the fraction on the other side - this only happens because we are multiplying the fractions by their denominators).

Sure, you can express this in terms of units and dimensions of you really choose to. You can also express it in terms of different definitions of +, or even of =. It is pretty unnatural to me to invent an ad-hoc measurement unit N1, "number of people at 1 table" and a different measurement unit, N2, "number of people at 2 tables", with the relation 1N2 = 2N1, and then correct the student's formula to 1/3N1 + 1/3N1 = 2/6N2. It is correct, but it is extremely artificial to me.

By far the most natural way to explain it is using the correct mathematical interpretation of the phrase "one third of the 3 people" - (1/3) × 3.

Inventing measurement units to describe exact quantities reminds me of a silly joke from Portal: "computer: 2 + 2 = 10 <pause to wonder if the computer is broken> ... in base 4". You can always find a way to make the formula direct by adding assumptions.

I could say that 1+1 = 2 or 1+1 = 10 but it wouldn't be right to say that 1+1=2 && 1+1=10 because while both are true if we're talking about decimals and binary, we're omitting the units and everything loses its' meaning if we do that.

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

Units in your example are unclear. Also results interpretation depend on question.

Here is my example:

Student should wrote 2 articles, but wrote 1 of 3 pages for the 1st article and 1 of 3 pages for the 2nd article.

Q: How many pages student wrote? - A: 2/6.

Q: What part of the whole task is done? - A: 1/3.

Q: How many tasks partially completed? - A: 2/2.

Q: How many tasks fully completed? - A: 0/2.

https://www.youtube.com/watch?v=bCoGMYV3UPk&t=2m29s

Units are absolutely necessary, and 'fractions' don't make sense without them.

'1/3' without units is not a 'fraction' but instead is 0.3333, a number on the number line, which is why it looks so wrong to see 1/3 + 1/3 = 2/6.

Girl A is 1/3 of Table A, which seats half of the students, so she is 1/3 * 1/2 = 1/6 of the total.

Girl B is 1/3 of Table A, which seats half of the students, so she is 1/3 * 1/2 = 1/6 of the total.

So girls represent 1/6 + 1/6 = 2/6 of the total.

1/3 * (1 table) + 1/3 * (1 table) = 2/6 * (2 tables)

or

1/3 * t + 1/3 * t = 2/6 * 2 * t

but 2/6 * 2 = 2/3 so you get

1/3 * t + 1/3 * t = 2/3 * t

which is correct.