Hacker News Reader

Comment by forbiddenvoid | Hacker News Reader

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.

View on HN

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.

This. What the student said was 'wrong' and they need to look at the whole. This "1/3 + 1/3 = 2/6" problem would be a great example to use for a lesson on units though.

What they said wasn't wrong. Their mental model was absolutely correct.

"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.

Separate from the discussion of what is or is not counterproductive / educative, the student's mental model was wrong.

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.

No. The mental model was exactly correct. Their assignment of notation to the model was non-standard when they wrote "+" for the operation of merging the wholes.

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.

Math is about learning how mathematical language is applied to problems. She is incorrect about how the math is applied. Ergo she is wrong. Her intuition is on the right track.

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

> The more I think about it, the more I think the right 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.

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.

The right response could then be expanded...

"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.

Ah, I like that! Yes, re-frame the wrong statement into a correct one, then keep going. Feels better than ignoring it. (I wouldn't introduce ratios though, that's a totally new concept.)

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.

I like the units explanation a lot more. Saying you can't put two tables together when you clearly can in the real world is seems deeply unsatisfying. Kids hate being told they can't do something.

mnsc's explanation shows that you can put the tables together, you just need to take both tables into account from the outset.

I don't think I agree with everything here.

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.

I'd absolutely agree if you were tutoring the child one-on-one. My concern is about leading the other 19+ kids astray with a confusing example before they're ready for it.

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

My observation of points at which even very smart people fall off the "math train" k-12:

- 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.

> My concern is about leading the other 19+ kids astray with a confusing example before they're ready for it.

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.

I agree. I think many people are thinking the main obstacle here is the mathematical mistake, when the main obstacle here seems to be explaining the mathematical mistake to 4th graders who are learning fractions for the first time. I'm also not sure what the best way to do that would be. If you start with one type of fraction (say, half circles), it's probably easier to intuit things like 5 halves and how much that is, but other things are probably going to be more difficult.

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

Not only to 4th graders.

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 of one table = 1 student.

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...

It’s 1/3 + 1/3 = 2/3, and 1:3 + 1:3 = 2:6 = 1:3.

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

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.

Another way of labeling this:

  : = 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

Somewhat aside, but in grade school I recall learning that ratios were written [group-A]:[group-B], not [group-A]:[total]. So the girl-boy ratio at the table would be 1:2. Different for you?

It really depends on the two things being compared. The ratio of boys to girls is 1:2, but the ratio of boys (implicitly: to the total) is 1:3.

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?

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.

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?!

Why? Each is a pure number without units. Since the units cancel in each term.

> You're right, but the tricky part is, how do you explain that to a group of children who is just being introduced to the concept of adding fractions, without leading them off track? That's hard!

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?"

This is interesting. I wonder if kids can then take that and transfer it over. Bottles and Pencils are obviously different but kids at table A and kids at table B are both kids. Even more problematically, the equation appears to work. I suspect there'll be some amount of dissatisfaction here and the kids won't understand.

Not sure if this is at the level of comprehension of these kids, but I'd explain it this way: "kids at table A" and "kids at table B" are both "kids", but different amount of kids. You can treat them as the same only if you have a conversion factor. So, if you know that there are 12 kids at table A, and 20 kids at table B, you can multiply your variables by these amounts, and now both expressions have the unit "kids", and you can add them together. But if you don't know the conversion factor, they're like Bottles and Pencils.

> you can multiply your variables by these amounts, and now both expressions have the unit

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

Since numbers are usually introduced to children in terms of quantities of things (e.g. 3 apples), I think it might suffice to stress the importance of keeping these things always attached to those numbers. In general, numbers very rarely go alone.

Sure. Might point is that you’re whole statement is way beyond the math comprehension of grade schoolers.

> 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!

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.

You are right, I think before jumping to different things, kids should learn on simple examples, that do not include different types, and then when they grasp one concept they should be gradually introduced to the other concept. table/tables would just confuse them even more.

The core concept kids need to know when adding fractions together is that the denominators need to be the same before you can add the numerators. Your use of units is on the right track, but I think you need to use units for both the top and bottom of the fraction:

(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)

(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)

Ah, yes, that would definitely be a better way to write it if you were to go forward with the unit approach.

There is no track!!! Of course there is, because what we call mathematics education is broken beyond repair, but that's a crime.

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?

there's not much evidence that most humans would ever discover the concept of fractional arithmetic by themselves.

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.

There's incredibly strong evidence that the way to teach mathematics and the only way to learn it is to foster the environment in which you can rediscover the key insight for yourself. Obviously (to the younger students, to me, Lockhart, Dewey, and others, but not to most teachers, administrators, voters) there is no track and no curriculum for this.

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.

You're stretching what I said. I should note that I'm a lifelong fan of radical education (particularly John Holt), and I don't think that the way we teach children most things has a lot to recommend it.

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.

Yes. Every child I've ever met is curious about something, and that's what they should be learning. The idea that in year X every student should learn Y during hour Z every day is one of the reasons I said the system was irreparably broken. That is how schooling gets in the way of education.

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.

I have a hard time accepting this. My daughter is in 3rd grade and seems to have a reasonable grasp of fractions. They teach her about them at (public) school and we've discussed them at home. Sure, "1/3 + 2/7" is outside her reach, but simpler stuff, things she can visualize, are well within her grasp.

> 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".

No disrespect, but you read HN and think that your daughter's take on math and the home context you provide for it is a sensibly representative starting point?

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 :)

I think the homework her school gives out is a sensibly representative starting point. Why wouldn't it be?

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.

I think she could have continued by adding more tables until they got to 4/3. That is obviously wrong - so it is easier to convince the kids that something is wrong. Then back track and explain what went wrong and explain the rules for adding fractions.

I think we really need to start teaching people paying attention to units. Even in math classes.

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.

Units is a real eye opener once you do it right. I remember when i studied and could almost verify what I did was correct, I would just check if the resulting unit was what I expected

Just a missing step in the explanation:

"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.

Yup, it's a typing problem. Do it with different things in the two sets and it becomes clearer: 1/3 apples + 1/3 oranges = 2/6 fruits.

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.

It's not really a typing problem, it's a problem of references. You can't add 1/3 apples and 1/3 oranges. However, you can add 1/3 * x people with 1/3 * y people and get 2/6 * z people, if x people +y people =z people.

you can. It's what you do in fruit salad.

Well, in fruit salad 1/3 apple + 1/3 orange = 1/3 apple + 1/3 orange. I can say 3 * (1/3 apple + 1/3 orange) = 1 apple + 1 orange.

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

You definitely can add 1:2 apples and 1:2 oranges, you just need to do so using a common base type (such as fruits or objects).

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)

The grandparent's note about units and your mention of ratios get at the same problem for me:

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.

You can't perform operations with apples, oranges and fruits unless you use them completely interchangeably ,i.e. 1 apple = 1 orange. Otherwise, you will break algebraic equalities.

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.

I think you both are saying the same thing, in this context at least.

There is a very important difference between types and quantities.

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).

    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"

This does nothing to elucidate the problem. 1/3 apples + 1/3 oranges could be 2/6 fruits, or 2/781 fruits, or any other number you want. In normal mathematical notation, 1/3 is interpreted as 1/3 of 1, and in that case 1/3 apple + 1/3 orange is 2/3 fruit.

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.

You're being dishonest in your reasoning.

> 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.

Interesting - I never thought of CS as being something to help with this. Instead, I’d think of other engineering fields where there is a heavy reliance on correct units and unit conversion. Pound of feathers vs pound of gold , etc...

F# offers a nifty "unit of measure" language feature to help with that. You might have

  [<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.

One of the article's comments:

"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."

That's totally wrong. It's not OK to add ratios. It's OK to add populations.

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.

One could explain the mistake to have different units on both side without fractions at all:

1+1=1

One shoe plus one shoe equals one pair of shoes.

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

This is a great baseline for the units answer.

I think it then needs an addendum on things that are the same dimension but expressed in different units.

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.

Or a different way to look at it is that if you put two things together, the mathematical operation is not always "+"; it totally depends on the things (and how you put them together). You use plus if you put together fractions of the same thing (e.g. fractions of the same box of crayons), but potentially some totally other operation if you put together different things.

Someone mentioned on this thread using CS to teach math. What you have just described here is the fact that the + operator can be overloaded.

Wonder if the kids can grasp that concept though.

The units aren’t the issue IMO. You said:

> 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.

I think you guys actually agree more than you think --

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...

I'm not sure we can really guess what the student meant, but I do know for sure that she was wrong.

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.

They sneakily replaced scalar addition by point-wise vector addition. [1,3] <+> [1,3] = [2,6]

This was what I came up with in the moment, but with drawing. Draw two circles on the board, each divided into thirds and compare it with a single circle divided into sixths. It demonstrates that the total has grown.

Also a fantastic way to represent this. The assumption in fractional arithmetic is that you're always performing arithmetic on things with the same type/unit.

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.

As I said elsewhere, this is not a problem of units or types. If it were, then the computation wouldn't make sense.

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) ...).

> The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other).

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.

You are trying to look at a different problem. It was absolutely correct that 1/3 of the students at one table of 3 plus another 1/3 of the students at another table of 3 is the same number as 2/6 of the 6 students sitting at the two tables. This is not disputable.

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.

If you are omitting the units, it would be a fair assumption that the values all belong to the same unit, unless stated otherwise. You're representing the fractions related to each other and you can't do that if the fractions are of two separate things.

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 units are missing

> 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.

I can't see the page (slashdotted), but that sounds just like the puzzle from an old kids' math show we used to watch:

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

Yes, 1/3x + 1/3x = 2/6(x + x).

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.

I would do piecewise addition like this:

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.

Exactly what I was thinking. Maybe it could be formalized this way? (with t = 1 table)

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

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.

or if students are in both tables

Reply on news.ycombinator.com