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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 t

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.

mywittyname 6y ago
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.
- forbiddenvoid 6y ago
  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.
  - tsimionescu 6y ago
    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) ...).
    - TeMPOraL 6y ago
      > 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.
      - tsimionescu 6y ago
        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.
p4bl0 6y ago
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.
- tsimionescu 6y ago
  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.
  - soperj 6y ago
    you can. It's what you do in fruit salad.
    - tsimionescu 6y ago
      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.
  - CydeWeys 6y ago
    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) https://en.wikipedia.org/wiki/Mediant_(mathematics)
    - jrumbut 6y ago
chpmrc 6y ago
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) 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.
dejj 6y ago
They sneakily replaced scalar addition by point-wise vector addition. [1,3] <+> [1,3] = [2,6]
- deleted 6y ago
  [deleted]
Wowfunhappy 6y ago
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.
- joshocar 6y ago
  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.
  - forbiddenvoid 6y ago
    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.
virgilp 6y ago
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.
- prmph 6y ago
  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.
ken 6y ago
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 https://www.youtube.com/watch?v=bCoGMYV3UPk&t=2m29s
tener 6y ago
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.
- veilrap 6y ago
  This is a great baseline for the units answer.
  - TeMPOraL 6y ago
    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.
mcguire 6y ago
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."
- lonelappde 6y ago
  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.
walshemj 6y ago
or if students are in both tables
microtherion 6y ago
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.
amluto 6y ago
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.
- heisenzombie 6y ago
  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#Dimensional_homogeneity https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...
jtsiskin 6y ago
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.
isanybodythere 6y ago
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.
app4soft 6y ago
> 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.
httpsterio 6y ago
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.

mywittyname 6y ago

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.

forbiddenvoid 6y ago
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.
- tsimionescu 6y ago
  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) ...).
  - TeMPOraL 6y ago
    > 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.
    - tsimionescu 6y ago
      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.

p4bl0 6y ago

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.

tsimionescu 6y ago
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.
- soperj 6y ago
  you can. It's what you do in fruit salad.
  - tsimionescu 6y ago
    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.
- CydeWeys 6y ago
  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) https://en.wikipedia.org/wiki/Mediant_(mathematics)
  - jrumbut 6y ago

chpmrc 6y ago

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) 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.

dejj 6y ago

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

deleted 6y ago
[deleted]

Wowfunhappy 6y ago

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.

joshocar 6y ago
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.
- forbiddenvoid 6y ago
  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.

virgilp 6y ago

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.

prmph 6y ago
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.

ken 6y ago

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

tener 6y ago

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.

veilrap 6y ago
This is a great baseline for the units answer.
- TeMPOraL 6y ago
  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.

mcguire 6y ago

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."

lonelappde 6y ago
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.

walshemj 6y ago

or if students are in both tables

microtherion 6y ago

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.

amluto 6y ago

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.

heisenzombie 6y ago
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#Dimensional_homogeneity https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...

jtsiskin 6y ago

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.

isanybodythere 6y ago

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.

app4soft 6y ago

> 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.

httpsterio 6y ago

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.