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This is a great example of the downside of over-relying on metaphors and analogies to teach mathematical concepts. The reality is that fractions are not like gr

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

bsaul 6y ago
Except mathematical concepts emerged from real-world observation and problems. We didn't invent fraction out of nowhere. The moment when you have to stop relying on intuition is a pretty delicate matter, but it's still interesting to try to rely on metampho, and then understand when and why a particular metaphor stops working. Much better (imho) than teaching math as a purely formal and transcendent topic that happens to apply to real-world problems, and start with axiomatic definitions (which is the way maths are often taught). Note : i'm sure you're not advocating for that as well, and i don't mean to contradict you. Just that i think it's better to start with a partially broken metaphor, then fix it using formal definition, than not try at all.
- macspoofing 6y ago
  >We didn't invent fraction out of nowhere. But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world. Rational numbers do not exist in the real world. Real-world does not have infinities. It does not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way that the fourth-grader applied the analogy. >The moment where you have to stop relying on intuition is a pretty delicate matter I didn't argue that metaphors and analogies shouldn't be used. I argued that analogies and metaphors are intrinsically flawed and this article provides a great example. At some point, you have to give up on the analogy and fallback on the underlying axioms. You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions.
  - jancsika 6y ago
    > You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions. It is pedagogically superior to choose the route implied by the comments about this being a type error. That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), you gift them a tool they can use to map between the real world and basic unitless mathematical notation. (I'd even add explicit operator definition to that.) For example-- such an educated student could hear your ascetic declaration that "it's against the rules" and quickly grasp something like the following: 1. "1/3+1/3=2/6" doesn't have any units, but it must somehow map to operations with units. 2. If unitless math can be applied regardless of units, then perhaps "1/3+1/3" may mean "1/3 blue table + 1/3 of the red table, where + means joining the two tables." That would equal 2/6 of the joined tables. But "1/3 blue table + 1/3 (same) blue table" would give 2/3 of that blue table, with + mapping to adding those two fractions of the same table. 3. 2/3 does not equal 2/6, so unitless math can't map to both operations. 4. macspoofing said that 2/6 is wrong. 5. Therefore, unitless fraction addition implies addition of things of the same units, and not joining two different things together and finding the new fraction of the new joined unit thingy. If on the other hand a student of your apparent method of declaring rules for unitless math came to a class that had practiced explicitly mapping unitless <-> unit math, they wouldn't have any tools to understand the mapping. (Well, at least if the teacher made a similarly ascetic declaration regarding mapping.) I offer into evidence this very article to show what happens when a student of your apparent method becomes the teacher and encounters the most trivial of unit -> unitless mapping errors. Edit: clarification
    - macspoofing 6y ago
      >That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), Sure. You can certainly ad hoc extend the analogy to bring it in line with the mathematical rules. But at this point, you do hit a higher level of complexity. Your simple analogy is gone and you're slogging through the weeds. That was my point. Analogies are flawed. The teacher started with a very simple rule that worked well and communicated the ideas under certain constraints and then those rules were extended in a logical but incorrect manner by a fourth-grader ... and now the complexity that was hidden in the abstraction is leaking out. Your explanation is more confusing to me and wouldn't be grasped by the vast majority of fourth-graders. At some point, simply stating that fractions have different rules is the most simple (and correct) explanation.
twic 6y ago
Fractions are like groups of pencils or tables of girls and boys. Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.
- macspoofing 6y ago
  >Fractions are like groups of pencils or tables of girls and boys. No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? I can multiply -1/3 * -100/3 and get a positive fractional value - if we're talking about tables of boys and girls - what the heck happened there? How about 1/0 or 0/1? >Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods. I think you have it reversed. The formal definition is the pure definition, which is ancillary to the application to the real-world but covers much much more.
  - simiones 6y ago
    You're taking a strange ahistorical perspective. Most Natural numbers, Integers, Rationals/Fractions, and even some Irrational numbers are much older than the concept of formal mathematics. Natural numbers were discovered as a solution to counting. Negative numbers appear naturally when you need to subtract quantities, especially when you end up with debts. Rationals appear from the need for division. Pi appears when you want to relate lengths of circles with lengths of straight lines. Square roots also appear from basic geometry used in construction, as the naming suggests. All of these are real-world concepts that were used in some form or another at least 6000 years ago. A lot of the time, the closest formal definitions they had at the time were actually based on geometry, as algebra was probably a few thousand years away from being invented/discovered. The modern formal definition is a post-hoc formalization of an existing intuitive concept. If the formal definition did not agree with the intuitive concept, then obviously the formal definition would not be useful and no one would use it. There are mathematical constructs that first appeared in mathematics, and don't have intuitive concepts attached, or even don't have any real-world interpretation. A lot of irrational numbers may be in this area, as would the transfinite numbers, and probably complex numbers as well, though even some of those are usually defined based on some real-world applicability. > I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? It's a bit contrived, but say I ask the pupils at the table to send someone away. They can compute that they will soon have 3 people + (-1/3 * 3 people) = 2/3 * 3 people at the table. A less contrived example is "I have 2/3kg of flour, and I owe you 1/3kg, while someone else owes me 1/4kg. So, I own 2/3kg + (-1/3kg) + 1/4kg = 7/12kg".

bsaul 6y ago

Except mathematical concepts emerged from real-world observation and problems. We didn't invent fraction out of nowhere. The moment when you have to stop relying on intuition is a pretty delicate matter, but it's still interesting to try to rely on metampho, and then understand when and why a particular metaphor stops working. Much better (imho) than teaching math as a purely formal and transcendent topic that happens to apply to real-world problems, and start with axiomatic definitions (which is the way maths are often taught). Note : i'm sure you're not advocating for that as well, and i don't mean to contradict you. Just that i think it's better to start with a partially broken metaphor, then fix it using formal definition, than not try at all.

macspoofing 6y ago
>We didn't invent fraction out of nowhere. But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world. Rational numbers do not exist in the real world. Real-world does not have infinities. It does not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way that the fourth-grader applied the analogy. >The moment where you have to stop relying on intuition is a pretty delicate matter I didn't argue that metaphors and analogies shouldn't be used. I argued that analogies and metaphors are intrinsically flawed and this article provides a great example. At some point, you have to give up on the analogy and fallback on the underlying axioms. You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions.
- jancsika 6y ago
  > You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions. It is pedagogically superior to choose the route implied by the comments about this being a type error. That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), you gift them a tool they can use to map between the real world and basic unitless mathematical notation. (I'd even add explicit operator definition to that.) For example-- such an educated student could hear your ascetic declaration that "it's against the rules" and quickly grasp something like the following: 1. "1/3+1/3=2/6" doesn't have any units, but it must somehow map to operations with units. 2. If unitless math can be applied regardless of units, then perhaps "1/3+1/3" may mean "1/3 blue table + 1/3 of the red table, where + means joining the two tables." That would equal 2/6 of the joined tables. But "1/3 blue table + 1/3 (same) blue table" would give 2/3 of that blue table, with + mapping to adding those two fractions of the same table. 3. 2/3 does not equal 2/6, so unitless math can't map to both operations. 4. macspoofing said that 2/6 is wrong. 5. Therefore, unitless fraction addition implies addition of things of the same units, and not joining two different things together and finding the new fraction of the new joined unit thingy. If on the other hand a student of your apparent method of declaring rules for unitless math came to a class that had practiced explicitly mapping unitless <-> unit math, they wouldn't have any tools to understand the mapping. (Well, at least if the teacher made a similarly ascetic declaration regarding mapping.) I offer into evidence this very article to show what happens when a student of your apparent method becomes the teacher and encounters the most trivial of unit -> unitless mapping errors. Edit: clarification
  - macspoofing 6y ago
    >That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), Sure. You can certainly ad hoc extend the analogy to bring it in line with the mathematical rules. But at this point, you do hit a higher level of complexity. Your simple analogy is gone and you're slogging through the weeds. That was my point. Analogies are flawed. The teacher started with a very simple rule that worked well and communicated the ideas under certain constraints and then those rules were extended in a logical but incorrect manner by a fourth-grader ... and now the complexity that was hidden in the abstraction is leaking out. Your explanation is more confusing to me and wouldn't be grasped by the vast majority of fourth-graders. At some point, simply stating that fractions have different rules is the most simple (and correct) explanation.

twic 6y ago

Fractions are like groups of pencils or tables of girls and boys. Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.

macspoofing 6y ago
>Fractions are like groups of pencils or tables of girls and boys. No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? I can multiply -1/3 * -100/3 and get a positive fractional value - if we're talking about tables of boys and girls - what the heck happened there? How about 1/0 or 0/1? >Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods. I think you have it reversed. The formal definition is the pure definition, which is ancillary to the application to the real-world but covers much much more.
- simiones 6y ago
  You're taking a strange ahistorical perspective. Most Natural numbers, Integers, Rationals/Fractions, and even some Irrational numbers are much older than the concept of formal mathematics. Natural numbers were discovered as a solution to counting. Negative numbers appear naturally when you need to subtract quantities, especially when you end up with debts. Rationals appear from the need for division. Pi appears when you want to relate lengths of circles with lengths of straight lines. Square roots also appear from basic geometry used in construction, as the naming suggests. All of these are real-world concepts that were used in some form or another at least 6000 years ago. A lot of the time, the closest formal definitions they had at the time were actually based on geometry, as algebra was probably a few thousand years away from being invented/discovered. The modern formal definition is a post-hoc formalization of an existing intuitive concept. If the formal definition did not agree with the intuitive concept, then obviously the formal definition would not be useful and no one would use it. There are mathematical constructs that first appeared in mathematics, and don't have intuitive concepts attached, or even don't have any real-world interpretation. A lot of irrational numbers may be in this area, as would the transfinite numbers, and probably complex numbers as well, though even some of those are usually defined based on some real-world applicability. > I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? It's a bit contrived, but say I ask the pupils at the table to send someone away. They can compute that they will soon have 3 people + (-1/3 * 3 people) = 2/3 * 3 people at the table. A less contrived example is "I have 2/3kg of flour, and I owe you 1/3kg, while someone else owes me 1/4kg. So, I own 2/3kg + (-1/3kg) + 1/4kg = 7/12kg".