32 ms·

6? It seemed so

393 points by ColinWright 6y ago

CydeWeys 6y ago
I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fractions, i.e. when you're totaling subgroups into a larger group. It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging. It's not as bad of a nightmare as trying to explain the answer to the missing dollar riddle: https://en.m.wikipedia.org/wiki/Missing_dollar_riddle https://en.m.wikipedia.org/wiki/Missing_dollar_riddle. That's an absolute nightmare.
- ColinWright 6y ago
  I never understood why the Missing Dollar Riddle ever confuses people. As soon as they say "Add the $2 to the $27" I say, "But why are you adding something someone has to a total that people paid?" That, in turn, is like the "Age of the Shepherd" problem[0] ... people just add/subtract/multiply/divide things randomly without thinking about what they mean. [0] https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff https://mystudentvoices.com/how-old-is-the-shepherd-the-prob...
  - wcarey 6y ago
    The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
    - ColinWright 6y ago
      As I recall, that study was debunked/retracted, so you may have trouble finding it. The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hump_retraction.pdf http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...
      - wcarey 6y ago
        Thanks! I hadn't seen the retraction. That's really interesting.
      - twic 6y ago
        It's a bit more complicated than "debunked/retracted", as that second document explains: > Dehnadi, to his credit, stuck to his guns and did the meta-analysis that showed that he’d discovered a phenomenon and that his test was a worthwhile predictor. The original paper contained several linked claims: that there is an ability to make consistent mental models, that it's intrinsic and fixed, that it predicts ability to program, and that few people have it, and hence few people can learn to program. AIUI, the debunked/retracted claims are that it's intrinsic and fixed, and that few people have it. It looks like the ability exists, but it can be learned, and it is linked to programming ability. Which i think does line up with wcarey's point: > The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. [...] The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
        ColinWright 6y ago
        Agreed. I won't go back and edit my comment, but what you say is true. There really is an interesting thing going on here, even if the original paper was an over-reach.
      - nyanpasu64 6y ago
        Excerpt from the retraction: > In autumn 2005 I became clinically depressed. My physician put me on the then-standard treatment for depression, an SSRI. But she wasn’t aware that for some people an SSRI doesn’t gently treat depression, it puts them on the ceiling. I took the SSRI for three months, by which time I was grandiose, extremely self-righteous and very combative – myself turned up to one hundred and eleven. I did a number of very silly things whilst on the SSRI and some more in the immediate aftermath, amongst them writing “The camel has two humps”. I’m fairly sure that I believed, at the time, that there were people who couldn’t learn to program and that Dehnadi had proved it. The paper doesn’t exactly make that claim, but it comes pretty close. Perhaps I wanted to believe it because it would explain why I’d so often failed to teach them. It was an absurd claim because I didn’t have the extraordinary evidence needed to support it. I no longer believe it’s true. A sad story :(
  - rcxdude 6y ago
    This reminds me of this problem which made the rounds recently[0], where a word problem essentially makes a lot of people commit a fencepost error [0] https://math.stackexchange.com/questions/379927/how-long-will-it-take-marie-to-saw-another-board-into-3-pieces https://math.stackexchange.com/questions/379927/how-long-wil...
- barbegal 6y ago
  And the "missing dollar riddle" can be exploited as the "change raising scam" https://www.youtube.com/watch?v=1Uaw88H0AmE https://www.youtube.com/watch?v=1Uaw88H0AmE
- didgeoridoo 6y ago
  Someone in the comments makes a good point that the best thing to do here may be to introduce ratio notation for proportions (e.g. 2:4) which CAN be added/combined according to the kids’ intuitions — 1:2 combined with 1:2 does indeed equal 2:4, which reduces back to 1:2. You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof, you’ve invented averages! No way someone would have come up with that approach on the fly, though.
  - CydeWeys 6y ago
    Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough options in Unicode that an existing operator should be suitable for this purpose: https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode https://en.wikipedia.org/wiki/Mathematical_operators_and_sym...
    - wcarey 6y ago
      And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples: 1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules. 1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular. 15% + 20% - who knows? 15% of what? 20% of what? (1,2) + (2,4) - can't be done. a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.
      - uryga 6y ago
        > (1,2) + (2,4) - can't be done. are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors > a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)
- yamrzou 6y ago
  The confusion comes from the fact that we attach the same label "3" to two different sets in the case of bottles, so we end up saying "one out of three plus one out of three". It should be "one out of group 1 + one out of group 2", and then as you said, we'll need a special operator for combining groups.
- bwestergard 6y ago
  Adding numerators and denominators to find the "mediant" is sometimes called "doing a freshman sum". In some countries I gather this is introduced as a separate operation (see page 23 in the translation of a Soviet primary school textbook by Gelfand linked below). https://en.wikipedia.org/wiki/Mediant_(mathematics) https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...
  - CydeWeys 6y ago
    Thanks for putting a name on this specific operation. It sounds like it definitely needs its own specific operator. And hi Bjørn!
- 6gvONxR4sf7o 6y ago
  > I'm surprised there is no such operator for averaging. You can't have an operator for combining portions of groups with fractions alone, because 1/3 = 2/6. Combining groups of B boys and N people total, you get B1+B2 boys and N1+N2 people. Let's use @ for that operator, just to not distract from the usual addition. a/b @ c/d = (a+c)/(b+d). Let's combine a group of 1/3 boys with a group of 2/3 boys. 1/3 @ 2/3 = 3/6. But 1/3 = 2/6, so that should be the same as 2/6 @ 2/3 = 4/9. But 3/6 isn't 4/9. You end up with this issue of a/b @ c/d = p a / p b @ q c / q d = (p a + q c) / (p b + q d), which can be anything. So we have the conclusion that if this operator makes sense, then all numbers are equal. You end up with a notion of numbers that is useless for the original problem of combining groups of people of different genders. What we should do here is define it on pairs of numbers rather than fractions. A fraction and a total (p, N), or the number of boys and number of girls (b, g). The latter is super straightforward: (paul, jenny) + (bob, alice) = (paul and bob, jenny and alice), so numerically, it's (b1, g1) @ (b2, g2) = (b1+b2, g1 + g2), but (1, 3) is not (2, 6) here, unlike with fractions. Real simple. If we want to connect this back to the world of fractions, (p1, N1) @ (p2, N2) = (number of boys / total, total) = ( (p1 N1 + p2 N2)/(N1+N2), N1+N2 ). It's just a weighted average, so you need to keep track of the weights.
  - deleted 6y ago
    [deleted]
  - sdenton4 6y ago
    Well, you're right so long as 1/3 = 2/6. This is a mathematical equivalence... Fractions are a 2d projective space: they are pairs (n, d) where (mn, md) is equivalent to (n, d). The suggested operation is totally fine if you are working in ambient (that is, non-projective) space... ie, if you continually keep track of the 'size' of the thing being measured. The equivalence operation is hard to master, and makes the arithmetic complicated.
- OscarCunningham 6y ago
  > It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging. This would also cause confusion, because combining 1/3 with 1/2 and 2/4 in this way would yield different results, even though 1/2 = 2/4.
- GavinMcG 6y ago
  You do this by including units. 1/3 + 1/3 = 2/3. And here, 1/3x + 1/3y = 2/6(x+y).
krm01 6y ago
That’s the same as saying: if you put 1/3rd pizza slice on top of an other 1/3rd pizza slice, you get 2/6 pizza. Layered math.
wcarey 6y ago
This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conceptual shear between those two things is murder for lots of students. Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that's a problem for many students. It's a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it's your first "mathematical" language so you're also trying to learn what the abstract objects the language manipulates are. Some other lovely examples: 3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input. x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell. There are many more.
- mkl 6y ago
  I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.
  - wcarey 6y ago
    Right. Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices. It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.
    - mkl 6y ago
      Well unnecessary parentheses are often used to indicate a substitution has happened. E.g. in a topic I just taught I would write things like ∫_{y=0}^3 x dy = [xy]_{y=0}^3 = x(3) - x(0) = 3x. In context I think it's perfectly clear and a good notational choice.
tyingq 6y ago
One third of one half, plus one third of the other half, is one third of the whole?
- Dumblydorr 6y ago
  Yeah the writer didn't have the words "of one half" or "of the other half", therefore it seemed a lot more confusing than it should have.
gilbetron 6y ago
"If we reduce 2/6, we get 1/3, so 1/3 + 1/3 = 1/3, that seems weird, what did we do wrong?"
JBorrow 6y ago
A great opportunity to introduce multiplications of fractions! 1/3 of the students at table A are girls. 1/3 of the students at table B are girls. What fraction of the tables does table A represent? A is one out of two tables, so A is 1/2 of the tables. Likewise, B is 1/2 of the tables, too. When we want to consider the whole here, we need to take into account what fraction of the whole each proportion represents. The question that we want to answer is 'what proportion of _all students_ at _all of the tables_ are girls?'. This is a combination of the question 'what proportion of students at table A are girls, and what proportion of students at table B are girls', and 'what proportion of all of the tables does each table represent'? That second question might seem quite convoluted but it is important! To do this, we need to multiply the fractions together like so: (Fraction of tables that A represents) * (Fraction of students at table A that are girls) + (Fraction of tables that B represents) * (Fraction of students at table B that are girls) = (Fraction of students at tables A AND B that are girls). So in this case we would have: (1/2) * (1/3) + (1/2) * (1/3) = (1/6) + (1/6) = (1/3). This is even clearer when we consider the case where there are two girls at table B. There, we can do the same thing: (1/2) * (1/3) + (1/2) * (2/3) = (1/6) + (2/6) = (3/6) = (1/2).
- quietbritishjim 6y ago
  I think this one of those situations where you do a bunch of working out and get to the end and see that, mathematically, the problem is fixed, but in your heart it still feels like the original problem is still there. (1/2) * (1/3) + (1/2) * (2/3) might seem like a small calculation to us, but if you've just encountered fractions for the first time I think that is a huge amount of abstract notation. Instead, I think it's better to follow CydeWeys's suggestion of saying that both are correct results of combining 1/3 with 1/3, but they're two different ways of combining them. Say that when you combine two fractions within the same group we call it "addition" and use a plus, but when we combine two fractions from different groups we call it "averaging" (and maybe make up your own symbol for it). Once you've talking about averaging a bit you can move on to multiplication, which in some ways is a more basic concept but, for fractions, is actually a bit less intuitive.
- danbruc 6y ago
  I would suggest to replace 1/2 with 3/6 since the number of students at each table is what matters, not the number of tables, which only yields the correct result if each table has the same number of students.
ValleZ 6y ago
I don't understand why this simple 2nd grade math is featured on hacker news. I guess the next topic will be a long discussion what would be 1/2 divided by 2/3.
- siberianbear 6y ago
  One of the things I've realized in life is the best teachers are people who were not very good at something and then became very good at it through sheer practice and grit. People who are naturally gifted at something because they can't understand the reason for someone else's failure can't teach well. I worked for a year of work study as a math tutor at a community college. "How to do integration by parts[1]?" one student asked? That's easy to explain. My hardest day was when I was trying to help this one woman understand that when you multiply two negative numbers you get a positive numbers. "Wouldn't that make it more negative?" she asked. I decided I would have been a terrible math teacher even though I was very good at math. And reading this article was interesting to me because it explained to me why I would be a terrible math teacher. [1] https://en.wikipedia.org/wiki/Integration_by_parts https://en.wikipedia.org/wiki/Integration_by_parts
- CydeWeys 6y ago
  A lot of people are at home right now trying to teach their kids who are also at home.
  - jpindar 6y ago
    I'm surprised I haven't been seeing more discussions about education recently on HN and everywhere else.
    - CydeWeys 6y ago
      I suspect we skew young enough on here that most of us don't (yet?) have kids.
      - LyndsySimon 6y ago
Spivak 6y ago
I can't imagine doing this to a class of fourth graders but I don't think her thinking is wrong. I think the correct way to views maths notation is that it's a language and we should treat people using it "incorrectly" as a grammar mistake and try to understand the idea they're trying to express. The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." It's hard when the language required to express the ideas they're having is just a little to advanced. And nothing about this stops when you get older. The "just a little outside your knowledge" keeps stretching on forever.
- macspoofing 6y ago
  >The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are not like tables of girls and boys. There are rules for how you add fractions that flow from the underlying axioms. Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of any intuitive reason, but because it's disallowed by the 'rules' of fraction addition - that's it.
  - wcarey 6y ago
    How would you explain that successfully to fourth graders? (Or, for that matter, seventh graders?)
    - bradjohnson 6y ago
      I would avoid multiplication entirely and say that if you are combining tables, you need to update the bottom with the new number of total students. So something like: (T1 girls) + (T2 girls) 1 + 1 _______________________ = _____ (T1 total) + (T2 total) 3 + 3 I think that's more intuitive than averages, and would still work if the tables were of different sizes.
kingkawn 6y ago
It’s 1/3rd of something. what that something is matters. Fractions express proportions of a whole, so you can’t add and subtract them without taking into account what whole they refer to.
macspoofing 6y ago
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.
jmmcd 6y ago
This is exactly why third-level teaching is so much easier.
JoeAltmaier 6y ago
No cat has nine tails. Every cat has one more tail than no cat. Ergo, every cat has 10 tails.
- twic 6y ago
  Half a loaf is greater than nothing. Nothing is greater than God. Therefore, half a loaf is greater than God.
jessermeyer 6y ago
Proportions are tricky to introduce since they are the first obvious move away from absolute quantities. We're taught that division is just fancy subtraction, but it's actually the more subtle idea of proportionality. Similar with multiplication as dimensionality. From here, it feels like the natural setup to show that you can't just 'combine' proportionalities without accounting for what portion these proportions contribute to the new whole.
- wcarey 6y ago
  Perhaps an argument for arithmetic followed by geometry using Nicomachus and Euclid. We sit on that until 9th grade, but I wonder how young you could go with it?
  - jessermeyer 6y ago
    Introducing line segments as alternative representations of numbers at this point feels very natural, and is already implied by most circulum with the standard 'number line'. As you say, we don't do anything with that until much later. Let's leverage that early on!
mynegation 6y ago
I would try to explain semantics of addition (without the word “semantics” of course). I would say, in the first group 1/3 is the girl so if we take out one boy and replace it with another 1/3 (another girl), the resulting proportion is 2/3. Once we mix both groups, that is not what “plus” is.
forbiddenvoid 6y ago
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) ...).
jnbiche 6y ago
In the comments, an educator references an excellent article related to this confusion, "When Can you Meaningfully Add Rates, Ratios, and Fractions" that implicitly suggests some pedagogical approaches: https://flm-journal.org/Articles/11019C10CF34E90DC5866E53E905E8.pdf https://flm-journal.org/Articles/11019C10CF34E90DC5866E53E90...
logfromblammo 6y ago
This is why I taught my kids to label units. ( 1 girl / 3 students at table L + 1 girl / 3 students at table R ) can't be added, because the denominators are different. To fix that, you can multiply by the factors ( 3 students at table L / 6 students at both tables ) and ( 3 students at table R / 6 students at both tables ) so you get ( 1 girl / 6 students at both tables + 1 girl / 6 students at both tables ) = ( 2 girls / 6 students at both tables ). To hammer it home, you point to the table that has 2 or 4 students at it, and ask someone how to add the proportion of girls at that table to the two already under consideration.
pfortuny 6y ago
> It’s hard to think and teach at the same time! So true... You do not see anything when you are at the blackboard.
dr0nkprogrammer 6y ago
Javascript: console.log((0.1 + 0.2) === 0.3) // => false C#: (0.1m + 0.2m) == 0.3m // => true see here: https://dotnetfiddle.net/zVnrNQ https://dotnetfiddle.net/zVnrNQ Javascript is such a poor language, there is no even a good way to work with currencies or exact numbers and you have to multiply numbers with a factor to avoid such problems. The worst language I ever used.
pdkl95 6y ago
As others have already pointed out, the question is ambiguous without more information. Always include the unit; bare magnitudes could mean anything! The student was describing the mediant[1], which was the correct solution when the problem is interpreted with forbiddenvoid'a units[2]. "Adding" fractions with the mediant leads to fun things like the Farey sequence[3] (related to Ford circles[4]) and the very interesting Stern–Brocot tree[5]. (Numberphile has a nice introduction[6] to the fun properties of the Farey sequence) [1] https://en.wikipedia.org/wiki/Mediant_%28mathematics%29 https://en.wikipedia.org/wiki/Mediant_%28mathematics%29 [2] "1/3 (of the students at a table) + 1/3 (of the students at a table) = 2/6 (of the students in the room)" [3] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence [4] https://en.wikipedia.org/wiki/Ford_circle https://en.wikipedia.org/wiki/Ford_circle [5] https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree [6] https://www.youtube.com/watch?v=0hlvhQZIOQw https://www.youtube.com/watch?v=0hlvhQZIOQw
- deleted 6y ago
  [deleted]
Jabbles 6y ago
Don't use tables of the same size: 1/3 + 1/4 = 2/X There would be no confusion as to what X should be. And as someone else said, use units: 1/3 of table A + 1/4 of table B = 2/7 of both tables.
pdevr 6y ago
The moment you add three more students, the first girl is no longer one girl out of a group of three students, she is one girl (out of the two girls) out of a group of six students. Same with the second girl. So, once you add 3 more students, 1/3 becomes 1/6. 1/6 + /6 = 1/3.
knappa 6y ago
From a 'not currently in front of the class' perspective, it's pretty clear what the student meant by 1/3 + 1/3 = 2/6. They were taking + to mean something like a general 'and' or combining action, not in the strict sense of standard fraction addition. It even has a name in mathematics, 'Farey addition'. The kids clearly want to write it in shorthand, so maybe the thing to do is to come up with another symbol for this similar but distinct operation. For example, ⊕.
andred14 6y ago
Apologies but I don't see the mystery here? A fraction describes a proportion ie. the amount of something(s) in relation to another amount of something(s) thus, if we double all of the things in a fraction the proportion and hence the fraction is equivalent.
zuminator 6y ago
1/3 + 1/3 does equal 2/6. 1/3 of one table + 1/3 of one table = 2/6 of two tables.
nurettin 6y ago
This is pretty simple. Put two tables together, 0/6 girls. Now put only one girl: 1/6 put the other girl: 2/6
joshocar 6y ago
This reminds me of a homework problem I did in numerical methods where I made a small mistake in the program but still got an answer that was super close the the actual answer out of pure luck. The professor spent a while looking at it knowing it had to be wrong before he ran out the decimals and showed that it didn't converge.
bjourne 6y ago
Many students at beginner university level math courses do the same kind of mistakes, albeit with larger numbers: 10/13 + 7/21 = 17/34. Decimals are much easier to deal with imo, but mathematicians does not like them because they are inexact.
xigency 6y ago
When you want to add fractions of a whole together. 1 1 2 --- + --- = --- 3 3 3 When you want to add wholes together and make a new fraction. 1 + 1 2 --------- = --- 3 + 3 6
prvc 6y ago
Her first example, "proportion of a six-pack of water" is actually very complex, with a lot of potentially distracting details. Nowhere does she mention that's what her fractions are measuring, either! The whole lecture was simply a mass of detail with no reference to the underlying concept it was putatively intended to explain. No wonder the children were confused. Why not start with cutting a pie? Guess that doesn't lead to nearly as much busy-work.
jfkienennd 6y ago
My favourite: I got 1/3 on the first test and 1/3 on the second test. So in total I got 2/6. So 1/3 + 1/3 = 2/6.
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rmtech 6y ago
"+" is the wrong mathematical operation for combining fractions of distinct sets. The right operation is a weighted sum: 1/3×1/2 + 1/3×1/2 = 1/3 The "1/2" is there because each of the two sets that we are combining is the same size.
ash 6y ago
One of the commenters gave a great suggestion: > I would use bar models to show how the whole changes. First, draw one bar (table) with 3 students inside; label the bar one whole, star one student and label that student as one third of that whole. Do the same thing beside the first bar model, again showing the bar as one whole, starring one student and labelling that student as one third. Then, push the two bar models together (I’d use a Smart Board) to show a new whole: the two bars together with 6 students in the one bar is labeled as the new whole. Point out that this is a NEW whole, with a different # of pieces. Now they would see the 2 starred students in the one new bar made up of 6 total students, so the “old” 1/3 student becomes the “new” 1/6 student once the whole changes. This is like a name change, once the size of the whole changes. You can also show them that by cutting each “student” in half, you would get an equivalent fraction of 1/3 = 2/6 of the first one whole bar. So 1/3 = 2/6, not two 1/3’s = 2/6. http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-so/#comment-27347 http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seeme...
wondringaloud 6y ago
I think this is a perfect example of why people find math hard. Not because there's something inherently difficult, but because we have teachers like this one in charge of open and curious minds at a young age. You get a teacher who doesn't quite understand what it is she's trying to teach, and this confusion is multiplied over and over again for each batch of students that pass through. Not too long afterwards you get kids who think they don't understand math, when in fact it was a failing of the teachers who couldn't correctly explain it in the first place.
adrianmonk 6y ago
The difficulty isn't with fractions. It's about understanding what "+" means. Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie"). His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about the word "plus" to the symbol. People even use "+" (rather than "plus") to mean combining things. Maybe Addison saw "Nature's Path Pumpkin Seed + Flax Granola" at the grocery store. So why shouldn't he try using it that way? Somebody needs to communicate to Addison that, in math class, "+" always means something specific. He doesn't know that yet, but he has been asked to use the symbol anyway. Math uses a whole lot of lingo. If it's not covered well enough, stumbling over the terminology can be an impediment to learning. This includes both new words ("quotient", "integer") and words that are used in everyday language but differently in math ("where" meaning condition or definition instead of place, "of" meaning multiplication, "real" numbers).
- Phlogistique 6y ago
  That's my favourite explanation so far. That's how I would put it to the student: Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples: * At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table. * Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables. The first operation is what mathematicians call "+". They write "1/3 + 1/3 = 2/3" The mathematicians do not have a good name for the second one, so let's invent one: "1/3 1/3 = 2/6" What's better with this explanation, compared to the "ratios vs. fractions" thing, or the units thing, is that you do not have to introduce a separate category of numbers that sound very similar but act differently.
  - wcarey 6y ago
    We explicitly encourage the "let's invent name for this similar but unnamed thing" with our students, and use whatever name they come up with for the thing for the rest of their time with us. Sometimes those student created names become legendary. They really enjoy the idea that they can have "Jane's relation" be used by students after they graduate. (Of course, sometimes Jane's relation is really Euler's totient function, and we have to encourage them that even though they found something a very good mathematician also found, we're going to call it by the common name.)
Wowfunhappy 6y ago
Something I missed the first time around—in the comments below the article, the teacher expands on what she actually did. This was originally one long paragraph, but I've added paragraph breaks for readability: --- > At the time, I let it go so I could think and regroup myself. There was more than fractions to think about: Is it possible for both equations to be true: 1/3 + 1/3 = 2/6 and 1/6 + 1/6 = 2/6. This was a long discussion that kept their interest, in which I learned that many (most?) of the students didn’t think of fractions as numbers. That was another hurdle. I kept going back to what we know about adding whole numbers. > Then I took another approach, once I was sure that they understood that 1/3 and 2/6 are equivalent, so how can I add a number to itself and wind up with a sum that’s the same as one of them? In all of these discussions, students would change their thinking. We looked at adding on a number line. I used pattern blocks to explore the same problem. I kept talking about keeping our attention on what was 1 whole. > My, this all takes time, and the time is important for students to develop, cement, and extend their understanding. What I didn’t do, that I’ve been thinking about now, is to make it part of a writing workshop on persuasive writing and have them choose a conjecture and write. I think I could spend most of the year on this with students.
08-15 6y ago
That's what happens when you try to "build an intuition" for a concept instead of defining it. Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!! The whole point of common fractions is to close integer arithmetic over division by introducing a new kind of numbers, precisely those fractions that don't correspond to integers. The question isn't so much "What is half a bottle of water?", it's "How do you calculate with one half (a bottle or whatever)." Math is beautiful, but only if built up from simple rules. It's amazing how much you can make from counting and a handful of convenient notations. Even children can recognize that beauty. But high school instead teaches math as a jumbled mess of things you have to memorize, without structure, without rhyme and reason. Needless to say, I hated it. (School, not math.)
- wcarey 6y ago
  As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that way. For lots of students, the process of moving from definitions and simple rules to other statements is extraordinarily difficult. I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from the definition of the exponent. About a third of the students say, "cool" and never miss anything on any assessment again because the answers flow necessarily from the givens. About a third work their way through it fine. About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite working many particular examples where that is manifestly false.
  - 08-15 6y ago
    > About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 ...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too. A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (ac)/(bc) = a/b, is it? Of course, she never asked why a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so. I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.
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alakadfs 6y ago
I would explain that the "+" sign is not in the right place in the equation and write (1+1)/(3+3) = 2/6 instead.
boomlinde 6y ago
Of course 1/3 + 1/3 = 2/6, and in this case you can make an error in reasoning that is probably common among children who are just learning fractions that will still arrive at the same answer. In this case I would have demonstrated a case where the error in reasoning that was possibly made does not lead to the correct answer, for example 1/8 + 1/4. The erroneous intuition of simply adding the groups and their subsets together breaks down when you can show that 1/8 + 1/4 ≠ 2/12 but 3/8
- boomlinde 6y ago
  And now I feel stupid for obvious reasons
0xfaded 6y ago
Not that I would have thought of this on the spot, but ... The numerator and denominator represent different things. The numerator is "how many things", and the bottom is "of how many spaces". Asking what fraction of the table is girls is dangerous, because the answer is (1 girl) / (1 girl + 2 boys). Still a stretch, rephrasing as "what fraction of seats are filled by girls" might have helped show the idea that the number of spaces is fixed.
tosh 6y ago
related: notation as a tool of thought https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=true https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=tr... 1/3 relates to one of the tables 2/6 relates to both tables
hathawsh 6y ago
This is fantastic material. I'm going to present this quandary to my kids. Some of my kids struggle at math, so it will help them understand that math is a language and, like any language, it has abbreviations that sometimes lead you down a confusing path until you spell out more. Some of my kids are good at math and it will help them relate with others.
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inimino 6y ago
This kind of thing fascinates me. I've always felt that you don't really understand a thing, even a simple thing like fractions, until you can find and resolve all the paradoxes that arise in using it naively. My take when reading the 1/3 + 1/3 = 2/6 equation is that Addison was right but the "+" here is a different operation than the "+" of 1/3 + 1/3 = 2/3. In one case we are adding disjoint parts of the same whole and counting how much of the parts of the whole are now included, and get 2/3, and in the other case we are adding two wholes and counting how much of the new whole is included. People who pass primary education without internalizing this distinction will have to do so later if they learn statistics or probability theory. The answer "what would you do now?" is the same one I would try to ask the students. First I would stop, confused, and write 1/3 + 1/3 = ? elsewhere on the board, and then wait for the students to see why I'm confused. They already understand that we want mathematics to be consistent, and getting different answers means we were wrong at least once, but they will have a hard time to find out or explain why 2/6 and 2/3 both are intuitively and obviously right. You have to compare the intuitions directly to each other to see that in one case you're adding parts of the same whole (set or group or table of kids) and in another case adding the wholes. From what I know of elementary school, we focus a lot on fractions and the formal manipulation of them as unitless values, and drill into kids that 2/4, 1/2, and 0.5 are all "the same thing". Bullshit! It's no wonder they have problems with word problems! Two-out-of-four apples and one-out-of-two apples and a half apple are very different things, as every elementary student knows well. How can we expect them to believe something that's obviously false unless we've introduced the narrow context in which it's true? When used as unitless "coefficients" or bare numbers that exist only to scale another value that represents a real quantity of something, then 1/2 and 2/4 are the same. This is what we should be teaching when we teach fractions, not some mindless rote symbol manipulation like cross-multiplying or what have you without any ability to intuitively derive those rules yourself. I'd spend the rest of the class time letting them work out all the different ways of looking at it, and the connections to everything else fractions and ratios and addition are good for. Of course we want to be able to have "+" be well-defined, especially in elementary school, and the students understand that! We've now found a perfectly intuitive meaning of addition for which our standard "+" definition does not fit! This gives an opportunity to explore the connections between mathematics and intuition and the symbols and the real world operations they stand for. It gives us an excellent chance to, as Polya puts it, "introduce a suitable notation" for the setwise addition of "marked" sets that we have just discovered. Then we can explore the properties of this operation and the ones between it and the familiar addition operation. We can show that the plus sign and the addition operation we use it to represent are distinct, and our use of one for the other is a choice, and one we could make differently. I saw some mentions of units in the comments, and yet 1/3 + 1/3 = ? is still ill-defined if we know only that "1/3" means "one out of three students". What we need to know is which students they are! So if one out of Alice, Bob, and Charlie has red hair, and one out of Charlie, Doris, and Evan has red hair, how many of Alice, Bob, Charlie, Doris, and Evan have red hair? Well, we made the question challenging even for adults if we ask it that way! And now you can introduce uncertainty and degrees of belief. (What if three people tell you they have one red-headed friend, in a class of 23 students? Do they all have the same friend, or is there more than one redhead?) Of course what most of us will actually do as the teacher in this situation is freeze and think "oh no, I've made a mistake and the teacher (in your mind) is going to point it out and the class is going to laugh at me!". Because that's what we've learned happens in these situations. If we interpret confusion and being wrong as painful and embarrassing, little wonder that mathematics seems a long and painful road that we avoid whenever we can.
karmakaze 6y ago
Use pie graphs to show the original tables split up into thirds of one girl two boys. Then show two ways of combining two tables. (>-) combined with (>-) is a table of six: 2 girls, 4 boys show all boys and girls moving from tables with 3 seats to the same table with 6 seats. Now show a table with three seats, one occupied by a girl. Show another table with same configuration. One girl moves from one table to join the first table. 1/3 full + 1/3 full (at the same table) is 2/3 full.
luord 6y ago
> What would you do now??? I would probably start crying, to be honest. But in all seriousness, this is the kind of thing that makes me dread teaching. I mean, I know what is wrong, I just can't wrap my head around how to explain it, specially without going into stuff that would be too advanced for these particular students.
wwarner 6y ago
I would express this in a diagram with a dotted line around the set under consideration. When dealing with thirds, make a diagram containing three objects with their respective properties, and end by drawing a dotted line around the set. When switching to sixths, add three more objects, erase the line around the first three, and draw a new dotted line around the whole set of six. Then go on to sevenths by adding one more object, erasing the dotted line for the six and drawing it around all seven. If I do it this way, my 4th grader can add fractions just like numbers without resorting to common denominators.
Geee 6y ago
I would have said that a whole table is 3/3 and that both tables together are 6/3 and not 6/6 (maybe they'll get it that 6/3 equals to 2). Probably a good idea to explain that the "whole" isn't all pens and bottles but the group of 3 or 6 or 12 or whatever. Maybe start with explaining from the start what happens if there are 7 bottles, when bottles are counted in the groups of six-packs. How many six-packs can you make from 15 bottles?
contextfree 6y ago
Wow, I had no idea this blog existed - I loved Marilyn Burns's children's math books as a kid, awesome to see her still in action.
Gravityloss 6y ago
You're taking a weighted average, not adding the fractions. It takes a while to learn what that is used for.
devit 6y ago
It's easy to see that this cannot be a valid definition for fraction addition because applying it along with the rule that a/b = (ak)/(bk) gives a contradiction: 3/7 = 2/4 (+) 1/3 = 1/2 (+) 1/3 = 2/5, absurd because 2/5 != 3/7 Once this is realized, it's easy to see that the correct formulation for the "table joining" operation used here is a weighted arithmetic mean, i.e. a(/)b (+) c(/)d = weighted arithmetic mean of a/b and c/d with relative weights b and d = ((a/b) * b + (c/d) * d) / (b + d) = (a + c)/(b + d). On the other hand, fraction addition is clearly determined as a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd given the a/b = (ak)/(bk) and (a/b + c/b) = (a + c)/b axioms.
- karatestomp 6y ago
  2/4 + 1/3 would clearly be 5/12. You halve each to put them in terms of the two wholes you're combining (which you must be doing, since you're adding them—else where do those go?) then combine those. Same process that gives you 1/3 + 1/3 = 1/3. Worked out (conventional meaning of + for clarity; we'd need a new operator otherwise): 1/4 + 1/6 = 6/24 + 4/24 = 10/24 = 5/12
carapace 6y ago
http://iconicmath.com http://iconicmath.com We probably haven't come up with the best visual abstractions yet, eh?
benlivengood 6y ago
I love this discussion and want to throw in my two cents. Why not teach beginning students group theory as the foundation to addition and subtraction? While it wouldn't solve this particular problem since fractions require fields, it would give teachers the tools to explain precisely why addition of two tables is not a group or field operation; they are from different sets and require measurement with units to define a new inclusive set. My own understanding of arithmetic and algebra was enhanced immensely by a book outlining the derivation from Peano axioms. Ultimately the concepts are very simple and kids intuitively do the steps without having the names for the mental processes that they're doing. The educational hurdles to me seem to be 1) identifying the memorized names of the sequence of natural numbers with a procedure for generating them; 2) distinguishing between procedures, objects, numerals, and numbers; 3) basics of set theory; 4) natural induction.
- GuB-42 6y ago
  Bad idea! They tried exactly that in France in the 60s and 70s. It was called Maths modernes (New maths). I think it was the same in most of the western world. It was a disaster and they abandoned it in the 80s. The problem was that while it may have helped the best students understand advanced concepts later on, it produced a generation of people who lacked practical counting skills. EDIT: By "bad", I don't mean "stupid". The people who came up with new maths were experts in their field, with the goal of having a more scientifically literate population. No doubt very smart people, but maybe a little too smart for their own good. Turned out these concepts were too much for most young kids, and too remote from the way math is used on a day to day basis.
savanpatel 6y ago
To fill the quiet, I said to the class, “When thinking about fractions, it’s important to keep your attention on what the whole is.” This is basic. There was no need to write entire blog of how you misunderstood this.
biddlesby 6y ago
I would say it was a mistake to introduce the concept of adding fractions from two different "wholes". Instead, teach that in order to add fractions, you first have to get everything into the same "whole". Like, you can add Jack / 3 to Ben / 3 because they were at the same table. But adding the boys from one table to another is quite a different thing. Instead, you should teach that you should first make the fractions with both tables as the whole. Only then are you allowed to add. This could come as a later concept
dgentile 6y ago
1/3 and 1/3 is 1/6 ;)
chx 6y ago
I happen to have a math teacher masters though I myself do not teach (but I do help with the program of a tiny, tiny reform school). This teacher here had painted themselves into a corner and it's hard to get out of it. Do not explain fractions with things you can't change the denominator of. Rather tell it with money. Say, 1/3 means if the table has 3 coins , one kid gets 1 coins. If there are 6, 9, 12, 15, one kid gets how many? If the other table has five kids and 5 coins then 1/5 means when splitting five coins a kid gets one. Figure out together what happens with splitting 10, 15, 20 coins. Now putting together the two tables we want to calculate 1/3+1/5, how many coins can we do that with? Step through it, we practice 1/3 with 3, 6, 9, 12 ... but can you tell what the fifth of nine coins are? You can't ... Then find 15 and then show them 1/3+1/5 means 8/15. This is all play and very smooth. To answer the question posed in the blog post: I would plan the class carefully to avoid the entire situation. But if I must, I'd point out 1/3 is a mere shorthand for division, 1:3 and writing "1:3 + 1:3" is adding two operations together and it does not even make sense. We can restore sanity but that has its own rules.
nartz 6y ago
If you make it extreme, then its easier to see the folly / error - 500/1500 students + 1/3 students = ?
sova 6y ago
Really great anecdote! That first plus sign that Addison wrote is actually more of an "And" instead of an arithmetic "plus"
fallingfrog 6y ago
I think that one key issue is that the students are confusing addition with the more general concept of considering two things together.
matvore 6y ago
I think the problem is the teacher moved from one type of "fraction problem" to another kind. The example of drinking 4 bottles of water to only have 2/6 left of the original pack is doing an integral problem and wrapping up the answer as a fraction. Same deal with the pack of 12 pencils. The students can grasp those problems as e.g. (1+1)/6 rather than (1/6 + 1/6). In other words, there is only one "whole" in the problem. When you're adding the fractions of desks filled with students, the fraction is counter-intuitive because both desks have to have the same number of students for it to make sense. (1/3 of a 5000-student round table is different from a 3-student table). And to say "the units are wrong" is kind of a limited way of explaining this. The units also need to have the table capacity as part of the unit identity (i.e. the units would have to be 3-student-table). That's a pretty sophisticated way of thinking about units. I think that after the pencil/water examples, transitioning from the pencil/water pack examples to a more "pure" fraction example would be better. e.g. one group of students eats 2/3 of a pizza, and another group eats 2/3 of a pizza. Now you can throw away one of the original pizza boxes and put the two remaining 1/3 in a single box which is 2/3 full. Now the "whole" for each fraction is no longer arbitrary (like 6 bottles of water or 12 pencils).
mintyc 6y ago
The and operation could be logical so the result of anding with one self is a mop
poleguy 6y ago
1+1=50.8 My forth grader understood that this equation makes the same mistake. Hint: units of inches and mm must be made explicit.
EmilioMartinez 6y ago
My take: Adding proportions is kinda "semantically incorrect". It's well defined as a value, it just doesn't mean what you think it means, akin to thinking that union of two sets is written A + B (which would be the set of pair-wise additions of elements). Now, dovetailing the equation with units that signify concrete quantities (as suggested by others) circumvents this by making it meaningful, at the cost of no longer working with the abstract concept of proportion. An example of semantically meaningful manipulation of proportions is the weighted combination of proportions. This here's a more verbose version of what the teacher wrote: (3/6)* (1/3) + (3/6)*(1/3) = 1/3 (that is, the weighted average of the proportions give the final proportion)
lucio 6y ago
When you put two tables together, each table has 1/2 of the total number of kids. By having 2 tables, now you have 6 kids, so: the original 1/3 in the first table becomes 1/6 the original 1/3 in the second table becomes 1/6 so adding the two tables means 2/6 are girls She needs to explain 1/3 of 1/2 (fraction of a fraction) before using an example where she joins two units (two tables)
djmips 6y ago
As the concept of fairness is innate in primates including humans in the 4th grade I wouldn't stop and correct the students but try and re-frame the problem in terms of a problem where fairness was involved and then immediately the ratios and the units would come into more clear focus. As an example I might say that if I promised to give you 1/3 of the pizza as a reward for cleaning your room. And then how much should you get if I had 1 pizza. How about 2 pizzas? Things like that.
imetatroll 6y ago
Have the kids stand up and count out numbers. Now they know how many kids are in the room. Then have them go and stand in two different corners. Count one corner. Then have them stand in three different corners. Count one corner. Finally the four corners. Count one corner. Then have the two tables distribute their students to the corners and figure out these elusive fractions. Have each corner be a Pokemon or something. That should do the trick.
ValorieO13 6y ago
I solved this by then introducing the kids to slices in a pie. Visually, the kids can see that 1/3 of a pie plus 1/3 of a pie is not the same as 2/6. Fractions are often introduced as "parts of a set" which is great for the first day. It's easy for kids to see and understand, but kids are more familiar with trying to slice the birthday cake for all the kids at the party. After the kids realize there are two kinds of fractions, we spend time at each problem thinking through is this a set (which we'll return to in ratios) or a whole (which can't be expanded) and visually representing fractions both ways to check their answers. Adding units is absolutely important, but when I teach it, that's what I add in fourth or fifth grade when I'm teaching them to reduce.

393 points by ColinWright 6y ago

I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fractions, i.e. when you're totaling subgroups into a larger group. It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging. It's not as bad of a nightmare as trying to explain the answer to the missing dollar riddle: https://en.m.wikipedia.org/wiki/Missing_dollar_riddle https://en.m.wikipedia.org/wiki/Missing_dollar_riddle. That's an absolute nightmare.

ColinWright 6y ago
I never understood why the Missing Dollar Riddle ever confuses people. As soon as they say "Add the $2 to the $27" I say, "But why are you adding something someone has to a total that people paid?" That, in turn, is like the "Age of the Shepherd" problem[0] ... people just add/subtract/multiply/divide things randomly without thinking about what they mean. [0] https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff https://mystudentvoices.com/how-old-is-the-shepherd-the-prob...
- wcarey 6y ago
  The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
  - ColinWright 6y ago
    As I recall, that study was debunked/retracted, so you may have trouble finding it. The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hump_retraction.pdf http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...
    - wcarey 6y ago
      Thanks! I hadn't seen the retraction. That's really interesting.
    - twic 6y ago
      It's a bit more complicated than "debunked/retracted", as that second document explains: > Dehnadi, to his credit, stuck to his guns and did the meta-analysis that showed that he’d discovered a phenomenon and that his test was a worthwhile predictor. The original paper contained several linked claims: that there is an ability to make consistent mental models, that it's intrinsic and fixed, that it predicts ability to program, and that few people have it, and hence few people can learn to program. AIUI, the debunked/retracted claims are that it's intrinsic and fixed, and that few people have it. It looks like the ability exists, but it can be learned, and it is linked to programming ability. Which i think does line up with wcarey's point: > The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. [...] The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
      - ColinWright 6y ago
        Agreed. I won't go back and edit my comment, but what you say is true. There really is an interesting thing going on here, even if the original paper was an over-reach.
    - nyanpasu64 6y ago
      Excerpt from the retraction: > In autumn 2005 I became clinically depressed. My physician put me on the then-standard treatment for depression, an SSRI. But she wasn’t aware that for some people an SSRI doesn’t gently treat depression, it puts them on the ceiling. I took the SSRI for three months, by which time I was grandiose, extremely self-righteous and very combative – myself turned up to one hundred and eleven. I did a number of very silly things whilst on the SSRI and some more in the immediate aftermath, amongst them writing “The camel has two humps”. I’m fairly sure that I believed, at the time, that there were people who couldn’t learn to program and that Dehnadi had proved it. The paper doesn’t exactly make that claim, but it comes pretty close. Perhaps I wanted to believe it because it would explain why I’d so often failed to teach them. It was an absurd claim because I didn’t have the extraordinary evidence needed to support it. I no longer believe it’s true. A sad story :(
- rcxdude 6y ago
  This reminds me of this problem which made the rounds recently[0], where a word problem essentially makes a lot of people commit a fencepost error [0] https://math.stackexchange.com/questions/379927/how-long-will-it-take-marie-to-saw-another-board-into-3-pieces https://math.stackexchange.com/questions/379927/how-long-wil...
barbegal 6y ago
And the "missing dollar riddle" can be exploited as the "change raising scam" https://www.youtube.com/watch?v=1Uaw88H0AmE https://www.youtube.com/watch?v=1Uaw88H0AmE
didgeoridoo 6y ago
Someone in the comments makes a good point that the best thing to do here may be to introduce ratio notation for proportions (e.g. 2:4) which CAN be added/combined according to the kids’ intuitions — 1:2 combined with 1:2 does indeed equal 2:4, which reduces back to 1:2. You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof, you’ve invented averages! No way someone would have come up with that approach on the fly, though.
- CydeWeys 6y ago
  Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough options in Unicode that an existing operator should be suitable for this purpose: https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode https://en.wikipedia.org/wiki/Mathematical_operators_and_sym...
  - wcarey 6y ago
    And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples: 1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules. 1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular. 15% + 20% - who knows? 15% of what? 20% of what? (1,2) + (2,4) - can't be done. a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.
    - uryga 6y ago
      > (1,2) + (2,4) - can't be done. are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors > a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)
yamrzou 6y ago
The confusion comes from the fact that we attach the same label "3" to two different sets in the case of bottles, so we end up saying "one out of three plus one out of three". It should be "one out of group 1 + one out of group 2", and then as you said, we'll need a special operator for combining groups.
bwestergard 6y ago
Adding numerators and denominators to find the "mediant" is sometimes called "doing a freshman sum". In some countries I gather this is introduced as a separate operation (see page 23 in the translation of a Soviet primary school textbook by Gelfand linked below). https://en.wikipedia.org/wiki/Mediant_(mathematics) https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...
- CydeWeys 6y ago
  Thanks for putting a name on this specific operation. It sounds like it definitely needs its own specific operator. And hi Bjørn!
6gvONxR4sf7o 6y ago
> I'm surprised there is no such operator for averaging. You can't have an operator for combining portions of groups with fractions alone, because 1/3 = 2/6. Combining groups of B boys and N people total, you get B1+B2 boys and N1+N2 people. Let's use @ for that operator, just to not distract from the usual addition. a/b @ c/d = (a+c)/(b+d). Let's combine a group of 1/3 boys with a group of 2/3 boys. 1/3 @ 2/3 = 3/6. But 1/3 = 2/6, so that should be the same as 2/6 @ 2/3 = 4/9. But 3/6 isn't 4/9. You end up with this issue of a/b @ c/d = p a / p b @ q c / q d = (p a + q c) / (p b + q d), which can be anything. So we have the conclusion that if this operator makes sense, then all numbers are equal. You end up with a notion of numbers that is useless for the original problem of combining groups of people of different genders. What we should do here is define it on pairs of numbers rather than fractions. A fraction and a total (p, N), or the number of boys and number of girls (b, g). The latter is super straightforward: (paul, jenny) + (bob, alice) = (paul and bob, jenny and alice), so numerically, it's (b1, g1) @ (b2, g2) = (b1+b2, g1 + g2), but (1, 3) is not (2, 6) here, unlike with fractions. Real simple. If we want to connect this back to the world of fractions, (p1, N1) @ (p2, N2) = (number of boys / total, total) = ( (p1 N1 + p2 N2)/(N1+N2), N1+N2 ). It's just a weighted average, so you need to keep track of the weights.
- deleted 6y ago
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- sdenton4 6y ago
  Well, you're right so long as 1/3 = 2/6. This is a mathematical equivalence... Fractions are a 2d projective space: they are pairs (n, d) where (mn, md) is equivalent to (n, d). The suggested operation is totally fine if you are working in ambient (that is, non-projective) space... ie, if you continually keep track of the 'size' of the thing being measured. The equivalence operation is hard to master, and makes the arithmetic complicated.
OscarCunningham 6y ago
> It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging. This would also cause confusion, because combining 1/3 with 1/2 and 2/4 in this way would yield different results, even though 1/2 = 2/4.
GavinMcG 6y ago
You do this by including units. 1/3 + 1/3 = 2/3. And here, 1/3x + 1/3y = 2/6(x+y).

krm01 6y ago

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

wcarey 6y ago

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conceptual shear between those two things is murder for lots of students. Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that's a problem for many students. It's a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it's your first "mathematical" language so you're also trying to learn what the abstract objects the language manipulates are. Some other lovely examples: 3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input. x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell. There are many more.

mkl 6y ago
I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.
- wcarey 6y ago
  Right. Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices. It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.
  - mkl 6y ago
    Well unnecessary parentheses are often used to indicate a substitution has happened. E.g. in a topic I just taught I would write things like ∫_{y=0}^3 x dy = [xy]_{y=0}^3 = x(3) - x(0) = 3x. In context I think it's perfectly clear and a good notational choice.

tyingq 6y ago

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

Dumblydorr 6y ago
Yeah the writer didn't have the words "of one half" or "of the other half", therefore it seemed a lot more confusing than it should have.

gilbetron 6y ago

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

JBorrow 6y ago

A great opportunity to introduce multiplications of fractions! 1/3 of the students at table A are girls. 1/3 of the students at table B are girls. What fraction of the tables does table A represent? A is one out of two tables, so A is 1/2 of the tables. Likewise, B is 1/2 of the tables, too. When we want to consider the whole here, we need to take into account what fraction of the whole each proportion represents. The question that we want to answer is 'what proportion of _all students_ at _all of the tables_ are girls?'. This is a combination of the question 'what proportion of students at table A are girls, and what proportion of students at table B are girls', and 'what proportion of all of the tables does each table represent'? That second question might seem quite convoluted but it is important! To do this, we need to multiply the fractions together like so: (Fraction of tables that A represents) * (Fraction of students at table A that are girls) + (Fraction of tables that B represents) * (Fraction of students at table B that are girls) = (Fraction of students at tables A AND B that are girls). So in this case we would have: (1/2) * (1/3) + (1/2) * (1/3) = (1/6) + (1/6) = (1/3). This is even clearer when we consider the case where there are two girls at table B. There, we can do the same thing: (1/2) * (1/3) + (1/2) * (2/3) = (1/6) + (2/6) = (3/6) = (1/2).

quietbritishjim 6y ago
I think this one of those situations where you do a bunch of working out and get to the end and see that, mathematically, the problem is fixed, but in your heart it still feels like the original problem is still there. (1/2) * (1/3) + (1/2) * (2/3) might seem like a small calculation to us, but if you've just encountered fractions for the first time I think that is a huge amount of abstract notation. Instead, I think it's better to follow CydeWeys's suggestion of saying that both are correct results of combining 1/3 with 1/3, but they're two different ways of combining them. Say that when you combine two fractions within the same group we call it "addition" and use a plus, but when we combine two fractions from different groups we call it "averaging" (and maybe make up your own symbol for it). Once you've talking about averaging a bit you can move on to multiplication, which in some ways is a more basic concept but, for fractions, is actually a bit less intuitive.
danbruc 6y ago
I would suggest to replace 1/2 with 3/6 since the number of students at each table is what matters, not the number of tables, which only yields the correct result if each table has the same number of students.

ValleZ 6y ago

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

siberianbear 6y ago
One of the things I've realized in life is the best teachers are people who were not very good at something and then became very good at it through sheer practice and grit. People who are naturally gifted at something because they can't understand the reason for someone else's failure can't teach well. I worked for a year of work study as a math tutor at a community college. "How to do integration by parts[1]?" one student asked? That's easy to explain. My hardest day was when I was trying to help this one woman understand that when you multiply two negative numbers you get a positive numbers. "Wouldn't that make it more negative?" she asked. I decided I would have been a terrible math teacher even though I was very good at math. And reading this article was interesting to me because it explained to me why I would be a terrible math teacher. [1] https://en.wikipedia.org/wiki/Integration_by_parts https://en.wikipedia.org/wiki/Integration_by_parts
CydeWeys 6y ago
A lot of people are at home right now trying to teach their kids who are also at home.
- jpindar 6y ago
  I'm surprised I haven't been seeing more discussions about education recently on HN and everywhere else.
  - CydeWeys 6y ago
    I suspect we skew young enough on here that most of us don't (yet?) have kids.
    - LyndsySimon 6y ago

Spivak 6y ago

I can't imagine doing this to a class of fourth graders but I don't think her thinking is wrong. I think the correct way to views maths notation is that it's a language and we should treat people using it "incorrectly" as a grammar mistake and try to understand the idea they're trying to express. The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." It's hard when the language required to express the ideas they're having is just a little to advanced. And nothing about this stops when you get older. The "just a little outside your knowledge" keeps stretching on forever.

macspoofing 6y ago
>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are not like tables of girls and boys. There are rules for how you add fractions that flow from the underlying axioms. Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of any intuitive reason, but because it's disallowed by the 'rules' of fraction addition - that's it.
- wcarey 6y ago
  How would you explain that successfully to fourth graders? (Or, for that matter, seventh graders?)
  - bradjohnson 6y ago
    I would avoid multiplication entirely and say that if you are combining tables, you need to update the bottom with the new number of total students. So something like: (T1 girls) + (T2 girls) 1 + 1 _______________________ = _____ (T1 total) + (T2 total) 3 + 3 I think that's more intuitive than averages, and would still work if the tables were of different sizes.

kingkawn 6y ago

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

macspoofing 6y ago

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.

jmmcd 6y ago

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

JoeAltmaier 6y ago

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

twic 6y ago
Half a loaf is greater than nothing. Nothing is greater than God. Therefore, half a loaf is greater than God.

jessermeyer 6y ago

Proportions are tricky to introduce since they are the first obvious move away from absolute quantities. We're taught that division is just fancy subtraction, but it's actually the more subtle idea of proportionality. Similar with multiplication as dimensionality. From here, it feels like the natural setup to show that you can't just 'combine' proportionalities without accounting for what portion these proportions contribute to the new whole.

wcarey 6y ago
Perhaps an argument for arithmetic followed by geometry using Nicomachus and Euclid. We sit on that until 9th grade, but I wonder how young you could go with it?
- jessermeyer 6y ago
  Introducing line segments as alternative representations of numbers at this point feels very natural, and is already implied by most circulum with the standard 'number line'. As you say, we don't do anything with that until much later. Let's leverage that early on!

mynegation 6y ago

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

forbiddenvoid 6y ago

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) ...).

jnbiche 6y ago

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

logfromblammo 6y ago

This is why I taught my kids to label units. ( 1 girl / 3 students at table L + 1 girl / 3 students at table R ) can't be added, because the denominators are different. To fix that, you can multiply by the factors ( 3 students at table L / 6 students at both tables ) and ( 3 students at table R / 6 students at both tables ) so you get ( 1 girl / 6 students at both tables + 1 girl / 6 students at both tables ) = ( 2 girls / 6 students at both tables ). To hammer it home, you point to the table that has 2 or 4 students at it, and ask someone how to add the proportion of girls at that table to the two already under consideration.

pfortuny 6y ago

> It’s hard to think and teach at the same time! So true... You do not see anything when you are at the blackboard.

dr0nkprogrammer 6y ago

Javascript: console.log((0.1 + 0.2) === 0.3) // => false C#: (0.1m + 0.2m) == 0.3m // => true see here: https://dotnetfiddle.net/zVnrNQ https://dotnetfiddle.net/zVnrNQ Javascript is such a poor language, there is no even a good way to work with currencies or exact numbers and you have to multiply numbers with a factor to avoid such problems. The worst language I ever used.

pdkl95 6y ago

As others have already pointed out, the question is ambiguous without more information. Always include the unit; bare magnitudes could mean anything! The student was describing the mediant[1], which was the correct solution when the problem is interpreted with forbiddenvoid'a units[2]. "Adding" fractions with the mediant leads to fun things like the Farey sequence[3] (related to Ford circles[4]) and the very interesting Stern–Brocot tree[5]. (Numberphile has a nice introduction[6] to the fun properties of the Farey sequence) [1] https://en.wikipedia.org/wiki/Mediant_%28mathematics%29 https://en.wikipedia.org/wiki/Mediant_%28mathematics%29 [2] "1/3 (of the students at a table) + 1/3 (of the students at a table) = 2/6 (of the students in the room)" [3] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence [4] https://en.wikipedia.org/wiki/Ford_circle https://en.wikipedia.org/wiki/Ford_circle [5] https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree [6] https://www.youtube.com/watch?v=0hlvhQZIOQw https://www.youtube.com/watch?v=0hlvhQZIOQw

deleted 6y ago
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Jabbles 6y ago

Don't use tables of the same size: 1/3 + 1/4 = 2/X There would be no confusion as to what X should be. And as someone else said, use units: 1/3 of table A + 1/4 of table B = 2/7 of both tables.

pdevr 6y ago

The moment you add three more students, the first girl is no longer one girl out of a group of three students, she is one girl (out of the two girls) out of a group of six students. Same with the second girl. So, once you add 3 more students, 1/3 becomes 1/6. 1/6 + /6 = 1/3.

knappa 6y ago

From a 'not currently in front of the class' perspective, it's pretty clear what the student meant by 1/3 + 1/3 = 2/6. They were taking + to mean something like a general 'and' or combining action, not in the strict sense of standard fraction addition. It even has a name in mathematics, 'Farey addition'. The kids clearly want to write it in shorthand, so maybe the thing to do is to come up with another symbol for this similar but distinct operation. For example, ⊕.

andred14 6y ago

Apologies but I don't see the mystery here? A fraction describes a proportion ie. the amount of something(s) in relation to another amount of something(s) thus, if we double all of the things in a fraction the proportion and hence the fraction is equivalent.

zuminator 6y ago

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

nurettin 6y ago

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

joshocar 6y ago

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

bjourne 6y ago

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

xigency 6y ago

When you want to add fractions of a whole together. 1 1 2 --- + --- = --- 3 3 3 When you want to add wholes together and make a new fraction. 1 + 1 2 --------- = --- 3 + 3 6

prvc 6y ago

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

jfkienennd 6y ago

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

deleted 6y ago

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rmtech 6y ago

"+" is the wrong mathematical operation for combining fractions of distinct sets. The right operation is a weighted sum: 1/3×1/2 + 1/3×1/2 = 1/3 The "1/2" is there because each of the two sets that we are combining is the same size.

ash 6y ago

One of the commenters gave a great suggestion: > I would use bar models to show how the whole changes. First, draw one bar (table) with 3 students inside; label the bar one whole, star one student and label that student as one third of that whole. Do the same thing beside the first bar model, again showing the bar as one whole, starring one student and labelling that student as one third. Then, push the two bar models together (I’d use a Smart Board) to show a new whole: the two bars together with 6 students in the one bar is labeled as the new whole. Point out that this is a NEW whole, with a different # of pieces. Now they would see the 2 starred students in the one new bar made up of 6 total students, so the “old” 1/3 student becomes the “new” 1/6 student once the whole changes. This is like a name change, once the size of the whole changes. You can also show them that by cutting each “student” in half, you would get an equivalent fraction of 1/3 = 2/6 of the first one whole bar. So 1/3 = 2/6, not two 1/3’s = 2/6. http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-so/#comment-27347 http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seeme...

wondringaloud 6y ago

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

adrianmonk 6y ago

The difficulty isn't with fractions. It's about understanding what "+" means. Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie"). His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about the word "plus" to the symbol. People even use "+" (rather than "plus") to mean combining things. Maybe Addison saw "Nature's Path Pumpkin Seed + Flax Granola" at the grocery store. So why shouldn't he try using it that way? Somebody needs to communicate to Addison that, in math class, "+" always means something specific. He doesn't know that yet, but he has been asked to use the symbol anyway. Math uses a whole lot of lingo. If it's not covered well enough, stumbling over the terminology can be an impediment to learning. This includes both new words ("quotient", "integer") and words that are used in everyday language but differently in math ("where" meaning condition or definition instead of place, "of" meaning multiplication, "real" numbers).

Phlogistique 6y ago
That's my favourite explanation so far. That's how I would put it to the student: Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples: * At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table. * Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables. The first operation is what mathematicians call "+". They write "1/3 + 1/3 = 2/3" The mathematicians do not have a good name for the second one, so let's invent one: "1/3 1/3 = 2/6" What's better with this explanation, compared to the "ratios vs. fractions" thing, or the units thing, is that you do not have to introduce a separate category of numbers that sound very similar but act differently.
- wcarey 6y ago
  We explicitly encourage the "let's invent name for this similar but unnamed thing" with our students, and use whatever name they come up with for the thing for the rest of their time with us. Sometimes those student created names become legendary. They really enjoy the idea that they can have "Jane's relation" be used by students after they graduate. (Of course, sometimes Jane's relation is really Euler's totient function, and we have to encourage them that even though they found something a very good mathematician also found, we're going to call it by the common name.)

Wowfunhappy 6y ago

Something I missed the first time around—in the comments below the article, the teacher expands on what she actually did. This was originally one long paragraph, but I've added paragraph breaks for readability: --- > At the time, I let it go so I could think and regroup myself. There was more than fractions to think about: Is it possible for both equations to be true: 1/3 + 1/3 = 2/6 and 1/6 + 1/6 = 2/6. This was a long discussion that kept their interest, in which I learned that many (most?) of the students didn’t think of fractions as numbers. That was another hurdle. I kept going back to what we know about adding whole numbers. > Then I took another approach, once I was sure that they understood that 1/3 and 2/6 are equivalent, so how can I add a number to itself and wind up with a sum that’s the same as one of them? In all of these discussions, students would change their thinking. We looked at adding on a number line. I used pattern blocks to explore the same problem. I kept talking about keeping our attention on what was 1 whole. > My, this all takes time, and the time is important for students to develop, cement, and extend their understanding. What I didn’t do, that I’ve been thinking about now, is to make it part of a writing workshop on persuasive writing and have them choose a conjecture and write. I think I could spend most of the year on this with students.

08-15 6y ago

That's what happens when you try to "build an intuition" for a concept instead of defining it. Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!! The whole point of common fractions is to close integer arithmetic over division by introducing a new kind of numbers, precisely those fractions that don't correspond to integers. The question isn't so much "What is half a bottle of water?", it's "How do you calculate with one half (a bottle or whatever)." Math is beautiful, but only if built up from simple rules. It's amazing how much you can make from counting and a handful of convenient notations. Even children can recognize that beauty. But high school instead teaches math as a jumbled mess of things you have to memorize, without structure, without rhyme and reason. Needless to say, I hated it. (School, not math.)

wcarey 6y ago
As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that way. For lots of students, the process of moving from definitions and simple rules to other statements is extraordinarily difficult. I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from the definition of the exponent. About a third of the students say, "cool" and never miss anything on any assessment again because the answers flow necessarily from the givens. About a third work their way through it fine. About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite working many particular examples where that is manifestly false.
- 08-15 6y ago
  > About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 ...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too. A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (ac)/(bc) = a/b, is it? Of course, she never asked why a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so. I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.

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alakadfs 6y ago

I would explain that the "+" sign is not in the right place in the equation and write (1+1)/(3+3) = 2/6 instead.

boomlinde 6y ago

Of course 1/3 + 1/3 = 2/6, and in this case you can make an error in reasoning that is probably common among children who are just learning fractions that will still arrive at the same answer. In this case I would have demonstrated a case where the error in reasoning that was possibly made does not lead to the correct answer, for example 1/8 + 1/4. The erroneous intuition of simply adding the groups and their subsets together breaks down when you can show that 1/8 + 1/4 ≠ 2/12 but 3/8

boomlinde 6y ago
And now I feel stupid for obvious reasons

0xfaded 6y ago

Not that I would have thought of this on the spot, but ... The numerator and denominator represent different things. The numerator is "how many things", and the bottom is "of how many spaces". Asking what fraction of the table is girls is dangerous, because the answer is (1 girl) / (1 girl + 2 boys). Still a stretch, rephrasing as "what fraction of seats are filled by girls" might have helped show the idea that the number of spaces is fixed.

tosh 6y ago

related: notation as a tool of thought https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=true https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=tr... 1/3 relates to one of the tables 2/6 relates to both tables

hathawsh 6y ago

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

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inimino 6y ago

This kind of thing fascinates me. I've always felt that you don't really understand a thing, even a simple thing like fractions, until you can find and resolve all the paradoxes that arise in using it naively. My take when reading the 1/3 + 1/3 = 2/6 equation is that Addison was right but the "+" here is a different operation than the "+" of 1/3 + 1/3 = 2/3. In one case we are adding disjoint parts of the same whole and counting how much of the parts of the whole are now included, and get 2/3, and in the other case we are adding two wholes and counting how much of the new whole is included. People who pass primary education without internalizing this distinction will have to do so later if they learn statistics or probability theory. The answer "what would you do now?" is the same one I would try to ask the students. First I would stop, confused, and write 1/3 + 1/3 = ? elsewhere on the board, and then wait for the students to see why I'm confused. They already understand that we want mathematics to be consistent, and getting different answers means we were wrong at least once, but they will have a hard time to find out or explain why 2/6 and 2/3 both are intuitively and obviously right. You have to compare the intuitions directly to each other to see that in one case you're adding parts of the same whole (set or group or table of kids) and in another case adding the wholes. From what I know of elementary school, we focus a lot on fractions and the formal manipulation of them as unitless values, and drill into kids that 2/4, 1/2, and 0.5 are all "the same thing". Bullshit! It's no wonder they have problems with word problems! Two-out-of-four apples and one-out-of-two apples and a half apple are very different things, as every elementary student knows well. How can we expect them to believe something that's obviously false unless we've introduced the narrow context in which it's true? When used as unitless "coefficients" or bare numbers that exist only to scale another value that represents a real quantity of something, then 1/2 and 2/4 are the same. This is what we should be teaching when we teach fractions, not some mindless rote symbol manipulation like cross-multiplying or what have you without any ability to intuitively derive those rules yourself. I'd spend the rest of the class time letting them work out all the different ways of looking at it, and the connections to everything else fractions and ratios and addition are good for. Of course we want to be able to have "+" be well-defined, especially in elementary school, and the students understand that! We've now found a perfectly intuitive meaning of addition for which our standard "+" definition does not fit! This gives an opportunity to explore the connections between mathematics and intuition and the symbols and the real world operations they stand for. It gives us an excellent chance to, as Polya puts it, "introduce a suitable notation" for the setwise addition of "marked" sets that we have just discovered. Then we can explore the properties of this operation and the ones between it and the familiar addition operation. We can show that the plus sign and the addition operation we use it to represent are distinct, and our use of one for the other is a choice, and one we could make differently. I saw some mentions of units in the comments, and yet 1/3 + 1/3 = ? is still ill-defined if we know only that "1/3" means "one out of three students". What we need to know is which students they are! So if one out of Alice, Bob, and Charlie has red hair, and one out of Charlie, Doris, and Evan has red hair, how many of Alice, Bob, Charlie, Doris, and Evan have red hair? Well, we made the question challenging even for adults if we ask it that way! And now you can introduce uncertainty and degrees of belief. (What if three people tell you they have one red-headed friend, in a class of 23 students? Do they all have the same friend, or is there more than one redhead?) Of course what most of us will actually do as the teacher in this situation is freeze and think "oh no, I've made a mistake and the teacher (in your mind) is going to point it out and the class is going to laugh at me!". Because that's what we've learned happens in these situations. If we interpret confusion and being wrong as painful and embarrassing, little wonder that mathematics seems a long and painful road that we avoid whenever we can.

karmakaze 6y ago

Use pie graphs to show the original tables split up into thirds of one girl two boys. Then show two ways of combining two tables. (>-) combined with (>-) is a table of six: 2 girls, 4 boys show all boys and girls moving from tables with 3 seats to the same table with 6 seats. Now show a table with three seats, one occupied by a girl. Show another table with same configuration. One girl moves from one table to join the first table. 1/3 full + 1/3 full (at the same table) is 2/3 full.

luord 6y ago

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

wwarner 6y ago

I would express this in a diagram with a dotted line around the set under consideration. When dealing with thirds, make a diagram containing three objects with their respective properties, and end by drawing a dotted line around the set. When switching to sixths, add three more objects, erase the line around the first three, and draw a new dotted line around the whole set of six. Then go on to sevenths by adding one more object, erasing the dotted line for the six and drawing it around all seven. If I do it this way, my 4th grader can add fractions just like numbers without resorting to common denominators.

Geee 6y ago

I would have said that a whole table is 3/3 and that both tables together are 6/3 and not 6/6 (maybe they'll get it that 6/3 equals to 2). Probably a good idea to explain that the "whole" isn't all pens and bottles but the group of 3 or 6 or 12 or whatever. Maybe start with explaining from the start what happens if there are 7 bottles, when bottles are counted in the groups of six-packs. How many six-packs can you make from 15 bottles?

contextfree 6y ago

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

Gravityloss 6y ago

You're taking a weighted average, not adding the fractions. It takes a while to learn what that is used for.

devit 6y ago

It's easy to see that this cannot be a valid definition for fraction addition because applying it along with the rule that a/b = (ak)/(bk) gives a contradiction: 3/7 = 2/4 (+) 1/3 = 1/2 (+) 1/3 = 2/5, absurd because 2/5 != 3/7 Once this is realized, it's easy to see that the correct formulation for the "table joining" operation used here is a weighted arithmetic mean, i.e. a(/)b (+) c(/)d = weighted arithmetic mean of a/b and c/d with relative weights b and d = ((a/b) * b + (c/d) * d) / (b + d) = (a + c)/(b + d). On the other hand, fraction addition is clearly determined as a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd given the a/b = (ak)/(bk) and (a/b + c/b) = (a + c)/b axioms.

karatestomp 6y ago
2/4 + 1/3 would clearly be 5/12. You halve each to put them in terms of the two wholes you're combining (which you must be doing, since you're adding them—else where do those go?) then combine those. Same process that gives you 1/3 + 1/3 = 1/3. Worked out (conventional meaning of + for clarity; we'd need a new operator otherwise): 1/4 + 1/6 = 6/24 + 4/24 = 10/24 = 5/12

carapace 6y ago

http://iconicmath.com http://iconicmath.com We probably haven't come up with the best visual abstractions yet, eh?

benlivengood 6y ago

I love this discussion and want to throw in my two cents. Why not teach beginning students group theory as the foundation to addition and subtraction? While it wouldn't solve this particular problem since fractions require fields, it would give teachers the tools to explain precisely why addition of two tables is not a group or field operation; they are from different sets and require measurement with units to define a new inclusive set. My own understanding of arithmetic and algebra was enhanced immensely by a book outlining the derivation from Peano axioms. Ultimately the concepts are very simple and kids intuitively do the steps without having the names for the mental processes that they're doing. The educational hurdles to me seem to be 1) identifying the memorized names of the sequence of natural numbers with a procedure for generating them; 2) distinguishing between procedures, objects, numerals, and numbers; 3) basics of set theory; 4) natural induction.

GuB-42 6y ago
Bad idea! They tried exactly that in France in the 60s and 70s. It was called Maths modernes (New maths). I think it was the same in most of the western world. It was a disaster and they abandoned it in the 80s. The problem was that while it may have helped the best students understand advanced concepts later on, it produced a generation of people who lacked practical counting skills. EDIT: By "bad", I don't mean "stupid". The people who came up with new maths were experts in their field, with the goal of having a more scientifically literate population. No doubt very smart people, but maybe a little too smart for their own good. Turned out these concepts were too much for most young kids, and too remote from the way math is used on a day to day basis.

savanpatel 6y ago

To fill the quiet, I said to the class, “When thinking about fractions, it’s important to keep your attention on what the whole is.” This is basic. There was no need to write entire blog of how you misunderstood this.

biddlesby 6y ago

I would say it was a mistake to introduce the concept of adding fractions from two different "wholes". Instead, teach that in order to add fractions, you first have to get everything into the same "whole". Like, you can add Jack / 3 to Ben / 3 because they were at the same table. But adding the boys from one table to another is quite a different thing. Instead, you should teach that you should first make the fractions with both tables as the whole. Only then are you allowed to add. This could come as a later concept

dgentile 6y ago

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

chx 6y ago

I happen to have a math teacher masters though I myself do not teach (but I do help with the program of a tiny, tiny reform school). This teacher here had painted themselves into a corner and it's hard to get out of it. Do not explain fractions with things you can't change the denominator of. Rather tell it with money. Say, 1/3 means if the table has 3 coins , one kid gets 1 coins. If there are 6, 9, 12, 15, one kid gets how many? If the other table has five kids and 5 coins then 1/5 means when splitting five coins a kid gets one. Figure out together what happens with splitting 10, 15, 20 coins. Now putting together the two tables we want to calculate 1/3+1/5, how many coins can we do that with? Step through it, we practice 1/3 with 3, 6, 9, 12 ... but can you tell what the fifth of nine coins are? You can't ... Then find 15 and then show them 1/3+1/5 means 8/15. This is all play and very smooth. To answer the question posed in the blog post: I would plan the class carefully to avoid the entire situation. But if I must, I'd point out 1/3 is a mere shorthand for division, 1:3 and writing "1:3 + 1:3" is adding two operations together and it does not even make sense. We can restore sanity but that has its own rules.

nartz 6y ago

If you make it extreme, then its easier to see the folly / error - 500/1500 students + 1/3 students = ?

sova 6y ago

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

fallingfrog 6y ago

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

matvore 6y ago

I think the problem is the teacher moved from one type of "fraction problem" to another kind. The example of drinking 4 bottles of water to only have 2/6 left of the original pack is doing an integral problem and wrapping up the answer as a fraction. Same deal with the pack of 12 pencils. The students can grasp those problems as e.g. (1+1)/6 rather than (1/6 + 1/6). In other words, there is only one "whole" in the problem. When you're adding the fractions of desks filled with students, the fraction is counter-intuitive because both desks have to have the same number of students for it to make sense. (1/3 of a 5000-student round table is different from a 3-student table). And to say "the units are wrong" is kind of a limited way of explaining this. The units also need to have the table capacity as part of the unit identity (i.e. the units would have to be 3-student-table). That's a pretty sophisticated way of thinking about units. I think that after the pencil/water examples, transitioning from the pencil/water pack examples to a more "pure" fraction example would be better. e.g. one group of students eats 2/3 of a pizza, and another group eats 2/3 of a pizza. Now you can throw away one of the original pizza boxes and put the two remaining 1/3 in a single box which is 2/3 full. Now the "whole" for each fraction is no longer arbitrary (like 6 bottles of water or 12 pencils).

mintyc 6y ago

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

poleguy 6y ago

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

EmilioMartinez 6y ago

My take: Adding proportions is kinda "semantically incorrect". It's well defined as a value, it just doesn't mean what you think it means, akin to thinking that union of two sets is written A + B (which would be the set of pair-wise additions of elements). Now, dovetailing the equation with units that signify concrete quantities (as suggested by others) circumvents this by making it meaningful, at the cost of no longer working with the abstract concept of proportion. An example of semantically meaningful manipulation of proportions is the weighted combination of proportions. This here's a more verbose version of what the teacher wrote: (3/6)* (1/3) + (3/6)*(1/3) = 1/3 (that is, the weighted average of the proportions give the final proportion)

lucio 6y ago

When you put two tables together, each table has 1/2 of the total number of kids. By having 2 tables, now you have 6 kids, so: the original 1/3 in the first table becomes 1/6 the original 1/3 in the second table becomes 1/6 so adding the two tables means 2/6 are girls She needs to explain 1/3 of 1/2 (fraction of a fraction) before using an example where she joins two units (two tables)

djmips 6y ago

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

imetatroll 6y ago

Have the kids stand up and count out numbers. Now they know how many kids are in the room. Then have them go and stand in two different corners. Count one corner. Then have them stand in three different corners. Count one corner. Finally the four corners. Count one corner. Then have the two tables distribute their students to the corners and figure out these elusive fractions. Have each corner be a Pokemon or something. That should do the trick.

ValorieO13 6y ago

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