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.
You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard!
> The confusion comes because no one calls out that they're talking about fractions of different things.
But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking about the two tables together."
Now, I think something closer to what you're suggesting is, the teacher could have written the following two equations on the board:
"1/3 of the students at the first table + 1/3 of the students at a second table = 1/3 of the students at both tables"
"1/3 of the students at the first table + 1/3 of the students at the first table = 2/3 of the students at the first table"
Accompanied by some drawings, maybe that would have worked. But I think it could just as easily end up confusing everyone—you've made the concept of addition much more complicated! And sure, the real world is more complicated too, but you've got to learn the basics first.
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The more I think about it, the more I think the best response might have been: "No, you can't do that, because those kids are at a different table. If we added another third of the kids at the same table...", and move on. Ignore the confusing example and refocus on the simple one.
Yup, it's a typing problem. Do it with different things in the two sets and it becomes clearer: 1/3 apples + 1/3 oranges = 2/6 fruits.
This is another instance of situations where teaching compsci or at least programming could help teaching maths rather than the other way around as it is traditionally thought.
"This is why students are confused and have misconceptions about ratios in middle school. When we teach fractions it is part(s) of a whole (Water bottles and pencils context) and when we teach ratios they are sets (boys and girls). It is actually okay to add ratios (as fractions) by combining the numerators and denominators, no common denominators needed. In my opinion, ratios should not be written like fractions until later after students have conceptual understanding and fractions should never be taught with sets in the 3-5 work. Many teachers are not even aware of this difference and misconception we are creating in student understanding."
Or a different way to look at it is that if you put two things together, the mathematical operation is not always "+"; it totally depends on the things (and how you put them together). You use plus if you put together fractions of the same thing (e.g. fractions of the same box of crayons), but potentially some totally other operation if you put together different things.
> What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those tables).
That’s not what ‘+’ means. Addition doesn’t mean “I have this thing and the other thing; please describe the result”; addition means a specific operation on numbers (or on elements of an additive group, or on numbers with units, etc). But you cannot fully describe 1 student at table of three people as 1/3. Sure, 1/3 of the students at that table are that one student, but if you want to add across tables, you need more information and a better description.
Explaining this in a classroom setting may be quite challenging indeed.
I'm not sure we can really guess what the student meant, but I do know for sure that she was wrong.
If you add up a sixth of a six-pack and a sixth of another six-pack you get a sixth of two six-packs -- two twelveths.
The student's misunderstanding comes from being taught fraction addition in terms of items in a collection -- which only holds if you keep to the same set (what you called scale).
This is a common choice -- "students already know how to add integers, so let's start from there", but as it did in this case, it doesn't always work as intended.
This is a great example of taking an analogy so far that the student didn't learn anything new at all. Everyone feels happy -- teacher's teaching, student's learning -- until you test what your knowledge on outside the domain of the analogy.
Fraction and integer addition are one and the same, yes -- but from the point of view of fractions, wherefrom integer addition is a special case. It remains challenging to teach and understand from the point of view of the integers, which is where the student stands.
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.
If you are omitting the units, it would be a fair assumption that the values all belong to the same unit, unless stated otherwise. You're representing the fractions related to each other and you can't do that if the fractions are of two separate things.
I could say that 1+1 = 2 or 1+1 = 10 but it wouldn't be right to say that 1+1=2 && 1+1=10 because while both are true if we're talking about decimals and binary, we're omitting the units and everything loses its' meaning if we do that.
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.
Why can't we have mathematics devoid of these ambiguities? One reasons is that humans have small working memories, and novice mathematics students have even smaller ones. The ambiguity of notation, while confusing, once mastered, allows us to write shorter expressions, whose meaning we resolve from context, and which become both easier to write and to understand.
A second reason is that while mathematical logic is rigorous and precise, unequal mathematical objects are similar to each - even objects that on first glace seem nothing alike. For instance, the two types of inverses you mention, or sets and linear spaces, or groups and tetrahedrons. And because of the diffuse nature of mathematical objects, it is inevitable that the same notation will be used for unequal objects. Because, it advances our human understanding of mathematics to use the same notation for two unequal but similar objects.
This second reason, once understood, is one threshold between the mechanical mastery of the intermediate student and almost artistic use of mathematics by the advanced student.
Another is sin^2(x) meaning (sin(x))^2, but by a more intuitive reading it should mean sin(sin(x)).
I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always.
Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant notation confusion plaguing much of maths. Perhaps moving to some type of machine-readable notation will be better for consistency and avoiding of much notational confusion.
> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.
That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.
> \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell.
I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.
Allow me to introduce modern theoretical physics, where comments like "X is obvious in context" never is, except to the person that wrote it. Don't worry though, they helpfully add that "When X isn't what it seems in context, this will be called out". Okay. "Sometimes it may not be." Wat?
Of course, every individual researcher, or at least each individual Physics department has their own conventions, and the conventions are critical to the meaning. It's like the programming paradigm where the naming of functions invokes "magic glue" instead of using strongly typed interfaces. It's unbelievably confusing to the uninitiated.
If you're trying to "cut across" a bunch of theories being worked on by different groups, you basically have no hope. Everybody ends up being super specialised not only to a specific sub-field of study, but to a specific research group.
I just watched a 1 hour lecture on extending GR by some physicist last night. He was reading the equations out loud, and at one point he was making noises like the following non-stop for about 2 minutes: "Eta mu nu, one minus one zeta nu mu, mu nu eta zeta one". It was ludicrous.
I also remember having constant confusion of what were assumed associations in written bigger formulas especially involving the functions and the practice of not writing the braces, and before that, if 3 1/5 meant three times 1/5 or 3 wholes and 1/5th.
I thought it was only on these low levels of educations that people settle on such confusing practices.
Later on the university I've discovered that it's the same on all levels. The notations used outside of programming are simply very context-dependent, and it's pity that that dependency is not made clear more often. Also, some practices directly come from some historical uses, and different paths to the current notations still end up to the similarly looking but different meanings.
I think it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time.
When you see something like.
f(x) = summation(x^n, n=0, 10)
We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.
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).
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.
I would say the issue with symbols is why I had trouble with linear algebra in college, specifically dot and cross products. The fact that they used the same symbols as scaler multiplication and are also called products confused my mind so much.
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)
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.
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.
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).
> 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.
> 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.
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.
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.
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).
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.
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:
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> 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.
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.
>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.
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/11019C10CF34E90DC5866E53E90...
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, ⊕.
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.
> 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.
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.
( 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.
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).
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
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.
[+] [-] forbiddenvoid|5 years ago|reply
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.
[+] [-] Wowfunhappy|5 years ago|reply
> The confusion comes because no one calls out that they're talking about fractions of different things.
But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking about the two tables together."
Now, I think something closer to what you're suggesting is, the teacher could have written the following two equations on the board:
"1/3 of the students at the first table + 1/3 of the students at a second table = 1/3 of the students at both tables"
"1/3 of the students at the first table + 1/3 of the students at the first table = 2/3 of the students at the first table"
Accompanied by some drawings, maybe that would have worked. But I think it could just as easily end up confusing everyone—you've made the concept of addition much more complicated! And sure, the real world is more complicated too, but you've got to learn the basics first.
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The more I think about it, the more I think the best response might have been: "No, you can't do that, because those kids are at a different table. If we added another third of the kids at the same table...", and move on. Ignore the confusing example and refocus on the simple one.
[+] [-] p4bl0|5 years ago|reply
This is another instance of situations where teaching compsci or at least programming could help teaching maths rather than the other way around as it is traditionally thought.
[+] [-] mcguire|5 years ago|reply
"Janelle Schorg says:
"This is why students are confused and have misconceptions about ratios in middle school. When we teach fractions it is part(s) of a whole (Water bottles and pencils context) and when we teach ratios they are sets (boys and girls). It is actually okay to add ratios (as fractions) by combining the numerators and denominators, no common denominators needed. In my opinion, ratios should not be written like fractions until later after students have conceptual understanding and fractions should never be taught with sets in the 3-5 work. Many teachers are not even aware of this difference and misconception we are creating in student understanding."
[+] [-] tener|5 years ago|reply
1+1=1
One shoe plus one shoe equals one pair of shoes.
Once they grasp that explaining the fraction issue should be easier.
[+] [-] virgilp|5 years ago|reply
[+] [-] amluto|5 years ago|reply
> What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those tables).
That’s not what ‘+’ means. Addition doesn’t mean “I have this thing and the other thing; please describe the result”; addition means a specific operation on numbers (or on elements of an additive group, or on numbers with units, etc). But you cannot fully describe 1 student at table of three people as 1/3. Sure, 1/3 of the students at that table are that one student, but if you want to add across tables, you need more information and a better description.
Explaining this in a classroom setting may be quite challenging indeed.
[+] [-] isanybodythere|5 years ago|reply
If you add up a sixth of a six-pack and a sixth of another six-pack you get a sixth of two six-packs -- two twelveths.
The student's misunderstanding comes from being taught fraction addition in terms of items in a collection -- which only holds if you keep to the same set (what you called scale).
This is a common choice -- "students already know how to add integers, so let's start from there", but as it did in this case, it doesn't always work as intended.
This is a great example of taking an analogy so far that the student didn't learn anything new at all. Everyone feels happy -- teacher's teaching, student's learning -- until you test what your knowledge on outside the domain of the analogy.
Fraction and integer addition are one and the same, yes -- but from the point of view of fractions, wherefrom integer addition is a special case. It remains challenging to teach and understand from the point of view of the integers, which is where the student stands.
[+] [-] dejj|5 years ago|reply
[+] [-] mywittyname|5 years ago|reply
[+] [-] httpsterio|5 years ago|reply
I could say that 1+1 = 2 or 1+1 = 10 but it wouldn't be right to say that 1+1=2 && 1+1=10 because while both are true if we're talking about decimals and binary, we're omitting the units and everything loses its' meaning if we do that.
[+] [-] wcarey|5 years ago|reply
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.
[+] [-] abdullahkhalids|5 years ago|reply
A second reason is that while mathematical logic is rigorous and precise, unequal mathematical objects are similar to each - even objects that on first glace seem nothing alike. For instance, the two types of inverses you mention, or sets and linear spaces, or groups and tetrahedrons. And because of the diffuse nature of mathematical objects, it is inevitable that the same notation will be used for unequal objects. Because, it advances our human understanding of mathematics to use the same notation for two unequal but similar objects.
This second reason, once understood, is one threshold between the mechanical mastery of the intermediate student and almost artistic use of mathematics by the advanced student.
[+] [-] billfruit|5 years ago|reply
I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always.
Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant notation confusion plaguing much of maths. Perhaps moving to some type of machine-readable notation will be better for consistency and avoiding of much notational confusion.
[+] [-] twic|5 years ago|reply
That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.
[+] [-] Denvercoder9|5 years ago|reply
I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.
[+] [-] jiggawatts|5 years ago|reply
Of course, every individual researcher, or at least each individual Physics department has their own conventions, and the conventions are critical to the meaning. It's like the programming paradigm where the naming of functions invokes "magic glue" instead of using strongly typed interfaces. It's unbelievably confusing to the uninitiated.
If you're trying to "cut across" a bunch of theories being worked on by different groups, you basically have no hope. Everybody ends up being super specialised not only to a specific sub-field of study, but to a specific research group.
I just watched a 1 hour lecture on extending GR by some physicist last night. He was reading the equations out loud, and at one point he was making noises like the following non-stop for about 2 minutes: "Eta mu nu, one minus one zeta nu mu, mu nu eta zeta one". It was ludicrous.
[+] [-] acqq|5 years ago|reply
I thought it was only on these low levels of educations that people settle on such confusing practices.
Later on the university I've discovered that it's the same on all levels. The notations used outside of programming are simply very context-dependent, and it's pity that that dependency is not made clear more often. Also, some practices directly come from some historical uses, and different paths to the current notations still end up to the similarly looking but different meanings.
[+] [-] mkl|5 years ago|reply
[+] [-] mamcx|5 years ago|reply
If we learn new syntax(langs) all the time then push for a new syntax(math) could not be that bad of a idea...
[+] [-] Spivak|5 years ago|reply
When you see something like.
We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.[+] [-] adrianmonk|5 years ago|reply
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|5 years ago|reply
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.
[+] [-] wjmao88|5 years ago|reply
[+] [-] pdkl95|5 years ago|reply
"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
[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
[4] https://en.wikipedia.org/wiki/Ford_circle
[5] https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree
[6] https://www.youtube.com/watch?v=0hlvhQZIOQw
[+] [-] CydeWeys|5 years ago|reply
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. That's an absolute nightmare.
[+] [-] ColinWright|5 years ago|reply
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-prob...
[+] [-] bwestergard|5 years ago|reply
https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...
[+] [-] 6gvONxR4sf7o|5 years ago|reply
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.
[+] [-] OscarCunningham|5 years ago|reply
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.
[+] [-] didgeoridoo|5 years ago|reply
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.
[+] [-] yamrzou|5 years ago|reply
[+] [-] barbegal|5 years ago|reply
[+] [-] GavinMcG|5 years ago|reply
[+] [-] JBorrow|5 years ago|reply
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).
[+] [-] wondringaloud|5 years ago|reply
[+] [-] Wowfunhappy|5 years ago|reply
---
> 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.
[+] [-] Spivak|5 years ago|reply
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|5 years ago|reply
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.
[+] [-] jnbiche|5 years ago|reply
[+] [-] xigency|5 years ago|reply
[+] [-] knappa|5 years ago|reply
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, ⊕.
[+] [-] chx|5 years ago|reply
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.
[+] [-] ash|5 years ago|reply
> 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-seeme...
[+] [-] jessermeyer|5 years ago|reply
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.
[+] [-] gilbetron|5 years ago|reply
[+] [-] logfromblammo|5 years ago|reply
( 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.
[+] [-] matvore|5 years ago|reply
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).
[+] [-] biddlesby|5 years ago|reply
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
[+] [-] karmakaze|5 years ago|reply
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.