Can 1/3 and 1/3 = 2/6? It seemed so
marilynburnsmathblog.com
marilynburnsmathblog.com
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.
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.
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) ...).
That is, precisely, the problem of units. Each object is its own unit here.
> The correct equation would have been 1/3 3 + 1/3 * 3 = 2/6 * 6.*
That's not how you add fractions. The correct equation would have been, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 (or, 1+1=2), if these 3 all truly had the same units. But they don't, so you can't add like that.
> 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals)
That's the point (but it's 1/3, not 2/6). Also, 1/3 of a meter, + 1/3 foot = 1/3 (1 meter + 1 foot). Different units, but same dimension, so if you know the conversion factor (here, 1 meter = 3.3 feet), you can change it into (1/3 meter * 3.3 feet/meter) + 1/3 foot = 1.1 foot + 1/3 foot = 33/30 feet + 10/30 feet = 43/30 feet = 1.43(3) feet.
You can do the same math with students at tables.
The way you can write this observation mathematically is as I did: ((1/3) × 3) + ((1/3) × 3) = ((2/6) × 6), or 1 + 1 = 2, after computing the fractions. The student's observation was perfectly correct, but he was missing the proper explanation, as it is not about the addition of fractions (it is almost a coincidence that the fractions used on one side of the equation happen to have the sum of their numerator and the sum of their denominators equal to the numerator and denominator of the fraction on the other side - this only happens because we are multiplying the fractions by their denominators).
Sure, you can express this in terms of units and dimensions of you really choose to. You can also express it in terms of different definitions of +, or even of =. It is pretty unnatural to me to invent an ad-hoc measurement unit N1, "number of people at 1 table" and a different measurement unit, N2, "number of people at 2 tables", with the relation 1N2 = 2N1, and then correct the student's formula to 1/3N1 + 1/3N1 = 2/6N2. It is correct, but it is extremely artificial to me.
By far the most natural way to explain it is using the correct mathematical interpretation of the phrase "one third of the 3 people" - (1/3) × 3.
Inventing measurement units to describe exact quantities reminds me of a silly joke from Portal: "computer: 2 + 2 = 10 <pause to wonder if the computer is broken> ... in base 4". You can always find a way to make the formula direct by adding assumptions.
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.
And then by way of analogy of course, (e apples)^(pi oranges) + 1 apple = 0.
Note that I'm using ratio notation because the answer for the above is not the same as adding 5:10 apples and 1:2 oranges; in other words, the exact numerator and denominator both matter, so it's not really a simple fraction; a simple fraction can be reduced to its lowest terms (e.g. 5/15 becomes 1/3), but you can't do that here and still support the mediant operation. https://en.wikipedia.org/wiki/Mediant_(mathematics)
The lesson mixes two of the physical world concepts that are often modeled by fractions.
One is an incomplete whole (the six pack) and the other is the proportions of a mixture (table demographics).
Fractions always have the same rules, but the rules map differently to the different territory being modeled.
You can convert apples to fruit and oranges to fruit and then do operations on fruit, but you can't go back from the result to apples and/or oranges. For example, 1/7 of (3 apples + 4 oranges) is 1 fruit, but we can't tell if it's one orange, or 1 apple, or 1/3 apple and 2/3 orange.
For example, 1/3 + 1/3 = 1 is true, if I mean 1/3 of 3 + 1/3 of 0. 1/3 + 1/3 = 2/6 can only be true if we add some quantities, we can't make it true by choosing the right types (we could fix it by choosing a different definition of +, or =, though).
abstract class Fruit:
prop name: string
prop rotten: bool
class Apple (Fruit):
prop name: string = "apple"
prop rotten: bool = False
class Orange (Fruit):
prop name: string = "orange"
prop rotten: bool = False
func add (a: List<Fruit>, b: List<Fruit>) -> List<Fruit>:
return foldl(lambda l, f: l.append(f), a, b)
func filter (p: Func<[Fruit], bool>, l: List<Fruit>) -> List<Fruit>:
return [f for f in l if p(f)]
func is_rotten (f: Fruit) -> bool:
return f.rotten
a1 = Apple()
a2 = Apple()
a3 = Apple()
a3.rotten = True
apples = [a1, a2, a3]
o1 = Orange()
o2 = Orange()
o3 = Orange()
o3.rotten = True
oranges = [o1, o2, o3]
fruits = add(apples, oranges)
print(len(filter(is_rotten, apples)), "/", len(apples))
> "1/3"
print(len(filter(is_rotten, oranges)), "/", len(oranges))
> "1/3"
print(len(filter(is_rotten, fruits)), "/", len(fruits))
> "2/6"And of course this only works if 1 apple = 1 orange = 1 fruit, which means that they are the same unit of measure, or, equivalently they are of substitutable types. It's even debatable of it's correct to say that 'if I have 1/3 of an apple and 1/3 of an orange, I have 2/3 of a fruit', so depending on what you want to do with your apples, oranges and fruit, your hierarchy may in fact break Liskov substitution. For example, if 1 apple + 1 orange = 2 fruit, so 2 fruit - 1 apple = 1 orange, so 1 apple = 1 orange, which is not true.
> In normal mathematical notation, 1/3 is interpreted as 1/3 of 1
Why would you deny the possibility of 1 being 1 list of fruits rather than 1 fruit?
In the original example, it was 1 table of students, not 1 student.
Everything you say after that is based on this broken assumption.
[<Measure>] type lbFeather
[<Measure>] type lbGold
[<Measure>] type lbLead
let convertGoldToLead ( weight : float<lbGold> ) = 1.0<lbLead> / 1.0<lbGold> * weight
// works
let barOfGold = 1.0<lbGold>
let barOfLead = convertGoldToLead barOfGold
// doesn't work
let bagOfFeathers = 1.0<lbFeather>
let leadDuck = convertGoldToLead bagOfFeathers
// Error reported: mismatched units.1/3 * (1 table) + 1/3 * (1 table) = 2/6 * (2 tables)
or
1/3 * t + 1/3 * t = 2/6 * 2 * t
but 2/6 * 2 = 2/3 so you get
1/3 * t + 1/3 * t = 2/3 * t
which is correct.
> The confusion comes because no one calls out that they're talking about fractions of different things.
But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking about the two tables together."
Now, I think something closer to what you're suggesting is, the teacher could have written the following two equations on the board:
"1/3 of the students at the first table + 1/3 of the students at a second table = 1/3 of the students at both tables"
"1/3 of the students at the first table + 1/3 of the students at the first table = 2/3 of the students at the first table"
Accompanied by some drawings, maybe that would have worked. But I think it could just as easily end up confusing everyone—you've made the concept of addition much more complicated! And sure, the real world is more complicated too, but you've got to learn the basics first.
---
The more I think about it, the more I think the best response might have been: "No, you can't do that, because those kids are at a different table. If we added another third of the kids at the same table...", and move on. Ignore the confusing example and refocus on the simple one.
"One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.'
Their written representation of the mental model was incorrect because their instruction was focused only on abstract numbers instead of concrete labels. If those fractions (or ratios or whatever) are labeled properly the equation is completely correct.
The student was taking the model for ratios and applying it to fractions. If I need to add 2 + 2, and I multiply instead, I did the wrong calculation. It does not matter whether I multiplied correctly, nor does it matter that, in this case, both operations equal four.
The confusion of what is wrong with what is correct but non-standard is in fact quite standard and quite wrong.
The confusion of the mental model with the notation is also a standard mistake.
The only "whole" in this case is the entire set of students (or seats or whatever it is that is being talked about).
Yea, the typing (or w/e) is the problem here. I like another posters examples of apples + oranges = fruits.
Otherwise you could further blow their minds. Where 1/3 + 1/3 == 2/6 and 2/6 is equal to 1/3 so 1/3 + 1/3 == 1/3. Hah. Understanding the incompatibility, or at least difference between the first, second and resulting fractions seems essential to understand.
"No, you can't do that, because those kids are at a different table. If you want to add Jack and Brad you have to go back and see their fraction _of both tables_. They are each 1/6 so together they are 1/6 + 1/6 = 2/6"
Then you could talk about if the ratios are same at both tables 1/3. Adding both tables together keeps the ratio. 1/3 + 1/3 = 1/3.
This is closer to what the teacher actually did—but then she became focused on explaining why the original supposition was wrong, as opposed to moving on to clearer examples.
Glossing over critical mental models while explaining new concepts is what creates the confusion. Absolutely using drawings would have made things easier. That's a good way to establish those correct mental models.
"No you can't do that" is probably not the right approach.
The kid's instinct was spot on. The calculation they did was correct, just out of context. Rather than telling them they did it wrong and disrupt their correct mental model, I think it makes way more sense to talk about how what they did is different.
Ignoring it does feel wrong, but I don't know that it is. There's only so much time in a class period, etc.
That said, I really liked where mnsc went with it: https://news.ycombinator.com/item?id=23312266
- Fractions
- Factoring
So yeah, I'd say great care is warranted in teaching kids fractions. Screw that up and there's a good chance you've lost them for life.
Except that the kids were all agreeing with her. Whether she did made that statement or not, they still believed that made sense.
I agree with you about the solution from mnsc, that was a good way to skip that without confusion.
So yes, telling kids that there are more complications when approaching certain problems that you can explain later might be a good approach.
I have a PhD in physics and the more I think about that, the less obvious it becomes (or at least makes you seriously think about it).
The average person is probably in the easiest situation because they learned how to add fractions without further philosophy and they can live happily after.
1/3 of two tables = 2 students.
1/3 of one table + 1/3 of one table = 2/3 of one table.
2/3 of one table = 1/3 of two tables.
1/3 of one table + 1/3 of one table = 1/3 of two tables.
--
It would take a lot longer to explain than to write, but that's how I'd be tempted to proceed.
As far as learning the basics, it sounds like this class was just getting them introduced to the concept of fractions, while the arithmetic rules would come later...
You add slices of pie thus /, and win a vote by ratios thus :.
: = Out Of
A = Table A
B = Table B
E = Every Table Added (A + B)
1:3 * A + 1:3 * B = (1:3 + 1:3) * E
2 out of 6 among Every Table Added1 (person at table A) / 3 (people at table A) can't be added to 1 (person at table B) / 3 (people at table B) without conversion of units.
I heard so many people complains that each year in maths they would essentially learn that everything they learned the year before was wrong... can we fix that please?!
She's already introduced bottles of water and pencils. So one interesting question is, "If you take three bottles of water, and add three pencils, do you now have all of the bottles of water?"
In grade school where these kids are, they won’t understand the concept of an “expression”, a “variable”, and barely the importance of “units”.
I think it's the opposite: if you are taught the procedure for adding fractions without understanding what's going wrong here, then you just know a procedure, not a concept. Taking the detour into the detailed discussion of what 'the whole' is takes longer, but cements that crucial idea, and so lays the foundation for understanding addition of fractions more generally.
(1 girl at table A)/(3 students at table A) + (1 girl at table B)/(3 students at table B) = (2 girls at tables A & B) / (6 students at tables A & B)
vs
(1 boy at table A)/(3 students at table A) + (1 boy at table A)/(3 students at table A) = (2 boys at table A) / (3 students at table A)
Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do.
Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing anomalies is the best habit you want the next generation of engineers or statespeople to learn as a reflex from an early age?
this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves.
there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning experience in general.
now, if you are homeschooling a child or somehow in a 1 on 1 (or at least, working with a very small teacher:student ratio), then perhaps a more freeform exploration of fractional arithmetic might be a wonderful thing.
doing so with a general class of kids? i strongly suspect you're wrong. they won't even understand that there is an anomaly, because they don't even understand the basic concepts that make it an anomaly. why shouldn't adding two fractions (never seen those before!) result in different answers? in fact, why is 2/3 different from 2/6 anyway? and so on.
this is not about shutting down curiosity IMO. it's about nurturing the basic concepts so that curiosity can grow amidst them in the (near) future.
They are not stumbling into it by themselves! They have a guide! The guide's job is to point out the works in the museum, and to make sure the kids don't get lost. It's not to stand in front of the art so the kids can't even see it and lecture about it!
There's nothing complex or sophisticated about recognizing that "+" has a meaning that we chose, and we could have chosen others, and in some cases other choices would be more natural.
You don't think students learning fractions know that there is one right answer, or that 2/3 and 1/3 can't both be right? Their short little lives have already been filled with enough test-taking to teach them that, at least.
And if they don't understand that 1/3 and 2/3 represent something different about the real world, and that 2/6 and 1/3 are different in a different way from 1/3 and 2/3, then why are we going on to teach them even more complex sophisticated concepts before they have grasped these most basic of basics?
There's never been a pedagogical program to shut down the curiosity of primary students. We do it by accident, by "nurturing the basic concepts" (the ones you learned at that age) so that curiosity can grow amidst them "in the future," which means, maybe, after they finish calculus, if they are lucky.
However, your conception of how this could work starts from the supposition that the kids are actually interested enough to wonder. I don't doubt that there's something that will get every child wondering (and likely, more than one thing). But if we're going to actually require the teaching of fractional arithmetic (implying that we're requiring the learning of it, really) then we need to accept that we'll be teaching it a great many children who are not interested in the conundrum, and who even in the presence of a great teacher will remain not interested in it.
Self-led discovery learning is without doubt the best kind, but its not compatible with the current goals of education in a (post)industrial highly structured society, because it will naturally lead to people who for their own reasons chose never to learn things that we consider vital. I might be entirely willing to agree that they are not vital, and even that a (post)industrial highly structured society may also be a bit of an issue, but pretending that every child will just be naturally curious about 1/3+1/3=2/6 vs 1/3+1/3=2/3 is, IMO, not ground in reality.
In the original article, the guide isn't "standing front of the art". She's just taken them over to a corner that has a piece called "no mammals lay eggs", and then noticed that right next door to it, there's a platypus. She's wondering what to say next, or whether to say anything.
Does every student learn to understand fractions under the current regime? Not in my experience.
Edit: And yes, you should say something about the platypus, because understanding that models are simplifications and incomplete can be enough of an escape hatch for the smart kids (the ones that usually hate math class) to notice that even the teacher knows that there's always more, and that can be enough to keep them from throwing it all away in disgust as a useless mishmash of arbitrary, conflicting, and incorrect rules.
> why is 2/3 different from 2/6 anyway
Because 2 pieces of a pie that you cut into 3 pieces is more than 2 pieces of a pie that you cut into 6 pieces. She understands that 4/8 == 2/4 == 1/2. Sometimes she needs to think about it a bit, but she does "get it".
My daughter's now 25 and has been quite the nerd herself through the years (eventually landing in linguistics and speech pathology), but I'd never assume that the fascinations she had instrinsically and that I helped foster as a parent were really typical. I wish they were - and hey, here I am reading HN too :)
That homework expects a reasonable understanding of fractions. Enough that the child doing them can understand the difference between the number of slices in a pie being representative of the bottom number of a fraction.
Sure, I do math exercises with my daughter that aren't representative of what they teach in school (square roots, the fact that parallel lines _do_ meet at the vanishing point in the real world, etc). But those things aren't what I'm basing my assumptions are; the expectations the school has of her are.
1/3 + 1/3 is not a correct mathematical description of the problem.
1/3 [students at table A] + 1/3 [students at table B] is.
Let's shorten this to: 1/3 sTA + 1/3 sTB. Then, you factor out the 1/3, to get: 1/3 * (sTA + sTB). And now it's it's impossible to give the wrong answer "2/3".
Or, in other words, it's best to "keep your attention on what the whole is" by keeping it explicitly written out in the equation.
"1/3 of the students at the first table + 1/3 of the students at a second table =
1/3 of 1/2 of all the students at both tables [ie. those students at the first table] + 1/3 of another 1/2 of all the students at both tables [ie. those at a second table] =
1/3 of the students at both tables"
In US math education I noticed there is a lot emphasis on specific patterns/formulas/etc., kind of "deus ex machina", and a lot of excercises to mechanize their application, yet too little is spend on where it comes from, manual derivation/proval of formulas, establishing logical connections, etc.
Wonder if the kids can grasp that concept though.
1+1=1
One shoe plus one shoe equals one pair of shoes.
Once they grasp that explaining the fraction issue should be easier.
Like, "students at table A" and "students at table B" are both of the same dimension as "students", but are different units. Like meters and miles. You can add them together, but only if you know the factors needed to convert them to a common unit. In this case, the conversion factors are, how many students are at table A and how many at table B.
"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."
You can't add 1/3 and 1/3 to get 2/6 if the first 1/3 was reduced from 3 of 9 and th second was 1/3. Well, you can, but that only works in the degenerate case where the items you add (actually, average) are equal and there's no point in adding in the first place.
Girl A is 1/3 of Table A, which seats half of the students, so she is 1/3 * 1/2 = 1/6 of the total.
Girl B is 1/3 of Table A, which seats half of the students, so she is 1/3 * 1/2 = 1/6 of the total.
So girls represent 1/6 + 1/6 = 2/6 of the total.
> 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.
You say "you cannot fully describe 1 student at table of three people as 1/3", which is true: What's missing is the unit (or dimension, I'm ignoring the difference here).
You can only add two things if they have the same units, as per "dimensional analysis" [1].
So this is an entirely meaningless statement:
[students]/[seats at table 1] + [students]/[seats at table 2]
But you can fix the units with some multiplication (because dimensions do form an Abelian group under multiplication):
([students]/[seats at table 1]) * ([seats at table 1]/[total seats]) + ([students]/[seats at table 2]) * ([seats at table 2]/[total seats])
Which simplifies to:
[students]/[total seats] + [students]/[total seats]
Now that's a statement with meaning!
Since I know that
[seats at table 1]/[total seats] = 1/2
[seats at table 2]/[total seats] = 1/2
I've just derived the calculation that I really wanted to do:
(1/3)(1/2) + (1/3)(1/2) = (2/6)
[1] https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...
Units are absolutely necessary, and 'fractions' don't make sense without them.
'1/3' without units is not a 'fraction' but instead is 0.3333, a number on the number line, which is why it looks so wrong to see 1/3 + 1/3 = 2/6.
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.
> The student came up and wrote 1/3 + 1/3 = 2/6.
Units in your example are unclear. Also results interpretation depend on question.
Here is my example:
Student should wrote 2 articles, but wrote 1 of 3 pages for the 1st article and 1 of 3 pages for the 2nd article.
Q: How many pages student wrote? - A: 2/6.
Q: What part of the whole task is done? - A: 1/3.
Q: How many tasks partially completed? - A: 2/2.
Q: How many tasks fully completed? - A: 0/2.
I 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.
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.
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.
edit: I suppose, what I'm trying to get at, perhaps too glibly, is that audience matters terribly much in mathematical writing. In the same way that Latin students don't start with Tacitus or Sallust, famous for their idiosyncratic grammar, math students shouldn't jump into the full context-dependent mess of the notation that experienced mathematicians use.
But I think we often thrown them in unintentionally because we're so used to it.
Btw, I wouldn't abuse latin comparisons on an american forum, I don't think it's quite in the culture ^^
Of course context would help resolve this if you have a function named x or not, or a variable named x or not, and if you have both a function and a variable named x, well, you worked for your confusion and you have obtained it; congratulations. :)
However, the notation is relatively clear notwithstanding. It almost has to be x as a function with an input of 3.
But I get your point.
I have also heard the objection that [x] is a 1×1 matrix whose only element is ‘x’.
There's other context too, based on what is known. Up to a certain point in first year at my university, most engineering students haven't ever seen functions named x and y, and so they'd mostly interpret x(3) as multiplication. Then we show them parametric curves, and suddenly x(3) looks like the x coordinate of a point on the (x(t), y(t)) curve.
(I have no mathematics background past K-12)
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.
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.
Maybe they did.
In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value.
There is some ambiguity, since multiplication of functions can also often make sense. This is usually resolved by context, or by being explicit where the context isn’t clear.
And, of course, multiplication is a function so one is just a specific instance of the other.
Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with s-expressions in the first place.
Why not?
Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.
No, I meant what I said. The fact that iteration is a kind of composition is true, but it's a tangent from the point I was trying to make, which is that infix notation is a Really Bad Idea. No one in their right mind would use it if they were not indoctrinated into it.
> "the successor function is not a composition of anything" is definitely false
That's news to me. I genuinely thought that successor could legitimately be considered a primitive. What is successor a composition of?
> the point I was trying to make, which is that infix notation is a Really Bad Idea
Not to be snarky, but, reading these two (from your two thread-successive posts https://news.ycombinator.com/item?id=23312725 and https://news.ycombinator.com/item?id=23314776) in succession, I still can't see anything about the first one that indicates the point that infix notation is a bad idea. Not that I'm disagreeing with the point, just that I can't find it in the first post. Could you clarify the connection?
> What is successor a composition of?
It's a composition of, for example, itself and the identity function, like everything else; or you could view it as a composition (-1) . (+2). Those kind of silly solutions are why I thought you meant 'iteration'.
Iterative solutions are easy if you don't restrict yourself to natural numbers—for example, (+1) is (+(1/2)) composed with itself—but that's clearly not what you meant. (As soon as you leave the natural numbers, even the idea that addition is iterated successor becomes false.)
If you do so restrict yourself, then it becomes true that the successor is not a non-trivial iterate. (I just skated the edge of claiming the opposite in my post, but avoided error by not specifying what domain I meant. That's just luck, though; I meant a particular thing, and I was wrong. To prove it, supposing you start your natural numbers at 0 and that f is a function such that f^{\circ k} is the successor function for some k > 1, then note that f is injective (because a composition power is). If f(0) = 0, then succ(0) = f(f(0)) = f(0) = 0, which is a contradiction. Put n = f(0) and note that f^{\circ n k}(0) = succ^n(0) = n, but n k > 1.)
By the way, you quoted (https://news.ycombinator.com/item?id=23314776):
> > you meant the more specific term
Just to be clear, what I said (https://news.ycombinator.com/item?id=23314530) was "I think you meant the more specific term". And I was wrong, but I intentionally didn't just assume I knew you what you meant!
I didn't make a very good argument for it. I really intended that to be more of a throwaway rant than a serious critique. But since you ask...
There are two problems with infix:
1. It's hard to parse. It requires precedence rules which are not apparent in the notation. In actual practice, the precedence rules vary from context to context and this causes real problems. It's an unnecessary cognitive burden that pays very little in the way of dividends (a few less pen strokes or key strokes).
2. It obscures the fact that infix operators are just syntactic sugar for function applications. It leads people to think that there is something fundamentally different about a+b that distinguishes it from sum(a,b) and this in turn leads to a ton of confusion.
> that's clearly not what you meant
Indeed not. I meant the successor operator as defined in the Peano axioms.
> you quoted
Yeah, sorry about that. When I first replied, I thought you were the same person who posted the grandparent comment. My first draft response turned out to be completely inappropriate when I realized you were a different person, but some of my initial mindset apparently leaked into the revised comment. My apologies.
Thus I thought your point was that, since multiplication is a function, it’s a form of function composition. But that wouldn’t follow, for the reasons I said.
As for multiplication being “composed” addition, addition being “composed” succession, etc.: The multiplication function (at least of integer arguments) is composed of addition. Function composition is a function that returns a composition of two other functions. The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing.
Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)). The function which does function composition is not itself a composed function, and the act of multiplication is the act of a composed function, but not an act of function composition.
Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. That is the whole point.
Numbers and addition and multiplication happen to be particularly important functions, but they are not structurally different from any other functions. Giving them special notation, especially when you are first introduced to them, obscures this fact. This kind of mental damage is very hard to recover from in later life. I believe it's one of the reasons so many people think they hate math. Math can be beautiful and elegant, but the standard notation used for school-book arithmetic is arbitrary and perverse, a bizarre accident of history with no actual merit.
IMHO of course.
[UPDATE:]
> Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)).
Function composition is a function, no different from any other function. There is no more reason to use infix notation for it than there is for any other function. In fact, if you drop the infix notation it immediately becomes obvious how ubiquitous and non-special function composition actually is:
compose(f,g)(x) = compose(f)(g)(x) = f(g(x))
And no matter what, there's still a difference between composing a with b, and composing b with itself a times, which is what I mean by the distinction between composition and iteration (sort of like the distinction between a brick and a brick wall).
At this point I feel we're going around in circles, so I'll bow out.
But they are not mutually exclusive. That's the whole point.
Raising to the power of "-1" means that we are inverting the operation in question, so inverting the application of function "f" to argument "x" will be to apply f^{-1} to x instead.
Now I'm not saying it's good notation, but some of the fields I've worked in, the notation is not only OK, it's genuinely empowering, as a good notation should be.
I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.
Indeed - I wish we could conceal those notations until the redemptive context became more apparent, in the same way that Latin teachers can conceal Sallust's inconcinitas until students are ready for it.
Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue. Within a single domain, (like high-school arithmetic) you will usually not have this ambiguous problem. But once you move past that, it is something to get used to.
As von Neumann said, very much something to get used to.
To be clear, I think you aren't saying "It's not because 'function powers' make sense" (which seems to apply that's not the reason it's done, and possibly that the reason isn't correct), but rather "It's not [an appalling notation], because 'function powers' make sense"—to me, that extra comma changes the meaning!
It sounds like an algorithm for computing the inverse of the square root function, so one might think it's for squaring non-negative numbers, or something along those lines. Not so. It's for computing the reciprocal (the 'multiplicative inverse') of the square root of a number. [0]
Related to this: the way a superscript '2' next to a function means the function shall be applied twice (that is, composition with itself)... unless it's a trigonometric function, in which case it means the square of the result. [1] [2]
[0] https://en.wikipedia.org/wiki/Fast_inverse_square_root
[1] https://en.wikipedia.org/wiki/Function_composition#Functiona...
[2] https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...
I also would expect f^{2}(x) to mean f(f(x)), not (f(x)) * f(x)) (which would be (f(x))^2).
When you see something like.
f(x) = summation(x^n, n=0, 10)
What do you think the polynomial is? I ask, because in all situations similar to this that I've encountered it made sense to define 0^0 as 1, not as 0. If you genuinely have a case where 0^0 = 0 makes consistent sense then I'd be interested in understanding it.
So, what do you think the summation actually is when expanded?
An example where 0^0 = 0 occurs when dealing with areas. The measure of the real line in the plane is zero but it's also a rectangle with sides (0, inf) and we define the area of a rectangle to be l*w.
You usually see this written as inf × 0 = 0 but you sometimes you see the interpretation as 1/0 × 0 = 0^0 = 0. And you know this is a an ad-hoc definition because you're not allowed to algebraically manipulate it at all.
0^0 = 0
1^0 = 1
0^1 = 1
1^1 = 0
exp(x) = summation(x^n/n!, n=0, inf)
Fair, infinite series are pretty esoteric. How about derivatives. The power rule:
d/dx x^n = n x^(n-1)
Eh maybe not, programmers don't use calculus that often. But surely statistics!
(1 + x)^n = summation((n choose k)x^k, k=0, n)
(1 + x)^n = (1 + 0)^0 = 1 = 0^0 = x^0 = x^0 + ... + x^n
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.
If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.
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.
If we learn new syntax(langs) all the time then push for a new syntax(math) could not be that bad of a idea...
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.
As an aside, df/dx treated a fraction is used as soon as second or third semester of uni when they learn how to solve differential equations using separation of variables, and physicists/chemists start using total derivatives for thermodynamics. I am not aware of notation different from df/dx where these subjects would be just as clear.
More than any of that, though, I think what you can see (even here in this thread!) is that, while we pretend that mathematics is a single cultural practice of rational humans communicating with other rational humans, it's really many smaller communities of mathematicians, all of varying skills trying to communicate with each other. My mathematical language as a teacher of 12-18 year olds is very different from my mathematical language when I did computational geometry for a living.
Because you have many communities of mathematicians producing new notation, you end up with dialects that all sort of meld together in the same way that reading Shakespeare is very different from reading Hemingway or Eco (in translation).
The closest programming analogue would be C++, where you have several mutually unintelligible dialects spoken by different communities of programmers with different concerns.
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.
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).
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.
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.)
"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
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.
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...
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.
The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf
Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...
> 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.
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.
> 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 :(
[0] https://math.stackexchange.com/questions/379927/how-long-wil...
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.
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.
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 ;)
The conversation with 16 year olds when you explain that their previous teacher who told them that you couldn't add points wasn't lying, but was, perhaps oversimplifying things to make their life easier, is a fun and fraught one.
I've had to reason kids through the fact that a^2 + b^2 is not equal to (ab)^2 or even (a+b)^2 more times than I can count. What's particularly difficult is that, confronted with the fact that 25 and 49 are manifestly different numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the "law of distribution", which they haven't learned as the "law of distribution of multiplication of monomials over addition, and only that".
That's just notational sugar on fractions though (15/100, 20/100) - so why is that less troubling than the general case?
15% of 8 times 20% of 10 isn't 35% of any nice arithmetic combination of 8 and 10. That's hard to communicate to many students.
Another common, and I suppose related, one (but that I don't bother correcting often) is 'percent' != 'percentage point'. Talk of 47% of something being '3% less than' half of it really winds me up - and it's stupidly common among journalists - but it's too common to bother pointing out IMO. Live and let get wrong.
I think the most basic answer to the “what do you do next” is unfortunately to explain they can’t do 1/3+1/3=1/6 but that why will be a future lesson.
The article gets a “fair” stopping point for the day at being mindful of the whole. And that you just can’t always add them.
Except that 1/3 + 1/3 = 2/3, not 1/6.
You probably just mis-typed, but in this context it's an interesting mistake.
> Except that 1/3 + 1/3 = 2/3, not 1/6.
That's why they can't do it.
I think better to address the problem directly in fractions by saying that they're two different ways of combining them, addition and averaging, like the parent comment says. Feel free to make up a different symbol for the averaging operator, just let the kids know that it's not standard.
Arithmetic average or geometric average, to start.
https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...
And hi Bjørn!
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.
The equivalence operation is hard to master, and makes the arithmetic complicated.
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.
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).
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.
---
> 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.
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.
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.
(T1 girls) + (T2 girls) 1 + 1
_______________________ = _____
(T1 total) + (T2 total) 3 + 3
Also, looking at maths from this pure beautiful abstract lens is something that some people (hiii!) enjoy but it's also the common language for modeling problems in the real world. And sure, we do sometimes intentionally confuse the fact that modeling systems with maths is supposed to be descriptive but we have no choice because otherwise we're betraying the fact that the attempt to describe these systems is where the notation came from in the first place.
All this is to say that teaching math is hard and that the abstract "rules" view doesn't do it justice. It's practical for sure but it's also the reason you have lots of students who are good at pushing symbols around but not much else.
That is absolutely incorrect. There is an intuitive reason why 1/3 + 1/3 != 2/6. That reason is that 1/3 + 1/3 = 2/3, and 2/3 != 2/6.
The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents. The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
Now, the example from the classroom is more subtle, because it deals with an improper translation of the English into mathematical notation. If I say 'one third of the students at table 1 are girls', that should be translated to 1/3 * 3, not 1/3. Applying this rule gives 1/3 * 3 + 1/3 * 3 = 2/6 * 6,which is perfectly correct. Similarly, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 is obviously correct.
>There is an intuitive reason why 1/3 + 1/3 != 2/6.
You and I have different ideas of what 'intuitive' means. It's 'intuitive' once you understand the rules of fractions and what they mean. It's not so easy to derive this rule if you're working in the space of real world things.
And sure, I agree you can ad hoc extend the analogy of tables to bring it in line with the underlying mathematical rules, but then your analogy is no longer as simple as it was. The complexity is leaking out of the abstraction you had it under.
>The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents
Sure, using abstractions and analogies is a powerful way of teaching. All I did was point out that analogies have limits and at some point they can become detrimental to understanding the fundamental concepts.
This is a common complaint by Physicists when doing public lectures on Quantum Mechanics and then having people extrapolate from the metaphors to derive incorrect physical rules (e.g. faster-than-light communication from a shallow understanding of quantum entanglement).
>because it deals with an improper translation of the English into mathematical notation.
It isn't just about the improper translation to English. It is also about the improper mapping of fractions to real-world things. 2 girls out of 6 kids in a table maps nicely to the fraction 2/6. But even though 2/6 is equivalent to 80/240, the latter is a little harder to map to a table of 6 kids and 2 girls - don't you think?
>The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
I disagree with that in context of learning how fraction operators work. The fourth-grader logically extended the analogy that they were given because conceivably, there could have been an operator defined that matched their intuition, for example, let's call it '@' and define it (not rigorously) as "a/b @ c/d = (a+c)/(b+d)". This operator, if existed, would work very well for combining tables of boys and girls and getting the fraction of girls to match the fourth grader's intuition. The fourth-grader is learning fractions for the first time, and that operation could have conceivably existed - so the only reason they were wrong is that they haven't been told what the rules of fraction addition are and NOT that they misunderstood the analogy. The problem is that the "+" operator does not work that way because it isn't defined this way as per axioms for fractions.
For example, 1/3 of an orange + 1/3 of an orange is actually 2/3 of an orange, not 2/6 of one. And 1/3 of 1 kg of flour is exactly 2/6 or 300/900 of that kg of flour. Sure, it's hard to talk about 1 Graham's number / 3 graham's number of 1kg of flour, so it does break down at some point, but unless you go overboard with quantities, all of the rules for fractions are in fact intuitive, and important for day to day things like cooking and money management. In fact, fractions and their operations are probably older than the idea of abstract rules, because they are fundamentally useful things.
The child in this example wasn't even making the mistake of thinking the rule for + is the rule for your @ operation. They were confused because they were trying to apply the intuition they had built up for how to translate real-world problems into fractions in the wrong way. Their result was in fact physically true: it was true that 1/3 of the children at one table + 1/3 of the children at the other table was equal to 2/6 of the children at both tables. This was confusing them because it suggested a different way of manipulating the numbers than they had just been shown.
The right solution, again, was to teach them how to translate 'a fraction of something' to rational numbers - that is, to multiply the fraction by the something, with only a special notational case when that something is 1. If they had known to do this, their intuition would have translated directly into the correct algebraic formula. No need to learn the abstract rules yet.
1 1 2
--- + --- = ---
3 3 3
1 + 1 2
--------- = ---
3 + 3 6The 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, ⊕.
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.
http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seeme...
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.
Let's leverage that early on!
( 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.
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).
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
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.
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.)
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.
...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.
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.
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.
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.
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.
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
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)
1/3 of one table + 1/3 of one table = 2/6 of two tables.
If I do it this way, my 4th grader can add fractions just like numbers without resorting to common denominators.
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
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.
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?
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.
We probably haven't come up with the best visual abstractions yet, eh?
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)
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.
(1+1)/(3+3) = 2/6
instead.
Same with the second girl.
So, once you add 3 more students, 1/3 becomes 1/6.
1/6 + /6 = 1/3.
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.
No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? I can multiply -1/3 * -100/3 and get a positive fractional value - if we're talking about tables of boys and girls - what the heck happened there? How about 1/0 or 0/1?
>Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.
I think you have it reversed. The formal definition is the pure definition, which is ancillary to the application to the real-world but covers much much more.
Natural numbers were discovered as a solution to counting. Negative numbers appear naturally when you need to subtract quantities, especially when you end up with debts. Rationals appear from the need for division. Pi appears when you want to relate lengths of circles with lengths of straight lines. Square roots also appear from basic geometry used in construction, as the naming suggests.
All of these are real-world concepts that were used in some form or another at least 6000 years ago. A lot of the time, the closest formal definitions they had at the time were actually based on geometry, as algebra was probably a few thousand years away from being invented/discovered.
The modern formal definition is a post-hoc formalization of an existing intuitive concept. If the formal definition did not agree with the intuitive concept, then obviously the formal definition would not be useful and no one would use it.
There are mathematical constructs that first appeared in mathematics, and don't have intuitive concepts attached, or even don't have any real-world interpretation. A lot of irrational numbers may be in this area, as would the transfinite numbers, and probably complex numbers as well, though even some of those are usually defined based on some real-world applicability.
> I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls?
It's a bit contrived, but say I ask the pupils at the table to send someone away. They can compute that they will soon have 3 people + (-1/3 * 3 people) = 2/3 * 3 people at the table. A less contrived example is "I have 2/3kg of flour, and I owe you 1/3kg, while someone else owes me 1/4kg. So, I own 2/3kg + (-1/3kg) + 1/4kg = 7/12kg".
Sure, but we're aiming to teach those 'post-hoc' formalizations. What fourth-graders are learning is supposed to be a foundation for future abstract mathematics. And we are using the operators and syntax of the modern formalizations as well.
Again we're not just teaching kids how to think about ratios and do practical arithmetic. If that's all we wanted, we could just give them an algorithm and an abacus and that's all they would ever need. Fractions can be used to think about ratios and things like that but they come with WAY more baggage because they are more than that. They are abstract concepts that underpin further abstract mathematics.
>It's a bit contrived, but say I ask the pupils at the table to send someone away. They
You can always add an ad-hoc extension to whatever the analogy you're using. I'm not saying that's wrong. I'm making no judgement call on how to teach mathematics to fourth-graders either (I don't know what the best way is). But I do want to point out that your extension to the analogy (negative fractions being equivalent to sending a student away) is because you've reached the limits of your initial abstraction. It's not a clean mapping either. For one thing, you just assumed the numerator is negative (student being away), but it is just as valid that the denominator may be negative (or both!). And it isn't clear that a negative numerator and a positive denominator is equivalent to a positive numerator and negative denominator. You either have to prove that from first principles using formal methods or you simply have to assert that to the fourth-grader and justify it with "it's just the way things are"
My belief is that is exactly how you end up with students doing fraction addition as a/b + c/d = (a+c)/(b+d); or the infamous shepherd's age problem[0] - they forgot the actual rules, and picked a different rule that makes just as much sense to them.
And regarding analogies, I think that the best approach is to pick a real-world problem, and translate that into math, as a starting step for explaining the formal rules and building this intuition. Doing thing the other way around is much more likely to lead to contrived examples. But math rules have good intuitive reasons for existing, and explaining these as you introduce the rules is likely to help rather than hinder.
Of course, I wouldn't advocate for having students go back to the analogy while solving more advanced exercises with the rules that they have internalized. But having lots of exercises initially that try to drive home the intuition behind the rules is going to be very helpful in my opinion.
[0] "A shepherd has 25 goats and 53 sheep. How old is the shepherd?" A lot of kids will give you an answer: if they do, they will probably say that it's 25+53 or 53-25, since they may apply some common-sense reasoning after they "do the math", but at the wrong end of the problem.
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.
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.
It is pedagogically superior to choose the route implied by the comments about this being a type error.
That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), you gift them a tool they can use to map between the real world and basic unitless mathematical notation. (I'd even add explicit operator definition to that.)
For example-- such an educated student could hear your ascetic declaration that "it's against the rules" and quickly grasp something like the following:
1. "1/3+1/3=2/6" doesn't have any units, but it must somehow map to operations with units.
2. If unitless math can be applied regardless of units, then perhaps "1/3+1/3" may mean "1/3 blue table + 1/3 of the red table, where + means joining the two tables." That would equal 2/6 of the joined tables. But "1/3 blue table + 1/3 (same) blue table" would give 2/3 of that blue table, with + mapping to adding those two fractions of the same table.
3. 2/3 does not equal 2/6, so unitless math can't map to both operations.
4. macspoofing said that 2/6 is wrong.
5. Therefore, unitless fraction addition implies addition of things of the same units, and not joining two different things together and finding the new fraction of the new joined unit thingy.
If on the other hand a student of your apparent method of declaring rules for unitless math came to a class that had practiced explicitly mapping unitless <-> unit math, they wouldn't have any tools to understand the mapping. (Well, at least if the teacher made a similarly ascetic declaration regarding mapping.)
I offer into evidence this very article to show what happens when a student of your apparent method becomes the teacher and encounters the most trivial of unit -> unitless mapping errors.
Edit: clarification
Sure. You can certainly ad hoc extend the analogy to bring it in line with the mathematical rules. But at this point, you do hit a higher level of complexity. Your simple analogy is gone and you're slogging through the weeds. That was my point. Analogies are flawed. The teacher started with a very simple rule that worked well and communicated the ideas under certain constraints and then those rules were extended in a logical but incorrect manner by a fourth-grader ... and now the complexity that was hidden in the abstraction is leaking out.
Your explanation is more confusing to me and wouldn't be grasped by the vast majority of fourth-graders. At some point, simply stating that fractions have different rules is the most simple (and correct) explanation.
This seems totally absurd to me. The concepts of proportion, ratio, etc preexisted any kind of formal definition of fraction.
They may have invented a definition of fraction, with the correct notation, and the correct set of rules after a long series of trial and errors (as with a lot of mathematical "rigourous definition"), but they always had in mind that this concept they tried to define should "work" when manipulating ratios / proportions, etc.
I may be wrong, but it seems to me that purely mathematical concepts spawning out of pure mathematical world exploration is a very modern (aka 19th century max) concept.
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
500/1500 students + 1/3 students = ?
It takes a while to learn what that is used for.
So true...
You do not see anything when you are at the blackboard.
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.
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.
> It’s hard to think and teach at the same time!
This teacher seems pretty bright. I'm sure if she had a few quiet minutes to think and regroup, she could come up with a stellar explanation. But instead, she has to proceed with the lesson, manage the classroom, and tackle an interesting puzzle at the same time.
At home, merely trying to manage my work and two kids with their remote learning often puts me in the same spot. I will often do some tutoring with them during a work break. But my brain power is already silently depleted, so if any kind of obstacle or creative question arises, I struggle to even begin thinking of an approach to it.
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.
console.log((0.1 + 0.2) === 0.3) // => false
C#:
(0.1m + 0.2m) == 0.3m // => true
see here: 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.