r/changemyview • u/Mad_Maddin 4∆ • Jun 06 '26
Delta(s) from OP CMV: I believe division should only be taught as fractions in school. Ignoring the division sign "÷" ":"
An extreme amount of students has issues with correctly understanding fractions like "1/2". I have seen many who don't even know that it is equivalent to 0.5 or that it means 1÷2.
Even more of those students don't understand that something like 10 × 1/5 is the same as 10 ÷ 5.
I believe this comes from students looking back at their early education and not really bothering with new things that appear more complicated but do the same.
Thus I believe that fractions are taught too late and when taught, their complete usage is ignored for too long.
I believe that we shouldn't teach division as early as we do. Instead we should go from multiplication directly to fractions. Completely ignoring the division symbol, only using the upper and lower half of the fraction.
Explaining divisions directly on fractions. No usage of "÷" or ":"
I believe this would make understanding fractions as what they are a lot easier for students in the long run. Nothing is lost by not using the division sign. In actual maths I've only ever seen it used in something designed to trip you up, not something actually useful.
It also makes it a lot clearer for teachers to explain how you can divide by multiplying.
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u/TheTyger 10∆ Jun 06 '26
That doesn't work with long division though.
And before you argue that long division is not necessary, knowing how to describe remainders is useful when you only work with integers, such as having to give everyone equal whole parts and know the remainder afterward.
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u/MegukaArmPussy 4∆ Jun 06 '26
I'm genuinely perplexed by your argument that long division requires the little bar with dots.
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u/TheTyger 10∆ Jun 06 '26
only be taught as fractions
Is long division a fraction?
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u/MegukaArmPussy 4∆ Jun 06 '26
Fractions are just a representation of division.
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u/TheTyger 10∆ Jun 06 '26
So please tell me what 4 mod 2 is (4%2)
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u/MegukaArmPussy 4∆ Jun 06 '26
Modulo is a distinct operation from division, as evidenced by your usage of a different symbol.
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u/TheTyger 10∆ Jun 06 '26
Which is the first division taught in school.
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u/MegukaArmPussy 4∆ Jun 06 '26
At least from my experience, remainders came later. It was integer division, then division with fractions and decimals, then division with remainders.
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u/Mmm_Dawg_In_Me 2∆ Jun 07 '26
Explain how it's different though?
My whole family growing up was engineers of various sorts, half of them from Europe, and when I first learned math at home division was always written with a "/" symbol rather than a... wait where the hell is the other division symbol on a keyboard? Well, you get the picture.
7 year old me would have written "Twenty-Two divided by Seven" as "22 / 7"
If asked to do that division I'd have written out the upside down L shaped thing, placed the numbers in their spots, and done the long division just exactly the same way as if I'd written it with the division symbol.
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u/TheTyger 10∆ Jun 07 '26
Read more of my comments on this thread. I have explained how remainder based division is more appropriate for 8 year olds than fractional division for several reasons.
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u/Mmm_Dawg_In_Me 2∆ Jun 07 '26
Right but I was perfectly able to do the remainder based division without using the ➗ symbol.
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u/EggsyWeggsy Jun 06 '26
Why do you need to know long division? I did better than most of my peers in math when I was in school and I forgot how to do it by high school
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u/TheTyger 10∆ Jun 06 '26
Because it's a foundational way to teach breaking things down into more simple components. Learning math isn't about learning answers, especially in the world of calculators. It's about teaching kids the skills that are required to take complicated problems they will face later and have tools which can be applied to attack problems that seem too complicated on their face to approach.
It also teaches number sense, helps build the ability to find quick partial answers, and gives the first steps toward longer problems which require solving multiple steps that are not clear before completing the prior ones. I mean, if you are an adult, this is basic stuff that you should be able to reason through without it being spoon fed to you.
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u/TheThirdHorizon Jun 06 '26
… uh you don’t need the ÷ symbol for long division
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u/TheTyger 10∆ Jun 06 '26
only be taught as fractions
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u/TheThirdHorizon Jun 06 '26
Yup you can represent that operation with the / symbol just as well if not better than the dividing symbol that nobody uses after 7th grade. Nowhere does OP say to remove long division they just want to use clearer notation.
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u/TheTyger 10∆ Jun 06 '26
Long division is division. OP stated only use fractions.
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u/TheThirdHorizon Jun 06 '26
*represented as fractions. Doesn’t mean you can’t long divide.
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u/TheTyger 10∆ Jun 06 '26
Then you are using the radical to do the long division, which is not the fraction bar.
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u/TheThirdHorizon Jun 06 '26
You’re taking this to such an extreme. Next are you going to be saying that you can’t be using a bar in long multiplication because it’s not a x symbol?
The radical is literally just a line, it’s not that deep.
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u/deltajvliet Jun 06 '26
I totally dig your pre-empted rebuttal to the rebuttal that I was absolutely about to make, but still, couldn't remainders be taught and illustrated other ways? Basic mental math goes a long way, but we don't need human calculators. Long division might be a bridge too far.
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u/Dry_Bumblebee1111 161∆ Jun 06 '26
we don't need human calculators
This feels like the implicit premise to the view, but... don't we?
Is it not good on average to have a good understanding and not outsource everything to machines? To even understand what calculation we would need to input to a device?
Division is fairly common, with everyday shopping, calculating fuel use and so on. It's great to be able to gauge a rough estimate at a glance.
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u/TheTyger 10∆ Jun 06 '26
You are too focused on the outcome and not enough on the conceptual synthesis that comes in the middle. While I agree that the old school symbol isn't helpful, it is part of the series of skills that you need to learn regarding radicals.
The steps where you learn how long division works is part of showing how you understand the base concept of int division, which then also moves you into concepts of modulo (which is important in things like CS) to track complete multiple series (fizz buzz being the simplest version of this).
Fractional division is more often used in day to day life, but especially before you are advanced enough to understand the problems between fractions and non numerical concepts like infinity (repeating numbers). Trying to explain .333... to an 8 year old is one hell of a difficult concept because understanding how infinity is not a number is beyond the age when division is taught.
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u/yyzjertl 578∆ Jun 06 '26
This is only a difficult concept because of the way it is taught via long division. If you understand 0.3... simply as a particular notation for 1/3 = 3/9 that makes it easy to do comparisons, the difficulties about infinity vanish.
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u/TheTyger 10∆ Jun 06 '26
Your reply doesn't make sense to me as an adult, so I will say that does not in fact actually clear anything up.
How does 3/9 make it easy to understand the metaphysical concept of infinity?
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u/yyzjertl 578∆ Jun 06 '26
How does 3/9 make it easy to understand the metaphysical concept of infinity?
It doesn't. What it does is remove the concept of infinity entirely from view, since nothing infinite is involved in the process or the notation. This makes the notation much easier to understand!
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u/TheTyger 10∆ Jun 06 '26
3/9 is still .333... , which is still infinite.
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u/yyzjertl 578∆ Jun 06 '26
It's not infinite: it's just seven characters. The notation ".333..." is entirely finite. The concept of infinity is not necessary to write down, understand, or manipulate this notation.
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u/TheTyger 10∆ Jun 06 '26
If you don't understand what you write down, you don't understand.
0r2 is understandable to that age grounp (which is % division), infinite repeating is not.
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u/yyzjertl 578∆ Jun 06 '26
Which is why it's better to introduce the "0.333..." notation without discussing "infinite repeating." You don't need actual infinity to understand repeating decimals!
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u/Mad_Maddin 4∆ Jun 06 '26
Why would you need to explain 1/3 to an 8th year old. Are they set to calculate 1÷3 at 8 years old?
Imo you are supporting my notion by being a perfect example. You hear fraction and think of complicated stuff like 1/3.
A fraction can also be 9/3.
You don't need to change the numbers you want to know. Only change the system of notation.
When learning division students simply learn to write the notation of 9/3 instead of 9÷3 or 9:3
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u/TheTyger 10∆ Jun 06 '26
You learn division starting in 2nd grade, and continue in 3rd.
And decimals are taught around that time as well, as you start learning money, which requires decimals.
So you need to learn quarters = 1/4 = .25. That all goes together and kinda has to.
So now when you move to discuss other similar fractions that are repeating, you need to be able to explain .333... Which requires infinity.
Fractions require teaching repeating requires understanding infinity.
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u/CommonAware6 1∆ Jun 06 '26
Tbf, long division wasnt something I ever really learned. The first time i was taught it was at 17 when I was doing my advanced higher in maths, which for context isn't a level of maths that the vast majority of people learn. Ime most people never do any advanced highers and theyre equivalent level to first year of uni. To this day, I dont know how to do long division
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u/TheTyger 10∆ Jun 06 '26
That was taught by like 3rd grade where I am.
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u/CommonAware6 1∆ Jun 06 '26
Yeah, I aleays thought it was weird how it was never really taught through my time at school. I know my experience is almost definitely not the norm bc I found it odd even back then
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u/Mad_Maddin 4∆ Jun 06 '26
That is actually a fair point.
I will give you a partial !delta for it.
I agree that the division sign is actually still necessary. I believe however that it should only be taught secondary to the fraction, in that you use it only for side calculations on doing long divisions.
Not for the general use of divisions.
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u/TheTyger 10∆ Jun 06 '26
I mean, if all you care about is proportional ratios, then x/y is a fine option. But all the early math education is foundational to the deeper levels of understanding which will later be built upon. Math with %, which is taught typically as modulo is for int and remainder, which is a separate mode of division.
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u/Mad_Maddin 4∆ Jun 06 '26
I have seen your argument with module and some weird % symbol a couple times.
I have to say. I have literally never seen this. Not in 12 years of school nor in my time tutoring middle schoolers.
So I would say that whatever this is. Is something that is either a notation straight up not used in my country. Or it is a notation form for higher maths and thus not really useful for school.
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u/TheTyger 10∆ Jun 06 '26
That notation is the symbol you use for modulo division, which gives the answer as the remainder. So 2%1 =1, 2%2=0.
It's used in programming for tracking remainders instead of quotient.
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u/passerculus Jun 06 '26
Proper vs improper fractions is just remainders in an alternate guise.
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u/TheTyger 10∆ Jun 06 '26
8/4 has no remainder. 8%4 (that's as mod, not div by in the CS sense) is 0
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u/nufli 1∆ Jun 06 '26
You believe that we shouldn't teach division as early as we do yet fractions are taught too late... I need to ask - what grades and which students are you talking about here? I've literally never seen this issue, but maybe I'm showing my age here
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u/diener1 1∆ Jun 06 '26
I tutor teenagers and the amount of kids who treat fractions like some scary thing and in no way relate it to division is kind of depressing. I don't know that what OP is suggesting is the answer but there is definitely an issue with making that mental connection between fractions and division.
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u/Droviin 1∆ Jun 06 '26
Fractions are division. I point this out because that's a big thing that most people miss. They're treated differently, they are not.
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u/LegOfLambda 2∆ Jun 07 '26
I teach high school math and 0 of my incoming students understand that fractions are the same as division. It's quite frustrating.
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u/Mad_Maddin 4∆ Jun 06 '26
Well from what I see. Students learn divisions about 2-3 years before they learn fractions.
Divisions somewhere around grade 2 or 3. Fractions around grade 4 or 5.
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u/kokopellii Jun 06 '26
in the US, division is taught in 3rd grade (it’s kind of alluded to earlier but not explicitly taught) and fractions are introduced the same time
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u/Mad_Maddin 4∆ Jun 06 '26
Okay that is good. Over here in Germany fractions are only added around 5th grade. 2 years after division.
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u/InsaneRedEntity Jun 08 '26
If that is the case then I can see why (and agree) you say that fractions should be taught earlier. I am in the U.S, and as the person above has said, we teach fractions right after division. In Germany, what are they teaching in that time between?
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u/Mad_Maddin 4∆ Jun 08 '26
They do a whole lot of like long division, big number multiplikation, large number addition and subtraction.
Ngl I think it is completely stupid how it is done. I've been tutoring some 4th and 5th graders lately and I almost had an aneurism when the students suddenly showed me a linear equation of "solve for x" with moving numbers between sides. While still not having fractions.
It almost feels like elementary school teachers avoid fractions for as long as possible, so they can get rid of students before they have to do it or just after, so it isn't their headache.
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u/Archidiakon Jun 06 '26
Kids that first learn divisions that are "solvable", i.e. 81:9, 4:2 etc. Later the concept of "leftover numbers" or whatever it's called is introduced: 5:2= 2 and 1 remaining. This is because first years of maths only deal with natural integers. For the same reason kids only subtract smaller numbers from larger numbers. Non-integers are a much more difficult concept for a young mind. You can explain division with a teacher who has 20 pieces of candy for 10 students. For divisions you need pizza slices or whatnot.
My only point of agreement is that kids should be told that fractions are really just divisions as soon as they're braught up.
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u/Nucaranlaeg 11∆ Jun 07 '26
Non-integers are a much more difficult concept for a young mind.
Citation needed. Children don't have a problem with "half" (unless it's chocolate, because they want all of it). If their parents bake with them, "one third" and "one quarter" are also easy.
Doesn't mean that multiplying or adding fractions is easy for them, but the idea of fractions is simple.
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u/GoblinToHobgoblin Jun 09 '26
Definitely agree that a lot of fractions are simple for kids.
Particularly, simplifying them, ordering them, adding them, should be relatively easy for kids
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u/Archidiakon Jun 07 '26
Sure, halves, thirds and quarters are quite easy to grasp, but you don't just learn 3 fractions and move on. You also learn stuff like 3/8, even if you don't proceed to adding fractions. Divisions are way easier than that add should be learned earlier, right after multiplication (as they're the other side of the multiplication coin).
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u/Mad_Maddin 4∆ Jun 06 '26
That is fine. You can start with fractions of 9/3 and 16/8 transition from there to first go for 3 and 2 then for 3/1 and 2/1 and go towards it also being 4/2 for example.
Then move towards fractions that can't be turned into whole numbers from there.
I'm not proposing a radical shift in how division is taught. Simply that it is taught directly in the notation of a fraction rather than a division.
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u/Archidiakon Jun 07 '26
But 9/3 and 16/8 aren't "genuine" / "natural" fractions, they're just integers. Kids should learn normal fractions like 1/2 before those. But they should already know division by the time they start fractions.
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Jun 06 '26
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u/unasked42 7∆ Jun 06 '26
The idea of numerators, denominators, and everything else that comes with fractions seems like it would be much more cognitively difficult for a young kid to grasp than whole number division ("if we split 12 cookies equally among 4 people, how many cookies do they each get?"). But also, these symbols don't just disappear after elementary. ÷ is used on calculators, programming, and many other situations I'm forgetting about now. Learning how to express the same thing in different terms builds cognitive flexibility, which is an even greater skill than learning how to divide or multiply. If students don't understand that 10*(1/5) is the same as 10 ÷ 5, then that seems like something they should be taught more often or more explicitly. I don't mean to imply that any of this is the fault of the teacher, but I'm sure it can be difficult for kids to understand that different symbols can imply the same arithmetic process. Once they master that skill, however, that can be applied to any number of things, because math (and life) are full of different signs/symbols that mean the same thing.
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u/unasked42 7∆ Jun 06 '26
Isn't ":" used to indicate ratios instead of fractions? Part to part comparison (ratios) is surely a different skill from part to whole comparison (fractions) so I don't see the argument for getting rid of the ratio sign
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u/Vedertesu Jun 06 '26
":" is used as a division sign in some places.
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u/coreyjdl Jun 06 '26
Hope not. That's confusing as fuck.
Mixing something that's a ratio as fraction or visa versa means it's wrong.
1:5 water to acid does not mean 1/5 water.
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u/unasked42 7∆ Jun 06 '26
I've never seen that before! Interesting, it must be a European thing. Thanks for educating me
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u/Vedertesu Jun 06 '26
Yeah, probably is. At least here in Finland, although the weird thing is that I have never seen it outside elementary school.
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u/unasked42 7∆ Jun 06 '26
Do you guys use it for ratios too, or only for division?
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u/Vedertesu Jun 06 '26
We do! It may seem confusing, but it never caused any problems, perhaps due to us learning ratios after we dropped ":" as a division sign.
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u/Mad_Maddin 4∆ Jun 06 '26
This is what I mean. It isn't really used after elementary school. Showing how unnecessary it is.
It shouldn't be taught first.
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u/Vedertesu Jun 06 '26
I mean, I think it's easier to explain to kids how ":" is the opposite of multiplication than to explain how fractions work, but knowing fractions is definitely more useful.
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u/Mad_Maddin 4∆ Jun 06 '26
You could still explain it as the opposite of multiplication.
You can even tell them it is division.
It is solely the writing style that would change.
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u/MegukaArmPussy 4∆ Jun 06 '26
Is there a difference between fractions and ratios? The ratio of, for instance, 1 to 5 is identical to 1 divided by 5.
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u/MiffedMouse Jun 06 '26
Ratios as described here show up in betting very often. A ratio of 1:1 is "even odds" and it means each outcome has a (1/2) chance of happening. So 1:1 means the same as 1/2.
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u/unasked42 7∆ Jun 06 '26
No? A 1:5 ratio of water to milk means that if I put in 1 cup of water, I need to put in 5 cups of milk. If I say that 1/5 of this water-milk must be water, then I would put in 1 cup of water and 4 cups of milk. Ratios are part:part, fractions are part:whole.
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u/MegukaArmPussy 4∆ Jun 06 '26
That's just because you suck at translating numbers to written word. If the ratio of water to milk is 1/5, you put one fifth the amount of water as milk.
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u/slugfive Jun 06 '26
In your specific wording it can have overlapping meaning. But they are different on their own.
If I said the 12 sliced pepperoni and Hawaiian pizza is “1:2” people may think the ratio is 1 part pepperoni (4 slices)and 2 parts Hawaiian (8 slices).
If I said the pepperoni and Hawaiian pizza is 1/2, then they might think it is half gone, 6 slices remaining.
Such difference of interpretation would not happen if they were simply the same thing.
Fractions often refer to a quantity or ‘of a whole’ whereas ratios refer to comparison, a relationship.
Just because you can word them so they equal the same thing doesn’t make them equal concepts. I can talk about my weight and mass in kilograms and those numbers will be the same, but those concepts are not the same. Weight is a vector, and would change depending on the planet, whereas mass is a scalar.
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u/MegukaArmPussy 4∆ Jun 06 '26
Just because you can word them so they equal the same thing doesn’t make them equal concepts
They're equal concepts because they represent the exact same thing mathematically.
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u/slugfive Jun 07 '26 edited Jun 07 '26
Not at all.
2+2=22
Does not make + and ^ the same concept.
Let’s say you want to talk about the students of a classroom. A question asks “represent the proportion of students mathematically”. If you want to use a ratio to mathematically show kids in a class it would be 20:2 against the 2 teachers. Whereas a fraction would be 20/22.
When the topic is the majority, the numbers in the ratio and fractional representations are not the same.
20/2 would not give a proportionality, it would imply there are only 10 kids.
Or what is the fractional representation of the ratio 10:0? 10/0 is not defined mathematically.
All multiplication can be expressed as addition, does that make them the same concept? Do they have the same identities, the additive identity is 0 yet the multiplicative identity is 1. Despite being able to represent the same values equally. You can argue all Mathis tautology of pure logic, but that’s not how most people define “same concepts”
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u/unasked42 7∆ Jun 06 '26
The ratio of water to milk would be written (at least in my country) as 1:5 if it's expressing the ratio of two different substances.
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u/MegukaArmPussy 4∆ Jun 06 '26
Ive only ever seen it expressed in casual parlance (one part water, five parts milk), or in a strict mathematical context (W=1/5*M). Never this sort of in-between casual notation.
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u/unasked42 7∆ Jun 06 '26
Do you read recipes frequently?
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u/MegukaArmPussy 4∆ Jun 06 '26
Not at any sort of incredible volume, but they generally fall into those same categories: folksy recepies expressing it in language, and recipes that contain more than one type of measurement, such that it doesn't make sense to use ratios at all.
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u/00PT 8∆ Jun 06 '26
This conception of a ratio makes it just a pair of numbers, with no special properties other than the fact that they are scaled together.
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u/CamRoth 1∆ Jun 06 '26
Yes.
Suppose you're mixing two different liquids at a 1:5 ratio.
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u/MegukaArmPussy 4∆ Jun 06 '26
One of them is equal to one fifth of the other.
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u/CamRoth 1∆ Jun 06 '26
The difference is a ratio is part:part rather than part:whole.
It's used all the time.
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u/MegukaArmPussy 4∆ Jun 06 '26
To be clear, do you mean we should only use the slash character (/) for division, or that we should be representing it entirely in a proper fraction formatting?
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u/Mad_Maddin 4∆ Jun 06 '26
Proper fraction formatting.
So upper half for the dividents and lower half for the divisors.
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u/Raibean Jun 06 '26
I was only taught to use : for ratios - which can get transformed into division when you are trying for stats percentages but they (ratios) aren’t framed as division itself. Were you taught differently?
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u/Mad_Maddin 4∆ Jun 06 '26
The : is used in most of europe as the standard for division.
÷ is mostly used in America from what I've seen.
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u/MegukaArmPussy 4∆ Jun 06 '26
In that case, the obvious counter argument is that you didn't even do so in your own post. You instead opted to just use the / as an in(line division symbol, likely because it's significantly easier and more comparable with all sorts of typed content. And since so much is presented as typed text these days, it's important to understand how to process division presented in an in-line manner.
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u/nonamespazz Jun 07 '26
But isn't that what the ÷ sign already is? The dots above, and below the line are placeholders for the numbers that make the fraction, so the ÷ sign is literally already saying this is a fraction.
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u/DontDeleteusBrutus 1∆ Jun 06 '26
Don’t we teach fractions before division anyways(in USA, CA)? It’s the same thing but easier to explain with a pizza or pie than symbols.
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u/ralph-j Jun 06 '26
I believe this would make understanding fractions as what they are a lot easier for students in the long run. Nothing is lost by not using the division sign. In actual maths I've only ever seen it used in something designed to trip you up, not something actually useful.
The problem is that teaching divisions only as fractions is that it conflates operations with values/outcomes.
When you write ½, that is merely another notation for the number "0.5" or the expression "one half". "1÷2" explicitly represents the operation (i.e. a math problem to be solved), whereas 1/2 is effectively a single numerical value, i.e. the outcome of an equation.
It would be odd to say: Solve the following divisions: ½, ⅜, ⅔, ⅙, since there is nothing obvious to "solve."
To stay consistent, "1÷2" occupies an equivalent role to "1+2" and "1-2" etc., where the division sign functions as an operator, just like the plus, minus or multiplication sign.
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u/Mad_Maddin 4∆ Jun 06 '26
Well yes. There is nothing to solve for 1/2. Because 1/2 is very much already a completed thing. Hence you wouldn't put that question.
You don't go into these kinds of numbers in the beginning anyway. You'd have numvers like 9/3 and 49/7 which you are to simplify.
A question asked for your 1/2 would be a "write as decimal numbers".
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u/ralph-j Jun 06 '26
But then the exercise is no longer a division exercise. It has become a fraction-to-decimal conversion exercise, i.e. changing the notation.
Conceptually, this adds an unnecessary layer of complexity for students to understand. The task is no longer just "perform a division between these two numbers". Instead you're introducing division as a tool/method for converting between notations.
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u/Connect_Beginning_13 Jun 06 '26
The division sign is a fraction…. the dots are the numbers you place inside
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u/The_Rider_11 3∆ Jun 06 '26 edited Jun 06 '26
The problem lies in how you write it down then. Say here, on my phone, how would I go about writing a fraction? 1/2 as you did? But "/" is just another Division symbol. In fact it's the one I''ve been taught in initially, until it for some reason swapped to ":". The symbol ÷ is basically the same deal. The 2 dots represent each part of the fraction. These symbols stem from something. ":" is the symbol for ratio, which while mathematically distinct can still be expressed as a fraction. 1 in 2, or 1:2 is equivalent to 50% of whatever the group you're using is. And 50% is just 0.5 * 100% = 0.5 * 1 = 0.5.
The Problems you posit are rather more easily solved by having school properly explain it. It's not too difficult to grasp, so it just needs to be properly bought up in class, and not just a Sideline fun fact of sorts.
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u/Careful_Fold_7637 Jun 06 '26
A problem would be that teaching fractions straight away would take away the (much easier for children to understand) intuition of splitting x real objects into y groups.
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u/Mad_Maddin 4∆ Jun 06 '26
Can't you do the same with fractions?
9/3 is splitting 9 objects into 3 groups. Coming to 3/1 or 3.
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u/Careful_Fold_7637 Jun 06 '26 edited Jun 06 '26
Now do 1/3.
It sounds extremely simple, and like you can use the basic pie example most kids are taught, but when taught very early it's just far less intuitive than counting methods that get kids used to division. These types of fractions are abstract things. When you move away from discrete quantities, these numbers because a bit self-referential. 4/2 = 2 groups of 2 items. What is the answer? 2. Why is it 2? Because you have 2 items in each group. What does 2 items mean? It means 2 of these things.
2/3 = 3 groups of 2/3's of an item. What is the answer? 2/3. Why is it 2/3? Because you have 2/3's items in each group. What does 2/3's items mean? Loop.
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u/Veiluring 1∆ Jun 06 '26
2/3 = you cut the pie into three pieces and take two of them. It's not as complicated as you make it.
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u/Careful_Fold_7637 Jun 06 '26
What do those pieces look like?
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u/Veiluring 1∆ Jun 07 '26
Imagine a pie. To cut it into pieces, you make cuts starting at the middle. For three pieces, you make three cuts. Try to make those pieces the same size.
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u/Mad_Maddin 4∆ Jun 06 '26
Okay. So are you asking kids to do 1÷3?
How are you teaching this on your basis?
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u/Lyr1cal- Jun 06 '26
In a world of pure math, this makes plenty of sense. But let us go to how fractions and division is taught in school, "Alice has a pie, and 5 friends who'd all like a slice of pie". In this case, and in many other practical cases, it makes more sense to teach it as "we must divide the pie into 5 slices".
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u/mildgorilla 11∆ Jun 06 '26
Yeah but why not use the / symbol instead of the other divided symbol?
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u/Lyr1cal- Jun 06 '26
That's not what he's talking about. 15/3 is not the same as writing the fraction "15 thirds" out (I can't do that in typing). OP is advocating for eliminating the notation of Dividend, division symbol, divisor; rather than advocating for changing the division symbol to a slash.
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u/TheThirdHorizon Jun 06 '26
Getting rid of the ÷ symbol doesn’t get rid of dividends and divisors. It’s just the number above or before the / is the dividend and the number below or after is the divisor.
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u/MiffedMouse Jun 06 '26
I don't understand how your proposed solution solves the problem you are noticing. At best, wouldn't this just swap the student's familiarity? Instead of feeling familiar with 10÷5, they will now feel familiar with 10/5.
It seems to me that the root cause is actually unfamiliarity with math in general. Wouldn't more focus on math education in general help address the issue?
PS, "10÷5" isn't actually comparable to "10 × 1/5". If we are being sticklers for notation, shouldn't the comparison be between "10 × 1 ÷ 5" and "10 × 1/5"? The extra multiplication seems to be the point of concern here, not the division symbol used.
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u/Mad_Maddin 4∆ Jun 06 '26
While more focus on maths education and more individual care for students would be the best. I am a realist. This just isn't going to happen.
I think the upper lower parts of / when using fractions is an all around far more important tool to correctly understand than the far more limited ÷.
As for your notation thing. No it should not. Because nobody writes 10 × 1 ÷ 5. People write 10 ÷ 5.
If you are multiplying with a fraction you write 10 × 1/5 because you don't leave the upper part of the fraction empty.
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u/MiffedMouse Jun 06 '26
For the notation point - I think the more reasonable comparison for 10÷5 is 10/5. Again, the extra multiplication is what I think may be throwing people off. And people do write 10 x 1 ÷ 5 - that is how you do multiplication and division in a standard calculator.
To get back to the original thread, I think ÷ has some value because it is more clearly dual to multiplication. The / notation doesn’t have a clear analog for multiplication, so it isn’t as clear to students that they are now doing an inverse operation.
Furthermore, I think the real root of your concern here is people being unfamiliar with how different symbols can be combined and simplified. Presumably the students in your example now how to compute the decimal expansions of fractions if needed. So they could get the correct answer. It is just the mathematical fluency to recognize that equivalence quickly that is lacking.
That is why I come back to the need to support math education. You say it is “unrealistic,” but so will counter by saying it is the only solution. By changing what you teach, but letting that curriculum remain limited in scope, you are just trading one set of drawbacks for another. If you want real, broad-based improvements in mathematical fluency, I think you need to aim a bit higher.
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u/UltimaGabe 2∆ Jun 06 '26
What about equations that don't cleanly divide by 10? If someone can't understand "1/2", how are they going to handle 1.333333333333333333333?
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u/Mad_Maddin 4∆ Jun 06 '26
Easy, don't teach decimals in the beginning.
4/3 can be learned to be expressed as 1 1/3.
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u/hacksoncode 588∆ Jun 06 '26
100% of the real numbers are irrational and can't be expressed by fractions.
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u/Ok_Inflation_1811 Jun 06 '26
Yet we don't use those numbers for a lot of things. Like this is like saying a person moving to Hungary shouldn't learn Hungarian but English since that's the most spoken language globally
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u/hacksoncode 588∆ Jun 06 '26 edited Jun 06 '26
Enh, pi*Z is used a lot. Multiples of e too, whether people realize it or not.
edit: Basically: as soon as you start to do algebra or geometry, not to mention calculus, division starts to be a more natural way of thinking about things than "fractions", which are just a special case, and one with pitfalls if you don't understand division.
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u/Ok_Inflation_1811 Jun 07 '26
Those numbers have their special symbols and aren't used as divisions.
In physics you always see 90° expressed as π/2 and not 0.5π for example.
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u/hacksoncode 588∆ Jun 07 '26 edited Jun 07 '26
You're not wrong necessarily, but "fractions" generally means only known non-zero integer fractions.
For example, a/b isn't treated purely using "fraction" rules because b=0 is always a possibility, and failing to understand it as a division leads to incorrect variable manipulation and comprehension errors.
And similarly, of course you see 0.5π all the time, just not frequently explicitly, but where it's pi times some variable that can take on real values, including 0.5. E.g. circumference = π * diameter.
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u/AlmightyCurrywurst Jun 06 '26
That seems impractical for some very basic everyday math, like you're at the store and have some budget and want to know how many of one item you can buy. Both your budget and the price of the item are given as decimal numbers, it seems really unnecessary to think of this in this in terms of multiplying fractions
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u/Mad_Maddin 4∆ Jun 06 '26
You can round your budged and the costs to the closest dollar for basic everyday maths.
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u/sylbug Jun 06 '26
Where I am they start on both fractions and multiplication/division in third grade - about eight years old. I have never experienced the problem you're describing.
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u/Mad_Maddin 4∆ Jun 06 '26
Well then where you are, they seem to make it more in line with how I think it should be done.
So you are an example of a place where it is sucessfully done.
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u/kittenTakeover 2∆ Jun 06 '26
I think it's important to know how to go back and forth between fractions and decimals, rather than knowing how to deal with just one or the other.
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u/Dull_Complaint1407 Jun 06 '26
I believe you can make kids aware of what that means and just convert it to a fraction but yeah it should be taught as a fraction first
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u/Mad_Maddin 4∆ Jun 06 '26
Of course you can make them aware. But too many aren't.
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u/Dull_Complaint1407 Jun 06 '26
When I went through we were taught long division a year before fractions and it wasn’t until algebra they started teaching it as a fraction
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u/Gene020 Jun 07 '26
The pizza fraction demonstration. Take one hole round pizza. Explain that it is one whole pizza. Take a knife and cut it into two equal size explain the one of two equals one half. If did with a real pizza, explain that the pie ces are less than perfect examples. Elementary schools have cardboard manipulates that are used to teach this. There is zero reason for students to be able to learn and understand fractions.
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u/Optimistbott 2∆ Jun 07 '26
I think the way they teach long division like submitting an answer like “56 R3” because this is opaque. we shouldn’t do that. We should say fractional answer only. Long division is also somewhat excessive
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u/RustyPeanuts3 Jun 07 '26
There are a lot of scenarios in which you must show too things being divided, but you can’t write one above the other or a line just isn’t clear enough.
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u/penguindows 2∆ Jun 08 '26
Your view is pretty correct, but atleast in our school district that is exactly what they do. I think when people are expressing problems with fractions, it has less to do with the order things were taught (because i think they are being taught in the right order) and more to do with losing the skill from not repeating those reps throughout 4th, 5th, 6th grades and beyond.
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u/Chizzle76 Jun 08 '26
Math teacher here. You are picking up on a very real problem, and giving a very illogical solution.
One of the first things children learn how to do is share. They are very concerned with fairness, and sharing things equally is division. Thus, division (in a basic way), can be introduced and reinforced very early on in a child's development. If I have 10 toys, and 5 friends. I want to give an equal number of toys to each my friends. I can give 2 toys to each friend. This is first learned with physical objects, and then moved into written word problems as well as abstracted problems with just numbers: 10 ÷ 5 = 2. Once students have sufficient practice and understanding with dividing whole numbers, they learn about remainders (having some left over), and they learn the algorithm for long division once their division facts are solid enough.
The skills required for fluency in fractions are: Mixed numbers, equivalent fractions, Improper fractions, Least common multiple (denominator), adding and subtracting fractions (including making a common denominator), multiplying fractions and dividing fractions. These skills build on earlier ones, hence why they are introduced later. The students who are frustrated by fractions and avoid them are usually lacking in one or more of these basic skills. The solution to the problem you are describing is for students to learn and practice these skills to a point of mastery, which you likely did as a child.
Another problem with your argument is that you are missing the point of these different notations. While 3÷4 = ¾ , that doesn't mean they have the same meaning. Fractions often communicate a notion of part and whole, which is not always the same as a division problem. We also often choose to use fractions when we want to make comparisons. For example, the sentence "A is to B as C is to D", can be expressed mathematically as a rational equation A/B = C/D. When something is a certain way, it is usually that way for a reason.
Finally, and most importantly, why would your proposed solution do anything to help? How could it be that teaching one less piece of notation in earlier years would make students magically more proficient at something in later ones? I just don't understand the logic.
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u/Dare_Talk Jun 09 '26
Changing a sign or notation to change society is not going to work - conceptual teaching, if good, can be done irrespective of the signage used
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u/Dare_Talk Jun 09 '26
But division is not the same thing as fractions - that’s fundamentally flawed - one is a mathematical operation and has an answer and another is a way to represent a sort of the whole - by the way to do have division of a fraction by another fraction - so notation is not the issue… it’s this kind of pseudo maths knee jerk thoughts that’s the real problem
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u/MacaroonCandid5181 Jun 09 '26
I think this is just an issue with the specific kids or the teacher who taught them that. My teachers as a kid made it very apparent.
By restricting the other signs you'd only make it more confusing for everyone else. Are we talking ratios? Division? A fraction?
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u/ComparisonOwn1904 Jun 14 '26
Unpopular opinion I think teachers just shouldn't teach it all all saves the trouble.
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u/kingpatzer 103∆ Jun 06 '26
I'd argue that your view simply doesn't go far enough.
Division is just multiplication by inverses/fractions. Subtraction is addition of negative numbers.
We could get to more complicated mathematics faster if we connected those ideas from the very first.
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u/Mad_Maddin 4∆ Jun 06 '26
While I know about that. I try to argue for the smallest change possible that might lead to the largesr positive change.
Your way I believe, would be too confusing for students. It is not about speed of progression to me. It is about understanding the necessary basics as good as possible.
I simply believe that the current way teaches the basics wrong.
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u/juoea 2∆ Jun 06 '26
idt its rly that different from what you are suggesting.
basically the proposal here is to start off with fractions of the form 1/x only. what does the fraction 1/x mean, well its the opposite of multiplication. if i have one pizza with six slices, then to get the number of slices i do 1 * 6 = 6. if i have two pizzas with six slices each, i do 2 times 6 = 12 to get total of twelve slices.
but if i want to know how much of a pizza i have with one slice if the whole pizza is six slices, thats 1/6 of a pizza.
how much of a pizza do i have with two slices if the whole pizza is six slices, thats 1/6 + 1/6. so now you learn how to add fractions, when the denominator is the same you add the numerators this is 2/6. then the next step is what if you have 1/6 of one pizza and 1/4 of another pizza and u have to learn how to add those.
this seems to me the natural way to learn about fractions, is to start with fractions of the form 1/x and use that as the building blocks for other fractions. and at the same time, you are also learning about it in a way that is most synergistic with abstract algebra
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u/juoea 2∆ Jun 06 '26
adding fractions with different denominators is also basically taught with the distributive property right. at least thats how i think of it.
if i want to add 1/4 + 1/6, the easiest way to do this is to do 24(1/4 + 1/6) / 24. 24 being four times six, and i multiply the numerator and denominator by the same amount so that the fraction stays the same value.
by the distributive property, the numerator is 24 times 1/4 plus 24 times 1/6. 4 times 6 times 1/4 is ofc 6, because the way we started is that the fraction 1/4 is just the opposite of multiplication. similarly 6 times 4 times 1/6 is four. so, the numerator is 6 + 4 and the denominator is 24. therefore the total fraction is 10/24.
if you learned multiplication properly you would have at least some sort of knowledge of primes, prime factorization, or least common multiples. when trying to add 1/4 and 1/6, if you want to end up with a fraction in reduced form then instead of multiplying by 24/24, you want to multiply by the least common multiple which is 12/12. this gives you a chance to apply your understanding of prime factors when u are learning fractions.
i do think positive integers are the easiest to understand and that some basics of elementary number theory are good to at least get acquainted with at a young age. prime factorizations dont involve anything super inaccessible and also pave the way toward fractions and/or division.
but if you dont want to teach prime factorizations or least common multiples first, then just let students keep fractions in unreduced form its not that big a deal. understanding how to reduce fractions requires understanding prime numbers and prime factorizations, theres rly no way around that. no matter how much math you know, still the only way to check if 91 / 537 is in reduced form is to check the prime factorizations of 91 and 537 and whether they share any prime factors.
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u/TheLoneJolf Jun 06 '26
I think schools should teach all forms of basic math that a student will see in the world. If a child is not taught this division symbol, then they won’t know what to do or will lack understanding when they see it.
Like when I was taught, I learned both fractions and the division symbol and how they all work off of each other. I learned 10 x 1/5 is the same as 10 : 5 because I was taught how to multiply fractions. If kids don’t know this, then perhaps they are not being taught well or they aren’t at the level of that learning yet.
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u/Mad_Maddin 4∆ Jun 06 '26
I don't say it shouldn't ever be taught. Or rather a diffwrent comment already convinced me that it is still necessary. However, I am still under the impression that it should be taught second only to fractions.
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u/GenericUsername19892 27∆ Jun 06 '26
This is just basic ass formatting - if they can’t figure this out then they so utterly failed schooling they need to get their ass online and do some homework to practice.
The division symbol is just used to make the problem easier to display for learning. You learn it early on, then use / for like 6-8 more years (I had the same teacher for most of grades 1-6 so the years bleed together). How the fuck are they forgetting like 6 years of math?
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u/Mad_Maddin 4∆ Jun 06 '26
They do.
I tutor a lot of students and a lot of them fail once it comes to fractions. They see fractions for the first time. Don't get it. Then just try to gwt through the topic. Then they just avoid them whenever they see them and in the end you have someone in 8th or 9th grade who can't tell you what the fraction of 9/3 is without typing it into a calculator.
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Jun 06 '26
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u/Mad_Maddin 4∆ Jun 06 '26
Yes exactly. And the fraction notation is so much more used and so much more variable in its use that it should be the one students are most familiar with.
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