More about shitty math problems
Nov. 18th, 2025 10:40 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
which_chickI'm still unhappy with math problems.
Given a fairly long stint in school learning about math, I've kind of decided that "math" is basically learning rules and abstractions related to numbers, which are themselves abstract things that PEOPLE MADE UP and which are only sort of real. Nobody ever explained this, like "What the fuck is math and why should I care?" and I feel kind of butthurt about it. So, I'm going to talk about my feelings.
Math starts with stuff like "What are numbers?" and "How do numbers work?" by exposing you to real-world positive integers and counting real items in the real world. Here you'll see number lines. Adding. Subtracting. Later you get multiplying and dividing, which are basically fancy addition and subtraction, but that's still arithmetic.
For what it's worth, I do not remember ever having it explained to me that multiplication and division were "fancy" addition and subtraction. In fairness, I learned about that stuff in like 1979. Much later in my life, while thinking about math, I kind of settled in my mind that multiplication was basically fancier addition and division was fancier subtraction but I do not recall this ever having been made clear to me back in the day.
I'm not sure that it would have helped, though, because learning my times tables was a fucking strugglebus of flashcards, brute-forcing the answers into my head. Six times seven is FORTY FUCKING TWO. Six times eight is FORTY EIGHT which rhymes and everything and I still couldn't remember it for ages. Seven times eight is FIFTY FUCKING SIX I HATE IT WITH THE POWER OF A THOUSAND SUNS. Honestly all the sevens can go fuck themselves.
The math taught in school is to get everybody aware of the parts of math that they might actually need to know and want to use in their own lives. (For most people, "math they might actually need to know and want to use in their own lives" tops out with basic equations in Algebra 1 plus the part of geometry about right triangles.) People going into math-heavy fields like sciences and engineering need more math to be able to do those jobs, so they get to study more of it, but still mostly as... a tool for doing the other things they are interested in, like making better bridges or developing more efficient turbines or studying effectiveness of a clinical trial. These folks do not do math to do math, they're still using it as a tool for other ends, just like ordinary people who want to know how much flooring they need to order for two bedrooms and a hallway. At the far end of the bell curve, some people like math enough to want to do original creative work in math and obviously you'd want such a person to be up to speed on "the story so far" so that they don't reinvent the wheel. These people take lots of math classes and then... extra math classes in graduate school. It's sort of a niche interest, original creative work in math, but people do become mathematicians and they do put forth original creative work in mathematics so I figured I would mention it.
Anyway, once basic numeracy (counting, knowing that 5 is more than 2) exists, you work on arithmetic, doing the basic four (add, subtract, multiply, divide) operations on numbers.
This is stuff like
16+27=___________
28/9 =___________
For these sorts of problems, the goal is to just do the calculation. There's not a whole lot of thinking involved and this sort of problem never becomes an argument on Facebook because it's very much cut and dried. It's add, subtract, borrow, carry-the-one, remainders, that style of thing. I resented learning how to do this stuff because even when I was in grade school, electronic calculators existed. We have machines for this! I pointed out, but it made no difference.
After someone can do arithmetic, math class advances to "more complicated" math problems that still only ask for basic operations. This is where "Facebook argument" problems start to appear. Here's an example I pulled from Facebook:
8/2*(2+2)=?
This problem produces arguments because a person can come up with a different "answer" depending on what part of the problem is completed first. However, there is only one answer you SHOULD get if you are mathing correctly, in accordance with the rules of math. This question is not testing your ability to do simple arithmetic. It's checking to see if you know the rule about what things to do in what order. The rule is PEMDAS or, as I was taught, Please Excuse My Dear Aunt Sally and it works like this:
1. stuff in parenthesis happens first. (PLEASE)
8/2*4=?
2. Next do Exponents (EXCUSE)
We don't have any exponents, so we just move on.
3. Multiplication & Division happen next. This is (MY DEAR) in the mnemonic. You do these ones working LEFT TO RIGHT, each operation completed just as you come to it. Our problem AS WRITTEN is now 8/2*4 (because we have dealt with the stuff in the parenthesis) so now it's time to do the multiplication and division, working left to right. So, that's 8/2 =4 (the first part you come to) and then take that result, 4, and multiply it by 4, so that we get 16 for the overall answer.
4. Addition/Subtraction (if you have any) comes next, working left to write, as written. (AUNT SALLY)
This type of problem represents a pretty clear step away from checking your ability at arithmetic and towards seeing if you know about the rules for playing math.
The reason we are supposed to use Aunt Sally is because it is the agreed upon convention so that everybody can look at a more-complex expression and evaluate it the same way. Like, it's one of the rules of math. It's helpful here to realize that on some level, math is a... thinking game that people have invented, kind of like chess. Bishops only move diagonally in chess and there's no cosmic reason for this. It's just how bishops move and everybody agrees to it so that they can play the game with other people. Similarly, there's no foundational rule of the universe behind PEMDAS -- it's one of the rules humans have made up for playing the game of math.
Side note: Unlike chess, math relates to the real world in ways that appear to have "rules" or properties that humans definitely didn't make and fuck are they weird. Look at π. If PEOPLE were making a constant for the relationship of a circle's radius to its circumference and area, they would 100% have picked 3 and not π. There is nobody on this planet who would have looked at π compared to 3 and said "Oooh, yeah, that's way better than 3. Let's use that." If we want integer numbers to be "real" and to represent exact amounts of countable things, then π is 100% how circles work. Every circle, everywhere, has the same irrational number describing the relationship of its radius to its circumference and area, a fork to the eyeball of logic and reason.
Anyway. After we meet Aunt Sally, we get to more complicated problems. More complicated problems come in two sorts. First sort is solving for X, equations. Equations covers both "equations" and "inequalities". They're very much the same effing thing except for the shape in the middle and you can treat them the same.
Let's look at an example:
X+3 > X/2 (This is an inequality. As I noted above, it's very like an equation and we're going to solve it exactly the same way we'd solve an equation.)
X+3 > x/2
2X + 6 > X (multiplied both sides by 2)
X+6 > 0 (subtracted an X from both sides)
X > -6 (subtracted 6 from each side.)
And it's done. The inequality holds for numbers greater than -6.
Here, the single rule to remember is if you do shit to one side of the equation or inequality, you gotta do it to the OTHER side of the equation or inequality. That's it. It has to balance like a see saw. This is the core concept of of Algebra 1. I'm not sure why it took all of eighth grade to impart this but... on second thought, I have met eighth graders. Yeah. That's why.
The OTHER part of more-complex math problems comes with hooking math to the real world. Generally this is done by way of reading problems. The student is presented with a text scenario and is supposed to come up with a math formulation to solve the problem and, eventually, an answer. An example might help illustrate this.
You are reflooring part of an apartment. (Not a fake example.) The part you are reflooring is two bedrooms and a hallway, all of which can be reasonably approximated as "rectangles" even though the walls aren't really all that square and there are closets and stuff. The rectangles are as follows:
Bedroom 1: 15' x 9'
Bedroom 2: 11' x 9'6"
Hallway: 14 x 3'6"
(These are "Freedom Units" because I live in the world of construction and not in the world of science. Science people get to use metric.)
You already have 2.5 boxes of flooring. Standard scrappage for this sort of job runs 10%. How many boxes of flooring do you need to order if there are 27 square feet per box?
This sort of problem isn't math-hard but it hopes to confuse the shit out of you with lots of steps and making you read carefully. People, generally, suck at reading problems or anything with more than three steps. The thing a student is doing here is breaking shit down and working carefully through the problem and its steps. The actual arithmetic isn't difficult.
First, find the area of each rectangle. (Here I made the executive decision to convert to decimal feet to make the math easier.) Bedroom 1 is 135 sq. ft, Bedroom 2 is 104.5 sq. ft, hallway is 49 sq. feet.
Next, add up all the sq. ft. The total flooring needed is 288.5 square feet.
As stated in the problem, allow 10% for scrap. Here, that's 29 square feet. Let's add that on, for 317.5 square feet of flooring needed.
But we need "boxes of flooring". So, convert needed flooring into "boxes of flooring" because the nice lady (Heidi) at the flooring store does not sell partial boxes of flooring. There are 27 square feet per box, so that's 11.8 boxes.
Recall that there are 2.5 boxes of leftover flooring that can be applied towards this project. So, subtract those from the amount needed. 11.8-2.5 is 9.3 boxes but boxes have to be purchased in whole box amounts. There is no "partial" box potential when buying flooring in the real world no matter what shitty lessons you may have learned from the Half-A-Dog math problem of craptacularity. So, it's 10 boxes.
Once you get past equations (and friends) and handling reading problems, then shit starts to get abstract and it's not at all about adding and subtracting and multiplying and dividing any more. You still have to do that stuff, but the focus shifts to being about something other than doing arithmetic.
In my education, after the bullshit slog of Algebra 1 and Algebra 2, we got a respite into Plane Geometry, which was my favorite math class, ever. It was the very best. I have never loved another math class like I loved geometry. We'll talk about geometry next time I feel moved to discuss math.
Given a fairly long stint in school learning about math, I've kind of decided that "math" is basically learning rules and abstractions related to numbers, which are themselves abstract things that PEOPLE MADE UP and which are only sort of real. Nobody ever explained this, like "What the fuck is math and why should I care?" and I feel kind of butthurt about it. So, I'm going to talk about my feelings.
Math starts with stuff like "What are numbers?" and "How do numbers work?" by exposing you to real-world positive integers and counting real items in the real world. Here you'll see number lines. Adding. Subtracting. Later you get multiplying and dividing, which are basically fancy addition and subtraction, but that's still arithmetic.
For what it's worth, I do not remember ever having it explained to me that multiplication and division were "fancy" addition and subtraction. In fairness, I learned about that stuff in like 1979. Much later in my life, while thinking about math, I kind of settled in my mind that multiplication was basically fancier addition and division was fancier subtraction but I do not recall this ever having been made clear to me back in the day.
I'm not sure that it would have helped, though, because learning my times tables was a fucking strugglebus of flashcards, brute-forcing the answers into my head. Six times seven is FORTY FUCKING TWO. Six times eight is FORTY EIGHT which rhymes and everything and I still couldn't remember it for ages. Seven times eight is FIFTY FUCKING SIX I HATE IT WITH THE POWER OF A THOUSAND SUNS. Honestly all the sevens can go fuck themselves.
The math taught in school is to get everybody aware of the parts of math that they might actually need to know and want to use in their own lives. (For most people, "math they might actually need to know and want to use in their own lives" tops out with basic equations in Algebra 1 plus the part of geometry about right triangles.) People going into math-heavy fields like sciences and engineering need more math to be able to do those jobs, so they get to study more of it, but still mostly as... a tool for doing the other things they are interested in, like making better bridges or developing more efficient turbines or studying effectiveness of a clinical trial. These folks do not do math to do math, they're still using it as a tool for other ends, just like ordinary people who want to know how much flooring they need to order for two bedrooms and a hallway. At the far end of the bell curve, some people like math enough to want to do original creative work in math and obviously you'd want such a person to be up to speed on "the story so far" so that they don't reinvent the wheel. These people take lots of math classes and then... extra math classes in graduate school. It's sort of a niche interest, original creative work in math, but people do become mathematicians and they do put forth original creative work in mathematics so I figured I would mention it.
Anyway, once basic numeracy (counting, knowing that 5 is more than 2) exists, you work on arithmetic, doing the basic four (add, subtract, multiply, divide) operations on numbers.
This is stuff like
16+27=___________
28/9 =___________
For these sorts of problems, the goal is to just do the calculation. There's not a whole lot of thinking involved and this sort of problem never becomes an argument on Facebook because it's very much cut and dried. It's add, subtract, borrow, carry-the-one, remainders, that style of thing. I resented learning how to do this stuff because even when I was in grade school, electronic calculators existed. We have machines for this! I pointed out, but it made no difference.
After someone can do arithmetic, math class advances to "more complicated" math problems that still only ask for basic operations. This is where "Facebook argument" problems start to appear. Here's an example I pulled from Facebook:
8/2*(2+2)=?
This problem produces arguments because a person can come up with a different "answer" depending on what part of the problem is completed first. However, there is only one answer you SHOULD get if you are mathing correctly, in accordance with the rules of math. This question is not testing your ability to do simple arithmetic. It's checking to see if you know the rule about what things to do in what order. The rule is PEMDAS or, as I was taught, Please Excuse My Dear Aunt Sally and it works like this:
1. stuff in parenthesis happens first. (PLEASE)
8/2*4=?
2. Next do Exponents (EXCUSE)
We don't have any exponents, so we just move on.
3. Multiplication & Division happen next. This is (MY DEAR) in the mnemonic. You do these ones working LEFT TO RIGHT, each operation completed just as you come to it. Our problem AS WRITTEN is now 8/2*4 (because we have dealt with the stuff in the parenthesis) so now it's time to do the multiplication and division, working left to right. So, that's 8/2 =4 (the first part you come to) and then take that result, 4, and multiply it by 4, so that we get 16 for the overall answer.
4. Addition/Subtraction (if you have any) comes next, working left to write, as written. (AUNT SALLY)
This type of problem represents a pretty clear step away from checking your ability at arithmetic and towards seeing if you know about the rules for playing math.
The reason we are supposed to use Aunt Sally is because it is the agreed upon convention so that everybody can look at a more-complex expression and evaluate it the same way. Like, it's one of the rules of math. It's helpful here to realize that on some level, math is a... thinking game that people have invented, kind of like chess. Bishops only move diagonally in chess and there's no cosmic reason for this. It's just how bishops move and everybody agrees to it so that they can play the game with other people. Similarly, there's no foundational rule of the universe behind PEMDAS -- it's one of the rules humans have made up for playing the game of math.
Side note: Unlike chess, math relates to the real world in ways that appear to have "rules" or properties that humans definitely didn't make and fuck are they weird. Look at π. If PEOPLE were making a constant for the relationship of a circle's radius to its circumference and area, they would 100% have picked 3 and not π. There is nobody on this planet who would have looked at π compared to 3 and said "Oooh, yeah, that's way better than 3. Let's use that." If we want integer numbers to be "real" and to represent exact amounts of countable things, then π is 100% how circles work. Every circle, everywhere, has the same irrational number describing the relationship of its radius to its circumference and area, a fork to the eyeball of logic and reason.
Anyway. After we meet Aunt Sally, we get to more complicated problems. More complicated problems come in two sorts. First sort is solving for X, equations. Equations covers both "equations" and "inequalities". They're very much the same effing thing except for the shape in the middle and you can treat them the same.
Let's look at an example:
X+3 > X/2 (This is an inequality. As I noted above, it's very like an equation and we're going to solve it exactly the same way we'd solve an equation.)
X+3 > x/2
2X + 6 > X (multiplied both sides by 2)
X+6 > 0 (subtracted an X from both sides)
X > -6 (subtracted 6 from each side.)
And it's done. The inequality holds for numbers greater than -6.
Here, the single rule to remember is if you do shit to one side of the equation or inequality, you gotta do it to the OTHER side of the equation or inequality. That's it. It has to balance like a see saw. This is the core concept of of Algebra 1. I'm not sure why it took all of eighth grade to impart this but... on second thought, I have met eighth graders. Yeah. That's why.
The OTHER part of more-complex math problems comes with hooking math to the real world. Generally this is done by way of reading problems. The student is presented with a text scenario and is supposed to come up with a math formulation to solve the problem and, eventually, an answer. An example might help illustrate this.
You are reflooring part of an apartment. (Not a fake example.) The part you are reflooring is two bedrooms and a hallway, all of which can be reasonably approximated as "rectangles" even though the walls aren't really all that square and there are closets and stuff. The rectangles are as follows:
Bedroom 1: 15' x 9'
Bedroom 2: 11' x 9'6"
Hallway: 14 x 3'6"
(These are "Freedom Units" because I live in the world of construction and not in the world of science. Science people get to use metric.)
You already have 2.5 boxes of flooring. Standard scrappage for this sort of job runs 10%. How many boxes of flooring do you need to order if there are 27 square feet per box?
This sort of problem isn't math-hard but it hopes to confuse the shit out of you with lots of steps and making you read carefully. People, generally, suck at reading problems or anything with more than three steps. The thing a student is doing here is breaking shit down and working carefully through the problem and its steps. The actual arithmetic isn't difficult.
First, find the area of each rectangle. (Here I made the executive decision to convert to decimal feet to make the math easier.) Bedroom 1 is 135 sq. ft, Bedroom 2 is 104.5 sq. ft, hallway is 49 sq. feet.
Next, add up all the sq. ft. The total flooring needed is 288.5 square feet.
As stated in the problem, allow 10% for scrap. Here, that's 29 square feet. Let's add that on, for 317.5 square feet of flooring needed.
But we need "boxes of flooring". So, convert needed flooring into "boxes of flooring" because the nice lady (Heidi) at the flooring store does not sell partial boxes of flooring. There are 27 square feet per box, so that's 11.8 boxes.
Recall that there are 2.5 boxes of leftover flooring that can be applied towards this project. So, subtract those from the amount needed. 11.8-2.5 is 9.3 boxes but boxes have to be purchased in whole box amounts. There is no "partial" box potential when buying flooring in the real world no matter what shitty lessons you may have learned from the Half-A-Dog math problem of craptacularity. So, it's 10 boxes.
Once you get past equations (and friends) and handling reading problems, then shit starts to get abstract and it's not at all about adding and subtracting and multiplying and dividing any more. You still have to do that stuff, but the focus shifts to being about something other than doing arithmetic.
In my education, after the bullshit slog of Algebra 1 and Algebra 2, we got a respite into Plane Geometry, which was my favorite math class, ever. It was the very best. I have never loved another math class like I loved geometry. We'll talk about geometry next time I feel moved to discuss math.
