Common core made an effort to teach kids to think about numbers this way and people flipped the fuck out because that wasn’t how they were taught. Still mad about that.
The problem with common core math was not that they taught these techniques. It’s that they taught exclusively these techniques. These techniques are born from the meta manipulation of the numbers which comes when you have an understanding of the logic of arithmetic and see the patterns and how they can be manipulated. You need to understand why you can you “borrow” 1 from the 7 or the 9 to the other number and get the same answer, for example. It makes arithmetic easier for those who do it, yes, but only because we understand why you are doing it that way.
When you just teach the meta manipulation, the technique, without the reason, you are teaching a process that has no foundation. The smarter kids may learn to understand the foundational logic from that, but many will only memorize the rules they are taught without that understanding of why and then struggle to build more knowledge without that foundation later.
Math is a subject where each successive lesson is built on the previous lessons. Without being solid on your understanding, it is a house of cards waiting to fall.
To add to this, people come up with math tricks all the time but you then have to check it against the manual method, and often multiple times with different numbers, before you can connect the manual process to the trick for later use.
In my opinion I don’t think you can teach just the trick side of it, if thats what common core is.
When I was tutoring, i had a few elementary-school aged kids. They’d have homework where they had to do the problems three or so different ways, using each of the methods that they were taught (one of which was always the way I was taught when I was their age). I actually feel like I learned a lot from them, as there were some interesting tricks that I didn’t know before helping with the homework. I think that’s a really good way to approach it, because a kid may struggle with some of the methods but generally was able to “get it” with one of them, and which method was “the best” was entirely dependent on the kid. For me, being able to see which methods clicked and which ones didn’t helped me be more effective as a tutor, too, since it showed me a bit more about how their individual little brains were working.
But I agree, if you’re not also at least trying to explain why the different methods get you the same answer, it can lead to problems down the line. Some of them saw the “why” for themselves after enough time working at it, and some needed a bit more external guidance (which, considering they were coming to me for tuturoing, I guess they weren’t getting at school). My argument would be that no one really taught me “why” when I was in school learning The One Way to do math either. I still had to figure out little tricks that worked for me on my own, since my brain is kinda weird. It may not have taken me so long to believe that i’m actually pretty damn good at math if I’d done those kids’ homework when I was their age, as i would have had more tools in the toolbox to draw from.
Yeah, no, the way we were taught was often lacking too. Definitely not advocating for the old school methods as a whole. It was still very prescriptive and the whole “show you work” mentality with a rigid methodology expectation meant that even though I could rapidly do stuff in my head by using these shorthand techniques, I still had to write out the slower longer methods to demonstrate that I was able to. For my ADHD ass, that shit was torture.
I think common core went in the right direction. Teaching shorthand techniques that may not have been naturally apparent to some students probably made doing arithmetic more accessible to some. But I think it was an over correction. They should have been teaching them the basics without the rigidity and prescriptivity, but following that up with giving them useful techniques/tools to make arithmetic smoother and easier for different types of thinkers. Instead, they skipped or breezed over the basics, went straight to the techniques and then maintained that prescriptive expectation of the “show your work” mentality to ensure and enforce the techniques are being followed properly.
I understand why they maintained that show your work mentality to an extent. The teachers need to be able to understand how you arrived at an answer, correct or incorrect, and identify mistakes in logic so that it can be fixed. But the entire point of those techniques is that you understand the underlying logic but find a method of thinking that makes it easier for you and makes sense. As demonstrated in this thread, there’s a number of different shorthand methods, and different preferences for them for every person. Teaching and practicing all these different patterns of meta techniques to add numbers and forcing them to write them out and explain them in weird esoteric ways is the literal opposite of the point of the techniques. I have to imagine it mostly confused their understanding of the basic logic as well.
Yes, i do think the biggest problem is shoving so many different tricks at them at once that it leads to confusion. There was also a bit of frustration from some of my tutees from having to solve the same problems multiple times. Some found it boring and tedious, and some found it confusing and made them less confident in their skills since not all methods they were taught “clicked”.
You answer the why in college talking progressively harder math classes until you say fuck it and accept that’s just how it works, or you become either a mathimation or a math teacher where you dumb everything down and let the next generation ask why, and you ask yourself why can’t I afford to live, I should have majored in computer science as you spend your summer as an Uber driver.
I stopped at calc 2 and became a software engineer. My math rival(ex gf in hs/best friends now ex wife who I took all my college math with) became a HS math teacher.
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You are literally responding to a comment about how our education system is now teaching kids to understand the basic fundamentals of mathematics instead of just rote learning methods.
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Mental arithmetic is all little tricks and shortcuts. If the answer is right then there’s no wrong way to do it, and maths is one of the few places where answers are right or wrong with no damn maybes!
Well, there are certainly wrong ways to arrive at the answer, e.g. calculating 2+2 by multiplying both numbers still gets you 4 but that is the wrong way to get there. That doesn’t apply to any of the methods in the post though.
Unsolved problems do not all fall into binary outcomes. They can be independent of axioms (the set of assumptions used to construct a proof).
I like your funny words, mathemagic man
Unless you consider probabilities. That’s a very strange field—you can’t objectively verify it.
You can’t objectively verify anything in mathematics. It’s a formal system.
Once you start talking about objective verification, you’re talking about science not math.
It is actually the opposite, since it is purely abstract everything in math is objective. There is literally no subjectivity possible in something that isn’t in the real world.
That’s also all common core is. Instead of teaching the line up method which requires paper and is generally impractical in the real world, they teach ways to do math in your head efficiently.
What is “common core” and what is the “line up method”?
Hmm, you seem to be completely discounting calculus, where a given problem may have 0, 1, 2, or infinite solutions. Or math involving quantum states.
In math, an answer is either right, wrong, or partially right (but incomplete).
Quantum states is physics, not math.
And mathematically a probabilistic theorem is still a theorem.
Yes, but physics is math with more variables.
But there’s plenty of math related to quantum states that can make sense, such as if you know a given machine will give the right answer 51% of the time, and you want to know how many iterations you’ll need to get a certain confidence that you are seeing the correct answer. That’s basic statistics, which is also math, but it’s relevant to quantum states in that you’re evaluating a computing system based on qubits.
Those are quite far from mental arithmetic though
Calculus is generally pretty easy to do mental arithmetic on, especially when talking about real-world situations, like estimating the acceleration of a car or something. Those could have multiple answers, but one won’t apply (i.e. cars are assumed to be going forward, so negative speed/acceleration doesn’t make much sense, unless braking).
Math w/ quantum states is a bit less applicable, but doing some statics in your head for determining how many samples you need for a given confidence in a quantum calculation (essentially just some stats and an integral) could fit as mental math if it’s your job to estimate costs. Quantum capacity is expensive, after all…
What does adhd have to do with anything?
ADHD is sometimes used as a catchall to mean a set of behaviors that does not coincide with the majority at school or work. Ive met a bunch of people on ADHD medicine, but it was usually because they wanted to force themselves to be good at or like something they didnt want to do normally.
In this case its called ADHD because the student has found their own way to solve it despite the method the teacher is teaching and that the rest of the class uses.
It’s because it’s stupid. The bottom answer is at least sort of similar to a simple rule for adding 9s. But the op is just so incredibly specific that it won’t help most of the time.
Well the OP is a joke form of a more serious example. Its meant to illustrate the point but not actually require much thought or calculation.
Nothing, it has become quite common to say ADHD causes every little odd behavior. I’m not sure if all those people are even actually diagnosed and not just lying for internet points…
I assume people with actual ADHD find it offensive their condition is made fun of by “quirky” idiots online.
Yep. Just because you do something in a nonsensical, stupid way doesn’t mean you have ADHD or that is what someone with ADHD would do. People with ADHD are also “intellectual.”
For me, this is how I’d solve 9+7:
Day 1: Fuck it, I’ll do it tomorrow
Day 2: Alright gotta do that problem now! Just gonna eat and take a walk to prepare my mind
Day 3: okay for real this time
Day 4: staring intently at problem for half an hour before getting incredibly inspired to do anything else
Day 5: anxiety
Day 6: paralyzed but anxiety
Day 7: Either I actually try to do it and it takes 30 seconds or I give up entirely and flunk the class
Not “hehe quirky look at me I’m so stupid because my brain does things differently, ur so smart I wish I was like you and not so dumb! x3”
I wanna be charitable and say that these sort of behaviors might be commonly associated with ADHD because for us they become a necessity to exist in the world.
While an NT person might have no problem adding 9+7 without breaking up the problem, it becomes much harder with ADHD. so ADHD people are more likely to develop them as a coping mechanism.
For me personally, the more steps a math problem has, the less likely I am to follow through. My mind prefers cutting corners rather than breaking equations up
For many of us it is cutting corners. Memory is hard, but I know my fives and anything less than five so really I just need two spots in ram instead of a bunch of tables on my tiny hard drive
Yes! This is true, for example, if I’m given something like 16 + 27, I’ll sooner make an educated (wrong) guess 3 times than stop and think about it. Not sure if that’s ADHD though!
Oh I see you’ve seen my leadership style
The problem here is that what you’re posting is accurate, realistic, and far more importantly, makes no use of italibold. Sorry, friend.
You know how sometimes you go outside and there’s a bird and you’re like, “cool”
classic adhd
If you said squirrel I would’ve called you ableist.
Absolutely fuckall, because apparently no one with ADHD can ever be (an) intellectual.
I don’t think manipulating an addition problem so you can equate it to a multiplication problem would be a normal action.
They are probably just using ADHD
(not even a diagnoses anymore IIRC - it’s all ADD now)as a shorthand for ‘funky brain thing goin on’. Not exactly good, but I don’t really think it does any meaningful harm either.Edit: had it the other way around. It’s all ADHD now, not ADD. Thankyou for the correction @JackbyDev .
ADHD (not even a diagnoses anymore IIRC - it’s all ADD now)
Other way around.
TY, my B.
Has nothing to do with ADHD.
Wouldn’t say nothing to do with.
Many neurodivergent students find themselves in situations where they haven’t fully absorbed the taught material. Many of them end up figuring problems out themselves, with varying degrees of creativity and successNeurotypical students do the same thing. It’s not like every neurotypical will internalize every piece of material they are taught.
Yup, I’m most likely neurotypical (never been diagnosed either way, just never had issues w/ traditional learning), and I generally ignored the teacher and did things my own way. I was always really good at math, so the teacher’s way was usually less efficient for me, so once I understood the operation, I’d create shortcuts.
We’d go over the same material a lot, so I’d usually just do homework while the teacher taught some new way to do the same operation. I’d get marked down for doing it differently from the instructions, but I’d get the answer right.
I would have done 10+6, but that’s effectively the same thing as the OP.
Aside from literally counting, what other way is there to arrive at 16? You either memorize it, batch the numbers into something else you have memorized, or you count.
Am I missing some obvious ‘natural’ way?
I’d argue memorizing it is the natural way, at least if you work with numbers a lot. Think about how a typist can type a seven letter word faster than a string of seven random characters. Is that not good proof that we have pathways in our brain that short circuit simpler procedural steps?
For my kids, apparently some kind of number line nonsense, which is counting with extra steps.
I just memorize it. When the numbers get big, I do it like you did. For example, my kid and I were converting miles to feet (bad idea) in the car, and I needed to calculate 2/3 mile to feet. So I took 1760 yards -> 1800 yards, divided by three (600), doubled it (1200), and multiplied by 3 to get feet (3600). Then I handled the 40, but did yards -> feet -> 2/3 (40 yards -> 120 ft -> 80 ft). So the final answer is 3520 ft (3600 - 80). I know the factors of 18, and I know what 2/3 of 12 is, so I was able to do it quickly in my head, despite the imperial system’s best efforts.
So yeah, cleaning up the numbers to make the calculation easier is absolutely the way to go.
A mile is 1760 yards, and there are three feet in a yard. Therefore, 1760 feet is 1/3 of a mile, and 2/3s of a mile is 3520 feet.
The imperial system is actually excellent for division and multiplication. All units are very composite, so you usually don’t need to worry about decimals.
Yup. The reason I went with yards was because I knew 1760 was closer to a nice multiple of 3 than 5280 (neither 5200 or 5300 is a multiple of 3; I’d have to go to 5100 or 5400).
But yeah, imperial works pretty well for multiplication and division, it’s just not intuitive for figuring out the next denomination. Why is a mile 1760 yards instead of 1000 or 1200? Why is it 5280 feet instead of 6000? Why is a cup 8 oz instead of 6 (nicer factors) or 10? Why is a pound 16 oz instead of 8 oz like a cup would be (or are pints the “proper” larger unit for an oz)?
The system makes no sense as a tiered system, but it does make calculations a bit cleaner since there’s usually a whole number or reasonable fraction for common divisions. Base 10 sucks for that, but at least it’s intuitive.
Metric would be perfect if 10 wasn’t such a dog shit number to base our counting off of. Sure it works for dividing things in half, but how often do you need to break something down into fifths? Halves, thirds, and quarters are 90% of typical division people do, with tenths being most of the rest since 10 is that only number that our base system actually works with.
It is not as if any other system of measurement used base 12 which would be the sensible choice by that standard (or base 60 but that might be a bit unwieldy in terms of number of digits required).
As in, visualizing a number line in their heads? Or physically drawing one out?
I could see a visual method being very powerful if it deals in scale. Can you elaborate on that? Or, like try to understand what your kids’ ‘nonsense’ is?
I think my 7yo visualizes the number line in their head when there’s no paper around, but they draw it out in school. I personally don’t understand that method, because I always learned to do it like this:
7372 + 273 =====And add by columns. With a number line you add by places, so left to right (starting at 7372, jump 2 hundreds, 7 tens, and 3 ones), whereas with the above method, you’d go right to left, carrying as you go. The number line method gets you close to the number faster (so decent for mental estimates), but it requires counting at the end. The column method is harder for mental math, but it’s a lot closer to multiplication, so it’s good to get practice (IMO) with keeping intermediate calculations in your head.
I think it’s nonsense because it doesn’t scale to other types of math very well.
You still haven’t told me what the number line method actually is. I know how to add up the columns bud
Number line is something like this:
100 | 200 | 300 ... | 10 | 20 | 30 ... | 1 | 2 | 3 ==================================================You write out the numbers that are relevant and hop by those increments. So for 7372 + 273, you’d probably start at 7000, hop 100 x 5 (3 for 372 and 2 for 273), hop 10 x 14 (7 for 72 and 7 for 73), and so on. It’s basically teaching you to count in larger groups.
To multiply, you count by the multiple (so for 7 x 3, you’d jump in groups of 3).
This article seems to explain it. I didn’t learn it that way, so I could be getting it wrong, but it seems you do larger jumps and and the jumps get smaller as you go. I think it’s nonsense, but maybe it helps some kids. I was never a visual/graphical learner though.
So, are you just talking about number lines in general?
I learned how to use those in grade school too. 20+ years ago. But the way you phrased it made me think there was more to it. Calling it nonsense is… shocking.
I guess we used it for an exercise or something a couple times, but never for more than indicating how numbers work. They’ve taken that idea and kind of run with it, instead of leaving it behind once the basics of addition have been mastered. I learned multiplication as just repeated addition, and there’s no reason IMO to get a number line involved because addition should already be mastered.
This is a 2nd grade class, and I expect them to have long since mastered addition. At that point, a number line feels like a crutch more than a useful tool. Sure, use them in kindergarten and first grade to grasp how counting works (and counting by 2s and 10s), but that should honestly be as far as it goes. But they still use it for fractions and larger sums and products.
My mental image is squishing the 7 into the 9 but only 1 is able to be squished in, leaving 6 overflowing
I’m also in 10+6 gang, and it’s more universal, as in a decimal system you will always have a 10 or 100 to add up to, and a “pretty” 8+8 is less usual
Theres more complicated ways for sure, but I think we have identified all the simple ones. Could break it into twos I guess.
Mental abacus. You visualize the beads to come to the answer.
Definitely not ‘natural’, that shit takes major training.
Why is everything ADHD?
Yeah, this has nothing to do with ADHD.
The “ADHD way” is literally what they are teaching in school.
Yup, this is what parents are complaining about when they say math has changed. Before, math was primarily about rote memorization. You just memorized that 9+7 is 16. There were multiplication tables you were expected to memorize and regurgitate ad nauseam. Sure you could count it out on your fingers, but that only works for numbers under 11. For anything above that, you just referred to your memorized addition, subtraction, multiplication, or division tables. But this also meant that numbers outside of those tables were really difficult to do in your head, because you were poorly equipped to actually calculate them out.
Common core math is attempting to make math easier to do in your head, by teaching the concepts (rather than promoting rote memorization) and helping students learn shortcuts to avoid getting lost. 9+7 is 16, but it’s also 10+6 or 8*2, which are much easier to visualize in your head without counting on your fingers.
Yep, and what happens is that when kids need help they can’t explain the “new” way from the beginning and only half remember stuff which is extremely confusing to hear as a parent so then the parents get mad at the method.
Admittedly I was in school multiple decades ago, but our teachers wanted us to memorize addition and multiplication tables. Which of course made anything outside the tables hard to do. I (and others apparently) thought it would be a great idea to use shortcuts like this.
So many failed tests. So many. When teachers saw us write down that we took the 21 apples multiplied by 7 bushels and just did 2x7, and tack a 7 on the end, they broke out the red pen.
“Show your work!”
“How? You taught me to memorize, and I did it from memory…”
Why are these posts always shitting on teachers? I don’t know what teachers you’re seeing, but I’ve never seen any teacher of any subject / age-group ever discourage anyone for thinking about something a different way. Quite the contrary, different ways of approaching problems are always encouraged.
My math teacher (at a private school) was just a random students’ mom. She had no higher degree and only taught the book. If you got the right answer by using a method not included in the book, it was marked half-credit because she didn’t understand and wasn’t interested in hearing your logic, because “that’s not what the book says”.
Being taught by people who have no drive for knowledge and just want to teach the standardized test answers SUCKS.
I had to memorize multiplication. We weren’t taught any other way. 3x3 = 9 because it just is that way, memorize it. I had stacks of flash cards. My mom struggled so hard for weeks to teach me my multiplication tables at home. In the end, I (somehow) passed the multiplication quiz or whatever and did my best to do as little multiplication as I could for the rest of my life. As a result, I still have never learned all the multiplication tables, and have a deep dislike of math or numbers in general.
I bet my education would have been a lot different if I could have learned how to multiply effectively from a young age.
Same here. Multiples are just a memorized table in my head. I look at solutions like the one in this meme and having never been taught anything like that, I just shrug and add 7 to 9.
You’re old school, like me. You’re literally describing the “new math” that boomers hate. Teachers are finally teaching kids to do it the way we’ve always done it in our head.
“8 + 7 is awkward, but if you take two from seven and give it to eight, now you have 10 + 5 and that’s easy mental math.”
And the reason they teach it that way is because it’s what the people who are good at math were already doing. Math isn’t about memorization it’s about understanding how numbers work and that’s how numbers work
No no no 10+7 = 17 and 9 is always one less so 16
If your teacher gets mad about breaking an addition problem into easier problems, then that teacher should be fired. Phony tale.
If anything, these are exactly the techniques that “New Math” was supposed to teach. Your brain doesn’t work math the same way as a computer. People who are good at math tend to break the whole thing down into simple pieces like this. New Math was developed by studying what they did and then teaching that to everyone.
I tend to add 9 to things by bumping the tens digit up by one (7 becomes 17) and then subtracting 1 (17 becomes 16).
Most of the arguments against New Math tended to prove the point; our mathematical education was in dire need of fixing.
But they posted in italibold, which makes it 420.69% leejit. pwned.
IT IS ILLEGAL TO WRITE LIES IN ITALLIBOLD.
Let’s make that 9 a 10 because it’s good enough, it’s smart enough, and goshdarnit people like it. Also, I don’t wanna add with a 9. So 10 + 7 would be 17, but we added 1 to the 9 to make it 10 so now we take 1 away, 17 - 1 = 16.
ezpz
9 plus a number? No. 10 plus a number, minus 1. Yis.
I just memorized any addition with 9 adds a 1 in front while reducing the other number by one. Same general step, but there’s no 10 in my head, just 9+7 -> 16. Basically, promote the tens column while demoting the ones column. I think of it more like a mechanical scoreboard (flip one up, flip the other down) than an operation involving a 10.
If it’s anything other than 9, I fall back to rote memorization, unless the number is big, in which case I’ll do the rounding to a multiple/power of 10.
Yeah that’s a more accurate description of what i actually do in my head to. I’m not “adding 10”, because I already would use a short hand method for adding 10 anyway to promoting the tens place or flipping the score card, as you said.
Whatever number is closest to 10 steals enough to make itself 10. Same goes for hundreds, thousands, whatever. Get your round numbers first, add in the others later. All numbers must become 10. In a pinch, a number may become a 5, but if so, it’s really just become a half-10, and it should feel bad about itself that isn’t a full 10 yet.
10 is just easier for me to work with so…
9+1=10 10+7=17 17-1=16
The second method is very chemistry-like. I do that too naturally
I thought that too, 9 is like a halogen, it wants to resolve to 10 anyway it can like fluorine wants one last electron. So allow the 9 to rip one off of the neighboring numbers and then perform the calculation.
I’ve never really liked the anthropomorphic description of chemical bonding, but maybe it’s actually similar to the addition thing. On the one hand, we can say 9 wants to resolve to 10 and takes a 1, and on the other hand we could say there are a bunch of different ways we could rearrange these numbers but the end result is the same as if we resolve 9 to 10 first. Maybe chemical reactions are similar, so there’s a bunch of configurations that could have happened, but the end result is the same as if we had said fluorine wants that last electron
Although, electron affinity is a thing… so the analogy does break down pretty quick. Rip
So do I! When I saw that 9 stealing a 1 and becoming a 10, I thought “Wait… These are electrons. This is chemistry. That 9 is a halogen!”
