Yo but hear me out. Because 7 ate(8) 9, 7 + 9 = 7
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.
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’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
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.
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.
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?
Theres more complicated ways for sure, but I think we have identified all the simple ones. Could break it into twos I guess.
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!
Unless you consider probabilities. That’s a very strange field—you can’t objectively verify it.
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
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).
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…
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.
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.
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.
There’s
peoplealiens who would add 9+7 instead of 10+6 or 8+8 in their heads?I do, because 9 plus anything is just a 1 in front of the other digit minus 1.
Weirdly enough, I just thought about using the methods here for the first time in my life earlier today. Weird.
9 plus anything is just a 1 in front of the other digit minus 1
This is also how it works in my head, but isn’t it the same as the other guy was saying, 10+6?
The difference would just be how you think of the process. I sometimes shuffle around the numbers to make math easier, but the shortcut for adding 9s just feels different. Instead of 9+7 = 10 + 6, it’s more like 9+7 = 17-1. It feels less like solving it with math and more like using a cool trick, since you didn’t really use addition to solve the addition problem.
Sort of, same numbers different logic. Its like mixing up the order of operations. You could learn both tricks but it seems redundant if they do the same thing. Like having two of the same hammer.
And it scales with multiplication too.
9*7is(7-1) and whatever adds to 9, so 63. This breaks down for larger numbers, but works really well up to9*10. I don’t know what “common core” teaches for that, but you can’t change the 9 to a 10 for multiplication (well, you could, but you’d need to subtract 7 from the answer).Treating 9s special makes math a lot easier. Doing the “adjust numbers until they’re multiples of 10” works for more, but it’s also more mental effort. 9s show up a lot, so learning tricks to deal with them specifically is nice. I just memorized the rest instead of doing “common core” math to adjust things all the time.
That said, I do the rounding thing for large numbers. If I’m working with lots of digits, I’ll round to some clean multiple of 10 that divides by 3 (or whatever operation I need to do) nicely. For example, my kid and I were doing some mental math in the car converting fractional miles to feet (in this case 2/3 miles to feet). I used yards in a mile (1760) because it’s close to a nice multiple of three (1800), and did the math quickly in my head (1800 - 40 yards -> 6002 yards - 40 yards to ft * 2/3 -> 1200 yards - 120 ft2/3 -> 3600 ft - 80 ft -> 3520 ft). I calculated both parts of the rounding differently to make them divisible cleanly by 3. I don’t know what common core math teaches, but I certainly didn’t learn this in school, I just came up with it by combining a few tricks I learned largely on my own (i.e. if the digits add to 3, it’s divisible by 3) through years of trying to get faster at math drills. If I wasn’t driving, I would have done long division in my head, but I needed to be able to pause at stop signs to check for traffic and whatnot, and just remembering two numbers w/ units is much easier than remembering the current state of long division.
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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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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.
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.
8+8 and 8X2 are literally the exact same thing, why did they feel the need to make that an extra step?
Probably because they were forced to memorize times tables, but not arithmetic so they wanted to show where they are leveraging that memorization from
9 is 3+3+3, 7+3 is 10, 3+3 is 6, 6+10 is 16. I’m also a fucking heathen.
What the fuck
Might as well do:
9 is 1+1+1+1+1+1+1+1+1, 7 is 1+1+1+1+1+1+1 therefore 9+7 is 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 which is 16.
I explained to a teacher one time this as my method, the get to ten version, and she looked confused as hell like why would anyone do that. She was cool with it though, gave me a whatever works for you kind of response.
Okay this is nice and all but how do people do 3974* 438 mentally, without paper? And bigger and some outright freaks seem to do it in an instant
For me:
3974 * 438 -> 4000 * 438 - 26*438 -> 4000 * 438 - 26*440 - 26*2 -> 4000 * 438 - 20*440 - 6*440 - 26*2And so on, and I’d do some of the intermediate calculations as I go (e.g.
20*440and6*440).But that’s only really needed if I need a precise answer. If I can get away with an estimate, I’ll simplify it even more:
4000 * 430 ~= 43 * 4 * 10000 = 86 * 2 + 10000 = 1,720,000Actual answer:
1,740,612.4000 * 440would be easier (I like multiplying 4s), but I know it would overshoot, so I round one up and the other down. Close enough for something like estimating how much a large quantity of something kind of expensive would cost (i.e. if my company gave everyone a hot tub or something).Not any great easy way I can think of to do that one but I would attempt to do 400 by 3974 and then add chunks of 438 x 10 or x5 until I got really close and then add individual blocks.
So like 400 by 3974, you can round to 4000 and remove 4 x 26 = 104 after doubling 4000 twice. So we have 4000 to 8000 to 16000 remove 104 is 15896, add zeros is 1,589,600. Forget all other numbers but this one.
We are missing 38 x 3974. We can do the same round and remove trick to add 10 x 3974 by changing it to 10 x 4000 - 10 x 26. We need four of those though, so we can double it and turn from 40000 - 260 to 80000 - 520 and then 160,000 - 1040 or 158,960. Need to remove 2 x 3974 though, so remove 8000 and add 52 so 151,012.
Hopefully ive been able to keep that first number fresh in my head this whole time, which involves repeating it for me, and I’d add 1,589,600 and 151,012. Add 150000 and then 1,012 so 1,739,600 and then 1,740,612.
That all said, I make way more mistakes than a calculator, and I was off by 400 or so on my first run through. Also its really easy to forget big numbers like that for me. I’d say if you gave me ten of these to do mentally I’d get maybe 2 correct.
That’s great but this is juggling numbers in memory and I simply cannot do this reliably. I will have this one current operation and put the other ones into the mental basket so to say and it evaporates and blurs as I calculate the other thing right so I wonder how these folks can do this and really fast. Not that I ever seriously tried other than some rare bored moments so maybe it is simply a matter of training?
Its very impressive though when you give these ppl two big numbers and they say result nearly in an instant
Over time those bigger numbers become more common too. Someone who can mentally do the type of problem I just did and get it right quickly likely have a ton of practice and will know quicker tricks, and be able to simplify it in a way.
Another part is they would be able to recognize a wrong answer more accurately as well. I didnt realize my answer was off by a lot until I put it in a calculator, but someone with more practice might know intuitively they were wrong.
I just don’t consistently do this type of math, I used to be good at it in school but its become mostly irrelevant for me outside impressing someone a slight bit. It is helpful to have the ability to do things manually but it just rarely comes up.
Depends how much neuron density you have in the part of the brain that handles this. It’s mostly about memory, being able to accurately and quickly remember all the little steps you have already done and what the results of those steps were. Then just keep going one digit pair at a time keeping in mind all the results so you can deal with the carry overs.
But the whole reason we can focus on teaching everyone shortcuts for smaller math now is because we do literally always have a calculator on us now. So while it’s still good to know how to do bigger math more efficiently, you’ll never catch up to a calculator anymore. It’s more important that they know the foundational concept well enough to move on to the next step now rather than practicing doing big math faster and faster. Can leave that to the individuals with talent in the area.
