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.
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.
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?
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.
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!
Unsolved problems do not all fall into binary outcomes. They can be independent of axioms (the set of assumptions used to construct a proof).
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.
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.
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).
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…
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.
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.
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.
You know how sometimes you go outside and there’s a bird and you’re like, “cool”
classic adhd
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.
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 success
Neurotypical 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.
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.