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.
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*7 is (7-1) and whatever adds to 9, so 63. This breaks down for larger numbers, but works really well up to 9*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.
I was very competitive in school like this, wanted to finish things first. I think maybe you make a good point about wanting to solve things faster leading to these types of tricks developing. Sort of puts math competitions in a new light.
Yep, there are many ways that people (some of whom may or may not be of earthly origin) have developed to perform various degrees of math all in their heads.
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.
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.
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.
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.
I was very competitive in school like this, wanted to finish things first. I think maybe you make a good point about wanting to solve things faster leading to these types of tricks developing. Sort of puts math competitions in a new light.
My brain did something similar, but maybe weirder.
7 + 3 + 6, rather than 9 + 1 + 6.
That’s basically what I do
Yep, there are many ways that people (some of whom may or may not be of earthly origin) have developed to perform various degrees of math all in their heads.
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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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