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It's first grade stuff in the curriculum my kids did.
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Miranda
Mountain mama to three great kids and one great grownup
Yes, first grade for our school, too. Fun that she's interested enough to figure it out!
Momma to 8 y.o. DS and 5 y.o. DD. Married to a Maker!
It is part of our 1st grade curriculum which is ages old 5/6/young 7 (depending on time of year my two were 5 at the start of the school year). My DDs class started it late last fall (start of 2nd semester).
We saw it in kindergarten, first, and second.
I can not figure out what you are talking about.Â What does that rule mean?Â It sounds likeÂ 3Â concepts to me  doubling and + 1s and 1s.Â I am sure I am missing something.
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Tjej
You can figure out 8+7 quickly if you remember that 7+7 is 14.Â So you figure out (7+7)+1 or (8+8)1.Â It's a commonly taught mental math technique.
Didn't know about this trick till now ...
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I can see doubling is easier than tripling, quintupling, etc  but we're talking about addition, not multiplication here.Â
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Back to the addition  in binary form, I can see that adding 2 identical integers is preferable to adding 2 arbitrary ones (just add 0, it's like multiplying by 10 in decimal)  but why the preference for base10?Â Why would one need a trick to add 2 singledigit integers anyway?
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They teach this at school?Â I'm a bit surprised, or shocked rather  this is nothing like what I think of as math, and is this really part of some formal curriculum?
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Edited for clarity.
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Have you worked with many elementary aged children in math?Â I think those that post in this forum based on their own personal experiences and their experiences with their own children sometimes lose sight of the path a lot of kids must travel to learn math.Â I did some mental math not dissimilar to this in class today, and it baffled my students.Â My college students.
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Teaching strategies like this seems like an excellent thing to be doing, and will serve as a bridge to multiplication for some, and a path towards memorizing math facts for others.
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Skills come to different kids in different ways.Â Some kids will struggle with learning to add 7+8 and get the answer right every time.Â Teaching them a lot of different strategies really helps.Â Not every kid will need every strategy, but many will make use of it.Â If you don't have single digit addition down, then moving into multidigit addition, multiplication and division will become quite laborious.Â The Learning at School board used to have long impassioned conversations about the necessity toÂ learn addition and multiplication facts.Â The fact of the matter is that the more of these micro steps you can make automatic, the more of the brain you can free up to focus on the higher level thinking skills.
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These lessons start on single digit addition, but I've noticed my kids both use mental math tricks like this on larger numbers.Â My daughter does her homework now without a calculator, and often in her head.Â Â She can fluidly move between addition and multiplication, picking whichever strategy is most efficient.
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I use strategies like this to calculate tips, figure my average running pace, and estimate a wide range of things in life.
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A lot of arithmetic is a bridge to higher level math for kids.Â Many would say that it's not necessary, but for many kids who require to go through the concrete stages before the abstract, these are pretty necessary.
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I'm not sure your point about different bases?Â Sure, you can add and subtract in base 2, but why?Â This is a topic my DD has seen in her middleschool "math circle", a program for kids who love math.Â It was hard for them.
Yeah  I agree arithmetic is useful  that's not what bothers me.Â What surprise me is how the trick seems to complicate things.Â Why do you want to double anything when all you want is to add 2 singledigit numbers?Â The trick only works for a pair of very small numbers with certain pattern  why all the trouble?
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But then I thought  hey, this actually can make things easier if the numbers are in their binary form and they can be arbitrarily large.Â For example, let's take 6 + 7, which is 110 + 111 in binary form.Â Doing long addition, one has to worry about carryover, this is prone to mistake for larger numbers.
Now, let's apply the trick above:
a. take the smaller of the 2 numbers
 which 6 or 110
 don't want to take the larger number, it's easier add 1 then subtract 1 later, in case of carryovers etc
b. double it
 we have 12
 but in the binary form, you just add a 0 because doubling in binary is like multiplying by 10 in base10
 so we have 1100
c. add 1
 we get 13 in decimal
 or in binary, 1100 + 1 = 1101
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I am all for learning the concrete first, you have to know your facts, mathematical or otherwise  but it's the overcomplication and lack of generality that got me  the trick only works for singledigit numbers with very specific constraint.
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Though, at the end of the day, this is probably a matter of esthetics  I don't know, perhaps this can work for some ...
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We must swim with different 6 year olds.Â ;)
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It's one of many strategies.Â Most children need to be taught through direct instruction at some point.Â
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Yeah  I agree arithmetic is useful  that's not what bothers me.Â What surprise me is how the trick seems to complicate things.Â Why do you want to double anything when all you want is to add 2 singledigit numbers?Â The trick only works for a pair of very small numbers with certain pattern  why all the trouble?
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It does not seem to complicate it for the 5/6/7 yr olds in Â a regular 1st grade class.
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For most kids they find the higher integer math problems harder and result in finger count/numberline/etc the math problems past +/ 1,2 and for memorization, doubles are fairly quick for most kids due to the pattern. Â Â so if a kiddo knows 6+6.....but honestly may struggle with 6+7 or 5+6. They can easily +/ 1 which for many kids this is easier than starting at 5and adding up 6, or finger counting, etc.
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The 1st grades here are also using the base ten to add/ higher numbers. If you know 1510.....then you can easily find 159 by  1. Same goes for 9+ 7 (you can easily figure it out if you know 10+7).Â
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The basic facts that have 6,7,8,9 seem to take longer to master. Using double +/ or base ten )=/ are great mental tools to help kids move past finger counting/number lines, etc.
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My kids DO use it for higher math since it is easy to use base tens with larger digits. They easily can mentally figure out what 500432 is (they say it is 70 Â counting up by tens to 500 from 430 then add 2) or 4021 they know there are 20 between 20 and 40 then to add one, but they would not know how to 'borrow' to find the answer at this point.
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Their general education 1st grade class regularly does mental math like what is Â 189 +1 or what is 6+62 or what is two tens plus one or what is double 7 and take away 2. It is really fun to hear the kids do this and they love it!
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"Doubles" are among the first addition facts that kids tend to learn naturally in our culture. I can say this with certainty because my own kids were totally unschooled in math and they learned them easily. Doubling and halving small numbers are extremely common reallife situations. Share 4 cookies with your brother. Make two teams out of 8 kids. Hold up two hands of five fingers each. Roll two 6's playing a board game and you have the highest roll you can get. Buy a pack of 4 pairs of socks and you have eight of them. Two front tires and two back tires makes how many? An 18wheeler truck has how many wheels on each side? If an egg carton has each row halffilled, how many eggs? Make sandwiches for all six family members and how many pieces of bread do you need?
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I could go on, but my point is that doubling/halving observations are extremely common in a small child's world. Far more so than, say, 5+6. As such, kids tend to naturally absorb these math facts without much if any overt memorization work. So when they're trying to answer addition problems that they don't have memorized, it is very natural for many of them to use those alreadyinternalized doublesfacts as touchstones. My kids, like the OPs dd, all used this strategy very naturally.Â
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Whether this strategy, if explicitly taught, will be helpful for any particular child depends in large part on their learning style and on the way they conceptualize numerical values. But I still think it's very much worth teaching as a mental math strategy, even if it doesn't click with every last person. A similar approach can help with other mental math problems too. Consider the problem of 378 + 199. I would look at that and think to myself "Well, I know what 378 + 200 is, so it must be one less than that!" This is very much analogous to saying "I know what 8+8 is, so 8+7 must be one less than that."
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Miranda
Mountain mama to three great kids and one great grownup
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yes, and part of math education is learning to THINK in different ways. It's like exercises to help the brain become more flexible.
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So much of success in life is about being able to think in a variety of ways.
but everything has pros and consÂ
Hm, I'm still thinking what specific things bother me  it's really about personal preference, I believe.
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About the overcomplication:
Addition of integers seems like a concept that a child can easily intuit.Â You have some stuff, add/subtract some, then you have more/fewer  math gives you the tool to say how many more/fewer.Â While the notion of division by 2 and doubling seems more complex  it involves the ability to discern some pattern, or matching pattern.Â Of course  who am I to judge what is simpler or not.Â If one assumes the notion of addition to be more elementary  then why obfuscate the intuition and simplicity with tricks?
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About the lack of generality:
The trick given doesn't seem to lead to more general concepts or ideas in mathematics. It applies to a very special case, which is how to add 2 integers that differ by 1.Â In my mind, math is not about special cases.Â Its power comes from how ideas can be applied in so many different cases  its power comes from its generality.
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It seems that the trick approach might lead to the impression that math has lots of tricks which are unrelated, with no bigger picture or more fundamental ideas behind them.Â This approach doesn't seem to teach what's beautiful about math  just my personal preference
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Originally Posted by MamaMunchkinÂ
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The trick given doesn't seem to lead to more general concepts or ideas in mathematics. It applies to a very special case, which is how to add 2 integers that differ by 1.Â In my mind, math is not about special cases.Â Its power comes from how ideas can be applied in so many different cases  its power comes from its generality.
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It seems that the trick approach might lead to the impression that math has lots of tricks which are unrelated, with no bigger picture or more fundamental ideas behind them.Â This approach doesn't seem to teach what's beautiful about math  just my personal preference
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I totally disagree. It isn't a trick.
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It's teaching that 7+8 is the same as 7+7+1 OR 8+81.
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That's not about doing tricks. That's teaching how numbers work.
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Learning to manipulate numbers is at least as important as the facts, if not more so.
but everything has pros and consÂ
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It's not the trick per se.Â Of course you can do any shortcuts  even professional mathematicians do that.Â But math is not about shortcuts  it's a lot more than that.
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Edited:Â don't want to change the subject ... sorry ...
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We posted at the same time.
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Yes, it's true they can learn that 7+8 = 7+7+1 = 8+81 but why not also learn that this is no different than 7+3+5 = 7+1+1+6Â = 7+10092 = etc.
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Knowing all these manipulations is fun and educational, but the method given only applies to the +1 case  it's not about it being wrong, but about being unnecessarily restrictive.
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Pro rights (vaxes).
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About the lack of generality:
The trick given doesn't seem to lead to more general concepts or ideas in mathematics. It applies to a very special case, which is how to add 2 integers that differ by 1.Â In my mind, math is not about special cases.Â Its power comes from how ideas can be applied in so many different cases  its power comes from its generality.
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I gave an example above of generalizing the skill. If you know (or can easily figure out) the sum of two numbers, then you can work out the sum of two related numbers by compensating for the difference between them and the original pair. eg. 378 + 199 = (378 + 200)  1. The reason it's taught in a simple "restrictive" context in first grade is because the kids are only dealing with addition facts to 20. It's like teaching anything: you start with the simple applications of a new skill and generalize as the basics are mastered. In 2nd grade you use the same principles to develop more complex mental math skills, and you keep building on that as the years pass.
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re: overcomplication.... I think you're getting hung up on the idea that by calling them "doubles" the children are having to learn about multiplication. That's not the case at all. What is meant by "doubles" is the set of addition facts that are made up of like numbers: 5+5, 4+4, 1+1. etc.. These are the addition facts that many childrenÂ already know, so the idea is to use what they already know to get at related information.
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Miranda
Mountain mama to three great kids and one great grownup
I think the cool part was that she figured it out on her own. Debating whether this is a useful way to learn the commutative property (Ha! I really hope that's the name of the property!) Â or whether it's advanced seems beside the point to me. A kindergartner who is sufficiently interested in arithmetic to do this, and who sees the pattern, is showing some signs of maybe needing math enrichment. You will have to pay attention to see what other things she likes and does on her own, to find out whether you have to give her more.Â
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I didn't start thinking that my son, who also did a lot of unassigned mental math, might be gifted in math, until I realized that everyone else was not doing what he did and I started to see that the school did not know how to enrich the curriculum for him.Â
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If the question is, "Should I be worried about this?" the answer is probably not, as long as she seems to be having fun and isn't bored. If the question is, "Should I be excited about this?" then I think the answer is yes! You don't have to worry unless school isn't serving her needs, but you can be excited because now you can ask her to balance your checkbook. (OK, that was a joke; I don't do that. But he does a lot of household math and it's totally fun.)Â
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For me, the thing that has been a challenge is that I'm not gifted in or interested in arithmetic or any other branch of math, and my kid is in love with it. It looks totally different than the gifts I had with reading and writing in the early elementary grades. With math it seems sort of binarythere are the kids who love it and then the rest. It's hard to see whether you have a gifted kid when she's good at math, and if you do, how to enrich. I think some of the other moms of gifted kids here are actually good at math themselves, and don't have to wonder.
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Divorced mom of one awesome boy born 232003.
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I gave an example above of generalizing the skill. If you know (or can easily figure out) the sum of two numbers, then you can work out the sum of two related numbers by compensating for the difference between them and the original pair. eg. 378 + 199 = (378 + 200)  1. The reason it's taught in a simple "restrictive" context in first grade is because the kids are only dealing with addition facts to 20. It's like teaching anything: you start with the simple applications of a new skill and generalize as the basics are mastered. In 2nd grade you use the same principles to develop more complex mental math skills, and you keep building on that as the years pass.
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re: overcomplication.... I think you're getting hung up on the idea that by calling them "doubles" the children are having to learn about multiplication. That's not the case at all. What is meant by "doubles" is the set of addition facts that are made up of like numbers: 5+5, 4+4, 1+1. etc.. These are the addition facts that many childrenÂ already know, so the idea is to use what they already know to get at related information.
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Miranda
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The simple restrictive context:
But why stop at double +/1?Â Why not do double +/2?Â What about double +/3?Â If not now, later?Â And if the numbers are not whole, say 1.75 + 1.5  then would they have to learn double +/0.25 then?Â Isn't plain addition simpler than this approach?
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Hungup on multiplication/double:
Nope, no problem with multiplication, or doubling.Â At the end of the day, the kids,orÂ any of us, will do whatever easiest, now and later.Â However, my personal preference would be that a more formal curriculum would take a more general approach.Â Specialized methods can help but they need to be presented with the bigger picture, where the more fundamental ideas are presented.
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So far, my DD has been in an alternative program in a public school  they have used the approach that aligns with my preferences.Â I was surprised that kids are expected to know the double +/1 ruleÂ  hence my initial reaction.
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This whole thread reminds me to something similar ...
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Talking about doubling/multiplication ... there's this cute and clever way to multiply 2 integers  the Russian peasant algorithm.Â I think the ancient Egyptians (??) figured it out too.Â Anyway, multiplication is done by repeated halving/doubling, and at the end addition.
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Anyway, I find it really cute and it'd be fun for kids to learn about different methods of multiplication.Â However, I'd prefer something like this not to be part of the curriculum  it'll make a nice supplement but not as part of the formal teaching.Â The reason is, this algorithm only works for integers, not for real numbers.
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It seems to me that math is such a good subject to learn how to abstract, extend, and generalize ideas  the very specialized methods without the bigger pictures don't seem to foster that approach.Â Again, just my personal preference.
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Originally Posted by captain optimismÂ
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With math it seems sort of binarythere are the kids who love it and then the rest. It's hard to see whether you have a gifted kid when she's good at math, and if you do, how to enrich. I think some of the other moms of gifted kids here are actually good at math themselves, and don't have to wonder.
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The kids who love it and the rest:
That may depend on how they were exposed to math, how math is presented.
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How to enrich:
Different schools of thoughts  that probably shows from this discussion so far :)
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The point of the mental math stuff is to teach kids how to break numbers apart and put them back together so that they can create numbers that are easier to work with.
He is currently a kindergartner who is doing all four operations (many of them in multiple digit problems), fractions, percentages, and basic ratios/proportions. clearly, learning doubles plus 1 early in his math life did not stunt or confuse him as he learned other numbers.
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He is currently a kindergartner who is doing all four operations (many of them in multiple digit problems), fractions, percentages, and basic ratios/proportions. clearly, learning doubles plus 1 early in his math life did not stunt or confuse him as he learned other numbers.
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Who said anything about stunting growth etc  huh?Â My point is that I am very surprised that a formal curriculum, such as in schools, would expect kids to learn this stuff.Â It's not whether or not the children will get the right answers, or how fast  if anything these special methods will help you get the answers faster.Â And yes, of course there are many ways to get the right answers.
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An analogy:
Suppose you're (general you here) teaching how to multiply 2 numbers.Â If you're familiar with some trick, you can do 38x42 mentally, for example.Â You can use the identity a^2b^2 = (a+b)(ab).Â So 4x6 = 5^21 = 24.Â This can come very handy.Â
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What I've been trying to point out is, using the analogy, I would expect  or rather, expected  a formal curriculum to teach how to do long multiplication instead of the above identity.Â The formal way to do it works all the time.Â The identity works very well sometimes  at the end of the day when you need to get the answer, you do what seems easiest for you.Â But to teach someone the identity above as multiplication seems like a convoluted way to teach the notion of multiplication.Â But what's convoluted to me may be easier for others  that's why I've kept saying this is probably esthetics issue at the end of the day.
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edited: wrote something false.
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I guess what surprised me was that my son started quizzing me on arithmetic facts in the summer before kindergarten (What? What? Why? ) finding fractions everywhere he could, seeking out math enrichment videos on Youtube, adding up the numbers on the digital clock to find out whether the time was a multiple of three, and so on, without my encouragement and certainly without any teacher's encouragement. He started getting into math before school and by kindergarten was completely smitten. His interest predated formal instructionwhich might be normal, but it didn't happen to me, and I think that's part of why the OP's question spoke to me.
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I can't speak to the question of whether teaching tricks like the doubling one we're discussing is good math education or not. Â I think my kid would enjoy the "double it and subtract one" trick if he'd been taught it instead of figuring it out for himself, just because there really has not been ANY math trick he hasn't loved! I don't know if this is something all kids enjoy, since I only have this one under the microscope of parenting, but he likes card tricks and slight of hand and math jokes, too, so probably he would have responded to the doubleandsubtractoradd very nicelywithout getting distracted from other methods of understanding addition.Â
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The OP's question speaks to me as the mom of a mathy kid, because it can be really hard to tell whether kids figuring out number relationships on their own is a sign of something. It seems like here the answer is, "maybe!"Â
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Divorced mom of one awesome boy born 232003.
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Who said anything about stunting growth etc  huh?Â My point is that I am very surprised that a formal curriculum, such as in schools, would expect kids to learn this stuff.Â It's not whether or not the children will get the right answers, or how fast  if anything these special methods will help you get the answers faster.Â And yes, of course there are many ways to get the right answers.
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An analogy:
Suppose you're (general you here) teaching how to multiply 2 numbers.Â If you're familiar with some trick, you can do 38x42 mentally, for example.Â You can use the identity a^2b^2 = (a+b)(ab).Â So 4x6 = 5^21 = 24.Â This can come very handy.Â
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What I've been trying to point out is, using the analogy, I would expect  or rather, expected  a formal curriculum to teach how to do long multiplication instead of the above identity.Â The formal way to do it works all the time.Â The identity works very well sometimes  at the end of the day when you need to get the answer, you do what seems easiest for you.Â But to teach someone the identity above as multiplication seems like a convoluted way to teach the notion of multiplication.Â But what's convoluted to me may be easier for others  that's why I've kept saying this is probably esthetics issue at the end of the day.
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edited: wrote something false.
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I'm not sure I'm understanding your concern. Do you think that formal math curricula are NOT teaching basic addition and long multiplication and other operations? I'm pretty sure that if the curriculum shows how to use doubling +/ 1, it's already covered basic addition of single digit numbers and expects that the students have some proficiency at it. Maybe I'm mistaken, but I don't think I've read that anyone has experienced otherwise.Â
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It seems like you are saying that it is a waste of time for a student to be exploring these concepts at school and trying these tactics, and that they should stick to learning basic operations. Perhaps I've misunderstood you though.Â
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BTW, when I looked at your example, I immediately multiplied 40x42 and subtracted 84 and had the answer in a few seconds, before I read on further. A robust mathematical education covers a variety of techniques and encourages a wide exploration of relationships between numbers. Simply teaching rigidly applied algorithms and formulae isn't likely to achieve that outcome.Â
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MamaMunchkin  Maybe you are caught up on the idea that this is a "rule".Â That was what confused me with my newborn mush brain.Â I understand the concept, but thinking of it as a particular rule in and of itself was confusing.Â To me, it isn't really a particular rule, but a general idea of how to manipulate numbers.Â That I can understand and appreciate.Â I think that kids can infer from the "rule" that you can do +/ other #s as well  those that are ready for that kind of mental math.Â And those who aren't ready can generalize the "rule" later.Â Or extend it on paper.
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captian optimism  IIRC, the commutative property is the one that says 3+4=7 and 4+3=7Â  position doesn't matter.Â I never really got the idea of this being a rule either.Â To me it was just stating the obvious of how #'s work.Â Â Like 3 buttons = 3.Â I don't think that needs a specialÂ button rule. :)Â
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Tjej
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