Virgil Showlion
Distinguished Associate
Moderator
[b]leones potest resistere[/b]
Joined: Dec 20, 2010 15:19:33 GMT -5
Posts: 27,448
|
Post by Virgil Showlion on Mar 28, 2014 13:32:27 GMT -5
Well geez virgil, you just start by taking away 70 billion from 80 billion. That leaves us with 16 billion. Now you take away 2 billion from 16 billion and you have 14 billion... Another simple subtraction method that many people aren't aware of uses only addition and a simple "flip". Suppose you have 86298301927 - 72634918274 To solve this, simply "flip" all of the digits on the bottom row by subtracting each of them from 9. Hence 0 becomes 9, 1 becomes 8, 2 becomes 7, etc., etc., up to 9 becomes 0. You get: 86298301927 + 7263491827427365081725 Now add the two numbers. You get: 113663383652 Finally, cut off the first digit (which will always be a 1), and add 1. You get: 13663383653 which is your answer. Great for kids that know how to add well but can't subtract. This also happens to be the way that computers perform subtraction, only they do it in binary rather than decimal. In binary, it's called the "2's complement" method.
|
|
Virgil Showlion
Distinguished Associate
Moderator
[b]leones potest resistere[/b]
Joined: Dec 20, 2010 15:19:33 GMT -5
Posts: 27,448
|
Post by Virgil Showlion on Mar 28, 2014 13:34:20 GMT -5
Once again, a number line is an excellent early tool for visualizing what becomes a mental process... And yes, its one way. And the new method is to teach multiple ways. But also to focus on numbers and relationships, etc. rather than rote memorization and algorithm... Conditioning the brain to handle algorithms, sequences of rules, etc. is important, though. And math is one of the only disciplines where algorithms are taught to kids.
|
|
973beachbum
Senior Associate
Politics Admin
Joined: Dec 17, 2010 16:12:13 GMT -5
Posts: 10,501
|
Post by 973beachbum on Mar 28, 2014 13:34:24 GMT -5
oped I think this actually gets back to a pet peeve of yours. We are trying to force kids to learn things before they are actually ready. Common core is a list of things students should have learned at the end of particular grade. Where I may have learned multiplication in 4th grade now they are teaching it in 2nd. Because of this IMO it needs to be way more broken up to help the kids understand things that if taught when they were older wonuldn't be quite that difficult. I am not talking about visual vs other types of learners, just the age that they learn it at. And I would LOVE to see you guys trying to do adding, subtracting and mult/dividing in Everyday math. I think a few heards would explode at the process of going from left to right. You don't get scratch paper? Stacking them is actually to make sure that the ones all stay in one column, the tens in another, and so on. It's another form of visualization. Ditto, with multiplication and even the dividing thingamajig. I think we could do it linearly, but it would slow us down in the same sense of typing on the non-Qwerty keyboard would. Once we got used to it, it would be probably be no big deal. But the horizontal method makes sense because it allows you to hyperfocus on one column at a time if the number of things to be added are many. They gave the kids as much scratch paper as they wanted. That didn't stop engineer DH's head from alsmot exploding at the sight of our daughter adding and subtracting from left to right, instead of right to left though.
|
|
Deleted
Joined: May 19, 2024 0:09:45 GMT -5
Posts: 0
|
Post by Deleted on Mar 28, 2014 13:41:37 GMT -5
I do very little in a linear fashion. Now higher level algebra, etc. needs a structure. But otherwise, math for me is very haphazard, much as i do everything else. It isn't about 'setting down the problem'... because that is so, not even an issue... setting down the problem should not be how one 'does math'... problem solving is messy. Numbers should be allowed to fly apart and together in the way that best fits the circumstances. In my opinion.
It always makes someone's head explode when they ask advice on teaching 'borrowing' and 'carrying' and i say i have no idea, i don't teach those concepts in the way in which they mean...
|
|
Miss Tequila
Distinguished Associate
Joined: Dec 19, 2010 10:13:45 GMT -5
Posts: 20,602
|
Post by Miss Tequila on Mar 28, 2014 14:35:44 GMT -5
Its probably a bit that Its also that i've always been a left to right kind of person in math and its finally nice to have that method acknowledged by something other than a blank look But I'm a right to right kind of person in math so you can appreciate my blank look at this method. It makes NO sense to me.
|
|
Deleted
Joined: May 19, 2024 0:09:45 GMT -5
Posts: 0
|
Post by Deleted on Mar 28, 2014 14:41:08 GMT -5
Oh, but i can do algorithms. I just find them clunky and time consuming for most day to day math.
I have no problem with you not wanting to use the 'stacking' method for yourself. But it is not the only way. AND it is not really an end product that i think we should be looking for, in terms of what i means to be able to 'do math'.
We need a much more elastic understanding of mathematics I think if we are going to remain competitive...
|
|
Virgil Showlion
Distinguished Associate
Moderator
[b]leones potest resistere[/b]
Joined: Dec 20, 2010 15:19:33 GMT -5
Posts: 27,448
|
Post by Virgil Showlion on Mar 28, 2014 16:10:40 GMT -5
Oh, but i can do algorithms. I just find them clunky and time consuming for most day to day math. I have no problem with you not wanting to use the 'stacking' method for yourself. But it is not the only way. AND it is not really an end product that i think we should be looking for, in terms of what i means to be able to 'do math'. We need a much more elastic understanding of mathematics I think if we are going to remain competitive... "Competitive" with respect to what? You realize of course that if the sum total of mathematical knowledge were a football stadium, the amount of it taught up to the grade 12 level in OECD nations would be about the size of a popcorn kernel. Using it as a kind of gauge as to how competitive your country is in mathematics is like using an athlete's skill in tying his tennis shoes as a gauge of how competitive he is in tennis. If you want to use math for anything more than "most day to day math", I'm sorry but you need the rigid, abstract, scalable, left-brained approach in your educational repertoire. You can certainly teach it in fun and creative ways, and supplement it with "elastic" right-brained conceptual aids provided the core material is still there, but ultimately a student will go nowhere in higher math without learning to embrace process, abstraction, syntax, procedural rigor, and other fundamentally inelastic concepts. There has got to be some left-brain pain. If you're constantly bending the lessons into something a child can easily visualize, you may be helping them in the short term with conceptual clarity, but you're also denying them that painful cognitive dissonance that forces them to respect the "clunky and time consuming" algorithms, algebraic axioms, equations, formulas, etc. These are the only tools that scale up and remain useful in higher mathematics. You cannot stack blocks or set up choo-choo trains on a number line to build a Reimmanian manifold in 5-space or to solve a constrained optimization problem. It ain't gonna happen.
|
|
Deleted
Joined: May 19, 2024 0:09:45 GMT -5
Posts: 0
|
Post by Deleted on Mar 28, 2014 16:42:37 GMT -5
Oped, let me make sure I understand you: If the problem is 12 +19 = ? You would add the 2 10s first to get 20 and then add the 2 plus the 9 to get 11 and then add the 20 and the 11 to get 31? That would be left to right, I am guessing. Even if the problem is expressed linearly, I would add the 9 and the 2 to get 11. I'd put a 1 (mentally usually)and add it to the other two 1s to get 31. So my method of addition is right to left even if the problem is expressed left to right. I think right to left is a natural progression of needing to fill the "ones" column first before anything rolls over to the "tens" column. But maybe that's because how I was taught. But going back to the finger form of counting, you would fill up your fingers first and then fold one down to mark the 11. I can't picture folding two fingers down and then counting to 11 again to fold the third finger down. I honestly feel that is where counting came from and why our system is based on ten. It corresponds to our ten fingers. That would imply it was always right (add ones first) to left (then the tens, etc.) . . . . I will admit, though, that that is possibly personal bias. I'm not saying that right to left is inherently "better" than left to right. It is just the more natural way in my opinion. If a culture was going to go left to right, I would expect 31 as we know it to be expressed as 13. Is there a culture that uses the left to right standard in the same sense we use the right to left as a standard? Would 31 to me be 13 to them with the same meaning? Am I missing something? It just seems very similar to word order in English. It isn't that you can't mess with "normal" word order in the sense that English is a SVO language. Writers mess with it all the time. But it is almost always conscious messing with it rather than embracing the idea that you should put the words together however you please. It then becomes a higher form of thinking to unscramble it so that it makes sense. Sounds a little bit like Latin now that I think about it. Or maybe I have just graded too many uninteresting research papers during Spring Break and am desperate to think of something else.
|
|
Deleted
Joined: May 19, 2024 0:09:45 GMT -5
Posts: 0
|
Post by Deleted on Mar 28, 2014 17:47:10 GMT -5
Actually, I'd take one from 12 and get 20 + 11 = 31, yes adding 30 then 1. My internal process would go something like 19, 20, 30, 31... Not that I necessarily verbalized the steps, but to illustrate my process for that problem. Depending on the numbers, till combine or deconstruct them in whatever manner is fastest/ easiest. Virgil... The point is that at any time an intelligent person may make use of algorithms and formulas... Or a machine can... But what we need to encourage is flexibility and elasticity of thought processes. This is difficult to do within a rigid employment of algorithms. I was very interested in the point someone made about girls doing generally better in math early on, but that eventually went away because they had, basucally, learned memorization of rules versus elasticity of thought... I'm paraphrasing
|
|
Deleted
Joined: May 19, 2024 0:09:45 GMT -5
Posts: 0
|
Post by Deleted on Mar 28, 2014 17:49:30 GMT -5
Why don't you SAY one and thirty then? Why read numbers left to right if you can only manipulate them right to left?
Algebraic equations, while they do need structure, are at least more flexible...
|
|
Deleted
Joined: May 19, 2024 0:09:45 GMT -5
Posts: 0
|
Post by Deleted on Mar 28, 2014 18:44:12 GMT -5
Actually, I'd take one from 12 and get 20 + 11 = 31, yes adding 30 then 1. My internal process would go something like 19, 20, 30, 31... Not that I necessarily verbalized the steps, but to illustrate my process for that problem. I don't get the "I'd take one from 12." No idea where that is coming from. But I don't add like you do. That is interesting why we don't say "one and twenty." I don't know. We do know "four and twenty blackbirds baked in a pie" (common nursery rhyme) is 24. It's not unheard of, which is my point. But it isn't common. I'm guessing that we ADD numbers right to left because of the ones/tens/hundreds column thing, but we actually read everything, including numbers, left to right. Let's just agree math is a foreign language with its own "rules." Like I said about SVO word order, you can get to the same place by a different method. It just isn't how most people express themselves. Most non-SVO word order sentences are beautiful because they are deliberate manipulations of language. Maybe deliberate manipulations of how we "usually" (notice that I am avoiding "normally") do mathematics are also beautiful. Or at least creative. Maybe it the same sense that some people see colors when they see names.
|
|
Deleted
Joined: May 19, 2024 0:09:45 GMT -5
Posts: 0
|
Post by Deleted on Mar 28, 2014 18:49:29 GMT -5
I mean the first thing I do is turn 19 + 12 into 20 + 11 (take one from the 12, leaving 11... Making the 19 into 20) Because adding 20 to 11 is easy.
|
|
Deleted
Joined: May 19, 2024 0:09:45 GMT -5
Posts: 0
|
Post by Deleted on Mar 28, 2014 18:58:49 GMT -5
I frequently both add and subtract in problems that are adding or subtraction.
For instance, If you had 316-157
I'd take 200 from the 316, leaving 116. From the 200 I take away 157, canceling the first hundred and knowing that 57+43= 100, so in taking 157 from 200, I'm left with 43.
116+43 = 159... And yes, I add 100, 150, 159..
That at sounds complicated, like the number line, but my mind does it quickly, honestly, it's like I read 316-157 as 116+43... Because that is the simplest mental math application of that problem... (Unless you notice that 157 doubled is 314, and proceed that way..)
|
|
violagirl
Familiar Member
Joined: Aug 17, 2011 11:04:54 GMT -5
Posts: 703
|
Post by violagirl on Mar 29, 2014 13:04:12 GMT -5
I frequently both add and subtract in problems that are adding or subtraction. For instance, If you had 316-157 I'd take 200 from the 316, leaving 116. From the 200 I take away 157, canceling the first hundred and knowing that 57+43= 100, so in taking 157 from 200, I'm left with 43. I would count up from 157 - my brain says add 100 brings to 257 then 43 + 16. I find it easier to add than subtract. I also play the piano and find I have no problems playing in Db+ but C#+ takes a lot more brain work. Dropping down a note is much easier on my brain than going up.
|
|