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#14442  04/23/08 08:33 AM
Re: Math intuition, math without books
[Re: doodlebug]

Member
Registered: 02/07/06
Posts: 802
Loc: Home :)

In effect, Dr. F. said that most peopleschools, teachers, homeschool parents, etc.mistake teaching arithmetic or showing kids how to "do" math for teaching mathematics, and that that's wrong. Doing math and getting math are two very different things. Instead of using canned problems out of a book, we should teach kids math only through natural methods, through science experiments like pendulums and bouncing balls, graphing changes in history, the weather, election results, etc. on Excel, and so on. Ask them how many oranges can fit in a box. Have them estimate the value of pi as closely as they can using only geometrical shapes and a ruler. Rather than teaching math facts or requiring memorization, we should encourage kids to derive their math facts every time they do a problem until they have internalized them. No memorization ever. If it takes longer to do the problems, then so be it; just do fewer, deeper, harder problems. Memorization kills intuition, and should be banned. Start with the big picture. Teach calculus to the littlest kids, but don't call it that and don't expect them to understand it all in one bite. Give it to them until you lose them and then move on to the next topic. It's the spiral method of teaching at its best: every 2 or 3 years, come back to calculus (or stats or trig or geometry or whatever), only with the next layer of complexity, picking up wherever the child stopped during the previous rotation of the spiral (if that makes sense, as I'm explaining it badly). Above all else, teach them that math is beautiful and encourage them to use their intuition.
To me the above is either a new age mumbojumbo or a very complicated way of describing problem solving. What is math, or why do we learn math? To spit out answers or to be able to problem solve? Rusczyk writes "true mathematics is not a process of memorizing formulas and applying them to problems tailormade for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems. We use the term “problem solving” to distinguish this approach to mathematics from the ‘memorizeuseforget’ approach."So why memorization without thinking (memorizeuseforget) is bad, you can't say that memorization per say is bad. It is a tool! Memorization of certain things in math is crucial. You have to understand "why", but once you understand, cetain things should just stick with you and you should be able to recall them right away, otherwise you will be lost in more complex problem solving. Kriston's son is 6 years old, so she is looking at a different math than I am looking at with 13 year old. Time tables are generally the first thing that kids are asked to memorize. Thay have to understand WHY, but if they don't memorize them, how are you going to do division? Intuitively? Then you will have ton's of mistakes, even though you do understand the principle, or it will take you forever... I would like to see an example of how do you teach multiplication problem using a swinging pendulum? Shouldn't you make math as simple as possible? I see memorization as a tool, not as a goal in math. Once you "have" multiplication, further persue of math should require you to memorize exponents, which will lead you to memorization of some logaritms. Having your factorials memorized is a huge advantage to problem solving  you are eliminating some brainless steps (once you know that is is a factorial you have to use, of course  and memorization won't take you there). So....I disagree with the statement that memorization in math is an enemy  blind memorization is an enemy, not memorization per say. Smart memorization is an excellent tool! Kriston is worried that her son will lose interest in math. Do word problems  no way you can get bored with those. Don't star calc yet :), unless you want to lose your son completely.
Edited by Ania (04/23/08 08:47 AM)

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#14447  04/23/08 08:58 AM
Re: Math intuition, math without books
[Re: Ania]

Member
Registered: 09/19/07
Posts: 6145
Loc: Midwest

Well, it's not newage mumbojumbo... I think it is problem solving. And you hit Dr. F's nail on the head here, I think: Memorization of certain things in math is crucial. You have to understand "why", but once you understand, cetain things should just stick with you and you should be able to recall them right away, otherwise you will be lost in more complex problem solving. It's learning the facts through use, not memorizing them to pass a test. I think you and Dr. F are on the same page, Ania. Please keep in mind that you're getting his take through me, and remember that I'm still trying to understand it myself. As I mentioned, a big part of this post is my own groping to figure out what it means and how to use it. "Problem solving" is accurate, I think, but is too simple to mean much to me personally. So why do you think doing calcaccessed through physics or history and at a level DS6 can understandwill lose him, Ania? You may be right, but I don't know why. Can you explain what's wrong with that take? Remember, it won't be calculus like it's traditionally taught. We're going out of the box here, big time! As for teaching multiplication through a pendulum swinging, he was talking about square roots there, I think. Apparently (???) the slowing swing of a pendulum corresponds roughly to square roots, and graphing the swing can be used for working with sqares and square roots. If I may...This was one slide of over 100, and it wasn't a presentation designed to teach me how to teach, and I am NOT a math expert, so I KNOW I don't understand completely yet. I FREELY admit that these nuts and bolts things need to be WAAAAAAAY clearer if I'm going to implement this teaching strategy. At this VERY early stage, though, I'd prefer to focus on the big picture, since I know I have LOTS of legwork and planning to do before I get to the practicalities of "How do you teach about a swinging pendulum?" I AM very glad that I have a good 4 months to figure it out! I think I'm going to need it!
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Kriston

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#14449  04/23/08 09:16 AM
Re: Math intuition, math without books
[Re: Kriston]

Member
Registered: 09/19/07
Posts: 6145
Loc: Midwest

Thanks, Dazey and Debbie. I'm looking at Living Math right now, and I'll check out Calc for Young People. I am feeling a little resistant to any curriculum right now, but if I find one that is more of a resource and less of a structured program, I'll probably grab it up so fast it will make your head spin! I really like what you said about thinking of it as guiding, not teaching, Debbie. I'll probably reread your post several times... I do use those teachable moments for all they're worth. I'm very comfortable with teaching that way...everything but math! LOL! But I feel like this is different from that, or at least I want it to be. I do want some sort of organized plan (with full freedom to deviate from it, of course! Ha!) for my own sanity. But I don't want the plan to focus on arithmetic and workbooks. I wanted to teach more science next year anyway, and this fits that goal nicely. But I have to plan the experiments and history lessons that require the graphing. I don't want himn to read about it, I want him to do it. I love all the help! Gosh, thanks everyone!
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Kriston

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#14453  04/23/08 09:27 AM
Re: Math intuition, math without books
[Re: Kriston]

Member
Registered: 12/14/07
Posts: 902

I think you need a good balance. You need problems which are challenging and which teach good math thinking, but you also need to learn the basic arithmetic because like Ania said, it's a tool.
So far our dinner math is usually full of new concepts and problems, but DS5 got lots of very useful drill in Montessori. You see we can and did let him figure out how to add two fractions like 3/5 + 2/3, but after that he needs to practice it. Practice is a big part of math and to be honest DS5 will do much better if he can start from simple problems and move up. He sometimes gets frustrated if he can do the logic but cannot do the arithmetic behind it and I believe there is a place for bunch of 3/4 + 5/6 problems.
I know lots of people who were extremely successful in math, who won prices in math and physics competitions and I hope to mimic the education they got.
_________________________
LMom

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#14454  04/23/08 09:29 AM
Re: Math intuition, math without books
[Re: Ania]

Member
Registered: 12/14/07
Posts: 902

I think that all you can do for a 6 year old at this point in regards to calculus is showing him that this is one of the ways to problem solve. In order to "get" calculus one should have an understanding of functions as well as some laws of physics. If you start without it, calculus makes no sense, IMO.
I never though that I would be the one to say that,(LOL, I am usually accused of speeding up the process) but why would you want to do it? Algebra is much more approachable without much prior knowledge of math, so why not look into algebra for elementary schools? Do you think that in order for your son to enjoy math he needs to be introduced to calc? I am not advocating waiting till college either, but there is so much more to math than calculus. Start looking into more descreet math, like number theory or probability . This should prove to be very enjoyble :) I agree. Calculus sounds way out of there. I am all for intro to algebra or logic, but calculus without algebra sounds strange to say the least.
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LMom

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#14457  04/23/08 09:32 AM
Re: Math intuition, math without books
[Re: Ania]

Member
Registered: 10/19/07
Posts: 175

I do agree with Ania that there are so many other beautiful math subjects that get ignored. Discrete math is wonderful and can be taught on many levels. Set theory, logic, probability, number theory, and graph theory are just a few of the topics that would appeal to elementary kids.
I have mixed feelings about teaching calculus to elementary kids. I think the theory is goodteach kids through problem solving and not through mindless memorization. I also think some elementary kids can grasp the basic concepts of rates of change and limits. However, problem solving in calculus is very dependent on algebra skills, and I'm not sure that many elementary kids would really be able to do it. I think that exploring limits and rates of change (the very basics) without getting into real differentiation and integration (which requires knowledge of algebra and functions) is worth a shot.

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#14458  04/23/08 09:36 AM
Re: Math intuition, math without books
[Re: LMom]

Member
Registered: 09/19/07
Posts: 6145
Loc: Midwest

Well, it's not calculus without algebra, exactly. Algebra is one of the math areas we would deal with along the way. But even that won't focus so much on the how to do it as the broad concepts.
I know I'm not explaining this at all well... *sigh*
I think you are thinking about it as traditional teaching, and it just plain isn't! I wish I had a better way to explain it, but it's teaching the broad concepts using physics so the young kids (like DS6) see calc in action, *not* teaching the equations. Does that make more sense?
Think about it as getting your feet wet, not diving into equations and howtos right off the bat. It's about seeing calculus (or algebra or geometry) as a way of solving real problems, not about getting everything about each branch of math at age 6.
Am I making it better or worse?
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Kriston

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