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    #14440 - 04/23/08 08:20 AM Re: Math intuition, math without books [Re: cym]
    Kriston Offline
    Member

    Registered: 09/19/07
    Posts: 6145
    Loc: Midwest
    Dr. F gave some examples of just this sort of intuition-deadening, cym. People who figure 100/8 by doing long division in their heads, for example. That's not the "natural" way to do it; it's learned. And sometimes that learning can kill the intuition. My dad had this happen to him, too. It's pretty tragic to see it happen.

    I'm the POGS secretary, so if you want, I could send you my notes from the meeting. It would be a long PM, and he talked REALLY fast, so there's a lot I missed, but I'd send it if you'd like. It's not the same as being there, but it might give you some food for thought. Just let me know.

    I'd say that my fear is not of math itself, kcab, but is of teaching math, if that makes sense. I have never felt that I have an intuitive grasp of math, though Dr. F would argue that it's just because I was "drilled and killed," as he's never met a GT person who wasn't FANTASTIC at math; they maybe just didn't know it. (I didn't do long division in my head to figure 100/8, so maybe there's some math intuition in there somewhere...) Math doesn't feel like it comes naturally to me, I never felt like I understood it, though I got A's in school because I could "do" it. That's not a great person to teach math, in my opinion. (Though I do think I'm probably more mathy than the average elementary school teacher, at least.)

    To tell you the truth, I'm scared of math because I'm worried I'll blow it. Math is the one area of homeschooling that I think I can really fail DS6, and that scares me.

    On the bright side, DH is a chemical engineer who spends much of his day writing computer code. Maybe this is one case where I hand off to daddy, eh, squirt? Each weekend, DH and DS6 do some big science project together, then I can help DS6 do the grunt work during the week to complete it?

    Hmmm...Maybe...

    I'm posting questions to a homeschooling group, too, since this sounds a lot like unschooling to me as well, Debbie. I didn't know about that Yahoo group, so thanks for that. I'm going there next!

    Good posts. Thanks all. It's helping me think this through.
    _________________________
    Kriston

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

    Registered: 02/07/06
    Posts: 802
    Loc: Home :)
    Originally Posted By: Kriston

    In effect, Dr. F. said that most people--schools, 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 mumbo-jumbo 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 tailor-made 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 ‘memorize-use-forget’ approach."

    So why memorization without thinking (memorize-use-forget) 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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    #14443 - 04/23/08 08:38 AM Re: Math intuition, math without books [Re: Kriston]
    doodlebug Offline
    Member

    Registered: 10/04/06
    Posts: 433
    Loc: Illinios
    Kriston: I think you'll like that group. My son much prefers "playing" with math than "learning" math - so the websites and fun stuff have been so helpful to us to keep his love of math alive.

    I find this a very interesting topic and one that most people just don't get. My son, at only 6 years of age, told me NOT to tell him answers or tell him how to do something. He said "I like to find out by myself." He was referring to math, but of course it applies to his intrinsic motivation for learning. He is an exploratory learner. Reading through this thread I think I've realized that this is what he HATES about school. He doesn't get to explore. He doesn't get to experience those "ah-ah!" moments where the lightbulb goes on. YKWIM - most of us here are probably SOOOOO motivated by those moments.

    I think that's what you are talking about, Kriston. Don't think of it as TEACHING math. Think of it as GUIDING the child to learning math. I believe this is something that I do just naturally so very often during the day. I bet most or all of us here do. You grab those teachable moments. You just put the information out there and the child absorbs it and learns.

    My 19 year old used to lament having to learn math and say "what good is it anyway? I'm never going to use this stuff." I started to point out what I called "functional math." It's now a big joke between he and I when we notice math in everyday life - one of us will smile and say "there's that functional math again." Kriston, I don't think this is a scary thing. This is really nothing more than baking cookies with your little one and letting them explore the relationships of fractions with the measuring cups. Then explore multiplication by putting the cookies on the sheets, Then explore division by figuring out how many cookies each person in the family gets. Then do doubling the recipe and try to determine how much more flour you are going to need.

    Unschooling really isn't so hard. I bet you actually do it already.

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    #14445 - 04/23/08 08:44 AM Re: Math intuition, math without books [Re: Kriston]
    Dazed&Confuzed Offline
    Member

    Registered: 04/05/08
    Posts: 1815
    Kriston - have you looked at livingmath.net? There is also a curriculum sold by Rainbow Resources called Calculus for Young People or something close to that. Now, it didn't get good reviews but it could be people were looking for more structure. There are also a few Teaching co. Courses on the Beauty of Math etc.

    I understand what you are saying and I see some of that with my son. I was hoping that Livingmath.net would more easily bring in that unit study approach. I do own it but w/ public school goin so badly, I've not gotten into it. It did fit nicely w/ our history but school takes up so much time.

    As far as math facts, the program I use, Rightstart math, doesn't drill them in the beginning. The facts are internalized through playing games. I found that approach was better for my boy. Later it does introduce using math facts work sheets to get quick at writing them but I like the game approach of figuring them out each time using sound strategies.

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

    Registered: 09/19/07
    Posts: 6145
    Loc: Midwest
    Well, it's not new-age mumbo-jumbo... wink

    I think it is problem solving. And you hit Dr. F's nail on the head here, I think:

    Originally Posted By: Ania
    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 calc--accessed through physics or history and at a level DS6 can understand--will 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!
    _________________________
    Kriston

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

    I really like what you said about thinking of it as guiding, not teaching, Debbie. I'll probably re-read 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! grin
    _________________________
    Kriston

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    #14452 - 04/23/08 09:24 AM Re: Math intuition, math without books [Re: Kriston]
    Ania Offline
    Member

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


    Edited by Ania (04/23/08 09:24 AM)

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

    Registered: 12/14/07
    Posts: 902
    Originally Posted By: Ania
    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.
    _________________________
    LMom

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    #14456 - 04/23/08 09:32 AM Re: Math intuition, math without books [Re: Kriston]
    kcab Offline
    Member

    Registered: 10/02/07
    Posts: 1603
    Loc: Sparta, apparently
    Sounds like fun! Last time that DS5 helped me bake he was doing a riff on the cookie sheet multiplication and division that Debbie mentioned. Baking has lots of opportunities for experimentation and math learning, and chemistry - isn't there a Kitchen Table Math book somewhere? Plus, then there is eating the results.

    Pendulums, that could be fun too - I'm thinking of swings. You could make a playground for Playmobil people (or something) and then do experiments with different lengths of swing ropes. Try changing the mass of the object too, how swing is started. Take notes...

    I think, stuff does get committed to memory, but there is more than one way to do that. If you're doing experiments, or playing, then you might be more engaged and likely to remember the result.
    _________________________
    kcab

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