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    #14408 - 04/22/08 08:30 PM Math intuition, math without books
    Kriston Offline

    Registered: 09/19/07
    Posts: 6145
    Loc: Midwest
    Please bear with me a bit on this post. It has two goals, which may or may not coincide neatly:

    1) I'm trying to digest a presentation my local POGS put on by a brilliant mathematician/inventor/teacher tonight so I can apply his methods for teaching math to my DS6.5, and

    2) I'm trying to share his insights with anyone here who is interested.

    I may have to muddle through 1) before I can share much that's useful to meet goal 2). smile
    • 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.

    I'm both excited and terrified by this notion. It lines up very neatly with what I'm seeing and feeling about my own experience of teaching math to my son--my fear that my approach is killing math for him, my dissatisfaction with "book-learning" (even the good curricula!) for math, etc. This gives me a totally different way to attack math, and a very child-directed way at that. It fits neatly with a unit study sort of approach, which I've been considering for next year, since the wholly hands-on tack lends itself to combining math with science, history, sports, etc.


    I'm still more-or-less terrified of math, and I'm not at all sure that I can teach calculus to a 7yo, even if it's very basic and I don't call it calculus! It's going to take a whole lot more effort on my part to make this work, and I'm not sure I have it in me to do it even a little bit well. It's a whole lot easier to cover a workbook than it is to really teach math.

    Of course, writing the situation out like that makes the choice obvious, doesn't it? I can teach math in a way that makes perfect sense to me and that my DS has literally been asking for, or I can be lazy and probably kill his love of math. Well, that's a no-brainer!

    I foresee a long summer of planning for me...

    #14416 - 04/23/08 04:38 AM Re: Math intuition, math without books [Re: Dottie]
    OHGrandma Offline

    Registered: 01/05/08
    Posts: 830
    What that speaker is proposing sounds like the selling points of "Investigations", the math program used in the elementary schools in our district, here is a website for Investigations.

    Take a look at what E.D.Hirsch says about this learning method, full story here.
    Dr. E.D. Hirsch, a brilliant educator and best selling author on the subject of education has written THE book on math curriculum. His book, "The Schools We Need and Why We Don't Have Them" is a landmark work taking every major educational study into account and listing the findings of each one. When viewed as a whole, Dr. Hirsch says, "experience has shown that 'discovery learning' is the least effective method in the teacher's repertory." (page 246). Discovery learning is the same thing as Investigations Math where the students "discover" their own solutions to problems. Opposite this line of thinking is where teachers actually teach students. Dr. Hirsch says "Saxon math's approach is reasonably close to what research is telling us about how students learn--much closer, than are the progressive methods advocated by the National Council of Teachers of Mathematics." (page 131)

    As Dottie says "there has to be balance between the two extremes!". Our teachers added 2 or 3 minute speed drills with little rewards to encourage the kids to memorize the facts after the kids understand the concept of adding groups of items. That has helped bring up school scores on the achievement testing.

    #14427 - 04/23/08 06:49 AM Re: Math intuition, math without books [Re: OHGrandma]
    squirt Offline

    Registered: 03/31/08
    Posts: 323
    Loc: Back in Texas, alas!
    Kriston - it sounds like an interesting theory. The hole I see in it is that *we* have to understand the math concepts deeply enough to apply them to everyday life. And when to spiral around to the concept again. Personally, it's been over 20 years since I did much (any?) calculus. So, for example, could I really seize the opportunity of having a popsicle to teach how to figure the volume? Well, I remember the concept and I remember it requires some kind of formula and it is calculus but it would take a lot of study on my part to learn all of that (most likely from a book). So, unless math is intuitive to one, it would make more sense to teach it to both parent and child from a book together and then incorporate it into other subjects and activities.

    Of course, I'm not an expert by any means. Just my initial thought. I think it's worth more consideration. And, by the way, I minored in math in college - really stuck, huh?

    #14428 - 04/23/08 06:53 AM Re: Math intuition, math without books [Re: OHGrandma]
    Kriston Offline

    Registered: 09/19/07
    Posts: 6145
    Loc: Midwest
    I'm a litle surprised at the response. I'm interested in why you both react so negatively. So far, I'm not persuaded that the guy's approach to math is wrong based on your arguments, though I appreciate the critical thought behind them.

    I think we have to start with the fact that I'm not worried about my son picking up arithmetic, but I am very worried about his love of learning math being destroyed. That's vital to understanding why this approach appeals to me. If down the road I find that DS6 is struggling with arithmetic, my view might change. But for now, I do not place a high premium on arithmetic. All year I have felt that we were only doing arithmetic and not math, and that has been a big problem for DS6, who is bored with it, and therefore for me.

    I think DS6 has the potential to be a real math whiz, and my biggest fear about homeschooling is that I will kill his love of math. (Not that school wouldn't have killed it, too, but now I'm responsible if it happens.) I believed--even before Dr. F said it--that arithmetic is not math, even though in elementary school, that's all that kids get. This is a way to avoid that problem, and it's the first potential solution I've seen.

    And for the record, Dr. F is not saying kids shouldn't learn arithmetic, just that arithmetic shouldn't enjoy the privileged place it has in the elementary school curriculum and that kids shouldn't be required to memorize it. He made the analogy that math is like music--large and varied--whereas arithmetic is just one subsection of math. Something like arithmetic is like playing the flute: it's nice, but it's not all music (though this is a flawed comparison--and he would agree--because everyone must eventually know how to do arithmetic while not every musician must learn to play the flute).

    He's the head of the math department at a local GT school, and he's really well-respected in our area. He was nominated for a Nobel Prize for one of his discoveries. Seriously, this is not some fly-by-night guy, and he's not selling anything, beyond the notion of helping GT kids to learn to love math. His talk was free (he didn't charge the group or those in attendance a cent) at a nonprofit GT group I belong to, and he has no product or book to sell. He recommended no particular curriculum, even when asked. The big point in his favor to me is that his approach matches my own experience with teaching math, not to mention the way I felt about doing (but never learning) math myself.

    DS6 will have to do arithmetic to solve the science and engineering problems. This is just a way to make the arithmetic *part* of math instead of *all* of math. It seems to me that it is a way to go deeper in the early grades, at the time when it is hard to go deeper. How many times have we seen the question asked, "How do you go deeper and not just faster in the early grades?" Well, here's a way! And it's experiential to boot, which is how kids learn *anything* best.

    If a kid learns his times tables by figuring volume or charting the swing of a pendulum, why is that any worse than memorizing the times tables? I don't see the problem there. Am I missing something?

    I don't really care about DS6's scores on achievement tests. To me, that doesn't mean he's learned math, only that he's learned what they're testing, which is going to be arithmetic. One of the reasons we're homeschooling is so that we don't have to teach to the test.

    And BTW OHG, I've heard lousy things about Saxon for GT kids. Generally, HSing parents of GT kids hate it because it is so repetitive and arithmeticky. At least all the ones I know. No offense to you or Dr. Hirsch (whose books I own!), but that quote just confirms for me that math teachers value arithmetic too highly and math understanding too little.

    I did look at the Investigations website you linked, but it looks nothing like what Dr. F was describing as a curriculum. It looks a lot like "Everyday Math," and we don't love that. Unless I missed something (and I may have), in 3rd grade the only experiments they do--if you can call them that--are reading a thermometer and a clock and some stuff with volume. It looks very book-bound and not tied to science and history and the child's interests as Dr. F was suggesting it should be.

    I certainly can't see why "many serious mathematicians" would be dismayed by this approach, Dottie. Can you tell me more? What's the fear? It's not boxed, which is actually one of the scariest things about it to me! It's *all* on me. No laziness allowed, no falling back on simple arithmetic workbooks. All REAL math! All MY responsibility to understand and teach. And the guy IS a serious mathematician! Not to mention an excellent teacher. We all know one when we see one, and this guy is a natural. So what's the problem? Can you give me more?

    Maybe I just didn't describe this very well...As I said, I'm still trying to wrap my head around it. It is certainly different than the norm, but given that I was not ecstatic with any of the usual suspects for math instruction that I've investigated, this appeals to me on all levels. It fits nicely wih unit studies, which I'm leaning toward for next year. It seems logical to me, and it also feels right for where DS6 is right now.

    So where's the harm in it? Is the danger solely that DS6 won't ever learn to multiply and won't ace his achievement tests in 2nd grade? (Things I'm not worried about.) Or is there something else I should worry about before I dive in?

    #14429 - 04/23/08 06:59 AM Re: Math intuition, math without books [Re: Kriston]
    Kriston Offline

    Registered: 09/19/07
    Posts: 6145
    Loc: Midwest
    We crossposted, Squirt. Sorry!

    Yes, that's my big worry--that I don't know math well enough to apply it appropriately. I know DS6 and I will be learning together. That's a given for this English major! But I think that can work. It certainly seems better than just doing arrithmetic problems ad nauseaum, unconnected to anything else as we have been doing.

    I think your're right that I'll have to adapt his method, since I'm not the natural that he is at math. But even so, I think that makes good sense and could be a lot more successful than what we've been doing. I just feel like I'm replicating all the mistakes teachers made with me. I need to try SOMETHING else! And I don't think a different book is the answer. This is radical, but it seems to be a step in the right direction, even if I have to modify it to my own limited math abilities.

    Maybe what I need is for Dr. F to teach me math so I can understand it and can teach it to DS6! wink

    #14435 - 04/23/08 07:30 AM Re: Math intuition, math without books [Re: Kriston]
    elh0706 Offline

    Registered: 03/08/07
    Posts: 353
    Loc: PA
    I am in awe of you smile

    I am a Poli/Sci major and there is no way I could pick up true math understanding to a degree that I could teach math other than as a process based on problems. (Considering I failed calculus once and dropped it 2 other times in college even with the aid of my math major husband...)

    The funny thing is DS9 takes those workbook problems and then tears them apart and comes up with really creative approaches to the problems and the solutions. Sometimes he is right according to the book. Other times he is way off base. However, when he is doing it, his eyes shine and his words run over themselves he is so excited to PLAY with math.

    #14436 - 04/23/08 07:39 AM Re: Math intuition, math without books [Re: Kriston]
    squirt Offline

    Registered: 03/31/08
    Posts: 323
    Loc: Back in Texas, alas!
    "Maybe what I need is for Dr. F to teach me math so I can understand it and can teach it to DS6!" (I still can't do the box quote thing.)

    I have a friend who's father-in-law is a rocket scientist - literally - who worked on the top secret rocket programs in the 60's and 70's. A true mathemetician and physicist. Anyway, when my friend's kids have questions about math, they get sent to Grandad. Also, grandad takes the time to teach the kids in everyday situations about physics and math. Not in a structured way but in a "oh, look, there's a leaf falling from the tree, how fast do you think it is falling?" kind of way.

    Maybe what you need is someone like that, maybe retired or maybe a whiz grad student, who will mentor your son after you have taught him the basics. You could pay him/her or maybe barter. I've thought about proposing this to my friend's FIL. You could keep a log of ideas he wants to explore and then have the mentor start with those ideas and see where they go from that. So YOU don't have to learn calculus in depth, just the concepts and then someone else works with him in depth.

    And, I understand not wanting to crush his desire to learn math. I also think it addresses the "how do you go deeper at age 7" syndrome with which I struggle.

    Don't know if this makes any sense but thought I'd toss it out there. I think you might be on to something that works for you but it might need to be tweaked to take less of the pressure off of you.

    I'm off to tour a private school - still trying to decide what to do with J for next year. This one is a Montessori so I don't have wild hopes, but we'll see. (oops, didn't mean to get off subject.)

    #14437 - 04/23/08 07:45 AM Re: Math intuition, math without books [Re: elh0706]
    doodlebug Offline

    Registered: 10/04/06
    Posts: 433
    Loc: Illinios
    There's a yahoo group called "mathing off" which supports the concept of unschooling math.

    Tons of great links and resources to take a more laid back approach with math. Might be something of interest if you want to try that approach. I've gotten lots of great tips.

    #14438 - 04/23/08 07:51 AM Re: Math intuition, math without books [Re: Kriston]
    kcab Offline

    Registered: 10/02/07
    Posts: 1603
    Loc: Sparta, apparently
    I like this approach. I can see how it might be difficult to implement for a large group and requires more from the teacher. But - that could be good, right? Don't you want to exercise your brain too, Kriston? (For that matter, why are you terrified of math? ) And I'm thinking too that I should search out that article on Finland's schools, this is reminding me of that for some reason ... will see if connects later.

    So here's some thoughts:
    - I think the point is to start with teaching ideas and to think, rather than teach formulas. Eventually you'll need to introduce notation, but the ideas come first as motivation for the notation. So, in the case of volume and popsicle, think about it with the child and figure things out. Try to think of different ways of figuring the volume out, of comparing it to other volumes. You can bring in thermo too .... compare frozen volume to liquid. What happens with different mixtures.... Could be a very rich topic and you could go in lots of different directions.

    - this method intrinsically provides motivation for math. One of the things that drives my DD(newly)11 crazy is word problems with helpless people in them - she'll rail, "Why should I figure out how many apples Mr. X has, why can't he figure it out himself!" or something of the sort. I think the point behind word problems is to show how computation might come up in the real world, but it would be better for it to actually arise in the real world.

    - Might take longer to achieve math whizziness, but the understanding might be greater.

    - Math questions come up everywhere, if you look for them. Part of the challenge to the teacher here is in recognizing the opportunities.

    - Prior to 1st grade, DH and I thought our DD had good math intuition. She wasn't way out there, but she had good conceptual understanding. However, having to do only 3 minutes a day of computation drill homework (plus whatever they did at school) killed any interest in math that she had. She's been gaining it back recently, but it took teaser introduction of more advanced topics before she would even consider caring about math again. Which is just to say that I think it's valid to fear that you could diminish your DS's love of math.

    Well, whatever. What do I know anyway! smile I do think it sounds like a more intense and challenging way to teach. I bet it could feel like taking away the safety net. Of course, that could be more exciting too.

    #14439 - 04/23/08 07:53 AM Re: Math intuition, math without books [Re: Kriston]
    cym Offline

    Registered: 05/01/06
    Posts: 865
    Loc: southwest
    Originally Posted By: Kriston
    Memorization kills intuition, and should be banned.

    Very interesting! I would have loved to hear his talk. I have secretly subscribed to this idea for a while now, but it's so exciting to hear an expert state it.

    The whole reason I pursued DYS for DS now 9 (then just turned 6) was that he "terrified" me with his math intuition. He could calculate problems significantly faster than I could. I was convinced he was this rare talent that needed nurturing. After a couple more years in elementary school working through 6th-7th grade math curricula, I saw he could no longer calculate as fast or as accurately because he was "bogged down" by the methodical, multi-step process taught and his compulsion to show his work, crooked columns making errors and he didn't automatically check to see if the answer made sense. I was very disturbed. Last summer we did Mental Math in hopes of regaining his intuition.

    I pressed for advanced math (algebra 1) this year and the teachers denied us. So we signed him up for AoPS and he's really enjoyed it. I find the instructor and classmates exude the "love of math" mentality and are not "bogged down" by the process. Many times the instructor asks them to "Guess! Don't think about it--first thing that comes to mind" type stuff. I feel that continued enrollment in advanced math classes, even if it's just for exposure to concepts, instructor/mentor, and excited peer group, rather than for credit and grades, is the needed solution for DS9.

    I have also seen that my oldest DS (13) cannot problem solve like DS9 unless it resembled recipe from his textbook. I really believe that DS13's Alg 2 class has not taught him anything new except the discipline of doing assigned problems every night. His math reasoning has not been strengthened--and may have been weakened by fall-asleep mode for a year.

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