Math intuition, math without books

Posted by: Kriston

Math intuition, math without books - 04/22/08 08:30 PM

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.

OTOH...

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...
Posted by: OHGrandma

Re: Math intuition, math without books - 04/23/08 04:38 AM

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.
excerpt:
Quote:
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.
Posted by: squirt

Re: Math intuition, math without books - 04/23/08 06:49 AM

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?
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 06:53 AM

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?
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 06:59 AM

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
Posted by: elh0706

Re: Math intuition, math without books - 04/23/08 07:30 AM

Kriston,
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.
Posted by: squirt

Re: Math intuition, math without books - 04/23/08 07:39 AM

"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.)
Posted by: doodlebug

Re: Math intuition, math without books - 04/23/08 07:45 AM

There's a yahoo group called "mathing off" which supports the concept of unschooling math. http://groups.yahoo.com/group/MathingOff/

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.
Posted by: kcab

Re: Math intuition, math without books - 04/23/08 07:51 AM

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.
Posted by: cym

Re: Math intuition, math without books - 04/23/08 07:53 AM

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.



Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 08:20 AM

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.
Posted by: Ania

Re: Math intuition, math without books - 04/23/08 08:33 AM

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.




Posted by: doodlebug

Re: Math intuition, math without books - 04/23/08 08:38 AM

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.
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/23/08 08:44 AM

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.
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 08:58 AM

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!
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 09:16 AM

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
Posted by: Ania

Re: Math intuition, math without books - 04/23/08 09:24 AM

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 :-)
Posted by: LMom

Re: Math intuition, math without books - 04/23/08 09:27 AM

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.
Posted by: LMom

Re: Math intuition, math without books - 04/23/08 09:29 AM

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.
Posted by: kcab

Re: Math intuition, math without books - 04/23/08 09:32 AM

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.
Posted by: pinkpanther

Re: Math intuition, math without books - 04/23/08 09:32 AM

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 good--teach 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.
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 09:36 AM

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 how-tos 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?
Posted by: OHGrandma

Re: Math intuition, math without books - 04/23/08 09:40 AM

Kriston, I don't think I expressed myself well in my first post due to my disappointment in our schools math curriculum, "Investigations in Number, Data, and Space". The person who spoke about it so highly is our son's gifted teacher, and her description sounded a good deal like you described.

In a homeschooling situation I can see it working extremely well. In a group setting I see a good bit of frustration. "Investigations" does use a lot of real life examples to discover the properties of arithmetic, geometry, and some beginning algebra notation. Where my frustration is coming from is how it's implemented in a group setting. For example, some of the third graders are still grasping how you calculate the perimeter & area of their kitchen table by measuring it with their ruler -- that really was a problem on his homework recently. GS8 is ready to calculate the perimeter & area of our pastures, multiply the estimated forage yield by the area, divide that by the estimated forage use per head of cattle(which he gets by multiplying 3% of an average estimated weight of animal), and estimate how we should subdivide the pastures into paddocks so the forage is removed in a proper amount in approximately 3 days, then move the cattle to the next paddock so the grazed paddock can regrow.

I had no concerns about "drill & kill" when making him take a couple days to memorize his multiplication table but I have a big concern about how many more times he's going to be asked to measure the length and width of an object before everyone else in the room 'gets it'. Right now I'm looking at these 'real life experiences' as being "drill & kill" for GS8. Unless there's a real application, like calculating grazing capacity, GS8's going to be working on his standup comic routine in class.

As a method for introducing new concepts, what Dr F recommended is great. Just understand it can be "drill & kill" when used repetively, too!


Posted by: LMom

Re: Math intuition, math without books - 04/23/08 09:50 AM

Originally Posted By: Kriston
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.


Do you mean that whoever can figure out 100/8 in their head or just whoever does it the same way like long division on the paper?

Just asking since I assume pretty much everybody can figure out 100/8 in their head (whatever way they do it), including DS5 and I don't think it kills his math intuition by any means.
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/23/08 10:23 AM

OK I haven't read this article in it's entirity - let me say that upfront - but it sounds like it might be relevant to this topic. Lockhart's Lament

Posted by: calizephyr

Re: Math intuition, math without books - 04/23/08 10:43 AM

I liked what the math guy said, but it's difficult to actually do, I think.
One thing I can say is there are a lot of kids I tutor who are decent at math, but have no idea what they are actually doing (in Calculus).
I do think the basic tenets of Calculus can be taught to interested children (but like others said, it may have to be someone who really understands this stuff in depth!). Calculus is about rates of change. One example off the top of my head, if one draws a curve on graph paper, if many rectangular boxes are drawn under the curve, we can show we have an estimate for the area under the curve, which is the integral.
Posted by: kcab

Re: Math intuition, math without books - 04/23/08 10:54 AM

Originally Posted By: Dazed&Confuzed
OK I haven't read this article in it's entirity - let me say that upfront - but it sounds like it might be relevant to this topic. Lockhart's Lament

and then there's the "think" method (as espoused by Professor Harold Hill in Music Man). eek

Sorry all, I've only read the first page, but couldn't resist!
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 12:14 PM

As a mathematician, I just wanted to post some random thoughts after reading this thread.

I decided to major in math because it was the easiest subject for me--nothing had to be memorized! I am not good at rote memorization but I could "see" how to derive formulas from first principles. I did learn my math facts in grade school but it wasn't exactly by rote. It was more that I repeatedly thought about them until the answers were clear. For example, 7+8=15 because 8+8=16 and 7 is one less than 8. This process builds upon itself--i.e. at some point I had to become convinced that 8+8=16 before I could use that to conclude that 7+8=15. After a while, I felt that I could skip the reasoning part and just go straight to the answer.

I remember having "aha" moments like understanding long division as repeated subtraction and how to find the area of a triangle. I was not discovering these things, however. The teacher was presenting that material. Still, there was a moment where I "got" it.

In order to get a concept, a student has to make a habit of understanding each step in the process. If we try to replace understanding with rote learning it introduces a gap in the chain of reasoning. Rote learning can be a useful tool for increasing math fluency. But it should never be a substitute for "getting" a concept.

Lately, I have been teaching third graders fractions. Their teachers had told them that a fraction represents a certain number of parts out of the total number of parts. This is true, but it doesn't seem to be intuitive to third graders. My approach is to look at math like a language to learn. When we cut something into pieces we give those pieces a name depending on how many pieces we made. (You would be surprised at how many kids are not making this connection.) I.e. if we cut something into 4 equal pieces we call each piece a "fourth". In math language we write that as "1/4". "A" means "1" and "/4" represents "fourth". Now if we have 3 such pieces we say we have "three fourths". That is written "3/4". This lays down the foundation for understanding how to add fractions. Now that we really "get" what a fraction means, the only thing that makes sense is to count up how many of each kind of piece we have by adding the numerators. If we need to cut some of the pieces into smaller pieces so that all the pieces are the same size, it makes sense to do that.

Now many of your kids are already beyond this kind of thing. My point (I think I have one smile ) is that verbal reasoning can be used to understand math as a language for representing real problems. This is SO important for kids to understand. The way math is taught in school you would think that the math and verbal domains were completely seperate.

I think that GT kids have the ability to intuit this connection. Mathy kids don't need to have things translated for them this way. Exposing kids like this to math is like immersing them in a foreign language. They will soak it up. Exposing them to calculus at a young age is like letting them read books with big words in them. They may not understand them right away but that's ok.

My son seems to be a mathy kid and he LOVES that Descartes' Cove program. He can't solve the problems on his own but it is still a good teaching tool because it is exposing him to what is possible. My own dad did stuff like that with me, like showing me how to use his slide rule, teaching me about logarithms and exponents and teaching me Newton's method for approximating square roots. We also did set theory and Venn diagrams which I loved. No, I did not completely get the stuff he was teaching me until I was older but I think that early exposure was very valuable and helped to lay down pathways in my mind for future understanding. It also whetted my appetite for more math! I teach a mathlab at my kids' school where my main goal is to expose the kids to stuff beyond arithmetic. This kind of enrichment is beneficial to kids at all levels--without it, math just seems like an arithmetic wasteland to them and they lose interest.

Cathy
Posted by: pinkpanther

Re: Math intuition, math without books - 04/23/08 12:19 PM

Cathy,
As a fellow mathematician, I just want to thank you for your comments. I completely agree, and I never had to memorize anything either.

I also agree that math is very verbal, and I think that's why my DD9 is so good at it. That's how I teach, too!
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/23/08 12:25 PM

Cathy - can I sign up for your class? Pretty please??

I was actually researching VCI and PRI and what do those indices mean in the real world. I was surprised to read somewhere that VCI correlates more with algebraic thinking and PRI with geometric abilities ie that math and verbal domains are linked as you stated.

There is a math website I came across and it explained fractions just the way you did and also addressed the importance of doing so. Coincidentally, w/out knowing why or the significance of it, it's how I taught my boys fractions at a young age.
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 12:35 PM

Originally Posted By: Dazed&Confuzed
Cathy - can I sign up for your class? Pretty please??

I was actually researching VCI and PRI and what does those indices mean in the real world. I was surprised to read somewhere that VCI correlates more with algebraic thinking and PRI with geometric abilities ie that math and verbal domains are linked as you stated.


Thanks, Dazey! It's so fun to teach people who are excited about learning (that's why I like third graders smile .) Teaching a bunch of jaded college students taking a required course that they hate is torture.

Quote:

There is a math website I came across and it explained fractions just the way you did and also addressed the importance of doing so. Coincidentally, w/out knowing why or the significance of it, it's how I taught my boys fractions at a young age.


I'm not sure how significant it is, but it sure makes sense! Otherwise, kids just get a look of panic on their faces when they see two numbers with a weird little line between them. What the heck is that about? It must be hard. If nothing else, exposing kids to higher math will get them accustomed to seeing different kinds of notation. Just like we expose toddlers to the alphabet without expecting them to read right away. Why do we (as a culture) feel like we have to keep math a secret? Why do we send the message that it's "too hard" or "too confusing"?
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 12:38 PM

Originally Posted By: Dottie
Excellent thoughts, but if my calc student doesn't buckle down and just memorize a few basic trig points, she is going to lose too much time deriving them on her upcoming test!


That's exactly why I said that rote learning can be a valuable tool for increasing fluency. And fluency is important, not just for tests, but to allow a person to glide over those concepts that have already been reasoned out. If you can't do that, it overtaxes your working memory when you're trying to reason out a new concept that builds on the old ones. Once the pathways are laid down in the brain by that initial "aha", they need to be reinforced by practice.
Posted by: kimck

Re: Math intuition, math without books - 04/23/08 12:39 PM

Cathy - that was wonderfully stated, coming from another mathematician.

I also never actually memorized anything. I can memorize things. And then immediately forget it 10 minutes after the test. How you would derive your math facts really hit home for me. I couldn't comfortably jump through teachers hoops until I really understood WHY we were doing something a certain way.

Anyway - thanks for your post.

Posted by: st pauli girl

Re: Math intuition, math without books - 04/23/08 12:45 PM

Cathy - thanks for your example! i printed it out for when we get to fractions. This so makes sense to me.

I am one of those people who was told that boys were good at math, and girls weren't, and i suppose i believed it. I always got good math grades, but only took the courses that were required. I was thrilled when my college accepted logic courses for math requirements! I do not even have the foggiest idea what calculus is. blush

But, I do know how to ask for help when needed, and DH is very good at math. So DS4 will be OK. wink
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 12:51 PM

Calculus sounds a lot more daunting that it is. Basically, it is a way of cutting curved things into little pieces that have practically straight sides so that we can add them up just like regular old geometric shapes. It's a clever concept which was developed as way to calculate phenomena encountered in physics. It has its own notation--you may have seen integral signs which look like elongated esses. They are for adding things up. The esses stand for "sums". It's actually a very visually based kind of math.

ETA: I wish more people would take logic!
Posted by: Ania

Re: Math intuition, math without books - 04/23/08 12:53 PM

Great thoughts, Cathy!

Can I ask you some questions about Descartes' Cove game, or has this been discussed in a different post already?
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 01:02 PM

Originally Posted By: Ania
Great thoughts, Cathy!

Can I ask you some questions about Descartes' Cove game, or has this been discussed in a different post already?


Sure. We haven't played all of it yet. DS is working on the easiest level (Measurement) which is about unit conversions, decimals and metric system. There is other stuff thrown in, though like formulas for volumes and areas of different shapes. I checked out the Algebra level and there were some reasonable tricky problems on there. Good for stretching your brain! smile
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 01:05 PM

Originally Posted By: Cathy A
I did learn my math facts in grade school but it wasn't exactly by rote. It was more that I repeatedly thought about them until the answers were clear. For example, 7+8=15 because 8+8=16 and 7 is one less than 8. This process builds upon itself--i.e. at some point I had to become convinced that 8+8=16 before I could use that to conclude that 7+8=15. After a while, I felt that I could skip the reasoning part and just go straight to the answer.


Exactly! This is exactly what Dr. F was saying! If you get it, then the doing is easy and you learn math facts just from daily use. If you don't get it, then you're forced to rely on rote memorization for the doing, and that's painful for a GT kid. Better to start with the concepts and let the knowledge of the math facts come naturally from there.

Originally Posted By: Cathy A
My point (I think I have one smile ) is that verbal reasoning can be used to understand math as a language for representing real problems. This is SO important for kids to understand. The way math is taught in school you would think that the math and verbal domains were completely seperate.

I think that GT kids have the ability to intuit this connection. Mathy kids don't need to have things translated for them this way. Exposing kids like this to math is like immersing them in a foreign language. They will soak it up. Exposing them to calculus at a young age is like letting them read books with big words in them. They may not understand them right away but that's ok.


EXACTLY! This is something Dr. F said, too, almost verbatim! That's why he believes ALL GT kids are naturals at math, even if they're highly verbal. Because language and math are not separate entities. Our brains don't divide math and language that way.

Oh, Cathy, you're explaining this SOOOOOOO much better than I did! Thanks! laugh

Originally Posted By: Cathy A
This kind of enrichment is beneficial to kids at all levels--without it, math just seems like an arithmetic wasteland to them and they lose interest.


Again, right on the money. Dr. F said we should, in effect, aim high with these kids. If they don't get it all at age 6, so what? They've got years more to pick up what they missed on the first exposure! But showing them what's out there, what math REALLY is--and it ain't workbooks!--captivates them, shows them that math is beautiful. We don't teach kids to read by diagramming sentences, so why would we try to teach math by starting with arithmetic. Teach them to love it first, then the nuts-and-bolts will come.

You rock, Cathy! grin You made that a whole lot clearer than I did!
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 01:10 PM

OHG: I see what you're saying. Thanks for the clarification. I think you're absolutely right that anything can get old if done to death. Good point. It's not like math facts have the corner on that market! LOL!

And I'm definitely not suggesting this style of math for a general classroom. I have no idea how that would play, and frankly, I don't really have to know. Dr. F is doing it with GT kids in a GT school, so there's that, I guess. But for my part, all I'm worried about is my own progeny!

I thought it might work for some of the afterschoolers out there, perhaps in some modified form.

...Or not. You know me--I'm not highly evangelical about anything I'm doing! wink

At least it's made for an interesting conversation! grin
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 01:20 PM

Originally Posted By: CFK
Granted, I can kind of be a laissez-faire parent, but I don't really understand the concern here. Kriston, or anyone, if your son is gifted, to the point of being accepted into DYS, is ahead of age/grade mates already in math, doesn't want or like to do math right now, and you are homeschooling so there is no need to be concerned about standardized testing or acceptance into academic programs, then.. why do math at all? Isn't the major benefit of homeschooling the ability to follow your child's lead?


Well, because I don't think that it's *math* he dislikes; I think it's *my approach to math* that he has disliked. He's fascinated by engineering principles and has always had a mind for patterns and mazes that astounds me. Dropping math altogether doesn't seem like it solves anything, and may in fact mean we'd be missing some opportunities.

An entirely new approach, however, may be just the thing to light his fire again, a fire that he's always had until he hit school age and workbooks--either the public school's workbooks or mine.

I can always back off if it looks like that's what's needed. I can appreciate some deschooling time as much as any homeschooler. But before I just give up and hope that he comes to math on his own, I'd rather try to SOLVE the problem, a problem that I feel is mine, not his.

Does that make sense?

He's also a lot less far ahead in math than he is in his other subjects. He's reading at the 7th+ grade level, but he's only doing 3rd grade math. I have no trouble with asynchronous development, but I think this has less to do with his abilities and more to do with my teaching. I feel like I'm letting him down, and this offers a different way to give him what he needs. It seems worth a try.

And BTW, geometry, which I've approached in a far more conceptual and far less arithmeticky manner than our previous math work led us to, has been pretty successful. He's having fun with it. Geometry is one of my data points suggesting going for higher-level, conceptual math rather than arithmetic is going to work better with him.
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 01:27 PM

Ooh! DOK! I wish we had such a club for HSers my son's age!

It may be time to start creating the things I want instead of just wishing they were around...
Posted by: st pauli girl

Re: Math intuition, math without books - 04/23/08 01:28 PM

Thanks for the simple definition, Cathy! I actually liked geometry -maybe I'll like calculus too.

I suppose now's a good time to also confess that for years as a child I thought engineers drove trains. (Not entirely my fault - my grandpa was a train engineer).
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 01:32 PM

Originally Posted By: LMom
Originally Posted By: Kriston
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.


Do you mean that whoever can figure out 100/8 in their head or just whoever does it the same way like long division on the paper?


It's about the method. Long division is one way to get the answer, but if you basically write the problem out in long division form in your head, then you're not using intuition because long division is a method taught to you, not one you would just come to on your own.

Yes, pretty much everyone can figure out 100/8 in their heads, I hope! smile The problem itself has nothing to do with intuition; it's just a way to check how you think about math.
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 02:40 PM

Originally Posted By: Dazed&Confuzed
OK I haven't read this article in it's entirity - let me say that upfront - but it sounds like it might be relevant to this topic. Lockhart's Lament


I just read all 25 pages. Thanks for sharing that, Dazey!

He makes many good points, I think, some the same as Dr. F and some that are more extreme and a wee bit scary...

But the focus of the article on the art and beauty of math, the DIY nature of math that current math instruction is lacking makes some sense to me. I like the notion of using the history of math to teach math. I think that gives me a way into math that is interesting and useful. DS6 loves history, and I find it less intimidating to go at it from that direction, so that might be very helpful to me.

The article also makes me think I need to go back to the geometry book and be sure I'm not making all the mistakes he named in my use of geometry with DS6! Yikes!
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 03:41 PM

Originally Posted By: Kriston
If you don't get it, then you're forced to rely on rote memorization for the doing, and that's painful for a GT kid.


It's not just painful, rote memorization takes a whole lot more effort (at least for me!) and is less "sticky" than understanding something. I think it's because when you just memorize you may learn a fact but it's just floating there in your brain, not connected to anything. It can be hard to retrieve that floating information. If I know how to reason something out it's like a trail of breadcrumbs to follow back to the memory. Each time I repeat the reasoning process it becomes easier to find and follow the breadcrumbs. If you think about something enough, the trail becomes like a superhighway leading straight to the answer.

Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 03:58 PM

Oh, Cathy! <swoon>

You made that whole argument against rote memorization that I was struggling to make so beautifully. the breadcrumbs analogy is ideal!

Okay, now I'm not only thinking that I'm having trouble with math, I'm having trouble with language... Wanna write my book for me? wink

Posted by: OHGrandma

Re: Math intuition, math without books - 04/23/08 04:10 PM

Originally Posted By: Cathy A
Originally Posted By: Kriston
If you don't get it, then you're forced to rely on rote memorization for the doing, and that's painful for a GT kid.


It's not just painful, rote memorization takes a whole lot more effort (at least for me!) and is less "sticky" than understanding something. I think it's because when you just memorize you may learn a fact but it's just floating there in your brain, not connected to anything. It can be hard to retrieve that floating information. If I know how to reason something out it's like a trail of breadcrumbs to follow back to the memory. Each time I repeat the reasoning process it becomes easier to find and follow the breadcrumbs. If you think about something enough, the trail becomes like a superhighway leading straight to the answer.



Haha, this reminds me of taking beginning Accounting in college. I did very well in Accounting but memorized very little except for terms. Just before a test I would review one note which reminded me when cash was a credit or debit. As soon as I sat down I would write on the test when it was a credit. Everything else was a snap because I could relate it to cash one way or another. Other students struggled to memorize individual transactions and never could grasp the relationships easily.
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 04:10 PM

Originally Posted By: Kriston
Oh, Cathy! <swoon>



Oh, Kriston! <melting> Do you know how great it is to have people be interested in and appreciate my pet theories smile ?

Ok, maybe that's enough of our little lovefest blush
Posted by: LMom

Re: Math intuition, math without books - 04/23/08 04:11 PM

Originally Posted By: Kriston

It's about the method. Long division is one way to get the answer, but if you basically write the problem out in long division form in your head, then you're not using intuition because long division is a method taught to you, not one you would just come to on your own.

Yes, pretty much everyone can figure out 100/8 in their heads, I hope! smile The problem itself has nothing to do with intuition; it's just a way to check how you think about math.


Good, I was getting worried that he didn't expect people to calculate 100/8 in their heads smile Honestly it must be a really hard work to do it in your head the same way like you do it on the paper. I am not going to try to figure out how DS5 does it. I am afraid he could loose me somewhere in the middle grin

Originally Posted By: Cathy A

It's not just painful, rote memorization takes a whole lot more effort (at least for me!) and is less "sticky" than understanding something. I think it's because when you just memorize you may learn a fact but it's just floating there in your brain, not connected to anything. It can be hard to retrieve that floating information. If I know how to reason something out it's like a trail of breadcrumbs to follow back to the memory. Each time I repeat the reasoning process it becomes easier to find and follow the breadcrumbs. If you think about something enough, the trail becomes like a superhighway leading straight to the answer.


I am the same way. I took heaps of math and computer classes and exams and it was such a relieve since I just had to get it so to speak. I didn't have to memorize it, as long as it made sense I was fine.

I hate when I have to memorize things, I don't usually see a point in doing that and the results show it.
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 04:15 PM

Originally Posted By: Cathy A
Originally Posted By: Kriston
Oh, Cathy! <swoon>



Oh, Kriston! <melting> Do you know how great it is to have people be interested in and appreciate my pet theories smile ?

Ok, maybe that's enough of our little lovefest blush


LOL! Well, I'm really thinking that maybe I can just get YOU to teach me how to teach math! Everything you've written could come right out of Dr. F's talk, and if I can make you melt just by telling you how brilliant you and your theories are, then maybe I can buy you off more easily than I can buy him.

Yes, I think I'm saying that you might be a cheap (math) date, but I mean that in the nicest way! grin
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 04:39 PM

I'm happy to help. I've never heard of Dr. F but this thread caught my eye. I have done some thinking about math pedagogy just for my own amusement, though. Mostly, I inflict my theories on my mom, who says that she has some kind of inability to learn algebra.

My mom is a very smart lady and she was a whiz at arithmetic in school. Geometry proofs were intuitive for her and she is a logical thinker. Somehow, she says algebra just doesn't stick in her brain. She can listen to someone explain a problem and it makes perfect sense at the time but she can't reproduce it on her own. I have always wondered if it is just a matter of finding the right way to explain it to her or if she truly has some kind of math disability. I often wish I could do experiments on her <rubbing hands together>....
Posted by: EandCmom

Re: Math intuition, math without books - 04/23/08 04:45 PM

LOL Cathy! Most people I know who were good at geometry weren't so good at algebra and people who were good at algebra weren't as good at geometry. I personally LOVED algebra and although I did well in geometry I hated proofs with a passion. I saw absolutely no point in them. Anyone else out here love one and disliked the other or is that just a phenomenon I've noticed in the few people I've discussed this with???
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 05:02 PM

It's supposed to be pretty common, I think.

Doesn't one of the books about being visual-spatial mention that liking geometry is one sign of many that you're a VS learner, while liking algebra is one sign of many that you're auditory-sequential?

I don't have that backwards, do I? Proofs always seemed pretty sequential to me, but the figures are obviously VS...
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 05:11 PM

Geometry has never been a passion for me, although I liked doing the proofs and writing them out neatly and logically. It seemed satisfying. Algebra just seems like a means to do calculus, which I love.

Abstract algebra (rings, fields, etc) I can do in a mechanical sort of way but it doesn't seem intuitive to me at all because all I can think about is blobs with arrows pointing to them. Algebraic geometry is similarly opaque smile My degree is in applied math and the way I think about math is mostly visual. Numerical analysis makes a lot of sense to me. By the way, that's something that can be understood with relatively little algebra skills. You need to learn about infinite series which are usually intriguing to kids and full of interesting patterns.
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/23/08 06:03 PM

EandCmom: I read that VCI (on the WISCIV) correlates with ability in algebra while PRI correlates with ability in geometry. It goes along w/ what Cathy said earlier about the verbal side of math. I don't know if that's accurate or not but could explain the cases you were referring to.
Posted by: squirt

Re: Math intuition, math without books - 04/23/08 06:13 PM

Originally Posted By: EandCmom
LOL Cathy! Most people I know who were good at geometry weren't so good at algebra and people who were good at algebra weren't as good at geometry. I personally LOVED algebra and although I did well in geometry I hated proofs with a passion. I saw absolutely no point in them. Anyone else out here love one and disliked the other or is that just a phenomenon I've noticed in the few people I've discussed this with???


I loved Algebra but saw absolutely no reason for the proofs in geometry. I mean, really, we know what the angles are and which is which and how they relate, so what's the point?

After having read all of this thread, I like the theories of teaching math but I have a question. If my guy LIKES doing workbooks, is there any harm in that? Will it stifle his love of learning or will he eventually get bored with them and want to move on to other things. I don't actually teach him out of them - just provide them.
Posted by: Kriston

Re: Math intuition, math without books - 04/23/08 06:55 PM

Nope, squirt. I think that if he's happy, let him go to it! Child-directed is what it's all about, I think. Especially at a young age.

Mine's just bored and antsy.

And Cathy, can you tell me more about numerical analysis. What is it? Can you point me to any resources for it? It sounds like something my infinity- and pattern-loving boy would totally adore!!! Thanks for the tip!
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/23/08 07:11 PM

YEs I'd like to hear/read more about numerical analysis as well.
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 07:35 PM

A quick and dirty description is on Wikipedia http://en.wikipedia.org/wiki/Numerical_analysis

Here's a description of Newton's method for finding roots of equations. http://en.wikipedia.org/wiki/Newton%27s_method
If you apply this to complex polynomials you can generate really cool fractal pictures like the one at the bottom of the page.

I love fractals, too. Check out the Mandelbrot set:

http://en.wikipedia.org/wiki/Mandelbrot_set
Posted by: snowgirl

Re: Math intuition, math without books - 04/23/08 07:40 PM

Barging in here, Kriston, you have the VSL angle correct, that Silverman says auditory-sequential learners are better at arithmetic and algebra while VSLs are better at geometry except for the proofs, which are sequential. Also keep in mind that the auditory-sequential person is supposedly better at rote memorization, while the VSL learns best by seeing relationships. And remember, the right brain is the place for intuition smile

You may have seen this article before, but I'll throw it out here for you just in case any of the points seem relevant to your current thought process: http://www.visualspatial.org/Articles/algebra.pdf

thanks for all the food for thought!
smile
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 07:45 PM

Originally Posted By: squirt
After having read all of this thread, I like the theories of teaching math but I have a question. If my guy LIKES doing workbooks, is there any harm in that? Will it stifle his love of learning or will he eventually get bored with them and want to move on to other things. I don't actually teach him out of them - just provide them.


I don't see how it could be harmful. It can be a kind of play and it can be satisfying to figure things out and fill in the blanks. Ask him about what's going on in his mind when he's figuring things out. It's a good exercise to verbalize your thinking process and you may be surprised at what he tells you. My son comes up with all sorts of ways to do problems. Sometimes it's a way of looking at things that I haven't thought of.
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 07:48 PM

Originally Posted By: snowgirl
... the auditory-sequential person is supposedly better at rote memorization...


My theory on this is that they are better able to retrieve the floating info by using auditory cues (like whispering to yourself, "6,8,48.") It gives them a hook to hang that info on and find it again. Eventually, they can just hear it in their minds.
Posted by: Cathy A

Re: Math intuition, math without books - 04/23/08 07:52 PM

BTW, my son is sitting on the living room floor doing a math workbook right now. It's way too easy for him but he gets satisfaction out of it and I suppose it's reinforcing those thought pathways... Oh heck, at least he's not watching the TV smile
Posted by: OHGrandma

Re: Math intuition, math without books - 04/24/08 04:25 AM

Originally Posted By: Cathy A
BTW, my son is sitting on the living room floor doing a math workbook right now. It's way too easy for him but he gets satisfaction out of it and I suppose it's reinforcing those thought pathways... Oh heck, at least he's not watching the TV smile


I think the satisfaction of doing arithmetic is why Sudoku is so popular. Some people like crossword puzzles, some like math problems.
Posted by: Kriston

Re: Math intuition, math without books - 04/24/08 06:21 AM

Originally Posted By: Cathy A
A quick and dirty description is on Wikipedia http://en.wikipedia.org/wiki/Numerical_analysis

Here's a description of Newton's method for finding roots of equations. http://en.wikipedia.org/wiki/Newton%27s_method
If you apply this to complex polynomials you can generate really cool fractal pictures like the one at the bottom of the page.

I love fractals, too. Check out the Mandelbrot set:

http://en.wikipedia.org/wiki/Mandelbrot_set


Thanks, Cathy, but oh, man! I read that stuff and I only hear the teachers in "Peanuts": "Wahwah-wah-wah." I think I need "Numerical Analysis for Dummies"! I may be in BIG trouble with this little scheme of mine...Much math to learn!

On the bright side, I spoke with DH about my grand plans last night though (he'd been out of town), and he's all in favor of handling a big math/science project with DS6 on the weekends next year as a matter of course. They'll ask the question and do the experimentation. Then hopefully I can help DS6 with any grunt work to find the answers to his questions. Worst case scenario: he has to ask dad. Not a huge deal.

There are definite benefits to an English major marrying an engineer! Together, we make one person who may actually be capable of effectively homeschooling an HG+ kid! wink
Posted by: AmyEJ

Re: Math intuition, math without books - 04/24/08 07:08 AM

Originally Posted By: Kriston

There are definite benefits to an English major marrying an engineer! Together, we make one person who may actually be capable of effectively homeschooling an HG+ kid! wink


This is what allows me to even entertain the idea of homeschooling in the future! My husband is SO good at math, and I'm SO not! I laugh about it being the subject that drove me to law school. The funny thing is that he's a lawyer too, but for completely different reasons. Anyway, I think he does math much like how Cathy A. has described it (I've loved the posts, by the way!). I've always been amazed at how quickly he can do arithmetic in his head, and finally I asked him about a year ago. He explained that he does it through rounding, usually to 10, then adds or subtracts as needed. I felt so dumb. I'd never even thought about math like that. It just came naturally to him, and I would be bogged down trying to carry my tens the way my teachers had taught me so many years ago. crazy He thought of it as kind of cheating, but I realized how I'd been focusing on something way too literal for my whole life, which is weird because I'm not a black and white thinker. But I think it was how I was taught math and I never thought to look beyond it.

I started disliking math in about the 4th grade when we had those math competitions to fill in the blanks (Math Factors, or something like that) as quickly as possible. Around that time I got it in my head that I wasn't very good at math, and it's always stuck with me. I will say, though, that despite "hating" math, I loved my AP chemistry class, particularly balancing equations. This wasn't math, though, right? wink That was chemistry, something completely different.

I feel a little silly now because at age 34 I'm finally beginning to see how a lot of math "works" and I'm actually delighting in it. As I'm explaining things to DD6, I find that suddenly I'm seeing patterns there that I had never really noticed before. I don't know if I was never taught math that way or if I just blocked it out but I would LOVE to take a class taught by Cathy A.! If I went back in time and learned math her way as a 3rd grader maybe I would have done a little better in my high school calculus class, or at least maybe I wouldn't be relearning it at my age now.

So Kriston: you go, girl! I love that you are exploring this way of learning about math. When you get it figured out maybe you can share it with the rest of us. I'd definitely be interested in that. And I'd love a PM of your notes, if it's not too much trouble.
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/24/08 07:15 AM

AmyEJ - I'm so with you!!! I've used Rightstart to teach my son math b/c I don't care for the Everyday Math he gets at school and he was begging to move ahead in math. I feel like I finally understand. It teaches math the way your DH does it. I explained it to DS this way. I can't remember the exact problem, but it was a problem where you could do it the long way and get the right answer or you could estimate to 10, subtract, and get the answer in 1sec. When I was in school, i was the kid doing it the long, methodical way. The kids who *knew* math, did it the quick and dirty way and were several problems ahead by the time I finished the one problem and had time left over to re-check answers whereas I did not. Teaching math this way, I finally feel like I've been let in on some secret! I too LOVED balancing equations in chemistry!


Kriston: I'd love a PM of your notes as well!
Posted by: Kriston

Re: Math intuition, math without books - 04/24/08 07:21 AM

Originally Posted By: AmyEJ
When you get it figured out maybe you can share it with the rest of us. I'd definitely be interested in that. And I'd love a PM of your notes, if it's not too much trouble.


IF I get it figured out, you mean! crazy Those Wikipedia pages scared me more than a little! Seriously, I had to look up *so* many words. It's more than a little intimidating. I did take calculus, both in high school and in college. I can do this, right?!

I keep reminding myself that that's just the notations and terminology that are throwing me. I don't have to know all that or use all that to teach DS6 that fractals are cool or that problem solving is about finding a fun puzzle of his choosing to figure out. At this stage, I just want to focus on the big picture. If I get bogged down in math the way I learned it, this won't work.

Outside the box...Outside the box...Outside the box...

Anyway, thanks for the moral support, Amy, and I'll send you and Dazey a copy of the guy's talk when I get my notes transferred to this computer.

laugh
Posted by: Lori H.

Re: Math intuition, math without books - 04/24/08 08:13 AM

For my outside-the-box, more verbally gifted than math gifted kid, math will never be his favorite subject. He usually tries to talk his way out of doing it. Yesterday, he told me that when he does math, he gets this strange sensation in the back of his head and it is very unpleasant and the only way he can find relief is to stop doing math and he thinks it could even be some rare sensory disorder and that it might be interesting to see a brain scan of his brain while he is solving a math problem mentally. I keep telling him he is only prolonging the agony, that it is better to just do it and get it over with, but he loves to argue.

But around the time he turned 5, he came up with his own way of doing subtraction using negative numbers because he found it easier to do it this way. Would this be considered intuition? A kid at acting class had told him about negative numbers and he just thought about it and came up with his own way of doing things. I found that I could do subtraction mentally a lot easier his way, so he actually taught me something. When we told the Kindergarten teacher about this, I think she just thought it was kind of weird. Teachers at this school think kids should only be allowed to solve problems one way- the way the book shows.

At almost 10, he is having no trouble doing the math in an 8th grade workbook and a pre-algebra workbook and a book with word problems. But he often dislikes the way the book explains things. For instance, the 8th grade workbook starts out with examples of subtraction and these instructions: "To subtract, start with ones. Rename 1 ten as 10 ones. Continue subtracting from right to left. Rename as necessary."

My son said they should have explained what to do if you have a problem like 1,111 - 7,777. He knows how to do this, but we couldn't find where it showed how to do this on paper anywhere in the book.

I usually have him do one problem the way the school would make him do it, just in case he ever had to go back to school, and the rest of the time he solves problems his own way. It works for him, for now.

But I never took calculus. I will probably need to find a math tutor, hopefully someone who really likes math and can convince my son that it is as fun as science and history and literature and arguing.



Posted by: squirt

Re: Math intuition, math without books - 04/24/08 09:02 AM

Kriston - I'd like your notes as well.

About 6 1/2 years ago (note that son is 6 3/4 YO), I got into a funk and gave about 15 boxes of books to the library. With them were all my college textbooks, which I had been saving for over 15 years. There went Calculus, Diff E, Latin, Logic, everything. Boy do I wish I had those back! Lessons learned: don't throw anything away in the midst of post-partum depression and EVERYTHING is worth keeping! Not that the fist part of the lesson will ever apply again!

Lori - my son figured out negative numbers when the teacher gave him a "2nd grade brain teaser". There were flowers and numbers on each petal and one in the middle. You were supposed to subtract the petal number from the middle number. He did them all backward and got negative numbers, which he didn't know how to notate, so he put "2 but 2 that is less than 0". I explained negative numbers to him and told the teacher about it at a conference as an example of his math skills. She told me "you shouldn't teach him something that we don't learn in 1st grade - that's a 5th grade concept". I read something about "subtraction with renaming" and had no clue what it meant. Now, you've described it for me. Totally counterintuitive to me - unless it's the same as borrowing? Okay, totally rambling now and not contributing.

This has been a very good thread and I'm glad to hear all of the thoughts, ideas, and theories of everyone.
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/24/08 09:08 AM

Yes renaming is the same as borrowing, back in the day.
Posted by: kcab

Re: Math intuition, math without books - 04/24/08 09:13 AM

Originally Posted By: Dazed&Confuzed
Yes renaming is the same as borrowing, back in the day.
Borrowing is the word that made me cry in math for first and second grade. I was glad to find out it had acquired a new name when DD started school.

(Because - "neither a borrower nor a lender be," taken literally.)
Posted by: Kriston

Re: Math intuition, math without books - 04/24/08 09:18 AM

Renaming also makes more sense, I think. The number hasn't changed; you're just renaming what's there.
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/24/08 09:35 AM

In Leping Ma's book on teaching Mathematics and comparing China to the US, she uses the term "decomposing" as in "decomposing a 10" when you don't have enough ones to subtract.
Posted by: kcab

Re: Math intuition, math without books - 04/24/08 09:41 AM

Ha, I like "decomposing" - as in "decomposing into its constituent parts." I just wish I didn't get a picture of roadkill in my brain.
Posted by: questions

Re: Math intuition, math without books - 04/24/08 10:12 AM

Yes, DS and I laugh when the EPGY lecture talks about decomposing. We think about dinner for vultures...
Posted by: Cathy A

Re: Math intuition, math without books - 04/24/08 11:41 AM

Here's something cool (and free smile )!

http://nlvm.usu.edu/

When I explain "borrowing" to kids I say that we take a 10 and break it apart into 1's or we take a 100 and break it apart into 10's, etc. Legos are a great way to demonstrate this.

When I teach subtraction I say things like, "You can make a 10 out of a 3 and a 7. If I take away the 7 part the 3 part is left." So many kids get stuck in counting backwards mode or drawing ten dots and crossing out 7 one at a time mode. They don't think of numbers as chunks. They are only thinking of them in terms of counting.

Kriston, sorry if I freaked you out with the wikipedia stuff. And remember, it's just notation! There are beautiful books about fractals and the Mandelbrot set. Kids love to study the pictures and look for patterns. And, guess what--that's what mathematicians do, too!

Today, with my math class I had them do some experiments on Mobius strips, cutting them in different ways. We also generated a distribution graph by rolling two dice and graphing the results. I try to use math vocabulary when I talk about these things. I say things like "surface" and "topology" and "distribution" and "probability". And I read to them from Penrose the Mathematical Cat.
Posted by: Cathy A

Re: Math intuition, math without books - 04/24/08 12:00 PM

You know, I don't want people to think that I'm against teaching the standard algorithms for doing calculations. It is important for kids to know how to do those things. The standard algorithms are an important tool because they always work, even on problems that don't have quick, elegant solutions. The standard algorithms should be part of every kid's arsenal of knowledge. But these algorithms are calculation tools; they are not the essence of mathematics.

We like poetry because it shows us unexpected and beautiful connections between words, images and emotions. Mathematics can show us unexpected and beautiful connections between numbers, objects and abstractions. I know it sounds corny, but I have always loved that Disney film "Donald in Mathmagic Land." It's from the 50's or 60's I think, and some parts seem a little dated, but it does a good job of communicating the poetry in math and the presence of math in our daily lives.
Posted by: Kriston

Re: Math intuition, math without books - 04/24/08 03:35 PM

Right on, sister! laugh That's what I want DS6 to get from my teaching. I know he'll get the usual algorithms along-and-along. That doesn't worry me. But I worry a lot that the poetry of math will escape him.

I loved that movie, too. How funny! I hadn't thought about that in years...
Posted by: incogneato

Re: Math intuition, math without books - 04/24/08 04:31 PM

I have strong vivid memories of HATING that movie...said the friend with dyscalculia!!!!!!!!!!!!!!!!
hee hee hee
seriously , I think that movie evoked strong reactions of anger when I was about 7. Maybe in school? I can't remember now.
We were made to watch it and I was crying on the inside.

smile
Posted by: Kriston

Re: Math intuition, math without books - 04/24/08 04:36 PM

Huh. Interesting. You don't happen to recall specifically what it was about the movie that prompted this reaction, do you, 'Neato? I'm curious.
Posted by: incogneato

Re: Math intuition, math without books - 04/24/08 04:46 PM

I don't know, Kriston, I really just disliked it intensely. I have a vivid emotional memory and some of the things I remember the best are things that provoked a really negative feeling, like for example, kindergarten.

Posted by: Kriston

Re: Math intuition, math without books - 04/24/08 04:50 PM

LOL! I just snorted at the screen, 'Neato.

It is possible that what I liked about those "edu-tainment" movies is that they were movies instead of regular class. Now that I think about it, I believe we also watched them only on the last day of school. Both may have factored into my enjoyment more than a little... wink
Posted by: incogneato

Re: Math intuition, math without books - 04/24/08 04:56 PM

LOL!!!

I'm so glad I could make you snort!

Wouldn't it be so great if their were a job available where you could be paid only to be sarcastic and glib?
Posted by: Kriston

Re: Math intuition, math without books - 04/24/08 05:05 PM

I think there is. It's called "mom," isn't it?

Oh, wait, we don't get paid...
Posted by: doodlebug

Re: Math intuition, math without books - 04/27/08 05:16 AM

Kriston: Back to the OP:
I know I'm coming back in really late in this thread, but wondered if you'd also searched Hoagies for help with ideas for introducing those more complex topics. I've used a lot of the info on Hoagies for finding websites, books and games that address math without actually teaching it.
http://www.hoagiesgifted.org/math.htm

http://www.hoagiesgifted.org/multiplication.htm

the physics link has lots of ideas that are math related:
http://www.hoagiesgifted.org/physics.htm

and games are a great way to introduce new thinking:
http://www.hoagiesgifted.org/smart_toys.htm

Take your time to surf around Hoagies. It truly is a goldmine.
Posted by: Kriston

Re: Math intuition, math without books - 04/27/08 06:12 AM

Thanks, Debbie! I've spent a lot of time on Hoagies, but you're right that I haven't tried it since I got going on this particular issue.

Great idea! laugh
Posted by: kcab

Re: Math intuition, math without books - 04/27/08 11:09 AM

I wanted to mention how timely this topic was, Kriston. I've been getting hit with posts about math education on my RSS reader all week. I thought this blog post by Math Mom might be of interest.
Posted by: Kriston

Re: Math intuition, math without books - 04/27/08 12:46 PM

It was, kcab. Thanks!
Posted by: kcab

Re: Math intuition, math without books - 04/27/08 03:22 PM

No prob! What I really liked was the idea of specialist math teachers, seems like a really good idea for a district to try. Unfortunately, I suppose it might also be more expensive.
Posted by: Kriston

Re: Math intuition, math without books - 04/27/08 04:10 PM

It does seem a shame to have math taught by people who think it's something to get through. And her point about what a good teacher can do with "Everyday Math" hit home, too.

Maybe the problem is not the "Everyday Math" curriculum, but is the fact that too many elementary ed teachers hate math.
Posted by: Dazed&Confuzed

Re: Math intuition, math without books - 04/27/08 04:22 PM

kcab - thanks for the link to the blog!!! I've been reading for over half an hour and I really need to do laundry. 8-)
Posted by: incogneato

Re: Math intuition, math without books - 04/27/08 07:33 PM

Yes, I bookmarked it, I like it alot too! I sent a link to my husband on peasant multiplication and asked him to check it out for DD8.
Posted by: kcab

Re: Math intuition, math without books - 04/28/08 09:09 AM

smile I'm feeling happy that you guys liked the link! (But I'll try not to get too rampantly emotional, LOL)
Posted by: Kriston

Re: Math intuition, math without books - 04/28/08 09:28 AM

Meh, go for it, kcab. RE all over the place! I'm feeling exceptionally supportive today.

:p
Posted by: Kriston

Re: Math intuition, math without books - 04/29/08 06:16 AM

I stumbled across this free math course from Annenberg Media for high school teachers and adult learners, and I fell in love. It looks incredibly interesting to me, so I thought I'd share.

I plan to use it myself this summer to learn more about the history of math and how to think math instead of "plugging and chugging," and then hopefully next year I can teach DS6 from what I've learned.

http://www.learner.org/channel/courses/mathilluminated/about/
Posted by: kcab

Re: Math intuition, math without books - 04/29/08 07:10 AM

That whole site looks like a great resource! I've bookmarked and am going to check it out. Thanks!
Posted by: Kriston

Re: Math intuition, math without books - 04/29/08 07:19 AM

Glad you like it. smile I'm still exploring the rest of the Annenberg site, but it seems like loads of good stuff!
Posted by: incogneato

Re: Math intuition, math without books - 04/29/08 09:13 AM

Thanks for sharing! I bookmarked it as well!