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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!

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.

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 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.

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 ) 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

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!

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.

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


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"?

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.

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.