One-Sixteenth

My plan, but don't tell Myrtle, is to drop off my daughter at her house and let her do the whole teaching this really hard math. I can maybe teach her kids poker or something. We'll work something out:)

Seriously though, I feel both frustrated (because of my own lack of math education) and hopeful when I go to her blog and when I get properly motivated, I'm going to seek out the things she plans on using with her kids.

Geometry.

Thanks! I think we are on the same track with RS and SM. My older will be doing SM after RS next year but I have added in the word problems this year.

Now, I've got to follow you on the proofs....I had proofs in school...it isn't in the curriculum anymore? Although I don't remember much about it!

Hi Lynx,

This may sound simple but answers to any problem at all should be written in a "math sentence" form that includes units e.g. X = 27 desks.
Then it is easier to write an English sentence answer.

I taught science to high school students for many years and most of them were pretty good at arithmatic but they couldn't for the life of them explain what they were talking about. It was like the numbers were some abstract form that did not relate to the real-world problem solving they did in science. Thus they couldn't "see" that an answer they gave was nonsense. Thus, students would do some "chicken scratches" and give me the answer that a density was less than zero! By making them write the answer as D = 1 gram/cc, they could see that they were talking about something real and better self-correct.

I am using Saxon math with my son and I don't know if it goes into proofs, but the problems are written in such a way as to encourage the student to write an equation and then solve for X and respond with real numbers. (This is not algebra yet, either).

I loved proofs. I thought they were much more interesting than plugging numbers into formulas, a la Algebra. A math in which there were multiple ways to come to a correct answer, where you could think outside the box (heh) and be creative, was so much more appealing than solving for x with only one right way to do it.

If I could make any suggestion, it would be logic puzzles, word puzzles and brainteasers (the aforementioned outside the box puzzle for one). They can help you learn to follow a logical thought process, while simultaneously learning creative solutions - essential traits for theoretical proofs.

The class Aaron is in (I think I've told you about it?) teaches four years of high school math in two years, to young kids who don't fit in to a regular math class. He is in his fourth semester, taking pre-calc, at age 13. The curriculum has been heavy on theory and proofs, and best as I can tell, the only real difference (besides the pace of the class and the fabulous instructors) is that they use college texts, not K-12 texts. I can look up the titles if you're interested, but his algebra book was something like "College Algebra" and the same with geometry/trig and pre-calc. These kids are all beyond plugging in numbers and the books they use have been heavy on the theory and proofs (which, interestingly enough, Aaron is not particularly fond of!).

Thanks for all the thoughts. I'm taking notes. I'm also stalking Myrtle around the internet bookmarking every book she mentions.

Angie, I'd love to know what texts Aaron is using. The titles are usually all the same, so authors would be helpful. He is so lucky to have such a great opportunity, and it sounds fantastic.

Penny, I didn't do proofs of any kind in school. And even if it is standard fare, we still don't do as much of that kind of thinking in American classrooms as many other countries do. We don't do it as well, either.

I have not dropped in for a while: I'm the theoretical physicist turned stay-at-home homeschooling dad.

I have some good news and some bad news. The good news is that I have known some extremely good engineers who never really got the idea of proofs. For most engineering work, you need math through introductory calculus and elementary differential equations, and those courses nowadays do not seem to heavily involve proofs.

So, even for engineers, from a purely vocational perspective, proofs may not matter that much.

Of course, if you intend to become a mathematician, proofs will be the center of your life! And, a field such as theoretical physics or some of the more challenging areas of engineering also requires a decent familiarity with proofs.

However, since we agree that education is not just vocational training, I think proofs do matter. The Greeks’ creation of the axiomatic method, especially as applied to geometry, is one of the great achievements of human culture. And most of the achievements of modern mathematics are inaccessible to anyone who cannot deal with proofs.

So, I have several concrete suggestions. First, proofs are not a goal in themselves: they serve a purpose. The purpose of a proof is to take some ideas that one is quite certain of (the axioms) and use them to validate some idea one is not completely sure of. One of the big problems kids have with proofs is understanding why they cannot be completely certain of something if they check out a couple of examples. Why bother with proofs? Why not just trust your gut?

The answer of course is that your gut is often wrong.

I think it is worth teaching that fact to young kids.

For example, as we all know, a Mobius strip is one-sided and has some other counter-intuitive properties. An early grade-school kid can check these out.

Another simple example shows why you cannot just take the commutative law for granted. Take two identical cereal boxes in the same starting position. Rotate one 90 degrees vertically and then 90 degrees around the east-west axis. Do the same to the other box, but do the east-west rotation first, then the rotation around the vertical axis.

The boxes end up in different positions.

Rotations in 3-D do not commute. (This is connected to the fact that multiplication of matrices does not usually commute.)

This helps show that the commutative law is not trivial: it really matters that multiplication of integers, fractions, etc. is commutative.

Another simple example is cancellation of factors. Take regular clock arithmetic. 2 * 2 = 4, on a clock of course. But 2* 8 = 4 also on a clock (since 16 o’clock is 4 o’clock). So, 2*2 = 2* 8. If cancellation worked, 2 would equal 8. (The cause of this is that there are “non-trivial divisors of zero” on a clock: 2*6 = 0, since 12 and 0 are the same on a clock face.)

There are actually some simple things with infinite numbers (so-called transfinite ordinals, which are simpler than they sound) that show that even the commutative law of addition can fail fairly easily for weird sorts of numbers.

So, if kids can learn why the common commutative laws, etc. are not as obvious as they seem, you can then start explaining how those laws make our ordinary arithmetic algorithms work, which is a basic introduction to the idea of proofs.

Anyway, that’s what I’ve been trying: my kids are at third/fourth grade level in Singapore Math.

Of course, I endorse Jeff’s pithy comment (“Geometry’). Many schools are abandoning the axiomatic method for geometry teaching, which seems to me a cultural disaster. While it would be possible to introduce the axiomatic method for number theory or abstract algebra, I doubt kids are ready for either before they are ready for geometry.

I think the important thing is to lay the framework, so that when they are ready for geometry and then higher math, they can understand why proofs are needed in the first place.

My own approach to this is still a “work-in-progress.” I hope others have some good suggestions.

Liping Ma’s “Knowing and Teaching Elementary Mathematics” is great for explaining how a conceptually-based approach and a mastery-based approach to grade-school math should really be the same thing. And, Irving Adler’s “Giant Golden Book of Mathematics” (long out-of print but used copies available on the Web) is wonderful for giving upper-grade-school-level kids an introduction to real math: Adler had a gift for explaining complex concepts in very simple terms.

Dave