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Tomorrow I begin Lesson 88 in Saxon Algebra 2.
I won't finish the book until the very end of the summer, which means I will have taken 12 months to get through the thing.
Next I start Saxon Advanced Mathematics, which will take me another year to get through if I'm lucky.
fyi: Saxon publishes four high school books:
Geometry and trigonometry are integrated throughout the first 3 books.
So I'll spend next year working my way through Advanced Mathematics while Christopher, who will be in 8th grade, finishes Math A: algebra and some geometry. (Math A is still a 1 1/2 year course, as far as I can tell, though people keep saying NY is going back to the old algebra 1 - geometry - algebra 2 sequence. His class began Math A in January.)
Freshman year, if he stays on the fully accelerated track which I expect he will, he'll be taking geometry - real geometry, with proofs. Or so I hear. (Is this a separate, souped-up geometry course that's neither Math A nor Math B? Don't know! I get all my info from my neighbor, whose son is a year ahead in school.)
Assuming you aren't hopelessly confused by now,* you may see the problem.
He's catching me.
I'm 2/3 of the way through Saxon Algebra 2, and I have yet to do a proof. Unless there are a lot of proofs in Advanced Mathematics (which there may be - don't know)** I'm going to be starting geometry-with-proofs the same time Chris does.
That's not good.
I need a royal road.
Does anyone know whether Tammi Pelli's Proofs Workbook might work?
At 64 pages, it's the right length.
(large image)
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**update: proofs in Advanced Mathematics
Just checked the scope and sequence for Advanced Mathematics:
Proofs
Elements of Proofs
Understand basic logic and reasoning
State the contrapositives of conditional statements
State the converses and inverses of conditional
statements
Do proof outlines
Do formal proofs
Theorems
Prove the chord-tangent theorem
Prove theorems about secants and tangents
Prove theorems about chord products
Prove the Pythagorean theorem
Prove similarity of triangles
Prove the law of sines
Prove that equal angles imply proportional sides
questions about Math A
Math A Regents exams (archived)
Math A Toolkit
Home Instruction Schools Regents Exam Review
sample Saxon Math lessons
Saxon scope and sequence
* I'm hopelessly confused, but I'm used to it
10 comments:
I think you may be overly paranoid about the next group of teachers, because (assuming NY has policies similar to the states I know about) secondary ed has much different requirements for certification than elementary or middle school does. Perhaps the best thing for your son at that point (9th grade) is to start teaching you! There is lots of evidence that teaching you will help him more than you teaching him. That is the whole point behind study groups and collaborative learning at the college level.
Besides, proofs are not all that different from what you have been doing all along. Any mathematical argument that leads to an answer for an algebra problem is, in one sense, a proof. What is being added is a new group of methods, such as proof by contradiction, and the basic rules of logic. Those two "State the ..." skills in your list are the foundation of all the rest.
In addition, I tried to e-mail this to you, Catherine, but the address I found via your blogger info bounced back.
You write often about fractions. I stumbled on this organization at Michigan State when I went there to look up someone's contact info and saw a front-page story about their work. Take a look at the PROM/SE research reports
http://www.promse.msu.edu/research_results/PROMSE_research_report.asp
specifically the May 2006 report on a crucial window for learning fractions. Carry on the fight.
I also got a link to a Seattle newspaper column back in April via a mailing list my wife is on ...
http://seattletimes.nwsource.com/html/opinion/2003674945_sundaymath22.html
You will not be surprised to see TERC listed as one of the usual suspects so you may have seen the article.
I am teaching me - I've been teaching me for 3 years - or were you talking about something different?
hmmm... I wonder why my address bounced back.
It should be cijohn @ verizon.net
"Perhaps the best thing for your son at that point (9th grade) is to start teaching you! "
I think he has a point. I'll tell you a story about a Chinese student.
He grew up in a single-parent household. His mother had only 3-yr or so elementary education. From the first day of school, his mother told him that she couldn't help him with his homework, and they couldn't afford tutors either. She told him to teach her everything he learned at school. She said if he could make her understand, then she would know that he'd learned it.
So they did this all the way through high school, the son went to QingHua university -- China's MIT.
Teach someone else is the best way to truly learn something. "Math Coach" said so too.
Where is this advice in Math Coach?
I'm not finding it....
Here's the opening to Chapter 5 Teaching Tips:
"Over the years, I have taught my children almost all the math in the school curriculum from kindergarten through precalculus. From such instruction, and their own studying, they were able to skip to higher grade-levels in math in school and win numerous math contests."
There's nothing about kids teaching each other or their parents in the Teaching Tips chapter (that I can see).
In "Ideas for Better Schools" he says, "almost all of the math and scientific knowledge students should learn must be taught directly by teachers."
I'm pretty sure Wickelgren didn't spend a lot of time having his kids teach him!
It's true, though, that teaching other people what you know radically changes what you know and how you know it.
Russian Math, in almost every chapter, asks students to explain concepts - concepts that have already been explained well by the authors.
On page 41:
He was talking about encouraging your child to help tutor peers or younger children:
---
Teaching requires the teacher to rethink the information he or she already knows, cementing the knowledge in the brain. This same process also spawns new insight about the material...
Tutoring can also boost your child's self-esteem and verbal skill....
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I guess I am assuming that his teacher teaches him the math in the class, and he teaches you at home.
No question this is true for me - true of writing, too.
Teaching other people also exposes the gaps in your knowledge.
Writing is brutal that way.
I once spent WEEKS trying to summarize a study by Richard Davidson. Absolutely could not do it. I appealed for help to John (Ratey); he couldn't help me, either.
It turned out that there was an inconsistency in his data that was tripping me up as a writer trying to write a logical summary.
I finally called him and said, "Is there an inconsistency in your data?"
He said there was!
He had contradictory data that he hadn't explained in the text due to the strict "genre" requirements of a scientific article. (At least, that's my understanding of why he didn't or couldn't do it.)
He and his colleagues had some ideas about why his data had turned out the way it had, but these ideas were purely speculative.
Since I didn't know, going in, that the data was inconsistent I had spent a very long time assuming I simply didn't understand the study.
I would have had no idea that inconsistency was there in the study if I hadn't tried to summarize it in writing.
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