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I was just sitting at the picnic table outside the kitchen, going over C's math homework, when I realized: we did it.
Christopher is taking algebra in the 8th grade.
I started this whole thing, back in June 2004, with that goal in mind: algebra in 8th grade. (You may have to hit refresh a couple of times.)
And now here we are.
39 comments:
congratulations!
Congratulations are certainly in order.
Wonderful! Well, you both stuck to it.
The children on the Trailblazers book cover still look a little scary to me. I'm not sure why, maybe it’s all the colors.
Reading the bit about the nursing students who washed out because they didn’t understand fractions reminded me why I do not give up with my daughter. She loves animals, and talks about working as a veterinarian or a marine biologist (her dream job is performing with the whales at Sea World). Her dreams may change a hundred times before she becomes an adult, but I never want fractions to be a roadblock to her aspirations.
Congratulations! I think the story itself would make for an interesting book: a middle-aged mom (re)learns math so that she can keep her bright (but not well taught) son on the path to Algebra. Along the way, she finds an on-line community that is willing to support her in this epic journey.
That's a wonderful story. I even envision a movie a la Atlas Shrugged.
To keep with the algebra theme, I'd like to post a d = rt problem and ask for a step-by-step solution:
C. walked to Alfie School of Excellence with Major Qualifications at an average speed of 3 mph and ran back in horror along the same route at 5 mph. If his total traveling time was one hour, what was the total number of miles in the round trip?
Silly math game:
http://www.learnenglish.org.uk/games/magic-gopher-central.swf
There is a different d=rt equation for each direction. I vaguely remember thinking (way back when) that you could use only one DRT equation for each path, or something like that. I don't have a sure-fire way to overcome that kind of thinking. The path and distance each way is the same, but the rates and times are different, so you need two different equations. I mention this because lots of things seem wrong when you are learning algebra.
My favorite approach is to let the algebra do the thinking for you.
You could say that going to school would be
d1 = r1*t1
and coming home from school would be
d2 = r2*t2
1 is going to school and 2 is coming home.
I have two equations and 6 unknowns, so I have my work cut out for me. Well, I know the rates each way - that's easy - so the equations become:
d1 = 3mi/hr * t1
d2 = 5mi/hr * t2
This is better. I have two equations and 4 unknowns.
What else do I know? I know that d1 = d2, so I will just call it 'd', to give me:
d = 3mi/hr * t1
d = 5mi/hr * t2
[I could keep the d1 = d2 equation, but I know that it's easier to combine them now.]
I'm getting close - 2 equations and 3 unknowns. I need to either eliminate one unknown or get another equation using the three variables.
What piece of data haven't I used? I haven't used the time for the total round trip: 1 hour. That gives me:
t1 + t2 = 1 hour
Now I have 3 equations and 3 unknowns and am ready to solve it.
t2 = 1 - t1
so
d = 5(1 - t1)
substitute for 'd' to get:
3t1 = 5(1 - t1)
t1 = 5/8 hour
t3 = 3/8 hour
Plug back in to find 'd' and then double it to get the round-trip distance.
By the way, I could have solved the problem by just thinking or using ratios. This is the sort of problem that teachers love to give students before they know anything about algebra. I don't like it.
I just wanted to show that with proper algebra skills, you (almost) don't have to think. And, thinking alone doesn't work when the problems get more complicated. Call it my motto. "Algebra requires less thinking." That's the point.
Before you know it Catherine, your son will be 12th grade, solving improper integrals, complex (in both senses of the word) differential equations, Taylor Series and be doing multivariable calculus.
All because he could do algebra I in eighth grade.
I am trying to get my sister (younger by 15 months) to take the AP Calc (AB) exam this year. I hate the Integrated Algebra stuff that was taught to her in 9th grade. She was supposed to be taking precal her 10th year, not her 11th!
So I have started tutoring her on top of everything a senior must do.
Being a parent is one thing, but my mother doesn't know much about calculus.
I NEED A RESOURCE FOR STUDENTS TRYING TO TUTOR THEIR SIBLINGS IN AN AP COURSE WHEN SAID SIBLING IS DEEMED BY THE SCHOOL NOT READY FOR SAID COURSE.
Also, preferably, a 5 in the exam when it comes in May.
The problem that I while I can help people solve problems I don't know how to teach someone an entire course. I'm sort of making my sister do it against her will. With parental support of course, but it's not like I have the authority to ground her if she doesn't do the stuff I give her.
For the best AP calculus stuff--go to APcentral.collegeboard.com. YOu can register as a teacher (they don't check) and download the acorn syllable. The Barron's AP calculus book is a good source of practice problems. The best wooks (best and D&S) are only available through bulk purchase).
Get yous sister familiar with graphing calculator and practice solving problems. NO two AP problems are ever the same. In order to do well, you really have to know how to think.
Congrats on Son's algebra in 8th!
SteveH,
Wow!
This is an amazing tour de force or perhaps tour de logique.
A method involving fewer steps could be:
t = d/r
1 = d/3 + d/5
15 = 5d + 3d
8d = 15
d = 15/8
Then double.
She loves animals, and talks about working as a veterinarian or a marine biologist (her dream job is performing with the whales at Sea World). Her dreams may change a hundred times before she becomes an adult, but I never want fractions to be a roadblock to her aspirations.
YES
This is what schools simply don't see (parents, too, really).
The whole system seems to focus solely on the child as he or she is right now. This minute.
The people around me -- this is a universal, not a criticism of a particular person or group of persons (i.e. "teachers" or "parents") .... the people around me universally assume, thinking about a particular child, that "he isn't going into math."
I do it myself.
I will think, "C. isn't going into math" or "C. isn't going to be a mathematician."
You simply can't think that way, although apparently it's natural to do so.
We can't predict children's futures, and when we DO predict, we determine the future!
My favorite late-bloomer story came from a dad here, who said that he is still friends with 2 other kids he went to school with for years.
One of those kids, always a good student but not especially interested in math, suddenly upped and became an econometrician as college & grad student. Apparently he is now internationally known, something like that.
Every time the various childhood friends get together to reminisce, they invariably ask themselves how this kid turned into a world-renowned econometrician.
One answer to that question has got to be that he had a sound K-12 math education regardless of the fact that he didn't look like he was headed toward a math-related career.
My sister "gatekept" herself out of business school because she didn't have the map.
She absolutely should have earned an MBA; she's a natural born businessperson.
She earned an M.A. in leisure studies instead. (And did well in property management....but you get my point.)
I couldn't do college math. Period.
Ed, on the other hand, coming out of his Levittown schools, could do engineering calculus at Princeton (with great difficulty, but he did it; I've told the story...)
My child is going to be prepared to take college-level math classes when he gets to college.
Period.
I'm going to repeat my oft-told conversation with the math chair here.
When C. was in 6th grade, and we were having very, very serious problems with the teaching, I said to her, in a friendly, help-seeking voice, "C. needs to be able to take math in college. That's our family goal."
She said, "He needs to take math to graduate from high school."
That was the end of her interest, the final comment she had to make.
"He needs to take math to graduate high school."
She expressed open disdain for our family educational goals.
Before you know it Catherine, your son will be 12th grade, solving improper integrals, complex (in both senses of the word) differential equations, Taylor Series and be doing multivariable calculus.
Maybe, maybe not.
The math chair is the calculus teacher.
Before you know it Catherine, your son will be 12th grade, solving improper integrals, complex (in both senses of the word) differential equations, Taylor Series and be doing multivariable calculus.
But thank you for saying so!
Where there's a will, there's a way.
He's going to be prepared to take calculus; if he can take it in high school, great.
If not, he'll be prepared to take it in college.
I think the story itself would make for an interesting book: a middle-aged mom (re)learns math so that she can keep her bright (but not well taught) son on the path to Algebra. Along the way, she finds an on-line community that is willing to support her in this epic journey.
Yes!
Actually, it's funny you say this, because I've started organizing all the old posts -- I've begun to create an annotated index to everything.
I don't know what sparked me to do this....it just struck me that it was "time."
I have no idea whether ktm will ever become a book, or whether it should be a story in a book --- but it's possible that it should.
Of course, if it does become a book, we'll have the middle school to thank.
The 4-5 school was completely supportive. (I'm going to put up a short, sweet post on this...)
It's been the middle school that has opposed us and thrown up obstacles every step of the way.
The middle school and the math chair, who teaches at the high school.
Going over the old posts.....
wow
The amount of conflict we've had with our school over the simple question of TEACHING MY KID SOME MATH is mind-boggling.
And, of course, that strife has led to all kinds of trouble over other issues in the district -- an ongoing process, believe me.
Taking a trip back memory lane doesn't make the saga seem any less surreal and crazy than it seemed living through the whole thing.
C. walked to Alfie School of Excellence with Major Qualifications at an average speed of 3 mph and ran back in horror along the same route at 5 mph. If his total traveling time was one hour, what was the total number of miles in the round trip?
lolllllll
PLUS the insane thing is.....here we are.
How did it benefit my school & district to fight us instead of doing everything in its power to make sure C. did make it to algebra in 8th grade?
The answer is: it didn't benefit the school and district at all.
It hurt.
They haven't learned their lesson, either.
Can't remember if I've posted this, but the problem now is that the first wave of Trailblazers kids is hitting the middle school.
WHAT DID THE PRINCIPAL SPEND LAST YEAR TRYING TO DO IN PREPARATION?
HE SPENT LAST SCHOOL YEAR TRYING TO CLOSE DOWN THE ACCELERATED 6TH GRADE CLASS.
He was blocked in this effort.
I NEED A RESOURCE FOR STUDENTS TRYING TO TUTOR THEIR SIBLINGS IN AN AP COURSE WHEN SAID SIBLING IS DEEMED BY THE SCHOOL NOT READY FOR SAID COURSE.
Also, preferably, a 5 in the exam when it comes in May.
The problem that I while I can help people solve problems I don't know how to teach someone an entire course. I'm sort of making my sister do it against her will. With parental support of course, but it's not like I have the authority to ground her if she doesn't do the stuff I give her.
now that is an undertaking
WHOA
I wonder how many sibling-tutors there are out there?????
There've got to be zillions.
The Glencoe Algebra reviews include one from a college student talking about his brother's lousy math textbook & what he's doing to remediate.
LAST BUT NOT LEAST:
COULDN'T HAVE DONE IT WITHOUT YOU GUYS!!!!
WOO-HOO!!!
wow: (pdf file):
Perhaps the most important finding is that parental involvement plays a critical role in students’ advanced
mathematics course taking.
The article defines "parent involvement" as parents talking to their kids about course selection.
Little do they know.
Little do they know.
No kidding.
Our little Trailblazer First Wavers hit the middle school, also. The district responded by restructuring the math departent to better "align" with the grade schools. Accelerated 6th grade pre-algebra has gone missing. Can't imagine why.
Many congratulations, and know that you are helping those of us behind you -- I am far more prepared for finding the right school for my kids because of reading of all of your experiences.
In the category of "no one knows percents" -- I was at my local children's museum yesterday, and the computer in the gift shop went down. I had to talk the counter staffer through how to apply a 10% discount and then put back the sales tax, and then explain why she should have guessed that the first price she quoted me was wrong (because it was bigger than the original price, and our sales tax isn't *that* high).
Percent-sense, anyone?
-m
When I was younger (i.e. in primary five) I used to be fascinated with lists of different products with a constant sum. (I don't know if anyone ever was similarly fascinated.)
The thought was, "what happens that makes 2x8 smaller than 3x7? Or why 6x4 is bigger than 3x7? Or 5x5 bigger than 6x4?"
It would then occur to me that the optimum (when the product was the largest) was when both factors were equal, because after 5x5 you went into 6x4 again.
Then I would think about the change in the product for every change in the factor. This was tricky (from a youngster's point of view) because the change in one factor was dependent on the other in order to get a constant sum. In changing from 2x8 to 3x7, one factor increases and the other decreases. I would try to find a way to describe this pattern.
So this stayed a while in my mind for some years until I had the idea to graph the relationship in Excel, and enter two lists A and B filled with values that added to a constant sum (nicely handled by the "fill column" feature.
I had a third column for the product.
A thought occurred to me. Why stop at whole numbers? I mean, what happens when you switch from 7x3 to 6.99 x 3.01? I ended up graphing in intervals of 0.001, as well as going into negative ranges (12x-2).
Then I had a fourth column where I measured the rate of change in the product as A and B increased and decreased, where I measured the change in the values for A and B.
Imagine my surprise when Excel spat out a ... parabola! I was like, "whoa." (Namely it felt rather profoundly eerie to me given we had just done some done some work on conic sections.)
I also had the surprise that the rate of change in the product was a line! But the relationship didn't seem that obvious before! Spurred by this, I struggled to find a way to rationalise the relationships that were brought before me.
I mean, it seems so obvious now. This was some years ago, and I had no idea that the term I was looking for (a formula that would describe the rate of change in my desired product, which I had not realised was a function) was "derivative".
It was also coolly eerie (in a good sort of way) because all the while I had simply thought my childhood problem was simply a "philosophy of numbers" issue, and didn't have anything to do with what I would later discover was called "calculus".
It seems really elementary now, but basically this whole tangential comment was triggered by the idea of applying a discount then putting back the sales tax. Actually, now I feel silly because I realise there's no difference by having a different order in the operations as long as they are of the same permutation group (multiplication and division -- such as applying percent, form a closed group of permutations or something -- I think -- my high school doesn't seem to teach group theory).
So anyway, that realisation made something click for me, because before then, I had known how to solve for parabolas, their roots, their directrixes or whatever, and could easily do problems like, "Pool Q is 1600 sq feet in area. One side is 40 feet shorter than the other. What is its perimeter?"
But until I did my little experiment in Excel I don't think I really fully grasped the stuff I had been doing. Previously, it had been solving for some root or solution. We didn't need to know the essence of what we were doing, which is what I suppose rate of change is all about.
Anyway, this post is getting kind off-topic now, but it was spurred by the idea of applying the same operation but getting a different result (due to one critical difference).
In the category of "no one knows percents" -- I was at my local children's museum yesterday, and the computer in the gift shop went down. I had to talk the counter staffer through how to apply a 10% discount and then put back the sales tax, and then explain why she should have guessed that the first price she quoted me was wrong (because it was bigger than the original price, and our sales tax isn't *that* high).
I have a collection of stories like this one!
This is what schools simply don't see (parents, too, really).
The whole system seems to focus solely on the child as he or she is right now. This minute.
When I was smack in the middle of all the rhetoric about math with my children's school last year, I mentioned something about my daughter perhaps needing math in her chosen profession someday. To which the principal answered something along the lines of "Really? You think that she might actually be a mathematician?"
I'm not sure what shade of red I turned when I responded that I had no idea what she would grow up to be, and that this was precisely the point. She could be just about anything she wants to be if she is prepared for it and she choses to work for it. Their job as a school and my job as a parent is to keep those doors open. It took everything I could muster to keep from losing it that day.
The district responded by restructuring the math departent to better "align" with the grade schools.
Now that we have Everyday Math K-6 our district restructured as well by replacing the traditional algebra/geometry track with Connected Math. Apprently it is now better "aligned" to our grade schools as well.
*sigh*
This is a case of the tail wagging the dog.
I suppose if math knowledge approaches zero you would have perfect alignment.
To which the principal answered something along the lines of "Really? You think that she might actually be a mathematician?"
That is EXACTLY the sentiment around here, from the school and from parents ---- AND, as I say, I believe this kind of thinking is "natural."
I've done it myself, Ed has done it many, many times, etc.
I've had to discipline myself NOT to see my own child's struggles with math as "natural" and "real" -- as indicating something about him, not something about the curriculum and pedagogy.
I believe that all educators need to consciously hold the principle that "any child might grow up to be a mathematician" inside their heads, in spite of the fact that it's not true!
I don't think "any child" might grow up to be a mathematician. I do think you have to operate under this assumption.
Here's a useful way to think of it, probably: my goal as a parent reteacher is to keep the doors open long enough so that it's my child who chooses which doors to walk through, rather than his K-12 school making the call.
The other problem is that most of us simply don't know what various careers involve.
You don't have to be a mathematician to need math in your life's work.
Far from it.
Last but not least, I don't think any of us has a good sense of the "advantage on the margin" a sound K-12 knowledge of math gives you in life.
(I have no idea whether "advantage on the margin" has any meaning at all!)
Here's an example of what I mean.
When Carolyn decided she and Bernie needed to leave academia and find work in the business world, it wasn't easy. The business world doesn't pay people to write proofs.
She finally landed a position largely because she told the interviewer that she liked to write.
She was hired, and not too long after that Bernie was hired, too.
So here are two mathematicians who are now working for Microsoft who to an important degree have their proficiency in writing to thank for this.
I would bet the ranch (just about) that the same principle applies for verbal types like me.
I would have been far better off as a writer if I'd been proficient in K-12 math and in statistics.
I plan to become proficient in statistics partly for this reason.
Being a proof-writer is rather like being an entrepreneur. You don't get paid for being an entrepreneur, either.
(i.e. effectively you have to employ yourself)
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