This is C. last summer at the picnic table outside our kitchen. We were probably working on percent (scroll down).
I don’t like math.
Why is there 1!?
1 + 1 equals 2!
Why is that?
I don't like math.
Words make sense. If I read “the dog,” I know what “the” means.
Math doesn't make sense.
- CHB summer 2007, 12 years old
I know you will all be impressed by the fact that I did not say, "You know what 'the' means?"
I give up. I am supposed to discover the answer to these algebra problems, through guess and check, if need be. I don't even know if I am lacking conceptual or procedural knowledge, or both. We went straight from What's your favorite number? and What color is math? to this. I now have a highly developed sense of math appreciation as a result. I sing the praises of math daily, especially in the shower, but can't I compute or understand much of anything in puncto math.
ReplyDeleteWho can help?
Solve for x:
x^2 -1 = x + 1
and this one:
x^2 - 1 = 0
My solution for this one so far:
x^2 = 1
So taking the square root at both ends would end up with x being either pos or neg one.
Will sing praises to whomever helps with the first problem.
Best to solve the first one by putting it in quadratic form. You will then end up with the equation:
ReplyDeletex^2 -x -2 = 0. Roots are x = -1 and 2.
If you try to solve it by factoring x^2 - 1 into (x-1)(x+1) and divide both sides by x + 1, you are eliminating x = -1 and will only end up with one root: x = 2.
cri de coeur
ReplyDeletesame story here
I realized nearly 4 years ago that constructivist math meant that math curricula is either far too easy OR far too hard. There are no boundaries; nothing is pegged to the level of the student.
C. was given college-level challenge problems in 6th grade. The parents did them all if they could; the folks around here did a couple for me.
I complained to the assistant superintendent who said he agreed that kids shouldn't be given problems they couldn't do, but added that the accelerated kids need to be "challenged."
Teaching is gone; challenging is the rule of the day.
Are the students challenged?
Give a 6th grade kid a problem in XXXX (have forgotten the term!) and yeah. He'll be challenged.
Hey---you guys.
What is the term for this kind of problem?
x^3+y^2+z
Carolyn told me the name for this but I've forgotten. (I think this was the problem...)
Anyway, you came to the right place. There are several people here who can do problems our kids have been assigned but have not been taught how to do.
ReplyDeleteI do wonder whether Kitchen Table Math had an effect on the "challenge problem" situation in the middle school.
ReplyDeleteI posted every one of those problems, usually in "real time."
I'm fairly certain that they've never been given again. I know for a fact that C's class never had them again.
TOTAL RECALL!
ReplyDeleteIt was modular arithmetic.
The 6th grade accelerated math class was given a "Challenge Problem" in modular arithmetic.
I still don't know what that is.
"What is the term for this kind of problem?"
ReplyDeletex^3+y^2+z
It's not really a problem. It's just a nonlinear expression in three variables. Is there supposed to be an "=" sign somewhere?
"Solve for x:"
ReplyDeletex^2 -1 = x + 1
I remember thinking for problems like this that you don't "solve for x".
When you first learn algebra, you start by solving linear equations like:
2x -6 = 4
You get the 'x' term by itself on one side of the equals sign and you move and combine all of the constants on the other side. Then you divide by the coefficient of the 'x' term. This is "solving for x" I thought.
Then you get an equation where the 'x' is squared, cubed, or something else, and they STILL ask you to "solve for x"! All of a sudden, solving for 'x' becomes something quite different. You don't isolate 'x' all by itself. You put it in standard quadratic form, you factor it, and then you set each factor to zero and solve.
I remember thinking that they (the book and teachers) took a lot for granted. It's not that I wanted a rote method. I wanted them to explain what they took for granted. They had some knowledge in their heads that they either didn't think was important, or they thought we knew.
SteveH said, "They had some knowledge in their heads that they either didn't think was important, or they thought we knew."
ReplyDeleteThere is a third alternative, probably the most likely, and that is they didn't realize there was something important that they took for granted. It is that a product of polynomials equals 0 if and only if one of its (prime) factors equals 0.
"Solve for x" in a linear equation means "find the unique value for x such that p(x) = 0"
So if p(x) = 2x - 6 then x = 3
"Solve for x" in a polynomial equation p(x) = 0 means "find all the values for x such that
p(x) = 0"
If the highest power of x in p(x) is x^n, then there will be exactly n such values of x, called roots. They may be duplicated -- the roots of (x - 1)^2 are 1 and 1 -- and they may not be integers or real numbers.
The roots of x^2 - 1 are 1 and -1
The roots of x^2 - 3 are sqrt(3) and - sqrt(3)
The roots of x^2 + 1 are sqrt(-1), usually written i, and -i
Proving these things is deep, but telling children explicitly and correctly what's going on does not require proof.
"...and that is they didn't realize there was something important that they took for granted."
ReplyDeleteI guess this is what I was trying to say, and I have made the same mistake with my son. I constantly have to evaluate all of my assumptions.
For example, when I look at an expression, I immediately "see" all of the terms, variables, and factors, and I know automatically what I can and can't do with it. I have to work very hard to break things down to the absolute basics for my son and always explain exactly why something can be done.
As for solving equations, I remember wanting some teacher to step back and give me an overview of the subject. I always felt that I was given the pieces, but then had to put the puzzle together myself.
"I have to constantly evaluate all of my assumptions."
ReplyDeleteI think this is one of the most critical components of effective teaching. I learned this my first year of teaching, and am constantly needing to remind myself of this. I have spent the last 20 years mastering concepts that my students may be encountering for the first time.
I often learn the most about what the students are learning when I work with them one on one. I can see where their holes or gaps are. I find homework assignments very valuable in this regard as well; I see where the gaps are.
"I remember wanting some teacher to step back and given me an overview of the subject."
I have felt that way myself on so many occasions. On the other hand, I have had outstanding teachers who provided that overview. I try to always remember to provide that overview (a reminder that this is the destination).
I will be the first to admit that I am not always successful in achieving those goals, but I think it's important to understand that it's what I should be striving for.
I don't mean to sound self-serving, because in truth, I am incredibly self-critical. I operate under the assumption that I can always do it better and that there is a constant need for learning and improving on my end.
ReplyDeleteI have a tendency to start in the middle and assume that people are where I am, and I have had to learn how to take complex tasks and break them down into smaller, more manageable components. That is, to provide the building blocks or base, first.
This comment has been removed by the author.
ReplyDelete"I think this is one of the most critical components of effective teaching."
ReplyDeleteThis point cannot be stressed often enough. It should be de rigueur in in PD.
Taking things for granted because we as teachers have done it so often and know it so well is natural and it takes extraordinary effort to step back and put oneself in the students' shoes.
This point was driven home to me again when I was teaching how to solve a simple linear equation like 2x+1=9.
I was introducing inverse operations and took it for granted that everybody knew that 2x contains an invisible multiplication sign and that the inverse operation would therefore be division. Big mistake. First show students the different ways multiplication can be represented. With x, dot, parentheses and, in the case of coefficients and variables, nothing.
It's not really a problem. It's just a nonlinear expression in three variables. Is there supposed to be an "=" sign somewhere?
ReplyDeleteYou know....thinking back I think it may have been an equation???
I'll have to dig it up.
Trip down memory lane.
For example, when I look at an expression, I immediately "see" all of the terms, variables, and factors, and I know automatically what I can and can't do with it.
ReplyDeleteThat reminds me. The ISEE prep book had a horrific-sounding word problem that stumped me. I couldn't tell whether it was doable or not. I think they created it just for entertainment value.
Taking things for granted because we as teachers have done it so often and know it so well is natural and it takes extraordinary effort to step back and put oneself in the students' shoes.
ReplyDeletebtw, there's a version of this that is one of the core skills in writing.
Writers somehow divine what readers do and don't know, what they'd like to know next, what questions are running through their heads, etc.
I read that in the Cambridge Expertise Handbook and recognized it immediately as something I'm constantly thinking about but take for granted. All the time, as I write, I have a Phantom Reader inside my head reading what I write and saying things like, "Why is that?" or "I don't follow" etc.
I'm so used to this that I don't think about it, but every once in awhile I'll think: who is that person inside my head and how do I know he/she has any connection to the real world of possible readers?
Apparently, at least according to people who've studied professional writers, writers are somehow tuned in not to a student they're working with one-on-one but to an audience of readers they've never met.
was introducing inverse operations and took it for granted that everybody knew that 2x contains an invisible multiplication sign
ReplyDeletethose invisible multiplication signs and factors, etc. HAVE to be pointed out a lot (I think)
To all who have helped out so generously, you have my gratitude.
ReplyDeleteI am convinced more and more that algebra requires some discrete skills and that guess and check alone won't do. It also helps to have been prepared properly in advance over the years, something that the cutting-edge programs didn't do very well, the incessant propaganda from the reformists notwithstanding. I find myself in the uncomfortable situation of having better metacognitive skills than actual math skills. This makes my deficiencies even more glaring. I wish I could squeeze my metacognitive skills back into the tooth paste tube.
x^2 -1 = x + 1
ReplyDeleteFor some reason I am just now absorbing the apparent fact that cri was asked to guess and check a solution to this problem.
Took me awhile.
What are the odds anyone guessing and checking this equation is going to come up with two answers?
Given my own experience relearning math I'd say slim to none unless you're a natural.