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I want to bring this up front.
LynnG said... Steve, it is always worse than you think.
Here's what EM's Teacher's Reference Manual (Grades 4 - 6) has to say about fraction division.
"Indeed, few adults ever need to divide fractions once they leave school.
Therefore, the main goal of division of fractions in Everyday Math is not to
give students practical skills . . ."
This is an exact quote?
This is an incredibly damning statement to make. I've heard this argument before from regular people, but never so blatently from a math program. I would consider this to be a smoking gun statement. There is no other interpretation.
The simple question is how on earth can they decide on this in K-6! This philosophy guarantees that kids will never need to divide fractions as adults.
Dividing rational expressions like
1/(x-5) divided by y^3/(X+5)
is a required and common skill, even for any high school college math track.
Invert and multiply. What's the big deal? Just think about all of the complicated tasks in life that most kids master. This doesn't rank very high, but oh no! Math is different. It's complicated! It's scary! You have to "understand" it.
It's not about understanding. It's about LOW EXPECTATIONS!
When I talk to other parents about our public schools, they might not know about the problems in math, but they sure know about low expectations.
Everyday Math is based on low expectations. According to Andy Isaacs, it's not for the "elite", and now we know what that means. It means that if you want your child to get into a college math track in high school, you need lots of outside help.
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Yes, it is a direct quote. P. 130 of the Teachers Reference Manual, direct from EM.
The paragraph continues with what EM believes IS the main goal of teaching fraction division:
"Complete the arithmetic system." (ed: you teach division because there are 4 operation of whole numbers, for balance you should show kids that there are 4 operations for fractions too)
"Anticipate future work with rational number expressions in algebra courses."
That's it.
The manual does not go on to explain how their method (find a common denominator, divide the numerators)will prepare kids for algebra. It isn't intuitive. In fact, I'll bet it wouldn't be all that hard for most of the math heads on this blog to come up with a few examples of the inappropriateness of using the EM method in algebra.
That's why a good curriculum thoroughly teaches invert and multiply. It is useful for algebra.
One divides factions all the time when cooking if the recipe is for more people than you plan on serving ...
-Mark Roulo
I hear there's lots of funny stuff in the EM Teachers' Reference Manual. They tell teachers to do this for kids with special needs:
"Utilize other available adults in the building during math time by arranging opportunities for children to play math games with the principal, custodians, cooks, specialists, and so on."
See that? It's not just the kids' time they want to waste. It's everyone's time.
I guess what I find most egregious is that they base what they teach on whether most adults will need to know that knowledge or skill. This is not checkbook math in high school. This is K-6 math. Talk about closing doors.
Another of my favorites, p. 123
"The authors of Everyday Mathematics believe that while it is important for students to learn the basic meanings and uses of division, it is not worth spending enormous amount of time mastering the intricacies of division with difficult whole numbers and decimals. . . . For this reason, the main approach to division with decimals in Everyday Mathematics is the same as for mulitiplication of decimals: Carry out the operation as though the numbers were Whole numbers and use an estimate to place the decimal point in the answer."
I found this out when my daughter came home in 4th grade with a decimal division problem. She had absolutely no idea what to do.
One could make an argument about not mastering long division by hand, but dividing fractions is something completely different.
The fuzzies try to force the discussion towards simplified understanding, discovery, and different basic math algorithms, but the real argument needs to be about low expectations and not providing the proper mathematical understanding needed for success in algebra.
It's only a smoking gun if someone is willing to press charges, and that person has some clout. So far, the only people willing to press charges are parents. And they don't count.
There are many smoking guns. They abound. EM probably took their cue from the NCTM standards; i.e., the PSSM. Check out p. 219 if you have a copy. In it, they discuss division of fractions.
NCTM thinks the rule of invert and multiply is too confusing to be mastered: “Some students forget which number is to be inverted” Apparently NCTM thinks dividing fractions is best done by drawing a picture and eyeballing the answer. "If we limit students’ mathematics experience to low- level, rote computation, they will certainly not be able to solve complex or higher- level problems on a test. Also, when a student has trouble with computation at the secondary level, we have found that spending more time on more computation does not often help."
The genesis is with NCTM who has denied culpability and claims that the PSSM was "misunderstood" so they had to come out with the Focal Points. What excuse will they be making 17 years from now when the next edition of NCTM standards hits the streets? Won't matter much at that point will it?
"If we limit students’ mathematics experience to low- level, rote computation, they will certainly not be able to solve complex or higher- level problems on a test."
This statement is incredibly manipulative and dishonest in many respects. I don't know where to start. Who wants to prevent students from solving complex or higher- level problems? Does NCTM really believe students' ability to solve complex or higher- level problems will be enhanced if they can't do computations. Their disparagement of computation makes it look that way. Just look at the inevitalbe pejorative terms such as "low-level" and "rote" that NCTM applies to computation.
The truth is that students who lack computational fluency and procecural knowledge never see the big picture necessary for solving complex or higher- level problems. I have seen it happen many times with CMP. So who is limting whom?
Linda Moran has these passages from EM at Teens & Tweens
"The authors of Everyday Math do not believe it is worth students' time and effort to fully develop highly efficient paper-and-pencil algorithms for all possible whole-number, fraction, and decimal division problems.
"Mastery of the intricacies of such algorithms is a huge endeavor, one that experience tells us is doomed to failure for many students.
"It is simply counter-productive to invest many hours of precious class time on such algorithms. The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator."
"Utilize other available adults in the building during math time by arranging opportunities for children to play math games with the principal, custodians, cooks, specialists, and so on."
See that? It's not just the kids' time they want to waste. It's everyone's time.
blinding arrogance
I guess what I find most egregious is that they base what they teach on whether most adults will need to know that knowledge or skill.
more blinding arrogance
this may be the core of my own radicalization
there is no effort here, none, to meet the needs and goals of the people paying for these books
this is the core nature of public school
parents and voters as a whole are disenfranchised
not only do we have no power, we don't even have due process
we don't even have formal, written explanations of our no-power and no-due process
our own school can simply impose on students and parents whatever it wants to impose whenever it wants to impose it
There are many smoking guns. They abound.
They've been abounding on the Irvington Parents Forum of late.
That's been a BIG innovation, posting excerpts from the Kendall Hunt advisory documents to a local, public forum one day after the administration refused to allow parents to ask questions about TRAILBLAZERS.
Timing is everything.
NCTM thinks the rule of invert and multiply is too confusing to be mastered: “Some students forget which number is to be inverted” Apparently NCTM thinks dividing fractions is best done by drawing a picture and eyeballing the answer.
That's funny.
Back when I first started trying to teach Christopher math, at the end of 4th grade, I found that his class had been studying fractions. He'd fallen off the same math cliff everyone else falls off.
I hadn't divided a fraction by a fraction (apart from mentally, in cooking) in maybe....25 years?
I remembered how to do it.
I wasn't sure I remembered, but in fact I did.
I divided one fraction by another, then checked the procedure in a book.
Invert and multiply is way easier than addition and subtraction with carrying.
I also had conceptual understanding of fraction division.
I read Liping Ma around that time and tried to do the challenge problem she posed to elementary school teachers.
iirc it was "Make up a word problem for 1 3/4 divided by 1/2."
Few American teachers could do this; a number of them really couldn't deal with 1 3/4 divided by 1/2 at all.
Other teachers interpreted 1/2 as 2, turning the problem into 1 3/4 divided by 2.
Twenty-five years after learning invert-and-multiply, having had a VERY procedural, fragmented education in K-12 math, I could write a mathematically correct word problem for 1 3/4 divided by 1/2.
I remember the problem I came up with is you have a dog who always gets 1/2 can of dog food at each meal, and you've got 1 3/4 cans of dog food left.
[The question of why exactly you would have 1 3/4 cans of dog food left if you're doling out servings of 1/2 cup per meal didn't occur to me at the time. Writing about it now, it jumped out at me. Which is good.]
In any case, my question was, "How many servings do you have left?"
You have 3 and 1/2 servings left, and the 1/2 serving = 1/4 can.
I hope.
I had a crummy education, and I could write that problem 25 years later when most U.S. elementary teachers, who are actually teaching fractions & thus using them frequently, apparently cannot.
"Few American teachers could do this; a number of them really couldn't deal with 1 3/4 divided by 1/2 at all."
I had trouble visualizing fraction division at one point. What finally helped me see the light was to ask (like your dog food example): How many times does one fraction of something fit into the other fraction of something.
Regarding 1 3/4 divided by 1/2 you could make a word problem with flour and cookies or buns. If you need half a cup of flour to make a cookie, how many can you make with 1 3/4 cups.
This form of visualization breaks down somewhat when the divisor is bigger than the dividend.
How many times does one fraction of something fit into the other fraction of something.
right
Singapore Math does quite a nice job, I think, of showing what's going on when the divisor is bigger than the dividend (what is 1/2 divided by 3/4?)
I think Saxon may do a pretty good job, too (can't remember..)
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