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We have approximately 30% of our 8th graders taking algebra. The figure at KIPP, in the Bronx, is 80%. ($10,000 per pupil spending, roughly)
They're not going to do it. They're so not going to do it they're not even going to say 'no.' They're just not going to do it.
While we're on the subject of well-funded school districts saying 'no,' I should add that the middle school is also not going to allow more students to take Earth Science in the 8th grade. Only forty-eight students, of 150 or so, currently take Earth Science, compared to 100% of students in Pelham. However, in the view of the school that is 48 students too many. As the chair of the science department told us, "If it were up to me, I wouldn't offer accelerated courses to any students in the middle school, but this community demands it."
The district argues, in meetings with parents, that learning depends upon maturity. Not all students are mature enough to learn Earth Science in the 8th grade. Or algebra.
Of course, maturity has nothing to do with ability to learn, as the National Math Advisory Panel reports. However, maturity has everything to do with a student being able to monitor his learning instead of depending on his teacher to perform this function. So, yes. It's easier to teach Earth Science to a high school sophomore than to a student in the 8th grade.
So why do parents continue to lobby school districts across the land to teach serious courses to younger kids?
What is the big deal, after all, about taking algebra in the 8th grade?
What's the difference when you take algebra so long as you get around to it sometime before college?
Brain Rules
It turns out there is a very good answer to that question.
It takes years to consolidate a memory. Not minutes, hours, or days but years. What you learn in first grade is not completely formed until your sophomore year in high school.
Rule # 6: Remember to repeat
John Medina
Bingo.
Here we have one of those facts of life many of us have picked up over the years but can neither verbalize in conversation with school officials nor defend as true, primarily because we don't realize we know it.
We don't know what we know. *
On the other hand, when we hear someone else verbalize it, we recognize it as true of our own experience. At least, I did, when I read this statement by James Milgram:
First of all, I claim that taking -- even asking to take it out of the curriculum -- shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they've been introduced. Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.
I'm going to guess that this is another reason why Singapore students are so far ahead of American students. Singapore students are doing simple algebra in the 5th grade. They're doing simple algebra in the 5th grade, and they're not dipping in and out of simple algebra, either; they're not being "exposed" to "algebraic thinking."
They're learning what they're learning to mastery.
At age 10.
* not to be confused with known knowns, known unknowns, and unknown unknowns.
58 comments:
It is sad when the advanced courses go. We lost the advanced math class in middle school when there was a budget crunch a few years back. 8th grade algebra is still an option at least. Advanced classes just don't get the attention they deserve in middle school.
I think this gets at one of the big obstacles to good education reform - often, you don't see the results of a given policy for ten years. So if a kid can't do algebra in 11th grade, is it because he received bad instruction in the 1st grade? 4th grade? 7th grade? If nobody is responsible for teaching (or, to use the educationist's lingo, if the student is responsible for his own learning), then there is no accountability, anywhere.
This also highlights what I think to be the most pernicious aspect of spiraling: when you have no measurable standards of progress, it is completely impossible to identify a student's abilities, or diagnose his deficits.
"That being: algebra in the 8th grade."
But your school does offer the high school algebra course in 8th grade, doesn't it? Actually, our school just started offering that course this year. Before that, they used CMP. This doesn't mean, however, that all of our Everyday Math students have any reasonable sort of way to get to this algebra course. The school believes it does a good job, but they are comparing it with the horrible (and disappeared off the face of the earth, thankfully) MathLand.
So our school could claim that they meet this standard. As far as percentages go, they would just say that you can't compare the US with other countries.
"What is the big deal, after all, about taking algebra in the 8th grade?"
The party line is that if many kids can't handle algebra in 8th grade, then the problem is with the kids and not the school, by definition. Some kids do quite well. Of course, they never ask parents why these kids do well. They don't want to know.
They don't want any accountability, so they hide behind developmentally-appropriate ideas and state that students have to be responsible for their own learning. This sounds like IQ-based education that will never close the academic gap that they so dislike. It's not just IQ, however, it's that the high SES parents see the huge problems and fix them. Schools think that all parents should help their kids, but they are blind to the level of help this entails.
One of our state education leaders said the following about poor 11th grade math scores.
"Is [our state], is the whole country ready to take on the responsibility for providing the support to teachers and kids to meet these standards?"
Denial and more money. That's how they see all solutions. It couldn't be that they are screwing up completely or that there is any validity to the math criticisms that they have heard for years and years.
So, schools want more money, smaller class sizes, and parents who do their part. Otherwise, don't blame them. And fix poverty too.
"What's the difference when you take algebra so long as you get around to it sometime before college?"
It doesn't matter if your teaching methods create huge problems as long students learn how to be self-lerners and get the job done before college? Their philosophy of education completely opposes any basis for accountability and analysis of their curricula and teaching methods.
Yes, the middle school offers Math A in 8th grade to 30% of students, many of whom have been tutored extensively in order to survive the "Death March to Algebra" we talked about back on the old site.
A friend sent me a school account of the way the 3-year accelerated track is structured. The math curriculum moves so slowly in K-5 that in 6th grade the kids have to do two years of content in one year; then in 7th grade they do two years in one again.
The survivors land in algebra in 8th grade, many of them with extremely shaky skills that have had no time to "consolidate."
They don't want any accountability, so they hide behind developmentally-appropriate ideas and state that students have to be responsible for their own learning. This sounds like IQ-based education that will never close the academic gap that they so dislike.
Unfortunately, I wasn't quick enough on the uptake to make this point last school year when the middle school put together a little ad hoc Earth Science committee whose purpose, as far as I could tell, was to shut me up. (I'd been writing about the Earth Science situation.)
That was the first time I heard the "not mature enough to take Earth Science" explanation.
I was dumbfounded; just didn't see it coming at all. I'd never even heard the argument before.
I came home and told Ed, "They're saying it's not IQ, it's maturity."
Ed said, "It's the same thing."
DUH
Schools think that all parents should help their kids, but they are blind to the level of help this entails.
The level of help required is extraordinary.
I've now reached the point of recruiting friends to help me teach Chris. Both Barry G & my neighbor worked the entire "review" sheet given to the kids for the quadratics test. I could do it, but I couldn't do it efficiently; I was doing crazy wild-goose-chase things -- and still getting some of the answers wrong.
The "consolidated knowledge" phenomenon explains exactly where I am now. I didn't learn much about quadratic equations in high school, in spite of my 3 years of straight As in math, so I've taught the concepts to myself.
I've probably done a decent job of it BUT I haven't "lived" with this knowledge long enough for it to be of much use to me -- certainly not when it comes to teaching it to my son.
The idea that knowledge requires years to gel is revolutionary; it's enormously helpful to me in terms of knowing what I need to do -- and what I can and can't do.
It also explains why the whole concept of having parents "help" with homework is bunk.
It supports Allison's doubts about homeschooling.
It supports the earliest possible sound teaching of content and procedures -- and it supports spiraling with mastery.
I wish I'd known about this research years ago.
It doesn't matter if your teaching methods create huge problems as long students learn how to be self-lerners and get the job done before college? Their philosophy of education completely opposes any basis for accountability and analysis of their curricula and teaching methods.
yup
I'm SO ready for teaching to become a profession.
>>What is the big deal, after all, about taking algebra in the 8th grade?
My college has shown statistics to interested alumni that say doing well in 8th grade algebra correlates with being successful in Calc AB, which correlates to being most successful in earning an engineering degree in four years as opposed to dropping out or taking five/six years.
I've run into the same thing as you did Catherine, with the unadvertised double courses in Gr. 6 & 7 in order to take Integrated Algebra I in 8th grade. The pacing means that it is all flashcard memorization of algorithms, with no in depth discussions. Division of fractions and mixed numbers in one 42 minute period, for ex.! The rote memorizers and tutored tend to crash in Gr. 7, leaving 8th gr. Integrated Alg. I for the math brains.
Really the best solution is for the district to get a backbone and start teaching the state standards in depth to all sections of each elementary cohort, rather than leaving so many '3' and '4' kids in the lurch while remediating all the '1's and '2's during the whole class lesson, however they keep coming up with excuses of 'elitism'. I'd rather they take a page from the 'excellence' book and put the money into all day pre-K for the poor instead of remedial for middle school. But I'm just one voter whose opinion is not heard through the shouts of 'elitism'.
"The idea that knowledge requires years to gel is revolutionary; it's enormously helpful to me in terms of knowing what I need to do -- and what I can and can't do."
I've mentioned before that I didn't feel that I really understood algebra until my junior year in high school, and I was a good student who took algebra in 8th grade. This time could have been reduced with better teaching (or effort on my part), but I've found that for most topics in math, understanding and mastery of skills are tightly entwined and can take some time.
Then there are diffrent levels of understanding. I learned about linear algebra in school and used it to do lots of real applications, but it wasn't until I had to teach a course in linear algebra that I fully appreciated linear spaces. But then again, that just brought me up to the next level.
Part of the shift I have witnessed has been in the dropping of topics that need to be mastered by the time algebra comes around.
Singapore introduces long division at the end of third grade. Many of the math reform curriculums don't teach it at all, leaving the middle school to cram in a needed skill in a very short period.
The operations of fractions (mastery, as well as understanding)are barely touched on, again leaving the middle schools to cover it in record time. It's impossible except for maybe the gifted/super-bright.
Understanding fractions is such a critical skill for success in algebra, but I still hear teachers say, "When do we ever need to divide fractions in real life?"
"The math curriculum moves so slowly in K-5 that in 6th grade the kids have to do two years of content in one year; then in 7th grade they do two years in one again."
In our schools, K-6 is all about developmentally-appropriate, low expectations. (Kids will learn when they are ready.) Then, 7th and 8th grades are all about the big push to high school. The teachers all specialize in their subjects and seem to take pleasure in telling kids that success is now their responsibility. Developmentally-appropriate transforms into sink or swim.
The district argues, in meetings with parents, that learning depends upon maturity. Not all students are mature enough to learn Earth Science in the 8th grade. Or algebra.
I'm not sure why the district is so very proud of that fact. Teaching a course that is inaccessible to all but the brightest (or "most mature") isn't very hard. Anyone can do that: no knowledge or skills required. I haven't studied earth science since eighth grade, and I bet that - with no preparation or anything - I could walk into any eighth grade earth science class in the country and "teach" a course that is AT LEAST as incomprehensible to most students as Irvington's course. But I'm not sure that is anything to boast about - it would just highlight my failure to properly teach.
The real trick is teaching a hard subject so that it is accessible to most students, even when they are not particularly "mature". As an example, Highlands Latin School uses a high school Latin text starting in fifth grade. They do go at a slower pace than a high school class would, but nevertheless this is a mandatory course - all their students take Latin. The teachers cite proper preparation (they start with an elementary Latin course in 3rd grade) and the clarity of the text used (Henle, although I think they've written their own text now) as the reasons they are able to do this. They also have a strong focus on mastery. I would also attribute their success to the quality of teaching involved.
This sort of success is something worth boasting about, but, of course, it is much harder than teaching incomprehensible "elite" courses, where any and all failure is by definition the student's fault.
-forty-two
Yesterday my husband and I got in a heated debate over what is inuitive and what is not. How long should it take for an average child to learn place value given the fact that the math curricula spirals the way it does?
This discussion evolved from watching our son forget place value when learning how to convert decimals to fractions and vice versa. He couldn't make the connection and it concerned my husband. My concern was were we making a mountain out of a mole hill? Is the issue truly a memory problem or the fact that he is still so young and has not been given enough practice to attain mastery? Is it a little of both?
So, I am relieved that "it takes years to consolidate a memory. Not minutes, hours, or days but years."
However, the school doesn't understand this and labels children unfocused or immature.
Paula,
I think you are confusing the original quotation with meaning that your child will take years to develop competency and mastery doing an operation.
That's simply not true.
Here's an analogy: every year around Christmastime, you decide to learn to bake a cake. you do this by watching a tv show or two on foodtv. in one case, it's a kind of normal two layer cake; in another, it's an angelfood cake. Then next year, you do something similar. This time, you watch the angelfood cake and watch one on souffles. Occasionally, you also go to the kitchen and experiment--but never with a recipe. You just take some of the ingredients and "see what happens" when you mix the flour without the eggs, or don't heat the oven. 5 years go by of this video watching and experimentation, and after that, what do you know about cake baking?
Nothing.
That's the spiral curriculum style of most math texts. They just glance on a topic, maybe show some oddities about it, or "interesting difficulties", and then add some "Constructivist" projects where you are supposed to derive deep truths. They do no actual math computations until mastery. Then they skip around to something else.
If you want to learn to bake a cake, you have to BAKE CAKES FROM THE RECIPE SEVERAL DOZEN TIMES.
Now, the quote "it takes years to consolidate a memory" means this: AFTER you have functional mastery of how to use your kitchen, of the cake recipe, of the equipment, then you can make a cake from memory. But the years won't help if you never do the several dozen cakes.
Going forward, what Milgram said about some skills taking a decade to develop, that's equivalent to this:
do you know WHY you need eggs in a cake recipe? At some point, after baking 100 cakes in your lifetime, and messing up the recipe maybe 10-20 of them, and baking 100 batches of brownies in your lifetime, you say to yourself, and baking cookies, and things, you are in the kitchen and you say "AHA!!!" The eggs stabilize the cake! When I leave out the eggs, it doesn't hold together and lift properly.
You don't understand that the first 100 times. But you've been USING the eggs the whole time.
Your child needs to be taught place value NOW to the point where he can follow the recipe, and eventually do it from memory. In a decade, he'll consolidate WHY place value matters. (He'll wake up one morning and say "AHA! How would you even Try to multiply 26 by 47 when you write them in roman numbers?
XXVI times XXXVII
Place value is amazing!)
but what he's getting now won't allow that to happen. Please intervene. Your husband is right to be concerned.
um, XLVII for 47....
---The idea that knowledge requires years to gel is revolutionary; it's enormously helpful to me in terms of knowing what I need to do -- and what I can and can't do...
It supports the earliest possible sound teaching of content and procedures -- and it supports spiraling with mastery.
To be clear, I'd like to elaborate on spiraling with mastery.
The truth is, you don't need to artificially generate a "Spiral" curriculum in the math and sciences. It's a naturally occurrring phenomenon!
You learn fractions; you learn algebra--you're learning about certain abstractions and mathematical operations. When you start to learn geometry, a real curriculum will build on your knowledge of fractions and algebra. You will see problems where you're expected to use your mastery of those-doing so NATURALLY creates the spiral phenmomenon (oh! I saw this before! THIS is what it meant!)
If you are in an even slightly technical field, this phenomenon continues to happen throughout life. Say you graduate from college and go into a field and occasionally do data reduction. You do something called "least squares" fit. You learned this technique in college, but never thought about it hard--you just did it because you had to. Now, you find that you understand why minimizing the square works--something that NEVER made sense before, and you never ACTIVELY thought about it at all.
Ever gone back to class notes or a textbook you remember being SO HARD a year or two later? And you can just read it? That's the same thing.
I don't have a lot of good humanities and lib arts examples of this, but it must be true that certain texts that were completely over your head at 18 are simply obvious at 30, right?
Back in the days of Webster's Speller, spelling (and phonics, the Speller teaches both) was taught and reviewed in every grade.
I had planned on spelling and phonics review through at least 3rd grade and possibly 6th grade...now maybe I'm thinking 6 - 12th!
They say it takes at least 10 years in an area before you're ready to truly innovate and understand within that domain. I've been tutoring with phonics (and spelling rules and patterns, the corollary of phonics) since 1994, I've really begun to understand things and come up with some new ideas in the last 3 or 4 years.
Your child needs to be taught place value NOW to the point where he can follow the recipe, and eventually do it from memory. In a decade, he'll consolidate WHY place value matters.
Eggsactly! And this highlights one of the biggest flaws I see in reform math: it doesn't always work to teach the concepts/understanding before the skills! I'm in a technical field, and I've had countless "aha" moments during my studies as the "whys" and "hows" of things I've learned have progressively (pardon the pun) dawned on me. It is that natural spiral that occurs as one applies background knowledge.
The problems with careless use of the concepts-first approach are exemplified in some of those multiplication "algorithms" taught in Everyday Math and TERC--they are so cumbersome, so clunky, a kid gets easily lost trying to wade through them. Most of them (lattice method excluded) very nobly illustrate place value concepts but this is lost on the kids, just lost.
In the end, the kids can neither do the math nor understand it.
PaulaV: Some fifth graders have the 'aha' on place value when they learn alien operations (aka operations in other bases), rather than using the manipulative base ten blocks or watching the person at the board chant 'this is the tens place, this is the ones place'.. etc as they evaluate an expression.
Prufrock Press has a resource titled 'Alien Math' that teachers could use, but it's probably simpler to take this approach: http://www.garlikov.com/Soc_Meth.html (be sure to read his linked paper)
Fun webquest on place value/number systems here that I'm thinking of using with my kids this summer: http://studenthome.nku.edu/~webquest/gabbard/index.htm
Allison,
If, according to rule #6, "you remember better by repeated exposure to information in specifically timed intervals," then perhaps my son isn't getting the "right" repeated exposure he needs to fix place value in his memory.
I'm not saying that learning place value isn't important or that we shouldn't be worried about him learning it. Of course, he should learn it. However, this would require the right approach and repeated exposures to fix it in his memory.
For instance, one will occassionally change a recipe because after following it verbatim the first few times, you still did not get the desired results. Then, someone may say add this or that or add more or less time. You can go through a few variables before finding the one you will use as the basis for committing the recipe to memory. Then you use this one over and over again.
I think it is asking a lot of some nine year-olds to be able to grasp some concepts on command. "You must be a problem solver so be one!"
I do not want my husband to fall into the trap the school has laid. They say kids need to be problem solvers and learn in the real world. How many nine year-olds do you know who are in the kitchen baking cakes or outside building houses? How many have checkbooks to balance?
How could an elementary school student have the background knowledge or memory of a 40 year-old? Yet, we are expecting them to learn things that many of us did not grasp until we were older and had much more experience under our belts.
"This discussion evolved from watching our son forget place value when learning how to convert decimals to fractions and vice versa."
It's not clear to me what the problem is or if there is a real problem. When I was learning math I knew lots of things, but not very well. Change the problem or approach and I would begin to question what I knew. This is common and why I always push the linkage between mastery and understanding.
I remember in algebra thinking that 'X' could be two numbers at the same time. (Can you imagine what the teacher must have thought?) It's stupid, and I don't know why I had that temporary crisis, but it's happened to me many times. I could do a problem one way, but I was at a loss when I tried it another way. I've mentioned before that it took me until my trig class in 11th grade before I stopped having these problems in algebra.
As for place value, I don't recall any class emphasizing this. I guess we did enough multi-digit multiplication and long division that it was never an issue. You could study other bases, but it's not clear that the problem your son is having has to do with that kind of understanding. The problem might be something other than place value. Maybe he is having one of my type of confusions and can't quite see how place value will get him unconfused. The best you can do is to make sure that he masters the conversion process to/from decimals to fractions. The problem with place value is that it's an easy concept to understand, but it's not easy to see how it can be used in any one situation. That kind of understanding takes time and practice.
One of the major goals of Kumon is called 'G By 5'. This is completion of Kumon Level G by 5th Grade. Kumon Level G is basic algebra. From the Kumon web site...
"Level G aims to develop students' skills in working with introductory algebra, e.g.
1) operations with positive and negative numbers.
2) simplifying algebraic expressions and
3) solving linear equations in one variable, based on the arithmetic skills that students developed-especially in calculating with fractions up to Level F."
I do find it amazing that schools take 9 years to get to algebra, when the primary skills needed for algebra are only addition, subtraction, multiplication, division, fractions, and decimals.
-Burnsy
lgm-
Thanks for those links. I especially enjoyed the first one. My husband has a comp sci degree, he'll be teaching this after we get through a little more addition and then again after multiplication.
At a request from our daughter who is learning to count by 2's, 5's, and 10's, I counted by 3's and then attempted 11's. I said, "this isn't as hard as I thought", my husband replied, "just wait til you get over 100." He was right, I let him take over after 99.
--I think it is asking a lot of some nine year-olds to be able to grasp some concepts on command. "You must be a problem solver so be one!"
That's right. It is UNREASONABLE to ask them to grasp the concept. Instead, they just need to LEARN THE TECHNIQUE.
---How could an elementary school student have the background knowledge or memory of a 40 year-old? Yet, we are expecting them to learn things that many of us did not grasp until we were older and had much more experience under our belts.
this is my point. The spiral curriculum leads them astray. Instead of teaching them how to do the computations, how to master the techniques, supposedly we're going to "spiral" our child into grasping abstractions that we didnt' grasp until we were adults. it's madness. Singapore math doesn't have this problem. They learn to do the computations to mastery. starting at age 6, continuing forward. you don't need to think about place value per se and what it means because you know how to convert back and forth.
re: decimals and fractions: wait for 3 more dayss. My exposition of Prof. Wu will get there, and maybe that will help you help your son to do the technique without getting bogged down in what "place value" means. It's likely your child is confused because he's been badly taught fractions. That seems to be true everywhere in the country now.
you said how many nine year olds can bake a cake? but any nine year old can. All they need is to follow the recipe. Your school's insistence that he learn from various ways or think about concepts is the problem. It's the equivalent of saying "bake a cake wITHOUT a recipe." In nearly all schools now, the students do that supposed "concept learning" at the expense of ACTUALLY learning the computations.
Nike is right: "just do it" . Your child is likely not getting enough computations done to just know how to manipulate the numbers in a spiral curriculum. If instead he was just drilled over and over again on converting fractions and decimals, he'd be okay doing that. would he "understand" place value? who cares! let that come when it comes! but GET MASTERY now.
Steve,
I think my husband is the one who thinks mastering place value will get our son unconfused. I certainly think it would make things easier, but our son just wants to get finished with the work.
My son is in level E in Kumon. Lots and lots of practice with adding, subtracting, multiplying and dividing fractions. Right now he is converting fractions to decimals and vice versa. It is slow going.
The beginning decimal lesson begins with the example 0.2 = 2/10= 1/5. You work several of these problems and it switches to 0.02= 2/100 = ? The answer is 1/50. Okay, so far so good.
Then you get to 0.004 = 4/1000. He wrote it equals to 2/40 = 1/20. A simple division error? Well, I thought so until I get to the third question which was 0.012. He wrote 12/100.
What just happened?
Allison,
Thanks so much for your comments! I am really trying to figure it out.
My son gets plenty of practice in Kumon, but then he goes right back into school and gets frustrated. His father and I are doing one thing at home and the school is doing another. Perhaps this is where the confusion lies. I'm afraid my husband is listening to the school about kids needing to learn conceptually first. He doesn't read what I do (KTM and other blogs) and so I'm trying to explain to him that I'm not sure it is a problem that our son is having some difficulty. He will get it and master it. I mean, the kid is in level E in Kumon and he isn't even in fifth grade yet.
Let's not make a mountain out of a mole hill by listening to the school. We need to keep him in Kumon and reinforce concepts at home. Are we teaching him fractions the right way? I don't really know. He seems to understand and is able to compute rather quickly.
I look forward to the rest of your exposition on Prof. Wu.
Is he frustrated?
Frustrated kids will write down any answer just to make the assignment go away. So will kids who are fearful or annoyed or bored at the work they are doing. I
start making dumb mistakes when I can't be bothered to read the assignment. Being bored is difficult to combat. cookies help me :)
I don't understand the lesson questions you posted. What is given, and what are you supposed to solve?
Do they do the decimal to fraction with a denominator that's the power of ten? Or does he?
To see if the issue is place value or something else, don't ask him to be simplifying the fractions. Just have him convert from the decimal to the fraction.
.2 = 2/10, .004 = 4/1000, etc.
can he do that? are you confident he can do that? is he?
I've run into the same thing as you did Catherine, with the unadvertised double courses in Gr. 6 & 7 in order to take Integrated Algebra I in 8th grade.'
Unadvertised is right!
No one ever told any of us that our kids were trying to learn 2 years of math in 1.
When they first brought Trailblazers in there was an entire YEAR where the district was unable to tell parents whether any of the kids would be able to take calculus in 12th grade!
Singapore introduces long division at the end of third grade. Many of the math reform curriculums don't teach it at all, leaving the middle school to cram in a needed skill in a very short period.
I learned this week that our 5th graders basically don't know how to do long division - not unless their parents have taught them.
They were taught "forgiving division" (aka partial products division); they spent a brief bit of time on long division.
And that's it.
They're going into middle school not knowing how to do long division.
Meanwhile we've got the new Tri State Consortium report saying K-5 math is fantastically good & the problems lie in the "traditional" environment at the middle & high school.
The administration has rather clearly signaled its intent to stop teaching "traditional" math altogether.
I've mentioned before that I didn't feel that I really understood algebra until my junior year in high school, and I was a good student who took algebra in 8th grade.
That's true!
You've said that a number of times.
I never knew what to make of that -- I had cognitive dissonance every time you brought this up. I'd think, "But Steve's a Math Brain! how can that be??"
It's absolutely true, though. Knowledge has to gel.
For me, this is one of those things that I've experienced all my life but never quite formulated consciously.
I'm not sure why the district is so very proud of that fact. Teaching a course that is inaccessible to all but the brightest (or "most mature") isn't very hard.
Well, needless to say, I have made this very point.
It fell on deaf ears.
This is an especially annoying topic to me because I write nonfiction books on fairly difficult subjects. I told the science chair that if I were writing "challenging" books I wouldn't be living here.
My job is to write accessible books on challenging topics.
To me, that should be the Prime Directive for teachers as well as writers.
Yesterday my husband and I got in a heated debate over what is inuitive and what is not.
Been there, done that!
One of the joys of being on the outs with your school is the spill-over heated debates at home!
Just this morning, as a matter of fact, I was ready to have a heated debate with the entire lot of them re: whether rationalizing the denominator by multiplying by a conjugate would or would not appear on today's quiz and, if not, whether a serious person goes ahead and learns how to do it "just in case."
He has always been a perfectionist so he gets frustrated easily when the answer doesn't come quickly enough. At the beginning of the year, he was having panic attacks so I had him tested by an educational psychologist. He scored low on working memory due to low motivation brought on by anxiety. His teachers have complained that he is unfocused, but this isn't the case. He has hard time with auditory memory. I cannot imagine what his school day is like...so many mixed messages and expectations.
The lesson is from Kumon. The first lesson asks you to rewrite the decimals as fractions. Example, 0.2 = 2/10=1/5. You have several types of these problems and then it gives you another example like 0.02 = 2/100 = ?.
The next page wants you to rewrite the decimals as fractions and reduce.
Each page is a little different than the last.
Yes, he can convert from the decimal to a fraction.
This discussion evolved from watching our son forget place value when learning how to convert decimals to fractions and vice versa.
I had a very upsetting experience along these lines last week.
The math teacher gave the kids a "Review" Sheet on quadratic equations that was an order of magnitude more difficult than anything they'd seen in class, been assigned to do for homework, or appeared in the textbook.
I could barely manage it myself, which was upsetting in and of itself. (I'm at the very end of Saxon Algebra 2.)
Barry and my neighbor had to bail me out; both of them did the problems and sent them to me.
When I saw what they'd done I realized that I had failed to notice that the first two mystifying problems on the Review sheet were simple algebra problems in two variables. All I needed to do was substitute the value I found through a simple knowledge of quadratic roots back into the original equation & solve.
Instead I was on a wild goose chase, trying to reason out what the answer would have to be given the coefficient of the 2nd term and on and on and on.
When I realized that I had failed to see that I needed to substitute a value into the original equation I was really upset.
Barry was great, though. He pointed out that that's exactly what happens when you're working beyond your limit -- you forget the stuff you already know. (I'm going to read Liz' cognitive load post after finishing reading comments here -- I was probably having a cognitive load problem.)
The truth is, you don't need to artificially generate a "Spiral" curriculum in the math and sciences. It's a naturally occurring phenomenon!
Yes, absolutely!
I'm sure this is why for a lot of us math is the subject we remember best from elementary school!
I don't have a lot of good humanities and lib arts examples of this, but it must be true that certain texts that were completely over your head at 18 are simply obvious at 30, right?
Well, I think this raises very interesting questions about humanities and liberal arts ---- because in those fields you NEVER see brilliant, mature works produced by people in their 20s as you do in math and physics.
I don't think anyone has any really good ideas why this should be so. Mostly people just assume that you need years of "life experience" or "wisdom" or.... years of experience with the thinking of others in the field...
No one knows.
I probably had my ah-hah experience with place value just a couple of years ago when I first heard about inverse operations....
Change the problem or approach and I would begin to question what I knew.
This happens to me constantly.
PaulaV-
"He has hard time with auditory memory."
Can he read the nonsense words babber, baber; latten, laten; bippen, bipen; loppet, lopet; zuttip, zutip? (bab-ber, 1st syllable closed so short, baber, 1st syllable open so long)
If he can't, you might want to have him work through my phonics lessons and/or Webster's Speller. Syllables are easier to hear than letter sounds, and learning syllables may improve his auditory memory if he doesn't have a good grasp of them already.
I should have divided baber for you: ba-ber.
ba, be, bi, bo, bu, by!
That's why they used to teach the syllabary.
"My job is to write accessible books on challenging topics.
To me, that should be the Prime Directive for teachers as well as writers."
That should go double for the folks who write software manuals, DVD manuals, etc.
I've YET to see one in anything that resembled English.
In fact, I remember hearing about a contest recently where you could submit your worst manual lines in a contest for the "best" (worst) paragraph.
I'm a big fan of finding out whether your child can read orthographically correct (if that makes sense) nonsense words.
Most kids can't, I'm pretty sure.
"I'm a big fan of finding out whether your child can read orthographically correct (if that makes sense) nonsense words.
Most kids can't, I'm pretty sure."
Most children taught with a good phonics program (the old open court or the only good public school program I know of today--Didax's school phonics, and most of the programs in use by private schools and homeschools) can do this. I have tested several children in this, though not as many as I've given regular reading tests to.
Most children taught in the public schools cannot, although a fair number have figured out the system based on the limited information given to them and can do this.
I'm not sure what the best term for word like these is, I personally like to shy away from words like orthography although I do know a lot of words like that. For example, I substitute "sound-spelling correspondence for "grapheme to phoneme."
Maybe a good term is a "plausible, regular nonsense word."
Sometimes there's no substitute for the word you want, though. And, you have to think of your audience. I bet that more people that read this blog know what the word orthographically means than an average person on the street.
I'm sure that more people here know what heteroskedasticity means than your average joe.
Paula,
--He has always been a perfectionist so he gets frustrated easily when the answer doesn't come quickly enough. At the beginning of the year, he was having panic attacks so I had him tested by an educational psychologist.
I am so sorry. I am 35, and have had panic attacks since I was a child. My first were all in school. I hated school. I cannot begin to tell you what despair and fear that school brought me. I sympathize with your son. All I know is that for me, school didn't get better until I was taken out of the panicking environment and put somewhere else.
---He scored low on working memory due to low motivation brought on by anxiety.
I don't know what this means exactly. I think I can guess what the working memory is, but I don't understand the "low motivation" bit, so my next comments may be off track, but they really rung a bell for me.
I completely understand how "working memory" can fail when panicked. I have NO ABILITY to remember ANYTHING when I'm having an anxiety attack. I cannot tell you the last word I read, the last image on the tv, the name of my friend when I'm panicking. I am just panicking.
So, if I'm panicking and trying to do a multistep math problem? I can't do it. I mean, I can't even tell you what the words I'm reading mean. I know lots of people don't believe in test anxiety, but for me, test anxiety was quite real.
My problem was also that in an anxious state, where I wasn't panicking but was simply anxious, I couldn't hold a lot of ideas in my head. especially mathematical ideas. because the way I know things is I talk to myself, and I couldn't remember the long conversation with myself when I was anxious. Writing everything down is hard. For sophisticated problems in grad school, you can't. You're supposed to hold them all in your head--you're supposed to have a compact schema for them. My not being able to mainpulate math without sub-vocalizing it to myself is probably anathema to most compact schema mathy folks invent. But it would work when I could keep this inner monologue running. But during anxiety states, I was left without that, and I would have to write down the simplest things in order to remember where I was at in a computation.
You said somewhere else how you couldn't really wrap your head around a bright kid with low working memory...how odd and confusing. I wonder, though, if it's really true all the time for your son. What if the low working memory comes under stress? What if the ed. psych testing was anxiety producing, too? What if this isn't so much a case of him never having a lot of working memory, but having it in highly stressed environments? And school is one of those?
Paula,
--Yesterday my husband and I got in a heated debate over what is inuitive and what is not.
Bumbling into your marital bliss is not my intention, but I wonder if your husband has observed your son very closely, and if you could get him to do so.
My husband and I are very different at observing. I am very aware of my son's mental states. I can tell you why he is doing what he's doing, even when he's doing "Weird" toddler things. As a teacher, I learned to start unraveling the puzzle of why a student would think a wrong answer was reasonable, etc. My husband just can't get to that overlap of consciousness the way I can. He can't really imagine why my son does what he's doing. He needs it spelled out for him. Maybe your husband is having trouble not just because he believes your son needs to know the concepts now, but because he hasn't really observed your son and seen the successes and the failues as often as you have?
>>The beginning decimal lesson begins with the example 0.2 = 2/10= 1/5. You work several of these problems and it switches to 0.02= 2/100 = ? The answer is 1/50. Okay, so far so good.
Then you get to 0.004 = 4/1000. He wrote it equals to 2/40 = 1/20. A simple division error? Well, I thought so until I get to the third question which was 0.012. He wrote 12/100.
-----------------------------------
From my classroom volunteer experience, he looks as if he's counting the zeros to figure out what the denominator should be, then using the rest of the digits for the numerator but is confused about the rules AND/OR he can't look at a decimal and say the name in words (see 0.058 and say fifty eight thousandths for ex.). Only he knows though.
Here are practice sheets that may help if you don't want to wait for the next tutoring session http://www.eduplace.com/math/mw/practice/4/reteach/21_2.pdf http://www.eduplace.com/math/mw/practice/4/homework/21_2.pdf
http://www.eduplace.com/math/mw/practice/4/practice/21_2.pdf
but if you find he can't read a decimal you could back up to the grid modeling
http://www.eduplace.com/math/mw/practice/3/reteach/20_1.pdf
http://www.eduplace.com/math/mw/practice/3/reteach/20_2.pdf
It does take a bit for young students to understand that 62 hundredths is 6 tenths 2 hundredths, just as it took a while for some to grasp that twelve hundred = one thousand, two hundred.
HTH
lgm,
Thanks so much for the Houghton Mifflin website and links. We will be using these over the weekend.
Paula
They're going into middle school not knowing how to do long division.
Yep. In our district they've implemented a computerized math scaffolding program to address the blaring deficits at the middle school level. The problem is that the majority of the students require it because they're arriving at the middle school unprepared and lacking mastery of foundational skills.
Instead of questioning the efficacy of Everyday Math and CMP, they're rearranging deck chairs by remediating entire cohorts. This signals a systemic problem that must be addressed at the elementary school level, which they refuse to acknowledge.
Meanwhile we've got the new Tri State Consortium report saying K-5 math is fantastically good & the problems lie in the "traditional" environment at the middle & high school.
Our Tri State Math is coming up next. I expect very similar recommendations as yours(despite everything the National Math Panel has said).
Everyday Math is like a cockroach.
The administration has rather clearly signaled its intent to stop teaching "traditional" math altogether.
It seems that if things don't change in response to the National Math Panel's findings (which just may take an act of God), just about every student will end up in CMP because they are working diligently to eliminate the pre-algebra/algebra opportunities in grades 7/8. In their eyes, CMP is the solution no matter what anyone else says-- even if they are mathematicians.
Allison,
Could you send my your email?
Paula
Allison's comment about hating school is important.
I've now seen enough kids go through school to believe that when school is a bad environment for your child you need to get him or her out.
The teen years pose a LOT of challenges, and my sense, at this point, is that if a child has spent years of his life as a square peg trying to fit a round hole, he heads into those years with some damage.
I think that's all I'll say.
BRAIN RULES probably has a good passage on working memory & anxiety.
I'll check.
I just went over to brainrules.net and you've got to take a look too! There are a series of video clips that discuss some of the 12 rules. There is one on stress that is very much on point with this discussion. The brain is not designed to function under prolonged stress, it shuts down and in extreme cases even atrophies.
Of course, I have got to have that book!
Hi Paula,
Email me at
greifer @ gmail dot com
without the spaces, etc...
"... just about every student will end up in CMP because they are working diligently to eliminate the pre-algebra/algebra opportunities in grades 7/8."
Our town got rid of CMP (finally) because even the most natural math brain kids were getting into high school unprepared for geometry. Parents demanded and got our school to offer the same algebra course/textbook that is used in the high school. As I've mentioned before, and as others have mentioned, that just shifts the big jump down to 6th or 7th grade, where kids have to do two years of work in one.
One unique aspect of our small town is that we send our kids off to another town for high school. It was common knowledge that kids from our town were bad in math. It's always nice to have an independent comparison, especially when it's better. The middle schools in the other town offer 3 levels of algebra in 8th grade.
PaulaV: slow down a notch and convert .012 the slow way
.012/1 * 1000/1000 = 12/1000
--rocky
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