Here's a snippet from Old Math, New Math : Everyday Math, aka Chicago Math by Roxane Dover at Silicon Valley Moms' Blog (who also blogs at Rox and Roll)
I had seen firsthand the effects of a problematic math curriculum on otherwise motivated kids. My daughter is a keen student who is interested in math and science to a surprising level. We want to encourage our daughter to stay her course – and hopefully, one day, she will be among those solving the energy crisis, curing cancer or healing the environment. Mastery of math is key to success in science. What I saw when my daughter experienced Everyday Math for three years in K-2 was a rejection of math out of frustration and a related inability to master basic mathematical concepts because of the Everyday Math approach of teaching every concept in too many different ways. This approach gives students a cursory understanding of several ways of addressing concepts at the expense of mastery.
Specifically, mastery falls victim to a concept called “spiraling.” Spiraling means that concepts are introduced but not necessarily mastered before new concepts are introduced, then the previously introduced concepts are revisited and built upon before something else new comes along, repeat. Mathematics learning, which should be progressive and built on a solid foundation, is replaced in this curriculum by a method of throwing a multitude of ideas at the kids without giving the kids time to properly internalize them to create that solid groundwork. It’s like cooking spaghetti and testing it by throwing a handful of noodles at the wall to see what sticks. Everyday Math doesn’t want it all to stick; it’s just concerned that some of it does. And that’s not good enough to build the solid mathematical groundwork that our children require.
From the San Jose Mercury News:
Critics, however, say the curriculum and its nontraditional algorithms are confusing. "Everyday Math" follows a "spiraling" method, where students move quickly through new concepts and may not necessarily learn them the first time around, but they revisit them over and over again in different formats or applications. They also say it encourages students to use calculators too often.
They never develop mastery, said R. James Milgram, a Stanford University math professor who sat on a state curriculum review committee in 2000 when the state rejected "Everyday Math."
"The mathematics these kids are seeing is hardly mathematics at all," Milgram said. "They learn a mush of things, most of which are just wrong."
Milgram said that while the program at its core makes sense, there are only 500 or 600 elementary teachers in the state with the expertise to teach it properly. He said he has seen enrollment in "Everyday Math" districts contract as parents pull out their students and send them to private schools, and said that could happen in Palo Alto.
One local preK-5 private school saw a 28% jump in admissions applications. This when local unemployment is at 11%.
I disagree with Milgram that there is a way to teach EM properly, unless by that he means ignore their sequencing and supplant it with something that really works.
ReplyDeleteYou just take "A Tale of Two Cities" and chop it up. Chapter 1 first grade. Chapter 2, second grade, and so on.
ReplyDeleteOf course it would be a bit difficult for the early grades but you can just change the words to one syllable and shorten the sentences. Voila! Early literacy and critical thinking of real literature. By the time the kids get the full boat they'll be in the eighth grade on chapter 8 and they can just discover the chapters that were changed for them earlier.
"I disagree with Milgram that there is a way to teach EM properly,..."
ReplyDeleteI do to. Everyday Math is fundamentally flawed. The only way to "supplement" Everyday Math is by changing it's basic assumption; that mastery of material is not necessary at any one time. Teachers have to ignore or fly over lots of material and spend more time ensuring mastery of fewer key topics. This takes coordination of the material covered and skipped in all grades. Proper teacher training is not going to magically make "Math Boxes" ensure mastery (finally!) in sixth grade.
Roxane's letter is very well done and it's from just a few days ago. I've been studying this for years and years and it's almost like nothing has changed.
ReplyDelete"Having attended the only meeting prior to the teachers’ recommendation of the Everyday Math curriculum to the board, I am concerned that there was no time allotted in that meeting for questions and answers based on the actual curriculum materials.Those of us who spent the time and who have major concerns lacked an open forum in which to have our questions answered – and now we’re in a position of having to try to change the board’s mind rather than feeling like we participated in a discussion that affected the outcome. Thus, my issue is with the process as well as with its recommendation."
Turf. You would think that these people might feel a little insecure based on their lack of knowledge of math. You would think that they would turn to the mathematicians, scientists, and engineers in their community to at least get some input. But no, they do what they want.
The schools know that there is a controversy. The schools know that this is not just about parents wanting only what they had when they were growing up. At least I hope they know that. Sometimes I wonder. K-8 schools seem to live in their own little fairyland.
I still think about the math open house our school had when my son was in first grade. Parents, many engineers, doctors, and scientists, sat in little chairs and were told about the wonders of MathLand, a curriculum so bad that you can now only find negative comments of it with Google.
Steve is exactly right and I've never been able to figure this one out. A curriculum leader once told me "we can't have 'them' telling us what to do" and I asked "why not?"
ReplyDelete(followed by silence...)
As a high school math teacher, I feel very strongly that it's my job to help prepare students for college. College math professors (not math ed) are the most equipped to lead. It's not intimidating to me. In fact, their direction would be greatly appreciated - if/when they had the opportunity.
We owe it to our students and to their parents who have entrusted their education to us to ensure that they are really prepared.
I just don't get it. Who created this chasm anyway?
I just don't get it. Who created this chasm anyway?The chasm was created by ed schools and the bureaucracy of school administrations, which work hand in hand. The conventional belief that I hear in ed school is "Mathematicians may know mathematics but they don't know how kids learn, nor how to teach them." To which mathematicians have responded "We may not know how kids learn but we know what content they need to learn and master to do well in the sciences, engineering or mathematics. How you teach them is your business, but what you teach them is ours."
ReplyDeleteThis hasn't gone over real well.
"The chasm ..."
ReplyDeleteAs I mentioned before, and as Paul brought up in another thread, there also seems to be a chasm between K-8 schools and high schools. The odd part is that if there is any sort of curriculum or philosophy push it's from the K-8 world towards the high schools. The math department of our high school seems very unwilling to disagree with the lower schools, even after they see the skills of incoming freshmen. Actually, I don't think there is any process for dealing with issues at this great divide. Our K-8 schools don't ignore how kids do in high school, but they see it more as proper placement, not whether the kids were properly prepared.
"The conventional belief that I hear in ed school is 'Mathematicians may know mathematics but they don't know how kids learn, nor how to teach them.'"
ReplyDeleteSo how do they respond to questions about what to learn? Do they assume that if they can find someone selling something they like (such as Everyday Math), that it has to be OK? I can also understand why they might not like the idea of parents having control over curriculum, but we're not even close to that line.
Schools decide with no input from parents even though they KNOW that there are significant issues. Their standard response that parents only want what they had when they were growing up is quite disingenuous. It would be one thing if they said that the selected math curriculum was better for their mix of students, but they have the incredible ignorance or arrogance to say that it's better for all.
I've been struggling for years to understand why the chasm exists in the first place, and why it's ubiquitous.
ReplyDeleteEd schools, turf, yes, but nothing explains why the particular poison: constructivism.
Is the real root of it the social justice agenda? The conviction that women and minorities can't learn math when traditionally presented, and that constructivism/discovery etc. is all about inclusion?
As long as this (false) myth is believed--that women and minorities need constructivist instruction in order to learn and compete, parents and others who want to see sequential, coherent, mastery-based instruction don't stand a chance.
Maybe what we need to do is get this out in the open and challenge the myth, like E.D. Hirsh has recently done. If we can show that historically disadvantaged populations are being hurt, not helped, will we finally get someone to listen?
I'm working on a Masters with about 15 other teachers in my district. The teaching style could fairly be called discovery although I'm not sure the college would call it that and they certainly don't push it.
ReplyDeleteHere's the odd thing. I'm not a proponent of constructivism but this program rocks. Honestly, I struggle with my belief system every time I come out of those classes. Where I am with this struggle right now might turn over a rock in this thread.
One thing that is very different with this program from a public school is that every single person in the room loves math passionately. Another is that every person in the room is an adult and lastly there is a lot of experience in the room. People are at all different levels of math ability (above and below the curriculum).
The setting is very informal. People sit wherever they like and groups coalesce (or not) in very random ways based on who is helping who. As one with more ability than some in the class I can say I've learned an awful lot from helping people and watching where they struggle so it's not a one way street.
The way the teaching works is for the teacher to set up a problem with some direct instruction/lecture (maybe 5-10 minutes). Then we practice that component (drill and kill)for about an hour. Next we are given a set of richer problems. Somewhere in that set is a wringer, a problem you can't solve with what you just got from the DI (if that's all you know). Finally, when the instructor senses that enough people are blocked by the wringer (another 30 minutes or so) they stop the exercise. This begins another round of DI/lecture on the thing that's blocking folks. Repeat.
I realize this is not a pure constructivist paradigm but it's close enough that somebody walking into the room wouldn't be likely to think otherwise.
To get back on thread with this, here's the conjecture...
Are schools and educational leaders projecting a style that works well in the environment I've described into an environment (middle and elementary schools) where it's unworkable? As theoreticians, have they failed to recognize the need for understanding their target environment?
There is a huge difference in the two targets. We don't throw stuff at each other when we're bored,we don't pitch a fit when we can't do it and most of us have lost more hormones than our students have acquired.
There are good and bad ways to conduct discovery type lessons. The recent article I wrote on discovery learning explores this in some detail. Paul B's example is one good way to accomplish it by exposing students to something beyond the "ZPD" and which then readies everyone for the next instruction.
ReplyDeleteExamples of bad discovery learning abound; just look at CMP, which PaulB has also described. They throw in the kitchen sink of information and expect the students to sort it out. Yes, it's all there, but there has been little or no sequencing or development of proficiency.
Ed schools, turf, yes, but nothing explains why the particular poison: constructivism.I think there's a very simple reason: constructivism allows teachers and administrators to judge and be judged based on something like social reciprocation, rather than on actual performance.
ReplyDeleteIt allows them to embrace mediocrity and convince themselves they are doing something positive or even transformative.
Not that I think they are conscious of this or making a deliberate choice, and I'm sure most of them believe that they are simply "doing good." But it's human nature (for most people) to take the easy, self-protecting choice rather than the harder, risky (in the sense of opening themselves up for possible failure), long-term-oriented choice, isn't it? And constructivism gives them an easy choice and allows them to fool themselves into thinking they are making the hard choice.
I guess I knew there's good and bad discovery. My example is probably one of the good ones because it's a good mix of instruction and pushing on your ZPD.
ReplyDeleteIn a way though, it's not discovery because there's no real expectation that folks will solve the wringer. It's more like a tease; bang your terrific math skills against this beast for a half hour then I'll show you a better way to do it. It seems to be designed more to provide a value proposition.
This is why I question whether this is a good format for kids with out the fundamental skills to enjoy what they are doing. There is no joy, nor is there value, in a discovery lesson delivered to kids without the available 'dots' to connect. In that sense I think it's even damaging because kids, over time, see the whole thing as hopeless if they don't get some joy at least once in a while.
"Is the real root of it the social justice agenda?"
ReplyDeleteI think that's one aspect of it. In our town, the main theme is full inclusion. To get this to work, you need to change your ideas about what and how kids should learn. This fits with a general belief in K-6 that the process of learning is more important than "mere facts" and "rote skills". Since they are so removed from the consequences, they don't have to deal with reality. In this sense, I think that these vague beliefs come first and the pedagogy comes second. The key ingredient is a belief in some sort of natural learning process. This translates to low expectations.
Constructivism is vague enough to provide them with intellectual cover for their basic beliefs. As many note, there is nothing inherently wrong with having students discover or figure things out. I do it all the time with my son. I want him to achieve the lightbulb moment. However, it's neither necessary or sufficient.
I could define a constructivist curriculum that is quite different. I would have students use constructivism to apply their mastered skills in new ways rather than use constructivism as a top-down approach to learning basic skills. In many cases, understanding how to use mastered skills is much more important than understanting everything about the basic skills themselves.
For example, when schools try to get kids to understand place value, do they talk about base systems, like binary or octal? Do they talk about place value in terms of algebra or orthogonnl linear spaces? Of course not. They are talking only about a simple conceptual understanding. There seems to be little discussion over different types or levels of understanding.
My son gave me another example. When he was in fifth grade, they had the 8th grade algebra teacher come in and work with the kids. He wanted them to solve:
12 * X = 60
using guess and check. As my son was describing the teacher's help with the problem, it sounded just like what kids (should) do when they solve a long division problem. This is the same school that detests the traditional long division algorithm. Their use of another algorithm is not about understanding, it's about lower expectations.
Schools worry about drill and kill, but they can't claim that mastery is not important. They're stuck. Rather than modify their beliefs, they try to work around the problem. That's why Everyday Math is so appealing. It gives them pedgogical cover. It's OK not to achieve mastery at any one point in time. Perfect, and it fits with full inclusion. How does EM do this? Spiraling (circling) and Math Boxes.
Somehow, magically, kids are supposed to fill in their gaps by themselves. There is no way that fifth and sixth grade teachers can identify and correct all of the individual gaps. They rely on circling to fix the problems. They might even say something about how kids will learn it easily "when they are ready". The corollary to this is that if kids don't learn, then it must be their own fault.
As I said, they are in their own little fairyland. All I do is look at the sample questions and bad scores on the NAEP tests and wonder what they do all day? They obviously have low and fuzzy expectations of learning.
The issue is really not about constructivism versus direct instruction. It's about low versus high expectations.
"...bang your terrific math skills against this beast for a half hour then I'll show you a better way to do it."
ReplyDeleteThere is no requirement to do this in groups in the classroom. I look at my son's algebra text and the homework sets are filled with carefully sequenced small discovery steps. When he was studying line equations, like:
y = mx + b
He was asked to plot:
x = 2
and determine the slope of the line.
There might have been some mention of this in the text in passing, but here it was in the homework. Proper math textbooks have done this for ages.
There are other reasons why constructivism is done in groups in the classroom, and they have little to do with understanding.
"We don't throw stuff at each other when we're bored,we don't pitch a fit when we can't do it and most of us have lost more hormones than our students have acquired."
If this is the case, then schools have to explicitly tell parents that low expectations drive the curriculum and that their kids have no opt-out choice. I don't want a curriculum designed for the lowest common denominator, and I don't want schools telling us parents that it's somehow better.
You can't let students who can't sit still and easily get bored define the curriculum. This points in one direction only; lower espectations. For individual teachers in a lousy situation, this can be a factor, but it's not solving the underlying problem.
"And constructivism gives them an easy choice and allows them to fool themselves into thinking they are making the hard choice."
ReplyDeleteI've struggled with this idea for years. What's really going on in their heads? I think that those drawn to K-6 tend to be nurturing types, and many of them want to work with special needs students. They are driven by caring and the desire to treat all kids as equals.
Unfortunately, reality has to be dealt with. Kids have widely different ability levels that can't be blamed on differences before they enter school. One way to deal with these issues is to claim that those abilities really aren't special. Facts become mere and skills become rote. Teachers told me this when my son was in Kindergarten and first grade. He really isn't smart. He just has a lot of superficial knowledge. Sure, he can read an encyclopedia, but what does he really understand? (Of course, they didn't bother to find out.)
Reality doesn't fit with their fundamental belief, so they define reality differently. That's why they redefine math. Constructivism and understanding just become cover for their fundamental core beliefs. This is not about setting higher expectations. It's about defining education in fuzzy terms so that they can continue to be nurturing and use full inclusion.
“So how do they respond to questions about what to learn?”
ReplyDeleteI think experts of all stripes should inform the process of discerning "what" our children should learn and parents should have tremendous input on "how" it is learned. But I'm confused why textbook adoptions fall into the "what" column. It’s been over a decade, at least here in California, since local schools were making decisions about "what to learn" in math. The state standards, especially in math, again here in CA, are extensively articulated and fleshed out with test blueprints and example problems. This, as we all know, is then well enforced by the state accountability machine. Consequently, little in the way of district textbook adoptions determines the pure content that is available to kids. The local textbook process is very much a "how" decision, ---a "how" that varies enough to greatly obscure or enhance the "what," true, but still a how. Or is it different in Connecticut?
“Is the real root of it [Constructivism] the social justice agenda?”
No. The real root of Constructivism, specifically, is Piaget and his view of the child's brain. But then again, you'd be surprised how little of education reform CAN'T be somehow linked to social justice.
As for constructivism, I think that it, like milk, is essential when very young but, after a little while, does not make a complete diet and will soon start to make you ill if consumed in very large doses. You still need it, though, to make your better, but bitter, beverages drinkable. With a little constructive struggle, as Paul points out, we're motivated to learn and remember what might otherwise be utterly abstract and meaningless.
"The Chasm"
In my view...
The chasm between experts and teachers comes from experts giving advice and teachers perceiving directives. Experts try to offer observations, teachers only hear criticism. Don't know who to blame for this one. They just want to help, we just want some respect.
The chasm between parents and teachers comes from parents wanting what they believe their child needs, rightfully, and teachers wanting, rightfully, to meet the needs of dozens, hundreds or thousands of very different children. When the needs conflict, as they must in anything but the Stepford Suburbs, so do the teachers and parents. Parents think, "The teacher isn't listening!" Teachers think, "How can I listen to everybody?"
A common thread is that no side actually sees the other at work. Teachers almost never go to workplaces to watch and hear the frustrations of managers with ill-prepared workers. Managers almost never come to schools to see the highs and lows of our public education system, let alone see the others in those supposedly menacing Asian and Scandavian countries. Teachers rarely sit down with each kid and family, until there's a problem, to understand what they want. Parents rarely observe the whole class in action before determining why, they think, their child's needs aren't being met. Add to the equation that administrators and policy-makers are also tremendously absent from the classroom and teachers from the data and you start to realize that there are chasms between everyone.
There are undoubtedly exceptions to this rule, but not nearly enough.
"I think experts of all stripes should inform the process of discerning 'what' our children should learn and parents should have tremendous input on 'how' it is learned."
ReplyDeleteBut that's the issue at PAUSD and most places; no process for input from any other stripe; parental or expert. The schools decide and then inform the parents and community.
CA sets better state standards than most in math, but as seen in PAUSD, it isn't enough. I don't look to the state for high expectations, only for low cutoff levels. Unfortunately, minimum cutoffs become maximum targets. Some people in our town are more than happy to translate success in getting most students over a minimun cutoff into "quality education".
Constructivism is whatever a school wants it to be, but it's always done in class in groups. I'd be surprised if that was a tenet of Piaget's work. No matter how effective it is, less material gets covered. The real goal is not constructivism, it's group work in class.
Most school constructivism seems more like catering to the kids rather than setting high standards and helping motivation. As I've said so many times and years ago, one student in a group might have the light bulb go on and then proceed to directly teach it (poorly?) to the rest. The classroom might look like a happy beehive of learning, but what's getting done?
"The chasm between experts and teachers comes from experts giving advice and teachers perceiving directives. Experts try to offer observations, teachers only hear criticism. Don't know who to blame for this one. They just want to help, we just want some respect."
ReplyDeleteI can't tell you how much I disagree with this. It's much, much more that that simple analysis. It has to do with fundamental differences of opinion over what constitutes a proper education. I once told a couple of members of our school committee that the school should hand out Hirsch's Core Knowledge series of books and tell parents that this is NOT the education their kids will receive. What kind of respect do schools show parents when they decide on fundamental assumptions with NO input from parents. Our town is supposed to have a Citizen's Curriculum Committee (I was supposed to be on it), but it has never held one meeting. The schools don't want it and the people in town haven't pushed the issue. Look at PAUSD. This is the norm. The chasm doesn't exist because some experts aren't going about the process in a constructive way.
" from parents wanting what they believe their child needs, rightfully, and teachers wanting, rightfully, to meet the needs of dozens, hundreds or thousands of very different children."
I've heard this so many times before. Do you think that's why I've been fighting this for about ten years? For just my son? Schools hide behind the different needs of so many kids. This is what I heard from the head of curriculum at my son's old private school. Everyday Math is better for our mix of kids. End of discussion. There is no process. Perhaps the problem is that the criticism is not simply a matter of offering helpful advice. It's about fundamental concepts and levels of expectation.
"Teachers think, 'How can I listen to everybody?'"
I don't believe this. They don't even try. There is no process. Many of the things discussed at KTM deal with the issues of all levels of ability. In fact, many of the curriculum issues cut across all levels. It has to do with proper math versus play math, not different needs for different kids.
"Parents rarely observe the whole class in action before determining why, they think, their child's needs aren't being met."
Parents know plenty about what's going on, and many issues have to do with basic assumptions, expectations, and curricula for the entire school, not just one classroom.
There is no process. That's the way schools want it.
I think that those drawn to K-6 tend to be nurturing types, and many of them want to work with special needs students. They are driven by caring and the desire to treat all kids as equals.That sounds right to me, and I also think many K-6 teachers believe (to greater or lesser extent) that the kinds of academic skills one can test directly are either fixed, or unimportant for young kids, or both. Which does make it almost logical to want to deemphasize performance--if you think that one can't do anything to alter performance (or that performance doesn't matter), it must seem particularly cruel to allow some students to feel inferior to others due to performance levels.
ReplyDeleteSo you have the double whammy of not really believing you can do anything directly to alter performance (making it a potent fantasy that you can do so indirectly, through "building the whole child") and of normal human nature that (reasonably, for most situations, but not for school) resists investing a tremendous amount of time and effort without being able to immediately see (or appreciate) a payoff.
That's how it seems to me, at least.
If the problem is educators having to cater to different sets of preferences from different parents, isn't the solution to allow those different parents to choose which kind of education they want? And if teachers have different philosophies on teaching, shouldn't they also be able to decide on which schools best fit their preferences?
ReplyDeleteThe fundamental question is: ketchup or mustard?