(Preconceived notions about place value)
Oh man! This is exactly what I've been noticing about the way reading gets taught these days. Reading teachers are convinced that there's tons of kids out there who read perfectly fluently but don't "comprehend". This is because they ask the kid a question like "what do you think will happen next?" and the kid says "I don't know." They think this proves the kid "doesn't comprehend".
ARGH! It's like they're just inventing problems out of thin air, as if nature doesn't provide us with enough.
I've never understood all this concern about teaching place value. It's not a particularly difficult concept and it is a crucial foundational concept that shouldn't be put off. I introduced my daughter to tens and ones at the age of 4. She's not gifted but she understands it. She is now doing online school with the California Virtual Academy and they cover place value in a K-Grade 1 online math class. If students really are not able to grasp place value, it can only be because schools aren't teaching it properly. If I'm not mistaken, most countries introduce place value in 1st grade. If their students get it and ours don't, we have to seriously look at how math is being taught.
That is most curious. There are, of course, children who at 4th and 5th grade are still remarkably shaky on numbers in a way that place value work (yes, even with manipulatives to some extent) is useful (I tutor some of those kids). But a gifted and talented conference? What? Those kids (almost) all have a great grasp on numbers (I'm a parent to some of those kids), and going back to manipulatives would be a serious waste of their time. Talk about not knowing your audience!
The idea that you shouldn't teach an algorithm before children can developmentally understand a concept is really dangerous. Often teaching the algorithm first will actually make it far easier to understand the concept later on. There is no harm in telling a child that these are the steps you need to solve a problem. Later on the why will make sense. We've got this problem now that kids don't understand the concepts because we don't teach them the algorithms and we're not teaching the algorithms because they don't understand the concept. It's like being stuck in an infinite loop.
I think an important idea touched on here is the connection between understanding math concepts and being able to verbalize those concepts. The connection is not necessarily very close. Some people would argue that "If you can't say it, you don't know it". I don't agree with that. And the contrapositive, "If you know it, you can say it" is obviously just as untrue. So what is true about knowing and saying? My perspective would be that saying is very valuable to knowing, and knowing is very valuable to saying. We should try to keep them together as much as we can. But we always need to remember that they can be separate, and often are. I have a few years experience teaching college freshman math. Here is something I have noticed again and again when trying to help a student in my office. I will have the student attempt a problem, and I will offer as much explanation as I can. But time and again I will be struggling to put together a string of words that concisely expresses whatever idea or information the student is missing, when the student will suddenly say, "Oh, I get it!" That's my cue to shut up. I am not a constructivist in the usual meaning of the term these days, but the constructivists are very much right about the central idea of the learner constructing his or her own knowledge. People who call themselves constructivists, however, don't seem to apply the really important meaning of constructivism. When a student comes to me for help, and after a bit of preliminary groping to understand the situation, I say, "Let's try this problem here . . .". In doing so I am setting up some elements that the student may use to construct knowledge. The student takes the problem and tries to assemble the elements of that problem into some sort of meaningful structure. I offer help as best I can, which typically includes verbalizing the essential math concepts that are involved. But my verbalizations are imperfect. The student's understanding of what I am saying is imperfect. Understanding comes when the student manages to take the elements of the problem and see how they are to be assembled together in a meaningful way as required by the problem. I think a very common fallacy for teachers is to think, "I explained it. They understand it. Therefore my explaining caused the understanding." That's not totally fallacious, of course, in most cases. A good explanation can be a powerful contributing cause to understanding, often even a necessary cause, but not necessarily a complete and sufficient cause. We don't just explain. We also assign problems. Every problem is a set of elements that the student must assemble into some form as required by whatever mathematical idea is being taught. Students must indeed construct their own learning. It's strange that adherents of constructivism seem to want to do anything but deliver to the student a carefully crafted "learning kit", which is what the combination of a good textbook, a good explanation, and a well chosen assignment is.
All this points to the idea that understanding, or the lack of understanding, is not necessarily easy to diagnose. Suppose I ask a student, "How many thirds make a whole", and the student looks a little puzzled. (And I have had a few occasions to suspect college students may not understand the basic meaning of a fraction.) Whether in third grade, seventh grade, or even college, a blank look may mean the student is wondering just what I mean and where it is leading, or it may indeed mean the student really doesn't understand the very simple primitive meaning of a fraction. Or, I may ask a student that same question and the immediate response is "three" with no hesitation at all. Can we take this to mean that of course that student understands the basic meaning of a fraction? I don’t think so. We should not take this as full and definitive confirmation that the student has the basic understanding of what a fraction means that we want that student to have. It is simple one bit of evidence Understanding, I would argue, is never easy to assess. It normally must come from observations and analysis over time. Unless you're an ideologue, of course. Then it's amazingly simple and easy, as examples given attest. And, if I may mention it, I have elaborated on what I consider real constructivism at http://www.brianrude.com/constv.htm.
"Discovery" or "constructing ones own knowledge" comes about through careful hints and prods as Brian Rude so accurately points out. In the examples he provides, what he is talking about is "scaffolding". Start with something the student understands and use that as the springboard for the concept/procedure you are trying to get across. If an assignment is constructed properly, sudents take away a good understanding of the material by the time they are done whether conducted in class or as homework Sometimes this manifests itself in an an "aha" experience brought about by procedural fluency as Brian Rude pointed out with his "Oh, now I get it" example. In tutoring students, I've seen that the procedural fluency resulting from the exercises helps clarify the concept, even if it wasn't fully understood before starting the problem set. I also agree with Brian's characterization and thoughts on "understanding". (See http://www.educationnews.org/commentaries/opinions_on_education/93277.html
October 31, 2010 10:56 AM
I recall walking along line of black cherry trees during first grade (1956-57) to pick up sticks. [Apparently we were too poor to buy "manipulatives", or our teachers knew when and when not to spend precious resources on such things.] The sticks were brought into the classroom where some were kept as “ones”. Others were bundled into “tens” with rubber bands. Some of the tens bundles were (without removing the tens rubber bands) grouped into ten and bundled with a bigger rubber band into “hundreds” bundles. These bundles could be grouped and regrouped, (carried to and borrowed from) much more easily than can be done with those yellow cubes and sticks and flats that cost a lot of money. Let’s see what happens when a student tries to “unbundle” (break) a stick or flat. But I digress. We learned place value both on the blackboard and with physical models that showed what we were doing on the blackboard. We combined rote with logic and visualization. It worked. Why do we have so much trouble remembering what works?
Friday, December 31, 2010
Favorite comments of '10: FedUpMom, Anonymous, Lsquared, FMA, Brian Rude, Barry Garelick, & Mark Bohland on place value
(Preconceived notions about place value)
(Math problems of the week: 4th grade Everyday Math vs. Singapore Math)
[Paul Sally] used to be director of the Chicago Math project-- though some googling just now says he left because he and the educators kept butting heads--they wanted to dumb down his curriculum....
Four years into the Chicago Math experiment, Sally departed as director, pointedly. “I got fed up with the educational bureaucracy,” he recalls, expressing the view that school leaders generally felt the best way to engage students in math was to make the math easier. He wanted to make it more challenging, in part by teaching the concepts behind simple mathematical operations—why any number multiplied by zero equals zero, or why the product of two negative numbers is always positive.I wonder what he envisioned as the ideal math program for elementary school-- as the homeschooling mother of a daughter who adores math (I told her she could do math whenever she wanted and she acted like I gave her unlimited nintendo!), I'd like to see what the Sally curriculum would be!
I have spoken with Jim Milgram, a math professor from Stanford, who knows Paul Sally and of his involvement with Everyday Math. The description you provided via Google is correct. What it leaves out is that the Chicago Math (i.e., Everyday Math) program as originally envisioned was for gifted and talented students. The lattice method of multiplication, which is a mainstay of the current incarnation of EM, was originally included as a sidebar type of discussion, not as an alternative algorithm. The sidebar was meant to provide some discussion of why the method worked--something notably missing from the current EM. Jim remarked that he can spot some math problems in the current EM which were part of the original, and it is interesting and disheartening to him to see how the problems are just left as problems with none of the discussion and development that Sally and crew had originally intended.
The forces of the ed school politics at U of Chicago prevailed at that time, and Sally was unable to push back against it. This may seem incredible given a mathematician of Sally's stature, but not so incredible when you consider that teachers with frighteningly little math knowledge and proficiency have told Jim Milgram that he doesn't know what he's talking about when it comes to how kids learn math. Jim would be the first to admit he is not an expert on pedagogy, but he does know what content students need to master and the proper sequence for presenting it. Content and sequence in the ed school perspective are viewed as obstacles that have prevented students from learning math.
(Is it bad to give children frequent tests?)
Too many educators are critical of grades and testing because they are mistakenly concerned about competition or hurt self-esteem. I have always found this odd. Even, as a child, I always saw grades and tests as an indication of how well I was doing and how much more I needed to do. My self-esteem was never harmed. I never felt like I was in competition with other students. I remember I used to always fail spelling tests in 4th grade. The teacher stepped in to help and with extra work, I started to score 100% on all of my tests. If I hadn't been tested and graded, I would never have know how I was doing. It would have been hard for the teacher to know what my weaknesses were and step in to help. I'm wary of high stakes state and national tests for young children but I think regular classroom testing is a necessity. I would also like to see multiple tests required for graduation like most other countries do. It creates a strong incentive to learn and work hard.
(Is there a left-brain learning style?)
You've kind of shifted the focus of the discussion, I think. It seems to me that cognitive theorists would have little trouble with the proposal that kids vary in their information processing capacity, although it would be hard to distinguish between predictions of the slow-serial/fast-parallel contrast you are proposing, and a more strictly quantitative contrast between slow/fast serial processing (with sufficiently fast switching masquerading as parallel). At the level of processing I'm familiar with (early perceptual processing of speech), this distinction wouldn't matter, but it would probably be important when trying to extend this to more complex processing, i.e. your history example.
There is definitely individual variation in such basic capacities as working memory/selective attention. I don't know much about research on variation within cognitively "normal" kids (i.e. kids who perform within age-normal criteria on standard tests) but there are definitely studies looking into apparent differences in working memory and sustained attention in specific clinical populations (i.e. a student I worked with found a small but significant difference in sustained attention in children with Specific Language Impairment). Similar studies have looked at working memory capacity, and probably other cognitive mechanisms as well, and I'm sure there are similar studies with kids "on the spectrum", but I don't know if anyone's looked at such variation within kids who are performing adequately in school.
It makes more sense to me to grade as follows:
Create tests that measure a lot of easy stuff, a lot of grade level stuff, and a bunch of stuff above grade level.
Take the raw scores, which can range from 1 (you get a point just so the math works), to several hundred (for some genius type kids) with "grade level" calibrated at 100. Then you take the natural log of the scores.
So Lenny from "Of Mice and Men" might score 40 on his tests, so his grade will be 3.689... Joe "trying hard" might get a 70 (4.25), On Grade level Sally would get a 100 (4.6) and perhaps, if this is a math test, Srinivasa Ramanujan might have scored a 536 as a kid (a mind boggling 6.28 grade).
If you want more granularity you can choose a different base for the log (2, or 1.5 or something)-- everything crams together if you use log10 though... 4.5-5 would be a reasonable thing to try to achieve on tests (there might be factors that make it difficult, but it is not superhuman).
I haven't thought it out completely, and I imagine things would need to be curved and given to a large number of students to figure out where the middle should lie, but its a system of nearly infinite granularity that is expressed with numbers less than 10...
Thursday, December 30, 2010
(Math problems of the week: traditional math vs. Reform Math)
I'm a high school math teacher in Texas. The CPM assignment made me laugh out loud. Who dreams crap like this up?? What in the world do "feelings" have to do with learning math???? If I only learned the things I felt like learning, I'd never go to another inservice as long as I lived.
(Barry Garelick on traditional vs. modern math instruction)
I thought it was standard in academia to provide citations -- NOT to tell people to do their own research.
(The humanely arranged classroom)
I often get marked down on my yearly evaluation because my students are seated in rows for the entire year unless we are doing an activity that requires partners or groups. I could care less about my evaluations. I know I am a good teacher, my students tell me they learn in my class, and every year parents request me.
I value the opinion of my students and their parents much more than that of my administration. No offense intended. The have all been brainwashed by the current educational trends.
(The year in review)
This has been the experience of several parents of children with autism in our community. Most of what they learn, they are taught at home. Parents don't fault the classroom teachers, they fault naive and over-optimistic assessments by the special ed department (then written into IEP's), which failed to figure out what specific practices and systems would make it possible to adapt the mainstream curriculum for their children. My take-away from this is that there will have to be a lot more nuts-and-bolts, practice-oriented research before children with autism (many of them) will be well-served in mainstream classrooms.
(Are humanities majors more compassionate than science majors?)
Med schools have never cared about student majors. They don't require upper division science coursework at all, although some California schools are starting to require upper division biology, and there have been discussions about dropping physics for at least the last ten years.
Med schools require organic chemistry for two reasons that have nothing to do with whether a student understands a Diels-Alder reaction or not.
1. It cuts down the number of applicants -- think about how many college students were pre-med until they hit organic.
2. It tests whether students can absorb a large amount of unfamiliar information. Every other premed course (General Chemistry, General Biology, and General Physics) is very familiar to a well prepared high school student. In fact, since med schools don't accept AP credit, many pre-med students are just retaking things they did already in high school. Organic is the only course that they are unlikely to have seen, and it requires learning an unfamiliar scientific logic and language very quickly. That's a marker for an ability to do well in med schools, but if they really cared about material, they'd require biochemistry.
At my institution we have a pre-med post-baccalaureate program, where students without science backgrounds complete the pre-med requirements in two years. These students get into med school at very high rates but they definitely are not science students. Our pre-med coordinator came back from a conference with medical school admissions officers, and was shocked at how little they cared about science coursework or research. In fact, a student who had done too much science research was at a disadvantage to students who had lots of "community engagement" that didn't have any science-focus. The basic attitude was "why don't they just get a Ph.D.".
Even the physics requirement shows that they don't exactly care about science knowledge as much as ability to memorize. Med schools don't require calculus based physics, which is typically much more systematic and which is based on deriving the formulas you need rather than memorizing. They'll take it, but they are perfectly happy with an algebra-based physics course (often called physics for life sciences).
Wednesday, December 29, 2010
(Math problem of the week: landmark numbers...)
I never heard of "landmark numbers". Can I make up my own, or are they defined somewhere? Is this concept like the new "compatible numbers" which I have seen defined as a "friendly number" that is easy to work with. It's very important to advise family members, friends, colleagues, other parents about landmark numbers - use of them may be a sure sign of a crappy math program!
(Where every child is labeled gifted)
It isn't just parent initiative. I know a couple of kids who are in gifted programs who were placed by the schools. The parents never requested that they be placed in gifted programs. Both of these kids are below grade level in Math. I don't know about reading. So, it seems that schools have some desire to place poorly performing students in these programs. I'm not sure why. I knew a mother who went through months of school visits to get her son into an honors Math program at his elementary school. They were very reluctant to put him into the program even though he got all A's in Math. That was in the mid 90s. Things have obviously changed a lot since then.
(How to ensure that autism is a tiny disability)
What, you think they don't have room for nonverbal kids like mine in these camps and social groups?? Oh, even the PUBLIC PRESCHOOL that is taxpayer-funded wants to run a daycare as well. Um, but children like mine are specifically excluded because they soak up too much staff time. Just burns me up. Can we please just QUIT PRETENDING we accept people with disabilities? Because we don't. I'm discouraged thinking that when my son is older, I won't be able to go out any more AT ALL because he can't go to the restroom with me... and I can't very well leave him outside the ladies' room and expect him to be ok.
July 13, 2010 10:53 PM
Nancy Bea Miller
I can't even tell you how many programs and studies my very autistic son was turned away from, only to later have enthusiastic people tell me about this great program for kids like my son, or how exciting it is that this new study (one that my son was turned down for) has found some exciting new news that will help people (like my son.) Oh the irony, and the injury to one's very soul.
(Math problems of the week: 6th grade...)
My response to their question would be: "I could tell you, but it would be much better if you discover the answer yourself."
(The logic of computer programming courses)
The high school my son is starting at next year has no computer science. The "computer" classes they have are web-page design and computer animation (neither involving programming, just use of WYSIWYG tools).
Obi-Wandreas, The Funky Viking
When I was a kid in the early 80s, I subscribed to PBS's "The Electric Company" magazine. In the magazine would often be a computer game of one form or another. But not on a disk - oh, good heavens no! On a sidebar they would have a description of the game, and then the source code in BASIC. You type it into your Apple ][ and boom! - there's your game. They never explained the code, but it was so simple that, as you typed, you could pretty much figure out what each line was telling the computer to do. If I had a link to download each program, I would never have given it a second thought. But when you do it yourself? - that gets a kid interested.
Tuesday, December 28, 2010
(Drawing the right lessons about creativity)
The problem I have with this is the fragmentation involved. Students are picking up bits and pieces of knowledge through these teaching methods. They aren't developing any overall understanding of the various sciences, math or writing. Unfortunately, few of these students will be able to go on to science or engineering majors because they lack a strong foundation in these subjects. I am all for teaching problem solving, creativity, and critical thinking in school. But it should be in addition to a knowledge curriculum that provides a strong basic foundation. Why not teach the basics of physics first and then assign real-world problems based on concepts taught? Why not put the horse before the cart instead of the cart before the horse? Knowledge is at the root of problem solving, critical thinking and creativity.
"Ted Schwarzrock...had been pushed into medical school, where he felt stifled and commonly had run-ins with professors and bosses. But eventually, he found a way to combine his creativity and medical expertise: inventing new medical technologies."
Schwarzrock combined his strong medical KNOWLEDGE with his creativity to become so successful. Medical knowledge alone would have allowed him to be successful. But creativity alone would not have got him where he is today. Both together are best. Schools need to realise that "skills" divorced from knowledge are meaningless in the real world.
It's interesting also that schools are using group projects as a means of promoting creativity. Studies have found that groups actually lessen creativity because groupthink tends to occur. It is often far better to send your team home and come up with potential solutions to a problem separately and then come together to afterwards to discuss those solutions. But this is for business purposes where each individual plays a certain role and so solutions can't be implemented by individuals. In schools, there is really little need for groups to deal with problems. It would make more sense to encourage more individual problem solving.
(Preventing bullying: yet more reasons for making students work in groups)
Nancy Bea Miller
One of my sons was bullied in middle school when he was assigned to be lab partner with two classic "mean girls": popular and cruel. How the teacher thought putting a shy, smart, softspoken boy into a threesome with two "heathers" would ever work I simply can't fathom. My son finally told me what was going on, after a miserable week or so and I got onto that teacher as fast as fast could be. To give the teacher credit, he responded immediately, but how about trying to avoid such situations in the first place? Kids are powerless in the hands of thoughtless adults. This educator must have thought all was well with the group he'd assigned...till I clued him in. How many other kids don't speak up?
(Math problem of the week: 3rd grade Singapore Math vs...)
Assuming I'm remembering correctly, I may know the answer to that question. At the end of C's 4th grade year, he placed into book 3B in the Primary Mathematics series. '3B' is the second semester of 3rd grade. C. was 1 1/2 years behind students in Singapore. Recently, a mom here whose child has had Math Trailblazers since Kindergarten told me that she did the same thing I did: she gave her daughter the placement test at the end of 4th grade. Her daughter placed into the middle of 2nd grade. (I'll check with the mom to make sure her daughter was the same age as C. --- but I'm fairly sure I'm remembering correctly.)
(More artsy science)
Science in and of itself is fascinating. There really is no need to do this kind of thing. A good teacher can generate excitement and amazement and teach real indepth content at the same time. With our modern education system, these teachers aren't really wanted anymore. What is wanted now is not teachers but facilitators of learning who will implement techniques like Discovery Learning, even though decades of research has found that these methods don't work. Not suprisingly, according to projections, America will have a serious shortage of scientists in the future. How can we remain an economic superpower when modern economies are based so much on math, science and technology?
(Do gifted programs gate off the truly gifted)
A mother of twin high school seniors told me earlier this year about how both boys were tested for the gifted (GATE) program in elementary school. Only one boy was accepted and out of curiosity, she asked to see the other's test scores. She and the Guidance Counselor were surprised to find that the one declined had higher scores, but he was more socially shy and awkward.
Good behavior was "rewarded" by being admitted into gifted classes. When I subbed in emotional support and autistic support classes, I would see lowered expectations and some very brilliant insights. When I taught in gifted classes, I would see well-behaved kids who were great at regurgitating concrete facts.
Just as teachers seem to have a different idea of what "gifted" means, I think they have a similar idea about the word "maturity".
Somehow, both of them seem to boil down to brain power that's average or a bit above combined with obedience.
Monday, December 27, 2010
(Gifted programs and obedient kids)
I had read our district's gifted policy and I am not sure exactly what my ideal description of an effective gifted program would be, but this isn't it. Here is a sample from our website: "The main goal of the senior high gifted program is to encourage the gifted students to challenge themselves and become self-motivated learners. It is also our desire that gifted students become producers of information and performers of artistic feats and services to society. The curricular framework for meeting the wide range of gifted needs and abilities include these essentials: Affective skills Leadership skills Communication skills Creative thinking skills Decision making skills Critical thinking skills Logical thinking skills Organization and management skills Research and independent study skills Specific content and career exploration"
I wonder why a public school should be interested in my kid's "affective skills" and what are they anyway? It seems to be from "Bloom's Taxonomy" which seems to be another educational atrocity from the limited research I have done: "Affective: growth in feelings or emotional areas (Attitude)". Includes "valuing" and "internalizing values or characterization" which seem to be the parent's domain, not the school's - but it supports our suspicion that certain kids with certain values are being excluded from opportunities, and that intelligence and academic ability are no longer the focal point of the gifted program.
(Sociability and academic distinction)
Math contests and science fairs don't require introversion as an entrance requirement. They are open to anyone who qualifies. It isn't fair to require extroversion as an entrance requirement for honors or gifted programs. The whole point of these programs should be to provide a more challenging curriculum for students who are unchallenged in regular classrooms. The ability to do higher level work should be the only entrance requirement.
I know students who aren't very bright who are in gifted and honors programs. They are being accepted into magnet schools. I would assume these not so bright students are achieving these things based on leadership skills and sociability alone, since academics are not their strong point. Of course, these programs are inevitably being dumbed down to accommodate these social but not so smart students.
So, the question is, what is out there for our highly intelligent students? Even our extroverted, highly intelligent students who can get accepted in these programs are going to go unchallenged due to the dumbing down of gifted and magnet programs.
(A Chinese marshall plan for U.S. math)
Niels Henrik Abel
I bet it never occurs to educrats that one reason why math people stay away in crowds from teaching is because of the pointless insistence on educational indoctrination as the only path to certification. As long as the powers that be care more about certification than qualification, we'll continue to have shortages of competent math (and science, too, for that matter) teachers.
(Neurologists argue for expanding group work)
There is a presumption in ed school that students have a natural inclination to collaborate. I had the opportunity to teach a lesson in an algebra class at a local high school--this was part of an ed school class I was taking. As an experiment, when I assigned a problem, I told the students they could work in groups, or not, it was up to them. Only one group formed; the rest were content to work by themselves. I had to report on my experience teaching the lesson in my ed school class, and actually have the class do one of the problems I had assigned. I gave the same instruction. Again, only one group formed: the teacher and the student sitting next to her. When I reported that the same thing happened in the class where I taught the lesson, the teacher remarked "Interesting!"
(April 1st Times article on the Blue School)
Yes! They can play all day with glo-sticks and send the math home in the backpacks!! It's so clever, and they get tons of money per student each year! Even the mob couldn't think of a racket this clever.
This doesn't bug me, because it is private. If parents want to spend their own money on this nonsense, then fine. My problem starts when they start using my tax dollars to duplicate it (since most elementary teachers I know would be charmed by this article) or when I am expected to send my kid to the public schools that follow this kind of philosophy.
Sunday, December 26, 2010
It can be even worse than you think - the anti-bullying programs can themselves wind up targeting some socially awkward kids. That kid is baited until he or she lashes out, and is then taken to the teacher. Since the original baiting seems like normal childish behavior, and often the socially awkward child can't explain exactly what set them off, it can be the bullied child who actually gets punished. One of many reasons we are planning to homeschool is that our families both skip between merely geeky and mildly autistic, but wherever you are in that curve, you are bully bait.
I have a suggestion to support the "tidy math students." Offer two curricula, messy math and tidy math. Parents can select either one. After all, if the argument is that tidy math doesn't work for a certain section of the population, then it follows that messy math won't work for the rest of the population. Allow the parents who work in quantitative professions to choose the math programs which, in their judgement, best prepare their children for quantitative professions. Don't refuse to allow lawyers' children access to the tidy math curriculum, but don't force tidy children to work with a messy math curriculum.
In the wealthier districts like mine parents get tutors for their kids, or become de facto tutors. This is never taken into account when these districts brag about their quality and test scores. This is our 2nd year of constructivist math, 3rd grade, and its been a nightmare for certain kids (and their parents), with hours-long homework sessions resulting in still-failing grades (or non-grades, we use "indictors" here, I suspect, to cover up for this math...). The school is very willing to refer parents to tutors who charge $40/hr while pretending each kid is the "only one" struggling, but even before this every teacher who wanted to tutor gets booked all summer. I cannot tell how constructivist our math classes are in practice, but the materials for the program definitely expects it to be taught that way. And it is everywhere -"socratic circles" are in middle school social studies and english classes, in lieu of teaching, and my shy daughter can never come up with anything to say. But then our high school mathematics program mentions using Bloom's Taxonomy, which I do not fully comprehend but its supposed to encompass cognitive, affective and psychomotor areas of learning. Huh? How about just teaching math and we'll worry about the rest. Home schooling is looking better and better...
My child's kindergarten teacher decided he didn't know how to read becauase he wasn't shouting out answers out of turn, demanding to answer every question, and jumping up and down like a Price is Right contestant. After he was finally tested at the end of the year,(she illegally disregarded the written request 3 months earlier) she found she had him in a group that was 2 GRADE LEVELS below his instructional level. American children do have manners, if they are from homes that teach manners. It is sad that veteran teachers with Master's Degrees can't figure this out.
The idea that the Asian educational process stifles creativity is a bit misguided, but the idea that the American educational system enhances it borders on senseless. If you can't teach something as straightforward as reading and arithmetic, how are you going to teach something as ambiguous as creativity? And particularly, how are you going to do it when most classroom teachers value their own rules over almost anything else? My fourth-grade son regularly gets in trouble in American school for all sorts of violations. He has never gotten into trouble in his Saturday (accelerated, full day) Japanese school.
By claiming to teach creativity, teachers give themselves an escape from accountability, since creativity isn't something you can measure. The same goes for critical thinking without content.
Students of wealthy parents don't do well because the parents bring them to museums, zoos, etc. They do well because wealthy parents usually see the deficiencies in the educational system and remedy them at home.
My theory is that everyone's complaints are justified. Progressives look at the public schools and say "this isn't a good progressive education!" and they're absolutely right. It's not progressive because there's no room for kids to develop their own interests. The homework overload and constant grading mean that it's almost impossible for a child to develop any sense of themselves as learners. Intrinsic motivation? Forget it. Traditionalists look at the schools and say "this isn't a good traditional education!" and they're absolutely right too. The curriculum is set at an incredibly low standard, and even that doesn't get met reliably. And if the kid didn't understand something, the school outsources the problem to the parents. There's a huge gulf between any theory and an actual public school classroom. In a classroom with 30 kids and one teacher, which has been "balanced" to include a couple of gifted kids, a couple of kids with learning problems, and a couple of kids with behavior problems, just keeping the peace for 6 hours a day is a tall order. This is why progressive theory finally gets implemented as graded coloring projects, and traditional theory gets implemented as endless repetition.
I come from Ireland originally and I was exposed to what would be considered a "drill and kill" education by many people involved in American education. But I enjoyed school and learning. I have an mp3 player filled with podcasts covering legal theory, science and other subjects. My educaton did not turn me off learning but actually increased both my desire and ability to learn. A certain amount of rote learning and drilling are required to produce a well-educated child. Rote learning and drilling aren't the only ways children should learn but they are necessary. People who think you can educate a child without a certain amount of memorization and drilling are out-of-touch with reality. I always hear that kids should be thought problem solving skills and critical thinking skills and not just a bunch of facts. But you can't solve problems without a base of knowledge relevant to that problem. You can't think critically about subjects you don't know anything about. I also hear that there is no point in teaching facts because students can use modern technology to look up whatever they need to know. But to actually make sense of what you have looked up, you need a pre-existing body of knowledge. There is no getting around it. Students have to be given a large base of knowledge built on a coherent curriculum.
Saturday, December 25, 2010
(How text to self connections might backfire)
This is so typical of the modern educational complex--they've taken out explicit phonics instruction and high-content material, the two things that actually teach a child to read. And then the thing that they pick to substitute for those efficacious things is not just a massive waste of time, but is actually training kids in a damaging and wrong-headed practice. It's not enough to waste the kids' lives--they have to actually teach them negative cognitive habits, too.
(Math problems of the week: 2nd grade Investigations vs. Singapore Math)
I was thinking about these TERC problems again. Look at the instructions: "Solve each problem. Show your work." The point to take away is that in 2nd grade the TERC authors think of adding two one-digit numbers as a "problem" for which work needs to be shown. I would consider that a "fact" that one should know by automatic recall by now, and let's get on to more and more and more (for example, adding numbers close to 100 to other two- and three-digit numbers). But the two problem sets are still related, because the TERC problem set is based (if I infer correctly) on the "making 10" strategy for learning the basic math facts that have to do with adding "large" one-digit numbers (9, 8, 7, 6). The Singapore problem set is for practicing the strategy of adding 98, for example, by adding 100 and subtracting 2. What Singapore does very well is they explicitly teach these kinds of strategies. Eventually kids don't need a strategy any more for adding 9+7, they will recall that; but they will always need a strategy for adding 98 (even if they get to where they can do it quickly). So this comparison of problems shows two things about TERC vs Singapore: (1) Singapore will get kids to automatic recall of simple things early on, and then move on to harder stuff; and (2) Singapore explicitly teaches kids strategies for computation (mental math, let's call 'em) and problem solving (think of the bar models and so on). It seems a waste of time and energy to be still worry about how to "solve" 9+7 when you're in second grade. In first grade, however, that counts as a problem that needs to be solved ... until you just know it. What counts as a problem is like the horizon. The further you go, the more it recedes.
(Enlightened exchanges about Reform Math)
I find that there are many people who mischaracterize traditional math as one that consists only of drill and "mindless rote". I went to school in the 50's and 60's and have looked at many of the textbooks in use at that time. Procedures are explained, in terms of what one is doing when executing a particular algorithm, as well as what types of problems are addressed by such procedure. Thus, multiplication of fractions is not simply left as an endless list of fraction multiplication problems, but includes word problems as well. The anonymous commenter that Katharine talks about is saying that inquiry-based teaching is getting a bad rap. I would say that inquiry based teaching done wrong has become prevalent, and not just in math. Student-led teaching takes the form of students working in groups and helping each other "construct knowledge"--supposedly with teacher guidance. There have been attacks by reformers on the idea of students "sitting in rows with the teacher at the front of the class, teaching", as if sitting in rows stifles knowledge, that no "ah-ha" experiences can occur via direct instruction, and that the teacher never asks leading questions or scaffolds students. (Sherry Fraser's testimony to the National Math Panel comes to mind; she's a principal of the IMP math series.)
(Artsy science; what about sciency art?)
The science of art is a fascinating subject and a good school could do a lot with it. The laws of perspective and the way a lens works (and how it relates to the way the eye works) are two subjects just off the top of my head. Art history can also be fascinating from a mathy perspective -- for instance, how did Philip Steadman prove that Vermeer must have used a camera obscura for his drawings? More recently, the painter David Hockney has floated the thesis that many of our favorite Renaissance painters were using mirrors, lenses and other optical tools much earlier than was previously believed. Can his theories be proven? As usual, it's all about the implementation. There's no reason that the intersection of science and art should be some dumb project like making a mole.
Friday, December 24, 2010
I think people also miss the mastery aspect of creativity: truly creative people have mastered all that has gone on before them. We underestimate the role that true mastery plays. For example, I was telling someone about Direct Instruction and its success. They, of course, asked "what about creativity?". They don't realize that true, high level creativity is the result of mastery of the basics.
(Discovery learning and politeness)
I have recently been teaching Emperor that even if someone else says that his doing something rude (hugging, getting too close, constant chatting) is "ok," that doesn't really mean it's "ok." No, not that they're lying, exactly (that was the question I got... why would they say it's ok if it isn't). Just more that they are telling you that it is "ok" in the same way that your answer to "How are you?" shouldn't be your entire life story... it should just be, "Fine, thank you. And how are you?" (Which of course makes no sense to him, either, but it's just what you do.)
"That's okay" as a response to an apology is sort of intended as smoothing, deflating reply, so that the child is not made to feel bad. In effect, though, it is belittling since it implies that is not worthy or mature enough to be culpable for what he did, and not significant enough as a person to take seriously. Saying "Thank you" or "I accept your apology" can mean a lot to a kid in that situation.
Favorite comments of '10: Matthew K. Tabor, Anonymous, and Bhammer on the freedom of speech of teachers
The climate in the administration of K-12 schools does not reward, let alone encourage or even condone, the type of conversations you and I would like to have with teachers. Those teachers willing to examine the issues are strangled by what's best described as fear imposed by their peers and admins. The professional climate is miserable - and then improvement comes at a snail's meandering, often misguided, pace. It's understandable when an individual teacher remains silent in a climate like this. Between the unwilling, those forced to compelled to remain silent and the do-nothing clowns in university ed departments, we're in trouble.
As a high school math teacher who felt "pushed out" by bringing these ideas to administrations, I am not surprised that many teachers avoid topics such as this. In many schools teachers, especially new teachers, are encouraged to just follow the line and don't make waves.
I have been trying to remove Investigations from my district for years, however district administrators are unwilling to engage the public in a curriculum review. It's tragic that the bad decisions of a few individuals can undermine the prosperity of an entire community. But that's exactly what happens when school leaders adopt programs that weaken the math skills of elementary students.
I'd be happy if students could possibly learn to write a short essay in which they took a position on a topic and supported their position with rational arguments. (Ban all sentences that include the phrase "I feel that..."!! Who cares how you feel? Tell me what you think.) As an added bonus, how about teaching students to use complete sentences with proper punctuation and spelling? Not using run-on sentences is nice, too.
(On Mathematically Gifted Boy Finally Gets What He Needs)
I know kids who are in GATE programs and magnet schools who aren't particularly smart and who have no interest in reading and learning. One boy I know was offered a place in a magnet school even though he is only profient in reading and basic in math. So, there really is no place for the gifted and advanced kids.
I have a cousin with a similar story. He was quite the behavior problem until his mom enrolled him in a programming class at the local community college when he was in fourth grade. From that point he did continue in his local school district and in the community college until he was a sophomore in HS. At that point he started university and finished with a PhD in some theoretical branch of math which he received when he was twenty two. His mom said the only reason he stayed in HS as long as he did was because he enjoyed playing bridge with the teachers.
Wednesday, December 22, 2010
Today's schools have joined the growing business of screening children for autism. And they've cleverly integrated their autism assessments into classroom activities and assignments. The increasing time students spend working in groups, for example, allows teachers more and more opportunities to assess a child's ability to relate to his or her peers. The increasing focus on realistic, social fiction and on social inferences in reading comprehension questions gives teachers more and more clues about a child's social reasoning skills. The increasing emphasis on sharing personal experiences and emotional reactions in writing assignments affords teachers ever greater insights into a child's ability to express him or herself emotionally. And the ever growing numbers of multi-step, multi-week, multi-disciplinary projects and heightened expectations of young children to take things home and hand things in without being explicitly reminded to do so, and to take detailed notes in class, helps teachers detect a variety of signs of executive dysfunction and attention deficit.
Monday, December 20, 2010
One of my comrade at arms in education reform was just telling me about one of her husband's colleagues, a high-ranking employee of a large, multinational corporation and father of a five-year-old boy. Apparently he recently relocated to a branch office in Lyons, France, largely because his son did not get into any of the elite U.S. schools they applied to. Public education in France, he'd heard, is much better than public education here.
Saturday, December 18, 2010
Even though it's always been one of his areas of greatest weakness, J is surprisingly willing to read--so long as the content interests him. And most things do interest him--so long as they make sense. This rules out many of those social, dialogue-intensive realistic fiction novels set in school yards and alleyways, but it still leaves history, geography, current events, and, of course, science and engineering. Recently, J was so taken with Eli Whitney and the concept of standardization of parts that he got up and walked around the house, lecturing me about all the different things it contains that are standardized--light bulbs, outlets, switches, hinges--and why their standardization makes life easier.
But because he's more of a High Functioning Autism child than an Aspie, reading more often provokes questions than lectures. Many of these questions are on target, but some stem more from his obsessions with ceiling fans and mischief making--and tend to arise as the reading gets tough and things stop making sense. So I've recently imposed a new rule that all the questions he asks during our reading sessions must be relevant to the reading.
Oh, but he's clever. So first he asks a question about the steam trains we're reading about. Then he follows this with a question about taking the train to see his uncle, and then a question about breaking the chains of his uncle's ceiling fans. When I point out that he's now violating our new rule, he grins and points out the chain of relatedness between the reading and each successive question.
He's thoroughly delighted with his ingenuity, but there's yet more joy to come. For then I point out that relatedness isn't transitive: I am related to him, and he is related to his (paternal) uncle, but I am not related to his uncle. He loves it! He knows full well what I'm getting at, but he's never encountered the term "transitive" before. In retrospect, it seems he's been aching for a word to attach to a concept he's long understood. Now that he finally has one, he springs up again and starts talking about all the relationships that are and are not transitive, putting objects inside and in front of one another ("contains" and "in front of" are transitive; "touching" is not); hypothesizing that all adjectives ending in -er are transitive; and working out that "like" and "friend of" are not.
Somewhere in the course of all this, he comes up with a metaphor for going off on tangents: "my boat has left the harbor and is going out to sea."
Indeed it has and is. And in a good way, too.
Thursday, December 16, 2010
"Real-world" measurement problems:
Tuesday, December 14, 2010
There's one way in which our schools are more old fashioned than ever before, and that's in the demands they place on parental time. So great are these demands that they assume a world of families in which at least one parent has, at most, a half-time job. Furthermore, because our society remains sexist in its expectations of men's vs. women's availability for unpaid labor, the assumption is that it's the mothers who should be volunteering their time.
Some complain that the system preys on maternal guilt and that it creates a sense that a mother’s worthiness is measured in how many hours she puts in at her children’s schools. Under the headline “Just Say NO to Volunteering,” Sarah Auerswald, a former PTA president in Los Angeles, wrote in June, “What I am about to say is not very PC, so get ready: Moms, stop volunteering so much.”
As local and state economies continue to struggle, budget cuts to rich and poor school systems are increasing the reliance on unpaid parent help, The need is so great that some school districts, like a couple of specialty schools in Prince William County, Va., have made it mandatory to commit to a small amount of volunteer time, and others are considering it. In San Jose, Calif., one elementary school district has been discussing a proposal that the families of its 13,000 students commit to 30 hours of volunteer work during the year.
Sunday, December 12, 2010
Of the 5 letters published in this week's New York Times in reaction to the PISA results, all five or which agreed that something needs to be done, only one discussed curricula.
Not one letter mentioned the most serious problem with today's U.S. curricula: those that begin as early as elementary school and leave many American students, including those at some of the most privileged schools, two years behind their international peers. Not one letter discussed how this curriculum compares, in specific ways and at specific grade levels, with curricula used in Shanghai's corner of the world.
Friday, December 10, 2010
Division word problems
Wednesday, December 8, 2010
No textbooks, no paper, no chalk, no desks, and no assigned seats.Just how newsworthy this sounds to you depends on whether you think chair mobility and table shape have a big influence on learning, on whether you've been following current trends in education over the last 50 years, and on how unusual you think it is for a teacher to "encourage his students to find answers to their own questions" and engage with them in exchanges like these:
Instead, students use laptops while sitting in rolling chairs at trapezoidal tables spaced out in hexagonal classrooms.
"Is this an obtuse triangle?" one student asks.As the Inquirer explains:
"Well, what can you tell me about an obtuse triangle?" Gaffey replies.
"One of the angles has to be more than 90 degrees," the student answers.
"Are any of the angles here like that?"
"Yeah. Oh, I get it now!"
This snippet of student-driven discussion is a glimpse of the style and approach that have earned Gaffey national and international recognition.
To fair, it wasn't this, specifically, that earned Gaffey his honors. Rather:
Les Foltos, one of the judges who reviewed Gaffey's work, was impressed by his emphasis on "actively engaging students in solving real-world problems." As Gaffey puts it, "If we want to teach math to learners, we should teach math how it is actually used. It doesn't matter how much you know. It matters what you can do."
In his classroom on a recent Tuesday, Gaffey's challenge to his "learners" - as students in the Parkside public school are called - was to estimate Earth's land area.
To solve the problem, the class first covered basic concepts about area and polygons - shapes with three or more straight sides.
Gaffey then asked, "If a shape has four sides, is it always a polygon?"
Learners who answered yes (the wrong answer) were asked to redefine what a polygon is, while those who answered no were asked to draw a four-sided shape that was not a polygon on the class "smart board."
Gaffey drew a shape with three straight sides and one curved side.
"Is this a polygon?" he asked.
"No," the class responded.
The class drew lines through each of the continents, chopping them up into complex polygons, then simple polygons.This sort of problem is not particularly new, as a quick survey through now-standard textbooks like Everyday Math and the Interactive Math Program makes clear. And it's been around long enough to have garnered some serious criticism--specifically in what Barry Garelick calls its "just in time" approach to teaching.
The final phase was to derive formulas for the areas of the simple polygons, and add up the areas.
Monday, December 6, 2010
A recent Globe article on gifted education unwittingly articulates a number of problems with today's versions of gifted education:
It's becoming clear that not every bright child needs a specially enriched program, especially as the educational mainstream shifts toward student-centred learning, which tries to take account of every child's particular needs and ways of thinking.Actually, student-centered learning is largely responsible for increasing the academic enrichment needs of gifted children. In math, for example, student-centered discovery learning--as opposed to teacher and textbook centered presentations of age-old mathematical concepts and algorithms--is a highly inefficient way to progress through mathematics, leaving mathematically gifted children bored and disengaged.
Instead, the kids who need help are those at risk of dropping out or failing because they are facing emotional and social problems.Why doesn't it occur to people that at least some of the emotional problems of gifted kids stem from boredom and disengagement in today's academically watered-down, group-centered, No Child Left Behind classrooms? Or that the social problems--bullying, isolation--might be ameliorated by allowing these children to work independently and/or to be accelerated into classrooms with intellectual peers?
Over the past few decades the definition of a gifted student worthy of special attention has been evolving away from the IQ-centred ethos that dominated the 20th century.The more heterogeneous needs are lumped together, the less likely for those with academic enrichment/acceleration needs to see those needs met.
“The cognitive assessment is only one part of the package,” says Deborah Lewis, a superintendent of learning support for the Calgary Board of Education. “There has to be a need. It's not just high grades.”
After her son began to withdraw from his Grade 3 class, Vancouver mother Erin Dyer pulled him out of school and sought a private assessment (his second – he had tested gifted in kindergarten). It got so bad, she says, his physical heath had begun to fail.The article seems to think the underlying problems are entirely emotional and largely a result of the pressures that come with heightened expectations, quoting one giftedness expert as saying:
“He seemed sick at the very thought of school,” she recalls. “He stopped reading and refused to respond to the teachers. He was shutting down, retreating into himself. He refused to participate in so many things that had once excited him. His enthusiasm for life and learning had vanished. He was skinny, pale and anxious. I felt desperate.”
“Gifted kids will often experience their giftedness as a big bag full of expectations. So there's some anxiety about being able to live up to those expectations.”
To help deal with such non-academic problems, Westmount teachers are now launching a pilot version of an “affective” (emotional) curriculum in their middle school.
For instance, teachers will lead kids into “supported failure” by asking them open-ended questions – such as whether euthanasia is ever justified – so that they can experience the frightening truth that there's not always a correct answer.
Some "gifted educators" go further, arguing that it's no longer necessary to target the academic needs of gifted children in particular because, or so they think, today's classrooms are increasingly imparting the best of gifted education to everyone:
At the Atlanta conference on Saturday, gifted-education expert Joe Renzulli's talk will lob out this challenge: Now that general education has appropriated techniques such as creative problem-solving, thinking skills and problem-based learning, what's left?
“We used to think these were the province of gifted education, but now all kids should develop them,” says Dr. Renzulli, the director of the National Research Center on the Gifted and Talent at the University of Connecticut.
Saturday, December 4, 2010
Photosynthesis may be an unlikely topic to inspire an opera or ballet, but in a 2nd grade classroom here recently, the children were asked to use dance to help them learn about that process.
“Do you think you’re ready to use your whole body?” teacher Katie Wright-Sabbatino asked near the start of the lesson, which featured learning objectives in both science and dance.
Small groups of pupils in this class at Fort Garrison Elementary School brainstormed to come up with dance movements to convey elements of photosynthesis, including water, sunlight, carbon dioxide, and chlorophyll. They leaned, they reached, they flowed, sometimes with surprising grace.
Photosynthesis changes the energy from the sun into chemical energy and splits water to liberate O2 and fixes CO2 into sugar.
[Photosynthesis] begins when energy from light is absorbed by proteins called photosynthetic reaction centers that contain chlorophylls. Some of the light energy gathered by chlorophylls is stored in the form of adenosine triphosphate (ATP). The rest of the energy is used to remove electrons from a substance such as water. These electrons are then used in the reactions that turn carbon dioxide into organic compounds. In plants, algae and cyanobacteria, this is done by a sequence of reactions called the Calvin cycle.
The general equation:2n CO2 + 2n H2O + photons → 2(CH2O)n + n O2 + 2n A
Carbon dioxide + electron donor + light energy → carbohydrate + oxygen + oxidized electron donor
The idea of integrating the arts, including dance, into the broader curriculum is not new, but it appears to be gaining a stronger foothold in public schools, proponents say, though national data are not available.
The growth comes as arts education advocates struggle to ensure adequate time and support for the arts in schools—whether music, visual arts, theater, or dance—amid the financial straits facing many districts and other challenges, such as pressure to boost test scores in core subjects like reading and math....
“It’s a way of keeping arts in the classroom,” said Laura M. Smyth, a senior associate at the Washington-based nonprofit Arts Education Partnership.
The argument for dancing across the curriculum is all too familiar. In Edweek's words:
It’s seen as a powerful way to promote creativity and critical thinking, among other skills.
The Government Accountability Office, the investigative arm of Congress, has found the overall research base regarding the impact of arts education on student outcomes in other subjects to be “inconclusive.”
Research examining the effect specifically of arts integration on student achievement appears to show mixed results as well. For example, a 2007 research overview of studies from 2000 to 2005 suggested that while there are “many advantages” to arts integration, there was a “lack of strong empirical research” to support the notion that it boosts student achievement.
The study in the International Handbook of Research in Arts Education, argued that focusing chiefly on standardized-test data is “misguided” and fails to fully capture cognitive gains and other benefits, such as improved student motivation.
For arts integration to succeed, it requires a strong commitment from classroom teachers and close collaboration with arts specialists, a point made by many dance advocates.
“You really need to have a dance specialist who knows what they’re doing,” said Ms. Bonbright. Effective integration, she said, should be done with “mutual support of both disciplines.”
Thursday, December 2, 2010
Mid-year multiplication and division word problems.