Friday, July 24, 2009
So here's a word most of us have a pretty good agreement on: The Socratic Method.
For now, we leave aside whether Socrates used the method the same way, and leave aside if he thought the method was for demonstrating ignorance or if it can be used to generate knowledge.
Let's just use it as it is used in teaching, such as in law school: the Socratic Method is both a learning theory and a pedagogy, where the teacher and the student engage in a discussion, or a series of guided questions, in order to reach the truth. The student is expected to familiarize themselves with the basic issues at hand (by reading and studying the assigned materials), and from there, the teacher begins a series of questions that are designed to expose errors in the students' beliefs or reasoning. Once exposed, questions are then asked that can only be answered by using correct reasoning. In this way, the teacher guides the student along to the right understanding.
The advantages of this method are pretty clear: first, the student has been forced, effectively, to engage more actively with the material and the ideas than in a listen-to-lecture setting. With practice, this engagement would lead to understanding how to use the appropriate material to prepare better in the first place. That means shoring up mastery of the underlying knowledge necessary to participate competently in the first place.
Second, this method uses students' misconceptions as the guidestones for learning. Misconceptions are the real reason for failure to understand material, and unrecognized misconceptions undermine all learning placed on top of them. By going to great lengths to determine and categorize the likely misconceptions in the first place, teachers are much more aware of stumbling blocks for their students, and are more likely to recognize when a student has foundational problems in their knowledge.
Third, while the method uses students' misconceptions, it uses teachers' corrective guidance to move past those misconceptions. It does not rely on the student making that realization on their own. The corollary to this is that the method pushes inexorably towards The Truth. That is: there is a right answer, that can be reached by knowing true facts and then using proper reasoning.
There are still some problems with this method. It's unclear that it scales well: can one lengthy conversation tackle a classroom full of misconceptions? Another issue is how to encourage participation. Is this exercise just one of humiliation? Can it be made to be encouraging rather than stifling for a group of young children? young teens? older teens? how so? Can a teacher keep this method from devolving into tearing students' ideas down without building up correct notions?
There are many resources to help teachers to use the Socratic Method. There are even some books that teach entire subjects using this, and not just philosophy! (My favorite is a computer science book called The Little Lisper, which teaches LISP to a child using this method.)
So could discovery learners and instructionalists agree on the value of using the Socratic Method for math and science teaching? If this were the discovery learning method being employed, what would be the objection? And to constructivists, if we granted you that we approved of the Socratic Method for discovery learning, would that be good enough? or not?
What do the teachers think? Anyone using this method to teach math or science? Anyone doing it with their whiteboarding? Just how difficult is it to use the Socratic Method in a classroom?
When I got the position there and was assigned to teach Physical Science, Biology, Anatomy, and Environmental Science, I realized that I came to an empty well - there was no curriculum plan (the list of objectives for science courses found in the office binder belonged to another school and was so vague that it could not help me), no teacher manuals for disciplines - nothing except the textbooks and a lab supply room full of 20 -year old "supplies". I knew the curriculum for biology but had no idea what should be taught in physical science (I don't think NY schools teach physical sience at all!), anatomy, or environmental science. So I worked from scratch and created my own curriculum maps. And as I did it, I understood that perhaps the most important course of all in this sequence, the one that required the most effort from me was Physical Science. So I wrote a letter to whoever will take my position to save that teacher some effort and prevent some common (as I saw in upper grades) problems growing from students not knowing the basics.
Here is the excerpt from this letter:
Dear fellow teacher:
As you are getting ready to teach sciences at xxxxx, even if you are a seasoned sciences teacher, please take few minutes to read this - it may help you to understand the major goals of the science courses sequence used in this school as well as to organize your planning and instruction. Also, it is some practical advice based on my experience of teaching here. Even though you will find the curriculum objectives (not really re-worked) and my syllabi, this may be still useful.
The sequence of science courses is as follows:
1) Physical Science – 9th grade
2) Biology – 10th grade
3) Chemistry – 11th grade
4) Physics – elective for juniors/seniors with pre-requisite of 85 in math.
5) Anatomy – elective, no pre-requisites
6) Environmental science – elective, no pre-requisites (not offered in 2009-2010)
Textbook (oldie but goodie!) – Focus on Physical Science by Heimler and Price. It is the best around, with great review exercises and good explanations and examples. I have ordered more through Amazon, but if they don’t come – make photocopies of the units for students. It also has a good lab manual.
As you can see, Physical Science is the first course in the sequence. It lays down the foundation for taking other science courses and, for majority of students, serves as a substitute for the physics course they don’t select to take. In addition, the freshmen come here from a variety of institutions – and many lack basic mathematical and logical reasoning skills not to mention the fuzzy middle school science that leaves them with no clear idea what science disciplines are and how to learn and study for science courses. That simply means that this course will require the most organized, structured, and clear instruction that you can deliver. It must give the students not just an understanding of methods of science, but provide mastery with basic chemistry and physics as well as logical reasoning and mathematics.
Overall, the advice is – leave the “discovery” for later and teach as directly as possible! Use the labs to practice and apply, not to introduce. Based on the attained mastery of Physical Science concepts and skills, the students (and you!) will have it much easier in upper courses from the sequence , where the student-designed and conducted investigations can be used.
Aligning with math.
Physical science involves mathematics. Shall we say that math used is on almost the elementary level, though you will see many students experiencing problems with applying math. And if these problems are not resolved now, they have little chances of experiencing success in chemistry and physics!. Understanding of concepts is absolutely inseparable from the ability to justify the reasoning with mathematics.
So, here are some “obstacles” that require more work on your part. It is also a good idea to ask the math teachers how they explain a particular math concept and skills to students, so you use the same language.
- In the very beginning, the decimal/fraction/exponent relationship must be reviewed.
Example: 100 = 100/1 = 10² and 1/10 = 0.1 = 1x 10-¹
- Metric system is based on 10. Ensure the mastery of conversions.
- Rounding to the nearest tenth, hundredth etc. should be reviewed as necessary (most likely – should)
- Fractions – basic addition, subtraction, multiplication and division of simple fractions.
- How to treat proportion as a fraction. (Absolutely essential for stoichiometry and some physics problems! ( If X relates to Y like this, then X1 relates to Y1 like that.)
- Simple algebraic equations (review before teaching oxidation numbers) as well as adding and subtracting positive and negative numbers.
Example: x +12 = 24
x = 12
12/x = 3
+5 – 7 = - 2
Note: have them write fractions in the standard form – with the horizontal line between numerator and denominator.
- Transforming formulas. a=b/c Find b. Find c. and so on.
- Practice treating units as numbers in calculations. It helps to see where the units of a variable come from and aids in remembering the formulas (and concepts).
Critical: Introduce the format for solving problems and stick with it. Require it even for the easy problems – then the students WILL definitely benefit when solving more difficult and multi-step problems in this course and in further courses. The format allows extracting and organizing information from the word problems clearly so the possible solution routs are easily predictable. I followed the format taught in soviet schools, which is this:
(I do it in the form of a table: divide the page in 3 columns
WHAT IS GIVEN in the problem:
Use appropriate variables and units provided in the texthat is asked (variable, units)
Basic formula(s) connecting what is given and what is asked + necessary derivatives
Solution (complete with units)
Answer, with units
From physics – Find the work done by gravity force when a 2-kg rock falls from the height of 1.5 m to the ground.
m rock = 2 kg
d = 1.5 mg=a = 9.8 m/sec²
W - ? (J)
W = Fd
(we were not given F, but we can find F if we know mass and acceleration)
Substitute ma for FW = mad
W = 2kg x 9.8 m/sec² x 1.5m = 29J
Example 2 (From Chemistry)
Find the masses of salt and water needed to prepare 300g of 10% solution.
V solution =300 g
% Conc. = 10%
m water -? (g)
m salt-? (g)
10g of salt in 100g of solution
x g of salt are in 300 g of solution
2)m of solution = m of salt + m of water, therefore
m of water = m solution – m salt
1) 10g/x = 100g/300g
x = 3000g/100g = 30g of salt
2) 300g – 30g = 270g of water
Answer: to make 10% solution one should take 30g of salt and 270 g of water
In terms of planning the program, I have found it more useful to clearly separate Chemistry part of the course and Physics part of the course. My syllabus reflects that. Chemistry part (including oxidation states, balancing of chemical reactions and stoichiometry problems) is done in the first semester, and Physics part - in second semester. It correlates with concept sequence taught by math department, so you can arrange the use of problems from your course in math classes and arrange with math teachers to provide more attention to necessary math skills.. It usually works well, and science and math are mutually reinforced.
Of course, you may choose to follow the sequence in the textbook or introduce physics first.
You will find the list of topics to be taught in the syllabus (in the Main Office, Science binder). However, let me emphasize the topics that will require more of your attention – the concepts and skills that usually cause problems for students in biology and chemistry courses if not mastered.
Metric System conversions
Symbols and names of major chemical elements; names of major acids, bases, and salts
Electron configurations and creation of bonds
Calculation and use of oxidation numbers to write the formulas of compounds
Balancing of chemical reactions
Simple stoichiometry – mass to mass problems
Calculation of concentration of solution; preparation of solutions
Properties of water
Reading and creating the graphs - speed, acceleration etc.
Light and Sound – electromagnetic and mechanical waves
Energy transfer, especially heat.
(End of excerpt)
Thursday, July 23, 2009
Core State Standards Draft
The problem I have with the document (in math) is that it's called a "College and Career Readiness Standards for Mathematics", but it treats all K-12 students the same. Their target readiness goal seems to be the end of high school. If a student meets all of these goals, there is no guarantee that he/she is ready for all college degree programs.
For careers that require a college degree, the requirements are already defined. You can survey all college departments and determine what high school math courses (and SAT/ACT scores) are required for their program. You can ask them what level of math they expect from new students, and what top level math course is required. Students and parents don't need some sort of generic, one standard fits all, document or education. Students need to know exactly what level of math they need going into college and coming out of college, and they need to know it when they are still in K-8. Reaching some sort of generic requirement by the end of high school is not enough.
This is what bothers me a lot about education; it's statistical, not individual. Their goal is some sort of statistical best average math goal by the end of high school. I would like to see some sort of individual, what I like to call door-closing, analysis. For better or worse, one of the key points is getting to a proper algebra course in 8th grade and having access to the traditional set of math classes in high school. Of course, this is usually preceeded by some sort of tracking test in 6th grade. Focusing on what an average student needs by the end of high school allows schools to ignore major problems in K-8 math. They can push all issues back to high school and put the onus on the students.
Tuesday, July 21, 2009
Digging deep and working for mastery builds interest, confidence, and a feeling of accomplishment in students. Students like what they have mastered and they dislike what they have only half learned. So slow down, stay awhile, let the student relish and enjoy learning the Latin grammar. The tendency to skim over, even to denigrate the lower skills in a rush to get to the higher ones is a characteristic of modern education. The result is superficial learning, which I think is the cause of student boredom and frustration.
This review persuaded me:
An Oldie But A Goodie
April 30, 2009
To put this TTC course in a historical perspective, it's course #102 with a copyright date 1994. If you can remember back then, there wasn't much of an internet, computer graphics were in infancy, and DVD technology was still being developed. Does this mean this course should be re-done and "modernized"? Well, that'd be nice, but it's really not necessary. But it does mean the visual aids, such as they are, are very basic, and the teacher makes extensive use of the simple old blackboard and flip-chart easel.
Now, I kind of liked that, because it reminded me of when I first took Algebra II in high school, 50 years ago. But if you - or the kid you got this for - get bored if there are no special effects, animations or related "eye candy" at every turn, then this course will indeed be a poor choice. This course is for those who not only want to learn algebra, but are willing to do what it takes to learn algebra. For those people, this course is an excellent investment. It's going to take some work on the student's part, first and foremost being concentration. To state it another way, if you want to learn algebra and how to work algebra problems, this course is a valuable asset, and if you don't, I'm at a loss as to what to recommend.
Most TTC Science & Math course are survey courses, meaning they "sell" the subject and its high points in an entertaining manner, and take you up to, but not into, problem-solving. That's NOT true with this one. It doesn't pretend, or even try, to get you fired up about algebra. This is a pure dirty-hands work-course. Frequently, the instructor will put a problem on the board, and ask the student to stop the "tape" and solve the problem. When you restart the course, the instructor will show the solution and what steps are involved. If you skip these problems, you have no way of gauging how much you've learned - or not. To learn algebra, you have to work algebra - there's no ducking this - and this course offers plenty of that kind of opportunity.
I'll also note that, although it's not on the web page, this course still comes with a workbook, aligned with the lecture topics, with answers in the back. If you're looking forward to a formal classroom course, working these problems is certainly in your best interest.
Another reviewer mentioned "Algebra II for Dummies". This is also a good self-study approach. There's a companion workbook, which I also recommend. (The price for both is only about $30.) Between this DVD course and the Dummies book and the workbooks, you should have all you need to fully move around at this algebra level. And if you'll be taking a high-school course, these two items will prepare you about as much as is possible.
Regarding the instructor - ok, Jay Leno he's not. But I enjoyed his "blue collar" kind of instruction, and I had no problem at all understanding him or the information he was trying to get across. And no, he doesn't look at the camera all the time, but this makes it "seem" like a classroom environment, where the teacher certainly doesn't (I hope) spend the entire period glaring at you.
I really enjoyed this course and got a lot out of it. It was well worth the price. For the intended audience, I definitely recommend it."
This is annoying. The chemistry course, which I bought just a few months ago, has now been revised.
How do the blind learn to read, or compute, or do any of the tasks we think of as reading/writing/math?
Has anyone ever read anthing about the cognitive workings of the blind?
I increase my working memory by writing things down on paper and looking at what I've written.
How do they maintain working memory? Are the brilliant scholars who are blind particularly adept at maintaing working memory using braille? or do they just have fantastically larger working memory than the rest of us? How do they organize their working memory--do they do it "visually" to some degree? Or do they use other senses somehow? Is their auditory loop for working memory MUCH larger than mine, e.g.?
Anyone ever read any research on this? Or even any anecdotal memoirs?
Well, I don't know but I'll chime in with some citations anyway. My guess is that the rate of incidence is dependent on a number of other variables. Thus, you would expect to see a higher incidence in some populations than others.
The evidence is mixed on this topic. Frank Vellutino is one of the major contributors to this line of research. Studies in several US states done by Vellutino (Albany, NY), Torgesen (Fla), someone in Texas -- I forget who-- and elsewhere came up with fairly consistent findings that explicit instruction in reading, including code knowledge, blending and advanced phonics skills reduced the number of students with decoding disabilities to under 10%. (The range was 3%-10%).
Vellutino's studies, and some others, did not include children with IQ's below 90. Decoding difficulties, per se, are not correlated with IQ.
Different programs were used, as well as ad-hoc instruction that drew on various sources and materials but was focused and explicit. No particular program or approach clearly surpassed others -- some were more suited to 1:1 instruction, others to group instruction.
Gough and others published a large-scale study that showed more than 95% of children in first and second grades could be taught effectively enough to score above the 25th percentile, which is considered in the average range.
One problem with all these studies, however, is that they lack longitudinal data. Children who successfully master basic decoding skills and are reading well in first and second grade may fall behind later for a variety of reasons (no single cause). Some continue to struggle with decoding skills at higher levels, others with syntax, language comprehension (including, but not limited to, vocabulary), fluency, auditory and working memory, and more. While poor readers in second grade rarely end up as outstanding readers in fifth grade, the opposite can happen: the good reader in second grade can be a failing reader in fifth and beyond.
Some of Vellutino's articles:
Vellutino, F. R., Scanlon, D. M., Sipay, E.R., et al. (1996). "Cognitive profiles of difficult-to-remediate and readily remediated poor readers: Early intervention as a vehicle for distinguishing between cognitive and experimental deficits as basic causes of specific reading disability." Journal of Educational Psychology 88(4): 601-638.
Vellutino, F. R., Scanlon, D. M., Snowling, M. J., & Fletcher, J. M. (2004). "Specific reading disability (dyslexia): What have we learned in the past four decades?" Journal of Child Psychology Psychiatry 45(1): 2-40.
Scanlon, D. M., & Vellutino, F. R. (1997). A comparison of the instructional backgrounds and cognitive profiles of poor, average, and good readers who were initially identified as at risk for reading failure. Scientific Studies of Reading, 1, 191- 215.
Vellutino, F. R., Scanlon, D. M., and Tanzman, M. S. (1994). Components of reading ability: Issues and problems in operationalizing word identification, phonological coding, and orthographic coding. In G. R. Lyon (Ed.), Frames of reference for the assessment of learning disabilities: New views on measurement issues (pp. 279-329).
Other sources make the point that low-achievers (who may find reading difficult for varying reasons) need much more instructional time -- at least twice as much, sometimes 3 times as much, and in a more intensive, fast-paced, interactive milieu. This is rarely provided and probably not even possible in most public school settings. The only longitudinal accounts of such success are from the UK, and the details of what was done and when and how in both the Clackmannanshire and W. Dunbartonshire studies are sketchy.
Nothing similar has taken place on this side of the Atlantic; the best overall results have been from DI schools, but even there, population mobility, staff turnover and limitations on instructional time lower the ceiling for student achievement.
Accelerating Reading Attainment: The Effectiveness of Synthetic Phonics by Joyce E. Watson and Rhona S. Johnston (pdf file)
One Man's Quest to Eliminate Illiteracy
West Dunbartonshire's Literacy Scheme
Check out this conclusion contained near the end of a 31-page case study of a 5th grade boy's reading:
John’s comprehension at his current fifth grade level is excellent when he is relieved of the task of recognizing words.
After my first visit to Hogwarts, on Sports Orientation Night, I came away thinking the place felt like a "happy military school." As it turned out, that was by design; the principal told us so on Family Orientation Day. I've forgotten how he put it, but the gist was that Jesuit schools have much in common with military schools.
Loyola himself, of course, was a soldier.
My point: happy military schools work.
Knowing what I do now, I take issue with this observation:
Behavioral problems are not an issue [inside military schools] due to the values students are taught at home.I would wager a very large sum of money that a person who has actually taught in a military school would read this passage and hoot.
Your typical classroom pits 20 to 30 kids against 1 adult. That's not quite the Battle of Thermopylae, but close enough.
Kids in groups are like dogs in packs* (I say that with affection for kids and dogs): they rev each other up, and beyond a certain point parents can't control their kids from home no matter how sterling their values. It is the school's job to keep order and a cheerful spirit.
Most schools don't: because of political correctness that dictates "radical inclusion;" because ed schools don't teach classroom management skills; because public schools seem to believe that classroom discipline begins and ends in the home.
It doesn't. The public schools in my district, where the school population is upper middle class children from well-organized families, are safe and reasonably orderly, but there are constant complaints about disrupted classrooms, kids not being able to hear because of commotion made by the 'hyper' kids, etc. At the high school, students and parents have been told that disruptive kids in honors courses "don't deserve to be in the class."
I think it's unlikely that military kids are better behaved than kids in my district when they're not under their parents' direct supervision, and I credit military school personnel, not military parents,** with the orderly environment inside military schools.
* "In 2008, 39% of the fatalities involved multiple dogs."
** Not a criticism of military parents or anyone else. All I'm saying is: if a parent isn't inside the classroom, he isn't the one keeping order.
Writing instructors everywhere (I include parents in this category):* save this Comment to show your students.
Here’s an African-American president, speaking to the NAACP, and arguing for reform in our schools and responsibility in our homes and community.
Notice the lack of any responsibility for schools. If school staff, the paid professionals, aren’t responsible for learning, why pay them?
And if parents are going to be responsible, then they should have the money, which they can then use to hire teachers if they think it’s worthwhile.
This is one of the most succinct rebuttals of an argument I have ever read.
It is a fantastically difficult feat to pull off.
As to Tracy's point, I would love to see a system where parents did the hiring, which is what is taking place in the world of homeschooling now.
I imagine that if parents did the hiring some teachers would establish 'private practices' while others might be employed by schools with an organizational structure similar to a community college, perhaps, or to a law firm. The school would employ teachers and provide health insurance, but would not assign students to classrooms. Parents would choose.
I can also imagine that these schools would have school-wide "positive behavior" systems in place, offering parents a happy and safe environment for their kids. For most parents, "happy and safe" is priceless, and such schools would be swamped with customers.
As in a law firm, teachers might have tenure; presumably they would also have a great deal more authority over what they teach and how they teach it.
* Mary Hake says that she helped all of her kids with their school writing assignments, and she advises parents to do the same. I'll get my interview with Mary assembled shortly.
The Liberal Arts of Logic, Grammar, and Rhetoric
Understanding the Nature and Function of Language
Monday, July 20, 2009
Catherine: "In other words, it takes two decades for writers to be able to edit their own prose well." (see: Kellogg, R.T. (2008). Training writing skills: A cognitive developmental perspective [pdf file] Journal of writing research, 1(1), 1-26)
Had you said something like, "edit their own prose and get anything useful out of the process", I might agree. I don't think that most writers ever get to the point where they can "edit their own prose well." The best that you can expect is that if you give your work a week or two before you look at it with an eye to editing you can sometimes see poor writing of various sorts.
Oh, you might be able to do a good job of basic copyediting. But I think that significant writing is too closely tied to each writer's idiosyncratic thinking process for the writer to notice flaws of logic, organization, and flow as often as you really need to. And absent those things, I don't think you can say that the writer can edit his own work "well."
Frankly, finding even a professional editor that can do a good job of structural editing of the work of others is a non-trivial task. Even good editors are often lured down the alleyways of comma placement, spelling, and a random selection editorial peeves of dubious validity.
IME, a good reader who is willing to say, "I don't follow this part", is as useful as (sometimes more useful than) most editors. If you can tell me where you're having a problem, I can often see all or part of the problem and rewrite to correct it. If you can articulate why you don't understand the part, you're invaluable.
But good readers are hard to find, too.
On outlining: In many ways, the formal outlining taught in the 70s was worse than useless. Until I managed to get past the cumbersome process involved in the sort I was taught, I actively avoided outlining at all, to the great detriment to my writing.
Now, I always outline any writing longer than a page or two, but I haven't used a formal outline since the last time it was required in high school. I make notes, draw lines, and rewrite the parts I can no longer read for all the scribbling. At the end, I have a plan that is functionally an outline, but it bears little resemblance to the roman numeral, capital letter, lower-case roman numeral (etc.) outline taught in some writing classes.
This is why Writers Workshop makes no sense. Editing is a separate process from writing; in the real world (the 21st century world!) the professional writer has a professional editor.
I'm now convinced that the goal of writing instruction should be to teach students to write the best first draft they possibly can -- because the first draft is pretty much going to be their last draft even if you force them to "revise."
The fact is: they're not going to revise.
They're not going to revise because they're nowhere near being able to read their own words the way a stranger reads them.
The Psychology of Writing
Placebo effects can be very powerful and many supposedly effective medicines do not in fact outperform the placebo. The sorry truth is that no one has compared modern education to a placebo. What if we just gave people lots of face-to-face contact and told them they were being educated?
Take a piece of paper and fold into quarters. Open it up and draw a rectangle in the center. Now you've got four quadrants and a rectangle which is not exactly 'four square' but in the interests of marketing I guess four square sounds sexier.
Read the problem twice. Then in the rectangle restate the question in the form 'find blah blah blah in units of xxxxxx'.
Read the problem again and in the upper left quadrant identify and define all the symbols (variables and constants) you'll use in your solution.
Read the problem again and use the lower left quadrant for a diagram/picture.
Read the problem again and use the upper right quadrant to define your strategy. This can be words or preferably a set of equations to solve.
The lower right quadrant is where you show your arithmetic.
Finally, the back of the paper is used to justify your answer.
It works well for entry level problem solving of the kind you would encounter through maybe grade six. If kids master this it should instill some good habits for more complicated things.
If 80% of student feedback is wrong, what % of teacher feedback is wrong? Are we just assuming that the teacher is 100% correct the 20% of the time they actually provide feedback?
I wish ---
Students have long believed (on good evidence) that if the same paper is submitted to two teachers in two different sections of the same course, the paper is likely to receive two very different grades. In 1961, Paul Diederich and his colleagues proved that this student belief is no myth. When 30 student papers were graded by fifty-three graders (a total of 15,900 readings), more than one third of the papers received every possible grade. That is, 101 of the 300 papers received all nine grades: A, A-, B+, B, B-, C+, C, C-, and D. Diederich also reported that
94 percent [of the papers] received either seven, eight or nine different grades; and no essay received less than five different grades from fifty-three readers. Even when the raters were experienced teachers, the grades given to the papers by the different raters never attained a correlation greater than .40. Diederich, P.B., French, J.W., and Carlton, S.T. "Factors in judgments of writing ability." Research Bulletin RB-61-15. Princeton, N.J.: Educational Testing Service, 60 pp.
The Schools We Need and Why We Don’t Have Them
E. D. Hirsch
As a teacher I see this in class every single day, kids wrongly explaining to others what they've done right and wrong. Especially what they've done wrong! Even if it's a subjective question the kids will authoritatively tell each other exactly how wrong they are when they might be onto something really special.
Remember my famous group work/peer review story? My son was in a group doing a math problem. He understood it and worked out the right answer; the four other kids were arguing for a different answer. He had been appointed group leader, who was supposed to give the group's answer. When called upon to give the group's answer, he gave his (correct) answer.
He was reprimanded for not giving the group's "consensus" answer. When he protested that it was wrong, he was told he should have done a better job convincing them of the rightness of his answer.
So you can see that the point of the whole exercise was far afield from the kids actually learning how to do the problem...
Peer review in my son's (3rd grade) class wass of the form, "It's got a great beat and you can dance to it, so I give it an A". Had this been managed at all by the teacher, it could have risen to the level of merely useless, but the teacher's classroom management skills needed remediation -- desperately.
From Bertrand Russell or was it Alfred North Whitehead...."the goal is to have no problems only exercises".
By expanding long term memory one can turn potential problems into exercises.
It seems that the goal of many pushing the discovery/inquiry approach is to have no exercises but only problems. *
I am still in shock from Dr. Ruth Parker's powerpoint given at NCTM national that the Standard Algorithm always harms conceptual understanding, which seems to be a recipe for lots of problems.
The person most everyone would like to hire is the one that has so much extensive background knowledge that most everything is an exercise.
Here's Ken on the subject of struggle:
* students must struggle