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As Nuthall (2007) has shown, 80% of feedback a student receives about his or her work in elementary (primary) school is from other students. But 80% of this student-provided feedback is incorrect!80%.
p. 4
UNDERSTANDING THE PROBLEM
First. You have to understand the problem.
What is the unknown? What are the data? What is the condition?
Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant Or contradictory?
Draw a figure. Introduce suitable notation.
Separate the various parts of the condition. Can you write them down?
DEVISING A PLAN
Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
Have you seen it before? Or have you seen the same problem in a slightly different form?
Do you know a related problem? Do you know a theorem that could be useful?
Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
Could you restate the problem? Could you restate it still differently? Go back to definitions.
If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem?
Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
CARRYING OUT THE PLAN
Third. Carry out your plan.
Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
Looking Back
Fourth. Examine the solution obtained.
Can you check the result? Can you check the argument? Can you derive the solution differently? Can you see it at a glance? Can you use the result, or the method, for some other problem?
My 3 1/2 year old "engages in reading-like behavior." He sits down with a book and proceeds to flip through the pages while telling a story that, while he made it up himself, is fairly plausible based on the illustrations. This is IMHO perfectly appropriate for his age.
The standard for a 6-7 year old in first grade, however, should be at minimum the ability to read BOB books.
To learn how to solve "all sorts of new problems" one has to start with basic problem solving techniques that are generalizable to a variety of different situations. What the ed school gurus were advising was what happened in Catherine's school as she described. Giving students problems for which they did not have the prior knowledge or skills to solve them.
If you throw a kid who doesn't know how to swim and/or is afraid of water in the deep end of a swimming pool with the instruction that he/she is to swim to the other side, one result is the kid may drown. The other result is the kid may do a combination of thrashings to keep himself afloat and somehow make it to the other side. If you asked the kid how he did it, a likely response may be "I have no idea what I did, but I never want to do that again!"
"Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation."Kids who know math can look at a problem and apply a straightforward, mathematical solution. For kids who know squat, EVERYTHING is a problem that has to be figured out. That is not a good thing. The route to complex problem solving does not bypass mastery of the basics.
Discovery learning in math (pdf file)
Historically, the achievement gap between America’s black and white students was widest in Southern states, where the legacies of slavery and segregation were reflected in extremely low math and reading scores among poor African-American children.
But black students have made important gains in several Southern states over two decades, while in some Northern states, black achievement has improved more slowly than white achievement, or has even declined, according to a study of the black-white achievement gap released Tuesday by the Department of Education.
As a result, the nation’s widest black-white gaps are no longer seen in Southern states like Alabama or Mississippi, but rather in Northern and Midwestern states like Connecticut, Illinois, Nebraska and Wisconsin, according to the federal data.
[snip]
By 2007, the state with the widest black-white gap in the nation on the fourth-grade math test (not counting the District of Columbia) was not in the deep South, but in the Midwest — Wisconsin. White students there scored 250, slightly above the national average, but blacks scored 212, producing a 38-point achievement gap. That average score for black students in Wisconsin was lower than for blacks in Alabama, Louisiana, Mississippi or any other Southern state, and 10 points below the national average for black students, the study indicated.
Wisconsin was the only state in which the black-white achievement gap in 2007 was larger than the national average in the tests for fourth and eighth grades in both math and reading, according to the study.
Racial Gap in Testing Sees Shift by Region
By SAM DILLON
Published: July 14, 2009
By way of introduction, I am neither mathematician nor mathematics teacher, but I majored in math and have used it throughout my career, especially in the last 17 years as an analyst for the U.S. Environmental Protection Agency. My love of and facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.
I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.
Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retire. I enrolled in education school about two years ago, and have one class and a 15-week student teaching requirement to go. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.
In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”—to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.
To set this in context, it is important to understand an underlying belief espoused in my school of education: i.e., there is a difference between problem solving and exercises. This view holds that “exercises” are what students do when applying algorithms or routines they know and the term can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, we future teachers are told that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.
As someone who learned math largely though mere exercises and who now creatively applies math at work, I have to question this thinking. I believe that students’ difficulty in solving new problems is more likely to be because they do not have the requisite knowledge and/or mastery of skills—not because they were given explicit instruction and homework exercises.
Those who make such a differentiation and champion “true” problem solving espouse a belief in having students construct their own knowledge by forcing them to make connections with skills and concepts that they may not have mastered. But, with skills and concepts still gelling students are not likely to be able to apply them to new and unknown situations. Nevertheless, the belief prevails that having students work on such problems fosters a discovery process which the purveyors of this theory view as “authentic work” and the key to “real learning.” One teacher with whom I spoke summed up this philosophy with the following questions: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?”
Discovery learning in math: Exercises versus problems (pdf file)
by Barry Garelick
THIRD EDUCATION GROUP REVIEW / ESSAYS, Vol.5, No.2, 2009
Despite this clear distinction between learning a discipline and practicing a discipline, many curriculum developers, educational technologists and educators seem to confuse the teaching of a discipline as inquiry (i.e., a curricular emphasis on the research processes within a science) with the teaching of the discipline by inquiry (i.e., using the research process of the discipline as a pedagogy or for learning. The basis of this confusion may lie in what Hurd (1969) called the rationale of the scientist, which holds that a course of instruction in science should be a mirror image of a science discipline, with regard to both its conceptual structure and its patterns of inquiry. The theories and methods of modern science should be reflected
in the classroom. In teaching a science, classroom operations should be a mirror image of a science discipline, with regard to both its conceptual structure and its patterns of inquiry. The theories and methods of modern science should be reflected in the classroom. In teaching a science, classroom operations should be in harmony with its investigatory processes and supportive of the conceptual, the intuitive, and the theoretical structure of its knowledge. (p. 16)
This rationale assumes
that the attainment of certain attitudes, the fostering of interest in science, the acquisition of laboratory skills, the learning of scientific knowledge, and the understanding of the nature of science were all to be approached through the methodology of science, which was, in general, seen in inductive terms. (Hodson, 1988, p. 22) "
EDUCATIONAL PSYCHOLOGIST, 41(2), 75–86
For several decades, educational psychology has been dominated by the view that direct explicit instruction is inferior to various combinations of discovery learning or “immersion” in the procedures of a discipline. This view was both attractive and plausible on the grounds that the bulk of what we learn outside of educational institutions is learned either by discovery or immersion.
Sweller, Kirschner, ClarkEDUCATIONAL PSYCHOLOGIST, 42(2), 115–121
"We teach children to mimic the behaviors of good readers without the conditions (background knowledge) that make those behaviors work."
It always amazes me how this blog seems to reflect conversations I'm having with a good friend of mine who is also a teacher. We talk non-stop about education. Just two days ago we were talking about this, which seems especially prevalent in high school remedial reading courses. Since these kids are perceived to be able to "read" (that is, say out loud the collection of letters on the page), the idea seems to be that they just aren't concentrating hard enough - or as mentioned mimicking the behavior of good readers well enough - and teaching them that should take care of the problem.
As for "Best Practices," I've often thought it was a very Orwellian phrase, and could go right up there with:
WAR IS PEACE
FREEDOM IS SLAVERY
WORST PRACTICES ARE BEST PRACTICES
A teacher from Australia shared a refreshingly honest "standard" for First grade reading there:
"engages in reading-like behavior"
Well, if it is "reading-like" it is, by definition, not reading, just as "catlike movements" are movements of some animal other than a cat.
I may have to start calling our reliance on “reading strategies” instruction “Cargo Cult Reading.” Its entire point is to teach children “what good readers do” and the habits of mind that are reflexive to able readers. It’s the exactly the same thing–you teach kids to mimic the behaviors that lead to comprehension–but without the background knowledge that actually makes it possible. Indeed, a staple of strategy instruction is to teach children that good readers ”activate their prior knowledge to create mental images, ask questions, and make inferences.” How exactly does that work in the absence of prior knowledge to activate?
[The Hogwarts summer assignment] list confirms 1000% that this is the school for E. and that public schools don't know what they are doing. When I would look at the work that my kids were given, from K - now, I would have NO IDEA myself what the teachers were asking for. And if I can't understand, I don't know how the kids are supposed to understand. I feel like taking the Summer Assignment list to my meeting with [the school] and just laying it down and showing them what CLEAR is. All of E's "needs break down of directions" and "chunking material" that's on his 504 plan is done on the Hogwarts Summer Assignment list without any "504 accommodation."I had exactly the same experience myself last summer, contemplating C's summer assignment list from Hogwarts.
Based on your reading of books 1-12 of the Odyssey, answer each of the following questions in essay form. Each answer must be no less than fifty words and no more than seventy-five words. Combine all three essays on one page in New Times Roman, 12 point, single-spaced format. Staple your page of answers to the Odyssey identification page you are submitting in September.I still remember, vividly, the feeling of relief that washed over me as I read this assignment.
1. Show three ways in which the challenges Telemachus meets in establishing his male identity are similar to the challenges certain students entering Hogwarts as freshmen meet in establishing their male identity. [have I mentioned lately that Hogwarts is highly boy-friendly?]
2. What do you consider the three most admirable qualities of Odysseus and why are they still needed today?
3. The Odyssey is filled with interesting female characters. Tell who is the most interesting female character and tell what are her three most admirable qualities.
This brings up something that's been rattling around in my head. Seems like in so many realms, the schools have completely lost track of developmentally appropriate strategies. They are applying a very simple model. Whatever they are striving for, they try to accelerate. Doesn't work that way.
Examples:
How to develop expertise:
Schools--put the kids in undefined environments and encourage them to think and act like experts.
In reality--provide kids (people) opportunities for knowledge acquisition, then synthesis and opportunities to apply their knowledge, over lengthy time periods (years), with the happy outcome of creating an expert (at the end).
How to make sure kids learn algebra:
Schools--teach algebra at earlier and earlier ages
In reality--teach basic math better, prior to teaching algebra
How to teach kids to handle long term/multiple projects:
Schools--give them long term/multiple projects at earlier and earlier ages. If this does not seem to work, go even earlier.
In reality--start with small nightly homework, several years of assignment books, parsed out assignments, frequent check-ins, short projects one at a time.
This is true in other areas too--
How to produce confident independent child:
Wrong--force child into independence early, cry it out, tough it out, figure it out
Right--meet child's many needs with constant love, attention, affection; kill boogey men; kiss away tears.
The general principle that I'm trying to articulate is that if you want to achieve a certain goal, you don't do it by prematurely acting as if you've achieved the goal. You don't do Z at an earlier age to achieve Z at a later age. You do X and Y at the earlier ages, which then allow you to move forward and achieve Z. The "X" and the "Y" are what developmentally appropriate education is all about.
For Secretary of Education Arne Duncan, innovation seems to mean grabbing the lessons from schools with records of high performance and grafting them on to problem schools. Finding “what works,” adopting it, spreading it around. Why not call that what it is: replication?
Replication is a worthy effort. But ‘new, here’ is not the same as ‘new, anywhere’...
In the South Seas there is a Cargo Cult of people. During the war they saw airplanes land with lots of good materials, and they want the same thing to happen now. So they've arranged to make things like runways, to put fires along the sides of the runways, to make a wooden hut for a man to sit in, with two wooden pieces on his head like headphones and bars of bamboo sticking out like antennas—he's the controller—and they wait for the airplanes to land. They're doing everything right. The form is perfect. It looks exactly the way it looked before. But it doesn't work. No airplanes land. So I call these things Cargo Cult Science, because they follow all the apparent precepts and forms of scientific investigation, but they're missing something essential, because the planes don't land. Now it behooves me, of course, to tell you what they're missing. But it would be just about as difficult to explain to the South Sea Islanders how they have to arrange things so that they get some wealth in their system. (RP Feynman, Caltech commencement, 1974)
The American public school system is vast. More than 15,000 school districts govern over 95,000 schools employing over 6 million people serving 50 million students and spending $500 billion per year (Hess & Finn, 2007). Public schooling, in short, is a colossus casting a very long shadow. Major reform efforts within the public education system will inevitably influence the private school sector, sometimes profoundly so.
Even as by far the nation’s largest system of private schooling, at 2.3 million students (McDonald, 2006) the size of the Catholic school system pales in comparison. Nevertheless, Catholic schools have a proud tradition of outperforming public schools, in particular with disadvantaged students. In vast swaths of urban America today, Catholic schools remain the highest performing schools available to inner-city youth.
THE IMPACT OF CHARTER SCHOOLS ON CATHOLIC SCHOOLS: A COMPARISON OF PROGRAMS IN ARIZONA AND MICHIGAN (pdf file) by Mathew Ladner
Investigations and EM both boast about how their programs emphasize mental math. Though they don't seem receptive to the idea of teaching long division. I wonder why not? Long division is an ideal way to hone mental math skills. A problem like 56,098 divided by 84 requires a lot in the way of number sense and mental math. The first step is looking at the maximum number you can have to multiply by 84 and stay under 560. Rounding, 560 divided by 80 would be 7, but a quick glance will tell you that 7 x 84 will be greater than 560, which quickly leads you to "6" for a choice. And so on and so forth. But reformers don't want to hear about all that. It's drill and kill, has no value, and anyway, calculators have made long division obsolete.
Suppose you find that your bucket leaks. Does that mean you need a bigger bucket? Not necessarily; you may just need one that doesn’t leak. With the best of intentions, President Barack Obama and U.S. Secretary of Education Arne Duncan are renewing the call for a longer school day and year—for a larger bucket. I believe this is premature.
Like most “structural” (vs. substantive) reforms, this one will consume our time and political energy, even as it postpones our encounter with a more vital, less costly opportunity: making good use the huge number of hours that currently “hide” within the conventional school day and year. If we recovered this time and properly redirected it, the impact on student learning would be greater than any reform ever launched.
There are, in fact, two to three months of learning time waiting to be recaptured within the existing school year. And though our communities and school boards might be surprised to hear it, a majority of educators are aware of this.
In the last few years, I have asked several hundred audiences of teachers, administrators, and union representatives the following question: Would you agree that almost all “worksheets” are a lamentable and unnecessary use of instructional time?* More than 90 percent of them agree unreservedly, by show of hands. Then I ask them to pair up and discuss what proportion of the school day or year students spend filling in such worksheets. After some give and take, the average response I get is a minimum of 25 percent to 30 percent—the equivalent of an entire grading period. More than two months.
I then say, “Gee … and I haven’t even mentioned movies.” This invariably provokes honest, almost relieved laughter. In the right setting, educators are refreshingly frank—and concerned—about the actual curriculum, which is starkly different from what the public (and many policymakers) imagine. Teachers and administrators know that the actual taught curriculum is rife with time-killing routines, worksheets, and often full-length films that add little or no value to the school day. Most interesting perhaps is that once educators have a chance to add it all up, they see that changes would add about 30 percent more learning time annually for almost every school in America. And it would cost us nothing.
We’ve known for decades that large chunks of class time are spent on ill-conceived group activities, on settling in or packing up at the beginning and end of class. Moreover, there has been an alarming increase, at all grade levels and in all subjects, of what the reading expert Lucy McCormick Calkins** refers to as classroom “arts and crafts.”
[snip]
Many parents suspect that some of this goes on in schools, but they hope against hope that it is rare or occurs only in low-scoring or inner-city schools. Would that this were so. I recently found these activities to be as, or more, prevalent in schools with their state’s highest academic designation. The fact is, students in most schools spend days at a time in academic classes on questionable group “projects,” on drawing and painting, making banners, castles, book jackets, collages, and mobiles. All of this is in addition to worksheets and movies.
[snip]
With a new administration and secretary of education in Washington, this would be an excellent time to honestly, and at long last, come to terms with how time is spent in schools. Then we would be in a better position to decide how much time and energy we should expend in fighting for longer school days and shorter summers.
Do We Really Need a Longer School Year?Education Week
Vol. 28, Issue 36
Published Online: July 7, 2009
It bugs the hell out of my kids, but I love saying it:I don't do mental math well at all. I was never taught, and when I taught myself how to do it in order to teach my little afterschool Singapore Math class, I didn't practice it enough to learn it well. (I think Saxon Math has some mental math exercises, right? Or am I confusing the two curricula?)
The fact that you're asking for a calculator is the surest sign that you shouldn't have one.
I have seen far too many kids so clueless about basic computations that they can't even recognize an answer which is off by several orders of magnitude. They add two numbers and get a smaller number. No clue.
Obi-Wan
Media carries with it a credibility that is totally undeserved. You have all experienced this, in what I call the Murray Gell-Mann Amnesia effect. (I call it by this name because I once discussed it with Murray Gell-Mann, and by dropping a famous name I imply greater importance to myself, and to the effect, than it would otherwise have.)Several of you mentioned the Murray Gell-Mann amnesia effect the other day, and after reading what seems like a definitive news article on French health care,* I decided to get it officially posted here on ktm.
Briefly stated, the Gell-Mann Amnesia effect works as follows. You open the newspaper to an article on some subject you know well. In Murray's case, physics. In mine, show business. You read the article and see the journalist has absolutely no understanding of either the facts or the issues. Often, the article is so wrong it actually presents the story backward-reversing cause and effect. I call these the "wet streets cause rain" stories. Paper's full of them.
In any case, you read with exasperation or amusement the multiple errors in a story-and then turn the page to national or international affairs, and read with renewed interest as if the rest of the newspaper was somehow more accurate about far-off Palestine than it was about the story you just read. You turn the page, and forget what you know.
That is the Gell-Mann Amnesia effect. I'd point out it does not operate in other arenas of life. In ordinary life, if somebody consistently exaggerates or lies to you, you soon discount everything they say. In court, there is the legal doctrine of falsus in uno, falsus in omnibus, which means untruthful in one part, untruthful in all.
But when it comes to the media, we believe against evidence that it is probably worth our time to read other parts of the paper. When, in fact, it almost certainly isn't. The only possible explanation for our behavior is amnesia.
