Update on 9-14-07: Part 2 of the interview is now posted!
[The comments and reactions are beginning to pour in here and on others' blogs. SteveH's and mathmom's give-and-take in the Comments section is a must read -- it's a blog of its own! Part 2 of the interview will be coming on Fri and Mon.]
As reported a few days ago, Prof. Steen, one of the most highly respected voices in mathematics education, graciously accepted an invitation for an online interview at MathNotations. He has been a driving force for the reform of school mathematics for many years and was on the development team that produced NCTM's Curriculum and Evaluation Standards for School Mathematics. For the last few years, he has been involved in Achieve's commitment to developing world-class mathematics standards for K-8 and ADP's similar commitment to secondary mathematics. He will have much to say about these standards and the new Algebra II End-of-Course Exam that will be launched in the spring of 2008. He is a man of great integrity and towering commitment to quality education for all of our children .
A few days ago, I emailed Prof. Steen a set of 19 questions that I felt reflected many of the concerns of my readership and, even more, confronted some of the major issues in mathematics education today. He agreed to reply to all of these, asking only that I publish his remarks in full. My role here was purely reportorial. This is not a debate. Once the questions were composed I stepped back and allowed him free rein. Prof. Steen replied thoughtfully and candidly within 48 hours. MathNotations will publish the interview in 2-3 segments to allow readers to absorb his replies and comment. If you've stopped here by the side of the road, tell your friends and colleagues about it. I invite fellow bloggers to spread the word across the blogosphere as well.
Philosophically, Prof. Steen and I have much common ground, although we diverge on some key points. What I truly believe is that honest dialog is the only way we can move forward, end the Math Wars and reach a strong middle-ground position that best serves the interests of our children. Whatever your ideology may be, Prof. Steen's comments are profound and thought-provoking. Enjoy!
I'll begin with my note of appreciation to Prof. Steen:
Thank you Prof. Steen for your prompt yet thoughtful replies. This has
been a new and rewarding experience for me and I know my regular
readers (and perhaps new visitors) will read your comments with great
interest, regardless of where their ideologies lie. You and I share
many common views and yet we can respect our differences. In the end,
we both want what is best for all our children. There are no easy
answers to difficult questions, however I do believe, as you do, that
dialogs like this will ultimately move us in the right direction.
Thank you again for contributing to this process.
Dave Marain
MathNotations 9-11-07
Math Notations Interview
Lynn Arthur Steen, St. Olaf College, September, 2007
1. Prof. Steen, your involvement in so many mathematics and science education projects is mind-boggling. At this time, what are your greatest concerns regarding mathematics education in the U.S.?
That in our stampede for higher standards we are trampling on the enthusiasms, aspirations, and potential contributions of many students for whom mathematics is best approached indirectly. There is plenty of interesting mathematics in areas such as medicine, technology, business, agriculture, government, music, and sports, but students don't get to see these until large percentages have already given up on mathematics. It is true that mathematics unlocks doors to future careers. But we also need to open more doors to the world of mathematics.
2. Over a dozen years ago, Professor Schmidt, Director of the U.S. participation in TIMSS, made his famous comment about our mathematics curriculum being ‘an inch deep and a mile wide’. He also stressed the importance of having a coherent vision of mathematics education. Since then, fifty states have independently developed sets of mathematics standards and assessments. Although similar in some respects, they lack overall coherence and consistency of high expectations of our children. What is currently being done nationally as you see it to remedy this situation?
Notwithstanding our constitutional tradition of federalism that leaves states responsible for education, some now suggest voluntary national standards as a cure for the incoherence and inconsistency that is evident in state standards. Indeed, Senator Dodd (D-CT) and Representative Ehlers (R-MI) have introduced just such a bill in the Congress. I rather doubt that there is sufficient political support for nationalizing education in this way. Nor do I think it would resolve the problem. It would simply shift the locus of inconsistency from written standards and assessments to teachers and students.
More promising are efforts such as the American Diploma Project Network which is an ad hoc coalition of states that decided to work together on a common education agenda. This is not a "national" effort, but it is more in keeping with the traditions of our nation. Public distribution of comparative data is another strategy for reducing unwarranted inconsistency. Recent studies such as Mapping 2005 State Proficiency Standards Onto the NAEP Scales (NCES, June 2007) that compare states to the common scale established by the National Assessment of Educational Progress lead naturally to improvement motivated by competition or, in some cases, by embarrassment.
Strategies that open more doors to mathematics are more likely to emerge in smaller jurisdictions, for the simple reason that innovation begins locally and the doors that need opening tend to have local roots. So I'm not terribly bothered by lack of coherence and consistency. I'd rather focus first on getting more students to learn more mathematics of whatever kind may interest them. What counts is that students gain sufficient experience with substantive mathematics—not just worksheets—to benefit from its power and, if possible, to appreciate its beauty.
3. What were some of the obstacles faced by Achieve’s Mathematics Advisory Panel, both at the K-8 level and for the secondary curriculum? Were many of the current conflicts in mathematics education (aka, the Math Wars) overcome by this Panel? If not, what issues remain?
This is not a simple question! First, Achieve's formal Mathematics Advisory Panel (MAP) was constituted to work only on the K-8 level and produced Foundations for Success, a report with outcome expectations and sample problems for the end of grade 8. When work moved into the secondary level, it became part of the American Diploma Project (ADP) and operated with an evolving set of advisors representing all levels of mathematics and mathematics education.
From the perspective of the "math wars," the original MAP panel was, for its time, a remarkably catholic forum. Strong voices from many different perspectives set forth conflicting views. Compromises were agreed to, and sometimes reversed after further discussion. Eventually a report emerged. No one was pleased with every detail, but I believe it is fair to say that everyone on the MAP committee agreed that as a whole it represented a good step forward.
We reached this point by agreeing to set aside issues of pedagogy and to concentrate only on content. We further agreed that lists of expectations were less capable of conveying our intent than were rich examples. That is why the final report had 8 pages of expectations and 130 pages of examples. It was far easier for the diverse MAP members—protagonists, witnesses, and victims of the math wars—to agree on the quality of a problem than on the wording of a standard.
We also choose to largely ignore the issue of calculators because it was one of the wedge issues on which we all knew that the panel could never agree. Some may view this as cowardice, and it may be that. However, it made possible the rest of the work and affirmed, in a sense, that issues such as this may best be left for local decisions.
Another wedge issue we faced head-on, namely the place of quadratic functions and quadratic equations. Here we compromised, setting an ambitious bar for end of eighth grade at completing the square with a deliberate mandate to not employ the quadratic formula until the next algebra course. The purpose, of course, was understanding rather than calculation, a goal that in this case everyone around the table could support.
Those on the panel with the most school experience worried that completing the square was much too ambitious. They were proved right in subsequent reviews from states who wanted to use the Foundations for Success as a guide for their own standards. Consequently, later Achieve documents dealing with the transition from elementary to secondary mathematics are much more realistic about just how much algebra can be expected for all students prior to ninth grade.
Secondary mathematics is part of Achieve's ADP effort; the benchmarks together with sample postsecondary tasks appear in Ready or Not: Creating a High School Diploma That Counts (Achieve, 2004). There the contentious issue concerned the quantity of mathematics, especially of algebra, that should be required of all students for a high school diploma. A compromise was reached in which certain benchmarks, marked with an asterisk, were described as recommended for all but only required for those "who plan to take calculus in college." Of course, this asterisk mildly undermines the nominal goal of the ADP enterprise, namely, to set a uniform standard for an American high school diploma.
These matters—the role of calculators, the amount of algebra—are but two of the issues that remain fundamentally unresolved both within the ADP networks and among individuals who care about school mathematics. Other sources of continuing disagreement concern the role of data analysis and statistics, the place of financial mathematics, the importance of arithmetic "automaticity" and a host of pedagogical issues that, as I noted, Achieve largely leaves to others.
4. I’ve expressed great concern on this blog about the lack of frontline teacher representation on these major panels, particularly the President’s National Mathematics Panel? I’ve reiterated my call for redressing this situation via numerous emails to the Panel and on this blog. To date, all such requests have been politely dismissed. How do you feel about the need for increased teacher representation on this and other panels? Was there more K-12 representation (current classroom educators) on the Mathematics Advisory Panel on which you served?
The names of all those who advised Achieve on its MAP and ADP projects are listed in the reports of these projects. Different individuals contribute different types of work: some meet in panels; some review drafts; some write standards or contribute problems. My impression is that quite a few of Achieve's mathematics advisors have taught K-12 mathematics, but relatively few were serving as "frontline teachers" at the same time as they were helping with the Achieve work. Frontline teaching doesn't leave that much spare time.
Generally, I find concerns about representation less important than those about relevant experience. Sometimes the complaint is about the lack of teachers, other times about the lack of mathematicians; often complaints are accompanied by qualifiers (e.g., "current classroom teachers," or "active research mathematicians") that appear to imply that those who do not meet the condition are somehow less capable. What matters is that a panel as a whole include individuals with a broad balance of experience, which for mathematics education certainly includes both mathematical practice and classroom teaching—but not necessarily all at the same time the panel is meeting.
5. Many critics of NCTM’s original 1989 Curriculum and Evaluation Standards for School Mathematics and the revision in 2000 have claimed there was not enough emphasis on the learning of basic arithmetic facts. In your opinion, is the issue primarily due to lack of clarity in the standards, or is there a real difference of position between NCTM and its critics on the importance of arithmetic facts?? What is your position on the relative importance of the automaticity of basic facts?
There is a range of opinion about the importance of arithmetic facts within NCTM, within the broader mathematical community, and within the public at large. I understood the 1989 Standards to acknowledge this fact. A chief insight of statistics is recognizing the importance of variation. Student and adult skills with arithmetic vary, so the goals of mathematics education must take this into account. Almost all disputes about NCTM's standards arose because the historic absolutes of mathematics were replaced by alternatives and variations. In this sense, the critics were right: the Standards made mathematics "fuzzy" by insisting that most problems can be solved in more than one way. In fact, they can be.
There is no dispute that knowing arithmetic facts is more desirable than not knowing them, and being quick ("automatic") is better than being slow. The issue is: how important is this difference in relation to other goals of education? It is a bit like spelling: being good at spelling is more desirable than its opposite, but there are plenty of high-performing adults—including college professors, deans, and presidents—who are bad spellers. They learn to cope, as do adults who don't instantly know whether 7 x 8 is larger or smaller than 6 x 9.
For what it's worth, my "position" is that every child should be taught to memorize single digit arithmetic facts because if they do so everything that follows in school will be so much easier. But failure to accomplish this goal should not be interpreted as a sign of mathematical incapacity. Indeed, both students who achieve this goal and those who do not should continue to be stimulated with equal vigor by other mathematical topics (e.g., fractions, decimals, geometry, measurement), just like both good and bad spellers continue to read the same literature and write the same assignments.
Part II is now published. I hope to hear from many of you!
Dave Marain
Posts
26 comments:
Thanks very much for putting this together, Dave. So far I agree with Prof. Steen's answer to nearly every question, in particular his concerns about federal control of education.
Dave,
First, thank-you for taking the time to orchestrate the interview.
I think I'm going to hold off on commenting in detail until the rest of the interview is posted.
"That in our stampede for higher standards we are trampling on the enthusiasms, aspirations, and potential contributions of many students for whom mathematics is best approached indirectly."
Baloney!
Indirectly = low expectations
There is absolutely no basis for this comment other than opinion. I teach my son the value of hard work and mastery because he doesn't get it at school.
If you base school on what makes students happy, they will never meet their potential. Top-down or thematic education, no matter how interesting, will never insure mastery of needed skills. The problem is that mastery of skills is hard work and there is linkage between mastery and understanding. Most schools devalue mastery.
The classic example is Everyday Math, which thinks that mastery will somehow magically happen over time. Lessons are interrupted daily with a hodgepodge of Math Box flashbacks that they hope will help kids master the material at their own speed. It doesn't happen. The curriculum doesn't ensure mastery at any point in time, so it doesn't happen for many kids. It then doesn't matter how interested in math they are. It's too late.
Educators are bound and determined to redefine math, but if they want to open career doors, they really need to take a good look at the Math SAT and work backwards. Mastery of skills is paramount. Spelling is to writers is NOT like math skills is to mathematicians. This is an ignorant analogy.
Some students might do better with a slower approach, but it still needs to be rigorous. Unfortunately, most schools can't even do this. You either get onto the AP calculus track in high school (in spite of the math in K-8), or you are on a track to checkbook (nowhere) math.
Many educators try to unlink mastery from understanding. It doesn't work. If educators can't find any other basis for education than their opinions, then please get out of my way. Choice is the only solution.
steveh--
I certainly expected that Prof. Steen's strong pro-reform views would evoke strong reactions. I encourage you to read ALL of his replies (I'll post the remaining ones on Fri and Mon) before forming definitive opinions. I hope that my views on math content, mastery, strong foundations, etc., have been articulated clearly on this blog. For this interview, I've chosen to neither defend nor refute Prof. Steen's positions. He certainly doesn't need me to defend him!
There is much of interest remaining in the interview. Gaining insight into the process of how K-8 standards were negotiated is fascinating to me and may serve as a blueprint for resolving the current conflict. In the end, I believe it's futile to debate philosophies. It's far more productive to discuss actual content and examples of what kinds of problems students are expected to solve. This has been the raison d'etre for this blog from its inception.
I also invite you to read Prof. Steen's commentary on the K-8 Achieve Math Standards. It may help you to understand his positions even better:
http://www.achieve.org/node/300
Download this commentary from the right side of the page. It was eye-opening for me.
I'm more than happy to get into details.
"What counts is that students gain sufficient experience with substantive mathematics—not just worksheets—to benefit from its power and, if possible, to appreciate its beauty."
Worksheets don't contain substantive mathematics? Students can't even begin to understand "substantive mathematics" without some level of mastery. I can give a talk to kids about the mathematics of computer games but there is very little they will understand without a lot of foundational math.
As far as standards go, they are driven by the lowest common denominator. They are driven by the status quo, not by international standards of what can be done. They can talk all they want about "understanding" or "substantive mathematics", but what they are talking about is not a slower route to math, but a worse one.
Education is not about cutoffs, and it's not about equal education. It's about individual educational opportunity. It doesn't matter whether standards are local or national. Low expectations are low expectations.
Affluent kids get private schools, tutors, and help at home. They get high expectations from their parents. Smart urban kids get low cutoff standards and low expectations. Rising low cutoff test scores make educators happy, but they don't help these kids at all.
No arguments here, Steve! I've been saying exactly this for months.
However, you need to read the K-8 standards adopted by Achieve. They
are not a lowest common denominator. Many were based on content from the highest-performing nations. If you've been examining the Grade 6B Placement math test from Singapore, you'd know what I mean. I've posted several articles about this in the past couple of weeks. These and related issues come up during the rest of the interview. The issue
of mastery is still a major concern and a sticking point however.
This has to be resolved for education to move forward. Thank you for your astute comments.
My goal in all of this is put the issues out on the table and perhaps invite more open dialog between opposing factions. I may continue to do this with other interviews representing all sides of the debate. Many parents, educators and professionals are angry and feel very strongly about what's wrong with American education and what needs to be done to fix it. However, there is also much misunderstanding. Only when representatives of all parties sit down and face other can there ever be any meeting of the minds. The lowest common denominator? Everyone wants what's best for their children.
Steve, a famous applied mathematician wrote an extraordinary book entitled, "Why Johnny Can't Add: the Failure of the New Math". Recognize the title? Recall the author? Morris Kline? I believe he was Professor of Mathematics, NYU. Know what year it was published? 1972 or 1973. See how far we've come! I will have more to say about his views and how they compare to the debate 35 years later!
Dave
"Only when representatives of all parties sit down and face other can there ever be any meeting of the minds. The lowest common denominator? Everyone wants what's best for their children."
I seriously doubt that there can be a meeting of the minds when it comes to basic assumptions and expectations. Mastery is a major issue. You can set all of the standards you want. You can force a school to use Singapore Math. If they don't believe in specific grade-level goals of mastery, then even that will fail. Mastery in math is not like spelling. This is not about middle ground.
Does there even have to be a meeting of the minds if you allow choice? Larger school districts can easily offer parents a choice in math curricula starting in Kindergarten. This should be the focus of the discussion, not middle ground.
"Everyone wants what's best for their children."
... their own children. When it comes to education, this translates to lower expectations and slowly rising cutoff scores. Individual kids are important right now, not slowly rising averages to meet minimal goals in 2014.
A slowly rising tide will lift all boats, but it will not teach kids to fly.
Steve,
I didn't take the comment you quoted ("...for whom mathematics is best approached indirectly.") as meaning low expectations. To me, this meant instead of teaching students that math is only a series of algorithms we must show students the beauty and application of mathematics. If they are so turned off by the brute force approach, does it matter how good they are at it?
Also, I interpreted "worksheets" as busy work. Something that gives the students 50 of the same type of problems to "solve" without extensions to include the meaning of the math, applications to real life, or synthesis with other concepts.
As for the issue of mastery, it is my goal for my students can achieve mastery of concepts. But what if, for whatever reason, they can't? Should they spend the rest of their time limited to arithmetic? Never get to study algebra or geometry or stats?
"... we must show students the beauty and application of mathematics."
You need to give me an example of this and explain how this how this substitutes for mastery. Besides, these topics are not incompatible with a process that emphasizes mastery.
"If they are so turned off by the brute force approach, does it matter how good they are at it?"
"brute force approach"? Have you found an efficient road to mastery that doesn't involve hard work?
"...50 of the same type of problems to "solve" without extensions to include the meaning of the math, applications to real life, or synthesis with other concepts."
This is vague. Please give me specific examples. Please explain how you can have a steady diet of "real life" or "synthesis" (whatever they are) and still achieve mastery. Your implication is that there is not enough time for "real life" connections. This isn't true. This is just another way of saying low expectations.
"As for the issue of mastery, it is my goal for my students can achieve mastery of concepts."
There it is. Mastery of skills is not important. You've broken the link between mastery of skills and understanding. You have no basis for this. This is math appreciation, but the students don't have the knowledge or skills to really appreciate anything about math.
"But what if, for whatever reason, they can't? Should they spend the rest of their time limited to arithmetic? Never get to study algebra or geometry or stats?"
If they can't master arithmetic, then you can forget about algebra or statistics.
How is this anything more than just your opinion? You can't redefine math and say that mastery of skills is not important. Math is cumulative and everything is based on mastery of skills. Skills are not rote. They are based on fundamental rules. Understand the rules and you understand math. THAT is the beauty of math.
Math is cumulative and everything is based on mastery of skills. Skills are not rote. They are based on fundamental rules. Understand the rules and you understand math.
I think to some extent, this depends on your definition of "mastery". When you talk about mastery of addition, are you talking about "understanding the rules" or about having instantaneous recall of all the addition facts up to a certain number? Same for multiplication facts. I've no argument with pushing kids to learn them. But I do have an argument with holding them back if they don't have them all memorized. Keep making them work on them, keep testing them, but in the meantime, let them move on to other things too (and perhaps they will see why knowing them would make their lives easier).
I personally have a bachelor's degree in Pure Math and Computer Science and a master's degree (from MIT) in Electrical Engineering and Computer Science. I counted some of my addition facts on my fingers right through high school. I might have learned them sooner if someone had forced the issue, but I might instead have become turned off from mathematics. But not knowing them certainly didn't prevent me from understanding higher level mathematics.
I have an 11yo son. Last year as a 5th grader, he scored in the top 5% on the 4th to 6th grade Math Olympiads for Elementary and Middle school students (and well above the 50th percentile, probably around 75th, for the 7th and 8th grade contest). Because he has always completed his math work (pre-algebra) quickly and accurately his school assumed he knew his times tables cold, but it turns out that when they tested them this fall, he couldn't complete 60 problems in 3 minutes. So, obviously, those need more work, but it doesn't mean he can't and won't move on to algebra at the same time.
Where I teach, I take each group of elementary students once every two weeks, and the middle school students twice every two weeks. (I don't say once a week, because I have them twice in a row, and then not at all the next week.) I teach problem solving topics such as combinatorics, probability, number theory, etc. based on MOEMS and MathCounts type problems. I teach them to all students, even those struggling with mastery of their basic skills. Kids at all levels are able to understand the basic concepts, and it helps spark their interest in math and motivate them to master the basic skills. In our experience over the past 6 or so years of working this way, allowing the struggling kids to participate results in greater motivation and success in mastering their basic skills.
I realize that this is all anecdotal evidence, but it's more than random opinion, and it works very well for us! I've watched kids who think they're "bad at math" blossom in this system, and go on to take honors level math in high school.
"When you talk about mastery of addition, are you talking about "understanding the rules" or about having instantaneous recall of all the addition facts up to a certain number?"
In that case, both. For other cases, there are different levels of understanding. For example, many math curricula push "understanding" of multi-digit multiplication in fourth or fifth grade. This is possible up to a point, but full understanding will have to wait until algebra. However, mastery in the lower grades requires instant recall of basic add and subtracts to 20 and the multiplication table to at least 10 X 10, by the end of third grade! No later.
"But I do have an argument with holding them back if they don't have them all memorized. Keep making them work on them, keep testing them, but in the meantime, let them move on to other things too (and perhaps they will see why knowing them would make their lives easier)."
But how does this work in practice? Everyday Math is based on no specific mastery at any point in time. The problem is that it never gets done. My opinion is that many schools and teachers don't think that it's important at any time. They never enforce mastery. At my son's school last year in fifth grade (Everyday Math), they had students who couldn't immediately give the sum of 7+8. Their mastery of the times table was worse. These are smart kids. There is absolutely no reason they could not memorize these facts. The teacher had to start an after-school program to fix the problem. (At least she realized that there was a problem.)
" ...perhaps they will see why knowing them would make their lives easier."
Self-motivation is always nice, but it's not a prerequisite. Either it's important to learn or it's not. Either they can handle the material or they cannot. If you leave it up to the kids, then that's the same as low expectations. New material in math requires mastery of old material. If you move along without mastery, you hurt both the kids who are behind and those who are up-to-speed.
"But not knowing them certainly didn't prevent me from understanding higher level mathematics."
But I bet you knew how to divide fractions and solve systems of equations. Small gaps or limitations in knowledge or skills (I have my share), aren't a guarantee of failure, but they should NOT be the basis of a curriculum.
"... but it turns out that when they tested them this fall, he couldn't complete 60 problems in 3 minutes. So, obviously, those need more work, but it doesn't mean he can't and won't move on to algebra at the same time."
Test him again with flash cards. It takes time to write. My sixth grade son tested in the top 1% in math in the country and might not (I'll have to check) meet the same goal, but now you're talking about cutoffs, not philosophy. My son's Everyday Math school (and very many others) don't enforce anywhere near that kind of standard because they don't believe in mastery.
"... and it helps spark their interest in math and motivate them to master the basic skills."
That's all very nice, but what do you do with the ones who receive no spark?
"...allowing the struggling kids to participate results in greater motivation and success in mastering their basic skills."
Compared to what? If you wait too long, the struggling kids cannot catch up, no matter what their motivation.
"I've watched kids who think they're "bad at math" blossom in this system, and go on to take honors level math in high school."
What about the kids who don't "blossom"? Is a "math brain" or a "spark" required to get an education in math? When my son goes to school, I expect him to pay attention and work hard even if he doesn't like the material. I check his homework daily and set much higher expectations than the school (that's not saying much). I don't tell him that he doesn't have to finish a writing assignment because he isn't motivated. I will try to motivate him and "spark" his interest, but failing that, I will apply (and the school should too) external motivation, like grades and flunking.
Self-motivation is a nice goal, but if it doesn't happen, then schools darn well better do something else. If you think that it's just a matter of 60 problems in 3 minutes, then you better look again. Standardized test cutoff levels are much, much lower than that. If schools want math to be a "pump" and not a "filter", then they need to do some hard pumping. Affluent parents do a lot of pumping. Unfortunately, poor kids have to wait for a "spark".
steveh, jackie and mathmom--
Thanks for keeping the lines of communication open. Civil dialog is exactly what is needed to move forward through all the confusion surrounding math education. I wish there were simple answers to these problems, but I don't think there are. But I do believe a thorough understanding of and respect for each others' positions are critical. Thanks again for making my efforts seem worthwhile.
Perhaps one day, "Everything will be Illuminated."
Just to clarify, I'm not talking about "Everyday Math" with which I don't have any direct experience.
That's all very nice, but what do you do with the ones who receive no spark?
Same thing we do with all the others. Keep teaching them math. Both basic skills and applications. I just don't believe that teaching applications, problem solving, etc. should be withheld until mastery is shown. It is also an important part of mathematics, and should be taught to all students, struggling or otherwise.
Steve, you have a good point about my son and the writing aspect of a timed math test. He does have writing issues. I'll try flashcards and see how that goes. I suspect he knows them, but not as quickly/automatically as we'd eventually like him to.
I will try to motivate him and "spark" his interest, but failing that, I will apply (and the school should too) external motivation, like grades and flunking.
I, and the private school I send my children to, have a difference in philosophy with what you've stated here.
Our school is "ungraded" in that the children are not grouped by chronological age, but work in multi-age groupings according to their individual needs. But it is also "ungraded" in that the students (K-8 age) do not receive "grades" but rather a long written report of where they are in their educational journey, where they excel, and where they still need work.
Students are expected to be self-motivated. There is no threat of "bad grades" or "flunking" or even a "bad" report (the reports are descriptive, not judgemental) to motivate students.
A student may not be motivated by interest in every subject; that is understood. But there are other reasons to work hard and do well than either interest in the subject or passing grades.
Students do not leave our school lacking basic skills or knowledge, despite the lack of "external motivators" such as grades. At all levels of ability, they take school seriously and are motivated to do their best. They go on to excel in both traditional and non-traditional high school programs.
If you move along without mastery, you hurt both the kids who are behind and those who are up-to-speed.
We have somewhat of a "spiral" approach -- kids work in multi-age groupings, and are not necessarily expected to "get" (master) everything the first time around. This is not "low expectations" -- quite the contrary. Kids are exposed to things we know they may not be ready to master (this includes Shakespeare in the original language, starting at 5yo), with the understanding that they will have these things presented again (and again, until they master them). Meanwhile, kids who are able to master them the first time around are given extensions and higher expectations, and eventually move on to a higher-level group where new concepts are presented. Those who did not fully master a topic that first time have laid the groundwork for understanding it better the next time they are exposed to it. You can "move on" without mastery as long as you plan to come back. With math, many topics are of course sequential, and you cannot move on to the next without mastering the previous. BUT, you can move on to orthogonal topics to give the brain a rest, a chance to process, and then try again.
"Drill and Kill" turns many kids off to math. They may (or may not) eventually acquire the drilled skills, but those kids who were held back and forced to play scales every year and never experience the music wind up hating math, and drop it at the first opportunity. Those who see what math may have to offer them beyond arithmetic, in my experience, stick with it more resolutely.
Now, I have no doubt that "spiral curriculum" is poorly done in many schools, that kids are never assessed with respect to mastery, that kids are required to re-visit topics either too many (for those for whom mastery came easily) or too few (for those who struggle) times. But, that's not a fault of the spiral curriculum, but rather of a poor implementation. With appropriate observation and assessment, (and a small group size certainly helps) the spiral approach can really work.
"Just to clarify, I'm not talking about "Everyday Math" with which I don't have any direct experience."
You really should. It gives spiral a very bad name.
"I just don't believe that teaching applications, problem solving, etc. should be withheld until mastery is shown."
You have to quantify mastery here. If you're talking about 60 questions in three minutes, I would agree. You have to realize that in most schools, mastery at even minimal levels is avoided. In cases like this, moving ahead with new material and applications will not work.
"I'll try flashcards and see how that goes."
I've added minus signs to some of our flash cards to make it a little bit more difficult.
"But it is also "ungraded" in that the students (K-8 age) do not receive "grades" but rather a long written report of where they are in their educational journey, where they excel, and where they still need work."
Obviously a private school. The issue is not what you think is best for your kids. The question is why do public schools feel they have the right to impose their opinions and expectations on everyone else. You got to choose, (a choice I wouldn't make, by the way) but most people can't. The question is why doesn't everyone get choice when opinion and assumptions dominate the discussion.
"At all levels of ability, they take school seriously and are motivated to do their best."
I think public school teachers call this "pre-selected".
"Drill and Kill" turns many kids off to math. They may (or may not) eventually acquire the drilled skills, but those kids who were held back and forced to play scales every year and never experience the music wind up hating math, and drop it at the first opportunity. Those who see what math may have to offer them beyond arithmetic, in my experience, stick with it more resolutely."
If you don't practice your scales, and if you don't work on Czerny, you'll end up in the audience, not on the stage. If you don't focus on mastery of the basics of math, you'll end up majoring in English Literature, no matter how much beauty you see in math. Practicing scales and going to concerts are not incompatible. Mastering math basics and understanding the beauty of math are not incompatible.
In fact, what kind of beauty can a sixth-grader understand about math? If I gave a talk to sixth-graders about the math of computer games or of rendering, they might oooh and aaah, but they wouldn't have a clue to the beauty of transformations, dot products or cross products. It might motivate them by getting them to say: "I want to do THAT!", but this sort of thing can be done with any lame curriculum. They still have to put in the hard work of mastering the basics.
"But, that's not a fault of the spiral curriculum, ..."
One could argue that all education is a spiral. A big difference in spirals is the steepness of the spiral and how much mastery is required before you move on. Everyday Math is known for it's very shallow spiral (some call it circling - one mother complained that she had kids in 2nd, 4th and 5th grades and they were all covering the same material!) and it's almost complete lack of mastery enforcement. I call it repeated partial learning. This is very inefficient.
You got to choose, but choice is the last thing that public schools want to offer. Why? Why not have the money follow the child so that even a child from the inner-city can choose to go to your private school? I wouldn't do that, but that would be my choice.
My sense is that you don't realize how bad the lack of mastery is in public schools.
I think public school teachers call this "pre-selected".
There is certainly an aspect of this. However, most children are admitted at 4 or 5yo when it is difficult to know what their attitudes and motivation will be. Others are admitted older when they public schools fail them. Certainly, they come from families that take education seriously and have enough disposable income to spend money on it (though tuition is on a sliding scale and most families are not "wealthy" at all). But the children themselves are not "pre-selected" to be the easiest to educate. Many children come to our school because their parent either fear, or know, that public school will (or has) fail(ed) their children. There are students with learning disabilities, with ADD, etc... even some kids adopted from Africa with minimal prior formal education.
What happens at our school is not due to pre-selection of the students. It is due to methods that really work. And yes, I would love it if public school parents had a choice of a program like this one. They do, to a smaller extent, have a choice of a very good multi-age program in the primary grades, but standardized test prep interferes with it in the higher grades. My older kids did attend this public program prior to moving to their current private school.
Public schools impose their "opinions" on others because they are required to get all students to pass certain standardized tests, and so they use the programs that they believe are most likely to allow them to do so. I happen not to agree that training students to pass those tests has much in common with providing a good education, but it's not the schools' faults that they are required to do this. :(
In fact, what kind of beauty can a sixth-grader understand about math?
I can prove to a 6th grader that 0.99999... = 1 and that 1/2 + 1/4 + 1/8 + ... also = 1. They think that is pretty cool. I can show them Gauss' "trick" to adding up arithmetic sequences, and how this relates to triangle numbers and "handshake" problems. I can show them how solving a simpler problem and finding a pattern helps them solve a problem that they didn't think they'd be able to solve at all, such as the locker problem. They can learn how thinking about parity helps them solve seemingly complicated puzzles. They can learn how thinking about numbers in terms of their factors can help them understand and solve all sorts of problems (while quietly improving their "numeracy").
All of this even if they need to sometimes count on their fingers, or be reminded how to "invert and multiply" or how to subtract negative numbers.... They'll master those things eventually too, in part by using them to solve interesting problems.
I can tell you on an anecdotal basis, that both strong and weak math students look forward to the days when I come to work with them, even though I am "mean" and don't allow them to use calculators (and also convinced their regular math teacher not to let them use them nearly as often).
I will also tell you that as touchy-feely as it seems to you, confidence in themselves as mathematicians is incredibly important to many kids at this age, especially the girls. Without it, they will track themselves into the "basic" track in high school, and close off a lot of opportunities to themselves. Kids need to be given opportunities to succeed and feel good about themselves, all the while developing the skills they really need to be strong mathematicians.
You asked about voucher programs which I think is another huge and complicated issue, perhaps best saved for another day.
"...Certainly, they come from families that take education seriously ..."
Don't get me wrong. I'm playing the devil's advocate here. My son started out in public school. We switched him to a private school in 2nd grade. Now he's back in public school for 6th grade. Our experience is that the public school sets very low expectations in the early grades because they see it mostly as socialization within a full-inclusion learning evironment. Our son was (and is) a sponge for knowledge and they were feeding him with a teaspoon. Now that he's older, there is less difference between the curriculum of the public school and the private school. I'm not saying it's great, but the public school finally accepts that they have to group by ability and not age starting in 6th or 7th grade. The private school provided more academics and expectations in the early grades, but this difference lessened in the later grades. With AP courses in high school, there are few academic differences between good public high schools and private prep schools. The biggest difference is in student support. Good prep schools give you all of the academic support you need. At public high schools, it's sink or swim.
"Public schools impose their "opinions" on others because they are required to get all students to pass certain standardized tests, "
Public schools have opinions whether or not there are standardized tests, and many parents strongly disagree with those opinions.
"I happen not to agree that training students to pass those tests has much in common with providing a good education, but it's not the schools' faults that they are required to do this. :("
The tests are trivial and all good schools should laugh at them. They define a minimum, but schools see them as a maximum goal. Some tests have weird questions, but many of those questions are based on these same teachers' opinions. Teachers create and calibrate the tests. If the schools aren't forced to meet some level of accountability, the education they provide will be worse, not better.
"I can prove to a 6th grader that 0.99999... = 1 and that....."
All of this takes time. I'd like to see what material your school is dropping to make time for it. If it doesn't take that much time, then it could be added to any curriculum and is unrelated to mastery.
"They'll master those things eventually too, in part by using them to solve interesting problems."
Top-down mastery. I don't buy it. Maybe your school (with serious students) can get it to work, but it would be a disaster in public schools. In fact many public schools are centered around top-down and thematic learning. Mastery never happens. It IS a disaster.
"... confidence in themselves as mathematicians is incredibly important to many kids at this age, ..."
Confidence derives from being able to do the problems. Doing the problems requires skills. As I said before, I think your idea of "enough" mastery has little to do with what's going on in public schools. We're talking NO mastery. You can't compare an idealized learning environment of serious students with what is going on in public school.
"voucher programs"
I'm not specifically suggesting this. Public schools could provide choice, but they fight charter schools tooth-and-nail. Public schools don't even want to offer alternative math curricula even though it would be very easy to do. You might think that what you do at your school is similar to what happens at public schools, but you would be be wrong. You might be happy with your top-down, delayed mastery approach, but at least parents aren't forced to use your school.
Top-down mastery. I don't buy it. Maybe your school (with serious students) can get it to work, but it would be a disaster in public schools. In fact many public schools are centered around top-down and thematic learning. Mastery never happens. It IS a disaster.
First of all, serious students aren't born, they're made. Children do not enter our school at 5yo as "serious students". Serious students are made by they expectations and atmosphere of the school. Students are respected and are expected to respect teachers, one another, and themselves. They are expected to put forth an honest effort, and when they don't they are called on it. They are exposed to the good example of older students who have been part of the environment for much longer, and who have become serious students.
Secondly, top-down mastery: I can certainly appreciate that a program based only on top-down mastery would be a disaster. That is not what we do, and not what I am advocating. Students in elementary school spend 1/6 of their math time on top-down activities. Students in middle school spend 1/3 of their time. The rest of the time is spent on what you would call "bottom up" learning of basic skills. There is more to math than arithmetic, and by developing students' problem solving skills, we produce stronger mathematicians than those who learn only the basic skills.
Spending time on varied "hard problems" is also an excellent check on mastery. The group may work on a skill until everyone shows mastery, but then that skill will come up again in a problem a month later, and many will have forgotten it. I can then alert their regular teacher that she needs to go back and work on a particular skill with certain students again.
All of this takes time. I'd like to see what material your school is dropping to make time for it.
Of course it takes time. It's a question of whether one considers it time well spent or not. Obviously we disagree on this issue, but I've seen it work out well for a wide variety of students over the past 5 years. As I said above, our school dedicates 1/3 of the middle school math group's math time to work like this. They don't drop anything to make time for it. They just spend a little less time on direct instruction in basic skills, and dedicate some time to this type of indirect practice.
A traditional math curriculum spends lots of time reinforcing taught skills, and in our case, some of that reinforcement is done via non-routine problem solving. This time is not wasted. Both the teacher and the students remark that the "word problems" in the textbook seem trivial after all the work on problem solving they do with me. When students who do not consider themselves mathematically inclined score in the top 50% on the elementary or middle school Math Olympiad, this gives them much more confidence in themselves as developing mathematicians than just getting a page of math problems correct.
Students leave our school well prepared for either 9th grade Algebra or 10th grade Honors Geometry, according how mathematically inclined they are, and how quickly they grasp the pre-algebra topics (how many times around that spiral they need).
They also spend math time on design work (tessellations, symmetry, etc.) and once every couple of years they do a "budget project" where they choose a career, get an income based on it, and have to budget for their necessities and luxuries, make a scale drawing of the house or apartment they would want, within their budget, etc.
And they still manage to learn pre-algebra, and in many cases algebra as well.
Can this work in a public school? Yes it can. There's nothing magical about our private school, though I will admit that smaller group sizes do help. Our state requires math problem solving portfolios from public school students beginning with 4th graders, and I've seen some of the work 3rd and 4th graders do on problem solving, and it is quite impressive. And these same students still manage to learn the basic math skills of their grade level as well. They still do "mad minutes" to encourage mastery of the basic facts. They still master long division, etc...
Spending time on non-routine math problems is not a waste of time. It helps stretch their "math brains" in other ways, improves their sense of numeracy, and ultimately leads to kids who are competent and confident young mathematicians.
By spending time on both bottom-up and top-down learning, we can have the best of both worlds. I've never advocated dropping the direct teaching of basic skills. I've never advocated for the adoption of Everyday Math (which I've heard can be excellent in the right hands, with plenty of differentiation, but which in reality usually is a disaster for students at many levels). I'm not advocating a program "centered around" top-down learning.
I'm advocating for a middle ground -- a position where math need not be only dull drill-and-kill mastery of basic skills. I believe in the direct teaching of basic skills, but I also believe in the value of using fun, interesting, and challenging problems as a tool to help spark more interest in mathematics, to help practice basic skills in a different context, and to give kids the thrill of the "aha!" moment that one gets when solving contest-style math problems, and the resultant confidence and self-esteem that comes from struggling with something difficult and succeeding.
"They are expected to put forth an honest effort, and when they don't they are called on it."
This doesn't happen in the lower grades of most public schools, by definition.
As you mentioned before:
"Our school is 'ungraded' in that the children are not grouped by chronological age, but work in multi-age groupings according to their individual needs."
Most public schools work on the basis of full-inclusion or what I call age-tracking. There is absolutely no grouping by needs, interest, or ability. You seem to think that you have some sort of correlation of education philosophy with public schools. That is NOT the case.
I don't think that what your school is doing is wrong because I suspect you enforce "enough" mastery. That is the key. However, I probably wouldn't send my son to your school. I don't want him to have any misconceptions about the need for mastery. In public schools, however, there is almost a dislike towards content, skills, and mastery in the lower grades. Things change starting in 7th and 8th grades, but for many, the damage has been done.
"...and in our case, some of that reinforcement is done via non-routine problem solving."
The implication is that the traditional, mastery first approach is more efficient, which leaves more time later on for non-routine problem solving. The downside, presumably, is that it turns some kids off to math. Your approach tries to make math more interesting by introducing non-routine problem solving even if kids are still counting on their fingers. This is less efficient but may (or may not) inspire more kids. However, you still haven't quantified the level of mastery you require before allowing kids to move on to the next level.
"..this gives them much more confidence in themselves as developing mathematicians than just getting a page of math problems correct."
This is a strawman. There is NOTHING about a mastery-centered approach that precludes "non-routine" problems, unless you're talking about covering different amounts of material each year. In that case, the traditional approach would be far, far ahead.
"And these same students still manage to learn the basic math skills of their grade level as well."
Are you saying that there are no problems with math scores in this country? Or is it the fault of the kids and parents?
"I've never advocated for the adoption of Everyday Math (which I've heard can be excellent in the right hands, with plenty of differentiation, but which in reality usually is a disaster for students at many levels)."
Everyday Math is structurally flawed. It advocates no specific expectations of mastery at any point in time. It contains so much non-essential material that there is no way to achieve any level of mastery. (My son's fifth grade class didn't cover 35 percent of the EM workbooks because they ran out of time - partly because the teacher had to try and fix mastery problems.) But EM thinks that's OK because students will see the material next year. It doesn't excite or turn on any kids to be constantly jumping from one topic to the next and always feeling that they don't know what's going on. There are NO "non-routine" problems in Everyday Math. It's just a sequence of tear-out worksheets; one to do in class and one to do at home. I just went through the latest edition of sixth grade EM this summer with my son to help him jump a grade to 7th grade Pre-Algebra, which uses a real textbook! I have issues with the textbook, but it's an order-of-magnitude better than the hodgepodge Everyday Math approach. By the way, my son was a model Everyday Math student because he could master the material each time through the spiral. Many kids couldn't. They didn't like math. It's not motivating to not know what you're doing. Everyday Math is repeated partial learning.
"I'm advocating for a middle ground -- a position where math need not be only dull drill-and-kill mastery of basic skills."
This is the same strawman. What you are advocating is the belief that lower expectations of mastery will increase motivation, mastery problems will disappear, and the net result will be positive. But your motivation comes from using non-routine problems. This is a separate issue because non-routine problems could be used with an approach that sets higher initial expectations of mastery. In fact, non-routine problems will take longer for those who are still counting on their fingers. Singapore Math expects a high level of mastery and also provides lots of non-routine or challenging problems; the best of both worlds and a very motivating formula.
The fact that many public schools currently use crappy methods to teach math does not mean that they couldn't use good methods to teach math. You keep telling me that the kinds of things our school does could never work in a public school, but there's not inherent reason why they couldn't. A good method does not have to mean "you do nothing more until you master a certain basic skill". Even Singapore does not require that -- the challenging questions are interspersed in the text, and are used to help kids cement their skills. There is no testing to qualify students to move on from the mundane problems to the challenging word problems.
I understand that most public schools work on an age-in-grade lockstep, but that doesn't mean that they have to. In K-3 my son was in multi-age classes, was appropriately challenged in school, worked with older kids when appropriate, worked on his own at other times, and generally thrived. It was only when 4th grade standardized testing came around that the school said they'd have to stop doing all of that and make him work on the same 4th grade curriculum he'd already completed in most subjects that we left the public school. If they'd had the faith to realize he would show proficiency on the 4th grade tests even if he worked with 5th and 6th graders on most subjects, and didn't get the "teaching to the 4th grade test" that practically defined the 4th grade year, he'd probably have stayed in public school. Since our HS has a good honors/AP program, he just re-started public school and appears to be thriving there once again.
I am not in charge of moving kids in our school up from one math group to the next, so I can't quantify the skills required to move up, but it is based on mastery of the skills taught at that level. But I will say that a child who has most of the skills but is lagging in a few will sometimes move if the higher group is a better fit overall, and they will just work with the child on those skills, often as homework, while moving on to other topics as well. Even children who have demonstrated mastery sometimes forget them later and need to review them.
There is no mastery requirement to take part in the problem solving sessions. The only exception to that is that in the youngest (K-2 approximately) group, I don't work on problem solving with the very youngest students until they have developed some concept of number, but I do work with the more capable students in that group right from the start. In the other groups, everyone works on problem solving. They take time out from their basic skills instruction to do so.
"And these same students still manage to learn the basic math skills of their grade level as well."
Are you saying that there are no problems with math scores in this country?
No, I am saying there is nothing wrong with the math skills of children at the school my children attend, where math skills are taught alongside interesting problem solving topics. And in fact, I am saying that, based on my observation and that of the principal of many years, that kids are leaving with better math skills, and better preparation for high school, now that we are formally teaching problem solving.
I'm not saying there is nothing wrong with the way many public schools teach math today. I am saying there is something deeply right about taking time out from basic skills instruction to also teach interesting applications and non-routine problem solving.
Math is more than arithmetic. While mastery of arithmetic is a necessary pre-requisite for higher-level mathematics, so are well-developed thinking skills. It is the latter that seem to be neglected in many basic skills curricula. (I'm not talking about Singapore -- I think that's a great curriculum. I'm talking about a lot of "traditional" American curricula that merely drill and practice on the basic skills and simple word problems.)
"The fact that many public schools currently use crappy methods to teach math does not mean that they couldn't use good methods to teach math."
Then why don't they? Because they think that what they're doing is fine.
"Even Singapore does not require that --..."
Since I'm a great supporter of Singapore Math (and used it at home with my son), I think you really misunderstand my position.
"In K-3 my son was in multi-age classes, was appropriately challenged in school, worked with older kids when appropriate, worked on his own at other times, and generally thrived. It was only when 4th grade standardized testing came around that the school said they'd have to stop doing all of that and make him work on the same 4th grade curriculum he'd already completed in most subjects that we left the public school."
This isn't the norm. Age tracking is normally used in the lower grades to facilitate full-inclusion and socialization. Most public schools don't provide any sort of ability grouping until 7th grade. (usually in math) My niece's school (in Michigan) starts grouping in earlier grades because of parental demand and to compete with charter and other public schools. Choice forces respect for parental wishes. This says nothing about whether they enforce mastery early or late. The key is flexibility and higher expectations.
Your example is the first I've ever heard of any school using testing as an excuse for eliminating ability grouping. I will ask around to see if this is happening elsewhere. I don't like standardized testing because the very low cutoff points soon become the best that the school will do. The problem is that if the testing requirement is eliminated, schools will get worse, not better. Most schools are better now because of the requirements, but that isn't saying much. Unfortunately, this is often done by shifting resources away from the more capable kids. But I can't imagine that (in general) removing trivial testing requirements will improve schools overall.
"I am saying there is something deeply right about taking time out from basic skills instruction to also teach interesting applications and non-routine problem solving."
The devil is in the details.
The problem is that most public schools say exactly the same thing, but their rhetoric hides a complete lack of emphasis on mastery. You may think it's the same thing as what you are talking about, but it's quite different. I know another parent (with engineering degrees) who really liked these same ideas - until he saw the real-life workings of Everyday Math. Is it just the implementation and not the theory? No, it's both. When I argued the case for Singapore Math (over Everyday Math) at my son's (previous) private school, they turned up their noses. Something else is going on here.
"I'm talking about a lot of "traditional" American curricula that merely drill and practice on the basic skills and simple word problems."
Arguing against "traditional American curricula" is a strawman. That's NOT the argument going on now in the so-called "Math Wars". It has to do with low versus high expectations. It has to do with with extremely low expectations of mastery hidden by talk of Higher-Order Thinking and Conceptual Understanding. All of the people I know who are fighting against "reform math" (aka low expectation math) would love to see all schools use Singapore Math. Nobody wants to go back to some sort of mythical "traditional" math curriculum. That argument is simply a ploy to avoid discussing the details and lack of mastery in reform math. Standardized tests and drill-and kill aren't the big problems in math. Low expectations, bad curricula, and poor implementations are.
I agree that Singapore is a great curriculum. My 6yo does the workbooks for fun at home. My 11yo will be using selected parts of 6A and 6B (based on his performance on their placement tests) to shore up his pre-algebra before moving into Algebra and enrichment topics. I don't remember if you've said how old your son is, but if you haven't already, you might want to look into the Art of Problem Solving texts for him to work from after finishing Singapore 6.
I think our main disagreement was that you took Prof. Steen's arguments to apply only to Everyday Math and the way you see it being applied in districts you are familiar with, and I took them in a more generic context. Also, I guess I've seen some different public school approaches to ability grouping and math teaching than you have. I've actually never seen a public school around here that didn't ability group for math and reading, at least within each individual classroom, but often sharing kids/groups between two classes at the same level, and at least minimal support for subject acceleration (though scheduling logistics often got in the way of that).
As to NCLB, I think it is making schools better for the kids just below proficiency, and worse for everyone else. A huge proportion of time and resources are spent on the so-called "bubble students" who are below the standard, but close enough that with work they might make it. Gifted programs are being cut, arts and music programs are being cut (short-term thinking!), and proficient students are being ignored. IMO, low expectations have come from NCLB more than anything else. I think, overall, that schools are far worse off because of it. btw, the public school still did ability grouping in 4th grade, but they would not allow any subject acceleration out of 4th because of the testing that had them so frenzied. :( This was in a school with over 50% of kids on free/reduced lunch, kids in foster care and all sorts of unfortunate situations. Getting the required test scores was far from assured. Virtually everything not designed to directly raise test scores for the bubble kids was marginalized. :(
" ...you took Prof. Steen's arguments to apply only to Everyday Math and the way you see it being applied in districts you are familiar with, and I took them in a more generic context."
I don't think so. I have seen his sort of arguments used for many, many years to hide low expectations behind a veneer of "understanding" rhetoric. They argue with generalities, but you never see the details.
"I don't remember if you've said how old your son is, but if you haven't already, you might want to look into the Art of Problem Solving texts for him to work from after finishing Singapore 6."
He's in sixth grade, but taking 7th grade pre-algebra. I will look into those texts. Thank you.
"As to NCLB, I think it is making schools better for the kids just below proficiency, and worse for everyone else."
They shift resources rather than think about fixing flaws in their assumptions. It doesn't change their overall level of expectations.
"I think, overall, that schools are far worse off because of it. "
But the converse isn't true; that if you get rid of testing, the schools will be better overall. The worst parts of testing is that resources get shifted, the goal becomes the low cutoff, and everyone thinks they're doing a good job - institutionalized low expectations. Is this worse? I could argue that case, but the solution isn't to go back just to help the more capable kids.
Great post, thanks!
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