This is the second and final part of my online interview with Prof. Steen. Part I is posted here. It includes useful background information about the new Algebra II End-of-Course Exam, its purposes, its content and its impact on districts that use a 3-year integrated math sequence. Prof. Steen also courageously tackles issues as diverse as proficiency with fractions, the role of factoring in the 21st century, AP Calculus as a model for a national curriculum, the linear mastery model of learning mathematics, gifted education, the critical factors needed to elevate mathematics education in our country, and attempting to resolve the Math Wars. He ends with advice for mathematics educators, restating the core message of the NCTM Standards.

Again I want to express my gratitude to Prof. Steen for taking the time to reply thoughtfully to some difficult and controversial questions regarding mathematics education. I'm hoping that this forum serves as a springboard for other bloggers to have further conversations with educational leaders and, perhaps, bring, opposing parties together at an 'online roundtable.' Regardless of personal ideologies, I hope those who have or will visit will find this interview as thought-provoking as I did. One thing is for certain. Both Prof. Steen and I have a new-found appreciation for how difficult it will be to resolve the major problems in education, mathematics education in particular. Again, I invite readers to post comments and keep the discussion alive. I would also be interested in reactions to the format of this interview. Suggestions for improvement? Perhaps make it more give-and-take?

**Math Notations Interview (continued)**

6. Many secondary teachers decry the lack of proficiency with fraction skills and fraction concepts demonstrated by their students. It’s always easy for each group of teachers from graduate school on down to place blame on prior grades. Do you believe that Achieve has addressed this problem adequately with their enumeration of K-8 mathematics expectations in their 2002 publication, *Foundations for Success*?

The expectations summarized in * Foundations for Success* certainly subsume the arithmetic of fractions and the relationships among fractions, decimals, proportions, and percents, but they do so quite concisely. Details are unfolded in Achieve's * K-8 Number Benchmarks*, especially throughout grades 4-6. However, no one associated with this project was so naïve as to imagine that the mere inclusion of an extensive discussion of fractions in a report will adequately address the problem of students entering high school—or college, for that matter—without understanding fractions. Setting out clear expectations is only a first step.

7. What is your position on the role of technology, calculators in particular, in K-4, 5-8 and 9-12 mathematics classrooms?

My view is that students should learn to use technology wisely, carefully, and powerfully. By wisely, I mean that they make conscious and appropriate decisions about when to use calculators or computers, and when not to. By carefully, I mean that they think enough about the problem they are working on to recognize when a calculator or computer result is beyond the realm of plausibility. By powerfully, I mean that they make full use of the most powerful tools available in order to prepare rich and accurate analyses. In this age, mathematical competence requires competence to use computer tools, so the use of technology must be an explicit goal of mathematics education.

It no more follows from students' widespread misuse of calculators that calculators should be banned than from students' widespread misunderstanding of fractions that fractions should be avoided. Use of technology is as important as use of fractions, and both need to be taught and tested.

8. I have stated repeatedly on this blog that the Advanced Placement Calculus syllabus from which I taught for over 30 years, is essentially a national curriculum for calculus and that I strongly endorse it as such. Do you agree with this characterization? Do you see projects such as ADP moving in a similar direction, working closely with states to achieve a common set of mathematics topics K-12 that must be covered at each grade level?

As AP courses go, AP calculus is one of the best. By intent of its sponsor (the College Board), it follows rather than leads national trends. For example, the most recent revision took place a few years after (not before) implementation of pilot projects supported by NSF's calculus reform program. The momentum for change was lead by college faculty, not by the College Board. ADP has a more ambitious goal, namely to lead the nation's K-12 schools to higher standards. In contrast to AP calculus whose syllabus is in the mainstream of college calculus courses, the expectations produced by MAP and ADP are on (and sometimes beyond) the leading edge of K-12 mathematics programs.

9. The types of problems Singaporean children, for example, are tackling seem more complex than their grade counterparts in the U.S. Do you believe that most mathematics curricula in the US, particularly in the area of problem-solving, are as challenging as those in other high-performing nations?

U.S. education clearly lags behind many other nations. This is not just a matter of curriculum but of teacher preparation, time in school, parental expectations, community environment, and perhaps funding. Some other nations (e.g., Japan) decided that their curricular expectations were too high and have reduced them. Others (e.g., England) have seen student performance fall. As I implied in my answer to the previous question, the MAP and ADP expectations, being calibrated to international standards, are well beyond what can be achieved at this time by most districts for most students. Their purpose is to set a target, but to reach that target we will need to change much more than curriculum.

10. The End-of-Course Algebra II exam will have a central core and 7 optional modules. Why were traditional topics such as log functions, matrices, conics and sequences/series pulled out of the core? Also, were the standards influenced by the Algebra II topics currently included on the SATs?

The traditional Algebra II course was developed as a stepping stone to calculus for the minority of students who felt they might want to study further mathematics. Two decades ago fewer than half of the age cohort took Algebra II. Today's course is intended for all students; it is a requirement for high school graduation in more than half the states. So it is natural that the "core" of Algebra II be rethought, with more specialized topics set aside into optional units. The new Algebra II may well be the last mathematics course ever taken by many of today's high school students, so I hope that the topics included in the new syllabus and test are well suited to the needs of all students.

I say "hope" because I actually know very little about the details of the test development process. In particular, I do not know if anyone has made any effort to coordinate topics with the revised SAT.

11. I’m assuming that school districts are already or soon will be receiving more detailed information concerning the new End-of-Course Algebra II exam. Will there be a full sample practice test made available? The Achieve web site will be helpful to Algebra II teachers, but could you suggest some additional resources they could use?

I am even more ignorant of these implementation issues than I am about the course goals. While it is helpful to see sample tests, the best way to prepare for an Algebra II test is to study a wide variety interesting and challenging problems. The internet is full of sites that offer enrichment and challenge problems for different high school courses. I'd suggest exploring the __Math Forum__ in the United States and the __Millennium Mathematics Project__ in the United Kingdom.

12. In your opinion, how will the End-of-Course Algebra II exam impact on those districts that use a 3-year integrated math sequence?

This is a very important question, and relates directly to the issue you raised earlier about what constitutes the core of the course. In my view, since passing the new end-of-course Algebra II exam will be a requirement for high school graduation for many students, it should be thought of more as an exam covering the third year of high school mathematics than as an exam covering algebra topics that are needed for calculus. Clearly there is much overlap in these two perspectives, but there are also some differences. I understand that the strategy of a core test with optional modules is intended precisely to reflect these two options. I remain concerned that the older calculus-focused view remains too dominant, at the expense of many newly-important topics that serve to introduce combinatorics, finance, probability, statistics, computer science, etc.

13. I still have a hard time when a student reaches for the graphing calculator to analyze the signs of the quadratic function f(x) = x^2-2x-8. Most textbook publishers have deemphasized factoring, relegating it to the back of the book. Educators have generally followed suit, although not all. How do you view the role of factoring in Algebra II and the secondary curriculum in general?

Factoring is one of the topics on the borderline of the two perspectives on Algebra II—preparation for life vs. preparation for higher mathematics. For life (e.g., citizenship and personal living) factoring is a relatively useless skill. For higher mathematics, the conceptual role of factors is crucial, but all real problems that may require factors are solved using computer tools (e.g., Mathematica). The only place where actual factoring of factorable polynomials is required on a regular basis is in mathematics courses. My advice is to be honest with students about this skill (and others like it). It is important for certain purposes, but not a life skill.

14. A recent article in Time magazine as well as a recently published book by Alec Klein make a strong case for gifted education and developing the talents of our brightest math and science students. Do you believe that our most talented math students are being adequately served? In particular, do you believe they can they flourish and develop equally well in heterogeneous classes as in fast-track accelerated classes?

This too is a very important and difficult question. Research and experience confirm that the presence of bright and intellectually aggressive students in a class helps propel all students to higher levels of achievement, so pulling these students out will in most cases make it less likely that the average students will reach their full potential. On the other hand, bright students whose mind has moved beyond the class syllabus—which is very common in mathematics—will be bored, resentful, and rebellious. Neither option is good; each short-changes far too many students.

Taking a clue from game theory, it seems to me that a mixed strategy is the best compromise: some work together, some work separate. In addition to raising the bar for average students, mixed groups help accelerated students learn to communicate mathematics—a skill that every client of secondary education—employers and professors alike—report is in very short supply. Separate groups help teachers and students focus on problems that are calibrated to match students' current skills.

However, even when students are separated by skill level, acceleration is not the only option. Mathematically able students should be challenged as much as possible by opportunities for horizontal exploration of optional topics that are not part of the mainstream curriculum. For many students, excessive acceleration is a great disservice. Except for the tiny minority (beyond three sigma) who need to take college mathematics while still in high school, most student who finish the school mathematics curriculum early wind up with a gap between high school and college mathematics, with rushed rather than deep mastery of high school topics, and with little or no opportunity to employ the mathematics they learned in parallel natural or social science courses. It is appalling how often students who receive a passing grade on AP calculus discover upon entering college that they need to take remedial algebra since they have forgotten whatever little they learned in their pre-calculus rush. Far better to slow down, spread horizontally, and dig deeper into the hidden corners of the regular curriculum.

15. Many mathematics educators I’ve spoken to and worked with believe that the learning of mathematics is essentially linear, i.e., one cannot be successful at level D unless one can demonstrate proficiency with levels A, B and C. What is your view on this model of learning mathematics? In particular, do you believe that students need to demonstrate proficiency in arithmetic skills and numeration before moving on to algebra?

The linear model of mathematics learning is wrong in almost every respect. Cognitive scientists remind us that the human brain learns by association, not logic. The history of science is full of examples of researchers who came to parts of advanced mathematics via some phenomenon or theory, not by a logical ladder of mathematical steps. Science students frequently encounter and use parts of mathematics in a physics or biology course well before they encounter it systematically in a mathematics course. Fields medalist mathematician William Thurston once described mathematics as like a banyan tree with branches that take root in different places, providing nourishment and growth along multiple pathways (Notices of AMS 37(1990) 844–850).

It is also extraordinarily counterproductive to our national goals. Dozens of reports have raised alarms about shortages of mathematically trained graduates from schools and colleges. Curricula and requirements based on the assumption that there is just one proper path to mathematics artificially and unnecessarily restrict potential mathematics graduates to those who find an intellectual kinship with that preferred approach. It cuts out those who might approach mathematics from other directions, be it from biology, or statistics, or computers, or finance, or construction, or energy, or environment, or any of a dozen other things that may interest students more than mathematics but which share a side door to mathematics.

16. Many states ‘talk the talk’ about higher standards and expectations, but translating these goals into reality in the classroom has proved difficult. Could you rank order the most important factors that are needed to accomplish these goals? For example, would you place teacher preparation above textbook quality?

Enthusiastic and imaginative teachers who are both mathematically and pedagogically competent are more important by far than anything else in the educational system. In particular, competent teachers need to be free to teach in whatever way is effective for them—which implies minimum constraints from state- or district-imposed curricula and tests. Imaginative teachers with minimum constraints would produce a lot of innovation; required standards and high stakes tests tend to stifle innovation. Clearly, some common expectations and assessments are important, but they should focus on the broad goals of education, not on narrow particulars.

Why do we get narrow particulars (that is, "standards") instead of imaginative teachers? The answer is obvious: money and political commitment. It is cheaper by several orders of magnitude to convene a consensus process to write standards than to attract, educate, and retain people with the interests and skills needed to teach mathematics well to all our nation's students. When you don't have enough teachers with the required competence, then the way politicians "make do" is to lay out specific standards and assessments for everyone to follow. I don't think we have much evidence that this strategy will work.

17. Hindsight is always 20-20, but if you could go back in time to the development of the original NCTM standards, what are some changes you would make, in light of what has transpired over the past two decades?

It is important to remember that at the time NCTM published its 1989 *Standards*, the very concept of standards was a subversive idea. Even the definition was in dispute: some viewed a standard as a banner to march behind, others as a hurdle that must be cleared. In this context, it was proper for NCTM to be somewhat cautious. Certainly there were places in the *Standards* where intentions were not adequately communicated, but nothing can ever prevent critics from selective reading. It is only human to read into a text what you want to find. Consequently, different readers read the *Standards * differently.

I read them as clarion call for eliminating the tradition, most evident in mathematics, to select and educate only the most able students and to provide others, disproportionately poor and minority, with only the illusion of education. For the first time a powerful national voice said that all students deserve a mathematics education. How this can be done, and how long it should take, are details that are still being worked out (as your earlier questions about MAP, ADP, and Algebra II attest). This commitment, that every student deserves an equally good education, is the one unequivocally positive aspect of the No Child Left Behind (NCLB) law.

If I were able to go back and make any change, I would highlight that central message more, and make clear that the suggested particulars were to be worked out through traditional American strategies of local innovation. The mistake NCTM made, if it can be called a mistake, was to let its critics define its message as the particulars rather than to keep the nation's attention on the central goal of providing all students with a meaningful mathematics education.

18. Here’s an innocent little question, Prof. Steen! The current conflicts in mathematics education are usually referred to as the *Math Wars*. In your opinion, what were the major contributing factors in spawning this conflict and how would you resolve it?

There are many factors involved. I think I can identify a few, but I have no confidence that I could resolve any of them.

One is the natural tendency of parents to want their children to go through the same education that they received—even when, as often is the case with mathematics, they admit that it was a painful and unsuccessful ordeal. This makes many parents critical of any change, most especially if it introduces approaches that they do not understand and which therefore leaves them unable to help their children with homework.

Another source were scientists and mathematicians who pretty much breezed through school mathematics and who were increasingly frustrated with graduates (often their own children) who did not seem to know what these scientists knew (or thought they knew) when they had graduated from high school. Our weak performance on international tests appeared to provide objective confirmation of these concerns, and they came to pubic notice just as the NCTM standards became widely known in the early to mid-1990s. Even though very few students had gone through an education influenced by these standards, the confluence of events led many to believe that the standards contributed to the decline.

A third source can be traced to the way in which the NCTM *Standards* upset the caste system in mathematics education. Mathematicians are accustomed to a hierarchy of status and influence with internationally recognized researchers at the top, ordinary college teachers in the middle, below them high school teachers, and at the very bottom teachers in elementary grades. The gradient is determined by level of mathematics education and research. So it came as somewhat of a shock to research mathematicians when the organization representing elementary and secondary school teachers, seemingly without notice or permission, deigned to issue "standards" for mathematics. Mathematicians would say, and did say, "we define mathematics, not you."

I could go on, but won't. But I do want to add that, as with any contentious issue, face-to-face dialog helps bridge differences. With some exceptions, I believe that has happened with protagonists of the math wars. Achieve was one of the first organizations to bring to one table people from all these different perspectives. Subsequently, other groups have made similar efforts, generally with good results. As mathematicians and educators roll up their sleeves to work together on common projects, each learns from the other and the frictions that led to the math wars begin to reduce.

19. Finally, I’ve observed considerable frustration among K-12 mathematics educators for the past 20 years. Each wants to do what she/he perceives is the best for her/his students but they are often mandated to follow new curricula and programs that come and go every few years and for which they often receive inadequate training. What message would you like to convey to these dedicated professionals?

I said above that teachers are the key to success in mathematics education, but that outsiders impose standards and assessments as a means of protecting students against soft spots in the system. This is not unreasonable, since in the K-12 sector the state is responsible for guaranteeing that children receive a proper education. It seems to me that the only way that teachers can regain control over their own affairs is for them to convincingly take on the role of ensuring quality education for all children. That will require much higher standards for initial licensure, for tenure, for professional development, and a commitment to post-tenure reviews. This is the regimen followed by most good colleges and with suitable modification, by hospitals. Self-imposed quality control is the sign of a true profession.

The problem teachers face is a severe mismatch between the needs of K-12 education, especially in mathematics and science, and available resources. But here teachers have an asset that they need to make better use of, namely, regular access to parents and school boards. What they need to do with that access is help the public understand the changing nature of mathematics and science, the unique value it offers their children, the challenges involved in keeping up with a rapidly changing discipline while at the same time teaching students of quite varied skills and preparation, and the concrete steps that teachers have taken to ensure that all students receive a sound education. Focusing on quality for all—the core message of the NCTM *Standards—*should gradually elevate the respect in which teachers are held and with it, the support they receive from the public.

## 42 comments:

Excellent interview, Dave! I'll post some reactions next week. (I think.)

mr. person--

Thanks! It has been tiring to say the least - I need a break for a couple of days to get ready for next week's carnival!

Actually, I hope Prof. Steen's astute comments have shed some light on the current state of math education. They have for me. I've always believed that some dialog is better than none! We need to keep the lines of communication flowing freely in both directions to move forward.

Thank you for doing this interview. I think there is a real audience for this sort of stuff, and I appreciate that Steen took time out to answer questions from a retired math teacher/supervisor. Also, he knows that the blogs will have people who disagree with him, so it took some courage to participate.

I take issue with some of the points raised. In particular, I am concerned about the role that teachers have played and will play in moving math forward.

I'll blog some of my ideas in the next few days. I think you are going to find that you have started some genuinely constructive conversations.

I'd like to address Prof. Steen's comments on ability grouping of gifted kids, where he said:

"Research and experience confirm that the presence of bright and intellectually aggressive students in a class helps propel all students to higher levels of achievement, so pulling these students out will in most cases make it less likely that the average students will reach their full potential. On the other hand, bright students whose mind has moved beyond the class syllabus—which is very common in mathematics—will be bored, resentful, and rebellious. Neither option is good; each short-changes far too many students."Substantial research shows that ability grouping, or at least grouping the highest achievers separately, is beneficial to high-achieving children without harming the rest of the students.

I'd direct readers to the Hoagies Gifted page on Grouping for pointers to research and article on this topic.

Steen's comment about how to use calculators:

"By powerfully, I mean that they make full use of the most powerful tools available in order to prepare rich and accurate analyses. In this age, mathematical competence requires competence to use computer tools, so the use of technology must be an explicit goal of mathematics education."

This is just the usual generic blather. I see calculators and spreadsheets used as "avoiders", not "enhancers", in grades K-8. I was in college when the transition was made from sliderules to calculators. Calculators allowed courses to tackle more difficult theories and applications. Shorter and simpler hand calculation assignments gave way to 20-30 page analyses that were made possible with the calculator. This doesn't happen in grade school. They use calculators to avoid mastery of the basics. This would be understandable if they replaced that time with more complex analyses using calculators. It doesn't happen. They could have students see how the average of a set of 10 numbers is greatly affected by the change in one number as compared to a set of 100 numbers. The underlying issue is hard work, not understanding.

"In this age, mathematical competence requires competence to use computer tools, so the use of technology must be an explicit goal of mathematics education."

Mathematical competence in ANY age requires the skills to CREATE the tools rather than just use the tools. In grades K-8, no assumptions should be made about whether any one child will or will not be a creator of tools, rather than just a user. I find it astonishing that many educators base their teaching approach on what is commonly needed by the average adult. "Math brains" might survive this approach, but for many others, I can hear the doors slamming shut by 8th grade.

"...students' widespread misuse of calculators ..."

Students don't misuse calculators. schools do.

"Use of technology is as important as use of fractions, and both need to be taught and tested."

They aren't equal. Fractions are much more important. They are a foundation. "Technology" is not.

"The new Algebra II may well be the last mathematics course ever taken by many of today's high school students, so I hope that the topics included in the new syllabus and test are well suited to the needs of all students."

" ..needs of all students"? All?

You mean the needs of those students who will never take a math class again. What about all of the stundents who may not need calculus in high school, but need a math background that prepares them for almost anything they want to do in college. Students don't fall into just the AP calc group and the no-more-math group. The common approach to math in K-12 starts separating kids by ability in 7th grade. Many of these kids, who might do well with almost any crappy math curriculum because of natural ability, parental support or tutoring, will get on the AP calculus track. You cannot assume that many others couldn't do it, or that they need a terminal course in math in high school. High schools shouldn't be closing doors on students. Many more kids want to go to college and most college degrees require some level of math courses. High schools need to categorize these different college levels and provide clear paths to those requirements. Many students avoid careers they might be very good at because they can't deal with the math requirements.

The wrong assumption here is that K-12 schools do a good job teaching math; all that is needed is different content. That isn't true. Most kids need rigor and high expectations, but at a slower and more deliberate pace. Why not define a curriculum path that leads to a rigorous, non-terminal, Algebra II course in high school, one that prepares kids for almost anything they want to do in college?

"The only place where actual factoring of factorable polynomials is required on a regular basis is in mathematics courses. My advice is to be honest with students about this skill (and others like it). It is important for certain purposes, but not a life skill."

Aaaaaaarrrrrggggghhhhh! Why not be really honest and show kids exactly what kind of math is required for each degree path they might want to take in college. Do a survey of colleges and list the terminal math course required for each degree. Show it to freshmen in high school. There is "life" after high school and for many, it includes college, and colleges require lots of math "skills". The world isn't broken only into math majors and those who need "life skills".

"Research and experience confirm that the presence of bright and intellectually aggressive students in a class helps propel all students to higher levels of achievement, .."

Only when the ability levels are fairly close. This is not happening in shcools. Full-inclusion means that kids are tracked by age and that borderline autistic kids are mixed in with the brightest kids. This is the primary requirement or assumption upon which all else is built. Everything else they do is an attempt to mitigate the associated problems.

One method is dual teaching where one teacher is geared towards special needs students. Another method they use to mitigate problems is "differentiated instruction". This is only useful in a limited sense because many students are ready for new material. Many schools will not do separate acceleration of content, so the differentiation becomes horizontal.

"In addition to raising the bar for average students, mixed groups help accelerated students learn to communicate mathematics—a skill that every client of secondary education—employers and professors alike—report is in very short supply. Separate groups help teachers and students focus on problems that are calibrated to match students' current skills."

These are just rationalizations to justify their governing desire for age-tracking.

"However, even when students are separated by skill level, acceleration is not the only option."

But acceleration can't be done. You can't have it both ways. You can't have very mixed-ability, child-centered group learning and acceleration of material for the more able kids. Something has to give. What gives is acceleration of material. It is replaced with enrichment. They try to make it sound really good, but it's only the best they can do given the primary assumption of age-tracking. For more able kids, schools try to make enrichment sound good with talk of "depth" and "understanding". When the school sent home a form this year (with the usual Multiple Intelligences junk) asking how my son learns, I wrote back and said "fast".

"It is appalling how often students who receive a passing grade on AP calculus discover upon entering college that they need to take remedial algebra since they have forgotten whatever little they learned in their pre-calculus rush. Far better to slow down, spread horizontally, and dig deeper into the hidden corners of the regular curriculum."

OK, I'll bite. What percentage of passing AP calculus (What did they get on the AP test?) students have to take REMEDIAL algebra in college? Is this even possible? Normally these kids would start college by retaking calculus because many colleges don't accept advanced placement. Second, how bad would high school math teaching have to be to produce calculus students who couldn't do algebra. If this happens, which I seriously doubt, it's NOT a problem with acceleration. This comment is just a way to bad-mouth acceleration and justify the lack of it in the early grades. They can't have full-inclusion and aceleration so acceleration, ability grouping, and talented students take the rap. When my son was in Kindergarten and first grade I got the sense that the teachers didn't like smart kids. At the very least, they wanted to maintain the idea that all kids are equal; it's just that they learn differently. This is done by reducing the emphasis on content and mastery of skills. By sixth or seventh grades they can't play this game anymore and are forced to start splitting kids by ability. Unfortunately for many kids, the damage has been done.

"The linear model of mathematics learning is wrong in almost every respect."

It's also a strawman. Almost all learning is a spiral. What matters is exactly how the spiral is done, exactly what level of mastery is required before students are allowed to move on to new material, and where the finish line is. This thinking is used to justify all sorts of low expectations. There may be lots of paths to go from point A to point B, but you darn well better get to point B. This can't be used as justification for changing the location of point B.

"The history of science is full of examples of researchers who came to parts of advanced mathematics via some phenomenon or theory, not by a logical ladder of mathematical steps."

But I bet they knew how to divide fractions. Besides, you don't want to build a curriculum around this concept.

"It is also extraordinarily counterproductive to our national goals. Dozens of reports have raised alarms about shortages of mathematically trained graduates from schools and colleges."

This isn't going to be solved with a terminal course in Algebra II in high school.

"Enthusiastic and imaginative teachers who are both mathematically and pedagogically competent are more important by far than anything else in the educational system."

Baloney. Curricula matter. Expectations matter. Mastery matters. This is the old "All we need are good teachers" approach. It isn't true and it won't happen, so deal with reality. Besides, your idea of mathematical and pedagogical competency is probably not the same as mine, especially if you don't agree with my point B.

"Certainly there were places in the Standards where intentions were not adequately communicated, but nothing can ever prevent critics from selective reading."

People weren't looking at the standards. They were looking at what was happening in the classroom. they were looking at what was coming home in backpacks. My son had MathLand in first grade. I wasn't looking at the standards, I was looking at the implementation. MathLand was the sorriest excuse for a math curriculum. Maybe the standards really meant something else, but it sure took them long enough to make their case more clear (backtrack).

"I read them as clarion call for eliminating the tradition, most evident in mathematics, to select and educate only the most able students and to provide others, disproportionately poor and minority, with only the illusion of education."

Don't pull out the poor and minority cards. So, traditional math was bad because of the curriculum, but the problem with reform math is the implementation. Do we now have a lot more poor and minority kids doing well in math? Math requires effort. It requires a clearly-defined path of content and mastery. There were problems before and there are problems now. You can't have it both ways. You can't demonize "traditional" math on one hand and then say that all we need are properly-educated teachers on the other. The illusion is that you are solving the problem. You are really just implementing your own opinions about mathematics and pedagogy.

"This commitment, that every student deserves an equally good education, is the one unequivocally positive aspect of the No Child Left Behind (NCLB) law."

The illusion is that the low cutoff standards of NCLB define an equally good education. Our public schools are all "High Achieving" schools. Almost all kids get above the low NCLB cutoff. Everyone is pleased as punch. In the meantime, affluent and educated parents know better. They set higher expectations for their kids. They work with them at home and provide tutors, if necessary. This is not to make them into super students. This is to provide the content, skills, and acceleration they will not get at public schools - by definition.

"One is the natural tendency of parents to want their children to go through the same education that they received—even when, as often is the case with mathematics, they admit that it was a painful and unsuccessful ordeal."

Baloney! We're not stupid. When my son was in pre-school I thought of all of the things I didn't like about my traditional math education. There were many. Then, the teacher told me that they used MathLand at our school! They're going in the wrong direction! MathLand has now disappeared from even the web, it was so bad. What parents want are high expecations for their kids. they want their kids to get to point B. Parents are smart enough to know that little Suzie won't get there by writing a report about her favorite number. The problem is not just how the school gets to the finish line, but where the finish line is.

"Even though very few students had gone through an education influenced by these standards, the confluence of events led many to believe that the standards contributed to the decline."

Lower expectations started long ago. The stadards just codified them and gave them a pedagogical foundation.

"Mathematicians would say, and did say, 'we define mathematics, not you.'"

Here it is. The justification that K-12 educators can redefine mathematics. Because they want to. Actually, it's no justification at all. It's a philosophical and pedagogical power grab based only on opinion and poor research. Unfortunately, students who want to go to college have to deal with this blatant power grab. K-12 educators do what they want and students have to pick up the pieces when they get to college. Is this better for the poor and minorities? they don't have parents who can see through the conceit and academic turf battles to get their kids prepared for college. The arrogance of K-12 educators is astonishing.

"As mathematicians and educators roll up their sleeves to work together on common projects, each learns from the other and the frictions that led to the math wars begin to reduce."

This is astonishing. You justify a unilateral decision to redefine K-12 mathematics, you laugh at college professors, and then say that the solution is to work together. Utterly astonishing. And parents see homework assignments asking their kids to write about their favorite number.

"What they [teachers] need to do with that access is help the public understand the changing nature of mathematics and science, the unique value it offers their children, the challenges involved in keeping up with a rapidly changing discipline..."

K-12 educators "understand the changing nature of mathematics and science" better than college professors and trained parents who have been working in those fields for decades? Go ahead and redefine math and science all you want. Just give parents their money to go someplace else. There are big reasons why public schools are so afraid of choice. They think that parents and kids will leave en masse; especially the poor and minorities.

One is the natural tendency of parents to want their children to go through the same education that they received—even when, as often is the case with mathematics, they admit that it was a painful and unsuccessful ordeal. This makes many parents critical of any change, most especially if it introduces approaches that they do not understand and which therefore leaves them unable to help their children with homework.Steve H addressed this nicely, but let me add to it. I did not find my mathematical experience (in the 50's and 60's) to be painful nor unsuccessful. I constantly hear about how the "traditional approach doesn't work". It depends on your definition of "traditional". The traditional texts in the 1800's were not all that good, and the traditional texts in the late 1970's through the 80's that came about because of the "back to basics" movement were also not very good. But some of the texts that came about in the 30's through the 60's were designed to address the age old complaints that there was too much memorization and not enough understanding. William A. Brownell and others developed arithmetic textbooks which provided good contextual explanations for the concepts, as well as providing the drills for practice. The sequencing was logical, and was cumulative; i.e., what was learned in previous lessons was integrated in later lessons. Brownell has been recognized by NCTM as well as math reformers as a pioneer in education reform.

The 60's new math led us away from these texts to a too-formal approach for grade schoolers. The high school math books that came out of the 60's new math is a mixed bag, but a bag worth looking at. Some of the texts that emerged as a result of that era were and are excellent: Dolciani's algebra textbooks, Moise and Downs' geometry text, Drayton and Wooten's algebra, and others.

There is the charge that these texts (the high school ones) were designed for an "elite class" who were destined for college. That is a matter of opinion. They were designed for anyone who wanted to learn math, and they enabled one to go to college. The algebra book I used in high school was developed in the 50's, and though perhaps not a direct product of new math, was certainly influenced by Max Beberman who helped revolutionize how algebra textbooks were written. I went to a high school that was 50% African Americans. I recall an African American in my Algebra 2 class who though he had an attitude and talked back to the teacher at times, was able to do any problem at the board he was asked to do. Whether he chose to go on to a productive path was not a function of a textbook being designed for an elite class. He was capable of mastering the material, and from what I could see, had done so.

While there were improvements that could have been made in the texts that I had, the math program served me well, and I ended up majoring in math. The texts were designed with high expectations in mind to allude to Steve H's post. Compared to the low expectations inherent in many of today's math curricula and texts, I would venture to say that even a book written in 1870 would do better than what we have today. At least students would know the procedures and do computations that for many of today's students are difficult to impossible (including taking 10% of a number).

Dave Marain wrote:

"We need to keep the lines of communication flowing freely in both directions to move forward."

It doesn't work this way. Educators do what they want and then try to "educate" parents. Prof. Steen made this quite clear. Parents have zero power or influence. There is no middle ground when it comes to assumptions and it's clear that many educators have mixed-ability groupings at the top of their assumption list. Second is lower expectations.

There are fundamental differences of opinion going on here that have nothing to do with research. The only option is choice, not middle ground.

About calculators, I've just written my own rant about them.

About "enrichment" for gifted kids, Steve is right that it's usually a joke, however, good enrichment may be better in some cases than additional acceleration for some gifted students, some of the time. (How's that for wishy washy.)

My youngest is substantially accelerated right now; but by middle school level, I think there is a place for branching out into enrichment as well, and that is our plan for my middle son this year. I'm hoping to blog about that later this week.

Steve, if you want choice, work to make it happen. In the meantime, work to make the schools your kids attend closer to what you would have chosen if you'd had the choice by volunteering to help them out. I suspect that if you volunteered to take rotating groups of kids out of math class to work with them on mastery of basic skills, you'd be unlikely to be turned down. (Some schools wouldn't let you do it for your own kid's classroom -- in that case find a like-minded parent from a different class, and team up with them, each working in the other's kid's class.) If you can show the school how your approach will improve their results, they will be a whole lot more likely to change their minds than if you just rant about their method sucking.

If you want to tell me that people shouldn't have to have/find time to volunteer in their kids' school for them to get a decent education, I will tell you that, ideally, you're right. But in the real world, you do what you have to do.

"...however, good enrichment may be better in some cases than additional acceleration for some gifted students, some of the time."

What happens is that K-12 educators hijack words (like spiral, understanding, and enrichment) and turn them into their own. You can't talk about them in a general sense. This is a favorite ploy by educators. Talk generalities to make parents go away and then decide on all of the details.

A classic technique is to talk about balance. I was at a meeting of teachers and parents (who were complaining about Everyday Math), when the teachers started talking about the need for balance. Who could be against balance? Teachers were going to provide more practice sheets. The discussion changed, parents were placated, but no details were discussed. The teachers continued to do what they wanted.

"Steve, if you want choice, work to make it happen."

I'm sorry, but you don't know what I have or have not been doing. You make it seem that the onus is on me or that I haven't been doing enough.

"If you can show the school how your approach will improve their results, they will be a whole lot more likely to change their minds than if you just rant about their method sucking."

"rant"?

Excuse me, but you're being extremely presumptuous here. If you want, I could go on and on about all of the committees I've been on, all of the teacher-parent meetings I've been to, and all of the after-school programs I've worked on. Perhaps I can tell you about all of the conversations I've had with our school's head of curriculum. Many of the issues are based on philosophy and assumptions and educators will do what they want.

Ranting is a pejorative term used to make people go away. Prof. Steen made some comments and I provided rebutting arguments. The tone might be harsh, but the arguments are specific and you need to address them directly if you have any comments.

"But in the real world, you do what you have to do."

Why do you presume that this is not what I've been doing ... because you don't like my comments? You don't like my comments. You don't (or can't) rebut them, so you call it ranting and assume that I am part of the problem and not part of the solution. This is a classic educator response.

Sorry Steve, I was not trying to presume what you have or have not done. But since you have said here several times, effectively, that parent choice is the only solution you will be happy with, then I was advising you to work toward it. If you already are, then great.

I also advised you to volunteer in you child's school. I am assuming you don't since you haven't mentioned in in several cases where it would have easily fit, but I could be wrong. It is easy to get the wrong idea about people on the Internet.

I'm sorry if you are unhappy with may use of the word "rant", however I feel that it accurately reflects the tone you have taken here. "Aaaaaaarrrrrggggghhhhh!" is generally a decent indicator of a rant. I'm not trying to make you go away, and I don't think your arguments are worthless. You and I agree on a great number of things.

You are complaining about what goes in "in public schools" in general, but as far as I can tell, you are familiar with only the schools in your area. I've seen a lot of things in a lot of public schools that isn't consistent with what you've said. When you say "public schools do X or Y" *that* is a straw man. Public schools do not all do the same things. I live in an area where the public schools are *not* particularly highly rated, yet they do not do most of the things you claim "public schools" do.

Many of the issues are based on philosophy and assumptions and educators will do what they want.Educators will do what they believe is best for their school, and the children in it, while at the same time not over-working the teachers. You can argue about assumptions and philosophy until you're all blue in the face, but if you can show them evidence that what you are saying will work for them, will give them what they want, they are much more likely to take it seriously. If you can make it easy for them, by offering to do some of the extra work for them, they are much more likely to try it.

Maybe you have already done all this, and if so, that's great. But if not, consider it some constructive advice toward getting what you want/need for your children and the others in your community. Your statement that "educators will do what they want" belies, IMO, a loss of hope, or giving up. Good luck!

"... but I could be wrong."

You are. I already told you that.

"'Aaaaaaarrrrrggggghhhhh!' is generally a decent indicator of a rant."

No it isn't. Look at everything else I wrote, none of which you address.

"Public schools do not all do the same things."

Just because you don't see it doesn't mean that it isn't going on. Nobody is talking about 100%.

"... but if you can show them evidence that what you are saying will work for them, will give them what they want, they are much more likely to take it seriously."

"Evidence"??? No they won't. Assumptions are assumptions. I've seen this over and over. Can I prove it happens in all schools? Of course not. Is this a minor issue? Of course not. You advocate separating kids by ability and trying to meet their needs. I mentioned before that this is happening at my niece's public school in Michigan (notice that I'm not saying that this happens everywhere), but this is an impossibility anywhere in our region. It can't happen. Grouping by ability happens only when you get to 7th grade. Twenty-five percent of the kids in our town get sent to private schools. The private schools have their own issues, but at least the kids get higher expectations in the lower grades. Parents have tried and tried and tried to get change in the public schools. It hasn't happened. All they get is enrichment.

"If you can make it easy for them, by offering to do some of the extra work for them, they are much more likely to try it."

No they won't. From your comments in the other thread, it appeared to me that you don't know what's going on in many other places around the country. You don't know what parents have tried to do. What I am saying is nothing new. The so-called "math-wars" aren't a matter of fussy college professors or nostalgic parents. The solution is not a matter of us showing the schools (the onus is on us?) that there is a better way when proof is almost impossible. This is another classic argument. Schools get to pick a math curriculum based on whatever ideas pop into their heads, but others have to prove that they have something better. It won't happen. Even in large districts where resources are plentiful, schools will not provide an alternate math curriculum in spite of demand. Things are happening out in the world and you seem to have no clue.

"But if not, consider it some constructive advice toward getting what you want/need for your children and the others in your community."

"constructive advice"?

This indicates that you really don't know what's going on around the country. After all of my comments in these threads (which you never address directly), you presume to give me "constructive advice", just like my son's first grade teacher (in her first grade teacher voice) telling us parents (sitting in little kids chairs) about the wonders of MathLand.

"Your statement that "educators will do what they want" belies, IMO, a loss of hope, or giving up."

That is why many (myself included) advocate choice. It's the only leverage many parents can hope for. That is the reason my niece was accelerated in certain subjects in 4th grade at her public school. Choice. In our region, however, this change is very unlikely. The public education system fights all charters (they got the state to impose a moratorium), there is zero money for gifted and talented education (I mean zero.), and they are very happy just meeting the low cutoff of trivial standardized tests. This attitude starts to change in 6th or 7th grades, but for many, the damage has been done. At that age it's easy for them to blame it on the child, the parents, or society.

Steve, assumptions change when evidence is provided to contradict them.

I have a pretty good idea what is going on in dozens of districts all across the country from various mailing lists I am on. Refusal to "ability group" at all does happen, but based on about 100 parents I correspond with on 3 separate mailing lists, parents who represent all different locations, it is the minority, not the majority, of public schools that work that way.

Also based on one of those mailing lists, parents who go into school and volunteer to do for the kids whatever it is they wish the school would do for them but they are not, are rarely turned away. I know many parents who go into public schools and offer enrichment for gifted students. (I have done this myself in the past.) I know parents who help drill kids on their times tables. I know parents who go into schools and tutor struggling students. I know parents who help supervise the "rest of the class" while the teachers do enrichment activities with the gifted learners. Once these things are facilitated and happening in the school, it is easy for teachers and administrators to see their benefits. Once educators can see the benefits of something, their assumptions change. That is what I mean by evidence, and it has worked in many districts.

Now, many people I know are "stuck" in Everyday Math districts, or other curricula that they don't like. In many cases they can't change the curriculum, which may be mandated from "on high", but they can facilitate differentiation within the curriculum (which Everyday Math is

meantto support), and they can facilitate a supplementary review of the basics for those who need it, etc. Parents can help raise the standards and the performance of kids in their districts by giving of their time.It's true that I don't know a thing about your district, and maybe it is so rigid that they wouldn't let you do anything, or get near any classroom, or anything else. But you cannot tell me that this is what "public schools do" as if that were the rule and not the exception.

The public education system fights all charters (they got the state to impose a moratorium), there is zero money for gifted and talented education (I mean zero.)Steve, there is no charter law at all in my state, and no funding for gifted and talented education either. Nor is there any mandate to provide it. However, this lack of funding is precisely why schools I've approached have been very grateful to have a volunteer come in and help them provide programming they do feel badly about not being able to provide themselves. Just because there is no funding for gifted education does not mean schools are obligated to turn down offers of free help.

Frustrated, concerned, passionate, educated,intelligent and "I'm mad as hell and I'm not going to take it anymore" , 'trying to make a difference'parents...

Passionate, talented, dedicated, and caring educators who are trying to make a difference voluntarily...

Without these dialogs, there will only more of the same rhetoric I've been hearing for the past 20 years.

Neither mathmom nor I nor hundreds of educators I've worked with or met convey a negative message of indifference to the needs of children. We are ALL frustrated by the current lack of direction in education. This is why I challenged the National Mathematics Panel. This is why I've challenged my state mathematics committee. This is why I started this blog.

No one is really going to change anyone's views in any significant way because of this or any other blog. Mathematics education is evolving and when one is part of the process, it's hard to imagine what the outcome will be. Read the Achieve benchmarks! Significant increase of expectations for K-8, stressing skill mastery, conceptual understanding and a higher-level of problem-solving.

Steve, get past your disgust over the history of reform and what you currently are enduring and focus on where we're heading. Choice is one way of thinking, however, I am not yet resigned to accepting that as the best solution. There are exciting changes taking place all over the country, changes, that I believe, are moving us in the right direction.

Steve, stop dissecting what I or mathmom is saying. Tell me exactly the nature of the curriculum and pedagogy you want, and perhaps we can move forward. Right now this dialog is stuck in limbo, exactly where the Math Wars are for most people. But I am not stuck there. I see the examples enumerated in the K-8 benchmarks by Achieve and I see hope. I read mathmom's, denise's, jackie's and others' comments and I see more than hope.

You can laugh at my optimism, but right now, I am in a position to talk to both parents and boards of education and tell them in what direction we should be moving. My opinions, of course, but some seem to respect those...

Dave

".. assumptions change when evidence is provided to contradict them."

Not in the case of acceleration of material or ability grouping.

"...it is the minority, not the majority, of public schools that work that way."

First you argue that it's not 100%. Now you argue that it's a minority. I think you have a different idea of ability grouping. I see lots of "ability grouping" in our differentiated instruction classrooms. But once again, details and definitions matter. Much of the ability grouping is on a horizontal or enrichment basis, and not a vertical, or acceleration basis. Lack of acceleration and low expectations is a major issue. It is not a minor problem unless you think that typical state standardized test levels are fine. You argue against standardized tests, but you never explain why it's so difficult for schools to meet those trivial standards and still have lots of time left over.

"I know many parents who go into public schools and offer enrichment for gifted students."

The issue is NOT enrichment! It's acceleration. It's higher expectations of content and mastery of skills. These are basic assumptions. They are not changed by providing after-school enrichment. How do you think a school would react to after-school sessions of the Singapore Math curriculum? You're basically telling them that the curriculum they use is wrong.

"I know parents who help drill kids on their times tables."

This is the teacher's job. I had to do it with my son to make sure that it got done, but what about all of those kids who don't have parents who will take up the slack? Teachers don't like drill (and kill), but it's OK if someone else does it?

"Once these things are facilitated and happening in the school, it is easy for teachers and administrators to see their benefits."

You're talking about being helpers for the teachers, not changing fundamental assumptions. You're talking about enrichment (or remedial mastery work), not grade-level expectations of content and mastery, let alone acceleration.

"In many cases they can't change the curriculum, which may be mandated from "on high", but they can facilitate differentiation within the curriculum (which Everyday Math is meant to support), and they can facilitate a supplementary review of the basics for those who need it, etc."

I've spent most of my posts talking about assumptions and structural flaws in K-8 education, not how to deal with day-to-day reality. My son is doing fine, thank you. I could laugh all the way to the SAT bank. I could explain to parents how to deal with the "system" (I've done that.), but you can't mix up the two discussions. They are separate.

"Parents can help raise the standards and the performance of kids in their districts by giving of their time."

I've given my time (including my share of after-school enrichment, like First Lego League robotics) and I have argued for fundamental changes like switching from Everyday Math to Singapore Math. These are two separate approaches. Providing enrichment does not change assumptions.

"But you cannot tell me that this is what "public schools do" as if that were the rule and not the exception."

That's because you've redefined the problem, just like K-12 educators have redefined math. If you don't like something, redefine it, and the problem goes away.

"However, this lack of funding is precisely why schools I've approached have been very grateful to have a volunteer come in and help them provide programming they do feel badly about not being able to provide themselves."

Enrichment, not acceleration. Helping them with their solution, not finding a better solution. The last thing schools want are parents on a curriculum selection committee.

"Without these dialogs, there will only more of the same rhetoric I've been hearing for the past 20 years."

It takes more than dialog. It takes a process. There is none now. It's not a matter of just listening to what parents have to say. The process has to provide some sort power or leverage for parents. If large school districts will not provide an alternate math curriculum in spite of demand, then talk isn't enough. Parents need choice of schools. The goal isn't necessarily one solution for all students.

"We are ALL frustrated by the current lack of direction in education."

But who gets to decide on this direction? Why are parents (the largest stakeholders) lowest on the list when it comes to input? Why is the goal of education to provide small statistical increases in low cut-off averages? Why is education about finding some middle (low) ground for everyone? (except for the affluent)

"There are exciting changes taking place all over the country, changes, that I believe, are moving us in the right direction."

You don't think I keep up with these changes? I'm not excited. Will these changes get rid of curricula like Everyday Math? Will these changes require schools to ensure proper grade-level mastery? Will these changes allow poor and minority kids to excel even if they don't have parents who make up the difference? By excel, I don't mean on standardized tests. I mean algebra in 8th grade. I mean getting on the AP calculus track. If Prof. Steen sees only "math brains" and "life skills" paths, there are still real big problems.

"Tell me exactly the nature of the curriculum and pedagogy you want, and perhaps we can move forward."

I thought that was clear.

Singapore Math with enforced grade-level expectations of mastery. This means no social promotion and hoping that kids will somehow figure it out later.

There should be no math wars. There is a simple solution. Choice. This doesn't necessarily mean choice of schools. It could mean choice of curriculum. You can't make everyone happy, but two approaches would go a long ways towards solving the problem. But this goes back to my core issues: acceleration and expectations of mastery. Schools have to drop their full-inclusion and enrichment-only philosophy. They have to drop their conceit that they are the ones who get to determine all of the assumptions. Like Prof. Steen, they can't unilaterally redefine math. I can accept that others have different assumptions from mine. But I'm not trying to shove my assumptions down their throats.

I think that teachers would love choice. I think that teachers would love higher expectations of grade-level mastery. Sixth grade teachers could focus on sixth grade material, not struggle to make sure that kids who don't know the times table are ready for standardized tests. Schools should laugh at standardized tests. They should take up only a small fraction of their time. Perhaps another track could take a spiral approach to mastery that includes all of the social promotion students. It's impossible to come up with a middle ground when fundamental assumptions are so different. Choice is the only reasonable solution, and it has to be parental choice.

I think you have a different idea of ability grouping. I see lots of "ability grouping" in our differentiated instruction classrooms. But once again, details and definitions matter. Much of the ability grouping is on a horizontal or enrichment basis, and not a vertical, or acceleration basis. Lack of acceleration and low expectations is a major issue.You and I have a substantial difference of opinion if you think acceleration is the only or even best answer for gifted learners.

Take a look at my blog for more about my take on acceleration versus enrichment for gifted learners.

Lack of acceleration is

notthe same as low expectations. Since you and I agree that the current standards represent "low expectations" particularly as compared to the abilities of gifted learners, I'm surprised that you wouldn't be happy to have gifted learners be taughtmore, be taught stuffoutsideandin addition tothat basic, minimal set of standards.They are not changed by providing after-school enrichment.Well, for starters, I am talking about in-school, rather than after-school enrichment. I believe that all students have a right to appropriate math education in math class, not only after school. And, if the enrichment is good, then they will be challenged by it.

How do you think a school would react to after-school sessions of the Singapore Math curriculum? You're basically telling them that the curriculum they use is wrong.I think you'd get few students to sign up for after-school Singapore Math, never mind what the school would think. But if, for example, you volunteered to take the top kids out of the class and do some extra problem solving with them while they helped the struggling kids master the basic curriculum, most schools would take you up on that. If you took those "extra" problems out of, say, a resource you "happened to" already have like, say Singapore Math, I doubt the school would complain. If you found that the students lacked the mastery of basic skills required to complete the problems, then you would naturally work on those skills with them as well. And mention it to the teacher when you handed the kids back. When I volunteered to not only take the kids but do all the work of preparing the "lessons" I found no difficulty getting takers. When I did this in the public school, I found teachers who were dying to give the top students more challenge, but frequently had trouble making the time to do it. They were thrilled for me to come in and help in that way.

If you really want to offer after-school Singapore Math, and you have the demand for it, use a room at the library, or in your house, or anywhere else. It will be harder to prove to the school that what you are doing matters to these kids, but you'll still be helping the kids, and at some point someone might indeed wonder why certain kids are suddenly doing better in math...

You're talking about being helpers for the teachers, not changing fundamental assumptions.I'm talking about taking one step at a time. First, by being a helper for the teacher. And slowly changing fundamental assumptions by the results of the help you provide. Because if what you're doing really makes a difference, it will show. And that kind of "evidence" can change minds. Also, as the old saying goes, "you attract more flies with honey than with vinegar." When you volunteer your time and effort to help teachers and schools, they get to know you better, get to trust you more, and may take your opinions, suggestions, requests and advice just that much more seriously.

I can tell you for certain that I have changed the mind of the principal and middle school math teacher of our current school about things like calculator use, and the value of contest math (for all students, not just the gifted ones). She could see the benefits of the things I was doing with the kids, and began to trust my opinion, and trust the evidence before her eyes.

"You and I have a substantial difference of opinion if you think acceleration is the only or even best answer for gifted learners."

I said neither, but many schools don't offer acceleration at all, by definition. It's not an option. Are you saying that no acceleration is fine? Enrichment can be good or it can be bad, but it can't be the only way to differentiate.

"Lack of acceleration is not the same as low expectations."

For most kids, it is. Enrichment cannot make up for a lack of acceleration.

"I'm surprised that you wouldn't be happy to have gifted learners be taught more, be taught stuff outside and in addition to that basic, minimal set of standards."

Once again, I didn't say that. The "more" I expect is based on acceleration of material, NOT enrichment. Enrichment is no substitute for acceleration, ESPECIALLY with minimal state standards. Our school is still trying to get kids to master their adds and subtracts to 20 at the beginning of third grade. What possible enrichment could you give the more able kids to make it OK not to move on to new material?

"When I did this in the public school, I found teachers who were dying to give the top students more challenge, but frequently had trouble making the time to do it."

You're still talking about something else. You're talking about dealing with a situation, rather than trying to fix the underlying problem. These are two separate issues.

"... and at some point someone might indeed wonder why certain kids are suddenly doing better in math..."

This is not a proper process for change. Besides, schools get plenty of kids who pass through Kumon and the schools don't know or care. Many parents help their kids at home and the school doesn't know or care. They're just happy that these kids make their scores look good. All of my posts have nothing to do with figuring out how to play the system. They are about fixing the underlying problem.

"Because if what you're doing really makes a difference, it will show. And that kind of "evidence" can change minds. Also, as the old saying goes, 'you attract more flies with honey than with vinegar.'"

So, I'm not allowed to challenge the system directly? Do you think that I haven't already done a whole lot of working within the system. This doesn't work for basic assumptions. Parents shouldn't have to play this game.

"When you volunteer your time and effort to help teachers and schools, they get to know you better, get to trust you more, and may take your opinions, suggestions, requests and advice just that much more seriously."

Do I have to spell it out for you? I've done these things. I've gotten along great with my son's teachers and principals. I've had long discussions with them, including those who were in charge of the math curriculum. We might as well be on different planets. I may have changed their thinking a little bit, but they still use Everyday Math.

This process for change is not acceptable.

The solution is not to change the subject and talk about how to best work within the system. I know all of that.

When I talk about fundamental changes like curriculum choice, you talk about being an education helper.

I think you'd get few students to sign up for after-school Singapore Math, never mind what the school would think.And you base this on what? Oh, excuse me, I entered a post a while ago about textbooks and parents' experience with their math programs. Thought I'd enter my two cents again, and really don't mind being ignored. Regarding after school programs, the Powell Elementary School in Washington DC started an after school Singapore Math program that was strictly voluntary. They got a pretty good crew of kids staying after, even up to the last week of school. The principal of the school talked to the new Chancellor of Education in DC (Michelle Rhee) about the success of the after school Singapore Math program, and Rhee agreed to have it be the official program in K-3 at that school starting this year. (FYI, DC Public Schools adopted Everyday Math in 2005. For more about that, see:

http://www.thirdeducationgroup.org/Review/Essays/v2n6.htm )

Besides the approach used in Singapore Math, and its effectiveness, the principal of Powell School also liked the simplicity of the language used in the books. In a school in which 90% of the school population are English Language Learners, the book has been very accessible. This is not surprising given that in Singapore, three languages are spoken there, but in public schools, all classes are conducted in English. Thus, the books are constructed with English Language Learners in mind.

Steve wrote:

Are you saying that no acceleration is fine? Enrichment can be good or it can be bad, but it can't be the only way to differentiate.Yes, actually, I

amsaying lthatno acceleration is fine. If enrichment is good, I think it's not only an acceptable way to differentiate, but an excellent way.Acceleration of material does not offer the kids "more", it offers them "the same, but sooner". Enrichment actually offers them "more" -- material they wouldn't otherwise cover at all.

Perhaps you and I are not using the word "acceleration" in the same way? What I have seen labeled as "subject acceleration" is allowing students who have completed the curriculum for 3rd grade to move on and work on the curriculum for 4th grade. The way I have seen this implemented most is by letting a couple of younger kids who are ready for 4th grade material go to a 4th grade classroom during math time. This generally doesn't cost the school anything, so most places offer this as a cheap way to write off the needs of the gifted kids. This is better than nothing, but by no means a great solution for gifted kids. You are offering the kids a some new content, but placing them in a class whose pace and teaching style are still aimed at average learners. It's still going to be too slow and too shallow for gifted learners.

The most important feature of a good differentiation plan for gifted learners is grouping them together, so that they can be taught at a faster pace and they don't always have to sit around waiting for the average and below average kids to get it, whatever "it" is.

A good math enrichment program is possible at all levels, even with kids who have only learned addition and subtraction. (And even with kids who haven't learned even that!) There is no particular reason that gifted learners absolutely must be taught multiplication or fractions as soon as they master addition and subtraction in order to offer them an appropriate level of challenge.

-----

Barry questioned on what I based my assumption that kid wouldn't want to sign up for after-school Singapore math programs.

I based it on the experiences of many parents of gifted kids I know who try to "after-school" their kids (with Singapore Math or anything else). Most people run into problems at the point when the school starts giving more than a few minutes per day of homework. At that point, most of the kids rebel against the additional school work.

I'm impressed with the fact that you got so many kids in your community to take a voluntary after-school math class.

Your story about how the success of those kids influenced a larger-scale change of curriculum for the district (?) is an example of exactly the kind of thing I was talking about -- demonstrate the value of what you are asking for and you are more likely to get it. Congrats to all involved in that effort!

"Yes, actually, I am saying lthat no acceleration is fine. If enrichment is good, I think it's not only an acceptable way to differentiate, but an excellent way."

For any low standardized grade-level expectations? What if the school is just teaching kids to tie their shoes in third grade? Silly? OK, what about my example of still teaching kids their adds and subtracts to 20 in third grade? Perhaps you assume that any content and skills level a school chooses is fine, but the relationship between grade-level expectations and the value of enrichment is not arbitrary. Enrichment (assuming that it's good) will always help at any level, but it's not a substitute for low grade-level expectations. Acceleration can mean just allowing kids to progress to the level of Singapore Math.

Also, enrichment, by definition, implies extra, or add-on. It shouldn't be necessary if the curriculum is good to begin with. Enrichment isn't necessary for Singapore Math. It may be nice, but it isn't necessary. You can't divide math into boring drill and kill and enrichment. There is a whole lot in-between the two.

"Acceleration of material does not offer the kids "more", it offers them "the same, but sooner". Enrichment actually offers them "more" -- material they wouldn't otherwise cover at all."

Acceleration is not more, but the same, sooner? If this material is so important, then it would be part of the curriculum, not enrichment.

I guess you have a funny definition of "more".

Enrichment is extra. Acceleration means allowing kids to progress at a faster pace through the material. Math curricula like Singapore Math is not a series of "the same, but sooner". Math is not built around enrichment. You can't redefine "extra" to mean "necessary". If you contort definitions, you can justify almost anything. If, however, a school offers (at least the choice of) a curriculum like Singapore Math, then one could argue against acceleration, but only from a practical or scheduling standpoint, not a pedagogical one.

"Perhaps you and I are not using the word "acceleration" in the same way? What I have seen labeled as "subject acceleration" is allowing students who have completed the curriculum for 3rd grade to move on and work on the curriculum for 4th grade."

And you call this "the same, but sooner", as if there is a whole lot more to math than what's in a math curriculum? Isn't a curriculum supposed to contain everything that's necessary? How can moving faster through a curriculum not be "more"? Are you just talking about "bad" curricula?

" ...so most places offer this as a cheap way to write off the needs of the gifted kids."

Strawman. I'm not talking about any such thing. Besides, many places use enrichment as a way to write off the faster pace needs of talented kids. And many G/T programs are about appeasing one group while not fixing the underlying problems for the rest.

"The most important feature of a good differentiation plan for gifted learners is grouping them together, so that they can be taught at a faster pace and they don't always have to sit around waiting for the average and below average kids to get it, whatever 'it' is."

I'm all for homogeneous grouping, but as I've said over and over this is anathema to most schools. It's an assumption. They will never do it, and no amount of after school volunteering will change it. But I don't understand you. Homogeneous ability grouping and a faster pace is NOT acceleration? Faster pace of what? The curriculum? Enrichment? If you separate kids by ability and if you use a good curriculum, like Singapore Math, then what's the big deal about enrichment? Acceleration becomes less of an issue ONLY for schools that provide multiple curricula, because the acceleration is built in. Of course, acceleration is not going to help a curriculum like the old MathLand or TERC. Faster-paced crap is still crap. But that's not what I'm talking about. Maybe you are. Even if you have a good curriculum, students can still benefit from acceleration or deceleration of the material, but this can only be reasonably done up to a certain point in one classroom. Enrichment is just extra, unless you're talking about a bad curriculum.

"There is no particular reason that gifted learners absolutely must be taught multiplication or fractions as soon as they master addition and subtraction in order to offer them an appropriate level of challenge."

This is another strawman, and a separate issue. I didn't say anything of the sort. Lots of methods can work, but it also depends on where you are going and what performance levels you expect, at least for some point in time. You can define your own expectations (perhaps with good reason), but there are certain key points where externally-defined tests need to be taken. The big ones are the SSAT, the SAT, and the AP Calculus tests. The other required expectation is a path to algebra in 8th grade that eveyone can manage to follow, NOT just gifted students. The solution is not just separating students by ability. The solution is to provide good math curricula and high expecations for all kids.

"Most people run into problems at the point when the school starts giving more than a few minutes per day of homework. At that point, most of the kids rebel against the additional school work."

That's because of the stupidity of doing double math work; because of the stupidity of not allowing Singapore Math as a choice during the day, not just after school.

"Your story about how the success of those kids influenced a larger-scale change of curriculum for the district (?) is an example of exactly the kind of thing I was talking about -- demonstrate the value of what you are asking for and you are more likely to get it."

This process is OK? This validates your smug position? Do all this work and hope that you can change minds? This is our fundamental disagreement. This process should not be necessary. All there needs to be is enough parental demand and a school with reasonable resources to meet that demand. We're not talking about teaching Creationism. We're talking about math. We're talking about years of opinion-based math and your position that the onus is on the parents to prove that something else is better. On top of it all, this process may not work. In fact, my son's school likes Singapore Math (in a general sense), but they think it's too challenging for most kids, especially in a mixed-ability classroom. Case closed.

Proof is not always enough. Their assumptions and no choice rule.

Steve, indeed you and I are using "acceleration" to use different things.

The definition I am using (which corresponds to the one used in literature about teaching gifted students) is allowing students who are ready for a higher grade level to move to it at a younger age. This can include things like "grade skipping", "compaction" (doing 2 grades worth of material in one year), or "subject acceleration" (doing a higher grade level material in a given subject), which is what I'm talking about here. With subject acceleration, if you're in an Everyday Math school and you get subject acceleration in math, you would do 4th grade (or 5th grade or 6th grade) Everyday Math instead of 3rd grade Everyday Math.

You seem to be using "acceleration" to mean "substituting a whole different curriculum". I haven't seen it used that way in the literature, so perhaps that's why we're talking past one another.

You seem to think that the sun rises and sets with Singapore Math, and it's so "complete" that gifted kids in Singapore Math would never need (or benefit from ?) further enrichment. I disagree. Singapore math seems to cover the basics in a rigorous way and offer a good deal of problem-solving practice. It's a great curriculum, but there are still tons of opportunities for enrichment. Meaningful enrichment that will allow a student to be a better mathematician and a better thinker down the road.

A curriculum, even a good one, prescribes what everyone must learn. Enrichment allows capable students to learn

. You asked why, if the topics are so "important" they are not included in the curriculum to start with. The answer is that the curriculum is developed for "everyone" not for gifted learners. Enrichment, effectively, adds additional topics to the "gifted curriculum". Many things can be beneficial for gifted learners to learn even if not everyone can or must learn them.moreNow, granted, if all a school taught was shoe-tying in math class, then it would be hard to offer mathematical enrichment to kids with no mathematical skills. But that's a strawman as well.

As to whether changing things in the way the DC group changed things is an OK process, sometimes when you can't walk into a building through the front door, you may find the back door more accessible. I've said earlier in this discussion that in an ideal world, parents shouldn't have to do things like this, but in the real world, you do what you have to do.

Mathmom, Steve--

I had been thinking of setting up an online interview between two articulate representatives of opposing forces in the Math Wars. Thank you for doing this for me. I believe you can respectfully agree to disagree. I would like to invite others to join in, but truthfully this debate is far too important and powerful to reside in a set of comments. I did have a set of 19 questions I would have liked to ask each of you, but, unfortunately, I seem to have misplaced them. I believe they are in the margins of one of my notebooks along with my proof of Fermat's theorem...

Seriously, I admire your tenacity and sense of purpose. Steve, you may not believe that I too have been extraordinarily frustrated with educational bureaucracies. Like you and Mathmom, I don't give up. I see merit in both of your positions even though you say there is no middle ground. I have been committed to providing enrichment for all of my students for all my years in the classroom. The curriculum, the textbook, the ancillaries never ever went far enough to develop student understanding. If you've read the investigations I've developed for the past 9 months, I'd like you to tell me in what math program you can find them, Singapore included. There is so much depth to plumb here. BTW, these investigation require strong skills. I don't believe I could ever convince you of any of this...

Math mom wrote:

Barry questioned on what I based my assumption that kid wouldn't want to sign up for after-school Singapore math programs.

I based it on the experiences of many parents of gifted kids I know who try to "after-school" their kids (with Singapore Math or anything else). Most people run into problems at the point when the school starts giving more than a few minutes per day of homework. At that point, most of the kids rebel against the additional school work. I'm impressed with the fact that you got so many kids in your community to take a voluntary after-school math class.

Your story about how the success of those kids influenced a larger-scale change of curriculum for the district (?) is an example of exactly the kind of thing I was talking about -- demonstrate the value of what you are asking for and you are more likely to get it. Congrats to all involved in that effort!

These were not gifted kids, however. They signed up because the principal of the school introduced it as a program. That's key. Someone high up in the school is advocating Singapore Math. Steve H and I can advocate for Singapore Math til we're blue in the face, and the best we can do is a response of "It's nice but too hard for our kids." There are not many principals like the one at Powell School, who are willing to take the chance she did. I had nothing to do with it. It was the principal who demonstrated the value of Singapore Math.

So far, it is only being implemented in Grades K-3 at the Powell School, thanks to Chancellor Rhee. No other school is using it. I am hoping that the Chancellor will agree to have Singapore Math used at other schools in DC.

Demonstrating the value of something is no guarantee, however. Politics plays a larger role. I am hopeful that the DC school politics are favorable to Singapore Math. They were not favorable to it in Montgomery County, MD where it was piloted in 4 schools from 2000 to 2003. Despite evidence of success, it was dropped. Everyday Math is now used in Montgomery County, MD. For more information on that, see:

http://www.hoover.org/publications/ednext/3853357.html

Dave, let me assure you, I don't "represent" anything other than my own unique opinions. (And anyhow, I thought I was somewhere in the middle ground, not really firmly on either "side".)

Steve and I seem, at this point, to be talking in circles. I don't think either of us will convince the other of much of anything new at this point. 'Tis probably indeed time to agree to disagree.

Let me know when you find that notebook. That Wiles guy's proof is a bit long, ya know?

Mathmom--

I knew you were more centrist like me, but, it's fair to say you and Steve are not exactly in the same place! I still think this ongoing dialog is publishable and perhaps should be required reading for all parents, teachers and board members everywhere!

Another thought -- why not a face-to-face on YouTube like the Point-CounterPoint segment from 60 Minutes...

I came across this interesting and semi-relevant blog post today about effecting change in schools. This particular article is about advocating for gifted kids, and the author has the advantage of being on staff at the school in question, but it still seems that there might be nuggets folks could take from that article and apply elsewhere.

"The definition I am using (which corresponds to the one used in literature about teaching gifted students) is allowing students who are ready for a higher grade level to move to it at a younger age."

We're not just talking about gifted students.

This all started on a previous thread in response to Prof. Steen's answers to questions about math education in general. The Math Wars is all about big differences of opinion over grade-level content and required levels for mastery of skills. All of this is NOT about enrichment. This is about educators like Prof. Steen who seem to get a kick out dissing parents and college professors and saying that they (K-12 educators) have the final authority over what math is or is not for K-12. I have said before that there is a large academic turf battle to this war and parents are ignored.

You seem to be hung up about enrichment. Who can be against enrichment? This is like the use of balance. Who can be against balance? Generally speaking, enrichment is good, but there has to be a trade-off, unless you demand that kids attend after-school enrichment. What are you giving up to get this enrichment? You seem to be saying that there are no trade-offs, that enrichment is always good. This is NOT the case.

Then, you go further away from the original discussion by assuming that the curricula are generally good (or can't be changed), so adding enrichment is always good. That's not what this discussion is all about. It's about curricula, not what you can add on to them if nothing else can change. But then you go further and claim that enrichment is always better than acceleration. You can't say this.

"Singapore math seems to cover the basics in a rigorous way and offer a good deal of problem-solving practice. It's a great curriculum, but there are still tons of opportunities for enrichment. Meaningful enrichment that will allow a student to be a better mathematician and a better thinker down the road."

Yes, more is better than less, but once again, we're not talking about more, we're talking about substitution. Besides, if all schools used Singapore Math, there would be no Math Wars. But what happens when the curriculum is worse, much worse? You seem to think that enrichment is still the best solution. Perhaps ONLY if there is nothing that can be done about the curriculum. (Again, this is not what the Math Wars is all about.) But I would still disagree with you. Acceleration can be better than enrichment, even with good curricula.

"Now, granted, if all a school taught was shoe-tying in math class, then it would be hard to offer mathematical enrichment to kids with no mathematical skills. But that's a strawman as well."

You deliberately ignored what came after my comment about shoe-tying. That was the example about our school which is still trying to teach adds and subtracts to 20 in third grade. This is based on their assumption of no separation of students by ability and full-inclusion. To get this to work requires lower expectations. Spiral curricula facilitate this assumption by giving a pedagogical basis for delayed mastery. But mastery doesn't happen, and enrichment provides no guarantee of a fix. It's also no excuse for lack of acceleration.

In fact, many educators trash the idea of skill mastery as being "drill-and-kill". They even go so far as unlinking mastery from understanding. They see mastery as only adding speed. You may use the example of 60 problems in 3 minutes, but the problem of mastery in schools is absolutely, positively nowhere near that level. We're talking about kids in 5th grade who have to think about the solution of 7+8. We have kids in 6th and 7th grades who still don't know their times tables. This problem is not just about basic arithmetic. It continues with fractions, decimals, and percents. This lack of mastery all adds up. Schools will only enforce mastery to the level of trivial standardized tests. This is not enough. Affluent parents get to send their kids to Kumon or to private schools where "serious" students can thrive with (or in spite of) almost any curriculum. Poor and minority students get low expecations at home and at school.

The fallacy is that reform math is somehow better; that they teach more understanding; that they prepare students for the 21st century. They do no such things. Enrichment is no solution. They have to change their basic assumptions. The onus is not on others to prove that there is a better way. The onus is on the schools to explain why they can't offer a choice of curricula. National groups seem incapable of defining content and mastery expectations that lead to algebra in 8th grade, so the only option is choice. The goal is not about raising low cut-off points on standardized tests. It's about giving every individual equal access to to curricula and expectations that match their abilities.

Steve, sorry, I still don't really understand what you're trying to advocate when you say "acceleration". You seem to be meaning "use a rigorous curriculum and require mastery". These are both things with which I heartily agree. They also have nothing to do with any educational definition of "acceleration" that I have ever seen.

I like Singapore math, and I have no problem with requiring mastery -- of course I agree that mastery should be required. This is not inherently incompatible with the use of a spiral curriculum. A good spiral curriculum implementation will have students re-visit topics

. At the same time, the seeds of more complex topics are sown. A good spiral curriculum implementation would be able to support children still learning adds and subtracts to 20 and also those who have mastered that and are ready for harder addition and subtraction. I wouldn't call that "acceleration" I would just call that proper implementation of a spiral curriculum, which includes different expectations for different children, based on their current degree of mastery of the skill being worked on.until they master themIf it's the case that Everyday Math never checks or requires mastery, then that is indeed a problem. It would be interesting to know if when folks observe EM to be lacking in that way, if that is inherent to the design of EM, or whether it is being mis-implemented, possibly due to a lack of teacher training?

Again, if it's the case that EM doesn't support differentiation of expectations based on individual students' levels of mastery, that is also a problem. Again, I don't know if that would be a fault of the program, or of those implementing it. And of course, even if EM does theoretically do these things "right", if it is so hard to implement correctly that it is being implemented incorrectly all across the country, then that too is a problem!

I also have no problem offering parents a choice between a Singapore-like curriculum, and an EM-like curriculum. But if you're really concerned about students who you claim get "low expectations" at home as well as at school, I don't see how "choice" is the solution. (If, on the other hand, it is not the expectations themselves that correlate with affluence, but rather the ability to do anything about them, then choice could help.)

There seems to be a mis-perception here that I'm defend "reform math" or to represent some "side" of an argument about particular reform curricula. I'm not here to participate in the "Math Wars" per se. I've made specific comments and arguments about specific statements and situations. I have never represented myself as a representative of any particular Math War position. It is you (and to some extent Dave) who seem to be interpreting my words more broadly than I wrote or intended them.

I'm sharing my own experiences and observations here. With a well-implemented spiral curriculum that's not related to any major "reform curriculum". With taking time to include non-routine problem solving in the curriculum for all students.

Mastery is good, choice is good (within reason), differentiation is good, ability grouping is good. We don't disagree as much as you seem to think we do, Steve.

I think our main differences are that:

1) I think that a spiral curriculum can be well implemented and work well for children with a wide variety of abilities and aptitudes.

2) I think that teaching problem solving, using challenging problem and investigations (even beyond what Singapore already includes) is an important and valuable part of math education for all students (even those still working toward mastery of the underlying skills), and that mastery need not be traded away to make the time to do it. (As a poor compromise, I'd recommend offering this as "enrichment" for the high achievers, at a minimum.)

I'm going to try very hard to shut up now. I've explained my positions, observations, arguments, etc. multiple times. If I have still failed to make myself clear, making additional attempts will make any difference. Anyone who was likely to be convinced by my arguments should already be convinced, and I doubt anything else I say will convince anyone else.

Steve, thanks for the civil discussion. Dave, and Prof. Steen, thanks for the opportunity and instigation.

"I still don't really understand what you're trying to advocate when you say "acceleration". You seem to be meaning "use a rigorous curriculum and require mastery". These are both things with which I heartily agree. They also have nothing to do with any educational definition of "acceleration" that I have ever seen."

Then you have a very narrow understanding of acceleration. I'll try to be as clear as possible.

1. The fundamental assumption of many K-6 schools is full-inclusion. Schools track by age and include kids who used to be separated and sent to other schools. This also means no separate gifted and talented programs or pull-outs. This is a noble idea, but it doesn't come without a price.

2. To get full-inclusion to work, they use team teaching, set lower expectations, and use a spiraling curriculum that is built around no set dates for mastery. This is the fundamental premise of Everyday Math. This is one of the biggest reasons for its popularity, not that it's such a good curriculum. (and it isn't. It is structurally flawed.)

3. This approach to spiraling allows schools to maintain full-inclusion and hide behind a veneer of "no drill and kill", "conceptual understanding", and "real-world" problem solving.

4. What this happy talk hides are low expectations and a slow pace. I told you twice about our school finally trying to finish up adds and subtract to 20 in third grade. The reason for this is full-inclusion. They can't expect more from many kids.

5. To solve this problem, they push "differentiated instruction". This is supposed to allow full-inclusion classrooms to meet the needs of all students. It can't because the expectations are too low.

6. The primary method of differentiation is enrichment. They can't allow "acceleration" of material as a way to differentiate (within a curriculum and within a classroom) because it makes mixed-ability, child-centered learning impossible, and THAT is the main purpose of full-inclusion.

Schools proceed to talk about the wonders of differentiation and enrichment, but what they are really saying is that they WILL NOT provide acceleration of content and skills because it can't work with full-inclusion - their fundamental assumption. That's why you see people like Prof. Steen saying that acceleration is not that important; that enrichment is all you need.

When you come along and talk about enrichment as the only thing you need, you sound just like them. I don't hink you are, but you don't seem to see what's going on here. This is NOT a small problem. You say that you like the idea of separating kids by ability, but for many schools, that cannot, will not, ever be a possibility, by definition, until 7th grade. Expectations are low, math curricula are bad, and they don't allow acceleration in their full-inclusion classrooms, only enrichment.

My son is in sixth grade and last week was in a group of three kids who were working on a social studies poster. (a collage in sixth grade!) The girl in his group was just cutting up tiny bits of paper all over their work and complaining to the teacher about the other two kids in her group. My son has no idea what is wrong with her and absolutely no preparation or training on how to deal with kids like this. This is what full-inclusion is like; a social experiment first, education second. As I said before, this is a noble idea, but there is a price. The price is low expectations, a slow pace, and a very fuzzy idea of what constitutes a proper K-6 education.

Acceleration is a term that can be used to mean much more than separating kids by grade or classroom for a particular curriculum. Acceleration versus enrichment is a common focal point in many discussions of full-inclusion and differentiated instruction. Parents want acceleration. The schools give them enrichment.

"If it's the case that Everyday Math never checks or requires mastery, then that is indeed a problem. It would be interesting to know if when folks observe EM to be lacking in that way, if that is inherent to the design of EM,..."

Never, or lacking? EM states cleary that there is no expecation of mastery at any particular time. This doesn't mean never, and many schools are smart enough to impose some level, but it's not required. If you've never seen it, you should. The Math Boxes are the worst. I could teach EM well, but that's not the point. Other (non-reform math) curricula are better.

"But if you're really concerned about students who you claim get "low expectations" at home as well as at school, I don't see how "choice" is the solution."

Schools can't do anything about parental help at home, so they better do something about expectations at school. This doesn't mean raising low cut-off standards a little higher for all. It means choice. Parents may not be able to help with math homework at home, but they (and the school) can push kids into better curricula. Individual educational opportunity is not improved by raising low cut-off standards.

"(If, on the other hand, it is not the expectations themselves that correlate with affluence, but rather the ability to do anything about them, then choice could help.)"

It's both; expectations and the ability to do something about them. Many poor have expectations too. They might not be as well-defined, but they have no way to do anything about them. If urban kids were given a free ride and the choice to go to a fancy private school, not many parents would say no, and that's the only expectation they need.

"It is you (and to some extent Dave) who seem to be interpreting my words more broadly than I wrote or intended them. "

It's because you don't understand how many schools pit acceleration and enrichment against each other. Many schools lower expecations and hide behind enrichment.

"1) I think that a spiral curriculum can be well implemented and work well for children with a wide variety of abilities and aptitudes."

As I said before, all curricula do spiraling at some level. There is nothing wrong with spiraling in general. The problem is that spiraling is used by reform math to allow full-inclusion and delayed mastery. Everyday Math is a classic example. It's spiral consists of repeated partial learning. Math Boxes are used in the desperate attempt to somehow get kids to eventually figure it out themselves someday. This is really nothing about using previously mastered material in more complicated situations.

"2) I think that teaching problem solving, using challenging problem and investigations (even beyond what Singapore already includes) is an important and valuable part of math education for all students (even those still working toward mastery of the underlying skills), and that mastery need not be traded away to make the time to do it. (As a poor compromise, I'd recommend offering this as "enrichment" for the high achievers, at a minimum.)"

"

Boy, I would too, but reform math (with full-inclusion) is so far away from this point that just getting the option or choice of Singapore Math (without enrichment) seems like a dream. Do you really understand how many educators despise Singapore Math? And it has nothing to do with any so-called lack of enrichment. You seem to be focused on some sort of perfection while the rest of the world is struggling with meeting trivial math standards. Have you looked at the NAEP tests and results. We're not talking Singapore Math-type enrichment here.

"(even those still working toward mastery of the underlying skills)"

You have to understand that this is just the sort of argument used to justify delayed (or never) mastery of skills. You have to be very careful how you define "mastery" and "delayed".

Apparently, I am not actually capable of sitting on my hands. :-/ But I'll keep it brief for a change.

1) In a class of 3rd graders where some students still haven't mastered addition and subtraction up to 20, what does EM say the other students who have mastered it are supposed to be doing?

2) What would you, Steve, like the kids who have mastered adds and subtracts up to 20 to be doing while the struggling kids work on that?

In the heterogeneous groupings I've experienced (and I'm talking about 3 grade levels worth of kids in one groups, so there's tons of spread in abilities), the kids who have mastered adds and subtracts to 20 would be working on things like 3-digit adds and subtracts with no re-grouping, and also learning carrying and borrowing. I now think that this is what you're calling "acceleration". I would call this "differentiation".

Once kids have mastered carrying and borrowing and other similar-level skills, they would move to the next group, and spiral on topics related to multiplication, division, fractions, etc... If they do this at a younger age than usual, I'd call this "acceleration".

When you say EM advocates "differentiation" but that it "can't work" because expectations are too low, that doesn't make sense to me. "Differentiated instruction" should by necessity include "differentiated expectations". Sounds like maybe your school just doesn't get how to do differentiation? Or again, maybe we're using the same word to mean different things.

Ok, that was only brief for large values of brief. :-}

" ...what does EM say the other students who have mastered it are supposed to be doing?"

In EM, everyone moves along at the same pace. Everyone is on the same page of the school and home workbooks. The kids who don't understand the material have to move on even though they haven't mastered the material (or even half-understand it). EM thinks this is OK because they will see the material again.

Spiraling in EM is not about using previously-mastered material in a more complex fashion. It's about seeing that same material over and over, whether you've mastered it or not. For those who have mastered the material, they consider it a review. For those who haven't yet mastered the material, they get to work on it some more.

You might call this differentiation over time or of mastery, rather than differentiation of material.

One of the biggest complaints of EM is that they introduce new topics without giving enough time to master previous material. There is little careful development of the material. This gets worse in the later years of EM. I might have mentioned before that I spent this last summer going over sixth grade EM (the new edition) with my son so that he could start taking 7th grade pre-algebra in sixth grade. By sixth grade, EM is desperately trying to make sure that eveyone is up to speed on mastery. Math Boxes are the main tools for doing this and they dominate the lessons. They still introduce new material, but some of it is more appropriate for 7th grade pre-algebra. Everybody has to do these problems, even if they are struggling with the review Math Boxes. It makes EM seem advanced, but it's not a careful introduction and development of each topic. They just throw new material at the students. It doesn't matter to EM because the students will see it again. EM has no mechanism for ensuring mastery except repeated exposure. That's the point. I call this repeated partial learning.

Inside of each EM lesson are two or three pages of Math Boxes. Each of these pages is broken into a number of rectangular boxes, each with a few review problems to do. These problems don't have anything to do with the current lesson, and each box has nothing to do with any other box. So, right in the middle of a lesson of new material to learn, students have to do these math boxes. There are so many math boxes and so much jumping around of the material in the math boxes, it's impossible for a teacher to spend class time to review the skills needed for any of the review problems if a student didn't master it the first time. The kids are on their own. In all of my son's previous EM classes, these boxes were self-corrected in class and not turned in. They just moved right along. Speaking of which, if You ever look at all of the books and workbooks that come with EM, add up the number of pages and divide by 180. There is way too much "stuff". It can't be done.

My son's fifth grade teacher didn't have time to cover the last 30 percent of the course. She ran out of time. She either had to fly through the course or skip part and take some time to help struggling students. The advanced students twiddled their thumbs and never got the material that they were ready for. It doesn't matter because EM throws it at them like splatter in the middle of page-after-page of Math Boxes. EM is not set up to allow kids to skip Math Boxes and move ahead to new material. Everyone is on the same page. Differentiation in EM means different levels of mastery, not different material.

"In the heterogeneous groupings I've experienced (and I'm talking about 3 grade levels worth of kids in one groups, so there's tons of spread in abilities), the kids who have mastered adds and subtracts to 20 would be working on things like 3-digit adds and subtracts with no re-grouping, and also learning carrying and borrowing. I now think that this is what you're calling "acceleration". I would call this 'differentiation'."

Three grade levels of kids in one group is not common, but this is not real acceleration. If you don't allow kids to move on to material in the next level, then it's really just compacting. Some schools like to fool parents and call this acceleration. Since most schools don't allow acceleration past the material defined for that grade, the only thing they can offer is enrichment. If the grade-level expectations are low (and they are), enrichment can't solve the problem.

True acceleration requires separation by ability. Even if you put three grade levels together, something has to give at the top end of the group. Differentiation, by definition, is used to group kids with widely different abilities. At least some of the time, these kids work together in mixed-ability groups. Prof. Steen argues that this is best for all kids. It is not. When ability differences get past a small range, separation and acceleration are necessary. Since many schools can't seem to provide math curricula that set high expectations of mastery and coverage of material, enrichment can't solve the problem, but that's what they claim.

"When you say EM advocates "differentiation" but that it "can't work" because expectations are too low, that doesn't make sense to me. "Differentiated instruction" should by necessity include "differentiated expectations". Sounds like maybe your school just doesn't get how to do differentiation? Or again, maybe we're using the same word to mean different things."

EM says that it's OK for each student to absorb whatever he/she can. It doesn't differentiate material. Expectations of mastery are low because that's the fundamental permise of EM. You might call this self-differentiation. There is no mechanism for teachers to decide whether students need extra time or a kick in the rear. For other subjects, the school will differentiate material or expectations explicitly, but it's easier to get away with this in non-math subjects. I don't agree with this, but it's less damaging than in math. In EM, kids who need a slower, more in-depth pace just get pushed along and told that they will see the material again. The problem is that later on, there is little or no time for explanations. There is so much "stuff" in EM, that you can't slow down unless you skip material. EM says that mastery will come (automatically?), but it doesn't. The best schools edit lots of junk out of EM, they slow down, and they don't allow any delay in mastery. They should just get a new curriculum.

All of this still doesn't deal with the underlying issue of separating kids by ability or level. Schools can't increase student ability ranges with full-inclusion classrooms and then say that differentiation will solve the problem.

This EM website says that each grade level comes with a Differentiation Handbook:

Grade-specific handbook provides that helps teachers plan strategically in order to reach the needs of diverse learners.Has anyone seen it and know what it really contains?

I wasn't trying to say that putting 3 grades worth of kids in one math group was "acceleration" but it certainly requires a great deal of differentiation! There are kids in the same group learning to count, add with manipulatives, understand place value, carry and borrow, all in the same group. They are not all doing the same work at the same time, of course. All I am trying to say here is that "differentiation" can be used to accomodate the needs of very diverse learners.

What

isacceleration is when a 6yo demonstrates mastery of the materials covered in that group and moves into the group that's working on multiplication, division, fractions, decimals, etc. And this does happen in our system, albeit rarely. Enrichment would be another reasonable (IMO) alternative for a 6yo who "finished" the K-2 curriculum, but it is a lot more work than letting them accelerate at that level. We do work enrichment in throughout as well.Nothing really gives at the top of the group, because expectations can be set on an individual basis, and eventually the student will move to a higher group. Our highest group covers the skills through pre-algebra. Kids who master those skills work on Algebra, but generally individually, with a tutor to touch base with them once a week. They could go beyond that as well, though we generally (in consultation with parents) prefer to intersperse more really good enrichment at this point and not accelerate them further than the standard honors stream, which expects freshmen to be ready for honors geometry.

By the way, I'm not trying to say that all schools should do math the way my kids' school does it. I'm only making the point that a skilled teacher can effectively differentiate over a wide range of levels. It's hard, but in a situation where ability grouping is not done, it's a necessary part of the teacher's job.

Ability grouping has many benefits, but if you can't have it, all is not necessarily lost.

1) All of my posts had nothing to do with making the best of an existing situation.

2) All of my posts had nothing to do with your school.

But, since you brought it up,

"Our highest group covers the skills through pre-algebra. Kids who master those skills work on Algebra, but generally individually, with a tutor to touch base with them once a week. They could go beyond that as well, though we generally (in consultation with parents) prefer to intersperse more really good enrichment at this point and not accelerate them further than the standard honors stream, which expects freshmen to be ready for honors geometry."

Normally, kids should get pre-algebra in 7th grade and algebra in 8th grade. Standard honors tracks in high schools usually require at least a 'B' average on a rigorous algebra course in 8th grade. It sounds like you think that all they need to do is to "touch base with them once a week" in algebra. Are you talking about 8th grade? Most schools offer two or three levels of math in 8th grade, including honors algebra.

Your school may be able to pull it off. I can't comment specifically, but this is a real problem in many other schools. Math curricula and decisions made by schools in 4th or 5th grade set kids onto non-honors ("life skills") tracks and parents don't figure it out until it's too late. In some cases, it's worse than that. Our school used to use CMP (stopped last year, finally!) which did not meet the requirements for entering honors geometry (or even algebra) in 9th grade. Very surprised ('A') students and their parents had to scramble to get ready for high school. Now, (like many other schools), they provide a class using the same algebra text that the high school uses. The only issue left is to prepare more kids for that track. This will not be done using a curriculum like Everyday Math, which leaves expectations (mastery) up to the kids or the state. Algebra in 8th grade should be the norm, not the exception.

Whether your school can get something else to work or not doesn't matter much. Some schools and parents like a more complete un-schooling approach. That's OK, but just don't force it on my child. In fact, I don't want to force my ideas or opinions (like a normal path to algebra in 8th grade) on anyone else. The only option is choice. Although most schools provide a choice of a rigorous algebra course in 8th grade, they don't provide a path (choice) to get there. I didn't get any help from my parents to get to a course in algebra in 8th grade, but that's unlikely to happen nowadays unless you're a math brain or get outside help.

It sounds like your district has problems with its math program. It's just not clear to me that those problems are caused by EM, as opposed to the refusal to ability group students, lack of in-class differentiation, etc.

"It's just not clear to me that those problems are caused by EM, as opposed to the refusal to ability group students, lack of in-class differentiation, etc."

Then you need to do your own research. Obviously, my detailed explanations didn't raise any doubts. This is common. Many teachers can't believe that EM is structurally flawed. They claim it's just the implementation. But this is always the case. Good teachers can teach math with almost any lousy curriculum. Imagine what could happen with a good curriculum.

1) EM doesn't allow ability grouping. Everyone is on the same page.

2) Their idea of differentiation relates to expectations and the different ways people learn, not differentiation of material.

It's very difficult to turn EM into something it isn't.

Ultimately, it's not my job to convince you or any school to change their opinion, and that's what it is. Opinion. There is a huge difference of opinion about what constitutes a proper math education. Many parents want something different, and many others would want it too when they see it in action. The onus is on the schools (not parents) to show why they cannot provide choice. Schools and teachers cannot force their own opinions of education and expectations on everyone and then offer no choice. People like Prof. Steen can't assume that parents are stupid and incapable of understanding the issues.

Schools and teachers don't want to admit that a large portion of what they do is based on assumptions and opinions. They get to pick a curriculum based on whatever they want, but then require "proof" from others who want a change, or even a choice.

The Math Wars is all about academic turf. Parents, professors, mathemeticians, engineers, and scientists have been arguing for (at least) choice in K-8 mathematics for years. Schools and teachers don't want to lose the right to force their opinions on others. They don't even want to allow choice.

I understand why you and I and other well-informed parents want choice for our kids.

I also understand why schools are reluctant to offer it without "proof" that what we are proposing to choose for our kids is at least as good as what the school thinks (its

opinion*, yes) is best. It's quite simple. In the final analysis, it's the school that is "on the line" to make sure their kids meet NCLB requirements, and whose reputation is on the line if their precious test scores drop. So, that's why theiropiniontrumps everyone else's -- they're the ones who have been made responsible for students' progress.[* for EM, that opinion is backed up by years of research. I believe, as I know you do, that that research may be severely flawed, but it is there.]

If some parents came along and made the school offer a choice of some inadequate program that they thought would be easier for their kids, and those kids did not meet state standards, it is the school that would get dinged for that. So, IMO, the school must demand "proof" that what anyone else is proposing or demanding be as good as what they want to do.

I know perfectly well that that's not what

youare asking for, that you're asking to be able to choose a particular curriculum that I agree would be better for most kids than EM, but once you start offering "choice", how do you draw the line as to what parents may choose, if not by requiring proof that what parents propose is adequate.If parents' opinions are to trump everything, it must be the case that parents are held responsible if kids don't progress, and that just isn't the way the system is currently set up (unless you choose to homeschool, of course). There needs to be some way for parents to at least share the responsibility if an "alternative" program that they chose for their kids fails to produce the required results, and I'm not really sure how you'd implement that.

"I also understand why schools are reluctant to offer it without "proof" that what we are proposing to choose for our kids is at least as good as what the school thinks (its opinion*, yes) is best."

Most schools allow choice in math in grades 7-12. Besides, K-6 educators don't want proof. They just don't want choice, by definition.

"It's quite simple. In the final analysis, it's the school that is "on the line" to make sure their kids meet NCLB requirements, and whose reputation is on the line if their precious test scores drop."

It's quite simple. K-6 schools don't want choice by definition. High schools allow choice. Most schools provide choice in 7th and 8th grades. The choice many are asking for in K-6 is for more rigorous curricula, not less. Besides, their reputation in math education (as reflected by standardized test scores) is not very good to begine with.

"So, that's why their opinion trumps everyone else's -- they're the ones who have been made responsible for students' progress."

Baloney! Their opinion is for low expectations. The standards are low.

"[* for EM, that opinion is backed up by years of research. I believe, as I know you do, that that research may be severely flawed, but it is there.]"

Flawed, but you'll use it anyways? Everyone else has to provide "real" proof? Check out What Works Clearinghouse. The very, very small percentage of positive results for EM are only for small relative changes. In fact, there is little good educational research on anything. Even WWC is grasping at straws to make it seem like their existence is of any value. In fact, many schools are using the insufficient data provided by WWC as justification of EM. In spite of all of the extra emphasis on data collection, statistics, and real-world problems in EM, it hasn't made schools smarter in analyzing data.

I had to explain to my son (his textbook didn't do it) that how one interprets a graph depends a lot on how you display the data. If you compress or eliminate the lower end of the vertical axis, you can make the data trend look flat or very steep. Many are scaling the EM data to make the benefit look good on an absolute scale. Sorry. They flunk even reform math.

"If some parents came along and made the school offer a choice of some inadequate program that they thought would be easier for their kids, and those kids did not meet state standards, it is the school that would get dinged for that. So, IMO, the school must demand "proof" that what anyone else is proposing or demanding be as good as what they want to do."

"Inadequate?" Singapore Math is "inadequate"? This is not about "some parents". This is about decades of complaints by professors, mathemeticians, engineers, and scientists. This is not about lowering standards. How can schools demand proof when they can't provide it themselves? This isn't about proof. It's about control. Schools don't require proof when they select curricula like EM.

"... but once you start offering "choice", how do you draw the line as to what parents may choose, if not by requiring proof that what parents propose is adequate."

See above. If decades of complaints by lots of professionals don't make any difference, then the issue isn't about a line. Besides, grades 7-12 provide choice.

"If parents' opinions are to trump everything, ..."

Strawman.

"... it must be the case that parents are held responsible if kids don't progress, and that just isn't the way the system is currently set up (unless you choose to homeschool, of course)."

Schools blame parents, kids, and society all of the time (not without some justification). But when 50% of fourth graders can't say how many fourths are in a whole (NAEP test), then how much worse can it get? This is about K-6 teaching philosophy and control, not proof. Schools can't have it both ways. On one hand, they use "responsibility" to prevent others from making changes or demanding choice. Then, on the other hand, they blame external causes for bad results. I'm more than happy to relieve them of their responsibility. I'll bet they wouldn't like the trade-off. Besides, before NCLB they weren't offerning choice anyways.

"There needs to be some way for parents to at least share the responsibility if an "alternative" program that they chose for their kids fails to produce the required results, and I'm not really sure how you'd implement that."

So K-6 schools really would like to provide choice, but they don't know how? I don't think so. No school or teacher in their right mind would say that Singapore Math (as a choice!) would be worse than EM or TERC. Still no choice. Schools don't want choice, by definition. They want full-inclusion. Absolutely no tracking is allowed. They see choice as a form of tracking and they would be right. Singapore Math is so much stronger than what they are currently using.

Lack of choice is not based on proof or responsibility. It's based on opinion and control.

Most schools allow choice in math in grades 7-12.Choice? Most schools offer some kind of ability grouping and/or acceleration in grades 7-12, but kids are placed according to placement tests and/or teacher recommendations, not parental or student "choice".

Besides, K-6 educators don't want proof. They just don't want choice, by definition.It's clear that that's your

opinion.Singapore Math is "inadequate"?I think you know perfectly well that that was not the point I was making.

Lack of choice is not based on proof or responsibility. It's based on opinion and control.

Again, in your

opinionAh, there will be no resolution here folks...

Actually, I'm going to be doing some consulting with K-12 math teachers in a school district in my area and I intend to begin our work together by having them read through Prof. Steen's interview and every one of the comments!I think it's a real 'page-scroller' that will set the tone for our work together. Steve, despite your insistence that everything can be reduced to choice, higher expectations and mastery, I just don't accept that there are simple answers to these issues. Public schools are part of a structure that needs change but this change will probably occur like most other changes in education -- very slowly. I do believe that we may not have Sputnik to spur radical change but we do have international comparisons of students and the realities of where our technological expertise will be coming from in the next few decades.

Steve, I do have one question for you, which is rhetorical. How many classified children of your own do you have? Children with a range of learning disabilities but who are capable of being mainstreamed in some classes? Inclusion is not a 'choice' in public education, Steve -- it's the law. We may not all agree that Federal mandates like these are in the best interests of the regular ed population, but there other points of view out there, different from yours, on this score. I personally have extensive experience in this area. It has helped to reshape my thinking about how incredibly difficult it is to educate 'all' of the children, but that's what schools are required to do. It does take extraordinarily talented and committed professionals to find ways to challenge all the children in her/his classes, but every day millions of wonderful teachers are trying their best to do just that.

Whether you choose to respond to these comments or not, well, that too is a matter of choice...

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