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.