I have just had the good fortune to receive an article so interesting, so full of pertinent information and ideas and so articulate that I would dearly love to reproduce it in the space for this column. Unfortunately that would require something like a 2 point font, which would be distinctly detrimental to its usefulness, so instead I shall hit the high spots of the article and wave gently in the direction of its details. The article is by Berkeley professor Alan Schoenfeld, one of the leading lights in the field of mathematics education. Its title is Making Mathematics Work for all Children: Issues of Standards, Testing and Equity.
Schoenfeld's first line of argument, set up by remarks from civil rights leader Robert Moses, is that in the present state of society mathematical literacy is an ever-increasing necessity for any form of employment. The combination of that increasing necessity with the fact that the children most likely to be defeated by mathematics are those already at a disadvantage - children of color and children of poverty - is fundamentally inequitable. More broadly, he argues that mathematical sophistication of the type championed in recent reform documents such as the national Council of Teachers of Mathematics' (2000) Principles and Standards for School Mathematics can be seen as a core component of intelligent decision-making in everyday life, in the workplace, and in our democratic society. To fail children in mathematics, or to let mathematics fail them, is to close off an important means of access to society's resources.
With this lens in hand, Schoenfeld inspects the recent history and current state of mathematics education in the U.S., both its strengths and its weaknesses, pointing up some areas of dire need, but also some reasons for hope and optimism.
Setting the scene, Schoenfeld cites some of the sources of the data correlating socioeconomic status, race and opportunity to learn - numerous sources from TIMSS to Jonathan Kozols. Then he traces the movements in mathematics education from the fifties to the present. Prior to the curricular reforms of the previous decade, he points out, the standard curriculum was geared for the college bound and mathematically inclined, with some room for very bright kids with non-standard backgrounds. (I suspect that includes much of this newsletter's readership.) For anyone not mathematically or academically inclined, or even with the inclination but without the confidence (notably girls and minorities), it served as a harsh filter.
It had a further weakness that is more rarely acknowledged, possibly because absences are less visible than presences: the entire focus of the curriculum was on the pure and the abstract, with calculus as the Holy Grail. Nowhere was there more than a nod to the real world; word problems were far more often a cover story for a computational problem than an exercise in modeling, and topics like statistics were completely swept under the carpet. Even the art of expressing oneself coherently in mathematics was at most marginally represented. This curriculum remains to this day the one most likely to be encountered in a large percentage of classrooms anywhere in the United States.
Look now at what mathematics is really needed by the students who are not going to wind up in graduate school, but rather are going to head straight from high school out into the world and the work force. Quantitatively literate citizens need to be able to make sense of a slew of information on subjects ranging from the national debt to medical risk, to distinguish between plausible and bogus statistical claims and, as communication becomes more and more vital in the job market, to convey their thinking effectively.
In short, there is a huge intersection between the set of elements of mathematics commonly lacking in the education of a graduate-school bound mathematics student and the set of elements required for the most basic quantitative literacy. Recognition of this intersection led to what Schoenfeld terms the most visionary of the recommendations in the Principles and Standards for School Mathematics (NCTM, 2000): that all students should study a common core of mathematics throughout their K-12 years. This was and is a radical and distinctly alarming thought- but one for which supporting data are now appearing. The consistent outcome is that a focus on conceptual understanding, applications, and problem solving, which are universal needs, does not produce a reduction in the procedural skills required for more advanced study.
The reason the data exist can be traced to the NCTM's 1989 Curriculum and Evaluation Standards for School Mathematics (the predecessor to the Principles and Standards). This astoundingly influential document had the same underlying philosophy as Principles and Standards, and triggered the production of several curricula based on that philosophy. A certain number of school districts adopted these curricula, and testing data have quite recently begun to emerge. The city of Pittsburgh has produced a particularly well documented set of data, and results from other areas confirm them: a well-implemented use of the "Standards based" curricula had no detrimental effect on results of students' tests of basic skills, and a consistently large positive effect on results of their tests of conceptual understanding and problem solving. In addition, racial differences in performance diminished substantially. These are spectacular results, about which Schoenfeld gives details and references. He also discusses some of the issues and obstacles that need addressing to insure the sustained improvement of mathematics instruction.
Among those issues and obstacles, he cites four as fundamental: 1. High quality curriculum; 2. A stable, knowledgeable, professional teaching community; 3. High quality assessment that is aligned with curricular goals; 4. Stability and mechanisms for evolution.
The issue of curriculum is already being addressed, with a number of curricula that are already strong and are being de-bugged and further improved.
The teaching situation, on the other hand, he characterizes as a national outrage and a national pathology. Partly because the skills required for teaching are underestimated and undervalued, teachers are generally sent into the classroom woefully underprepared, and then given appallingly little opportunity for professional growth. Data from Pittsburgh and Michigan back up what seems intuitively reasonable: when teachers are treated like professionals and given the opportunity to develop their skills and understandings over time, the result is significant gains in students' mathematical performance. Unfortunately, other places where professional treatment and opportunities happen are rare indeed in the United States. This is something that must change. Many people, in fact, subscribe to the idea that it must change, but how to do it is totally unclear. Schoenfeld's view, analogous to one of Robert Moses, is that all of us should provide any support we can, but that the basic responsibility for change must rest with the teachers themselves. Moses sets up an analogy with the political situation in Mississippi in the 1960s: many people had been supporting and speaking for sharecroppers, day laborers and the like, but the point when they got the vote was when they themselves rose up and demanded it.
The assessment situation is more mixed. On the one hand, Schoenfeld documents some of the pernicious effects of high-stakes, low-level testing. On the other hand, there is an increasing recognition of the damage this kind of testing can do, and an increasingly clear and articulate message coming from many branches of the educational community contrasting this testing with the kind of assessment which can be beneficial. The situation is fluid, and has at least some encouraging aspects.
The issue of stability and mechanisms for evolution is one of the diciest. In the U.S. our track record is not good. Over the past decades we have done pendulum swings with large arcs. We are a nation accustomed to quick fixes, and furthermore our politicians, who have a lot of influence, are strongly attracted by visions of change happening before the next election. This provides us with a daunting challenge. On the other hand, the current movement has now been sustained for long enough for solid data to begin to appear, and for specific knowledge of what needs to be done to begin to take shape. It is a process that began with research on mathematical thinking and learning twenty-five years ago - and this is not the kind of scale on which the public is accustomed to thinking for educational issues. Among the mandates we now face is that of educating the public to as many of the issues and of the possibilities in the current situation as we can.
Fortunately the possibilities really are there. The current situation, as Schoenfeld finishes by pointing out, offers tremendous opportunities. We have made what he describes as astounding progress, both in terms of students in general and in terms of traditionally underrepresented minority students. We have a solid base on which to stand. It is time, he concludes, to stay the course, build on what we know, and work in evolutionary fashion toward the improvement of (mathematics) education for all students. --