In response to one of my previous posts on the issue of meaningful open questions about mathematics pedagogy, an anonymous contributor wrote in part: "How can I teach a student at level N who hasn't mastered all the work of level N-1 (or maybe even N-2 for some things)? What can be done to help with the learning of the current content while remediating gaps in learning?"
This is a particularly nagging problem for most mathematics teachers in K-12 and beyond, though at the college level it is easier perhaps for instructors to be blithely condescending and dismissive of the problems of students who come to class ill-prepared for the level of coursework expected of them. College is, after all, a choice, and professors are not obligated to provide remediation, although there are generally non-credit classes offered that do just that, or at least purport to do so.
I wrote on email@example.com in response to the above question:
The first question is nearly ubiquitous in mathematics education and of course there is no simple answer to it (not that we won't likely here some in this venue from non-teachers).
I suspect that we'll never see a time where teachers don't have to ask that question for the simple reason that there will never be a "system" that eliminates differential intellectual development, readiness, or individual variability in motivation for learning any given subject. It seems inevitable that kids will not be equally ready in all sorts of ways for any given mathematical topic at a given age/grade level.
As long as we try to mass educate under the current model, with bizarre expectations that we can legislate kids to level N or intimidate schools, parents, teachers, or kids to some pull rabbits out of hats in an attempt to pretend that all kids are at level N at the appointed age/grade level, we'll miss the boat. A saner approach is to do the best we can to get kids ready for school and then teach them where they actually are, using differentiated instruction and methods to brings them as far as we can given where they start. That requires more flexible ideas about content and sequence, about instruction and tools, about the nature of classrooms, and about meaningful, useful assessment than we can reasonably expect to see given conservative/ reactionary opposition to sensible approaches to public school in particular, and education in general.
Richard Strausz replied:
Michael, I agree with what you say. Some critics use such real-world observations as more ammunition for their crusade against public education.
However, in talking with math teachers in Catholic and Jewish high schools, I hear similar situations in their classrooms.
And I commented:
I'm not surprised in the least. Didn't mean to suggest there was something unique going on in secular schools. I suspect we'd hear similar issues and concerns from private, non-sectarian classrooms.
Of course, the usual suspects in mathematics education, right-wing ideologues who pretend that all was once well in math teaching back in some imagined day, and/or that it will all be well again if only we had "real standards," (who decides and why they are qualified to do so always turns out to be those same ideologues and their like-minded brethren), "real accountability" (to whom is never quite clear, but it generally means to people who are at best marginally involved in actual teaching), "real textbooks" (and of course, nothing BUT textbooks, generally of the skill-based, routinized kind best exemplified by "teacher-proof," learning-proof Saxon Mathematics products), "real methods of instruction" (direct instruction with the teacher firmly in the middle, doing most of the talking and most of the mathematics), then all would be well.
Wayne Bishop Chimes In
In that light, the following predictable nonsense was offered up by the reliable anti-reformer, Wayne Bishop:
You are right about nonpublic schools but you seem to have misinterpreted the real problem with your identification of this problem being "more ammunition for their crusade against public education". It's ammunition for "their" crusade against colleges of education, the problem so well identified by Reid Lyon. The aversion against standards-based education (prior to the collegiate level where, at least in mathematics, standards tend to be used and accepted) is a direct consequence of the teaching of schools/colleges of education. For inexplicable reasons, almost religious in appearance, the industry prefers to place students by age-level even when it is obvious that students are being placed into situations where they are doomed to fail if honest performance standards are maintained.
It's always fascinating to hear Dr. Bishop cite, directly or otherwise, the execrable Reid Lyon, a former Bush education "expert" who was in part responsible to the multi-million dollar boondoggle known as READING FIRST, a project that appears to have been both rife with corruption and utterly ineffective, neither of which comes as a shock to Lyon's critics or fair-minded observers of how the current administration has lowered to every opportunity to help those most in need of its assistance. It was of course Mr. Lyon who said "You, know, if there was any piece of legislation that I could pass, it would be to blow up colleges of education" (McCracken, Nancy. "Surviving Shock and Awe: NCLB vs. Colleges of Education." English Education, January 2004, 104-118.) Scratch a right-wing ideologue these days, you may just turn up an anti-democratic terrorist with his hands in the public trough up to his shoulders.
In any case, here is my response to Dr. Bishop's latest screed:
For those of us who actually teach K-12, the issue that is far more onerous is not what book was used, what teaching methods were most prominent (well, that's a concern if one is using eclectic instructional approaches with students who've been taught in totally lecture-driven, teacher-centered classrooms in which student passivity is so ingrained that ANY requirement that they become actively engaged in mathematical thinking and mathematical discourse is useless until the teacher shows students HOW to become active in mathematics and convinces them that it is necessary and productive to do so), or even to some extent whether the previous class covers every topic that s/he would have. Rather, it is the passing of students from previous classes with grades of D and, generally, C. Frankly, I'd say that in my experience, unless we're talking about an honors course of some sort or an extraordinary school, anyone coming into Algebra II with a grade less than a solid B in Algebra I is going to have more mathematical holes in his/her head than it takes to fill the Albert Hall or are found in the typical argument from members of Mathematically Correct and NYC-HOLD about any aspect of mathematics education. And that's a lot of holes, let me tell you.
The obvious solution is to stop making grades the joke they are and instead go to minimum exit/entrance exams with multiple sorts of assessment employed to determine whether students are ready to move on. I suggest both exit and entrance exams so that there is a double check in place: one on leaving level N-1 and one on beginning level N. And also so that those who marginally fail to gain exit at level N-1 can get another shot at the beginning of the following year. Should they have gotten themselves up to speed by then, they should not be denied a chance to prove themselves ready to proceed.
Naturally, I'm not calling for a nationally-normed testing instrument here, especially if it's strictly some vapid multiple-choice test that values only one narrow set of skills. But within each state's standards, if there is a real attempt at meaningful formative assessment that leads directly to re-teaching areas of weakness before allowing students to take a crack at more challenging mathematics that builds directly on previous knowledge, then such a system would potentially reduce the number of students who are simply pushed further down the rabbit hole with nothing to hold onto.
For such a system to work, however, it can't be linked to a bunch of politicized punishments intended to help ideologues and politicians use teachers, schools, and kids as footballs for making propaganda.
Minimally, that would mean that people like Wayne Bishop would need to be keep as far as possible from influencing the assessment process. Instead, knowledgeable assessment experts who actually understand and stick to psychometric principles must have major input into each state and district's development of assessment tools. Close attention must be paid to some of the issues raised in two papers by SUNY @ Stony Brook mathematician Alan Tucker about cut scores, the theory of performance standards, and related issues. A balanced approach to assessment, regardless of the religious objections of the nay-sayers, must be used, and any structure that has been proven to promote wide-scale cheating due to absurd political pressure on educators, parents, and students, must be closely examined and, if found incapable of improvement, abandoned.
Is authentic, meaningful assessment expensive? You bet. But then, finding out what's really going on in a complex world is always dearer than looking for surface data that tells you what you already believed to be the case beforehand. And crafting real solutions rather than destroying public education so that the rich can get richer is always unpopular. . . among the powerful and their lackeys and mouthpieces.