From: hot@SOE.Berkeley.Edu (Henri Picciotto)
Subject: how changes happened at the Urban School of San Francisco
Newsgroups: geometry.pre-college
Date: 23 Feb 1995 10:00:25 -0500

Steven Means and Kathryn Garrett asked for more info on the changes that happened at my school. So here is a bit of an answer, departing a bit from the more theoretical tone of many recent debates.

Substantial change is a many-year project, which can only be undertaken one step at a time. Here is how it happened for us.

• Our first change (many years ago) was to move to the Stein series of textbooks, (available from Sunburst). These books were a breakthrough for us. They allowed us to move to a cooperative learning, student-centered model in all our classes. That was probably the most significant change. In some respects, these books are a bit out of date now, though the geometry book and the trig book are in my opinion still the best on the market. We no longer use their Algebra 1 and 2, except for a few ideas here and there.

  Around the same time, we started using computers, specifically software I wrote in Logo (available from Terrapin.) Again, by today's standards, this was primitive, but it allowed us to appreciate the value of a good lab.

  Next, we expanded the lab idea beyond computers. In Algebra 1, we started using the Lab Gear (algebra manipulatives I developed, available from Creative Publications.) In Geometry, we use pattern blocks, pentominoes, tangrams, supertangrams, straws, polydrons, patty paper, and so on. In Algebra 2 we measure the bounce of balls, make slide rules, simulate exponential decay with 10-sided dice, etc.

  So far, this did not involve substantial curricular changes. But the pedagogical changes led to a change in classroom atmosphere which was palpable. Students became much more engaged and self-confident, especially girls.

  Actually one great curricular change from that period was the introduction of the "mathematical modeling" problems from Foerster's Algebra 1, Algebra 2, Trig, and Precalculus books into various classes.

• The next change was to add other ways to assess the kids besides the usual test-homework routine (which we did not abandon.) We try to assign reports, projects, and problem sets to get the students' best work. This is tremendously revealing. It turns out that some kids who do very well on tests are not as good as we thought they would or should be when it came to expressing themselves, communicating about math, making connections, etc. And some of the students who we thought were very weak turn out to know and understand more than tests would indicate.

  Reports: explanation of material covered in class, with applications, illustrations, etc. Often with an imaginary audience. ("You are writing about bank accounts for a consumer magazine", "You are giving a lecture on the golden ratio to a group of architecture students", "Write a science fiction story involving exponential growth", etc.)

  Projects: each kids pursues their own project on a topic related to the material covered in the class. They present it to the class, and/or make a poster or other display. Analysis of student-collected data is a great arena for projects.

  Problem sets: very difficult problems, which would be too difficult for a test, but students have several days to do them. They can (or can't) get help from each other and the teacher, but must write things in their own words.

• Curricular changes: we introduced interesting electives. The longest lived has been a course called Mathematical Modeling and Probability-Statistics MMaPS. That course is based in part on the Quantitative Literacy Series (Dale Seymour) though it also includes material from Foerster's Precalculus (Addison-Wesley) and Stein's Algebra 2 on Pascal's triangle (Sunburst). The class includes a mix of students: for some it's the last math course ever, for others it's a side trip on the way to calculus.

  Other electives have come and gone depending on various things: Infinity, Geometry 2, Math and Art, Logic, etc

• Changes to the core program: we are gradually replacing Algebra 1, Geometry, Algebra 2 with Math 1, 2, 3, and trying to bring in what we learned in the electives into the core program. (e.g ideas from Math and Art into Math 2, ideas from MMaPS into Math 3, and so on.)

  We started using the _Algebra_ textbook I co-authored in Math 1. (The book is available from Creative Publications.) We are currently building a Math 2 out of: computer programming in the (not yet available commercially) language called Boxer; intermediate algebra topics from my book; and geometry through labs and the Stein book.

  Math 3: we use the UCSMP Algebra 2. We don't really like it. It does not fit our school's pedagogical style, but we like other books even less.

  For calculus, we use the new Harvard course, which is very consistent with our program.

I hope these notes help.

--Henri

From: hot@SOE.Berkeley.Edu (Henri Picciotto)
Subject: more on my school, since people are asking
Newsgroups: geometry.pre-college
Date: 23 Feb 1995 18:10:42 -0500

Is the Urban School public or private? regular kids or outlier kids? if outliers, what kind? The name seems unusual, which is why I ask. (And you may have said what it is earlier, in which case my apologies.)

Judy Roitman, Mathematics Department
Univ. of Kansas, Lawrence, KS 66049
roitman@math.ukans.edu

As far as math levels, we don't have as broad of a range as public schools do, but many of our students would not be in the college prep tracks in other schools. I would guess that we have in one untracked program kids who would be in at least three tracks in many public schools, with most of them in a college prep, but not honors, track.

The name of the school comes from a school philosophy tenet that since we're in San Francisco, we need to act accordingly. Our biology classes spend time in Golden Gate Park, at the beaches, and in other ecological niches in the Bay Area. Our constitutional law class goes to court. Our physics classes go to the Exploratorium (one of the great science museums in the known universe). Our art program has collaborated with the various museums in SF. Our students do a lot of community service. Our math program is a little weak on the "Urban" dimension -- we pretty much stay at school. Though one or two of our teachers have taken kids to play pool at a neighborhood Boys and Girls Club to look at how the balls bounce. I have had kids measure the slope of neighborhood streets, or the velocity of passing cars.

I was so excited to read Henri Picciotto's description of the innovative curriculum and changes which have been going on at the Urban School of San Francisco. The diversity of methods of teaching and activities set up for kids, as well as the diversity of methods of evaluating them, make a great deal of sense to me. I was just wondering if Henri could clarify one thing for me, though. When you say that you are reaching more kids now, how have you determined that? Mostly I am wondering if the students have taken part in evaluating the success of this program, or if the success has been gauged on the basis of improved student enthusiasm and performance.

Hilary Gehlbach Swarthmore College

For one thing, it's not easy to maintain the variety I advertised. When you feel overwhelmed or overworked, it's easy to slide back into "do numbers so and so on page such and such."

The only reason I believe we are reaching more kids is that so many more stick with math beyond the requirements, including mostly girls. Perhaps this sort of change is happening everywhere, and perhaps it is due to other factors than changes in our math program.

When we first starting to make these changes, I remember a student-organized "feminism month" (Not only is this is San Francisco, but we are the most progressive school in town, so something like this is not unthinkable!) The math dept. contributed to the festivities by having a lunch discussion on the topic of girls and math. We couldn't believe the turnout. Lots of kids came, and many expressed their gratitude for what they saw as a really unique math program. I don't know if the same thing would happen if we held such a meeting now. Many kids seem to not be aware that our math program is not your typical program. I don't know that they are that enthusiastic, frankly. We still have kids who say they hate math, we still get asked "when are we ever going to use this?" when the work gets difficult, we still have kids who don't do their homework, or copy it from others, etc. In other words, this is not utopia, it's a school which has implemented some worthwhile changes. But we're still part of the broader culture.

I don't know how you would judge improved student performance. Our SAT scores have had ups and downs, in a way seemingly unrelated to the math program. We know our current students can do things our students a few years ago could not do. But the reverse may also be true: we've changed our emphasis, and so our students' strengths have changed.

I don't mean to be cynical. Participating in this process has been exhilarating, and there are those great moments in the classroom. But education is a long road, and no matter how much progress you make, you're still at the beginning of the journey.

--Henri

From: mmaraff1 (Michelle Maraffi)
Subject: Re: more on my school, since people are asking
Newsgroups: geometry.pre-college
Date: 26 Feb 1995 19:04:46 GMT

I realize that the changes in this school have been bumpy and that progress in always slow when it comes to education, but I still believe that this is a wonderful program. I feel as if it embodies a lot of Dewey's (Dewey on Education, Experience and Education) ideas. In the different classes kids get to experience what they are learning about -- I think, as did Dewey, that experience is the best learning tool.Therefore, I think the field trips and so forth are terrific!

Also I think it is wonderful that you do not track kids in math. I think tracking can potentially be a dangeous thing because it can sometimes cause kids to only work to the potential that the teacher or the track expects instead of their true potential. Also, consciously or not, in many tracking programs the district put more minority students in lower tracks (in difficulty) and puts white students in upper tracks. Thus, it becomes an issue of race and at times socioeconomic status, which clearly is bad because it trains white students for better jobs in math than minority students.

Yet, it is a shame that this is a private school. I understand that you have a financial aid program -- yet, I wish more students could benefit from your program.

Michelle Maraffi
Swarthmore College

From: Marksaul@aol.com
Subject: Urban School and "tracking"
Newsgroups: geometry.pre-college
Date: 28 Feb 1995 22:18:58 -0500

I was concerned about Michelle's comments on what she calls "tracking" (other-than-heterogeneous grouping), enclosed within her notes on the Urban School. I cannot agree, and fear that the comments give in, perhaps unwittingly, to the worst sort of political correctness.

Two problems with homogeneous grouping are mentioned.

The school I teach in is a public school, but since it serves a wealthy community, our student population is more or less the same as the description of the Urban School. It's a very traditional place, and we track students -- honors, regular, and remedial. Michelle's comments about minority kids being put in lower tracks simply doesn't apply -- as it wouldn't to the Urban School, probably, because there are so few of these students.

As for her comment on students working only up to the teacher's expectations, that's an old saw, and I see it as nothing less than teacher bashing. I teach honors as well as remedial students, and have high expectations for each. I find, for my teaching, that it is very convenient to separate them -- I can pay closer attention to the needs of every student.

The problem I have experienced is that administrators sometimes consider my lower-level classes "dumping grounds," and I have to defend my work there. I also know that in many districts the lower track classes get the less experienced, or less talented, teachers, so that teachers are tracked as well as students.

But let's not throw out the baby with the bathwater. Heterogeneous grouping, whatever this means, is politically correct, but I see no evidence for any educational value in it per se. It all depends on how it's done.

I know many people, for example, who don't want to see calculators being used in elementary schools, or even in high schools, because the can be misused. I view these people as distrustful of the teaching profession to make the right use of calculators. Sure there will be abuses, but I don't think anyone can mention a system or tool which is proof against abuse.

Let us sort out grouping for instruction, and use of calculators, and any number of educational issues, on the basis of what works in the classroom, not what is politically correct.

From: appelbaum@zodiac.rutgers.edu (Peter Appelbaum)
Subject: Re: Urban School and "tracking"
Newsgroups: geometry.pre-college
Date: Tue, 28 Feb 95 18:33:55 GMT

Could you say more about how grouping helps you attend to your students?

Peter M. Appelbaum
Curriculum and Instruction
The William Paterson College of New Jersey
(201)595-3123 appelbaum@zodiac.rutgers.edu

From: hgehlba2@cc.swarthmore.edu (Hilary Gehlbach)
Subject: Urban school and "tracking"
Newsgroups: geometry.pre-college
Date: 1 Mar 1995 09:21:56 -0500

I have to disagree with the assertion that heterogeneous grouping is merely a "politically correct" move, based both on research and on personal experience. Literature reveals discrepancies in the rate of graduation of kids in lower tracks, discrepancies in the curriculum of different tracks, discrepancies in the manner in which subjects are taught, and more. There is evidence that learning opportunities are decreased in lower tracks because more time is spent on discipline, with the less attention to teaching (Snow, 1986).Kids in lower tracks also tend to have lower self-esteem, since they are compared to other, higher tracks. I could go on, but suffice it to say that there very definitely is research which shows significant, detrimental effects of tracking. (Much of the literature is based on High School and Beyond National Surveys - authors include: Gamoran, A.,"The variable effects of high school tracking.", Byrne, B. "Adolescent self-concept, ability grouping, and social comparison: Reexamining academic track differences in high school," and Snow, R. "Individual differences and the design of educational programs.")

As far as my own experiences go, in my last two years of high school, my school began mixing classes (it had been a highly tracked school). I had always been an advocate of tracking -- it certainly gave me advantages. However, when I began taking classes which were mixed my ability levels, I began hearing opinions I had never heard before, and seeing new ways of thinking which had not existed in my upper track classes. I believe strongly that mixed grouping is a beneficial and significant step in education. There is no question that it takes a talented teacher to handle mixed-group classes effectively, but I have seen it done, and it can definitely work.

From: Marksaul@aol.com
Subject: Re: Urban School and "tracking"
Newsgroups: geometry.pre-college
Date: 1 Mar 1995 12:34:16 -0500

- Could you say more about how grouping helps you attend to your students? [asks Appelbaum]

Sure. For example, this year I have an honors ninth grade class and a remedial eleventh grade class, both studying geometry. In both classes I use MIRA.

In my honors class, I gave them simple exercises and they did them easily: draw this perpendicular bisector, draw three altitudes in a triangle and observe that they coincide, etc. Then I gave them more difficult stuff--the billiard ball problem, relating this to the triangle inequality, and so on. They worked on these cooperatively and answered many of them (some they could not do).

In my other class, about half the kids couldn't do the simple problems. That is, these were really not simple for them. They had to learn that perpendicular does not mean vertical or horizontal. I had to show them how to turn their papers to draw the three altitudes of a triangle. They then had to practice "turning the paper in their heads". I still worked cooperatively--no two students work at the same rate. And two of these kids finished early and started thinking about the billiard ball problem.

Had the kids been mixed up, I would have had much more difficulty differentiating the curriculum. I would probably have ended up assigning the faster kids to teach the slower ones. The faster kids would never have gotten to the billiard ball problem, except a few. Maybe the slower kids would have learned from the faster and maybe not -- a kid who is good at math may not be good at teaching.

I have had cases where kids don't want to do more work, play dumb, and get themselves placed in a slower class. Once in a while I can work with such a kid, giving him or her more challenges. But in the few cases this has happened, I find that the kid has opted out (or maneuvred out) of the regular tracks because of severe personal problems that won't allow him to concentrate, and which are probably a priority over math in the kid's life.

This is the only sort of case I meet with regularly in which a kid in a slower track is not achieving his potential. Other cases, if there are any, are very much my fault (if I've misjudged what a kid can do). I try not to make these mistakes. I am more likely to make the mistake of overwhelming a kid with something more difficult than to withhold his or her "opportunity to learn."

The point about this that the researchers don't seem to understand is that you have to look in detail at how a teacher uses the grouping situation. It's not enough to get massive statistics.

Years ago, when teachers were first organizing into unions, much research came out saying that class size didn't make a significant difference in achievement. Having moved from the city to the suburbs, where my class size was cut in 1/2 or in 1/3, I can agree with the research. If you teach a class of 15 in the same way that you teach a class of 30, there will be no difference. But if you know how to take advantage of the small class size, it makes all the difference in the world.

The aspect of education that researchers can tell us least about is the art of teaching, which is the art of relating the math to the students.

From: Marksaul@aol.com
Subject: Re: Urban school and "tracking"
Newsgroups: geometry.pre-college
Date: 1 Mar 1995 12:56:11 -0500

I don't find research in teaching very enlightening, and the issue of tracking is no exception. There's even more, and more recent, research than the work that Hilary Gehlbach quotes, but these people have trouble distinguishing between correlation and causality.

Certainly there are "discrepancies in the curriculum of different tracks". That's the point of separating the kids. I will only teach a kid something that I can tell he or she is ready to learn. And I can usually tell. Aside from this, it's the use to which curriculum topics are put, and not the topics themselves, that really tells in teaching. Maybe a kid doesn't know the multiplication tables. If you drill him to death for five years he still won't know. But if you use the multiplication tables with him to discover patterns, he'll learn -- even if his peers in another class are doing algebra already.

"Discrepancies in the manner in which subjects are taught." Ditto, with the caveat that often the less gifted teachers are assigned the lower track. See what I mean by correlation vs. causality?

Lower self-esteem? I'm not convinced, and statistics will not convince me. I hate sports. I can't shoot a basket. I can't climb a rope. I don't want to learn. If I have to play a baseball game, I want to be with other nerds, not with olympic stars--and not even with merely competent baseball players. I won't get embarrassed. I'll relax and have fun. Maybe I'll come again. I'll be safe from ridicule. And I won't have to endure the condescension of others telling me when to swing for the ball.

And by the way, why do we extoll tracking in sports but think it evil in academics? Untrack the football team, and I'll untrack my class...

No, the research is not convincing.

But I would very much like to hear more about Hilary's classroom experience. Had she taught lower-track classes before untracking began?

From: Marksaul@aol.com
Subject: Re: Urban School and "tracking"
Newsgroups: geometry.pre-college
Date: 1 Mar 1995 16:34:22 -0500

- If you feel that tracking is such a great idea, then back yourself up! - Convince me!

No. I'm not at all interested in convincing you to teach my way. There are lots of kids, lots of schools, and lots of ways to teach.

I have all kinds of hard evidence of my success at teaching, which I need not go into here. I don't doubt yours for a minute. And there is nothing at all to be gained from seeing which of us is the more successful. It's not my idea of a good time--I'd rather go out to dinner.

I am very glad that you have success in teaching heterogeneous groups. I prefer other ways to group. My lower level kids have more self-esteem, not less. Those few "rigid thinkers" that struggle to survive in my honors classes do so at great personal cost, and I while I would not ask them to move, I do think they are misplaced. I struggle daily to loose the sometimes formidable intuition of the kids in my lower track.

And my class is often also non-traditional, based on problem solving and student communication--although sometimes I "load" the honors class with lecture notes for them to figure out over the weekend, which I rarely do for lower track kids. I don't want to reject any technique that works, traditional or non-traditional.

I was not recommending that the Urban School, or any other school, adopt one particular method of grouping for instruction. My argument is that heterogeneous grouping per se is not the answer. Good teaching and caring for students is the answer.

This is why I am concerned about the Political Correctness of heterogeneous grouping. I have met many people who assume that because I don't agree with the "research" on grouping that I care more about theorems than students, or that I'm "elitist," or that I pull the wings off flies in my spare time. There are circles in which rejection of heterogeneous grouping will bring an accusation of racism or gender discrimination.

The point is that there are many ways of teaching and learning. Grouping heterogeneously will not solve the world's problems. The arguments pro and con must stand on their own merits, and be tested against experience. It is experience that proves my ideas correct--for me.

I don't ardently love Chairman Mao. I would like to see a thousand flowers bloom.

- Many, though not all, "politically correct" positions actually have
- some strong merit to them. They are "politically correct" for a
- REASON!

No. They may or may not have strong merit to them. But they are labelled Politically Correct because of the erroneous confounding of different ideas and values. One of the great paradoxes of the grouping discussion, for me, is that I share many of the same values and assumptions of others who argue for heterogeneous grouping, but these lead me to completely different conclusions.

From: hgehlba2@cc.swarthmore.edu (Hilary Gehlbach)
Subject: Re: Urban school and "tracking"
Newsgroups: geometry.pre-college
Date: 1 Mar 1995 20:32:43 -0500

Gary -

I appreciate your interest in my post. Just for your info, I wrote a research paper a couple of years ago on the effects of tracking -- so that's what the following references are from: (and for anyone else who is interested, Gary asked for my full references)

1. Byrne, B. Adolescent self-concept, ability grouping, and social comparison: Reexamining academic track differences in high school. Youth and Society, 1988, 20, 46-67.

2. Gamoran, A. The variable effects of high school tracking. American Sociological Review, 1992, 57, 812-828.

3. Gamoran, A. and Mare, R. Secondary school tracking and educational inequality: Compensation, reinforcement, or neutrality? American Journal of Sociology, 1989, 94, 1146-1183.

4. Kerckhoff, A. Effects of ability grouping in British secondary schools. American Sociological Review, 1986, 51, 842-858.

5. Kilgore, S. The organizational context of tracking in schools. American Sociological Review, 1991, 56, 189-203.

6. Klaczynski, P. Sociolcultural myths and occupational attainment: Educational influences on adolescents' perceptions of social status. Youth and Society, 1991, 22, 448-467.

7. Lipsitz, J. Successful Schools for Young Adolescents, chapter 5, pp.93-128. Snow, R. Individual Differences and the design of educational programs. American Psychologist, 1986, 41, 1029-1039.

8. Tobias, S. Tracked to fail. Psychology Today, 1989, 23, 54-58ff.

9. Tygart, C. Strain theory and public school vandalism: Academic tracking, school social status, and students' academic achievement. Youth and Society, 1988, 20, 106-118.

From: hgehlba2@cc.swarthmore.edu (Hilary Gehlbach)
Subject: Re: Urban school and "tracking"
Newsgroups: geometry.pre-college
Date: 2 Mar 1995 00:37:28 -0500

Marksaul@aol.com asked me:

Well, what were you able to do better in heterogeneous groups than in homogeneous groups? If you read my comments to others, you will see why I find "research" more or less useless, particularly in this situation. But I'd be very interested in direct classroom experiences.

Thus, from a social standpoint, I thought the mixed grouping was very significant in that it facilitated interaction between all kinds of kids. (and, though you cite the kids of millionaires in your lower track classes, the tracks in my school were almost exclusively divided along socio-economic lines) I know that there were distinct divisions in my school -- typical high school cliques -- which were closely related to academic tracks. I and my friends were perceived (as far as I could tell) as snobby over-achievers. I believe that mixing grouping levels can alleviate a lot of these kinds of problems.

From a more academic standpoint, having different kinds of kids in the classes allows for the expression of more diverse viewpoints and ideas in the classroom. I was really amazed in my first mixed class to be hearing opinions that I had never heard before in all of my previous homogeneous classes. Particularly of note was the day that we discussed tracking -- and I got to hear how much kids in the lower tracks disliked it and perceived it as favoring those in the upper tracks.

Admittedly, the class I refer to above was an English class -- but I think that, if approached thoughtfully, the same can work in a math class. Undoubtedly, kids have different types of strategies they employ in problem-solving -- it seems to me that mixed grouping could be similarly effective if these sorts of things were tapped into.

And just to respond to a few more comments:

These can be impressive. I fear you miss the point, an important one, about less gifted teachers being assigned the lower track. This happens all the time. I have seen this happen in many schools. There are statistics, this time meaningful ones, showing that schools serving lower socioeconomic groups are staffed by less well prepared teachers. And so the correlation between low achievement and low tracking might just be due to the teacher being assigned to the lower track, not to the fact of its existence: correlation must not be mistaken for causality.

You haven't stated, but I suspect you will agree, that we are not very good at measuring ability in almost anything. So the serious question, if we factor out political correctness, is: what are we sorting on when we group by "ability?"

Hilary Gehlbach

From: appelbaum@zodiac.rutgers.edu (Peter Appelbaum)
Subject: Re: Urban School and "tracking"
Newsgroups: geometry.pre-college
Date: Thu, 02 Mar 95 07:18:40 GMT

I would like to thank Mark for reminding us again that most teachers teach the way they do because of professional, sound motivations that help them achieve their goals. I get very angry at "reformers" who see "good" teachers as those who quickly adopt the latest fad, rather than as those who make careful decisions based on a serious concern for their students.

I wuold think someone wanting to convince a particular teacher to give up on grouping would have to show the teacher how non-tracked/non-grouped classes could achieve the goals the teacher has.

Peter M. Appelbaum
Curriculum and Instruction
The William Paterson College of New Jersey
(201)595-3123 appelbaum@zodiac.rutgers.edu

From: Marksaul@aol.com
Subject: Gehlbach on grouping
To: geometry-pre-college
Date: Thu, 2 Mar 1995 21:55:36 -0500

I was amazed to read Hilary Gehlbach's response to my question about her experience of heterogeneous grouping. I had no idea that she was speaking as a student, rather than as a teacher. My kids go to New York City public schools in New York City, where they meet all kinds of other kids. I could bring them to school with me, in the homogeneously settled suburbs, but prefer them to have the urban experience (I'm not talking here about grouping for instruction within the schools). Since less is spent per capita on education in the city than in the 'burbs, my kids get a bit less in other ways. This is my compromise with America.

Hilary speaks to the very real problem of tracking as isolating groups of students, and of reinforcing socio-economic differences. In the school I teach at, this is not the case. It's an upper middle class community, so the separation of the classes is accomplished by the realtors, and we in the school can wash our hands of it. If anyone remembers, my comments about grouping were a response to a description the Urban School in San Francisco, which seems to have a similar student body. My school has a different solution to the problems posed by these kids.

But I still don't think that heterogeneous grouping is an answer, much less the answer, to inequities in our society. From where I sit, America is driven, for better or worse, by differences in class. The schools reflect this, tracked or untracked. Maybe heterogeneous grouping will help. But maybe prejudices based on social class, ethnicity, or other factors will continue into the heterogeneous classroom.

In either case, this reason for heterogeneous grouping is certainly not educational, but social and especially political. It's a pretty strong argument: perhaps in order to invest in the future of our society we must sacrifice the education of some students. I can live with it. But this is different from saying that it's always better educationally for kids to learn math in a heterogeneous environment. I can't live with this.

And there remains the serious and unremitting problem of grouping for living, through real estate sales, (which has been upheld often by the courts as the American Way), and the resulting inequities taxation, and therefore funding, and therefore staffing and equipment.

Other comments:

And so the correlation between low achievement and low tracking might just be due to the teacher being assigned to the lower track, not to the fact of its existence: correlation must not be mistaken for causality.

I agree that many of the problems with the tracking system do stem from the teachers' preparation/abilities, but, if this is true, isn't this another argument for mixed groupings?

Teaching is -- or should be -- a profession. A teacher should be held accountable for what happens in his or her classroom. Teaching in a "lower track" class must be of the same quality as teaching in an honors class -- all students deserve this much respect for their adaptations to the arduous tasks of adolescence.

So the serious question, if we factor out political correctness, is: what are we sorting on when we group by "ability?"

Again, I agree -- if we're not very good at measuring ability in almost anything, then how can we justify placing students in different tracks to begin with?

But the end result is that some people learn math more quickly and some more slowly. Some learn by reading and others by doing (although most learn by doing). Some learn by handling, and some learn by thinking. And all of use use different modes at different times. So we can clump people together for modes of instruction.

And we can group them by interest. By high school, kids develop different interests. Some may not be as interested in math as others. My honors kids will get interested in math for its own sake -- no motivation other than intellectual curiosity is needed. Other groups need intensive motivation, sometimes rational and sometimes emotional. Still others are motivated, but not in math: poets or actors who learn math just fine, thank you, but would rather not spend the time.

We can also group them by achievement, which is much easier to measure than potential. A "dull person" who can program in Pascal should not have to suffer through a simple course in introductory computers, even if surrounded by "geniuses". And vice-versa. I don't consider myself dull at all, but I would have to start with the most elementary course in Hindi should I decide to learn this language.

Looking at my classes, I can find cases of sorting in all three ways. For the most part, it all works out quite wellin my environment. Other environments are different, and may demand different sorting -- or no sorting at all.

I don't think of justifying grouping practices on the basis of theory ("natural ability"). I have no trouble justifying them on the basis of practice (what works for me and for the kids).

From: roitman@oberon.math.ukans.edu
Subject: tracking & remediation
Date: Fri, 03 Mar 1995 10:34:23 -0600 (CST)
To: geometry-pre-college

Do I have an opinion on tracking/remediation etc? Not fixed in concrete -- I like what Mark Saul says and I like what I see in good heterogeneous classes too.

But my kid is dyslexic and I am incredibly grateful for his being in a special LD English class right now in junior high, and very concerned about inclusion doing away with these classes next year. Some kids have knowledge deficits, sometimes needing a lot of work (like LD), sometimes only short term (didn't pay attention when fractions were taught). Remediation -- doing the same thing as before -- obviously doesn't work. But someone needs to give these kids special help in a situation where they don't feel like outliers, and with the attitude (which Mark obviously has) that they're not dumb, not doomed, but expected to eventually learn what everyone else is learning, just maybe not in the same order and in the same time frame.

Years of having essentially the same expectations RIGHT NOW as the other kids in elementary school have destroyed my son's motivations and his belief that he can do well in school. While LD is obviously an extreme case, it's still on the continuum, and there are a lot of kids with similar, but less dramatic, problems.

How do we meet the needs of all kids? That's what the Standards asks us to do. And I think flexibility is what's needed. This kid is appropriately in a heterogeneous situation here but not there -- that kind of thing. If all of our classes are taught with the goal that all students will (not just can, but will) learn at a high level, and if none of our classes are designed as dead-ends, then we don't have to be doctrinaire but can really look hard at individual kids and what's best for them.

Thanks for listening.

Judy Roitman, Mathematics Department
Univ. of Kansas, Lawrence, KS 66049

From: Steve Means
Subject: Re: tracking & remediation
Date: Fri, 3 Mar 1995 10:24:08 -0700 (PDT)

Thank You, Judy Roitman,

I have found that the power of personal experience is often helpful in developing better teaching strategies. By making TIME a variable, educators can help many special learners become good mathematicians.

One insight that has come to me with this change is that QUICK learners are not always LONG-TERM rememberers and not always ABLE to apply the knowledge in differing situations. Another insight is that a student who is QUICK to learn in my view may have learned something quite different from what I thought I was teaching.

I appreciate this list discussion in particular because it helps me keep an ongoing awareness of the differences in our needs and abilities.

from Steven S. Means means@belnet.bellevue.k12.wa.us
Math and Technology Teacher at Sammamish High School

From: Judith Haemmerle (judithha@u.washington.edu)
Subject: Re: tracking & remediation
Date: Sat, 11 Mar 1995 10:10:04 -0800 (PST)

Thanks for your comments on this tracking issue. In my second year of teaching, I got the "walking wounded" math class - kids with a steady record of failure and underachievement. (I teach at a private prep). Because I had no idea what to do with these kids (I'm a science, teacher, not a math teacher!) (Yes, these low level classes DO go to the unprepared and unqualified...) I started the year with a packet of word problems of varying sorts, and because they (both problems and kids) were such a mixed bag, there was great success for almost everyone. A student who has gone on to be a playwrite did especially well with the verbal tricks in (with apologies) John Conway type posers.

But the next packet I worked with was all Cuisenaire rod spacial visualization puzzles. The dyslexic students in the class - and there were several - ripped flawlessly through the packet at three times the speed of the non-LD kids. They were astounded. This was child's play for them, and for all of them it was the first time they had outperformed all their classmates in any academic class.

I started to question our geometry teachers, and found that in conceptual geometry, the pattern held true. Some years later there was an extract in Science News about dyslexic children. It suggested that because they have some processing deficits, dyslexic students expend more energy on pattern recognition and develop the pattern recognition centers of their brains more than non-dyslexics. Hence, compared to language, geometric shapes are a snap.

This made me a passionate advocate of never ever tracking geometry classes. For the years our school used a conceptual and visual geometry text, we never tracked. But alas, we have new teachers, new books, and tracking in geometry. Just for the record, I don't allow my science classes to be tracked, and I don't think it makes any sense in humanities, either. But algebra... well, I'm still reading...

On a personal note, I am the parent of a child who got 750 on the SAT at age 12, and who has never actually been in a math class. (No, I didn't teach him. Most of how he knows what he does is a mystery to me.) How do you track him? And how do you put him in an age-appropriate untracked class?