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Reviews of the proposed framework and its supporting
materials (A)
Review by Sanford Segal
The following review was prepared by Sanford Segal (Professor of Mathematics, University of Rochester), a research mathematician in the areas of number theory, complex analysis and history of mathematics, who has worked in mathematics education. In his role of external reviewer for the NSF-funded project "Supporting middle school learning disabled students in the mainstream mathematics classroom" (award #TPE-9153812), Professor Segal observed in its entirety one of our Summer Institutes and also examined selected artifacts from other components of that in-service program. Professor Segal's review provides the perspective of a mathematician on the experiences comprising our framework and, more specifically, on the illustrative units on Tessellations and Area that are at the core of that framework.
NSF Project #TPE-9153812: Overall evaluation
These are remarks on the program "Supporting Learning Disabled Students in the Maintenance Mathematics Classroom" -- NSF Project #TPE-9153813 based upon a comprehensive survey of the materials developed by the project thus far and a week long observation of the utilization of those materials and others in training teachers of mathematics. [...]
Learning disabled students learn best, it would seem, from an "inquiry approach". While this has special advantages for the learning disabled, it seems to me that all students benefit from such an approach for many, if not all, topics. This is because an "inquiry approach results in the students "owning" the material they have learned. While some students will acquire the necessary "ownership" in a more traditional lecture approach which attempts to penetrate a student's skull by years of the "osmotic pressure" of reiteration and drill, many do not. One especially good thing about the inquiry approach is that it gives meaning to the mathematics--an important desideratum too often neglected in teaching. What the "inquiry approach" does for the learning disabled student, is that it addresses the fact that motivation is not the problem for such students, but perception is. I might almost say that initially most students do want to learn, learning disabled or not. The small child is naturally inquisitive. However, stultifying classrooms can quickly stifle that desire. By involving the students continually, the "inquiry approach", at least as demonstrated by the modules prepared and their observed implementation, avoids this danger. This is certainly not the only way to keep students interested, but it seems an effective one. [...]
One primary benefit of the "inquiry approach" is that students learn from one another; peer learning is often excellent. The teachers, by being made to be students in an inquiry situation, learned both the benefits of such an approach to other students, and by observation of the group leaders and discussion, how as teachers they could facilitate such learning.
Examples. Rather than a dictated definition to be memorized and drilled, consideration was given to the issue of how does one make sense of a definition; how might it be changed to capture better one's original intentions; concision and reliability of a definition. Instead of drill, students are invited to conjecture, and to investigate how to test, those conjectures. Inventing deductive strategies for oneself, discussing them with the group, and testing them are encouraged.
The different group dynamics of small groups and large groups were elicited by experience and discussion rather than by instruction. How to value every student's issues while not attempting the impossible task of engaging all simultaneously was exemplified and then discussed. The teachers in the program learned how these kinds of dynamics are approaches that can enhance mathematics teaching and learning. [...]
The implementation of the project that I have seen (and I expect also to observe follow-up sessions) makes everyone into a participant, thus blurring the hierarchical teacher-student relationship--and thus demonstrating to teachers that it is possible to do this while maintaining classroom authority. This is a beneficial side effect of the investigators' style. [...]
In sum, the demonstration project encouraged teachers to make their classrooms into a learning community, by allowing them to experience the benefits of such a community. The importance of the process of learning for all students was realized by the teacher-students. The project showed how to embed traditional material in a more meaningful context. It showed how a secure teacher can give meaning and purpose to student work instead of demeaning it to routine drill. Some "facts" do have to be learned by memory as a basis for proceeding, but much can be done better in different ways. All students, even the learning-disabled students can successfully do with less structure provided their work has meaning to them.
Excerpts from detailed report, addressing the illustrative inquiry units.
I. Tessellation.
This activity is the one likely to be least familiar to both teachers and
students. It is also the one because of initial unfamiliarity) that teachers
are likely to be most dubious about introducing in class, or convincing
parents of its value. Therefore, it is the activity in which teachers are
most like learners, and can re-experience and be resensitized to be learners
most completely. Therefore, it is important that it be the first unit (as
it was). This made the "classroom learning" aspect real for teachers,
an aspect which could then carry over to other units. Thus the "Walk
Down Memory Lane" activity provided, by my calculation, in experiential
recall from the previous activity, the teaching sequence:
(i) Homework and initial presentation (or definition resulting therefrom);
(ii) small group discussion and conclusions;
(iii) small group presentations to the large group;
(iv) alteration of conclusions in (i) on the basis of (ii) and (iii) and
using these to formulate a new categorization (definition, presentation);
(v) conjectures and questions raised by (iv);
(vi) Homework based on (v);
(vii) Discussion of (vi) including ideas of "proof";
(viii) further work on (v) and (vii) ("at home" or "in class").
(ix) inclusion of the assistance of "manipulatives" in "at
home," "in class, small group" and (possibly) "large
group" activities focused on (viii);
(x) discussion of conclusions from (viii) (and open questions(!))
Of course, some of these steps might be iterated before reaching (x).
While this is a "mathematics-teaching sequence," what is striking
about it is not only that it applies to all mathematics learners, but that,
with suitable modification of some of the nouns, might well be used in learning
in many different disciplinary areas.
As already indicated, the discussion of implementation is absolutely
essential, and can be carried over to administrators and parents who need
"convincing." In a properly conceived and implemented tessellation
unit, students learn not only about geometric matters such as length, area,
and angle, but also about such things as pattern recognition, inference,
conjecture, and verification; abilities which carry over not only to all
mathematical learning, but, in fact, to all learning.
For students to truly learn anything, they need to "own" it,
to "feel it internally," especially if it initially seems exotic.
(This is one of the reasons for the oft-made observation that the way eventually
to learn anything firmly is to teach it.) Thus, it is important for teachers
to "own" the material. The experience of "owning" (or,
alas, "not owning") new material is one which students in the
classroom (learning disabled or not) experience daily--the learning disabled
perhaps experience failure of "ownership" even more intensely;
therefore it is important for teachers to re-experience that feeling. Thus,
to my mind, it is important for pedagogical purposes, that the most "foreign"
unit, tessellation, be the first present in the teacher workshop; whichever
unit is eventually presented first in the middle school classroom. (In the
classroom, there may be a pedagogical argument for presenting the most previously
familiar unit first.)
The above also justifies the additional time spent on tessellation in
the summer institute.
II. Area.
The content of this unit is prima facie more familiar-until "the fish"
appears. That is, the idea of area is, of course, familiar, but finding
it by some of the methods necessary for the "the fish" less so.
Just as some of the teachers, many of the students in the classroom situation
will approach such a problem first by guessing approximate fractions of
squares inside "the fish," but some (one hopes--and, if not, guided
hints will help) will discover that there are better "indirect"
ways which will give an exact answer. Of course, these others depend on
knowing at least how to find the area of a triangle. Again, the small group--large
group exchange proves exceedingly useful here. A student working alone feels
isolated, and, if in difficulty, has no resources; a student in a large
group, and in difficulty, can get left behind. Thus the small group--large
group exchange as with the tessellation unit, is ideal for helping achieve
learning for the maximal number.
One idea I would mention when "the fish" and similar irregular
figures have been treated adequately is the finding of areas by the imposition
of a suitable grid (analogous at this level to the choosing of suitable
axes for a problem in analytic geometry). While this will not always lead
to an exact answer, it will help in approximately an exact answer more nearly.
One can also discuss the effects an approximation of imposing finer or coarser
grids (if students get to "own" this idea in middle school, they
will be familiar with it if and when such later they come across its significance
in integral calculus!). Also in discussing grids, one should mention how
the giant figures in the Peruvian plains were drawn (not by creatures from
outer space).
To make the transition from unfamiliar and complicated "unique"
figures like "the fish" to applying some of the same techniques
to classes of figures not previously considered like stars is neat. That
is, first the area of a figure which is an instance of the class is found
(simple after "the fish"); then another and another will the students
(in this case the teachers, but eventually their classes) are led to generalize
for themselves. This again establishes "ownership," and is far
more effective than a teacher designating a rote formula to be learned (and
far more important mathematically).
I am slightly dubious about using parallelograms initially as they are too
familiar and often incorrectly recalled. Do them (and trapezoids) after
the more difficult stars etc. and they will be trivial. Of course, for students
in the classroom, parallelograms should be early. This is not the only case
when the arrangement of activity for participants in the institute is likely
to be different from what they use in the classroom. For the less imaginative
teachers, perhaps it should be stressed that there is no fixed "best
order" to items. Incidentally, I can think of at least three different
ways in which the area of parallelograms can be derived from that of rectangles
and triangles--this is thus a good, simple, example on which students can
experiment and be led to their own solutions, while large group sharing
will emphasize that there may be more than one route to a correct solution.
Once again, the area unit provides the opportunity of making tacit mathematical
process manifest in context, without prescriptive statements, and in a way
which will give students appropriate "ownership" (cf. (i)-(x)
above). Thus we have, to be more summary than earlier, the sequence of:
defining a figure (if necessary); breaking it up into known figures; the
search for counterexamples and "playing" (what every professional
mathematician does with problems); the small group development of strategies
and then sharing of them; verification of conjectural formulas; justification
of the formula (which involves returning to the original figure-definition);
possible conjectural generalizations which starts the sequence all over
again.
It cannot be emphasized too much that understanding where a formula comes
from both helps in remembering the formula and aids its rederivation should
it be forgotten.
Derivations devised by students, in addition to providing "ownership,"
give a powerful feeling. In addition to showing the multiple ways to a formula,
they may show different forms of equivalent formulas, thus again demonstrating
that mathematics is anything but the cut-and-dried procedure of memorization
and instrumental application it is often made out to be.
The main point is the class, instead of becoming a vehicle of purely
dictatorial instruction, becomes a learning community in which everyone
is validated for making a contribution, and the wrong contributions are
led as much as possible to self-correction. If this last is not possible,
errors are ultimately corrected by the whole group acceptance of the discovered
answer rather than the imposition of an answer by the teacher. However,
this requires the teacher learning a more subtle teaching style.
The area unit (like the tessellation unit) is a good example of embedding
traditional material in a meaningful context which gives (novel(!)) purpose
to student work.
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