As a guide to curriculum and pedagogy, I am impressed with the parts
of the Principles I read. However, without a second stage of detailed
curriculum materials and guides to implement them, the actual influence of
Principles in the classroom likely will be less than hoped.
[Mercedes McGowen]
There is a dilemma of how to teach a rich mathematics program which
requires time-intensive instructional methods and the incorporation of
reflective practices given the typical class time periods in most of our
schools. Does the document offer guidance to make hard choices about
content? Does it provide clear, unambiguous benchmarks in given content
areas?
[Ladnor Geissinger]
I have not yet read all of PSSM, but in several of the standards it seems
to lack the cohesiveness of a comprehensive view; the standards are there
as organizing principles but a sense of being guided by a fundamental
unifying theme is missing. A very good place to find one such theme is in
the talk by Hy Bass, "Algebra with Integrity and Reality", at the NCTM/MSEB
Algebra Conference in May 1997, and I wish that the PSSM writers would have
paid more attention to that in their scaffolding for school mathematics.
The presentation addressed the question of "what sense of the real numbers
is appropriate and useful for school math"; it was thoughtful and down to
earth, with wide-ranging implications. His approach is to take the point
of view that real numbers ARE the points on the geometric real number line,
which is a "primordial object of mathematical experience on which we can
build our early mathematical learning".
[Deborah Tepper Haimo]
The document should be substantially shortened if it is ever to be read.
There is far too much repetition as well as confusing, unnecessary
elaborations, and unneeded "puffery". Concise statements would tighten up
the document and make clearer what is meant. It should be a major, serious
task undertaken by others before the document becomes official.
I would like to see the document make it very clear that it deals with
MATHEMATICS and not any other subject. It should thus emphasize the
characteristics that distinguish mathematics from other disciplines. The
document points out the theoretical nature of the subject to some extent.
It stresses, however, mathematics' utilitarian role without stating clearly
that this is based partly on its very important abstract nature. With
various interpretations, for example, the same procedures can be used to
solve problems in seemingly totally unrelated areas.
[Henry Alder]
I could go on for several more pages listing some of the great concerns I
have about the Draft. Many of these have been covered by other reviews of
the Draft I have seen. I want to emphasize that I am not saying that the
Draft is a bad document, but rather, as again Steve Rodi says so very well,
it "would make a wonderful syllabus, indeed textbook for a course, in a
university mathematics teacher training program", but this is not what I
understood to be the goal of this document.
In my view - and those of a number of reviewers whose comments on the Draft
I have seen - this Draft falls far short of meeting the goal which I and
others have envisioned for it. For example, we had envisioned a document
that would contain a lot of helpful advice to teachers whereas this one
contains hardly any. It does not tell teachers what mathematics they should
cover at the particular grade level they teach, what mathematics they
currently teach should be omitted to allow for coverage of the additional
material recommended for coverage in the PSSM, to indicate what additional
mathematical content is needed for students going on into careers where a
lot of mathematics is needed, what should be done when calculus is covered
in high school, etc. I share the view which I have heard expressed that,
in anywhere near its present form, this Draft is likely to do more harm
than good.
CONCERNS
CONCERN #1
To whom is PSSM addressed?
[Henry Alder]
My most serious objection to the Draft is that it is not clear to whom it
is addressed, that is, what the intended audience is. I had always assumed
that it would be addressed to teachers of mathematics ands others involved
in designing school mathematics curricula. If that is the intention, then
the Draft provides essentially no guidance. If I were a sixth grade
teacher, I would want to know what mathematics I should cover in sixth
grade and NOT what mathematics should be covered altogether in sixth grade
and middle school without any indication how this material should be broken
up between sixth grade and middle school.
CONCERN #2
There is considerable disagreement on our task force as to
whether PSSM's use of bands is appropriate or whether the Standards should
be prepared grade-by-grade.
[Mercedes McGowen]
What level of expertise are students expected to attain by a given time?
What are students expected to "master" by a given time, and what are they
expected to have been introduced to, but not necessarily mastered by a
given time?
[Henry Alder]
The conclusion is clear: if this document is to be of any use to what I
assume to be the intended audience, it should give standards for EACH grade
K-12, rather than by bands. That this is possible and desirable is
demonstrated by what some states, for example, California, have recently
done.
Many of the recommendations made by our Task Force to NCTM in its various
reports have not been followed. Let me cite one example:
I quote from the Task Force report of January 1997, item 3 of the part on
"reservations and Concerns about the Standards":
"All (note: ALL) members of the Task Force want the revision of the
Standards to be more specific, with regard to both skills and expectations
for students' intellectual growth from one grade level to the next. Many
teachers, both those teaching from "reform" materials and those who
continue to use a more traditional approach, would like the Standards to
specify what skills are really needed for each level of mathematics".
This point is listed as item a) in the summary of the report among
"specific items that we wish to call attention to the NCTM Commission on
the Future of the Standards...." as follows:
"Specificity in terms of mathematical skills and understandings needed at
each grade level".
Instead of giving grade level recommendations, as recommended by our Task
force and so many other groups, all the Draft has done is make
recommendations for four bands instead of three in the previous NCTM
Standards.
[David Kullman, repeated from above]
Second, I like the overall structure of PSSM, with ten standards that are
common to all grade levels, elaborated for each of four different grade
bands (although I would have preferred only three grade bands.)
[Henry Pollak]
I think it is unrealistic to expect specific performance standards at each
grade level. We do not have a national curriculum, and the variability
across states and provinces is great, and for good reason. What SHOULD be
said, however, is that we expect each state or province to set and publish
grade-level goals and standards that fit their own more local
circumstances. It would be good for NCTM's "Principles and Standards"
document to say what it does NOT intend to be as well as what it does
intend to be. I agree with what is said on pages 13 (lines 36 ff), 27
(lines 12-18) and 37 (lines 24-28).
CONCERN #3
There is general concern that PSSM only addresses issues that
pertain to all students.
[David Kullman]
These standards are intended for ALL students. While this approach has
merit, who will set the standards (particularly in the 9-12 grade band) for
those students who will eventually become significant users of
mathematics?
[Alan Tucker]
It is important that calculus in high school be addressed. There are key
issues here, e.g., students in AP calculus classes should be required to
take the AP exam, rather than treat the course as a warm up for calculus in
college.
[Kenneth Ross]
I found the "all students should" preamble most unbelievable at the Grade
9-12 level in Standard 5, Data Analysis, Statistics and Probability (pages
306-311). With this accomplished, ALL students (NOT JUST COLLEGE-BOUND
STUDENTS) would be ready to skip our introductory college course in
statistics for business majors!
[Mercedes McGowen]
There is a dilemma with "math for all" and "math for our best and
brightest." Does PSSM discuss how to meet the differentiated needs of
various groups of students, including the needs of those who will
eventually take advanced mathematics courses in college, as well as the
overwhelming majority of students who do not? Does the revision address the
issues of what skills, when and for whom, as well as what level of
understanding, when and by whom?
[Stephen Rodi]
As I read PSSM, the repeated phrase "...all students should..." eventually
impressed itself upon me. My first instinct was to ask "What is the
contrast here?" Where is the section that says "...some students
should..." or "...students of this or that kind should..." That sent me
searching outside my assigned reading range to eventually find a relevant
comment on page 17, lines 11 - 18. (I also read pages 23-27.) On page 17
it is pointed out that the original Curriculum and Evaluation Standards
"has been interpreted by some as neglecting the needs of students who have
deep interest in pursuing mathematical and scientific careers." This
paragraph on page 17 responds that "this draft attempts to elaborate more
fully the issues and classroom-specific possibilities for meeting the needs
of students with diverse experiences and interests, including those who are
especially inclined toward continued study of mathematics."
Let me add that Principles also should offer direction for instruction of
students who show more aptitude than usual in mathematics and the ability
to absorb more mathematics and more sophisticated mathematics faster, not
just those who eventually might study more mathematics. The underlying
issue here is broader than just students who "might be inclined toward
continued study of mathematics."
In my view the "attempt" to include such diversity in this version of
Principles fundamentally has failed. I read very little in the draft which
differentiates according to student need, ability, or interest. While some
small elements of such differentiation might be "buried" here and there in
particular examples or supplementary comments, those are in essence
non-existent in comparison to the constantly repeated heading "...all
students should..." If one is serious about presenting guides for
diversification (as page 17 promises), the diversification should get the
same layout emphasis and sequential development as the norms for "all
students." Without such equity of presentation, the new Principles likely
will fall subject to the same criticism the old Standards faced on page
17.
CONCERN #4
There was concern about the material proposed for Grades 6-8. In
particular, many felt that algebra I is a suitable course for the 8th
grade, though Rodi provides another point of view.
[Alan Tucker]
A point that I would like to see made at the start of Chapter 6 (grades
6-8) is the importance of moving forward and not repeating material from
grades 3-5.
[Henry Alder]
The coverage of the needed mathematical content is made even more difficult
in the Draft by not requiring Algebra until grade 9. The reason why this is
a big mistake has already been commented on by others, so I will not dwell
on this. I would like to note, however, that starting Algebra I in grade 9
leaves only four years to cover the content of Algebra I, Geometry, Algebra
II, and Mathematical Analysis or Pre-Calculus (consisting of trigonometry,
analytic geometry, etc.) whether covered as separate courses or integrated
ones, and thus no time for covering additional topics, such as statistics
and probability. The Draft is completely silent on how all this is to be
done, thereby leaving it completely to the teacher who may not realize that
what the Draft advocates is impossible to implement in an adequate
fashion.
[Stephen Rodi]
In Grades 6-8, the presentation of measurement had a very strong geometric
flavor, which I thought appropriate. As I read this material, the
following thought struck me. One way of resolving the dilemma "Should
algebra be studied in 8th grade?" would be to start the formal study of
algebra in 9th grade but to substantially beef up the study of geometry in
Grades 6-8 with some deductive geometry and a little informal algebra
embedded. This might prepare students better to deal with the abstract
symbolism of algebra which seems to cause so many students trouble these
days. In fact, from what I could tell, Principles was recommending a
course similar to this.
[Mercedes McGowen]
Once again, the PSSM recommendations explicitly challenge the traditional
practices of when and how symbolic notation is introduced. They also raise
the question about what the 9th grade formal algebra course should consist
of. What is envisioned as content for middle grades students is much of
the foundation many students lack AFTER completing two years of high school
algebra as currently taught.
[Stephen Rodi]
In connection with Standard 5, I very much like the statement in line 28,
page 237: "Students' work with data analysis and statistics in grades 6 -8
draws on and integrates their knowledge of ratios, fractions, decimals,
percent, graphs, and measurement." This sort of integration, in my
opinion, is exactly what should be occurring in the middle school grades,
rather than a rush to algebra for all students. As I have said above, I
think geometry should be the other main support of the middle school
years.
CONCERN #5
There was concern as to what material will be replaced by the
substantial addition of statistical topics.
[Henry Alder]
Standard 5 (Data Analysis, Statistics, and Probability) spread over grades
K-12 includes the material that is typically covered in a semester course
in probability and statistics at either the high school or beginning
college level. I know this since I once co-authored a textbook on this
material. If this material is to be presented to all students, then a
semester's worth of other mathematics content has to be eliminated from the
mathematics curriculum. There is no indication whatsoever given in the
Draft how this is to be done. Let me make clear that I have no objection -
in fact, would greatly welcome - high schools offering such a course in
probability and statistics for all those interested in and properly
prepared to take it, but to REQUIRE this material for ALL students in lieu
of some other (unspecified) mathematical content makes no sense at all.
[Stephen Rodi]
In general, the sections on data analysis, statistics, and probability
were thorough and well-outlined. However, I was left with one major
question, perhaps to be addressed in another document. This Standard
offered a substantial amount of material over the course of Pre-K through
Grade 12, outlined no doubt by professional statisticians who want to see
proper treatment of their important subject matter.
However, it seemed to me that if all ten Standards were as thoroughly
outlined for all grade levels as this one and its predecessor on
measurement were, some choices will have to be made on content emphasis,
and even inclusion, in the curriculum if all students are to cover all this
material, as well as all the material in all the other Standards. In
addition, mathematics will not be the only subject competing for curriculum
time. Finally, more time-intensive instructional methods, like group work
and projects involving data collection, will reduce the scope of what can
be covered even in mathematics classes themselves.
I did not find in Principles any guidance on how these important coverage
choices should be made. This will be a matter of some importance in the
final implementation of Principles in curricula. It is useful to
"philosophize" about what should be, as the Principles document does. But
the Devil is in the Details of pragmatic plans on how to make philosophy
happen.
[Alan Tucker]
There is one very important forward-looking aspect of these Standards that
provides an INTELLECTUAL CHALLENGE which may trouble some, but must be
faced. This document has two mathematical goals that compete with the
traditional goal of school mathematics as a preparation for physical
science mathematics (e.g., calculus). These other goals are development of
statistical reasoning and development of discrete mathematics skills and
reasoning. The document does an impressive job of weaving discrete
mathematics needs into the traditional curriculum. Statistics seems to
stand alone a bit more. The challenge is how to do more when the current
single thrust is not done well. I don't have an answer for that, but the
reality is that in twenty years, enrollments of schools of computer
computer/information systems will likely outnumber those in schools of
engineering. In most of modern computing, calculus is of very little
importance. Statistics likewise is used by many more people than calculus.
At the collegiate level, discrete mathematics and calculus go their
separate ways. In school mathematics, they can and should be tied
together. I think we collegiate mathematicians need to give a lot of more
thought to this issue and play a major role in resolving this challenge in
school mathematics.
CONCERN #6
There is still concern that mastery of skills is not addressed
sufficiently.
[Alan Tucker]
I hope that we can agree to urge the Standards writers to aggressively
support the role of drill in school mathematics instruction -- as PART (not
but ALL) of the school mathematics experience. The current document seems
to be almost trying to "pick a fight" with traditionalists by downplaying
drill. I strongly feel that the Standards should try to accommodate the
views of as broad a spectrum of concerned stakeholders as possible.
Mastery of skills is aggressively downplayed, especially in Chapter 2 on
Principles (discussed below). Learning arithmetic facts and practice drill
is discussed but always in a negative light. e.g., in connection with the
Learning Principle, the following is stated on page 35, line 1: "A focus on
learning mathematics with understanding should not be taken to mean that
there is no place in school mathematics for the learning of routine
procedures." The word "skills" is virtually absent from the document.
As friends we should strongly urge the PSSM's authors to make strong
affirmative statements about the importance of mastery of skills. The
Standards are dead on arrival if they do not acknowledge the importance of
arithmetic and algebraic skills and the deep concern that a significant
portion of the population has about this issue. Everyone knows that skills
are important. Because some may overemphasize skills is no excuse for NCTM
to reactively downplay them.
CONCERN #7
The most important issue that hasn't been sufficiently
addressed concerns teacher education of the future.
[Mercedes McGowen]
What mathematics would an elementary teacher need to know in order to
competently teach the curriculum as envisioned in the Standards? Many of
those electing to become elementary teachers have weak mathematics
backgrounds characterized by instrumental learning and procedural
knowledge, with little or no conceptual understanding of the mathematics
they have learned. State certification requirements, while they have been
strengthened in the past few years in some states, are woefully inadequate
in many other states.
Recruitment of mathematically able students to become elementary teachers
is of major concern, as is the mathematical background of many teachers
currently in our classrooms and of those intending to become teachers.
Even students who graduate from high school with three or four years of
mathematics find themselves placed into remedial courses in college upon
enrollment. The need for warm bodies in classrooms in the next decade will
exert enormous pressures to turn out more, not fewer teachers, and the
number of those we would want to see teaching mathematics is nowhere near
what will be needed. This past fall at Harper College, 73% of students
enrolled in the pre-service content course for elementary teachers scored
less that 60% on an arithmetic competency test at the beginning of the
course. More frightening was the fact that nearly three-fourths of the
students who were in the content course began their college program
enrolled in a REMEDIAL mathematics course, with many of them having taken
TWO remedial mathematics courses prior to enrolling in the pre-service
course.
Will the mathematics skills and understandings these students acquire in
one 16-week course, or even in two 16-week courses, be sufficient
preparation (along with their methods courses) to teach the program of
mathematics envisioned by these standards?
I'm certain there are other assumptions stated or implied throughout the
document that need examination and discussion. It was, to some extent, the
lack of discussion of issues such as these which has led to some of the
misinterpretations and misunderstandings about the 1989 Standards. I would
hope we would direct our attention to the beliefs and assumptions on which
the recommendations contained in the document are based and identify those
we are in agreement with, those we have reservations about, and those we
have strong disagreement with.
[Stephen Rodi]
The breadth, depth, and specialization of the Grade 9-12 material reminds
me that a vast part of the secondary school teacher corps will be unable to
deal with this material in an insightful way, if it is dispersed throughout
four years in an integrated mathematics curriculum. Will this stark
reality in effect move these topics into separate courses taught by
specialists and effectively undermine the goal of all students seeing these
topics? If not, how will the ability of the teacher corps to teach these
topics be managed?
[Susanna Epp]
My other main concern, repeated lo these many times, is that in order to be
successfully implemented in the classroom, many of the suggestions in PSSM
require a level of teacher knowledge, insight, and ability to reason
logically that is simply not present in the general population of K-12
teachers. Nor does there seem any prospect for significant improvement for
many years. One partial countermeasure is to make sure that curricular
materials are clear and complete enough to serve the needs of teachers as
well as students.
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