Enclosure 2 Constructions in Plane Geometry Revisited
Enclosure 3 Crisis in school mathematics due to social conditions for which Standards provide no remedy.
Enclosure 4 The NCTM's own Research Advisory Committee says that Standards recommendations lack research support. (July 1988, report evidently ignored by Board)
Enclosure 5 Kilpatrick's balanced statement on the "trade-off between proficiency and comprehension."
Enclosure 6 Comments on the NCTM's Standards on Assessment (contains oath, "I solemnly swear that I have used the procedures I am about to advocate under normal teaching conditions for a period of at least two years. Moreover, I used these procedures in all five of my mathematics classes.") Sent to the Leadership Conference on Assessment which preceded the ICTM annual October meeting in Springfield, Illinois.
Enclosure 7 My vita with addendum
Enclosure 8 Excerpt from the Report of the NCTM's Secondary School Curriculum Committee (1959) showing (a) statement on "deductive geometry" and (b) list of personnel.
Enclosure 9 Professor Deborah Tepper Haimo's retiring presidential address on the need to preserve proof as the distinguishing feature of mathematics.
Enclosure 10 Kennedy on "Dumbing Down American Students".
[Note: This enclosure could not be reproduced in text format due to the inclusion of symbols not part of the ASCII character set. The enclosure contains a list of the essential components of the language of mathematical exposition including: 1 Set language - Intersection, union, inclusions, proper inclusion, and set builder notation 2 Definitions and assumptions 3 Quantification - some, all, there exists, etc. 4 Conjunction, disjunction, and exclusive or 5 Contradictory statements 6 Implication, converse, inverse, contrapositive 7 Equivalence 8 Contradictions of conjunctions, disjunctions and implications 9 Distributive laws 10 Tautologies 11 Argument, premise 12 Proof 13 Inductive reasoning, conjectures, mathematical induction Five examples of flow proofs follow
Historical Perspective: Prevailing philosophy on the reasons for studying school mathematics and demonstrative geometry as indicated by the following quotation from Schultze, copyright 1912, from which your speaker studied as an undergraduate in 1928. "The study of geometry should be primarily a course in the solution of originals and general methods of attack. The regular textbook propositions should be treated as exercises, with this difference, that the facts stated by them should be remembered. Exercises, however, should be studied not in order to be remembered. Exercises, however, should be studied not in order to be remembered, but in order that the student may familiarize himself with geometric working methods, which will enable him to do other and more complex reasoning. The student’s ability and progress in the subject can be measured only by his ability to solve exercises that are original to him, and not by his ability to repeat well-known facts."
The construction problem:
Tools: Compasses and straight edge
Geometric Competence: Basic theorems, transformations including translations, rotations and
reflections (Schultze, p. 239-244).
Ten Basic Constructions (see below)
Objectives: To develop an appreciation of the aesthetic properties of geometry, as shown in
beautiful proofs of ingenious constructions. To encourage the spirit of inquiry as probing
questions lead to new and open-ended problems. To develop the student’s capacity to be
enthralled (as we were) with geometry for its own sake.
Ten Basic Constructions
1. A circle having a given center and a given radius ("Scribe and arc").
2. A perpendicular to a given line through (a) a point in the line, (b) a point outside the line.
3. A line through a given point parallel to a given line.
4. An angle congruent to a given angle.
5. The bisector of an angle.
6. The perpendicular bisector of a line segment.
7. The fourth proportion to three given line segments.
8. The mean proportion between two given line segments. [Stone-Millis p.209]
9. The tangent to a given circle through (a) a point on the circle (b) a point outside the circle
[Stone-Millis p.18]
10. Upon a given line segment as a chord construct an arc in which an inscribed angle shall equal
a given angle (Figure 8) [Stone-Millis, Benj J. Sanborn, Copyright 1916, p.l82]
Suggested format for Constructions Problems:
Examples to be considered as time allows.
Two Beautiful Quotes
"Blindness to the aesthetic elements in mathematics is widespread and can account for a feeling that mathematics is dry as dust, as exciting as a telephone book. Contrarywise, appreciation of this element makes the subject live in a wonderful manner and burn as no other creation of the human mind seems to do". Davis and Hersch, The Mathematical Experience, Houghton-Mifflin, 1981.
"For indeed what good do we seek -- Certainly it is not to introduce them (students) to a collection of more or less ingenious theorems about the bisectors of the angle of a triangle or the consequence of the prime numbers, but rather to teach them to order and link their thoughts according to the methods mathematicians use because we recognize in this exercise a way to develop a clear mind and excellent judgment. It is the essence of the mathematical method that ought to be the object of our teaching, the subject matter being only well-chosen illustrations of it." Jean A. E. Dieudonne [Note: The remainder of this enclosure contains references and notes on the flow proof format.]
The standards writers have failed to provide an analysis of existing conditions in order to show a need for these changes. While they refer briefly to "A Nation at Risk" they do not seem to realize that conditions deleterious to learning have intensified by at least one order of magnitude in the last 30 years. Consider that not all pupils are clear-eyed seekers of truth from Central Casting with unlimited time for data gathering and individual conferences. A very substantial number of our "students" come to school in no condition to learn. It is difficult to teach students who, due to social conditions outside of school, are bone-tired, terrified, drug sodden or absent. As Clarence Page says, "The public schools have become a dumping ground for the problems of the larger society." Maybe--just maybe--this has something to do with lower test scores.
Plans to reform our schools overlook that about half of our youngsters grow up in families that are not adequately instilling traits that are pedagogically essential. Frequent divorces, a bewildering rotation of significant others, and parents who come home from work exhausted both physically and mentally have left many homes with a tremendous parenting deficit. Instead of providing a stable home environment and the kind of close, loving supervision that character formation requires, many child-care arrangements simply ensure that children will stay out of harm's way. As a result, personality traits essential for the acquisition of math, English and various vocational skills are often lacking. Children come to school without self-discipline, and they cannot defer gratification. Nor can they concentrate on, or mobilize themselves for, the tasks at hand.*
But an influential group of our math educators, who are insulated from these unpleasant classroom realities, seem to be completely unaware of the appalling school conditions that are prevalent in many depressed areas of our country including our inner cities. They have seized upon the present emergency, that was largely caused by these conditions, to proclaim a need for far-reaching changes in teaching and curriculum that have little substantive support in either research or experience and are not relevant to the problems that beset us. (The Titanic has hit an iceberg and is sinking. Let's rearrange the deck chairs.) The suburban schools in affluent areas don't need these changes. Their math departments are staffed by an elite group of highly trained, highly skilled professionals. While some suburban schools are beginning to have their problems (the rise of gangs and the sentiment that only "geeks" do homework) they are already incredibly good insofar as curriculum and instruction is concerned. The problems confronting schools in our depressed areas are, for the most part, not amenable to the changes proposed in the Standards.
Since the Standard's writers have misapprehended the cause of the present crisis in school mathematics, how can we trust their solution?
Frank B. Allen
*From an article entitled, "First, educate the reformers" by the nationally syndicated writer Amitai Etzioni. (Chicago Tribune 7/9/91
In the section on "Next Steps" in the Standards (p. 251- ) we are told that our teachers and mathematics educators must now trace out the "coherent network of relationships (that) exists among the identified topics" in order to "develop curricula based on the Standards." We are assured that the "nodes" of this network have been identified. (What more could one ask?) We are also informed that new tests must be devised to "assess" these new curricula. In short, after all this fanfare, we have no curriculum for school mathematics and we have no tests defining mathematical competency. Nor do we have "a scope and sequence chart" or a "listing of topics by grade level". These mundane but rather essential items are pointedly omitted from the Standards (p. 252) which offer only "a framework for curriculum development". (What kind of charge was given to the Standards Committee by the board?) Those who must produce the new curricula and accompanying tests face a gargantuan task.
The Standard's writers did the easy part, namely produce a precis' of Math Education 405. Moreover, impartial review reveals that they really didn't do this easy part very well. Their framework and their sweeping, loudly proclaimed recommendations are largely baseless insofar as cited research is concerned. The following statements by the NCTM's own Research Advisory Committee support this view. They appeared in our "Journal for Research in Mathematics Education" for July, 1988. While they pertained to the draft version of the Standards, they apply with equal force to the final version. "The Standards document contains many recommendations, but in general it does not provide a research context for the recommendations even when such a context is available." The Committee asks "For which curricular and instructional recommendations made in the draft version of the Standards document does there exist substantial research support?" The committee continues "Of course, it is also important to consider research evidence that might refute any of the recommendations made in the document. For example, the research base needs to be identified and clarified both for curricular recommendations, such as delaying and decreased emphasis on fraction computation, and also for instructional recommendations, such as the use of calculators with students at all grades K-12, extensive utilization of cooperative learning groups, the importance of work with manipulative materials, and emphasis on student inquiry and investigation".
I submit that these same remarks would apply to many of their other recommendations, such as the integrated curriculum, suppression of oral exposition by the teacher and their grotesquely complicated assessment procedures which Paul Greenberg might describe as "One more clarion call for vague incoherence."
To summarize: Most of the major recommendations in the Standards have nothing to support them other than the consensus of the authors and the conventional wisdom harbored by some of our more vocal mathematics educators. How dare these writers propose sweeping changes including a complete restructuring of the school mathematics curriculum on such flimsy evidence?
Readers will find the Standards replete with statements like these:
To these readers I commend the following statement by Jeremy Kilpatrick, which is also found in the July 1988 issue of our Research Journal.
"One of the most venerable and vexing issues in mathematics education concerns the trade-off between proficiency and comprehension, between promoting the smooth performance of a mathematical procedure and developing an understanding of how and why that procedure works and what it means. The trade-off is obviously not either-or; rather as William Brownell pointed out over 30 years ago, some balance needs to be found between meaning and skill. Amid today's arguments that technology has modified, and sometimes supplanted, the skills students need, the issue has grown into not just achieving a balance but finding a balance point. The working draft of the NCTM's Curriculum and Evaluation Standards for School Mathematics argues forcefully for a de-emphasis in skill instruction and for a change in the apparently widespread view that proficiency needs to precede, and perhaps to dominate, comprehension and problem solving. Although researchers may agree with the draft position--and many undoubtedly do--they should not dismiss too lightly the questions of how and where skill development fits into the school mathematics curriculum. Recent research in cognitive science suggests that a strong knowledge base is needed for problem solving, and surely some of that base should be composed of procedural knowledge. Furthermore, conceptual knowledge both supports and is supported by what Brownell termed "meaningful habituation," the almost automatic performance of a routine that is based on understanding.
A neglected yet critical item both in implementing the NCTM standards and in gaining a better grasp of the role skill development plays in learning mathematics concerns the folk wisdom in today's school practice. Why is it that so many intelligent, well-trained, well-intentioned teachers put such a premium on developing students' skill in the routings of arithmetic and algebra despite decades of advice to the contrary from so-called experts? What is it the teachers know that the others do not? What we often forget when we look at classrooms is that they are a place in which teachers too develop mathematical meanings. Although teachers often teach as they have been taught. At least some of our research needs to take them seriously as informants on the wisdom of practice."
When we consider this thoughtful, balanced statement which applies to the final draft as much as to the working draft, we have to wonder if some of the recommendations handed down in the 9-12 Standards are perhaps a bit too sweeping--a bit too general. Certainly they require and deserve intensive, impartial scrutiny, to which, so far, they have not been subjected.
The ten assessment procedures recommended by the NCTM "Standards" (pp 193-237) are so subjective and make such heavy demands on the teachers' time that they are not practical for a teacher working under normal teaching conditions. "Normal teaching conditions" implies five math classes averaging about thirty, plus at least one extra-curricular assignment. Therefore, let every speaker at the forthcoming ICTM "Leadership Conference" place the left hand on a copy of Stone-Millis "Plane Geometry", (Benj. H. Sanborn and Company, 1915) or some other strong geometry text from the period before they were emasculated , and take the following oath: "I solemnly swear that I have used the procedures I am about to advocate under normal teaching conditions for a period of at least two years. Moreover, I used these procedures in all five of my mathematics classes."
If this decimates your speaking corps, as it will unless they resort to wholesale perjury, you can spend your time more profitably constructing examinations and tests which are valid measures of the students understanding of school mathematics. A teacher who can neither devise or select such tests should not be teaching mathematics.
Brief Comments about Certain Criteria
Alignment. We have sense enough to know that our tests should "measure the content of the curriculum" (p 193). Standards 4,5,6,7,8,9. We have always tried to ensure that our evaluation procedures (exams, tests, bids, etc.) measured mathematical power, ability to communicate (write clear explanations), reasoning, mastery of mathematical concepts, and proficiency with mathematical procedures. Standard 10. Mathematical Disposition. We have always sought to inculcate a positive attitude toward mathematics. However, it never occurred to us to use our perception of how well we had succeeded in this regard as a component of the students course grade. We thought that if the student had a favorable disposition toward mathematics this would cause him (generic) to study harder and would be reflected indirectly, but effectively, in his test grades.
It is irresponsible for the NCTM to advocate assessment procedures which are devoid of any validation under normal teaching conditions in either research or experience. Moreover, if it were widely adapted it would so drastically change our procedures and criteria for measuring student progress that all comparisons with past records would be invalid. It is something like what would happen to baseball records if each batter were given four strikes instead of three.
Lord Stockton once said, "The Liberals have once again come forward with many new and good ideas. Unfortunately, none of the good ideas is new and none of the new ideas is good". This analysis applies pretty well to the 9-12 Standards. The NCTM assessment plan is a new idea that is not good. Drop it.
Frank B. Allen
I wish to convey the idea that, since my retirement in 1979, I have maintained my interest in both school and undergraduate mathematics and have kept myself informed about recent developments in these fields.
I attend meetings of local math organizations with reasonable frequency and speak occasionally. I visit local math departments where I have seen the TI82 and "Geometer's Sketchpad" used in the classroom.
In 1995, I purchased an IBM-compatible computer. It has the EXL language for mathematical symbols and the latest version of Geometer's Sketchpad. I have gained access to the Internet with this letter to President Price.
[Note: The only the relevant parts of this document are included here]
"It is evident that this provision of a variety of courses is likely to produce some courses which do not meet the requirements of a college-preparatory sequence in mathematics. A course the presents only the facts of geometry and pays little or no attention to the essentials of deductive proof certainly does not meet these requirements. While such courses may provide valuable learning experiences for some pupils, they should not be described as deductive geometry. This description should be reserved for those courses which are designed to advance the pupils’ understanding of the nature of proof."
Among the Committee Members listed are:
In a period of great change in mathematical education, she continued, we need to be careful that valuable new approaches are not followed at the expense of equally valuable old ideas. Experimentation, conjecture, and problem-solving are all highly important, she agreed, but so too is the one thing that, among the sciences, is particularly unique to mathematics -- the notion of proof.
She regretted that an emphasis on proof is "all to often ignored in our educational process," reminding the audience that, "Problem solving is not complete until the results have been firmly established. Proofs are an integral part of mathematics and must not be overlooked."
"By emphasizing the importance of realizing that more is required in mathematics than mere experimentation and conjecture, and by expecting the ablest students to prove assertions and validate them, we can educate all to the full extent of their abilities. This will not only enrich their lives, but prepare them to take their place in, and contribute to, modern society, and assist the continuation of the discipline as a major force in the world."
In short, the schools are unabashedly following a political-social rather than an educational philosophy. "Because many educators," the Chronicle tells us, "are uncomfortable with the idea of an academic elite . . . high achieving students are less likely to be given special classes and enriched curricula than in the past. The result is that fewer people do exceptionally well on standardized tests."
These results suggest that a number of American educators are quite deliberately attempting to average out the country’s intellectual class, to fashion a lumpen intelligentsia so that the country will finally look like its schools in which everybody passes with a C grade.
This may be the first time in the history of education that policies seek to defeat rather than encourage the higher mental, psychological and moral aspirations of young people.
When teachers no longer think that it is important to bring out the best in students, morals and culture eventually follow the same downward arc as SAT scores.