The Beleaguered Mathematics Teacher
Presented November 15-16, 1997
A Talk Presented by UCSMP Director Zalman Usiskin at the Thirteenth Annual UCSMP Secondary Conference, November 15-16, 1997. This article was published in UCSMP Newsletter No. 22: Winter 1997-98.
Mathematics education today is far more complicated than it has ever been in the past. When the President of the United States has a mathematics test as one of his agenda items, there is little question that what we teach is viewed by those outside our profession as important. Along with this increased visibility comes added responsibility and increased conflicting demands from our varied constituents (students, parents, colleagues, administrators, politicians). The end result of all this is what I would term the beleaguerment of the mathematics teacher.
Teaching is a Hard Profession
All teachers are beleaguered, every day. It comes with the job. Parents have every right to demand the best education for their children. School boards and the administrators they hire are under pressure to see that their schools, for which they have legal responsibility, provide that education. Students have the right not to be bored, to be given appropriate content, to be told why they are studying what they are studying, to be graded and otherwise treated fairly. In the midst of this, teachers are the critical pivot point, carrying both the freedom and the duty to deliver to these diverse interest groups a wise program of curriculum and testing day in and day out.
The difficult job of the teacher is often made more difficult by individuals within schools who lack understanding and respect for the teacher's job. In too many schools, announcements come over the P.A. at random times within class periods, students enter late and leave early for reasons that may or may not be valid, free time that teachers might use for planning or student help is instead taken up by monitoring doors or halls or lunch rooms.
Teachers survive because of the light bulbs that they see turn on when a student finally gets something, the delight that students have in learning, which sometimes they even show you, and the plaudits that one occasionally gets from colleagues or from graduates for work well done. Those ups far outweigh the downs.
The beleaguerment I wish to talk about today does not come from these traditional sources that plague all teachers. It comes from within the mathematics education profession and from other well-meaning people with the authority to affect what is taught to our students. Mathematics teachers today are being bombarded with contradictory advice or with conflicting suggestions regarding what and how to teach, how to deal with students, how to test, and on what criteria to judge their success. I wish to identify some of these difficulties and recommend some ways to deal with them.
Conflicting Views Regarding What to Teach
The decision regarding what to teach appears to be easy: either follow the NCTM Standards or don't. The fact of the matter is that it's not so easy.
In late September I was at two meetings back to back in which diametrically opposed views were both seen as following the Standards. The first group consisted of people from the National Science Foundation Curriculum Development Projects. The second group was the Mathematics and Science Advisory Committee to the National Assessment of Educational Progress.
Many in the Curriculum Development Projects of NSF view eighth grade algebra as a bad idea, because it leads to tracking and because it violates the notion that mathematics is an integrated subject. On the other hand, the National Assessment Committee looked upon the increase in eighth grade algebra enrollments as a major advance in school mathematics. They had data from 1986 and 1996. I have also added earlier corresponding data from the Second International Mathematics Study of 1981.
Percents of 8th graders studying various types of mathematics curricula
% of classes
|NAEP - 1986% of students||16||19||61||5|
|NAEP - 1996
% of students
The "typical course" in 1981 and the "regular mathematics" course in 1986 and 1996 are each devoted primarily to arithmetic. Notice the remarkable increase in the percentage of eighth graders studying a course that goes beyond that. In the 1981 Second International Mathematics Study, the content of 24% of mathematics classes was either an enriched eighth grade course or algebra; in the 1996 NAEP study, 56% of students had such a course.
The NSF group also interpreted corresponding increases in mathematics enrollments in geometry, second-year algebra, precalculus, and calculus as advances. But the NSF projects deal with integrated curricula, and in 1995-96, integrated mathematics as is done in these projects or in the state of New York, was used in too few schools to make a sizeable impact on the National Assessment data. So the NSF projects tend to ignore this information.
The Standards for grades 9-12 do not support the same curriculum for all students. For example, consider the standards on algebra, reasoning, geometry, and functions. These standards clearly distinguish between what are called "college-intending students" and others. College-intending students are expected in their study of algebra to "use matrices to solve linear systems, and demonstrate technical facility with algebraic transformations." In the reasoning standard, college-intending students are expected to "construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction." In their study of geometry, they are expected to develop an understanding of an axiomatic system through investigating and comparing various geometries." With functions, they are expected to "understand operations on, and the general properties and behavior of classes of functions."
Yet, there are some who believe that these differences violate the Standards. The "mathematics for all" slogan has been viewed by some as the same mathematics for all, and at the same time. There is a tendency for people to say that what they believe – regardless of what it is – agrees with the Standards, while what they oppose does not. None of us is immune from this disease.
There are also standards dealing with approaching geometry algebraically, as well as with approaching trigonometry, probability, statistics, discrete mathematics, the conceptual underpinnings of calculus, and mathematical structure. There are at least five years of content here, and this does not include either AP calculus or AP statistics. For all students who do not enter high school having finished both a year of algebra and a year of geometry, the pressure on senior high school teachers to churn out content is enormous, and it forces major decisions regarding what not to teach.
Frankly, the Standards writers were wise in writing Standards that could be interpreted in a variety of ways. But this causes major problems when one wants to implement any sort of Standards. Regardless of how you wish to implement the Standards, there will be those who tell you that you are not implementing them. These curricular difficulties cause great beleaguerment for the teacher.
Does Anyone in the U.S. Teach Well?
How are we doing with respect to instruction? In a TIMSS study of eighth grade classrooms in the U.S., Japan, and Germany, researchers videotaped 81 classes from the U.S., 50 from Japan, and 100 from Germany. They made transcripts of 30 tapes from each country. Then summaries of the transcripts were made and doctored so that you could not determine the lesson's country of origin. From examination of these summaries of transcripts, other researchers distinguished the percent of those concepts that were developed from those concepts that were only stated. Here is one of the things they found. Only 22% of the topics in U.S. were judged as containing concepts that were developed. This is in contrast to 83% of the topics in the 30 Japanese classes and 77% of the topics in the 30 classes from Germany.
The TIMSS researchers found very few classes in the United States that seemed to do anything substantial. The average level of the mathematics taught in U.S. eighth grade classes was evaluated to be at grade 7.4, while the Japanese classes were at 9.1, and the classes from Germany were at grade 8.7.
Finally, from these summaries the researchers judged whether the content of the lessons was of low, medium, or high quality. The criterion of quality was as follows: On the basis of the content in the summary, would what was presented increase a student's understanding of mathematics?" [Alfred Manaster, phone conversation, November 11, 1997.] The result was as follows none of the U.S. lessons had content judged to be of high quality. [Source: Nanette Seago et al., Moderator's Guide to Eighth-Grade Mathematics Lessons: United States, Japan, and Germany. Washington, DC: U.S. Department of Education, 1997.] In contrast, 30% of the Japanese lessons and 23% of the lessons from Germany had content of high quality.
You can obtain videotapes of 6 classes that we are told are representative of classes in these countries, a lesson with algebra content and a lesson with geometry content from each country. The low-level nature of the curriculum in the U.S. is illustrated by the tape of the U.S. eighth grade class studying geometry, where an entire class period is spent finding measures of vertical, supplementary, and complementary angles when one of the pair's measures is given.
Interestingly enough, when the videotapes of the U.S. algebra class and the Japanese algebra class were previewed at an NCTM Board meeting, some of us thought that the U.S. class, with its cooperative learning and active student engagement and with the teacher going around the classroom helping students as they needed it, was far more in line with the Standards than the Japanese classroom, which had the teacher leading a class discussion and calling on students one by one. The TIMSS researchers seemed to consider the Japanese class rich because it discussed one problem in great depth for almost the entire period. Perhaps the TIMSS researchers viewed the U.S. class as shallow because many problems were being discussed, some teacher-invented, others from homework; and often it seemed that due to the cooperative groups, more than one problem at a time was being discussed. So many ideas were mentioned but relatively few were covered in depth.
Was the content really low level? Here is the warm-up from the algebra class on the videotape:
- What is the largest integer n for which 2n > n! ?
- Find the number of cubic inches in the volume of a rectangular solid if the side, front, and bottom faces have areas of 12 in2, 8 in2, and 6 in2. (Hint: Draw a picture.)
- Find an ordered pair of integers (a,b), a > b, such that ab + ba = 100.
- What is the quotient when 6x2a+b–c is divided by 2xa+2b+3c?
You and I might not agree that these are all good problems. But it is hard to believe that students would not increase their understanding of some mathematics if they did these problems. This was clearly a class of high-achieving eighth grade algebra students, and it seems that the teacher expected quite a bit out of them.
From the summary of the class that is shown in the published Moderator's Guide to the videotapes, it seems that the evaluators, who worked not from the tapes but from transcripts purposely doctored to cut out references indicating the country, were not aware of the cooperative nature of the class. It also seems that the analysis was not sensitive to the greater importance that homework has in U.S. classrooms.
In a conversation with Alfred Manaster, one of the evaluators of the transcripts, he pointed out that an important element the evaluators were looking for was mathematical closure. Teachers in the U.S. tended not to summarize, not to state what was important about a lesson or part of a lesson. Summary is important, and it is often something particularly lacking when students are working on their own or in groups. However you organize your class, it is wise to bring the class together at the end of the period to focus the attention of your students on what was important to learn during that period.
Regardless of the reasons behind it, this result of TIMSS, that no U.S. classes had content of high quality, has been widely cited. It carries with it an incredible teacher-bashing. It suggests that no one is teaching well anywhere in the U.S., while large numbers of teachers are teaching well elsewhere.
There is a different way to interpret the TIMSS teaching data that suggests that U.S. teaching is not that bad. Recall that the content of Germany's eighth grade classes was at the 8.7 grade level on average, and ours was at the 7.4 grade level. Despite being over one grade ahead in content, the average TIMSS score of students from Germany was 509 and from the U.S. was 500. Not only is the difference between 500 and 509 a statistically insignificant difference, but the German schools excluded 11.3% of their students from participation, a quite high non-participation rate, whereas the U.S. schools excluded 2.1% of their students, so one wonders if the U.S. would not have scored at least as high had it the same non-participation rate. It suggests that our students know what they have learned as well as students from Germany who have studied the equivalent of an extra year's worth of mathematics. We can only believe that our students would even be better a year later.
This interpretation is backed up by the results of the First in the World Consortium, that group of school districts outside of Chicago that took the TIMSS test as if they were a country and whose eighth grade scores were exceeded only by those of the eighth graders from Singapore. These school districts felt that one reason for their success was that over half of their students take algebra in eighth grade. Still, on the average, the students in this consortium had not studied as much mathematics as their counterparts in Japan. Yet they scored as high. We could conclude that the teaching in the U.S. is more effective.
For these reasons, I believe that the problem with our performance is not with teaching but with curriculum. Since about one quarter of eighth graders now take algebra, which is approximately twice as many as did so in the early 1980s, about three quarters of students in the country do not. Furthermore, the texts used by many middle-school students within the latter group, though given titles which suggest a major break with the past, are simply the warmed-over sixth to eighth grade basal texts of a decade ago.
Prepare Your Students for All Kinds of Tests
Another reason for teachers' beleaguerment is the increasing number and importance of external tests. The grade most affected by these tests is eighth grade. Many states give external tests in eighth grade. And high schools get bothered when their ninth graders are not able to do all the things they would like them to do, so often tests are given so students can be placed in appropriate mathematics classes in ninth grade. Other major activities eat into the school year: graduation takes up at least two or three weeks at the end of the year, and in many schools the eighth graders take a trip to Washington or some other place which also takes a week or two. It is easy for a quarter of the year to be taken up by these sorts of activities. Thus, in many places, there is more pressure on eighth grade mathematics teachers than on any other teachers in the system. There will be even more pressure if the National Voluntary Mathematics Test becomes a reality. For this reason, the President might be well-advised to change the grade level of his recommended test from eighth grade to sixth or seventh grade.
External tests at the high school level tend to have more significant implications. In some school districts and some states, there are mathematics tests for graduation from high school. These particularly affect students who have had trouble with mathematics. Because they tend to cover applied arithmetic and very little algebra or geometry, they conflict with many aspects of a richer curriculum.
College-entrance tests—that is, the SAT and ACT—do not cover the same curriculum as the multitude of college-placement tests. The SATs and ACTs fortunately allow calculators, which makes them in alignment with all of the recent recommendations; however, the college-placement tests often disallow technology. College courses are mixed: a college freshman who is taking one of the newer calculus courses will likely be allowed to use calculators both on homework and on exams, but on the same campus the traditional calculus course will likely not allow any calculators on a test.
What is a teacher to do? Everyone knows that you can raise scores on a test by having students take sample tests and going over them carefully. Teachers are already spending weeks preparing students just for tests; additional tests can only take more time away from the development of concepts on which the TIMSS researchers say we are already too weak. If you choose to spend your time on problem solving rather than factoring polynomials, and if you make extensive use of graphing calculators in your study of functions, your students' SAT scores are likely to increase but their scores on college placement tests may drop. We at UCSMP developed both Functions, Statistics, and Trigonometry and Precalculus and Discrete Mathematics precisely because we felt that two years were needed after Advanced Algebra to cover all the objectives of the various tests a student is likely to encounter ?after ? (Zal-do you want "during" or "in the course of his/her high school experience" here?) high school.
Bombardment by Those at the Helm
The TIMSS report, "A Splintered Vision", begins with the widely-quoted statement, "There is no one at the helm of U.S. mathematics and science education." What is a helm? Here is what is found in a popular dictionary:
helm n. 1: a lever or wheel controlling the rudder of a ship for steering; broadly, the entire apparatus for steering a ship 2: a position of control
No one at the helm? With either meaning it's just the opposite. The helm is crowded with people! Everyone seems to want to steer the ship and to have a piece of the control of the classroom. At the national level, there is the President and there is the National Science Foundation. Of course, there is NCTM. And there is also the College Board and the ACT people. They want a say in what goes on because they have built huge corporations revolving around the testing of students, and they want their tests used. The commercial publishers are still another national influence.
In the twenty-two state adoption states in this country, there has always been control over the textbooks available to be purchased. Now, in many places, there are also state tests, and these tests are not always tied to the textbooks. What is a teacher supposed to do then?
Whatever you do, you have to teach something. So, if you are just a typical good teacher, you are faced with the following dilemma: You want to follow the NCTM Curriculum Standards, but you are receiving mixed messages as to what that means. While some of those messages match the books you are using, perhaps none of those messages matches the tests that are being given in your state. You want to follow the Assessment Standards, but they don't match the high-stakes ACTs or the SATs your students will take. You know that technology makes everyone's job easier, and you want to allow calculators on all your tests. You are helped in this by the ACTs and the SATs, but you still hear about college placement tests that don't allow calculators. You want to follow the Professional Teaching Standards by having your students engage in projects, having in-depth class discussions about mathematical ideas, and working in groups, but there is no time, and besides, the colleges don't care about this and almost no one tests on it. And when you decide to do some innovative projects, some parent may object that their child is not doing mathematics or that they do not understand the mathematics their child is doing, which means it must be bad.
It is not that no one is at the helm. Too many people are at the helm. There are so many at the helm that they are rocking the boat and many teachers, who are the passengers, are seasick.
The Danger of Beleaguerment
We have all been in situations where we are bombarded from all sides for one reason or another. It may be in a class we are teaching, when we are bombarded with questions. The usual response is to call for silence, to calm everyone down. When we are beleaguered we naturally retreat to ponder how to deal with the situation. We do nothing or very little and wait for the bedlam to work itself out.
The problem with this is that teachers and students are the pawns in this situation, and waiting does no one any good. Every student deserves an education that prepares him or her for the future, not the past, that utilizes all of his or her powers and makes him or her feel good while doing it, and that presents an accurate picture of mathematics in all its breadth.
Textbook publishers are no different from teachers in this regard. Many of them are also sitting around and waiting. They have put a nice amount of data and some applications in almost all their books, but do they test on these things? Except for statistics, these newer ideas are often window dressing on an old, cracked, and cloudy window.
But textbook publishers are in a worse position than teachers. For financial reasons, they must create books that satisfy a plethora of interpretations of what should be in the curriculum. Whether they are traditional or trail-breaking, in order to be used throughout the country in the large majority of classrooms, they need to contain enough content for people who disagree with each other to agree on the book. And so each book contains a huge amount of content. As a result, textbooks themselves become an additional source of pressure on mathematics teachers.
Teachers who use UCSMP texts are not immune from these pressures. Our books are used by all sorts of students, from the very well-prepared and conscientious to the ill-prepared and lazy. In some schools, all the teachers finish the books, while in other schools, teachers wonder how anyone could finish them. But all teachers, even those who finish our books, realize they could do more. There are nice technology activities or manipulative activities or lesson masters or any number of suggestions in the teachers' editions that they wish they had time to do. One of the purposes of our conferences is for teachers to meet with and hear from other teachers how they have sorted out what to do and what not to do. We want to avoid the danger that you will see more than you can do and then not do anything.
I have organized this talk around the three areas of the NCTM Standards. Five years ago at this conference, the title of my talk was "What Changes Should Be Made for the Second Edition of the NCTM Standards?" At the time of the talk, there were no definite plans for a second edition of the Standards, but now a revision is supposed to appear in draft form next fall with the final form in the spring of the year 2000. I still believe in the proposals I made then, but events since 1992 lead me to five recommendations about what should be done with curriculum, teaching, and testing. I begin with curriculum.
Recommendation 1. Stay on the path set by the Curriculum Standards
The current movement in mathematics education began either in 1975 with the NACOME report, in 1983 with A Nation At Risk, or in 1989 with the NCTM Standards, depending on how you want to look at things. It was, and continues to be, fueled by three major goals: to improve student performance; to update the curriculum; and to increase the number of students who take mathematics beyond algebra and geometry.
On all three fronts, the national indicators are positive as can be seen in the following graph of a long-term study of National Assessment scores from 1973 to 1996: Show graph Where is this?. They hit their lowest points in the 1970s, during the back-to-basics era to which some people with short memories want us to return, but show slow but steady increases throughout. A significant fact about this trend is that the longitudinal study is based on the 1973 procedures and the questions have been unchanged even in wording since 1986. The long-term assessment is different from the National Assessments which are now being administered at grades 4, 8, and 12. The long-term assessment has to be considered as an assessment on traditional mathematics.
Another measure of performance is the SATs. Here are the SAT average scores for college-bound seniors for the past ten years, on the re-centered scale:
|1975||< 500||> 509|
|1980||< 500||> 509|
(Source: The World Almanac and Book of Facts, 1997, p. 255)
Because of the recentering of the scores, I cannot easily compare the 1997 scores with 1975, but they are higher, because there was a slight improvement in mathematics mean scores from 1975 to 1986. Scores were constant until 1992. Then improvement begins to occur and is steady and significant. Notice that the improvement in mathematics has come despite a general long-term steadiness or perhaps slight decline in verbal scores.
Average mathematics scores on the ACT peaked during the time of the new math and only in the 1990s have the scores returned to those levels. The gains since 1985 are similar to the SAT increases but are from a different perspective. Although both the ACT and SAT are college-entrance exams, the ACT is more closely built around traditional courses and content than the SAT. Together the ACT and SAT trends establish rather convincingly that student performance has improved significantly in recent years.
(Source: The World Almanac and Book of Facts, 1997, p. 255)
The curriculum is being updated, though perhaps not at the rate some of us would like. But TIMMS results tell us that in every advanced industrialized country of the world 97% or more of eighth grade students (98% in the U.S.) have calculators at home. [Albert E. Beaton, Ina V.S. Mullis, Michael O. Martin, Eugenio J. Gonzales, Dana L. Kelly, and Teresa A. Smith. Mathematics Achievement in the Middle School Years: IEA's Third International Mathematics and Science Study. Boston: Center for the Study of Testing, Evaluation, and Educational Policy, 1996.] Only 10% of U.S. eighth grade teachers report never or hardly ever using them, while 44% report using them almost always. Real data appear in all the newer books, whether traditional or not, and the required use of graphing calculators in more and more high school courses is a significant trend that is likely to continue.
Finally, towards the goal of increasing opportunity for students, enrollments in high school courses are up. Here is National Assessment data from 1978 and 1996 about the highest level of mathematics taken by our seventeen year-olds. Since some seventeen year-olds are juniors, this presents a conservative view of the highest level of mathematics taken by high school graduates. I have also indicated cumulative percentages. This tells us the percentage of students who have taken the equivalent of each course. In this I assume that all those who have taken a second year of algebra have also taken a year of geometry. This enables the table to be interpreted more easily, though it may be off by a percent or two.
Highest course taken at age seventeen
|%||Cum %||%||Cum %|
|Precalculus or Calculus||6||5||13||13|
|Prealgebra or General Math||20||96||8||99|
|No data 4||100||1||100|
Source: NAEP 1996 Trends in Academic Progress
Furthermore, the gender gap has essentially been eliminated. It might even be argued that we need some programs to get more boys in advanced courses.
Highest course taken at age seventeen in 1996
|%||Cum %||%||Cum %|
|Precalculus or Calculus||13||13||13||13|
|Prealgebra or General Math||9||100||7||99|
Source: J.R. Campbell, K.E. Voelkl, and P.L. Donahue. NAEP 1996 Trends in Academic Progress. Washington, DC: National Center for Education Statistics, September 1997, p. 85.
These data, taken together, present a remarkable picture of achievement by the nation's mathematics teachers in the recent past. Scores are up. The curriculum is getting updated. And enrollments are up. If any sort of fundamental changes are made in the Standards, we may even inadvertently deviate from this successful path. Furthermore, changing course will only add to the beleaguerment of teachers. Stay on the path.
Recommendation 2: National standards should continue to specify the curriculum by grade bands, not by individual grade; states and large districts should follow suit
The purpose of this recommendation is to give the teachers in a school or small group of schools, influence over how they approach the curriculum. It is to ease that aspect of beleaguerment that occurs when one is powerless.
It is important to have an agreed-upon core curriculum because so many tests and educational materials are national and our population is increasingly mobile. States and large school districts are too large to specify curricula by grade level, because they contain a diverse population in which different communities have different expectations for their children. Because of this variety, states and large school districts should not even attempt to specify a curriculum by grade level.
Furthermore, by making such detailed decisions at the top, government officials take away the voice of teachers in deciding what is most appropriate for their students, and thus remove a major aspect of professional growth and responsibility. We at UCSMP, and many others across the nation engaged in curriculum reform, work on such reform not because we want to control mathematics education, but because we want to influence it. I cannot have any less faith in teachers than I expect them to have in their students. Given a choice of ways in which to operate, if teachers are well-informed, then they will, on the whole, make good choices. If we take this choice out of the hands of teachers, then we discourage professional growth and we encourage decisions about the lives of students to be made by people who do not know the particular students at all.
I think it is significant that UCSMP materials are most used in those school districts that can choose what they want to use from all materials available, and not from a selective state list.
A local community may have a diverse set of students, but it accommodates that diversity in a few schools in which the teachers know each other. The teachers can make curricular decisions based on their local needs. And it is important that they do so, because, within a school, the curriculum should be consistent. To be most effective, it must build carefully on previous years.
Recommendation 3: We need strong specific recommendations concerning what to teach in algebra, geometry, and statistics at each of the grade bands K-2, 3-5, and 6-8.
By this recommendation, I mean that we should not be content with the rhetoric of doing strands K-12, but be rather specific about what we expect to be introduced, studied, and mastered in certain age ranges. The current rhetoric — let's do geometry K-12, for instance — makes it too easy to avoid the topic. We should also be clear that building a concept requires skills. It is not sufficient to argue for activities that involve "algebraic thinking" at grades K-5; there need to be specific goals such as the algebraic description of numerical patterns and the solving of simple equations. Finally, we now have enough experience dealing with data in the classroom that we can be more specific about what we want students to be able to do with statistical ideas by the end of eighth grade.
This recommendation is designed to alleviate the pressure of the mountain of curriculum that is currently expected of senior high school teachers. We need to continue to move the study of algebra and geometry into the middle school so that senior high school teachers can cover the full gamut of the curriculum that is in the Standards. We cannot tolerate a situation where so many students enter algebra with so little knowledge that they cannot get past linear equations and linear systems, or enter geometry with so little knowledge that they cannot understand why proof is such an important way to test and organize that knowledge. UCSMP users know that the algebra and geometry studied in middle school makes it possible to do much more in senior high school mathematics.
This puts UCSMP in agreement with the conclusions of a white paper prepared last month for U.S. Secretary of Education, Richard W. Riley. Some of the conclusions of this study which are most relevant to this discussion are:
"The eighth grade is a critical point in mathematics education. Achievement at that stage clears the way for students to take rigorous high school mathematics and science courses......Algebra is the "gateway" to advanced mathematics and science in high school, yet most students do not take it in middle school......The results of the Third International Mathematics and Science Study reveal that the middle school mathematics curriculum may be a weak link in the U.S. education system." [Mathematics Equals Opportunity, Richard W. Riley, October 20, 1997.]
Recommendation 4: Do not specify a best approach to any particular topic, either mathematically or pedagogically.
The term "pedagogical content knowledge" is used by many researchers today to refer to the variety of ways by which one can approach a mathematical topic. Some of these ways involve different mathematical ideas than others. As one who has been developing curriculum for virtually his entire professional career, I am quite sensitive to anyone who is prescriptive about the curriculum. I know of the many ways to develop mathematical concepts and of the myriad of sequences and interconnections that can be made.
For instance, a couple of the NSF projects approach beginning algebra through functions. This development so bothers mathematicians that they have felt it necessary to remind mathematics educators that much of algebra does not deal with functions. The properties of the operations from addition to powering, the various algorithms for solving equations and inequalities, the conversion from one algebraic form to another (such as in rewriting a(x – h)2 + k as ax2 + bx + c or vice versa), the binomial theorem, and many other important topics in algebra are not about functions.
I do not intend here to demean the idea of functions. Functions are very important and very useful in algebra. In FST and PDM we use functions to help us conjecture and confirm trigonometric identities, but the proofs require an algebra that is not about functions. In PDM we have what I think is a beautiful and unique approach to solving equations and inequalities that utilizes function ideas, but that is done only after students have studied functions for at least two years.
In UCSMP we approach geometric congruence through transformations, and here we find ourselves in the minority. For a quarter-century, this mathematics has been commonly seen by mathematics educators, a quarter-century of Escher drawings and other tilings, of size changes, of reflections, rotations and translations in both elementary and high school textbooks, of applications to the graphing of functions and relations. It is astonishing to me that after all this time there are still authors who somehow view this conception as a frill, as something either to be in the last chapter of the course or ignored altogether. I think it is a crime not to approach congruence of figures using transformations. But I do not want to see someone dictate or attempt to legislate the approach to congruence that should be taken.
If the mathematical approach should not be specified, it is even more important that a particular pedagogical approach not be recommended. Specify the goals of teaching — the ends — but do not specify the means. If group work is considered so important, then make a group project one of the ends, but do not tell me as a teacher that I am not teaching my students well if I don't put them into groups. Maybe I have a class that works better as a full team. Maybe I have a class that works better individually because of the personalities in it. Maybe I work better in a different way.
It is obvious that it is better for students to be active rather than passive in the classroom, but there are many ways to be active. A long and boring lecture is horrible teaching, but no worse than a senseless activity. It is obvious, and confirmed by research over decades, that if a student discovers an idea, then that idea is better learned by the student, but that does not require that one believe in constructivism. It is also obvious that students learn mathematics by talking about it, but this kind of discourse is not foolproof, for if no one is listening to question or correct wrong impressions, the discourse will not achieve its purpose.
We ask those of you teaching from UCSMP texts to attempt to get students to read before you explain, but we never say to do that always, and we are careful to give the goal that students become independent learners. But even instructions that are flexible do not work for everyone. in some classes, students cannot read or do not do homework, and the reading must be done in class. In other classes, we hear of teachers who assume that if students are expected to be independent learners, then they do not have to teach. Prescribing methods of teaching is dangerous because those who prescribe do not know the classroom and are ignoring the wide variety of conditions of teaching in the classroom.
Recommendation 5: There needs to be a "testing summit" at which national, state, and local school officials work out a sensible distributed plan of testing.
This recommendation is not meant to eliminate tests, for they are both a practical necessity and a political reality. It is designed to bring some sense to the current situation in which tests are laid upon one another without an overall plan. The goal should be at most one external test in a subject each year.
Teachers are being beleaguered about the curriculum they teach, about the way they teach it, and by more and more tests because everyone is crowding to be at the helm. If some sense is not brought to the scene in each of the areas of curriculum, teaching, and testing, then we are likely to retreat to a period of stagnation. This would be unfortunate because the data indicate that we have made significant gains in mathematics education in recent years.
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