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Champions of Mathematics Education Reform

Robert A. Mead, Plymouth State College, NH

Spring, 1998

”Singing is an impulse to move. If your mind is only on pushing [the right] button, there is no singing. You have to have something that you want to say.”
- Phyllis Curtin, world-reknown soprano [Zinsser, p.230]

“Overall these problems make you think, as does most math. This is very useful in itself. Thinking makes you grow and in my opinion it keeps you alive. Thought separates us from the chair in a corner or a piece of chalk sitting near the blackboard.”
- Lily Marquette, College Algebra student


Something was different about the NCTM (National Council of Teachers of Mathematics) national conference in Boston in 1995. I noticed many enthusiastic people in attendance, and rather than the enthusiasm being about a presentation some had witnessed, or one about to happen, the spirit seemed to center around teaching. Maybe there was no real difference in the crowd. It could have been my peculiar perception, but then, what had changed my perception? I know that 20,000 math teachers do not a movement make, or do they?

The hubbub appeared to be about “Standards” teaching, or “math for everyone.“ The NCTM Curriculum Standards [1989] had been out for six years, and frankly, the initial reaction to them, of myself and others, was one of pure puzzlement. Following on the heels of those standards, however, was a sustained effort to interpret them and help teachers retrain in order to implement them. NCTM followed up with addenda which contained grade-level as well as content specific guides to a new kind of curriculum. [1991, 1992, 1995] The NCTM journal articles were being written in a new spirit that I would characterize as “my class and I have some terrific math we’d like to share with everyone.” Prior to this you would find many articles written by people showing off a few messy formulas. At the conferences, workshops were now being offered in active learning, writing across the curriculum, problem-solving, and new assessment methods. Judging from the furious debate now raging in the mathematical and education communities over these reforms, I believe they constitute a serious hope (threat?) that schools can make math the relevant and much-enjoyed subject it always should have been. Much of the vocal opposition to them comes from California where authorities are threatening to take calculators away from students, reinstituting such tasks as the paper and pencil extraction of roots [Appendix 1], while at the same time Virginia and Texas appear to want to put technology in to every student’s pocket. For a strong opinion in defense of reform see Appendix 2.

This report is not about the Standards, however, because we do a disservice to children by arguing their validity, to the exclusion of making needed changes. “The scores are down -- it’s the Standards fault.” “The scores are up -- it’s the Standards fault.“ While we make merely superficial judgments, I don’t see any hope for improvement in mathematics instruction. On the other hand, I don’t suppose anyone would object to improvements. What we need to do is examine specific changes we are contemplating, and judge them on their own merits. Who are the champions of those changes, and what gains have they observed? My research investigated those specific changes which many believe need to be made in mathematics classes, and who said what about them. For example, the new emphasis on problem-solving, I believe, stems from the constructivism theory advanced by Jean Piaget. This paper will look briefly into theories which preceded Piaget, his ideas, and the thoughts of critics who were to follow. It will outline the works of Polya and Alan Schoenfeld regarding teaching strategies which support the problem-solving effort. It will touch on Edward Zinsser’s justifications for using writing to improve learning, and it will mention sources for other promising reforms which appear to support the thinking and reasoning efforts of teachers and their students. Finally, I will have a few ideas for how we cool the tenor of the debate over Standards, and find the common ground and shared beliefs from which improvements can be fostered.

Piaget and Constructivism

Before Piaget

Edward L.Thorndike in 1922 investigated the application in education of stimulus/response behaviors, observed in animals performing simple tasks. He created very detailed lesson plans which would have students react “correctly” to the stimuli provided by the sequence of individual steps in an algorithm. As early as 1928, a skeptical William Brownell posits that children who respond “12“ to the stimulus “7 + 5“ have not necessarily demonstrated an understanding of the fact or the operation of addition. He went on to say that given the proper understanding of math principles, students could better apply them in novel situations. Swenson in 1949 found straight drill the least effective of three teaching strategies. In the late 70s there was disagreement on the cause of poor arithmetic performance, with some blaming inadequate mastery of basic facts, and others blaming mislearned or misapplied algorithms. Studies confirm both postures. [Resnick, p. 12-19]

Behaviorists dominated the educational psychology scene until the 30s and 40s, when a group from Europe, the gestalt psychologists started to analyze how problems are solved. Wolfgang Kohler in 1925 observed an ape assemble a long stick from two shorter pieces for the purpose of reaching food. The gestaltists then began to investigate the idea that we solved problems only when we recognized the situation (structure) as a whole. Wertheimer had a demonstration using a parallelogram where students who learned an algorithm well were unable to modify it for a slightly different problem. Resnick and Ford conclude that the algorithms are not at fault, but that they should be learned in the context of the underlying structures. [Resnick, p.132-8]

Jean Piaget

In Piaget’s view, knowledge is not being absorbed passively from the environment, is not preformed in the child’s mind and ready to emerge as the child matures, rather it is being constructed by the child through interactions between his mental structures and the environment. Intellectual development takes place when the child restructures his knowledge to compensate for events that disturb current understandings, in a process Piaget calls equilibration. [Labinowicz, p. 35]

Maturation, physical experience, social interaction and equilibration are all necessary for intellectual development. This development takes place as a continuing cycle of adjusting to new events. All children must pass through the developmental stages called pre-operational, concrete-, and formal-operational. They do at different rates. It is estimated that only one-half of the American adult pop has reached the formal operation stage. [p. 87]

Teachers can help by sharpening the discrepancy between the child’s framework of ideas and the feedback he or she is getting. Teachers must plan and structure the lesson to raise those timely discrepancies. Piaget strongly recommends looking at what the children know to decide course content, rather than consulting scope and sequence charts. [p. 165] He thinks that another flaw in formal education is the propensity to begin teaching with language instead of action. “The goal of education is not to know how to repeat or retain ready-made truths (a truth that is parroted is only a half-truth)”. The goal according to Piaget is learning to master the truth on your own and taking the time and trouble to perform the activities needed to know it for certain. [p. 171]

William Dunham in The Mathematical Universe [1994] gives us a sterling example of Piaget’s theories in action, when a rather gifted adult intellect tried and failed to comprehend a new idea. He writes that the idea of analytic [coordinate ] geometry is quite recent. It is "younger than, for instance, logarithms, Romeo and Juliet, and Boston." In 1637, Rene DesCartes attached an appendix to his paper Discours de la Methode, in which he presented this unification of geometry and arithmetic. He deemed it unnecessary to explain it clearly or deeply. His intention was to have the reader work it out. Sir Isaac Newton was one such reader who could make neither head nor tail of it. This suggests to me that because nothing like this concept had ever been uttered prior to this, giants like Newton were not prepared to internalize it. It was so brand new that he had no constructs in his vast understanding that would deal with it. It was a timely discrepancy. Keep in mind that Newton was to go on to write the calculus, a most analytical branch of mathematics, and he made liberal use of DesCartes’ system eventually. One of his saving graces was the edition by the Dutchman Frans van Schooten, who bothered to include a commentary which made DesCartes accessible to a wider audience, including Newton and Liebniz. [p. 273-5]

Piaget did much of his clinical studies using mathematics, a subject in which relationships are constructed by people and exist only in their minds. Children need time to interact with materials (manipulatives) in order to form those concepts for themselves. As interpreted by Ed Labinowicz [1980], Piaget issues cautions against teacher manipulated models, or too short an exposure time for the students. The manipulation of objects is critical in the eleven years leading up to formal operations (keeping in mind that it happens at varying ages or may never happen) Piaget would provide for active work with peers. The verbal interactions with peers and adults also promote cognitive growth, he believed. In summary, the teacher must find a way to provide the time [p. 188-223]:

Is our society prepared to nurture logical and critical thought in or out of the classroom? Piaget’s first goal in education is to change the existing culture. Since the world changes rapidly, we must respond to an increasing number of new problems. Piaget wants to have learners who have vision, who can foresee problems, and long-term effects, and who can see through propaganda. If he is right about the developmental stage that many Americans achieve, we teachers have serious work to do. The parents need to be educated about the quality issues in education that run counter to shallow thinking about acceleration, basic skills, or behavioral objectives. A competitive grade-assigning school may be preparing some achievers for success in a commercial world, and may be turning others into disinterested, disenfranchised scofflaws, but it is preparing hardly anyone to learn.

Outdated educational concepts have made the teachers merely transmitters of knowledge, preventing their initiatives, colleagueship, or the motivation to research and discover for themselves. Piaget questions the professional status of teachers in the United States since so many vital decisions are made without their knowing. Subtle long-range indicators of child development are hidden in short-term paper and pencil competence tasks. Universities are finding they have had to adapt to the needs of students. Should they have been setting more of a progressive example in the first place? They have not provided teacher prospects an opportunity to construct their own conceptual framework for learning and teaching, nor do they provide laboratory one-to-one experiences with children to develop interpersonal skills and an appreciation of research techniques [p. 264-276].

Piaget’s Critics

Jerome Bruner is another psychologist who has a deep understanding and interest in the learning of mathematics. He has written that the child in recent years has come to be regarded as a social being and uses social interactions as a framework for interpreting experience. The child seems more competent as a social operator rather than as a lone scientist coping with a world of unknowns. Language regains its primary role in describing reality and in transmitting meaning. These are two significant departures from Piaget. Bruner notes another development in more recent research: the use of collaborative activity and its capacity to develop language and cognitive skill. He asserts, “At the very least the child’s development must be mediated” [Bruner, Haste, 1987, p. 7-8]. Bruner tends to agree with Vygotsky that another component which contributes in a totally different way than peer-parent interactions, is the cultural-historical impact on the child’s value systems, and thus on the child’s intellectual functioning and growth [p.9-10, 17, 93]. Piaget seems to have naively disregarded outside influences other than teacher and parent with regards to direct impacts on the child’s development.

Bruner aligns with Piaget in believing that to teach structures, we must first assess what cognitive structures individual students bring to the class; then we need to study how the students interact with the presentations. This calls for much prescriptive work and carefully designed lessons [Resnick, p. 111]. This paper discusses Bruner’s contributions shortly.

Dunn, Butterworth, and Light [Bruner, Haste,1987] find fault with Piaget’s age ranges for the developmental stages. They object to his heavy dependence on using the child’s early high degree of egocentrism as a gauge for those stages. New research has shown that very young children are able to acknowledge the viewpoints of others. In this pre-linguistic stage it is very difficult to ascribe specific cognitive developments to children. It seems there is much to learn about learning.

Donaldson describes an incident with a child of almost three, who was used to going down stairs for a session, and who was told she would be going upstairs that day. She verbalized “not going down,” not disputing but realizing that one action precluded the other. Of interest is that they were not in the act of going up, so the child made an inference. Donaldson differentiates this from posing the question, “What is the consequence of going upstairs, given that you can do one or the other,” a question entirely beyond the child’s comprehension even though the answer is there in the gut. The problem is doubly removed from the concrete – distant in terms of time and space, but not embedded in anything the child is doing or possibly ever will do. Achievements in mathematics, logic, and science depend on developing this kind of distance thinking [p 102-7].

Robert B. Davis studies learning by analyzing errors [1984]. He says that from the study of error patterns, one can observe two types of consistency. One is the individual’s unique error tendencies, which can be identified and corrected. The other type is a pattern of errors which seem to be common to all learners. These patterns can transcend cultures, distances, and generations. What seems at fault is the persistence of frames of memory, constructed by the learner, which persist after their applicability has ceased to be reasonable. One example he uses is in the learning of basic addition facts, and the frames the student constructs to remember the addends and the sum, but that don’t pay sufficient heed to the concepts of addition and equality. This frame later causes consistent error patterns in other tasks, for example, in learning the multiplication facts.

Davis criticizes mathematics curricula which do not apprehend these problems. He finds most harmful the presentation of small, non-accumulating bits of information, the slowness of the pace, the superficial verbal learning, and the rote imitation of procedures. He asks the reasons why solutions are not being implemented, and he states a few probable ones: poor teacher preparation, low expectations, divergent views toward innovation, the personal risk involved in teachers and learners getting into each others heads, and the resilience of traditional curricular goals, such as the belief that students should learn some “disciplines” as sets of rules which must be obeyed, whether the subjects are understood or not [p. 339-49].

So Davis thinks that society needs to get in touch with itself, and that takes us back to Piaget. Despite the refinements or out-and-out changes many would make in Piaget’s theories of development, they have not addressed Piaget’s major social questions. Do we want a society of critical thinkers or do we need many to be good soldiers, followers of authority? Do we want teachers involved in the clinical study of their field? Do we want them to be partners in the educational decision-making process? I think it is very interesting that these questions are finally coming into open debate at the same time that there has been an explosion of electronic communication via the internet. Will participatory democracy find itself not nationally, but worldwide? Will discussion and decision-making come to be based on reason rather than emotion? Stay linked.

Jerome Bruner

The “New Math“ movement of the 60s, one which is closely identified with psychologist Jerome Bruner, emphasized fundamental structures. Its justifications were: efficiency, in that generalizations once mastered could enable students to solve whole families of problems; esthetics, in that math would be presented as the beautiful and exciting subject it is; and propriety, in that it recognized [and leaned heavily on] the intellectual capacity of young children. According to Resnick and Ford “Most lay persons have never considered mathematics as anything more than a collection of procedures for solving computations.” [1981, p. 13-8, 104-5] I have observed that most lay persons report being positively slain when they reach first year algebra. Perhaps there is a connection.

Bruner, in his writings in the early 60s, also raised some major issues which I think center on the idea of human dignity. He was identified with the now widely-discredited (and therefore probably worthwhile to some degree) New Math movement in that decade. Assuming for a split second that perhaps the total condemnation of those reform attempts was at least in part based on emotion rather than reason, let us look at what Bruner had to say. “It is not sufficient to merely be useful. The servant can pattern himself on the master -- and so he did when God was Master and man his servant.“ [Bruner, 1962] That was one thing he said, and he also observed that in the technological era, man is the master of machine and there is no dignity in patterning after a machine. Bruner values the creative act as a dignified one, and he analyzes it: “A creative achievement produces effective surprise.“ He says much of it grows out of novel combinatorial activity. This is not done blindly or by a non-interested algorithm, rather there is a heuristic that avoids senseless combinations and has a feel for promising ones [Bruner, 1979, p 18-26].

Bruner draws comparisons between mathematics and art as modes of knowing: “There is a deep question whether the possible meanings that emerge from an effort to explain the experience of art may not mask the real meaning of a work of art.” [p59] The same applies to math analogously. Are we disguising the meaning of math by showing many more exercises than problems to students? This leads Bruner to talk about the benefits of learning through discovery because that would increase intellectual potency, shift the student’s focus from a system of extrinsic to one of intrinsic rewards, and conserve memory”[p83]. He cautioned that the mathematician’s job is not mere puzzle-mongering, rather it is to find the deepest properties... so that he may recognize the puzzle as an exemplar of a family of puzzles.” [p98] Manipulation and representation in continuing cycles are necessary conditions for discovery. They are the antithesis of passive, listener-like learning.” [p. 101]

These all resonate with my own teaching experience. I began in an idealistic way, trying to excite my students by having them discover and learn for themselves. Bruner said, “I have never seen anyone succeed in inquiry by another means.” When immediate results were not forthcoming, I reverted to telling. After all, I had a job to do. I was never satisfied, however, with the results of this teaching style.

When returning graded papers I have often noted that there were two types of reactions: many low achievers were typically upset and ready to blame sun and moon and teacher for their woes; on the flip side, high achievers would look only at the mistakes and be blind to the successes (they were never satisfied). There were also crossovers: students with an A-minus who would be upset about any criticism at all. Bruner’s reward system theory makes much sense. Finally, I think he is right about the memory idea – that if you are not learning, but instead memorizing, and using ineffective models, and not connecting new information with experience, you are putting a drain on your memory capacity, regardless of how vast it is (or isn’t). In other words, episodic thinking [my term] requires remembering all the episodes.

Like Piaget, Bruner objects to the premature use of language as the formal medium in which to do math. It creates the impression the child knows no mathematics. In selecting content he urges teachers to opt for depth and continuity. He claims there is too much content, and insists that each piece of information serve the worthy purposes of causing delight, helping piece together the whole, and being useful. Some ideas of pursuing excellence and avoiding the trivial and preparing children for the future come from John Dewey in the early part of this century [p 104-122]. I wonder if the necessity for repeated military build-ups in modern times have made us forget about Dewey, and frown upon Piaget and Bruner.

In support, however, the National Sciences Education Board report entitled Reshaping School Mathematics [1990] declares that “learning is not a process of ... absorbing information and storing it in ... retrievable fragments.” The report goes on to advance effective criteria for a math education philosophy of the future. It also recommends the transition away from ability-tracking and paper-and-pencil drill, and toward significant mathematics for all students and full effective use of technology.

Eleanor Duckworth in The Having of Wonderful Ideas [1987] says that, in teaching, curiosity needs to be captured, resourcefulness fostered, knowledge made interesting. “The more we help children have their wonderful ideas... the more likely it is ... that someday they will happen upon wonderful ideas.” Are we structuring the learning of math as an activity that encourages having ideas?

I believe that all students deserve a dignified education. Kohler’s ape solved a problem in 1925, and we should expect students to do no less. I believe that some of our major societal problems and many individual woes we face today are caused by people making poor choices, or worse, not realizing there is a choice to be made. Bruner may have agreed that we have trained people not to solve problems.

Polya, Schoenfeld, and Problem-solving

Polya is often mentioned as one who brought new emphasis to problem-solving. He wrote that the secondary mathematics teacher has a great opportunity. Because it is not encumbered with complex details or subjective material, mathematics is the discipline in which students can first learn to solve problems scientifically [Krulik, ed., 1980]. If, however, the teacher fills the allotted time with drills, he kills interest, hampers the intellect, and loses the opportunity [Polya,1957].

Polya wrote How To Solve It [1957] in an attempt to revive the study of heuristic, vaguely defined as a discipline of philosophy and logic. The aim is to study the methods and rules of discovery and invention. Polya’s book provides a modern application of heuristic in the problem-solving emphasis in the mathematics curriculum. He outlines four phases in the act of problem-solving: understanding the problem, making a plan for its solution, carrying out the plan, and looking back at the complete solution. He provides an extensive “glossary” of the strategies often used in solving problems, and many problems which illustrate the effective use of them. Polya encourages the collecting of a bank of problems which illustrate various strategies, and then to organize that list around the mental stages students will typically need to pass through in solving a difficult problem.

Alan Scoenfeld is a mathematics teacher and a leading researcher into the effects of assigning problems and teaching heuristic. He has evidence that students can benefit from such instruction and become better thinkers and solvers. When he read Polya’s book he realized that strategies can be organized as to their usefulness at various stages of the solving process. The strategies can and should be discussed often in the classroom. As a mathematics student, he wondered why no one bothered to teach him the very strategies he would need to solve most math problems. “Very few adults ever have an overt use for, say, the quadratic formula or for proving a theorem in geometry; what they can and should have, as a consequence of their education, is the ability to reason carefully and to make intelligent and effective use of the resources at their disposal when confronted with problems in their own lives.“[p.15]

Echoing Bruner and Piaget, Schoenfeld presumes that “coming to grips with any subject, and in particular, learning to think mathematically, involves a great deal more than having large amounts of subject matter at one’s fingertips.” [p. xii] He presents a framework for investigating mathematical thinking. The issues he regards as vital are “resources,” meaning the store of content, algorithmic and conceptual knowledge that a student can bring to a problem; the heuristic, a term revived by Polya [1945 p. 112-3], meaning the correct and timely application of strategies to discover or invent an attack on a difficult problem; the control issues, which are the course-correction decisions made in the solving process; and the “belief system” that a student possesses which impacts on what the student thinks are appropriate tools for the problem. Control behaviors are exemplified by making plans, setting subgoals, and monitoring the process for needed shifts in strategy. This is associated with the term metacognition in the psychological literature.

Schoenfeld declares that, “in the best of all possible worlds the three behaviors, knowledge, heuristics, and control, should be sufficient to analyze mathematical thinking. He observed that what students believe about math, in spite of what they have been taught and remember about math, often determines the directions they take in problem-solving. He cites many sources for support of this opinion. For one, Perkins looked at informal reasoning and found that many people make a plausible argument, one that cannot be immediately discarded, and they believe it suffices. There is no attempt at a rigorous defense. Numerous studies show that decision-making can be highly biased and rigid in many situations. ”The point here is that purely cognitive behavior ... characterized by using resources, heuristic, and control, is rare.” [p.35] One of the harmful beliefs is that discovering or creating math is a task for geniuses only. People who believe it see a need to remember a vast amount of content knowledge, and when memory fails them, they believe it is time to give up. Another harmful act occurs when they accept at face value the procedures and logic handed them by the expert. This behavior feeds on itself.

Schoenfeld presents a detailed framework for analyzing the control behaviors, and illustrates its use [p294]. He takes the dual role of practitioner and researcher in his own classes. He creates the methodology with which to investigate mathematical behavior, and also tests the framework and elucidates it. He videotapes the classes so that he and they can chart how much time was spent on planning and modifying the plan, as opposed to carrying out the plan. He adopts Polya’s definition of a problem, “finding an unknown means to a well-conceived end,” which has been widely adopted by institutional reform movements such as the NCTM Standards and the Vermont Portfolio Project. This refers to a task for which the solver has no easy access to a solving procedure; it does not refer to “template” problems which the student “solves” by imitating a teacher-demonstrated example.

Schoenfeld has adopted many of Polya’s exemplary problems. He has taken Polya’s recommendation that all teachers need to develop a bank of meaningful problems. He has taught undergraduate courses in problem-solving that equip students with the heuristic, and the problem sets with which they practice it. He finds in general that even though students possess the content knowledge, and in some cases choose promising strategies, the issues of monitoring and controlling the process, and the impact of their beliefs about the nature and usefulness of mathematics, often slow or stymie the progress. For example, many students do not believe that a logical argument is necessary when they draw conclusions. Even some of the more successful students held very anti-mathematical beliefs that detracted from the problem-solving effort. Many failed to use the knowledge they possessed because they did not regard it as useful. Beliefs persist even after instruction. The examples he uses are from physics, where undergraduates who had either high school or college physics, or both, still in great numbers believe that without any external forces, an object can travel in a curved path [p. 146]. As a way to deal with belief systems, Schoenfeld shared selections from Polya and assigned geometric construction problems before and after. He found a shift in some behaviors away from reliance on careful drawings and toward mathematical arguments[p 352-3].

Schoenfeld is also concerned about the prevalence of form over meaning, exercises over real problems, and passive over active mathematics learning. He regrets that students are regularly assessed with items of no more than two or three minutes duration [p369]. Students are hardly likely to place any value on heuristic [or any math for that matter] if they never get to practice and be assessed on those skills. He argues for the teaching of heuristic in any effort to improve problem-solving because students are going to develop their own heuristic regardless. Why should they have to discover them all? Why not monitor the completeness and reasonableness of the strategies that students are using?

Knowing the heuristic is not enough. In addition, one must practice selecting, abandoning, and adopting new strategies. These are the control behaviors. Schoenfeld blames the oversimplification of heuristic, and the failure to take into account the control issues, for the lackluster results of problem-solving efforts in the research [p.73]. As an illustration of the complexity in the heuristic, Schoenfeld presents a problem of inscribing a square in a triangle. He identifies six ways to ease the problem constraints in order to make a start on the problem. Only two of the six prove fruitful in this problem [p. 85-91]. Learning to be selective and look for positive links between the problem and the simplified models is a complex skill. Schoenfeld is not pessimistic, rather, he wants us to be realistic about the complexity of the problem-solving effort.

He raises an interesting concern about computer-based instruction. If it is important for students to learn to monitor and control the processes, will they be getting too much assistance from computer tutorials that automatically deal with the heuristic and control issues as envisioned by one “expert.”[p. 134-9] I think distance-learning efforts, enabled by satellite TV, now coming into vogue, must take great care not to rob much of the processing experiences from the student by paving over them. It could be argued that many traditional classrooms do a good job of imitating the distance-learning model, with the anticipated consequences.

Schoenfeld supports Vygotsky’s view that social interactions are important in the development of control skills. I notice, however, that in some of Schoenfeld’s classroom transcriptions, the social skill of accommodation, that is, in giving proper recognition to the contributions of all, has a way of stalling the process at times. This difficulty can only justify teachers providing more opportunities for students to work together cooperatively, not fewer. In summary, Schoenfeld draws the following conclusions from his own classroom research. More exposure to problems is desirable but not sufficient (a control group did not solve as well as the group which was taught strategies even though they were exposed to the same problems). Heuristic can be learned, and knowing heuristic can make a difference when the environment supports the use of it once mastered [p209-212].

In more recent writings, Schoenfeld talks of the importance of changing the classroom culture (remember Piaget?), which can foster either constructive or destructive control and belief systems. He wants his students to demand reasons when someone claims to have a solution. Both he and his students analyze the videotape to improve their solving skills and the classroom dynamics. As a teacher, I would prefer to read what students think about problems. Schoenfeld knows that problem-solving ability cannot be assessed with two-minute responses. Extended tasks and sufficient solving time are essential, and, in addition, I think the time should be devoted to giving students the opportunity to reflect and then express in their own language the reasons behind solutions. These actions solidify the learning and help with language deficiencies which may in some instances be the reason for learning difficulties.

Zinnser and Writing to Learn

William Zinsser would favor writing activities in all subject areas, and he has written about teachers who are putting this strategy to work. Zinsser visits Gustavus Adolphus College in rural Minnesota, which is trying to implement “writing across the curriculum.“ He finds many teachers from widely varying disciplines who are enthusiastically committed to the effort. In his preface, he puts the well-known math phobia shoe on the other foot saying, “students with an aptitude for science and mathematics are terrified of the humanities, meaning all those subjects like English, Philosophy, and Arts which can’t be pinned down with numbers and formulas. He insists, however, that every discipline has a body of literature which will serve as a model for good writing.

He visits Joan Countryman, a teacher at Germantown Friends School in Philadelphia. She espouses constructivist mathematics, active learning, and the writing of the process that goes into problem-solving. Zinsser also talks to expert teachers in fields as far flung as chemistry and music, the visual arts, and physics. A common theme is present in all their reports, and it is that writing assists in the creative and learning processes. Since good writing almost always involves good rewriting, the emphasis for these teachers has shifted from product toward process. It seems that better products have resulted as well.

Zinsser talked to Professor Clair McRostie who teaches Economics and Management. He devotes his first three lectures to writing and reasoning – skills that, in addition to content, will be the basis of grading his students‘ three papers. His message: “economics and management are important but not as important as clear reasoning and writing”

History professor Kevin Byrne regards reading, writing, and thinking as an integrated whole. Teachers have a tendency to give their students writing assignments and then let them sink or swim. He designs his course to take time from history in order to explain how good writing is done. This takes commitment; it’s painful because he finds the semester too short to cover all the history he wants it to.

Zinsser himself has taught writing, and he once believed, “surely if I assailed my students with my sacred principles of clarity and simplicity and brevity...they would go and do what I told them to do. No such transfer takes place. Writing teachers are lucky if ten percent of what they said in class is remembered and applied.” The writing problems can be cured only by “[surgically] operating on what the writer has written.” [Zinsser, p.47]

Zinsser had a revelation when he saw that words could penetrate mathematical ideas: “It had never occurred to me that the teacher was not the sole possessor of mathematical truth.“ [p.149] He and Joan Countryman had the following exchange about math in the real world, specifically, in the architectural points in her house. For the sake of brevity, I paraphrase:

C: There are all sorts of interesting mathematics in the way this house is constructed. Most people don’t see the world that way because they’ve been told that math is something apart from what they’re able to do.
Z: So what is mathematics?
C: What are any of the disciplines but a way in which people try to make sense of the world? ...How much fabric do I need to cover this chair?
Z: When you ask a question about fabric, does it become something beyond arithmetic?
C: It does because you begin to make connections.
She believes that if the class does not make connections, the students will never think of math as part of themselves. “When I got into teaching I had serious questions about the relationship between what one does in school and what one does in life.” She saw that children were learning math in isolation, apart from other disciplines, and apart from other people. She immediately put her students into groups and had them teach and learn from each other – and take tests together.

What about the possible drawbacks for the gifted students who are forced to slow down and explain things? In time, hers were saying things like “I’m learning math better.” The kids with an aptitude would come up with a fairly quick answer, but they would not have thought about it much. Other students would not have to think at all – they could just wait for someone to come up with the answer. Countryman says, “writing engages the imagination, the intellect, and the emotions... powerful aids to learning.” She uses journal writing to enable her students to suspend judgment while they work out their problems in understanding the math. Zinsser concludes, “the engineer is as frightened of my language (writing) as I am of his. Scared away by the probable difficulty of learning the language, he never gets to its literature, the purpose for which it was created.“ [p. 149-161]

As a student, I would have to plead guilty to Zinsser’s assertions that number-oriented students fear words. I was a slow reader through high school, I was assigned very few meaningful writing tasks, and I took great solace in the fact that I did have an aptitude for mathematics. In the first half of my undergraduate studies, I took a minimum of courses that would develop my areas of weakness. It was as an older undergraduate that I began to enjoy writing and appreciate its importance in communicating what you know and what you want to say.

Marilyn Burns and Classroom Techniques

Marilyn Burns is a math educator who insists that we teach for understanding. She has taken her message to teachers and to the general public as well. She is the developer and author of numerous teaching aids and materials, and she has written a popular series of books for the families of students who are interested in mathematics [1975].

In A Collection of Math Lessons: from Grades 6 through 8, activities are developed based on her assessment of what is actually happening in the classroom. To teachers who express reservations about time, established curriculum, standardized tests, and continuity, she answers that assessment and learning can be served simultaneously in problem situations. Musicians in performance, she asserts, do not demonstrate their individual skills, rather they perform, and show their musicianship. The skills are one necessary aspect, but the students perform music; they do not perform scales and arpeggios. Likewise, proficiency in math skills should not be mistaken for proficiency in math. When do our students get to perform mathematics? How did we get the impression that every student needs the same sequence of skills? Music learners are permitted to demonstrate their abilities no matter what their level of competence.

Burns claims that important elements are missing from algebra. Students are not learning how it fits with the rest of math. Algebra can be introduced as an extension of geometry and arithmetic in a problem context [Burns, 1990, p. 23]. She has found that not all students will master the generalizing. Knowledge must develop over time, and it’s a different clock for each student. The problem-solving approach allows for multiple paths that can lead to a solution. This is what students of varying abilities need to be successful.

In spite of an overwhelming emphasis on calculating skills, the traditional curriculum produces students whose fraction skills are weak and whose understanding is understandably weaker. Breaking the topic into small parts does not help students put the whole picture together. Activities need to be accessible to all students and rich enough to challenge the more capable ones. Burns cautions instructors to resist the panic urge to teach by telling. She recommends talking to colleagues about strategies when unusual situations arise in the classroom.

The goal for her lessons in percents is that students will use them effectively in problem situations. Burns pre-assesses and then builds on what students already know. She uses a preassessment activity called “Sense or Nonsense“ [Lane County, 1982] to preassess student understanding of percent. The activities also focus on students’ thinking, and they provide the opportunity for students to create their own understandings of how to reason with percents. “Structuring an instructional sequence requires learning about what students do and do not understand, and about what sorts of activities and topics interest and motivate them.“ [Burns, p137]

Burns does not focus the lessons on the three algorithms for percent problems because, she says, it takes the focus away from making sense of situations. Her response to teachers concerned about their students‘ ignorance of the algorithms is that she is more fearful that students who know the algorithms would fail to reason when solving a given percent problem. Is the misapplication of an algorithm any better than the ignorance of same?

Burns thinks that “writing gives students a chance to reflect on their own learning and provides information to me about their thinking.” [p.170] She learned, for example, that some students believed that 100 was the maximum percent, and she was able to make a plan to deal with the problem. She lets students write problems for one another. This lets her know the areas in which they are comfortable solving problems. I have used this strategy to help students prepare for a teacher-made assessment.

Burns’ ”students were challenged to make sense of the situations, create methods they thought were useful, [and] explain why their methods were sensible.” [p. 182] She believes that the decisions about what to teach directly, and how to improve the lessons for the future, are matters of the craft of teaching and should be responsive to different groups of students – decisions which should not be given by default to a textbook.

Burns’ techniques suggest a curriculum always on the move. All teachers want to improve their service to the education of children, but I must ask why, in the four hundred or so years that we have been teaching Algebra, we are not more successful? Many would agree with the old saw that students learn their algebra while taking Calculus. Does this suggest that in order to internalize it they need to see it in use and connect it with something?

Cooperative Learning

Many of the educators mentioned have endorsed peer interactions as valuable in the development of intellect. New learning theories from Noddings [Davis, ed. 1990], Vygotsky, and others have supported this position. Activities that foster cooperation can provide enriching learning opportunities. Positive outcomes have been documented by proponents such as David and Roger Johnson.

Cooperative learning theory is very different from the practice of grouping students for the purpose of accomplishing individual tasks. Most cooperative models have the following features in common: students must know they are part of a team that has a common goal and expects a shared reward; individuals are accountable for meeting their responsibilities; and face-to-face interaction is vital.

Other sources for classroom ideas for using this strategy come from the Johnsons in Circles of Learning and Cooperation in the Classroom, and from summer courses they offer across North America. Another book from Bennett and Stevahn entitled Cooperative Learning: Where Heart Meets Mind, has many practical suggestions. Neil Davidson of the University of Maryland has given presentations of cooperative strategies that work in the high school classroom.

I have used group cooperation effectively in my own teaching. Most students, when challenged with a difficult and unfamiliar problem, are receptive to the idea of working together. They tend to spend more time on task, and in time they learn to self-start. By their interactions, I am able to observe and assess what knowledge the student brings to the task, what skills to reinforce, and what questions will elicit progress while enhancing independent thought. The philosophy behind my approach, ironically perhaps, is that in math as well as in life, there is only one person that can solve an individual’s problem, and that is the individual in question.


The goals of constructing learning, solving problems, and explaining process seem to require different modes of assessment than we have seen used in many mathematics classes. In the Assessment Standards for School Mathematics, the NCTM [1995] has created some guiding principles for those programs which are contemplating new modes of assessment and reporting. These standards call for evaluating process as well as content knowledge. They present guidelines for a fair and open process for evaluating student work. Finally, they present suggestions for how to incorporate classroom assessments into a system of total program assessment.

NCTM has also published How To Evaluate Progress in Problem Solving [Charles,et al., 1987]. This work presents four models for evaluating students. It has suggestions for fitting these into a total evaluation program.
Several states have adopted assessment programs which use new models. Vermont [1991] and Connecticut collect a sampling of student portfolios in order to measure progress in mathematical problem-solving. I have found that the new assessments align much more closely with the goals of understanding concepts and having problem solving skills, and with the type of work I have my students do.

Where from Here?

After doing much of this research, I decided to consult an old friend. This is a textbook that I used in a general math class, and which made more sense to those students than any other text in my and perhaps their, memory. It is Harold Jacobs‘, Mathematics: A Human Endeavor. In the introduction Martin Gardner writes, “I am not against the new math. The trouble is that it is being taught in the same old dusty-minded way as old math.” He opines that a poor teacher will probably do better with old math. He tells of a student who had finished an algebra assignment and had begun to analyze the game of tic-tac-toe. The teacher snatched the sheet away, proclaiming that only mathematics would be studied in [that] classroom. I would like to disagree with Mr. Gardner (for the first and probably the only time). He was speaking of the 60s “New Math,” and I think in these times it would be better to have an incompetent teacher trying to cope with a class of students who did their own thinking – those who did not take the word of an authority as truth.

Jacobs writes that the true significance of math does not depend on its practical use. Authorities insist, however, that to have a relevant curriculum it must be useful in the real world, impacting the bottom line if at all possible. I say we should level with our administrators and our students: point to your head and say, “math is up here.” The profound and simple concepts are being preempted by a few measurable manipulations. Jacobs: “If art were taught this way we’d spent our time learning to mix paint and chip stone.“ [p. xiii]

In the Journal of Research in Mathematics Education: Monograph Number 4 [1990, p. ix], editor Robert B. Davis observes that a common theme of the papers therein is that the ways that math instruction will reform depend significantly on individual views of the nature of mathematical activity. Davis and Maher present an irksome narrative of a teacher discussing a fraction problem with two students. The teacher not only had the wrong solution, but also could not see any reasonable model of the problem other than the teacher’s own. Davis and Maher assert the importance of a teacher knowing the child’s thinking in order to challenge and extend it. Interestingly, the mis-teaching in the story had no long-term effects – the students had no way to incorporate the teacher’s view of the problem into their long-term math framework, even though they were forced by the social situation to temporarily agree [p. 80-9]. This is indicative of the lack of long-term gains in the traditional approach to math instruction.

This lesson has practical implications for schools. Given the thousands of teaching assignments that need to be filled every year, and knowing that not all of those hired have mathematical credentials, and surmising that some if not many are flawed in their own mathematical understanding, what learning environment for the classroom would better equip students to continue to make intellectual gains, the absolutist/authoritarian environment, or an inquiring/reasoning one?

It is widely acknowledged that there is a shortage of qualified mathematics teachers. The problem may worsen. The December 1997 News Bulletin [NCTM, p.1] reports that student enrollments and teacher retirements are both expected to increase from now through the 2003-4 school year. The United States could lose 40% of its current high school teaching force while the high school population grows 13.2%. The U.S. currently has 34% of its math teachers working with neither a major nor a minor in the subject [p. 6].

More recently, researchers looking into the progress of math reforms in schools were inclined to think that teachers who believe that the basic skills need to come first are putting a concern about the futures of their students ahead of what they believed about the subject of mathematics. Teachers are also concerned about the security of their jobs because of the continued use of traditional assessment tools. [Ferrini–Mundi, p.112] I have heard many teachers report concern over the continued demand in some career paths that students show proficiency in paper and pencil arithmetic skills.

What is the right course of action? What should an education professional do? Perhaps a new idea can be found in the Building on Strengths project sponsored by the Educational Development Center of Newton, Massachusetts and funded by the National Science Foundation. In collaboration with the American Mathematical Society, they interviewed mathematicians and teachers from around the country. The main question they asked was “When you consider what high school graduates should understand about mathematics, what do you care most about?” A second component of the project was a national colloquium held in March of this year in Washington, D. C. Participants took part in presentations, discussions, and break out sessions organized around teaching themes. For a summary of the proceedings see [Scher]. Despite widely varying opinions on some issues, there were many areas of agreement. The intent of the project is to enable educators to focus on the commonalities and build improvements from them. In an on-going phase, both the EDC website and Notices are publishing fuller transcripts of the opinions expressed both in the proceedings and in the surveys.

I can see good reasons to have a unified profession knowing where it wants to go. Of course there will always be disagreements, but they should not be used as brakes by any person or faction for the purpose of halting progress. With all the literature, electronic information and sharing opportunities, and professional development activities, there is no reason to make uninformed or unsubstantiated decisions with regard to classroom practices. If everyone agrees that improvements can always be made, we should see no one standing still. We should all be moving when we do mathematics just as Phyllis Curtain does when she makes music.


Artzt, Alice F. and Newman, Claire M. How to Use Cooperative Learning in the Mathematics Class NCTM, Reston, VA, 1997.

Bruner, Jerome Haste, Helen eds. Making Sense: The Child’s construction of the World Methuen & Co. 1987.

Bruner, Jerome On Knowing: Essays for the Left Hand Belknap Press, Cambridge, MA 1979. (1st ed. 1962).

Burns, Marilyn The I Hate Mathematics Book Little, Brown & Company, Boston, 1975.

Burns, Marilyn and McLaughlin, Cathy A Collection of Math Lessons from Grades 6-8 Cuisenaire Co. of America New York 1990.

Charles, Randall et al., How To Evaluate Progress in Mathematical Problem Solving NCTM, Reston, VA, 1987.

Davis, Robert B. Learning Mathematics: The Cognitive Science Approach to Mathematics Education Ablex Publishing, New Jersey, 1984.

Davis, Robert B. et. al. JRME Monograph #4 Constructivist Views on the Teaching and Learning of Mathematics. NCTM Reston, VA 1990.

Dunham, William, The Mathematical Universe John Wiley & Sons, NY 1994.

Duckworth, Eleanor The Having of Wonderful Ideas Teachers College Press, New York, 1987.

Ferrini-Mundi, Joan and Johnson, Loren JRME Monograph 8: The Recognizing and Recording Reform in Mathematics Education Project: Insights, Issues, and Implications NCTM, Reston, VA, 1997.

Jacobs, Harold R. Mathematics: A Human Endeavor W H Freeman, San Francisco, 1970.

Krulik, Stephen, ed. Problem Solving in School Mathematics NCTM (yearbook) Reston, Virginia, 1980.

Labinowicz, Ed The Piaget Primer Addison-Wesley, Menlo Park, Calif. 1980.

Lane County Mathematics Project, Problem-Solving in Mathematics Dale Seymour, 1982.

NCTM, Curriculum and Evaluation Standards Reston, VA, 1989.

NCTM, Developing Number Sense 1991.

NCTM, A Core Curriculum 1992.

NCTM, Assessment Standards 1995.

NCTM News Bulletin ”Impending Teacher Shortage Might Hit Math Hard“, Vol. 34:5, Dec 1997.

Polya, G., How To Solve It Princeton University Press, New Jersey, 1957.

Polya “On Solving Mathematical Problems in High School” (ed. Krulik) NCTM, 1980.

Resnick, Lauren and Ford, Wendy The Psychology of Mathematics for Instruction Erlbaum Associates, Hillsdale NJ, 1981.

Reshaping School Mathematics: A Philosophy and Framework for Curriculum
MATHEMATICAL Scinces Education Board, national Research Council, National Academy Press, Washington, D.C. 1990.

Scher, Daniel, “The Building on Strengths Colloquium” Educational Development Center, Newton, MA, 1997.

Schoenfeld, Alan H. “Heuristics in the Classroom” (ed. Krulik) NCTM, 1980.

Schoenfeld, Alan H., Mathematical Problem Solving Academic Press, Orlando, 1985.

Vermont Portfolio Project, Teacher’s Guide VT Department of Education, 1991.

Zinsser, WilliamWriting to Learn Harper & Row, New York, 1988.

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Appendix 1

LA Times:
Tuesday, December 2, 1997

"State Endorses Back-to-Basics Math Standards"
Education: Board would stress fundamental skills, discourage calculators
and develop new test. Rules sacrifice understanding of key concepts,
critics say.

By RICHARD LEE COLVIN, Times Education Writer


SACRAMENTO-- The State Board of Education on Monday endorsed a controversial set of no-nonsense standards for math education from kindergarten through seventh grade that emphasize correct answers and lots of practice while discouraging the use of calculators. The first statewide math standards, the target of attacks by critics who say that they sacrifice thinking for rote memorization, will guide the development of a new state test aimed at monitoring how well California's public schools are teaching key subjects.

..Even the mathematicians in the audience Monday could not agree on whether the board's standards were superior to those put forth by the standards commission. Ralph Cohen, a math professor at Stanford University who helped the board write its draft, said the board's document was clearer in stating what students must know while also presenting the skills in a logical progression from kindergarten to seventh grade. Students who master its contents will be far better prepared mathematically than most California pupils are today, Cohen said. "Their skills will be strong, their problem-solving for sure will be strong because they will have the skills with which to solve problems, and certainly their conceptual understanding will be strong because you can't
ask kids to understand concepts without giving them the tools," he said.

But Dan Fendel, a mathematician at San Francisco State, insisted that students who master the standards will be adept at only one part of mathematics-- number crunching. The document approved by the board "shifts the focus to a very computational look at what math is and that's not what math is about," he said. He also opposed the board's decision
"For anybody to take a square root without a calculator is the height of absurdity," he said.

The vehemence of the views of both sides reflects the fevered debate over math education that has been building in California and across the nation. Responding to the fact that relatively few California students take advanced math classes-- only one in six now takes more than one year of algebra and only one in three completes geometry--math educators have been trying for decades to find a way to teach the subject in a way that was both interesting and rigorous. For the past decade or so, support has been growing among math teachers for stressing the ways math is used outside the classroom, or trying to make lessons more concrete by using blocks or folded paper to illustrate concepts, for instance, or enlivening lessons with games. But these so-called reform approaches have been met with growing skepticism among parents and many mathematicians who worry that pupils today are failing to internalize the basics.

Attempting to settle the dispute, the state board last year adopted an advisory policy that sought to balance the competing views. On Monday, though, state Supt. of Public Instruction Delaine Eastin said the new standards tip that balance too far in the direction of skills.

Copyright Los Angeles Times

Appendix 2

MAA President's Column
by Ken Ross

Back to Math Education Reform
[reformatted from the internet]

The past ten years have seen a remarkable amount of progress in improving mathematics education at all levels. The goal is to enable all students, including those from all racial and ethnic backgrounds and both genders, to master and appreciate mathematics. The emphasis is on understanding mathematics rather than thoughtlessly grinding out answers. For various reasons, there is now an increasing amount of resistance to what is usually called "math reform," which reflects some serious concerns that need to be addressed.

Pre-college math reform, based in large part on the NCTM Standards, and college math reform, usually labeled "calculus reform," are compatible in their goals and are now facing similar resistance. I believe that we are all in this together and that we need to work together to maintain momentum and establish better mathematics education for all. Parents, teachers, and the general public need to realize that the new approaches make sense and will empower the young people for the next century.

Unfortunately in the past, much of mathematics has been presented as a bunch of rules - rules for manipulating numbers and symbols. Underlying principles, general problem solving techniques, and serious quantitative thinking got lost. Certainly much of the interest, beauty, and fun vanished. A major thrust of the current reform movements is to present mathematics in a much broader context. It encompasses ideas and techniques that aren't even seen in traditional treatments of mathematics,
and they are interconnected. Mathematics isn't just a sequence of isolated topics that are to be struggled with, learned (or not), and forgotten.

At the college level the emphasis has been on "calculus reform." An excellent overview of calculus reform can be obtained by reading the articles in the January 1995 issue of UME Trends. As a starter, I especially recommend Alan Schoenfeld's article titled "A Brief Biography of Calculus Reform." A more formal and in-depth report can be found in the just published MAA report Assessing Calculus Reform Efforts, edited by J. R. C. Leitzel and Alan Tucker. This is a very readable and interesting
account of the history and current status of calculus reform. Where there's hard data, these reform efforts have been largely successful.

Many people within the MAA and other mathematical organizations are working hard to improve teacher education programs, develop new curricula, and help collegiate mathematicians get involved in the schools. The term "calculus reform" is misleadingly restrictive, because the changes at the post-secondary level extend far beyond calculus. A glance through the programs of the past few national mathematics meetings, especially the minicourses and sessions of contributed papers, shows that there are parallel changes in the way abstract algebra, linear algebra, differential equations, and precalculus are taught. More specialized courses - such as dynamical systems, Fourier series, and modeling - are also being taught in new, exciting ways that involve taking advantage of the new technology. The current push for calculus reform got its jumpstart from the now famous Tulane Conference in January 1986. During the same period the NCTM Standards were being created. They were published in 1989 and have been very widely accepted and used. The current political and sociological climate has led to some backlash. The NCTM is aware of this serious threat and has appointed a task force that will seek appropriate responses. The MAA representative on this task force is Naomi Fisher; her email address is u37158@ uicvm.uic.edu.

The goals of the NCTM Standards, which address mathematics education for K-12, are to "Create a coherent vision of what it means to be mathematically literate both in a world that relies on calculators and computers...and in a world where mathematics is rapidly growing and is extensively being applied in diverse fields," and "Create a set of standards to guide the revision of the school mathematics curriculum and its associated evaluation toward this vision." The vision calls for changes in the curriculum, including new content such as probability, statistics, and discrete mathematics, as well as for different approaches to some of the topics in the existing curriculum. It is envisioned that students will (1) learn to value mathematics; (2) become confident in their own ability; (3) become mathematical problem solvers; (4) learn to communicate mathematically; and (5) learn to reason mathematically. Each of these goals is elaborated on. For example, (3) states that "students need to work on problems that may take hours, days, and even weeks to solve . . . some may be relatively simple...others should involve small groups or an entire class working cooperatively. Some problems also should be open-ended with no right answer...." The most vivid changes in teaching have involved technology, but the real focus has been to improve the learning of students and to make sure that a wider group of students is able to benefit than has in the past. Changes in instructional practice include hands-on experiences using technology, increased focus on conceptual understanding, cooperative learning, student project activity, extensive writing, and less reliance on timed tests in assessment.

It's easy to detect flaws in any movement as broad as the reform movement and to overlook the progress. In February I attended an NSF/DOE conference on systemic reform in science and mathematics titled "Joining Forces: Spreading Successful Strategies." It became clear at this conference that a large number of people across the country provide excellent education in various creative ways. The focus of the conference, as its title suggests, was the daunting but vital task of identifying those programs that really can be duplicated throughout the country, without losing their
effectiveness, and then implementing them nationwide.

Statistics from the Department of Education (the Condition of Education, 1994) show that we are making progress. For example, substantially more high school graduates in 1992 are taking mathematics courses at the level of algebra I or higher than their counterparts in 1982. Thus in 1992, 56.1% of the high school graduates took algebra II and 7O.4% took geometry, while only 36.9% and 48.4%, respectively, took these courses in 1982. During the same period, the percentage taking remedial or below-grade-level math dropped from 32.5% to 1 7.4%.Another table shows that these dramatic shifts are happening for all racial/ethnic groups. We don't hear much about such statistics hidden in dusty government tomes, even when they are positive! With such big changes nation-wide in ten years, something right must be happening.

So what are the concerns that are leading to resistance to these changes? One is that the laudable focus on understanding has led to some decline in mathematical skills. Since it is easier to measure and spot deficiencies in skills than understanding, this problem can easily be over-emphasized. on the other hand, this is a serious problem, especially since our future scientists, engineers, and mathematicians must obtain both substantial understanding and substantial skills. The reform movements need to address this issue.

For teachers who are following the NCTM Standards, there's no doubt that it is more difficult to determine (or at least quantify) students' knowledge, understanding, and skills. This is now leading to serious challenges as the mathematics community faces assessment issues. Similar challenges are faced by post-secondary faculty as they change their instructional practices. I have no wisdom here except to acknowledge the difficulties tempered with the belief that they can be overcome, though it won't be easy. To steal a quote, "Tests should measure what's worth learning, not just what's easy to measure."

Another concern is largely political. There is a natural American resistance to centralized control. Some fear that the NCTM Standards are subverting local control. Wide-spread reform is hard to accomplish in such an atmosphere. We are still suffering from the bad taste that so-called "New Math" left in America's mouth thirty years ago. That was an effort that focused entirely on the curriculum. An equally serious problem all along has been in the pedagogy. Finally, in the United States, we are suffering from a widespread case of anti-intellectualism wherein all of us are experts on the schools because we once attended schools. We need to continue to learn from the successes and failures of other countries where many of the same problems are being faced.
In the long term, the college and pre-college reform efforts are intimately linked. Students and parents expect pre-college mathematics education to be a preparation for college-level mathematics. This is an important endeavor and everyone needs to be involved . A future column will discuss the role of post-secondary faculty and the MAA in addressing this interface. Students' mathematics learning should be seen as seamless as they progress K-16. We've come a long way in ten years. We have a long way to go.

This article was taken from FOCUS Vol.15, Num.3, Jun '95, pgs.3-4.

Math Forum
11 January 1996

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