Untitled

Devin Sullivan
76100o
Sauer
4/25/97

MATHEMATICAL ABSTRACTION IN THE PUBLIC SCHOOLS:

Advantageous or Unnecessary?

In the late nineteenth century a mathematician named Georg Cantor developed a new mathematical theory called Set Theory. This new branch of mathematics is based around the premise that all mathematical ideas, relations, operations and values can be formulated into sets that relate them. Instead of doing operations on values, operations can be done on sets. Even the operations themselves can be grouped into sets, so the backbone of set theory is the logical manipulation of sets. Set Theory was accepted worldwide in the early twentieth century and revolutionized the world of mathematics. Rigorous proof of conjectures, originating from a set of axioms, became the foundation of modern mathematics. A distinguishing characteristic of Set Theory and other modern mathematical ideas is that it is extremely abstract. Set Theory, Number Theory, and Group Theory are examples of branches that are considered to be "higher" mathematics (the name is misleading; it doesn't mean that the math itself is any harder, it actually can be simpler.) These branches of mathematics constitute the theory of the theory, in that they lay the foundations of other, more applied mathematical theories. One of the most pressing issues of mathematical education now and for the last fifty years has been the controversy over whether and when to begin to introduce the ideas of set theory and modern mathematics into the primary or secondary school classroom. Proponents of early introduction of Set Theory say that it is vital to the true understanding of the principles behind mathematical problems (Varga 58, Piaget in Goffree 34). An early understanding of the roots of mathematical principles can lead to a deeper understanding later in schooling. If one understands the theory behind mathematical equations, they are more likely to be able to manipulate the equations if called upon to do so by a real world application. One of the major influences on proponents of this program is Jean Piaget (a revolutionary French psychologist, philosopher and cognitive scientist). His cognitive model of a child's learning process was used to support the New Math program of the 60's and 70's (Goffree 36). On the other side of the issue are those who do not support early abstraction in mathematics education believing mathematical abstraction to be unnecessary for those who do not plan to become mathematicians (Crawford 14). These people believe math classes should focus more strongly on the applications of mathematical ideas rather than the theory behind them. Again, many of the supporters of this position base their arguments on a cognitive model of a child's learning methods. The model is that of Lev Vygotsky, a russian psychologist and philosopher. There also exists a middle ground of this issue. Many argue that mathematical abstraction is valuable in a child's education, but it must come after a solid foundation in practical mathematics because children can rarely comprehend that which has no pertinence to their lives (Khinchin 66, Freudenthal 152-153). Can a successful program be formulated that incorporates abstract mathematics into early schooling or would such a program be impractical?
Jean Piaget dedicated a portion of his work to the psychological processes involved in the learning of mathematics. This led him to many ideas on the way mathematics should be taught. Piaget once said:
"Experiments that we have been able to carry out on the development of mathematical and physical ideas have demonstrated that one of the basic causes of passivity in children in such fields, instead of the free development of intellectual activity they should provide, is due to the insufficient dissociation that is maintained between questions of logic and numerical or metrical questions.' (Piaget in Penrose 23)

Piaget was for the most part, a proponent of logic taught early in the development of students, but he believed that logic was an acquired skill that came with maturation (Penrose 22). Only when a child reached a level where he or she could think logically could the child be introduced to the more abstract nature of mathematics (Penrose 22). When the child reached this level, Piaget believed, he or she was ready for an almost complete separation of "questions of logic and numerical or metrical questions" (Piaget in Penrose 23). In this way, Piaget concluded, children are free from the hassles of number manipulation and can concentrate on the abstract ideas and theoretical relations. As a result, Piaget proposed that early math be taught on a geometrical level (Goffree 36). Geometry is both abstract in its foundations but also tangible to a student who needs some type of physical representation of a problem. Piaget's work was so influential that it shaped the curricula of the "New Math", a program prevalent in the 60's and 70's that introduced abstract mathematics to students at every level.
"It is only possible to abstract from the concrete" (Varga 58). Tamas Varga, a Hungarian mathematics educator, argues that an early introduction to abstract mathematics would be valuable later in a child's mathematical development. The usual first steps of a young child's mathematical education are arithmetic and geometry without any concepts of set theory. What would happen if a mathematics course with some level of abstraction was given at the initial level of education? This is the study that Tamas Varga performed in Hungary in 1961, well before New Math got off the ground.
Case Study: Varga did not want to fill his experimental course with subject material that was too difficult for the seven-year-olds that took part. He structured the course around some of the flaws that he believed to exist in the mathematics education that is usually given at that level. Some of the changes he made were:
1) The children were led through exercises that were meant to establish "=" as a symbol meaning "the same number as" rather than "gives the result" (for example, 3 plus 4 is the same as 7, 3 plus 4 does not give the result of 7).
2) Equality was considered to be a "border case" between "greater than" and "less than".
3) The definition of place value was stressed by doing arithmetic in bases other than 10.
4) Using rods, addition and subtraction was shown to be geometrically independent of the numbers that are involved.
The results of the experiment were encouraging (Varga 83-84). Children who took part in the experimental group performed better than children of the control groups (who received the standard teaching approach) on standardized tests given to children of that age in Hungary. Varga stressed that even though the results of his experiments were promising, it was not to mean that abstract mathematical education should go into widespread use in every corner of the globe. He insisted that the goal of his experiment was to show that it is possible to teach the fundamentals of abstract mathematical ideas in place of the usual manipulation of numbers. He wished that every math teacher in the world be given the opportunity and if deemed appropriate to teach up-to-date mathematics in an up-to-date way (Varga 83-85).
Piaget and Varga tend to have similar views of the necessities of the mathematical education of children. Both are proponents of early abstraction in the education of a child (possibly as early as age seven as in Varga's experiment). Piaget's ideas are based more on his cognitive models of young children rather than educational experience. (Goffree 38) Piaget was a brilliant analyst but in his later years, he let his experiments involving children become more arcane and theoretical, dedicated to topics such as the cardinal nature of the natural numbers rather than order or counting relations. (Cardinality is a measure of value without any representation of order. The integers have value and order, unlike cardinalities. It is used to develop congruencies between sets.) Piaget therefore let mathematical concepts (completely foreign to the children in his study) define their mathematical cognitive processes (Goffree 38). Varga's conclusions are much more well-founded than those of Piaget because they take into account a year long case study. Varga's results have to be considered as strong evidence supporting abstraction at a very early age.
A. Ya. Khinchin, a well-known Russian mathematician and originator of many mathematical teaching techniques represents the middle ground of the abstraction in education debate. He argues that although it is valuable for children to be introduced to modern mathematics at an early age but it lacks practicality. Khinchin believes that there must be motivation for the children involved (Khinchin 66). He states that whether or not children are intellectually ready for abstract math at a given age, they will almost definitely not be willing to learn pure math solely for the sake of pure math (Khinchin 67). He uses the inclusion of complex numbers into high-school math curriculum as an example (Khinchin 67). The complex number is an important part of pure and applied mathematics but it has little meaning to an 11th- grader. A complex number is a perfect example of an abstract mathematical concept in that it is a number that, by definition, has no real value. As a result, the average 11th grader who wants to do well in that particular math class will memorize rules and relationships taught by the teacher without having a clear idea of the theory behind them (Khinchin 67). The student, having never truly understood the material, will forget the relationships soon after the class is over. The math teacher knows that the students do not understand the material on a fundamental level but that isn't addressed because the students "are required" to have the introduction to complex numbers as dictated by the course curriculum (Khinchin 67). Khinchin's main objection to the introduction of abstract mathematical ideas into the early classroom is one of method. Only in very specific ways can such a course be taught. If an early abstract math course is structured anything like the type of classroom that he believes pervades the public schools, then it is not beneficial to the students and hence, Khinchin would not be in favor of such a course.
Hans Freudenthal is possibly the most influential figure in 20th century mathematics education. Freudenthal has always had mixed views on the ideas of higher level mathematics in the younger student's classroom. He once told a story of being in the company of a room full of "intellectuals" at a party. Someone performed a long, elaborate, card trick that culminated with the subject picking his original card out of a deck of 27 (Freudenthal 95). The mathematician in the group was able to predict the exact position of the original card before the card was picked up. The rest of the crowd responded with indifference, believing that the mathematician had "known the trick" refusing to believe that the trick was in effect, a mathematical problem with a definite solution (Freudenthal 95). This is Freudenthal's greatest difficulty with the state of mathematics instruction. Freudenthal did not expect anyone but the mathematician to be able to solve the problem but "it is disappointing if intellectuals who learned a bit of mathematics at school look upon this trick as if it were juggling or an empirical fact." (Freudenthal 95) Freudenthal, like Khinchin, warns of what he calls formalization, or the use of mathematical rigor almost to a fault. The result of this effect is that "a stream of formulae supersedes mathematics." (498) As a result, the ideas presented can be obscured by the formality of the presentation or even poor organization. Freudenthal uses the term "mathematization" in much of his academic writing and has made it one of the key buzzwords of the mathematics education community. Mathematization is defined by Freudenthal to be the use of proper mathematical rigor combined with optimal presentation order to maximize the understanding of the concepts of a mathematical model within the student's mind (134). Freudenthal's major point is that mathematizing does not occur in the schools, and he uses the card trick example to support that point. If students were being taught to think mathematically, they would have no problem understanding that the card trick has a mathematical solution. Freudenthal was an advocate, like Khinchin, of modern mathematics in the schools but he too believed that it must be introduced at an age where it could be appreciated (153). He fought "New Math" for all of the years of its existence due to its lack of mathematization and its overabundance of formalization (Freudenthal in Goffree 39).
A study performed of first year university students (Crawford 1993 in Crawford) supports what Freudenthal disdains about much of the mathematics education that exists in the world. Crawford concluded of her study, "...more than eighty percent of the sample studied viewed mathematics as a set of rote learned ruled and techniques and approached mathematics learning in a fragmented fashion with the intent to reproduce, using pencil and paper, axioms and standard techniques for examination." (Crawford 19) Even though the scholars mentioned thus far differ on certain issues, they would all agree that this is not the conception that students, albeit college students, should have about mathematics.
All of the authors mentioned so far have been somehow tied to academic mathematics (with the exception of Piaget) yet they have all made claims as to the optimal way to teach children in high school, junior high school, and even elementary. It is a fascinating "coincidence" that none of these scholars have ever taught children of the ages just mentioned (with the exception of Varga's experiment). How do teachers in general view the issue of greater levels of abstraction in the public schools? William Carroll is high school teacher at Roosevelt high school in Chicago. He claims that through years of teaching mathematics to children who were, to say the least, not enthralled by the subject, he has fallen back on a plan that uses a small amount of theory (which he feels is necessary for a basis) on which application can be piled to increase the theory's tangibility. He uses a simple format that is employed by many text books and teacher worldwide: use detailed examples that show the progression from theory to theory application to the solution of a given problem. This method, says Carroll, is the only way that students (who do not plan to pursue upper level mathematics, which is a vast majority) are able and willing to absorb the material. Where do set theory and other modern mathematical ideas fit into his curricula? The answer is nowhere. This is one of the reasons that New Math failed. It simply isn't practical for classrooms like Mr. Carroll's.
During his short but influential life, Lev Vygotsky, philosopher, and psychologist, made remarkable changes in the view of human development and consciousness. Vygotsky wrote that there was a fundamental difference between "activity" and "operations" within the content of human brain functions. (Crawford 18) He defined "operations" as relatively subconscious behavior that is stimulated by the existence of a goal (Crawford 18-19). "Actions" were just the opposite, conscious efforts en route to a goal (Crawford 18-19). A study performed in 1986 (Crawford 1986a 1986b in Crawford) found a direct correlation between "active" brain functions and mathematical problem solving. But in the same study it was found that mathematics instruction was traditionally organized in a manner that favored "operation" related brain functions (Crawford 19). The students int he study showed no spontaneous will to partake in any mathematical tasks. This and other results have shaped the Vygotskian view of mathematics education. "A Vygotskian view suggests a new set of questions for research in mathematics education which reflect the fact that mathematical knowledge increasingly shapes and is shaped by human activity and communication in new technologically sophisticated contexts." (Crawford 14) In such a view, Crawford believes, mathematical education is dependent on those who use it. It must be "active" in the idea that it must push along technology, and technology must pull along mathematical education. In such a view, mathematics education must evolve to fit within its time period. This is in contrast with many of the authors mentioned up to this point. Varga, Piaget, Khinchin and even Freudenthal believe that the introduction of simple "higher mathematics" and more abstraction could be beneficial to students still in high school or younger. But basic set theory, group theory, and field theory (modern mathematical theories of which the specifics are unimportant in this situation) are very static ideas and will remain so indefinitely. This is due to the fact that these theories are in essence defined by their initial assumptions. If these initial assumptions are changed, they cease to be sound mathematical theories. The Vygotskian view insists that mathematics education be dynamic and thereby reflect the applications that are prevalent in the time period (and location for that matter) in which the math is being taught. This is in stark contrast with static, abstract base which has been suggested to begin mathematics education. The views of Crawford and Carroll are indicative of the educators, who, for many reasons, would be the last to actually accept the changes in curricula sent down by policy makers and administrators. First of all, very few elementary school teachers even have a basic understanding of set theory, and other modern mathematical devices (Varga 81). If a policy change was made that incorporated the conclusions of Varga's experiment, it would be virtually impossible to educate all of the elementary school teachers in a thorough manner (Varga 81). (The re-education could be uselessly brief, as in the New Math program.) This also holds for the junior high level. More often than not, teachers of mathematics do not have degrees in mathematics and would have to learn the new material (Varga 81). Only when one looks at the high school level do "mathematicians" begin to find their way into education. "To reform mathematics against the wish of teachers would be just as absurd as teaching mathematics to children against their wish." (Varga 80) Inherent in the problem of mathematics teaching reform is the question of whether the teachers want to be reformed. The answer is decidedly no. A program that forces teachers to change their style (which abstract concepts will almost definitely do) is unwelcome with most educators who have established their methods (Freudenthal 168). Even though these factors are legitimate and would pose a serious threat to any attempt to modernize the elementary and junior high math curriculum, in the early sixties a global program called New Math was finding its way into the math curricula of many countries of europe (especially Holland, where the movement originated) (Goffree 28) and even the United States. The purpose of the program was to introduce the modern ideas of set theory and number theory into the classroom in hopes that it would spawn greater understanding of mathematical structures in young students (Goffree 29). New Math was fraught with problems and as a result it had begun to fizzle by the late 1970's. Ironically, New Math's downfall was due to one of the effects that it was supposed to rid. New Math was supposed to allow a student a proper basis from which all mathematics could be derived, thereby removing the need for memorization (which is not the correct way to learn mathematics). "One does not learn mathematics in order to forget it immediately, although New Math has certainly given that impression to many." (Goffree 39) It was found that children learn mathematics by doing. (Goffree 39) New Math simply did not take this into account.
What about abstraction at the high school level? The consensus among experts in mathematical education is that abstraction is a natural and advantageous tool to help students understand the nature and fundamentals of mathematical thought. All of the authors analyzed in this paper with exception of Carroll and possibly Crawford would agree that high school age students are ready for a more abstract mathematical education. Due to the fact that many high school math teachers have degrees in mathematics, they have undoubtedly been well trained in modern mathematical techniques. Even at more rural schools, it seems probable that at least one teacher would have the mathematics background to teach an introductory modern mathematics course. This makes the idea feasible. Would it be advantageous? The experts mentioned in this paper would say yes and I would agree, in theory. But could it be possible that the ideas that these authors have about students is out of date? I am a student at Carnegie Mellon University. The cross section of students here represent some of the most motivated and intellectually gifted in the nation. All four authors mentioned in this paper who support mathematical abstraction in education at the high school level (at the least) also stress that student interest must also be incorporated into the structure of a mathematics course (Piaget in Penrose 23, Varga 80, Freudenthal 62, Khinchin 77). There exists an introductory modern mathematics course at Carnegie Mellon University that is required of all mathematics majors, information-decision systems majors, computer science majors, statistics majors and electrical and computer engineers. Every year, the university prints a "Faculty Course Evaluation" of all faculty and courses at CMU. This introductory abstract mathematics course (called "Introduction to Modern Mathematics") ranked near the bottom of not only all of the math courses, but all courses offered at the university for the 1996 fall semester. Obviously, this course is not structured in a way that encourages students to pursue further abstract mathematics courses. Remember, this is at the college level! (a good college at that) One can't help but believe that an abstract math course structured like Introduction to Modern Mathematics would be even more unpopular at the high school level.
Is another New Math program feasible in the near future? In the public schools, the answer is no. A private school can cater to a specific cross section of the population but the public schools can not. As a result, the average child of age 5 or 8 or 11 is not ready for the mathematical abstraction that would be involved in such a program. It is almost definitive that the public school system does not have enough money to train every elementary school teacher in the country in such a way that it is completely clear to them (this involves more than just a two week seminar).
Could a New Math type program be successful on the high school level? The answer is a resounding maybe. Once the mind is ready to grasp the concepts of modern mathematics it is nothing but beneficial to absorb them. This is agreed upon by all of the experts. The program would, of course, have to be structured differently than Introduction to Modern Mathematics at CMU. There is no point in creating a mandatory class that high schoolers will dislike more than the current one, even if it is beneficial. A new generation of students who were required to take a well planned, well organized, mathematized modern mathematics course would be better equipped to enter an economy where the problem isn't something that's been seen before. If nothing else, the course would teach a generation of young students how to think logically and formulate solid argument and proof where only conjecture and opinion had been before.

works cited: