Math 629 - History of Mathematics

Teaching by example ala the Ahmes papyrus.

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I find it very valuable to teach mathematics by example. Many times, teachers give a general example and have students work the problem using specific criteria. With this method, students become good at plugging numbers into equations/formulas and chugging out an answer. This is a necessary skill for students. However, if material is always presented as going from the general to the specific, then there is no growth other than computational skills.

When mathematics is taught by example, as in the Ahmes Papryus, the concepts change. Students have to go from the specific to the general, which is a very valuable tool in becoming life-long problem solvers. When I teach my classes, I always present examples and expect the students to find similarities. I introduce each unit with a graphing calculator lab where examples are given and the students are to generate graphs and try to discover the patterns.

In terms of my preparation, it would be much easier for me just to teach the lesson, telling them what they need to be able to do. I also think it would be wrong. Part of my job is to prepare students for what comes after high school. Whatever path a student takes after high school, there are skills that are necessary. Business leaders tell us that they are looking for problem solvers—kids that can think out of the box. Teaching by example forces students to think for themselves, to look for patterns and generate their own ideas, instead of using someone else’s. This method is very hard for a lot of kids. But they’re better off because of it.

Having taught in high school for a year and a half, and teaching at a junior college at the present time, I have found that although I myself prefer and appreciate the abstract approach and representation to problems and concepts, it is somewhat difficult to explain this in a simple manner, and that it is difficult to motivate a lesson involving this unless there is some use that the student sees in it (the applications are not always obvious). Using examples is a great way to motivate or drive a lesson, in that students can get a feel for what this abstraction, generalization, or concept can do for us. A typical example of this comes from high school geometry, when trying to compute the area of any polygon, just introducing the formula will do no good, except for the few students who are already at ease with abstract representations. Most students need to see the polygon, with its dimensions labeled, and see the computation performed, and see some non-examples. In this way, the concept sinks in, they know how and when to apply what they have learned.

A graduate who has learned in this fashion would be prepared (we hope) to be what we would call an engineer, or perhaps a person in some other applied field. The person would not necessarily be concerned with why the tool (formula or generalization) works, but with how and when it works, and with this knowledge, things get done (such as the construction of structures).

Having been taught math in an abstract manner, it is difficult to fully understand the mindset of a student in Egypt at this time. It appears that the method of teaching by example only would be frustrating. As indicated in the text, there may have been some other teaching methods (abstract) used in Egypt, particularly ones that drew some well-known Greek mathematicians – but for the purposes of this question, I’ll address the example method.

Objectively, it would seem that teaching by example would be simpler, but with no flexibility. Again, it’s hard to think about learning this method without actually comparing the fact that I know the abstract method. However, one could imagine how using abstract numbers like b and h for base and height of a triangle (or other shape) could be confusing to some people. Going straight to using numbers to show a student, by example, how to say, find the area of a triangle, seems to be an easier method. But, without the abstract lesson, a student would not really understand what he is learning. This inherent inflexibility would be frustrating to a student when trying to solve a problem with different conditions. A student would not know the abstract equation for area and know which numbers (or measurements) to substitute.

Graduates of this example method would tend to be somewhat capable but extremely inflexible. These graduates may be able to develop an innate sense of abstract with a lot of experience – but it would take a long time and the ensuing time would prove to be without fruit, depending on the profession. For example, a construction foreman working on pyramids, etc may be able to learn through experience how to do necessary geometry for the job through time. But, an engineer would not have the abstract mindset to further develop new ideas or extrapolate current ideas.

This method is then good and bad depending upon the future of the graduate

Teaching by example has the benefit of allowing the student to learn steps and then apply those steps to other like problems. Once the steps are learned – which requires mainly memorization – successive problems of the same type can be worked out without much difficulty. A limitation however is that if the student is having difficulty remembering the steps, there is no underlying understanding there to help the student figure out what the next step is. But as long as no new type problems or variations arise, this method is probably the easiest and most efficient, at least for learning how to do a certain thing. Teaching by example does however require that many more examples be presented and practiced as each unique situation must be covered. There are 87 problems in the Ahmes Papyrus, probably for this very reason. Everything had to be, and was claimed to be, covered. This abundance of work is still probably easier than the effort required in understanding a process, which is initially more difficult, for both teacher and student. It takes a different level of effort to teach and learn the theory behind things. But in the long run, the student will be able then to use their knowledge to work any type of problem in the field or to find answers to new questions. However, since the Ahmes Papyrus purported to be a “complete and thorough study of all things…”, new problems may not have been anticipated by the Egyptians at that time.

Teaching by example leaves the student capable computationally, but it has the drawback of leaving the student less competent for any situation that is out of the ordinary.
By bypassing the fundamental understanding of concepts that is necessary for both adaptability to new problems and development of the discipline, teaching only by example limits the graduate’s ability to expand his or her knowledge into new aspects of the field. Because of this, only those really drawn to or predisposed for the discipline would ever be able to advance the field to a new level. In fact, Egyptian mathematics failed to progress very much over the course of a few thousand years and their “teach by example” approach may well be a major reason why.

What was education like for a scribe and what are they able to do after their education? In his book, Swetz discusses the life of a scribe in ancient Egypt or Mesopotamia:

He (and it is almost certain that it would be a boy; girls, while not forbidden a scribal formation, are almost entirely absent from the records) would of course first go to school.
What would the scribe learn during his schooling, which lasted at least ten years? We possess, from both civilizations, examples of school exercises including mathematical texts…
…They are two kinds, table texts and problem texts.
…Egyptians used their tables of multiplication, square roots, and sums of fractions. This is also the way in which they used the problem texts. A typical example, from an Egyptian papyrus of the middle of the second millennium, begins with a statement of the problem to be resolved…
Here the data are presented in the form of concrete numbers rather than abstract variables, followed by a step-by-step, solution with the answer at the end. Each step makes use of either the result of a preceeding step or of one of the data given at the beginning of the problem.
No argument is given to justify the procedure nor is any explanation offered for its form. But even with the numerical values included, the nature of this form is quite clear. Thus the student would be able to solve any other problem of the same type. Moreover these problems are often grouped in such a way that the techniques learned can be immediately applied in other cases…
…The main purpose of mathematical school exercises was to give the student scribe practice in the mathematical techniques used in solving problems. Technical drill, not direct application, was the main point. For this reason many of the apparently “practical” problems in these texts are far removed from real life…
The pedagogical purpose of all this is clear. Moreover, the structure of the problem and table texts permits an alternative approach to abstraction and generalization in mathematics. Rahter than take the path of incresing symbolization marked by a heirarchy of “levels of generality”, the Egyptian and Babylonian approach is to create a network of typical examples in which a new problem can be related—by a form of interpolation—to those already known.
…There were two paths that a young scribe could follow. A few would become teachers of mathematics themselves, perhaps exploring further problems that could be presented to a new generation of schoolboys…
…The graduate could also become an accountant—a calculator of work, rations, land, and grain. (Swetz 108-110)

This type of teaching would be simpler in a sense that only a specific type of problems was taught and solved, but would be more difficult in the fact that not all problems that arose could be solved in the manner of the problems shown. This would result in graduates having a limited knowledge of how to solve “real world” problems. If it was not something similar to an example that had been shown to them, or something that could be derived from an example shown to them, the would have difficulties solving it, if they could solve it at all. Boyer has the following to say about this method of teaching:

A number of deficiencies in pre-Hellenic mathematics are quite obvious. Extant papyri and tablets contain specific cases and problems only, with no general formulations, and one may question whether these early eivilizations really appreciated the unifying principles that are at the core of mathematics. Further study is somewhat reassuring, for the hundreds of problems of similar types in cuneiform tablets seem to be exercises that schoolboys were expected to work out in accordance with certain recognized methods or rules. That there are no surviving statements of these rules does not necessarily mean that the generality of th erules or principles was missing in ancient thought. Were a rule not there in essence, the similiarity of th eproblems would be difficult to explain. Such large collections of similar problems could not have been the result of chance. (Boyer 41)

The Ahmes Papyrus teaches mathematics by example only. There are no written rules to be generally applied to all problems, no principles applied to a wide range of problems. Mathematics can be learned in this manner but it brings up many issues that we not thought of in earlier times. Teaching in this manner can be simpler than by current methods of exploring rules and applying them to slightly differing problems. Teaching by the Ahmes Papyrus involves teaching students a set of steps to follow that will produce the correct answer to that particular type of problem with no variation. The steps never change so the student has nothing to reason out or think about. However, the Ahmes Papyrus problems do not offer different situations, so teaching the same type with different examples can get tedious. If a student does not understand how to solve one problem, there is no way to re-teach with out simply doing another example that was just like the one the student didn’t understand. Students that master a type of problem from the Papyrus are very capable of solving any type of problem they have encountered while studying the Papyrus. Many of the problems seem to be based on real-life situations that would come up often in daily life of early Egyptians. While students of the Papyrus are capable of performing mathematical tasks, they will have little luck at solving a type of problem of which there is not an example. Today’s students are expected to take what they know about one subject and apply it to another. This way of thinking and teaching was not possible in the times of the Papyrus. Those students were not flexible in what they could solve. In this way they were very limited, but they did have a wide range of practical knowledge to apply to real life situations.

Based on the Ahmes Papyrus, many historians have conjectured that the Egyptians taught math by example, rather than by theoretical concepts. If this is true, then the mathematic students would need to be good at copying steps, rather than using analytical skills to solve problems. When all that is required of a student is the process of copying steps, no real understanding of the actual problem is necessary. Therefore, from the students’ aspect, this would be an easier way to learn. But, because no real understanding is taking place, a student that learns only a set of steps that work for a given problem may not be able to properly adjust those steps as varieties of the problems present themselves. It would also be reasonable to expect that the student would only know to use the process learned in very controlled, relicas of the original problem. Therefore, graduates would say be very adept at finding volumes of truncated pyramids, but would not necessarily be able to apply their knowledge about volumes to other shapes. It would seem this would lend itself to machine-like graduates; graduates that can crank out a necessary answer when all of the information has been input in an exact manner.
I have seen this type of thinking in my students on many occasions. One area in particular would be in solving linear equations. If they learn only the process of say solving x+b=c by subtracting b from c, they can only solve problems in the same exact format and will be completely confused when a problem of the form ax +b = c is presented. Those that have learned by example only will not know what to do, while those students that have understood the idea that solving for the variable is like finding a missing piece of a puzzle (and that they must work their way back to the piece) will understand the correct order for undoing the operations on the variable and thus be able to solve for x.
In my opinion, a student that is taught by example, is not a truly ‘learned’ student. I am sure that there are some students that when taught by example can look beyond the exact numbers to the concepts and ideas and generalize the information so as to be able to use and adapt it in other situations, but on the average, it would seem that students taught by example only would be fairly useless (unless one only needed a calculating machine).

There are many problems that arise when teaching mathematics by example. Although teaching by example is often easier to explain, it creates a group of students which cannot connect the concepts taught to more complicated mathematical problems. Many students have a difficult time understanding the theories that support the methods necessary to solve all problems within a particular concept. If the instructor is purely teaching by example, the students are not forced to deal with the theory, thus they will not be able to apply the theory when working as an individual.
Teaching by example can be easier on the instructor in the sense that they would not have to try to explain the theory, which is often the most difficult part of the job. They would also not be challenged with the infamous question “Why?”. However, at the same time the instructor is creating excessive amounts of work for themselves. If the students cannot connect from one problem to the next, the instructor would have to stand over the shoulder of the students as they worked each individual problem.
This will eventually lead to the type of student or graduate that all instructors fear. This being the student who has not been taught to think on their own, has never questioned why, and, more importantly, hasn’t been taught to care why. These graduates we would be producing would not be capable of furthering mathematics. They would only be able to reproduce what they have been taught. Therefore, they would be very inflexible to new ideas and concepts that may arise in our future.

This is an interesting and practical question in that much of the mathematics taught today has no relationship to real life problem solving. The Egyptian method was more hands on, whereby real life problems were solved in a practical matter, no theory behind the calculations.

Teaching in this manner would be simpler in that there would be a physical object, not an abstract idea, so that the student would be able to get an instant visual feedback on what was being discussed. For example, in today’s world this could be applied to shop mathematics, whereby the student would be able to visually see the relationship between the circumference of a pulley and rotational velocity to linear velocity on a manufacturing conveyor belt. Or see the actual effect of increasing loads as moments on beams.

Teaching the Egyptian method could make it harder to develop theory that would transport across to other applications. By focusing on only one application the advantages of learning would be limited.

Graduates would be capable in the particular field that they were studying. For example, if a high school student were to decide to pursue carpentry as a career, it would make much more sense to train him on practical examples related to carpentry, versus astronomy. They would become a specialist in that trade. I also think that they would be flexible, in the sense that what they would learn with experience would help them solve problems in other areas. Hands on experience is more useful in solving everyday problems.

A great example of practical versus theoretical studies was shown by my Uncle every summer. He was a field supervisor for Atlantic Richfield in the Freer, Texas oilfields and every summer he would be host to two Petroleum Engineering students working as summer interns. After showing them around he would take away their calculators and ask them to work several problems related to their studies. Nine times out of ten they would be lost without the “theoretical tools” and my Uncle would show them how to do it in the real world. Indeed, there are many such “engineers” who never had the benefit of classroom studies.

Teaching by example ala the Ahmes papyrus.

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