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The study of deductive geometry in high schools is almost inexistent. Time and the students' level of abstraction make it almost impossible to teach. I have also met math teachers who "hate proofs". Is deductive geometry worth salvaging or do we move towards other areas: optimization, probabilities etc. . . ?From Philip Spencer, University of Toronto:
I have seen many students who cannot follow the course of a logical argument, and who cannot make the connection between a series of symbol manipulations by which one arrives at "an answer" and the progression of ideas that turns those manipulations into a convincing proof.From Edwin Sherman on December 18, 1996:
My own personal opinion is that it would be highly desirable to restore to the curriculum either deductive geometry, or some similar subject, in which students can see the relationships between tightly interconnected ideas, learn the basics of axiom systems and proofs, and encounter rigorous reasoning.
However, perhaps the demise of the subject is an indication that improvements need to be made to it rather than restoring it wholesale to its original state. Certainly Euclid's Elements, while an admirable feat of logic and reasoning, leaves much to be desired pedagogically. There's a vagueness of definition and lack of motivation, leaving students with the feeling that calling something an axiom is simply a crutch when you can't think of a good way to define or justify it, that the laws of geometry are a dry collection of incomprehensible theorems without any relevance, and many other reactions which have contributed to the dislike of the subject by students and teachers alike.
Perhaps our challenge is to rework the material in a way which is less susceptible to these misconceptions; that highlights geometric theorems as things known to be true with greater certainty than simply because they've been true in all the examples we've happened to observe, being instead necessary logical consequences of other things we know to be true; that establishes this way of thinking as useful and necessary in many other unrelated fields as well, with geometry merely a particularly cogent and complete instance of it; that illustrates the process of discovery of new truths through reasoning; and that is more appealing to student and teacher alike.
Does anybody have any thoughts on how this can be done?
I am really sorry to hear that Question even being asked but it is good that it was because it shows the sorry state of the teaching. I do agree that it is difficult to teach this subject and it always will be as long as it is left unmentioned until you get to Geometry in High School. Deductive reasoning is a BASIC part of all Math, (and all life) not just Geometry. For the last 50 years, and probably earlier, Math has been taught by memorizing tables whether it be, addition, substraction, multiplication or division. What has happened is students know 2+2=4 but don't know why.
Deductive reasoning must be started right at the beginning in the first grade. It starts with What is zero, and why does one plus one equal two. As long as math is taught with memorization only, we will have lots of people that will be able to obtain the answer to very complex problems but won't have the slightest idea why it is so. The grade and high schools believe that deductive reasoning is used only in higher math so they leave it to the colleges for teaching. Meanwhile, the colleges are expecting all entering students to already have a sound foundation in it. It goes back to the old adage, you can feed a hungry man or teach him how to fish. Feeding the students formulas, no matter how complicated, is still the feeding end where the deductive reasoning is the teaching of how to fish. This doesn't mean you start with the teaching of geometry although that would help but it can be taught also with simple basic math.
For example, if you have 24 apples in a box, 10 boxes on a pallet and three pallets of apples, the deductive reasoning part is teaching you can't multiply the apples times the pallets without including the boxes. That is where you teach students how to think. That is deductive reasoning.
Until this is recognized we will continue with the sorry state of math general knowledge that we have. Trouble is, we have traveled this path so long that even many of our grade school teachers cannot now answer a student's question of "Why is two and two, four?" Too often it is just answered, "Just because it is."
Deductive reasoning is used in every part of life. It is used in deciding what to buy, what to wear, even what to eat. It is basically simply how to think and knowing how we come to that conclusion and why. That should be the main purpose of education, not memorizing dates, names or even what happen but why it happened. What has happened is, we now have a generation that know how to use a can opener but are perplexed by not knowing why it won't work on a 55 gal drum.
No, deductive reasoning should not be dropped. Instead it should be started in the first grade and it wouldn't hurt if it was a separate course. Sorry to rattle on but this is to important to not speak out.
I am currently taking geometry. I think that deductive reasoning doesn't need to be further developed in this course because I feel that deductive resoning is a skill that can be naturally attained through your everyday life. Geometric Proofs only confuse students which furthers their personal beliefs of the mundane nature of geometry.
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