Is Deductive Geometry Worth Salvaging in the High-School Curriculum?

[Stibbymicto]

From Alexandru Pintilie (teacher), Bayview Glen School:

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.

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?

 

From Edwin Sherman on December 18, 1996:

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.

 

This has already been removed from my county. Just wondering what everyons thought on it was.

Edited March 18, 2009 by Stibbymicto

[NBVegita]

I agree that deductive reasoning and/or deductive geometry is sorely needed.

[Bak]

tightly interconnected ideas, learn the basics of axiom systems and proofs, and encounter rigorous reasoning.

Why not just teach first order logic in high school?

 

Geometry is less useful than other things they teach. Try to think of a non-academic context where you used it.

[NBVegita]

I would say teach them both. I mean unless you're an engineer how often do you use calculus in real life? I think the lessons are better shown in multiple ways than simply one.

[Dr Brain]

Geometry is less useful than other things they teach. Try to think of a non-academic context where you used it.

 

I've used geometry and trig countless times in engineering projects, carpentry and some of the more artsy programs I've written (I've even used it developing Hyperspace in a few spots). Yeah, people who go on to be historians and CEOs probably don't care (even then the deductive skills will probably be useful at some point), but any future scientists or engineers need geometry badly.

[NBVegita]

And to go along with that, you should prepare kids for any profession, not just for a select few so they have to waste time in college learning what we consider basics now.

[»doc flabby]

I wish i had learnt more geometry at school...learning 3d programming with very little geometry knowledge is challenging....I've spent more time learning the maths than programming the 3d parts so far.

[Drake7707]

It's not the subject per se that's important, it's how people get insight in problems that are similar. Geometry is 1 example to learn abstract thinking and it sometimes requires you to think outside the box and try solving mathematical problems in a different way. Without teaching that children will miss out vital parts of information and methods that can be required later on. Any scientific course (informatics, math, physics, engineering, electronics, ...) requires this knowledge, so it should be considered basics.

[Samapico]

It's not the subject per se that's important, it's how people get insight in problems that are similar. Geometry is 1 example to learn abstract thinking and it sometimes requires you to think outside the box and try solving mathematical problems in a different way. Without teaching that children will miss out vital parts of information and methods that can be required later on. Any scientific course (informatics, math, physics, engineering, electronics, ...) requires this knowledge, so it should be considered basics.

Learning to learn

[FMBI]

Figuring out how to do this is certainly important, but you can't just say "Oh, if only we taught our children how to prove things in math, the world would be better!" This is just one more Oh-For-The-Good-Old-Days lament, with the small twist that it makes more sense than most of its brethren.

 

Besides - while mathematical proofs tend to be rather useful, I personally never got much of a thrill from geometric proofs. A lot of them are nonsensical - "This is itself, therefore, That is itself" - and the rest are too complicated for most people to really get in the first place.

 

And let's not forget one more thing - logic can really suck ass when you're dealing with real world situations.

 

http://upload.wikimedia.org/wikipedia/en/f/f5/Principia_Mathematica_theorem_54-43.png

[NBVegita]

First off,

 

This is just one more Oh-For-The-Good-Old-Days lament,

 

It's not really a move for "the good ole days" it's a simple educational concept that is sorely useful. We expect to be on the forefront of technology and engineering yet we keep pushing the fundamentals of this proficiencies away from our children labeling them either too difficult or unnecessary.

 

http://news.bbc.co.uk/2/hi/uk_news/education/6589301.stm is a fair example.

 

Besides - while mathematical proofs tend to be rather useful, I personally never got much of a thrill from geometric proofs. A lot of them are nonsensical - "This is itself, therefore, That is itself" - and the rest are too complicated for most people to really get in the first place.

 

Math proofs are only nonsensical if you either don't understand them or may never use them. The problem is how can you tell a young teenager, I took this at 13 in 9th grade, that they won't need this in their lives, or that the basics don't impact us, when you have no idea what a 13 year old kid wants to do with their lives? Logic proofs work exactly the same way as do "First order logic", as Bak put it, does. It's just showing different applications of using it, which is very important. Math tends to be the easy subject to beat up on because most people find it difficult, but it is ESSENTIAL, not just basic math, for students. I know schools like to pad their grade books and all, but sometimes you have to think about what will help the students in the long term not just what is good for the teachers and schools. Deductive geometry sets the foundation for so many things it would be a travesty to take it away.

 

And let's not forget one more thing - logic can really suck ass when you're dealing with real world situations.

 

99% of "real world" situations can be solved with the proper application of logic.

[»Ducky]

I've worked construction for about 5 years and have never come across an instance where I needed extensive geometry knowledge. The majority of non 4 year careers are so dumbed down that most of the subjects taught in the high schools around this area (Western Pennsylvania) are simply lost to students.

I can't actually think of any under $30k jobs offhand that would even require a high school diploma.

The majority of people can receive on-site training and be competent granted the have the motivation.

 

Yes, you can give any example of an everyday trig or geometry scenario, but all of those problems can be broken down into simpler math or even logical means.

Granted the person may take an additional step in discovering the answer, the answer is still the same.

 

If you haven't been able to tell, I'm a strong supporter of pushing children into the real world around the 10th grade and letting those that wish to finish school do so at their own leisure.

Edited March 26, 2009 by Ducky

[NBVegita]

As a construction worker, you simply build the building, an architect designs the building. (Not meaning to belittle building the building, but the building doesn't require heavy geometry, the desining does.

 

Let's just say that I would not want an architect who is weak on his geometry.

 

I do agree that for most untrained jobs, which is what they are called, you don't need 90% of what you learn in school. School is there to prepare you for it you want to go further.

 

Again we're not talking about everyday simple things, the engineers who construct bridges, airplanes, cars...ect need that information.

[»Ducky]

So, my question becomes; why can't these types of lessons be taught in a college setting where the career path is already chosen rather than a highschool setting where it won't necessarily be applied to everyone.

[NBVegita]

My answer to that is college is supposed to prepare you for an advanced degree. High school is supposed to teach you basics.

 

For example say I were to be a math major in college. Instead of a 4 year degree program I'd be in college a minimum of 6, at a very minimum of 2 years to make up for what I've missed in college, just to stay competitive with other countries who make their kids learn this stuff in school.

 

Another answer is the majority of college students will change their major at least once before they graduate. Teaching them a good base in ALL areas allow for them to be more diversified in their backgrounds, thus giving them more options in their future.

 

Myself for example, from the time I was 12 I was positive I was going to be a programmer. I went to college as a comp sci major. I ended up graduating in 5 years instead of 4 as a physics engineering student. The only reason I even graduated in 5 years was because of the classes I took in hs.

 

Our education system is already failing beyond belief because we tell kids they don't need to be smart, more so even try, and that's ok. Well everyone needs someone to park their car and flip their burgers, but we can't cater a whole system to under achievers.

[Bak]

the majority of college students will change their major at least once before they graduate.

Source? Also why hate programming?

[NBVegita]

http://www.msnbc.msn.com/id/10154383/

 

Part of the problem was that I had started programming at 12 and when I entered college they had me learning basic programming in C. They customized the courses for students who had never programmed before ever, very similar to the way Ducky thinks it should be. I got so bored, well this should say enough; In my entire time in HS I never fell asleep in class once, not once. In college I spent more time sleeping in my computer science classes than being awake. As a joke one of my friends always used to bring a pillow into class for me. It was so sad that in my Comp Sci 201 class, attendance wasn't manditory. I showed up for the first day, 1st test, mid term, 2nd test and final. Got an A+. In every 3 hour lab I took the longest it took me to program the lab was 23 minutes. Most of that was because I was talking to other people. That is when I changed majors to the most difficult major my school had to offer.

 

After that, programming has never been the same for me. I still do it when I have to, or when I find it useful.

 

Also just an FYI, I didn't go to some crappy college, I went to Syracuse University.

[»Lynx]

I agree with NB, I think it's important that people are well rounded and knowledgeable in more than just the most basic of subjects, at their most basic levels.

 

-L

[»Ducky]

I can completely relate to the boredom aspect of high school when it comes to the ease of courses and I generally understand what you're saying about needing to be well rounded.

What I don't understand is why you couldn't be pushed out into the world after the sophmore year and use those 2 extra years going to college instead. Not only could you be taking the subjects you would have taken in highschool to fill your criteria, you could be taking additional classes that actually matter in your field of training. Why, if becoming a programmer would you want to take advanced courses in history in high school. Not every school offers fundamentals of advanced subjects you may necessarily need. There just seems to be a lot of wasted years for many people.

I would much rather have a specialist who was dedicated 6 years to something than someone who is more rounded who only has 4 years of specialized training.

 

When it comes down to it, I think high school basically scarred me on educational support. It became less of a place of learning as you ended up repeating many of the same classes year after year being held back by your lazy unmotivated class.

We took what was basically world cultures almost 4 years in a row. History, I understand. Civics, that's fine. Two years of what amounted to filler cross subject learning was utterly useless though.

 

I've never understood why people can't decide on a course of action early in their lives. Nor could I ever understand how people can change majors after 2 years of college. It seems a rather simple feat to me to just pick a career path and stick with it.

 

Keeping on topic though, I suppose I'm less determined to keep additional courses as I am to simply eliminate all additional courses to save years off of a persons life. I've never been motivated in life by career, so this seems a pretty given standpoint. However, I wouldn't fault the idea of keeping such courses in place if they are used by the students and the schools can easily support the criteria. I show full support for people who want to learn, I just hold no reservation for those forcing students to do so.

Edited March 28, 2009 by Ducky

[NBVegita]

Not even tackling the if kids would be prepared to make it in the real world at 15 and 16, you want to make kids have 6 years of college instead of 4? Most parents and kids struggle with considerable debt, or can even make it through 4 years of college because of the money, now they have to do 6? If they want a masters now they have to pay for 8+?

 

It became less of a place of learning as you ended up repeating many of the same classes year after year being held back by your lazy unmotivated class.

 

We must come from different educational systems because where I went to HS what the rest of the class did, didn't really effect your education.

 

I just hold no reservation for those forcing students to do so.

 

I would say that they problem with that is that the overwhelming majority of kids would much rather spend all day sitting around playing video games, or doing any other activity that they term "fun" instead of learning. You would have an overwhelming rate of children taking minimum wage jobs and in 4-5 years when they're tired of working awful jobs for little money, they don't have the resources or the skill sets to get better jobs.