https://www.researchgate.net/publication/291333791_Volume_One_Geometry_without_Multiplication
Very few of the top students in high school take an interest in geometry. After acing Algebra I, they typically feel that geometry is wasting their time and they are anxious to put it behind them and move onto Algebra II. As seniors looking back on their studies, they feel that geometry was just a review of Algebra I, which they had already proven their mastery of by getting an A, and which they now consider very basic. They cite Algebra II and pre-calculus as important and interesting classes, but never geometry.
On page 31 of Geometry-Do, I write:
"Glencoe Geometry (p. 256) declares two triangles congruent, one with all its sides and angles labeled: a=38.4 mm, b=54 mm, c=32.1 mm and α=45°, β=99°, γ=36°. The other triangle has the side corresponding to a labeled (x+2y) mm and the angle corresponding to β labeled (8y-5)°. Glencoe solves x + 2y = 38.4 mm and 8y - 5 = 99° simultaneously to get x=12.4 and y=13. What is the geometric interpretation of x and y? What are their units? Nobody knows. The chapter on congruent triangles is just one long lesson in the algebraic solution of simultaneous equations; the congruent triangles are used only to set up the two equations to be solved. They never solve three equations in three unknowns nor introduce Cramer’s Rule, making it all a very Algebra I lesson."
Instead, I attempt to make geometry relevant in a way that an algebra review cannot be. I do this in three ways:
On page 25 (end of green-belt chapter) of Geometry-Do, I write:
"Proving these two theorems [Guinand’s and Franzsen’s] is the work of sam-dan (3rd degree black belt) geometers, but green belts should take note of the publication dates, 1984 and 2011, respectively. In 2001, Varilly proved that the Fermat point is in the same circle that Guinand contained the incenter. These recent advances put to rest the notion that geometry is just a stale rehash of the work Euclid did 2300 years ago. Indeed, a brilliant teenager who hopes to publish in a refereed journal while still in high school is more likely to do so in geometry than in any other science."
On page 35 (end of red belt chapter) of Geometry-Do, I write:
"Cut the Knot has compiled no fewer than 115 distinct proofs of this famous theorem [Pythagorean], many of them first proven by 20th century high school students – the Aspirant’s grandfather’s classmates when he was their age – and published in Mathematics Teacher. Until 1974, when Gerard Debreu went down in flames and took deductive logic with him, geometry was the one science in which a bright teenager could publish original research. Now it is just a review of Algebra I."
On page 36 of Geometry-Do, I write:
"I also write poetry and have accompanied high school students to poetry competitions where they must recite a famous poem from memory before a panel of judges. I propose that high school teachers contact a university’s math and poetry departments and organize a competition where the students must prove – by any means but without notes – the Pythagorean Theorem before a panel of judges consisting of both mathematics and poetry professors, the former scoring accuracy points and the latter scoring style points."
1) Do you agree that students are inspired by learning of recent research in geometry?
2) Do you agree that students are inspired by learning of teenagers who invented their own proof of the Pythagorean Theorem and got it published?
3) Do you agree that the contest I propose (proving the Pythagorean Theorem by any means but without notes) would inspire students?
4) How would you recommend inspiring top students to take an interest in geometry?
Book Volume One: Geometry without Multiplication
According to B. B. Mandelbrot, Euclidean geometry is said to be 'cold and dry'. The students need 'living geometry', a term coined by Christopher Alexander in the Magnum Opus The Nature of Order. I would suggest new fractal geometry that can deal with things in our daily life:
https://www.researchgate.net/publication/270634544_HeadTail_Breaks_for_Visualization_of_City_Structure_and_Dynamics
https://www.researchgate.net/publication/282272777_A_City_Is_a_Complex_Network
https://www.researchgate.net/publication/263316490_The_Fractal_Nature_of_Maps_and_Mapping
In addition to teaching geometries (both Euclidean and fractal), the students should be taught the distinct ways of thinking behind the geometries; see more details in the following paper (Table 1 in particular):
https://www.researchgate.net/publication/282310447_A_Fractal_Perspective_on_Scale_in_Geography
Article The Fractal Nature of Maps and Mapping
Article Head/tail Breaks for Visualization of City Structure and Dynamics
Chapter A City Is a Complex Network
Article A Fractal Perspective on Scale in Geography
Maybe high school students don’t seem interested in geometry because they imagine its only about circles and straight lines in a Euclidean plane. When I was introduced to geometry (aged 11) I was immediately excited by Chapter 1 of the text book, about how to make models of the five Platonic solids. But then lessons began, starting at Chapter 2, and from then on it was all theorem, lemma, corollary; theorem, lemma, corollary…, presented in a pedantic way that didn’t stir the imagination. Geometry should excite the visual imagination and stimulate intuitive thinking as well as logical thinking. What about the Platonic and Archimedean solids and their duals; Euler's formula; symmetry; tiling patterns and “wallpaper groups”; Penrose patterns; the logarithmic spiral (seashells and galaxies); Fibonacci numbers and the golden ratio; hyperbolic geometry (Escher’s graphic art…); spherical trigonometry; map projections; four-dimensional geometry; projective geometry (perspective drawing, theorems of Desargues, Pappus, Pascal, etc…),... And, as Bin Jiang suggested: fractals!
The respondents so far have suggested teaching many things other than Euclidean geometry. This is also the approach of Glencoe Geometry. I can open this 973-page textbook to almost any page and find photographs of scuba divers and all kinds of things.
Opening the book at random to page 93, we find a photo of a man cutting a girl’s hair and learn that stylists must attend cosmetology school and obtain a license. Colorful charts and graphs display how many customers a salon is seeing on the weekends over a six-month period. Glencoe concludes, “Survey data supports a conjecture that the amount of business on the weekends has increased, so the owner should schedule more stylists to work on those days.” Ahem!
The net effect is to reinforce in the kids' minds the idea that geometry is stupid and boring and just a delay between Algebra I and Algebra II. If it were interesting, they why the need for photographs of scuba divers and multi-colored fractal designs?
But my question was how to inspire top students in high school to pursue geometry. I didn't ask what we could replace geometry with to make this boring interlude between Algebra I and Algebra II more fun. Can we please restrict our answers to the question that was asked?
I hope you are not implying that the topics I mentioned are not “geometry”!
My suggested list of topics was in part a response to your question “Do you agree that students are inspired by learning of recent research in geometry?” My answer is a resounding yes.
From your response, it now seems clear that the source of your problem is that the syllabus restricts you to teach nothing much beyond what Euclid was teaching over two thousand years ago! Or to reduce geometry to algebraic exercises (“…seniors looking back on their studies, they feel that geometry was just a review of Algebra I….”).
I get the impression that you are already doing all that can be expected of you to infect your students your own enthusiasm for "good old Euclidean geometry", which, I agree, is a fascinating subject. But not everyone can appreciate the beauty of it. I feel that it’s quite natural that only a small percentage of people can be ever expected to appreciate it. Be thankful for that small percentage of students, and encourage them.
Sorry I have nothing more constructive or encouraging to say.
Colin James-
Regarding worked problems, I intend to provide these, though in a supplementary study guide. You are right that it helps the students learn, but I also feel that it can add a lot of clutter to a textbook, making it difficult to follow the flow of logic or even to find the theorems when reviewing. (Glencoe is 973 pages and it took me hours to go through the index to catalog all their named theorems - no student could be expected to do this.) So the solution is to physically - but not conceptually - separate the proofs of theorems from their applications.
I agree that thinking, "if this angle or side is congruent to that then certain things follow" is helpful. But this is largely provided by properly organizing the theorems so each one is used in the proof of the following theorem and that those that are not needed in later proofs are assigned as exercises to avoid breaking up the flow of logic. Then, once you learn that certain angles or sides are equal, you immediately use this to prove the next theorem. My biggest complaint against extant textbooks is that they never prove anything but just present all the major results as factoids for the students to memorize. The students are expected to learn what a proof is by referencing these unproven results in their two-column proofs, not by seeing the results themselves proven. By the end of the semester, the major results of geometry are such a jumble in their minds that they just freeze up when sitting for the final exam - a little organization would have helped tremendously.
What I am mostly interested in hearing from Research Gate participants now is comment on the organization of my book. Later, when I write the study guide, I will be interested in comments on the worked problems that accompany each theorem.
Eric Lord-
I did not mean to imply that the topics you mentioned are not geometry; I meant to imply that they are not high school geometry. Penrose tiles, hyperbolic geometry, fractals, etc. are far beyond a teenager's ability. He can be a spectator of these things, but not a participant. Let me clarify what I expect of teenagers:
Cho-Dan or 1st degree black belt geometry is circle theorems (e.g. the intersecting chords theorem or the triangle similarity theorem) that are expected of any A student, though it is possible to pass the class without any familiarity with circle theorems.
E-Dan or 2nd degree black belt geometry is the nine-point circle.
Sam-Dan or 3rd degree black belt geometry is advanced theorems like Guinand's proof that, for any non-equilateral triangle, the incenter lies strictly inside and the excenters lie strictly outside the circle whose diameter joins the medial point to the orthocenter.
It can reasonably be expected that every high school graduate learns everything in my book up to the mid-segment theorem (halfway through the green-belt chapter) and selected topics from the red belt chapter, like the Pythagorean Theorem, and selected topics from the 1st degree black belt chapter, like the triangle similarity theorem. This is passing with a C. Mastery of everything through 1st degree black belt is passing with an A.
E-Dan and Sam-Dan geometry is within reach of top students, but not as part of the geometry class that everybody takes. These chapters would have to compose an honors course for seniors, and one that is worth five points towards the student's GPA. It could also work as an lower-division university course for math majors. But the subjects that you mentioned are beyond Sam-Dan geometry and even beyond undergraduate university students - you are talking about graduate level math.
Being a spectator of advanced mathematics is not inspiring - it is disillusioning. I had this experience in college. I was the top student in high school and, in the interim (because of money problems, I was 22 when I went to college) I studied on my own and so already knew things like linear algebra when I enrolled as a freshman. I took Calculus I, II and III and Abstract Algebra all my freshman year. So I was feeling pretty good about myself - until I attempted to read a book by John von Neumann on my own. I was shattered to realize that I could not get past page 3. I remember sitting in the library with the book in front of me and my head in my hands thinking, "I will never be like von Neumann." And the next logical conclusion was, "If I can't be great, then why bother being good?"
Fortunately, I got over this hump and became a good - but not great - mathematician. Now, decades later, the lesson to be learned from this experience is that exposure to really advanced mathematics is not inspiring to good students - it is disillusioning. So let's keep things at the high school level, okay?
In my opinion, studying geometry and geometrical meaning of some notions from other fields (such as analysis and functional analysis, convexity, etc.), might be of help in easy understanding definitions, statements, proofs, at a high level, in many other areas.
The question you ask, here specifically for geometry, is an age-old issue that all instructors worth their salt have wrestled with. I think one of the best commentaries I've heard on this is Richard Feynman's "fishing" metaphor of catching the most number of students on different kinds of hooks (he discusses this briefly, for example in this YouTube clip https://www.youtube.com/watch?v=PVob_tATVRI from the BBC documentary "The Pleasure of Finding Things Out). This philosophy is obviously applicable to geometry, e.g. augmenting the textbook proofs of SAS et al with sprinklings of the history of geometry, age-appropriate examples of its generalizations to the 2 non-Euclidean realms, connections to literature (e.g. Flatland or Heinlein's The Number of the Beast), connections to physics (e.g. Riemann's work that enabled Einstein's general theory of relativity, and so on and so forth. Of course the challenge is to put all of these things into age-appropriate form. My personal experience in teaching many technical courses is that this works and it works because students can relate to how the material is connected in larger ways to the world. Best of luck.
Dear Victor,
I am afraid your question cannot have an answer since you like retain the classic program centered on the classic Euclid's geometry. In fact in the Euclid's geometry lacks a general method to solve problems. Any exercise therein is a new theorem that must be proved without following some general method. This is just the difference with Algebra taught to the same students and at the same ages. The lack of a general method in the Euclid's geometry stresses students, since it is request a continuous inventive effort to solve problems. This shadows the beauty of intrinsic coherence of the Euclid's geometry, that instead appears to students like a sad exercise of philosophic arguments without any possibility to learn a general method to be applied to solve specific problems. Of course this general method will be learned many years after with the analytic geometry ... too late !
' ... l'important c'est la méthode ...'
(René Descartes)
My best regards,
Agostino
Vinoo,
your post does not interest this thread ... In fact, this Victor's question does not concern the historical validity of the Euclid's geometry. Really this thread should answer to the question whether for students in high school to learn intrinsic geometry, namely the affine geometry where the objects are points, and lines obtained by rods and compasses, can be made more attractive for such students. In fact they in general find it non very interesting. With this respect it is important to underline that Euclid's geometry is not the Euclidean geometry, but it is more correctly called intrinsic geometry.
In my opinion in order to
'inspire top students in high school to pursue geometry'
it should be better to introduce Euclidean geometry by means of Algebra, namely as an analytic geometry. In other words, from the pedagogical point of view, to introduce students in high school to the intrinsic Euclidean geometry is an useless effort.
I just now updated my book proposal with a more detailed description of black belt geometry. In particular, I have added material that specifically addresses Agostino Prástaro's claim that Euclidean geometry lacks a general method, beginning on page 42 (RG page 44) with this sentence:
"We now present the method of similitude, a general method used by geometers to construct figures that satisfy given conditions."
I quote from College Geometry by Nathan Altshiller-Court, which has a lengthy section, including many example problems, beginning on page three titled, "General Method of Solution of Construction Problems.”
I contrast the general method of Euclidean geometry with coordinate geometry, which only solves problems about triangles whose vertices are given numerically. When confronted with “any triangle” problems, what they actually do is use a geometry theorem without citing it or they dazzle us with anecdotal evidence of their prowess by solving one contrived example.
I challenge Agostino Prástaro to take the final exam using the "general" methods that he boasts of having learned from his study of coordinate geometry.
No Victor !
With your last post and your nonsense challenge to me, you are going in a wrong direction ...
You aim to confute Mathematics History. Algebraic methods were introduced in Geometry just to find general methods to solve geometric problems. Therefore it is silly that you now try to state that this is not true.
On the other hand I can understand that for mathematicians your ilk, namely fans of intrinsic geometry, Mathematics ends with Euclid. But unfortunately (for you) this is not an universally shared opinion. Furthermore, with respect to your question in this thread, I given a suggestion to stimulate a more students share. To continue insisting with the intrinsic geometry it will retain the actual situation that you reported...
All the best,
Agostino
Agostino-
I have a question that perhaps you can answer. Like all American geometry textbooks, Glencoe Geometry, published by McGraw Hill, is based on the axiom set of George Birkhoff.
Glencoe Geometry (p. 256) declares two triangles congruent, one with all its sides and angles labeled: a=38.4 mm, b=54 mm, c=32.1 mm and α=45°, β=99°, γ=36°. The other triangle has the side corresponding to a labeled (x+2y) mm and the angle corresponding to β labeled (8y-5)°. Glencoe solves x + 2y = 38.4 mm and 8y - 5 = 99° simultaneously to get x=12.4 and y=13.
My question is: What are the units of x and y?
@Victor Aguilar
I do not know the book you quoted ... but I am surprised that you have addressed to me a so stupid question !
I am sure that this cannot be caused by that book, and neither by Birkhoff's axioms that refer only to real numbers.
But you ask about the units of x and y that are only real numbers ! ...
I am afraid that you did not understand Birkhoff's axioms ...
@Victor Aguilar
I see that you have changed your last post addressed to me, by adding units for sides of the first triangle ... Why ?
Birkhoff's axioms are measure units independent.
In this way you confuse ideas in high school students !
You did not understand Birkhoff's axioms ...
Agostino-
Millimeters, inches, whatever. These units are isomorphic. I don't care which one you use as long as you do not measure length in degrees; that is impossible.
x + 2y = 38.4 mm implies x = 12.4 Seriously?
8y - 5 = 99° y = 13
What do you get when you add a length to an angle? A lengle! ROFL
Giving top students textbooks that make no sense is certainly NOT the right way to inspire them.
I would add a semantic point here that angles do not actually have physical units...they are dimensionless. This is not much of an issue in secondary school, but it does create a bad habit that has to be broken at the university level, particularly in physics and engineering students. I introduce this to my undergraduate fluid physics class every year about the time we get to the subject of dimensionless analysis by asking what is the difference between energy (a scalar) and torque (a vector) if they both have the same physical units, e.g. Newton*meters, the answer being that there is work (energy) associated with the torque if there is a unitless angular displacement.
No Victor !
Whether you use Birckoff's axioms you must work only with real numbers ... Therefore you can also consider algebraic relations between them, ... as you reported in your post ... (no in mine). In a previous post you have claimed that you use Birkhoff's axioms in your book and you refer to a textbook adopted in American schools.
On the other hand it is well known that the Euclid's geometry, is nowadays encoded by the affine geometry. In a my first post in this thread, I suggested you to introduce just these algebraic methods ...
I do not suggested you to follows Birkhoff's axioms ... this was an your idea. Really by considering all geometric objects identified with real numbers, one loses their geometric meaning. Instead by working with the methods of the affine geometry it is possible to underline the geometric meaning of fundamental objects. Unfortunately, in the schools the affine geometry is completely ignored and one prefers to adopt old points of view that instead can produce misleading interpretations (Birkhoff's axioms), or useless (intrinsic Euclidean geometry).
Let me also add that your referring to physical measure units in Euclidean geometry is not appropriate.
At this point I must insist: Why you address so stupid questions and remarks to me ?
“Angles, shmangles! Who cares what the units are as long as we do not have to learn anything new but can just review algebra we already know?”
Anyway, I think we have spent enough time discussing the mysterious lengle that appears in textbooks that turn every geometry problem into an algebra problem. So, can we please get this thread back on topic?
Specifically, I hope to hear from scientists and engineers who can advise me of geometry theorems that are particularly useful in their field.
For instance, my book includes a discussion of Fermat's Point and I prove that it minimizes the sum of the distances to each vertex of a triangle. This is very useful to electrical engineers who are trying to minimize cabling, especially of direct current, to avoid loss of power to resistance in the wires.
But I would like to know of other applications of geometry in engineering so that I can better convince the students that Euclidean geometry is not "useless" as the proponents of coordinate geometry would have us believe.
No Victor !
With your last words
'... so that I can better convince the students that Euclidean geometry is not "useless" as the proponents of coordinate geometry would have us believe.'
you show do not understand what is affine geometry developed on the basis of linear algebra. You confused affine geometry with the Birkhoff's axioms for Euclidean geometry. I suggest you to be more informed before to spend nonsense words on things that you show do not know ...
All the best,
Agostino
I have updated my book proposal, adding a few theorems and bringing some theorems forward to make white- and orange-belt comparable to first -year geometry in Russia, as taught by Pogorelov.
Also, I made the white belt chapter more of interest to tradesmen. I conclude it with a new section titled "Geometry for Construction Workers." I write:
"Teachers: Put construction workers and others who come to geometry with an open mind on Team Euclid. Put those who have closed their minds to deductive logic and believe only in coordinate geometry on Team Prástaro. In two classrooms, push the desks to the corners, staple butcher paper to the ceiling and draw a chalk line on it. Give each team a yardstick, two spools of chalked string and two ladders. A team that can draw a chalk line on the floor directly underneath the one on the ceiling gets an A, else an F. Test their answers with a plumb bob."
@ Victor Aguilar
You have written:
'... Put those who have closed their minds to deductive logic and believe only in coordinate geometry on Team Prástaro.'
No ! This does not correspond to what I posted. Therefore I must insist:
You show do not understand what is affine geometry developed on the basis of linear algebra. You confused affine geometry with the Birkhoff's axioms for Euclidean geometry.
To be more explicit. Affine geometry developed on the basis of linear algebra is also an intrinsic approach to Euclidean geometry, starting from some axioms, say (AA), that differ from Euclide's axioms, say (EA). Therefore (AA) develops its construction with the deductive logic as well as (EA). But (AA) allow to obtain also a relation with the set of real numbers, namely with Birkhoff's axioms, without forgetting the geometric meaning of objects identified with numbers.
By conclusion your last interpretation is completely wrong and definitively offensive ...
You ought to refrain from spreading gossip on my posts !!!
It is not your post, it is mine. Go away.
Everybody else: How can I inspire top students in high school to pursue geometry?
@Victor Aguilar
The last your post proves that you do not understand what you write.
1) You confused RG posts with RG questions. The question of this thread is yours, instead my contributions in this thread answering to your question and your replys are mine !
2) 'Go way' you ca say to your dog not to me ! You are unable to take a serious academic behaviour in RG !
3) You pretend to write on Euclide's geometry, but you do not know that nowadays this geometry is definitively encoded by the axioms of the affine geometry developed with the methods of the linear algebra.
4) Your mathematical incompetence and your attitude to deform the meaning of my posts is incompatible with your choice to be a my RG follower ! Therefore I invite you to cancel your name from the list of my RG followers.
'Asinus asinum fricat ...'
One way I am trying to inspire top students is by including more practical problems from the construction trades. Here are some that I pose and then solve:
Problem #1: Two poles are erected perpendicular to the level ground. We wish to pound an anchor into the ground between them with guy wires to the tops of each pole to reinforce them. Where should we position the anchor to use the shortest possible wire?
Problem #2: From a house in the country, construct a dirt road to a straight paved road, the latter twice as fast as the former, to minimize travel time to a nearby town on the paved road.
Problem #3: Two country roads intersect at an arbitrary angle. We wish to pave an arc connecting them with as large a turning radius as possible and going around the corner of a farmer’s field, which is on the angle bisector of the two roads.
Problem #4: Two country roads intersect at an arbitrary angle. We wish to pave an arc connecting them; use the turning radius mandated by the Highway Department.
Problem #5: A highway and a country road intersect at an arbitrary angle. We wish to pave an arc connecting them with as large a turning radius as possible and going around the corner of a farmer’s field. Because the highway has a wider right of way than the country road, the corner of the field is closer to the country road than it is to the highway.
Fermat Problem: Given a triangle with angles all less than an exterior angle of an equilateral triangle, find a point that minimizes the sum of the distances to the triangle’s vertices.
Fagnano Problem: Inscribe a triangle in an acute triangle with the smallest possible perimeter.
Can anybody here think of any other problems that might be interesting to the students?
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