The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school geometry classes even today.
What I'm getting at is this: Are the rules of construction just arbitrarily imposed restrictions, like a form of poetry, or is there a meaningful reason for prohibiting, say, the use of a protractor?
Okay, I seem to be ranting too much in comments, so let me try to put forth my points and opinion here (as a community wiki, since it is opinion).
First, to address the question: What is the significance of straightedge and compass constructions within mathematics? Historically, they played an important role and led to a number of interesting material (the three famous impossibilities lead very naturally to transcendental numbers, theories of equations, and the like). They are related to fascinating stuff (numbers constructible by origami, etc). But I would say that their significance parallels a bit the significance of Cayley's Theorem in Group Theory: although important historically, and relevant to understand the development of many areas of mathematics, they are not that particularly important today. As you can see from the responses, many find them "fascinating", many find them "boring", but nobody seems to have come forth with an important application.
Now, addressing the issue of teaching it at K-12. Let me preface this by saying that I am a "survivor" of the New Math, which came to Mexico (where I grew up) in the 70s. I would say I became a mathematician in large part despite having been taught with New Math, rather than because of it. Also, I did not attend K-12 education or undergraduate in the U.S.; I am a bit more familiar with undergraduate at the level of precalculus and above thanks to my job, but not very much with the details of curricula in sundry states in the U.S. So I may very well have a wrong impression of details in what follows.
Now, one problem, in my view, is that when we talk about "math education", we are really talking about two different things: numeracy and mathematics. This is the same phenomenon we see when we think about "English class". I suspect that English Ph.D.s are nonplussed at people who think they spend their time dealing with grammar, spelling, punctuation, etc, just as mathematicians are nonplussed that people think we spend our time multiplying really big numbers by hand. English education has two distinct components, which we might call "Literacy" and "Literature". Literacy is the component where we try to teach students to read and write effectively, spelling, punctuation, etc. Making themselves understood in written form, and understanding the written word. On the other hand, Literature is the component where they are introduced to Shakespeare, novelists, book reading, short stories, poetry, creative writing, historical and world literature, etc. We consider English education at our schools a failure when it fails in its mission with regards to literacy: we don't consider it a failure if students come out not being particularly enthused with reading classic novels or don't become professional poets, or even if they don't care or like poetry (or Shakespeare). The professional English Ph.D. engages in literature, not in literacy. Literacy is the domain of the grammarian.
Mathematics education likewise has two components; numeracy (to borrow the term from John Allen Paulos) and mathematics. Numeracy is the parallel of literacy: we want children to be able to handle and understand numbers and basic algebra, percentages, etc., because they are necessary to function in the world. Mathematics is the stuff that mathematicians do, and which we all find so interesting and beautiful.
"Mathematics" also includes advanced topics that are necessary for someone who is going to go on to study areas that require mathematics: so the physicists and engineers need to know trigonometry; business and economists need to know advanced statistics; computer science needs to know discrete mathematics; etc. Much like someone going on to college likely needs more than simple command of grammar and spelling.
Part of the problem with math education is that it so often conflates numeracy with mathematics; another part of the problem is that mathematics was included in the "classical" curriculum for much the same reason as Latin and Greek were included: historical reasons, because every "gentleman" was expected to know some Latin, some Greek, and some mathematics (and by "mathematics", people meant Euclid). To some extent, we still teach geometry in K-12 because we've always taught geometry. But it is not part of the numeracy curriculum.
It would be wonderful if we had the time in K-12 to teach students both numeracy and mathematics; it is a fact that today we are failing at both. We don't perform better in teaching students numeracy by teaching them beautiful mathematics, even if they understand and appreciate them, just like making them really understand and care for Shakespeare's plays (through performance, say) will make them able to understand a set of written instructions, or write a coherent argument.
Trying to excite students about mathematics is all well and good; but numeracy should come first. Trying to excite students about the wonderful world of books, plays, and poetry is all well and good, but we need them to be able to read and write first, because that's part of what they will need to function in society.
Constructions by straightedge and compass can become an interesting part of mathematics education, just like geometry. I just don't think they have a place in numeracy education. But they are taking the time we need for that numeracy education. Trigonometry used to be part of the numeracy education that people needed; this is no longer the case today. Trigonometry, today, is a foundational science for more advanced studies, not part of numeracy. But we are still teaching trigonometry as a numeracy subject (hence the rules, recipes, mnemonics, and the like). We could try to turn trigonometry into a mathematical subject, sure; or we could postpone it until later and only teach it to those for whom it is an important foundation.
Added. To clarify: I don't mean to say that the "solution" is to do less math and more numeracy. I think the solution is likely to be complicated, but the first step is to identify exactly what parts of what is currently branded as "mathematics" are really numeracy, and which parts are mathematics. Trigonometry is taught as part of "advanced mathematics", but it is taught as rote and rules because it was really numeracy. We don't need to teach trigonometry as numeracy any more, so we shouldn't. If it is to be taught, it needs to be taught in the right context. Geometry is similar: geometry used to be taught as basic numeracy because "every educated person should know geometry"; (of course, "educated" at the time meant "rich and land owner, or with aspirations in that direction"). We don't need most of geometry as basic numeracy, we want it now as mathematics. So teaching geometry as numeracy is a waste of time, and it takes up the numeracy time that should be spent in other things. (It can still be taught during the mathematics time). I certainly don't say "drop all the math, concentrate on the numeracy". I say, "when dealing with numeracy, concentrate on the numeracy, not the math, and don't confuse the two." Nobody seems to confuse spelling rules with reading novels, because we separate literacy from literature. Too many people confuse arithmetic with mathematics, because we don't separate them.
The reason I talk about dropping trigonometry and doing some basic statistics is precisely that: trigonometry is being taught as part of the numeracy curriculum, when it shouldn't. Basic statistics, say at the level of the wonderful How to Lie With Statistics, is not taught as part of numeracy. But in today's world, there is a far better case for statistics being part of the basic numeracy education than trigonometry. Everyone coming out of High School should know that taking a 10% pay-cut and then getting a 5% raise does not mean you are now at 95% of your old salary (go do a spot check, see how many people think you are). They need to know the difference between average and median, so they are not misled by statements about "the average salary of the American worker". They should understand what "false positive" and "false negative" means. They should be able to interpret graphs (even the silly ones on the cover of every USA Today issue) and be able to spot the distortions created by chopping axes, etc. These are numeracy issues.
Likewise high school geometry: it is trying to be both numeracy and mathematics, and I think it generally fails at both. There are some components of geometry that are part of numeracy, they ought to be treated that way, but the parts that are mathematics should be separate.
One problem I have with Lockhart's lament is that he does not make clear the distinction between numeracy and mathematical education. The nightmare he paints for the musician and the artist is precisely that musical education is being turned into the equivalent of numeracy/literacy education, thus doing a disservice to music-as-an-art.
The science of mathematics that we all know and love has some intersection with numeracy and arithmetic, but we all know it is a limited intersection; just as the study of literature has some intersection with the study of grammar an spelling, but the intersection is limited. The main purpose of K-12 education (or at the very least, K-6 or K-8) should be numeracy, with some limited forays into mathematics (just as the main purpose will be literacy with some limited forays into literature). The way to teach numeracy is necessarily different from the way to teach mathematics. Numeracy requires that we memorize multiplication tables, for all the horror this will cause to modern education people; this of course is a far cry from teaching the mathematics of multiplication, which may very well be very interesting and awaken the child's curiosity and wonder at the world. That can be done within the context of mathematical education, but it shouldn't be done in the context, and at the expense of, numeracy.
Trigonometry used to be part of numeracy; it no longer is. It is now either mathematical or foundational for advanced studies, so it should be treated as such. Geometry used to be taught for reasons which no longer hold, and to some extent we continue teaching it as a historical legacy; we shouldn't. Those components which are numeracy should be taught as that, and we could move the rest (including constructions with compass and straightedge) to the more creative, mathematical education side of the equation.
Anyway, I've ranted long enough, and probably made myself a few detractors along the way...
I personally think straightedge and compass is one of the most important tools i acquired during high school, regarding mathematics.
Other then it being, as you said, a fascinating and entertaining subject- It was the first time before college I ever had to work under severe restriction- and got unbelievable results. Later on, when I discovered Galois and Field theory the whole thing became even cooler.
What I'm getting at, is that I think subjects like Construtions are extremely important for math education. A few days ago someone posted Lockharts Lement- It mentions the fact that math became a subject of industry- kids learn math as a technical and terrifying subject. Telling them to use protractors and calculators and other stuff that make "short cuts" to solutions of problems makes math into paper-work.
Putting on restrictions, and teaching how to come up with beautiful stuff under them- makes it into a game. I remember when I was a kid I wasn't very much interested in paperworks. But games, I liked games. And that kind of things is what got me to enroll in a mathematics degree. At least that's my opinion on the matter :)
The beauty of straightedge and compass constructions, as opposed to the use of, say, a protractor, is that you don't measure anything. With ruler and compass you can bisect an angle without knowing its size, whereas with a protractor, you would have to measure the angle and then calculate the result.
In other words, the point of this form of geometry is that it can be done independently of calculations and numbers. I think this is an important idea to teach: mathematics is not about numbers, but about objects adhering to certain rules (axioms).
Throughout the history of mathematics there has been a tension between the issue of showing that something exists and actually constructing what one may know is there. One can, using Euler's polyhedral formula show that there are 5 combinatorially regular planar graphs, and since these are planar and 3-connected graphs, there have to exist convex polyhedra that correspond to these graphs. This does not, however, show that these polyhedra exist as "regular polyhedra." Euclid actually constructs the 5 regular convex polyhedra though since no mention is made of convexity the claim that these are the only "regular polyhedra" is problematical. Using a broader collection of "rules" one gets the four Kepler-Poinsot solids, which are not convex but are "regular."
There are also other rules for "construction." For example, one can carry out constructions using folding rules in the spirit of origami constructions:
http://www.langorigami.com/science/hha/origami_constructions.pdf
One can also look at constructions with compass alone or ruler alone. All of these lead to interesting mathematics.
I think it is safe to say that the practical use of straightegde-compass constructions is close to zero, just like that of playing or listening to music, painting, or reading literature. So those who suggest that playful mathematics should be dropped from the curriculum should also advocate the abolition of music and arts lessons at school (see also Lockhart's Lament linked to by Qiaochu above).
Why specifically straightedge-compass constructions? I guess the main reason for why these classical problems are still there is because it is an area in which one can quickly get to problems that require a mix of creative thinking and training without having a lot of technical baggage. It can also be regarded as a piece of history education, since these problems were so important to people in the past (Greeks, motivation for Galois theory, etc.)
I am sure different people have had different experiences with maths lessons at school, but for me, teachers that were not focused on mindless training of techniques and on practical applications were a welcome exception, rather than the rule. It is clear that a balance between different requirements on mathematics education must be found, and that will surely include practical applications as well as playful and creative mathematics, such as straightedge-compass constructions.
A side note: showing students that maths is fun and creative, as well as developing their problem solving skills, is a good practical investment even if these students don't become mathematicians.
Finally, let me address the argument put forward e.g. by Arturo Magidin, that since students are not taught enough basic skills in mathematics, one should sacrifice some of the more playful elements of maths education in favour of those basic skills (numeracy). I claim that doing that will result in students that learn less numeracy, not more. The reason is that the playful elements are fun and can help persuade the students that it's worth listening and participating in the maths lessons. If the entire maths education consisted of "numeracy", students would just doze off, and I wouldn't blame them.
Well.. since I came upon this post because my wife ( an eighth grade teacher ) was puzzing about a 'challenge' problem involving cones in a curriculum guide she was using. She had solved the problem by one means and I solved it by another. When I was in high school (early sixties), we studied Euclid in the 10th grade. The entire first half of the year was Euclidean geometry ( impossible without compass/ruler constructions) and the second half was Algebra. Our numeracy was far from universally competetent, just as today, but on the average, you could usually take us out of school and we didn't need a calculator to make change at the store which you can't do today, so I think there was enough time spent on numeracy, yet we learned ( by DOING!) mathematical proofs, symmetry, angles, and so on.
A lot of students struggled with the idea of mathematical proofs and a lot didn't. I personally found them exciting and challenging, never wanting to 'look up the answer' until I had proven the proposition. Back to the cone... I solved the cone problem easily by seeing from memory an actual image of a page from my tenth grade Euclidean geometry book involving 'similar triangles'. I realized that my mind had long ago made the connection between similar triangles and scaling, and between scaling and ratios. I solved the problem very simply because I could see that. Now if that ain't the basis of every numeracy challenged student's nightmare, I don't know what is. ( Fractions ! ). I now understand that my ability to estimate and solve lots of problems is linked to my understanding of scaling but that wouldnt have been one of the Euclidean geometry curriculum guide's 'targeted outcomes' in 1963. It all helps the brain build the ability to create visual representations of problems.
'Playing' around with compass and straight edge, then, is learning about fractions and more through your hands and brain. Same thing goes for using protractors and messing around with triangles.
There is the 'counting and numbers' representation of magnitude and there is the 'distance' representation of magnitude. Numeracy and mathematics require both representations.
From the beginning of the seventh until the end of the tenth grade, we were also required to take 1/4 of a year of each of the subjects metalwork, woodwork and drafting. This hands on practice moulded the minds of many students to allow them to grasp the idea of size, ratio and relationship, again, the instruction entering their brains through the hands rather than the printed page. My own children's modern school instruction was woefully deficient in these opportunities.
I guess, in short, I would be very careful about branding these practices as 'legacy' and dropping this instruction only to replace them with even more time in front of a screen to engender 'computer literacy' when the students already spend all the time they need to develop numeracy, in front of a TV screen or mobile phone.
Rather than drop so called 'legacy' material without understanding it's full impact on learning, perhaps we could try returning to 'individual mastery' as an instructional method in math. Continuously sending students on to the next level of instruction when they have not actually mastered even their times tables is much more of a waste of instructional time than learning a bit about Euclid.
I think trying to divide the pre-college instruction of mathematics into either numeracy or mathematics is slightly specious. One of the hurdles I have experienced as a High School math teacher is the notion that our content is nothing more than a series of unconnected algorithms that students either master or not. The beauty to mathematics IS the interconnectedness of it's many facets. What I try to get across to my students is that ALL of this stuff is mathematics. Some of it is rote and smacks of being tedious; other parts are almost esoteric in their complexity and beauty.
I want my kids to experience both in their time with me. Certainly the risk is that, if not handled well, we could lose computational rigor in a more explorative effort, but my job as teacher is to ensure we keep both.
As for geometric constructions, I am of the mindset that the potential insight and understanding that comes from dealing with a rigid structure that can produce such amazing complexity is well worth the time and effort it takes to do it well with students. In general, the study of shapes and relationships in structure is hugely beneficial for everyone. The key, as always, lies in the teacher's ability to connect the rigor and the beauty with relevance and engagement.
Well, I don't know how mathematics is taught in other countries (or in Germany today), but when I went to school, geometry was the very topic where we learned to do proofs. Since the topic was so different from everything we had learned before, there was no way to "shortcut" via prior knowledge. All the proofs had to be done using only what we had already learned in that year, so there was no temptation to use as of yet unproven stuff. On the other hand, geometric construction as such has a very low level of abstraction; everyone can understand what it means to draw a straight line through two points, or to draw a circle around two points. So it's an ideal environment to build upon.
As one who was taught in the old school of euclidean plane geometry, the whole apparatus of proposition and proof, gave a basis to the understanding of shape which more modern students do not have: the idea of a proof is something foreign: they are equal because they lkook it, does not answer, and the basis of calculus ( differentiation from first principals ) can only be fully comprehende by a mind trained in the rigours of proof: Then when the mathematician comes to study e.g. Rolle's theorem in elementary analysis, the need for the intermediate steps, and the preconditions ( continuous, closed, differentiable) which are all prerequisites for the reault to be rigorous and valid, are comprehensible: Always remember that it was that training which made the late great Bertrand |Russel able to work out his proposition that "1" is the first ossible whole number!! Moreover, the great innovaters iun ther fields ( e.g. Nijinsky & Nureyev in Ballet) built their advances on the solid technique nd practice of previous generations: innovatore who try to ignore the rock of the past come to grief: hence the mathematician, MUST know his foundation!!
The reason is because it’s the only way you could recreate a construction and measurement of some distance (without recourse to trig.)
Try to measure out some large distance in the direction of a visible landmark and you’ll realize how often the line wobbles from walking.
To remove the error of walk-wobble:
Take out a large rope: make a small circular arc roughly in the direction of the landmark.
That’s your compass.
Then test for incidence: from your original point to some friend on the circle. Move your friend until lined up with the landmark.
That’s your straightedge
It’s the most primitive but effective way to measure the earth in demonstrably straight lines
Algebra becomes “find the distance to this mark” which we denote “x”
As trig develops you add the concept of angles to leverage angle measurements in lieu of distances. Angles less costly to measure.
There is a reason why Euclid's Elements are so revered. It builds logic, critical thinking/reasoning, spatial skills, and some. Also, it is beautiful. It has been as influential as any one book in the history of time. I think the undertones of conversations like this usually imply one's (perhaps subconscious) preference for quality or for quantity. Are you a Journey person or a Destination person? So many modern ideas are so obviously sprung from this foundation work. The thinking of so many that shaped our society was very much shaped by Elements (Government, Law, Architecture, Building Construction, City Planning, Science). It deserves more than a cursory glance for ourselves and our kids. Euclid's Elements was such a QUALITY work that it was taught in more or less its original form for two thousand years. Getting to your question, the axiomatic structure used in Euclid's straight edge and compass constructions has helped move the idea of proof along. Without the definitions, common notions, postulates we are talking about a bastardized version that is less mathematically significant yet still important in the development of spatial skills. Remember, math is not reality and is all about imposing arbitrary restrictions to find the truths they bring about (and often beauty). As far as not using a protractor, this just creates an extra set of constraints to make the subject more challenging and make you work/think a little bit for the answer. Why not use a computer or a geometer for that matter? Why are you not allowed to phone a friend on Jeopardy? Constraints often require you to use more creativity to solve a problem and exist in the real world in spades.
