Is ST_GeomFromText better than providing direct geometry?

How do you teach problem solving in geometry?

• I tutor a high school student in mathematics. He's in his second last year of 'VWO' (A track in Dutch secondary schools, it's the highest level). The topic at this moment is geometry and I find it hard to teach him this. He just got into 'VWO' from a lower level where he didn't have learn geometry. The schools method is 'Getal en Ruimte'. With many other mathematical problems I can teach him to understand principles and such. But with providing proofs in geometry, it doesn't work like that. He can understand all the axioms and theorems but to find a proof, you somehow have to 'see' it (at least, that's how it works for me). There seems to be no logical step by step solution for finding it. How can I help him? Show him many examples of how to do it? Or let him figure it out for himself? I'm more in favor of the latter, but he then often will get stuck or move in directions away from the solution. And giving hints is often difficult without giving too much clues away.

• Answer:

  The study of geometry is both a blessing and a curse in that many of the facts encountered in introductory courses are 100% intuitive.  That is, at least in the Euclidean setting, basic statements in geometry are easily explained with a picture or using a few toothpicks and marshmallows, which makes it difficult to justify the need for rigorous proofs.  This is even true at the more advanced levels -- as a graduate student arguably studying theoretical geometry, I spend much of my time cutting models out of paper and examining interestingly-shaped objects on my desk! To me, there are two somewhat orthogonal issues that come into play when teaching high school geometry: 1.  Writing proofs In the traditional sequence of high school math classes (at least in the American system -- something to the effect of basics --> Pre-Algebra --> Algebra I --> Geometry --> Algebra II --> Pre-Calculus --> Calculus), often times Geometry is the first time students encounter the formal concept of a "proof."  The concept can be quite confusing -- after all, if the theorems of simple Euclidean geometry are so obvious, it's not clear why their proofs should not be as simple as "just look at the darn thing!" As Barry notes, there are books upon books providing theories on the best way to teach a student the art of writing rigorous proofs, and I cannot attempt to summarize (or claim any sort of knowledge of) this work.  In my personal experience working with geometry students, a few basic pointers might be as follows: When students are introduced to rules from formal logic,  it seems the first thing they do is forget that proofs can have a game plan -- writing a proof is hardly an exercise in applying breadth-first search using all the formal rules you can think of.  In fact, it is provable that this strategy won't get you far!  Instead, marking clear diagrams illustrating what you "know"/"don't know" can help encourage a more intuitive or informed approach. Similarly (perhaps just rephrasing!), I try to separate proof strategy from formal logic.  Both are difficult to understand on their own, and the interplay between them is somewhat complex.  Try to have students state their strategy in a few sentences before doing anything formal (but don't neglect the latter part either). It is important to remember that logic is not necessarily a geometric field -- you can write many highly nontrivial proofs without a single diagram!  Students in most geometry classes know some algebra, and simple exercises showing how one might axiomatize algebraic manipulation may help make this difference clear.  No need to derive the crowning results of Galois theory, just a few simple facts. To me, silly exercises writing down full sentences to the effect of "'If I play the cello, then I have an instrument' is not the same as 'If I play an instrument, then I have a cello' but is the same as 'If I don't have an instrument, I cannot play the cello'" actually helps make formal logic rules clear and more approachable.  A short philosophical/practical discussion of why we bother to codify these axioms may be useful as well. I've found that many of my students sit back in Geometry class and let the teacher do the proving.  I cannot emphasize enough how bad this mindset truly is.  One admittedly pedantic exercise I do quite often is to ask a student to reconstruct a proof covered in class.  If they get stuck, I'm happy to help.  Then when it's done, we throw away that page and try the same proof again.  You'd be surprised how many students can repeat this process a number of times before getting everything 100% right!  If they can't reconstruct proofs they've been given, it's hard to expect them to write new ones. Make sure your students know the basics, especially when covering trigonometry and certain aspects of analytic geometry.  At one point I discovered a student of mine really struggled with trigonometry simply because his understanding of manipulating and doing calculations with fractions was frighteningly low! 2.  Geometric intuition A dual aspect of being an effective problem solver in geometry is to get a strong intuition for the problems at hand.  Fortunately, for me at least, this process is much easier for geometry than in fields like abstract algebra or even topology!  In particular, while we may get caught up in the formality of "Geometry" as a discipline, we're bombarded with arguably more complex geometric problems when navigating a cluttered room than when doing high school math homework. So why do people struggle with this intuition?  One theory of teaching/learning that might be useful here is that of "types of learners."  For instance, many folks describe themselves as "kinetic," "visual," "aural," etc. -type learners.  For all but the second, figures in a geometry book are likely not the best way to gain geometric intuition.  Get people building, drawing, and searching for physical examples!  The more diverse senses you can appeal to, the more likely it is you may hit someone's preferred learning style by accident -- and as you work with them more, you'll be able to parcel this out more clearly. One particularly useful skill that is encountered in geometry, analysis, and other fields is that of testing the limits of a hypothesis or conjecture.  In particular, all too often I'll ask a student "Is this statement true?," and they'll draw a single example to verify my claim.  Always remember:  it takes one counterexample to disprove a statement, but a rigorous proof to justify one beyond any doubt.  Having students work on generating counterexamples to subtly false claims (logical, geometric, or otherwise) can be a great way to encourage a stronger approach to solving problems. Hopefully with some patience you'll be able to see concrete improvement in your student's work.  Remember that meticulousness rather than efficiency is the most important, and that the latter will come with practice and time.  Pretty soon they'll be proving Gauss-Bonnet without your help!