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I've encountered classes of 25 or so undergraduates in which more than half -- maybe two-thirds -- of the students claim to have had a high-school geometry course but none has seen a proof of the Pythagorean theorem or the proof of any theorem in geometry.

It used to be that some students who didn't remember that circles or triangles or lines or planes were ever mentioned in geometry nonetheless remembered that geometry is the subject where you prove things. That's the principal content of the course. So it was when I was in high school, and so it is in Euclid, which was the standard textbook before and during the time of the Roman Empire and from the middle ages until the middle of the 20th century.

I have the impression that some high school courses present a bunch of formulas about areas and volumes in a dogmatic way and call it "geometry". And some others do intelligent things, e.g. thinking about groups of rotations, reflections, and translations, or combinatorial things about convex hulls or polyhedra.

Is there some kind of standard secondary-school geometry curriculum today?

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    $\begingroup$ It depends on which country $\endgroup$ – Paracosmiste Feb 18 '17 at 19:30
  • $\begingroup$ see also matheducators.stackexchange.com/q/2074/127 $\endgroup$ – Gerald Edgar Feb 18 '17 at 21:41
  • $\begingroup$ Also this matheducators.stackexchange.com/questions/5778 $\endgroup$ – Paracosmiste Feb 18 '17 at 21:44
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    $\begingroup$ In the UK GCSE (public exam taken by most 15 to 16 year olds)-level geometry is called "shape". It is taught as a collection of properties of plane figures such as circles and triangles, parallel lines, regular polyhedra and simple prisms. The IGCSE includes questions on solid triangle based objects such as inclined planes or pyramids. There is no attempt at proof. The only "new" geometry at A-Level (18 year old, usually) is co-ordinate geometry of (mainly) circle and tangents. There is no attempt at proof. $\endgroup$ – Clive Long Apr 7 '17 at 21:50
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I'm going to say mostly the same thing as Ben Crowell, from just a slightly different perspective. I'm at an urban community college on the U.S. East Coast, teaching college algebra, statistics, etc. To the question, "Is there some kind of standard secondary-school geometry curriculum today?", then yes, the best answer currently is Common Core (Geometry).

Addressing a few of your specific items:

  • A proof of the Pythagorean Theorem is an expected standard for earlier in the 8th grade (8.G.B.6: "Explain a proof of the Pythagorean Theorem and its converse.").
  • Proofs of theorems are an expected part of high-school geometry (HSG: "Prove geometric theorems", "Prove theorems involving similarity", "Prove that all circles are similar", etc.)

In this regard, I'm modestly optimistic about what I've seen in Common Core mathematics, and that it's something of a re-alignment to solid, traditional principles. However: It's hotly debated, not implemented in all states, not yet fully implemented in the states that have signed on in the last few years, possibly not rigorously tested, not yet representative of what graduating high school students would know, etc. To the extent that schools drifted in a bunch of different directions, it would (at best) take time for Common Core to wrangle them back on a sensible path.

I will say this: Within the last year I've been giving a 1st-day introduction to all my courses including bullet points such as "This class will include proofs". As part of that I've been asking, "In what high school course did you talk about proofs a lot?", and gratifyingly every time so far I've gotten a common group shout-out of "Geometry". So for me that's actually been a bright spot recently, and a point from which I can start making connections and deeper understanding.

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I teach physics and a little calculus at a community college in California. My own kids went to local public high school, which is unusually good, and took the IB curriculum.

Sometimes in my physics classes, when discussing a topic like Newton's laws or DC circuits, I will use proof-based high school geometry as a reference point for the kind of formal reasoning and rigorous logic that is required. When I ask for a show of hands, about half my students say that they did two-column proofs in high school geometry. (This is usually accompanied with groans.)

My kids used a high school geometry textbook that did include a heavy dose of theorem proving, although proof was not emphasized as much as it would be if one was taking one's agenda directly from Euclid. To my taste, the approach was a little ugly and baroque, but the beautiful simplicity of Euclid was definitely embedded somewhere in there. The real number system was treated as something separate from but connected to geometry -- a strange and ugly approach, IMO.

I don't know if you're familiar with the US system. Our schools and curricula have traditionally been under completely local control, but over the last 40 years or so, control has gradually been shifting to the states and the federal government. We have something called Common Core, which is a set of national standards, which showed up around the same time that a well intended but sometimes farcical national political initiative called No Child Left Behind began to collapse under the weight of its impossible promises. Depressingly, Common Core says:

During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms.

If you try to parse this closely, it really doesn't make much sense. How does one do "careful proofs" without starting from "a small set of axioms?" It smells like language that was worked out as a compromise by a committee, taking into account the fact our educational system is neither able to provide rigorous educational opportunities for all, nor willing to make the attempt.

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I can share with you my experience. When common-core started to take off, I had just started Geometry. We were expected to prove things regularly, triangle congruency, similarity, angle-sum, parallelograms, Pythagorean theorem, etc. And we were graded on our reasoning.

Here's a list of what I remember we went through, proofs found there way in every part except transformations and constructions though we were expected to still explain what we were doing and why:

  1. Parallel lines cut by a transversal
  2. Triangle congruency, and proving theorems about triangles
  3. Transformations in the plane
  4. Constructions
  5. Triangle similarity (This is when we started to prove Pythagorean theorem)
  6. Trigonometry (sin,cos,tan,law of cosines, law of sines)
  7. Coordinate Geometry
  8. Proving and using theorems involving quadrilaterals
  9. Proving and using theorems about circles
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This is really a comment on some of the other answers but was too long to add as a comment to other answers, specifically about "short" axioms systems.

I believe that "proofs" tend to be given too much attention in lower grades but let me say something about "axiomatics." In America when people look to do Euclidean geometry "rigorously" they turn to Hilbert's axioms. The first version of Hilbert's axioms were not independent but in the often used independent version there are 20 axioms, which is not so small a list. And then there are his 6 primitives. Also, Hilbert axiomatizes 3-dimensional geometry rather than plane geometry and for the plane geometry one has to "worry about" non-Desarguesian planes.

https://en.wikipedia.org/wiki/Hilbert%27s_axioms

If one one wants to give the "spirit" of axiomatic systems and proving theorems within them one can select a small list of "more powerful" axioms and/or ignore issues that Hilbert calls attention to. I would rather show samples of proofs about what for me are more interesting geometric phenomena. Two of my favorites are Euler's traversability theorem for connected graphs (Euler circuits) and Euler's polyhedral formula ($V + F - E = 2$). I feel both the proof aspects here, and the "content" of the geometry is more important than a lot of the proofs done in axiomatic geometry contexts.

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