Proof barely exists in the Australian Curriculum. This is nuts, of course, but leave that be. Even when it is accepted that proof should – must – have a significant role in the curriculum, there are questions to be asked:
In which topics should proof be introduced and emphasised, and at what stage(s)?
Should “proof” be more a topic(s), or more an omnipresent concern?
How proofy should teacher-presented or students’ proofs be?
These questions, particularly in relation to Euclidean geometry, arose in comments to the post on the recent “trigonometric proof” of Pythagoras. (Also relevant are the excellent, thought-provoking articles by Hung-Hsi Wu, particularly this and this, deserving of their own post for discussion.)
It’s rare, at least for Australian students, to see anything proper of Euclidean geometry, except a piece here and a piece there. Which is exactly not the point. It is difficult to imagine anyone still using Jacobs’ beautiful Geometry, but perhaps there is some backwater where it has not been forgotten.
In their great (and free) book, The Essence of Mathematics Through Elementary Problems, Alenxandre Borovik and Tony Gardiner outline what they refer to as a “semi-formal” approach to school geometry (Chapter V). They introduce their outline with an extended discussion of the teaching of geometry (pp 169-173). A couple excerpts:
… in many instances, the “essence” that is captured by a problem requires that the problem be seen within an agreed logical hierarchy – a sequencing of properties, results, and methods, which establishes what is a consequence of what – and hence, what can be used as part of a solution. In particular, we need to construct proofs that avoid circular reasoning. …
… the cumulative architecture of [Borovik and Gardiner’s outline] conveys a rather different aspect of the “essence of mathematics”, deriving not just from the individual problems, but from the way a carefully crafted, systematic arrangement of simple “bricks” can create a much more significant mathematical structure.
There is no question that an extended logical hierarchy is more valuable than the sum of the individual bricks. But is there still value in presenting isolated bricks? What about the proof, or “proof”, of Pythagoras pictured below (Proof #9 in cut-the-knot’s list)? When should this proof be presented to students, and what hierarchy, if any, should accompany it?
Or, should it not be presented at all? Surely, that cannot be right.
Uh, the whole high school geometry course is proof after proof. Side angle side out the yingyang. Actual mensuration and types of shapes and the likes is what is underdone. And solid geometry gets kind of a too quick run at the end. But God damned, we have similar triangles and vertical angle theorem for almost a whole year. That and construction, which I never saw the point of, but we did a lot of it. It is fun to poke people with the compass though.
In the rest of the algebra to calculus rush, proofs make sense as occasional explication and motivation and really serve the purpose of derivations. But I push back hard core on people who want to turn calculus into real analysis class. It needs to be learning how to differentiate and integrate. All the functions and tricks. And with applications like related rates and work and volumes and the like. Yes…WITH all the algebraic manipulations too. (That is a feature, not a bug to solve partial fractions in algebra 2 class, then in second semester calculus, then in ODEs for Laplace transforms.) Little bit of parametric equations. Some series stuff…NOT a whole real analysis course, but some exposure to different convergence tests. (For one thing, series are just kind of algebraically detailed and it is nice to be exposed since you’ll need them later in ODEs class.) A primer on ODEs through second order constant coefficient heterogenous thingies. But please…emphasize the actual calculus through Euler’s time. Not 19th century analysis. Three quarters of a loaf done well, with lots of drill and the like is a better result than a whole loaf that is all screwed up.
Whose “high school geometry course”? (And no sermons, please.)
Umm…a high SES suburban country public high school in the US in the 1980s.
[Sermon] But I had the impression our course was pretty stereotypical in content across the country. Maybe our teachers were a little better and the students stronger. But it was a pretty classic American geometry course.
I looked at my Dad’s books from the 30s and the 40s from the War Department– they actually had 3 books to cover geometry, very progressive (in the sense of gradual not librul) and they were very similar. So probably even stereotypical across the decades. If anything our textbook seemed to have less cute photos and pictures than my Dad’s which had a few naval officer and airplane and automobile motivational pictures. Ours was just labeling angles after angles after angles.
It’s not Euclid 100%. Like they don’t list and number his axioms. And go word for word from his book. They label the theorems with names instead. And have drill problems. But you’re basically working through the same stuff. SAS and SSS are OK. But ASS makes an ass of you. I remember that.
I remember the teacher saying this course would be different than algebra 1. Or the rest of our courses through calculus (algebra 2, trig, functions, analytic geometry, calculus) in that we’d mostly be doing proofs, not calculations. It was maybe 90-10 proofs to calculations in geometry. The reverse in all the other x-pushing classes. [I get the impression this is still the case since a lot of teachers and students say that it is sort of a sidetrack from the race to calculus.]
P.s. But my perspective is that of an ex student, not a teacher. So maybe I’m missing some subtleties. But it’s not like I took the course in the 50s or something. I mean it was post New Math. Post hippies. Post Viet Nam.
I have a sense of what US geometry used to be. I don’t know what it is now. Australian geometry is close to non-existent.
It probably varies. But I get the impression it is mostly still an orgy of similar triangle proofs. Not that different from your father’s Oldsmobile. Here’s a 2021 post complaining about too many postulates and theorems and giving mensuration short shrift:
https://news.ycombinator.com/item?id=26330240
I explained how I try to introduce proof to my Year 9 students here in the comments:
As an example, attached is an assignment that I will give to my students this coming week. The students will do this in class: we have a double lesson. I showed this on Marty’s blog sometime ago and a correspondent kindly suggested a correction which has been incorporated.
assignment2
Geometry in VCE is now virtually non-existent, and even though proof is assessable in Specialist now (woohoo!) geometry is not too-long for the bin at 7-10 I would suggest.
Yes, there is all the congruence and similarity as Anonymous pointed out above, but there was (were?) once also circle theorems and their associated proofs in Year 10 for a while.
That not many teachers I know have noticed their absence probably says the proof part was never really taught.
I have a dim dim memory of something to do with chords and arcs. Were those the circle theorems? Probably the reason I remember the chord and arc was because they were cool words. Don’t recall what the theorems were though. Don’t remember them being needed later like the triangle is for related rates problems with sliding ladders (or trig). Or area of a trapezoid for Archimedes method.
Maybe you don’t need a lot of Euclidean geometry when you have analytic geometry to help guide you, also. I mean nothing wrong with marching the kids through a hard core geometry class. Get to see it once. But realistically algebra is much more vital. Like if you kick ass at algebra and suck at geometry, you’ll still be mostly fine in calculus. But if it’s the reverse? No way. Not unless the calculus actually helps firm up your algebra (which can happen…if you make the little monsters work problems and not push CAS buttons!). [Crap, I seem to be giving the eeful librul ed reformers cover for gutting geometry and using the time for more analytic stuff…or even stealing some of the time for more baby statistics…]
What really bugged me was not learning more about the solids. It was so late at the end of the year. Like it would be very cool to learn about truncated pyramids and prisms and cuboctahedrons. But it seems like we never did that much. Had to learn some of it in minerology classes.
The solids are harder to draw, too. I’ve heard that people learn it in hand drafting classes. My guidance counselor was right…I screwed up not taking drafting. https://www.youtube.com/watch?v=byTsfsWcmY8
Are Euclidean and Analytic geometry opposites though?
I am of the opinion that you Euclidean Geometry and then if you choose to replace the 5th postulate with something else you have non-Euclidean geometry. Either approach can be analytic if you choose, surely?
Pedagogically and in terms of how our minds work practically, I think they are rather different (not “opposite”) fields of knowledge. You are using a pretty different mental muscle when you use the distance formula on cartesian coordinates. If some Freeman Dyson type wants to equate the two (like with Feynman diagrams and Schwinger’s Green’s functions stuff) then fine. But in terms of practical learning/working/using, they sure are not the same thing.
I have no clue about non-Euclidean geometry. Haven’t worked it in any context. Nor will the vast majority of students. You don’t need it in undergrad science or engineering. So I don’t really see it as germane to the needs of students. I do remember my teacher in high school mentioning that you could vary the parallel lines axiom and get new geometries (and this sorta explained why what really looks like something we could solve as a theorem was an axiom). But that’s all I know about it…that it’s a field of knowledge. Never needed to actually learn/use it. I mean I know spherical trig is a field of knowledge also. But I never learned it. There’s a lot of stuff like that.
Note, I’m not against Euclidean geometry. It does (or did, or still does in much of the non-VCE-corrupted world) give an interlude of attention to proof versus algebraic techniques. It’s just one year off the pre-algebra to calculus rush. But let’s not strain our brains for justifications and distort reality. No way are the pointy compasses and pencils developing algebraic ability. No way is SAS doing that. And A = b*h/2 is already much simpler than the CDs and DVD problems we learned to disentangle with simultaneous equations in first year algebra. [Ugg…I think I am coming up with justifications for the eeful reforms of the bloodsuckers!]
I enjoyed reading the second essay by Hung-Hsi Wu that you linked and I agree with a lot of it. In particular, that what we need is not so much formal proof as very clear definitions, from which we can logically explain and derive why things are true and why subsequent definitions make sense. I think there’s plenty of room for very simple proofs in the sense of precise explanations of why one thing proceeds from another.
I think this should be a common occurrence and start with things that aren’t too complicated – simpler than the proof of Pythagoras theorem. I imagine this happens naturally so long as teachers have the mathematical knowledge necessary to understand the relevant proofs themselves and recognise flaws in their logic so they don’t make up false explanations.
For example, I find that as Red Five says, students don’t seem to learn much geometry anymore in Victoria. In particular, they’ve often never heard of similar triangles. Yet somehow, they’re expected to understand what trig ratios mean and why they should exist. (I work as a relief teacher, so I’m usually helping students who are confused about something they’re already supposed to have learned. I guess the teacher must explain? But they students say they don’t know about similarity and yet have their trig ratio notes out.) I find it really helps to preface any explanation of trig ratios with a preamble about similar triangles. At least, “how do you know two right-angled triangles are the same shape?”, “what can you say about right-angled triangles that are the same shape?” Then the definitions follow more naturally as something meaningful that students might be interested in and they get it. Without that, it seems pretty confusing.
The CAS button cares nothing for your shapes.
@wst: I am very interested in your experience as a casual relief teacher. I might be doing this next year. If you wish, you can tell me more about it by email. Marty will give you my address.
Done.
Not many texts actually gave similar triangles before trigonometry.
It does not seem pedagogically correct.
Not surprised students are confused, do not retain the routines.
Y9 trig chapter is not that much different from y10 trig chapter. What is the point of doing the same thing over again? Second chance?
When I first saw that image of Pythagoras’ theorem in a book somewhere as a child (or maybe it might have been somewhere else, I can’t remember), I didn’t understand the proof. Probably because the proof requires these concepts, which I might not have fully understood or “ingrained” at that stage in my education:
(a) that you can simply “copy” a triangle and get a congruent triangle
(b) that the sum of the angles of a triangle is 180 degrees, and that a line segment can be created by two line segments connected at a point by an angle of 180 degrees.
And probably, most importantly of all, this proof shows the Pythagoras theorem working for one specific right-angled triangle, and I wasn’t convinced that this image proved it for all right-angled triangles. I am still not entirely convinced on this point.
Some authors are critical of Euclid’s Elements because often he includes a diagram in a proof. For example, they say that he has used only one triangle. His proof is based on a sample of 1. What about all the other possible triangles? I do not agree with this criticism. My view is that the diagram makes it easier to follow the proof. You could present the proof without any diagrams and it would be equally valid. Still, the reader must bear in mind that Euclid has used only one diagram.
That is true, in the context of Euclid’s elements, where the diagram is just an additional supplement, and the real content of the proof is in the words.
But the proof of Pythagoras’ theorem presented in the original post is just a diagram alone.
Perhaps all those interested in pursuing a career in teaching mathematics should take a semester devoted to Euclid’s “Elements”. It’s a beautiful piece of work.
Suggest it to your mates at Deakin. Let us know how you go.
Mathematics departments in universities should offer subjects that will be useful to prospective teachers. After all, there are many opportunities for mathematics teachers. A course on “Elements” might be useful in a degree in mathematics.
2023-jobs
Sure. So suggest it to your mates at Deakin, and see how you go.
I don’t have any mates at Deakin. They have all moved on since I was a student there.
Fine. You just talked highly of Deakin. But the reality is you don’t have a snowflake’s chance in hell of convincing anyone in a maths ed faculty to do this.