Suppose you’re teaching something to your class – for argument sake, let’s say it’s that the incenter of a triangle is the intersection of the angle bisectors – and a student interrupts you to ask “Yah, but why does that work?”. So you reply with something more general you’ve been talking about recently – like, say, angle bisectors in general, or properties of triangles. But then the student says “yah, but why’s that work too?”. So you keep going back and back into the curriculum, trying to justify this answer of “why does this work?”. Essentially, you’re trying to answer the question of “Why does Geometry work?”. My question is: What is the intuition that you eventually stop at?

4 situations in Geometry which, in my opinion, get to the heart of this and are difficult to answer:

(1) Why is Side-Angle-Side a valid justification to show two triangles are congruent?
(2) Why are Corresponding Angles of parallel lines congruent?
(3) Why are the base angles of an isosceles triangle congruent without using a SAS triangle congruence proof
(4) Why is the longest side of a triangle opposite the largest angle? (Seriously – try and justify this with a proof or construction that doesn’t involve drawing every single triangle and observing that this property is true).

For the most part, these answers rely on some sort of “Well… it has to be. It can’t be any other way”. I have a philosophical problem with this justification because the advent of Non-Euclidean Geometry showed that justifying something as ‘It can’t be any other way – I’ve always observed this to be true’ has its holes. If I were to teach the above 4 questions to my students, I would eventually rely on ‘Well… you’ll either have to trust your intuition that these are true, or trust me that these are true”. But the question is – how do we go about finding the right intuition? How do we eventually satisfy the question of “Why does Geometry work anyway?” and why are we allowed to play Geometry?

Answer #1: Solve it the same way the Mathematicians of the 20th century did: turn Geometry into an axiomatic system with undefined terms and accepted postulates. Geometry works because we created the right set of rules and the right set of definitions, and it just so happens that these choices let us model some real-world situations. Our justifications are theorems and proofs derived from our choice of definitions. In other words, Geometry works because I can prove it. The questions above are all answered by a slightly fuzzy proof by contradiction.
Textbook which espouses this and is common in Arizona: McDougall-Littell

Answer #2: Solve it the way Euclid solved it: construct everything. If you’ve ever read (or skimmed) The Elements, everything is proved by construction – I can assert this theorem because I can describe a general procedure to construct the object which proves the theorem. This, to me, is a very computer-sciency approach to Geometry: everything we do is the result of an algorithm which will create the object we’re looking for with the properties we’re looking for. Geometry works because I can construct it. The questions above are all answered by the fact that every time I perform the necessary construction, the property I’m looking for is satisfied (this is a more inductive way of proving things, because it technically requires me to construct every possible triangle to satisfy question #4).
Textbook which espouses this and is common in Arizona: Discovering Geometry

Answer #3: Solve it the way the Common Core Standards do: Base everything on the notion of transformations in the coordinate plane. Quoting from the introduction to the Common Core Geometry section: “The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation”. Two shapes are congruent if and only if there is a sequence of rigid transformations which superimposes one shape precisely on top of the other (paraphrasing some of the POs). Symmetries in polygons are used to ‘discover’ many of the properties of triangles, quadrilaterals, etc. In other words, Geometry works because I can draw shapes and move them around. I’m not sure yet how the questions above are all answered – it seems a combination of induction (draw the shape, move it around, see that the property is satisfied) and proof.
Textbook which espouses this and is common in Arizona: NONE

A comment on the above line: As a new teacher, this annoys me – I still use the textbook as a resource and as the seed for an idea. As someone who went through a 4-year teacher education program and has done a lot of teacher observations, this scares me when I think of the number of teachers who blindly teach through their textbook without considering state standards. The Common Core, in my opinion, completely shifts the intuition of why geometry works which should also result in a curriculum shift – for example, Transformations should now be the very first thing taught in a course, as opposed to half-way through the year. If teachers don’t understand this and continue to teach with old textbooks, this will be a giant disservice to students when it comes time to take state exit assessments based on the Core Standards.

Since I’m teaching Geometry this year and will be teaching Geometry when the Common Core is officially implemented, I’m preparing for this intuition shift. And honestly, I like the shift a lot – I think it’s exciting and a better starting point for students. They come in with an understanding of shapes and lines and an understanding of movement and rotation and translation (think of every kid who ‘flicks’ and ‘pinches’ the screen of their iPod or iPhone or iPad) – why not use these understandings right away? I even like the intuition tied to congruence – I’ve always casually justified congruence by saying you could take one shape and flip it onto the other one, and I think most students do so too. There are other reasons I like the Core Standards, but mostly I find it interesting how it shifts the intuition of geometry away from many traditional justifications.

Anyway – those are some thoughts on intuition and the Common Core. And it’s all I have for now. Thanks for reading.

Edit 8/20: Removed a digression on intuition-based arguments and added several links in the text for curious parties.

From → Curriculum, Math