Mathematics in Real Life: Teaching the Elements of Proof

[Jamie Poff]

For the past week or so at my normal job (teaching high school geometry), I've been trying to get students to recognize the real-world value of one of the most "dreaded" skills in the class: writing proofs about the congruence of triangles.

The problem with this subject is that very few people have a need to show that two triangles are congruent...ever. Honestly, it's not all that important for day-to-day living. However, a little creativity seems to make some difference.

Why learn proofs? Well, something I try to get across to students is that the geometry is not the important part...it's the development of rational thinking skills. In proofs, rationale for an argument springs forth from three main resources:

-- postulates, ideas that are accepted as truth without any real evidence. In math, the numbers go in a certain order, points are part of all objects, stuff like that.

-- theorems, ideas that accepted only because they have been previously proven through research. example: the Pythagorean Theorem, which has been proven countless times in as many ways.

--definitions: ideas that develop by giving a term a meaning, and using that meaning to express an idea. For example, a bisector divides something into two congruent parts.

Using a combination of these things in a logical order, one arrives at a new idea, which can then be used for more advanced thinking.

And here's the part that too few students understand: every argument ever made, in any field or real-world situation, uses those same three things. So, today we did a little role play, making ourselves a prosecuting attorney, and imagining how we would find convict an idea into truth.

Attorneys do this all the time -- and they use the same tools:

Postulates ---> Ethics. We just KNOW that some things are right, others are wrong. We don't need to prove that stealing is wrong, at least, most of the time. (Tricky thing about ethics, I guess).

Theorems ---> Precedents. Attorneys are always looking at previous trials for evidence of how Court X ruled in a similar situation, then using that information to suggest a course of action.

Definitions ---> Definitions. Attorneys always have to know the key differences between terms to persuade someone about the relevance of a grievance. Example: the differences between manslaughter and murder, petty larceny vs. grand theft, etc.

And this kind of parallelism exists in almost every walk of life. Ideas we just know, ideas we know because of past knowledge, and ideas that express meaning of all the various forms of jargon in different life situations. One simply cannot live without learning to use these!

So, maybe my students will never need to compare two triangles for congruence in real life. Hopefully, today we started training ourselves how to reason and think. Maybe we'll all be better researchers, future doctors, businesspersons, etc. someday as a result.

I'm always about trying to find these connections to the real world for my not-so-real-world math lessons. If you have ideas or thoughts, feel free to chime in.

[Amy Pritchard]

And a commendable post at that Jen! I think my first post was "How does this thing work?"

[Jen Klein]

Hey, I'm geeky enough that I was excited recently when a new prime number was found.

(my first post -- hi, all!)

[Jamie Poff]

I think it's important to recognize that the math we learn in high school, if dissolved down to mere step-by-step processes, is, for the most part, something few people need.

However, I think there's a much bigger picture -- learning the "how" of math -- breaking down provided information, analyzing each part and drawing new conclusions, then organizing those thoughts into something greater -- is why everyone needs to go through the process. Even as a teacher, I'm seldom asked (outside of work) to DO math I learned in high school...or college. However, I'm the decent-to-above-average project manager I am because I CAN do those things.

Personally, I love doing math, especially when I find a decent use for it. But I try to look at it from a real-world perspective -- most adults don't DO that, unless you look at the bigger picture.

[Michael Pinkerton]

Wow, math was always my worst subject, but that really makes sense!

[Amy Pritchard]

This is brilliant. I just forwarded it to my husband who has a MA in Philosophy.

I think someday, I will write this in the opposite for the legal community. Giving you ALL the credit of course!