You probably already know how much I LOVE proofs. It's my absolute favorite thing to teach.
But this post is focused more on why I argue that we should never "water down" or cut back on explicitly teaching formal proofs in Geometry class.
The formal proof is a staple of the geometry curriculum. It has also been the center of debate among educators for quite some time.
Some educationalists believe that the proof should be abandoned for less formal ways of understanding geometric ideas, while others believe that the emphasis of the formal proof is an integral part of learning geometry.
However, any decrease in proof based lessons is an extreme disservice to our students.
What Is a Proof in Geometry?
Basically, a proof is an argument that begins with a known fact or a “Given.” From there, logical deductions are made through a series of conclusions based on facts, theorems and axioms. This will finally prove the proposition at hand, for example, the sum of the angle measures in a triangle equals 180˚. By writing out a proof, the answer is undeniable.
Why Are They So Important?
As powerful as our brains are, they can miss key facts and be fooled. There are times where things seem perfectly reasonable and they turn out to be wrong. That’s why we need to learn how to PROVE things. When you go through step by step, with the deductions laid out, you know what you’ve done is absolutely correct.
When mathematicians first began to form rules to prove valid mathematical statements, they did so through trial and error. This allowed congruence in learning. One person could show another person a mathematical rule and prove it through reproduction, which in turn made it valid.
However, proofs aren’t just ways to show that statements are true or valid. They help to confirm a student’s true understanding of axioms, rules, theorems, givens and hypotheses. And they confirm how and why geometry helps explain our world and how it works.
What Harm Does It Do When Proofs Are Removed?
All mathematics are rooted in problem sets, however the problems in geometry that require proofs of propositions do more than apply a theory. They are a part of it. When students learn how to postulate and prove concepts, they are tapping into a deeper stage of mathematics.
Geometrical proofs offer students a clear introduction to logical arguments, which is central to all mathematics. They show the exact relationship between reason and equations.
More so, since geometry deals with shapes and figures, it opens the student’s brains to visualizing what must be proven. It rounds out their knowledge, building upon the concepts of basic algebra.
Students have to combine all of their acquired knowledge. They have to develop a mental list of steps that will lead from the given to the conclusion. Then, they have to find ways to show algebraically that it all works out while simultaneously following along in a diagram. They must combine two lines of logic to create a new one and flow from one step to another. It can take some deep planning and thinking for a more challenging proof.
It’s a whole new way of thinking that develops entire new brain connections for them!
Benefits Beyond the Classroom
Basically, proofs do have a very important role in the geometry classroom. They offer:
- a means of communicating your reasoning with others,
- a justification for the way things work,
- and a basis of developing other applications of logical reasoning.
Sound Overwhelming? Read This for Help:
If proofs intimidate you as a new Geometry teacher, or even if you’re a veteran and your kids always struggle through the first weeks of proofs, you are not alone! It’s such a tricky new way of thinking for them.
But don’t skip them! Instead, go about it in a better way. Check out this twist: It’s a key step that I added into my introductory unit on proofs for a much smoother transition into teaching proof writing. It made a world of difference for my students.
I wrote up this post to guide you through a smoother way to set students up for success with proof writing. Click the image to read how to teach it and to download the free files to get you started!
Looking for even more support?
The complete unit I developed includes a presentation and printables to lead your students from the basics (properties, postulates, etc.), through a special revamped Algebra proof that scaffolds their learning all the way up through writing their first batch of Geometry proofs!
This method is the smoothest way to introduce this challenging unit. Check out what's included in the full unit.