What Is a Proof in Geometry?
If you’re jumping into your first round of teaching geometry, a quick refresher – Yes, we are talking about the two column proofs that we learned while we were in school. But do you remember exactly why or what they were for?
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?
Well, logical reasoning and deduction are central to understanding not only geometry, but mathematics as a whole. Being able to tell the difference between obvious mathematical concepts and ones that need to be justified is a new level of understanding in math. It shows comprehension of deductive logic and the ability to structure arguments to make mathematical conclusions. All of these skills are paramount to reaching a more mature and complete knowledge of geometry and arithmetic.
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?
Limiting the amount of substantial and challenging proofs in a geometry curriculum pretty much defeats the purpose of the course. Now, that may sound a little exacting, but it is true. The reality is that geometry is different than other math courses.
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
Reasoning is a skill that has a multitude of applications. Whether you’re proving a geometric postulate, working through a detailed word problem, navigating facts in a debate, or even making a monthly budget, you will need reasoning. While we do learn reasoning outside of geometry, students that practice proofs strengthen that skill even more. You learn how to reason carefully and find links between facts. This is something that is important for everyone, not just mathematicians.
Basically, proofs do have a very important role in the geometry classroom. They offer:
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