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. |

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.

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.

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.

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.

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.

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!

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:

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.

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. |

I send out some free resources for two-column proof writing to my email subscribers as part of a welcome kit! Enter your email here to get those downloads delivered right to your inbox:

These booklets are my favorite resource to leave for a substitute day, because they are completely self-guided and they even correct the students as they go through independently. The self-correcting features make it so easy to differentiate without needing the teacher for a day! They take a lot of paper to prep, so the best way to do it is to get them ready in the days before school starts (or when you have a pre-planned sub day). The kids LOVE them because they make their own choices throughout the story and get to see what happens! Check out this dive into how the booklets work: |

Click the button to get the version that will be best for each of your courses!

I hope that your students enjoy these, and that you can relax knowing that they are each on their own perfect learning track, not wasting time while you are out! (Even if you aren't absent, you can always use these for a quiet work day!)

]]>This collection includes apps that focus on what I believe are the best possible ways to integrate tech in the math classroom (with the inquiry-based approach that I’m so passionate about).The goals here are: - VISUALIZING with virtual manipulatives
- INVESTIGATING tricky math concepts
- INTERACTING & EXPLORING by dragging components, tweaking variables, and finding patterns
Through these methods with these apps, students can make discoveries for themselves and visualize and understand properties that are difficult to see as clearly without technology. |

Although I just posted recently that I strongly believe that the classroom needs to be rooted in pen and paper, that does not mean that our students need to be stuck in an era before technology. There is nothing wrong with allowing eager learners the ability to take advantage of living in the 21st century. In fact, that is part of what education is all about. Improving ways to teach, learn, and grow is how we make the world a better place.

I think the most effective way to integrate technology in a math classroom in the 21st century is to use interactive apps that allow students to**explore, investigate, manipulate, and visualize properties.**

Allowing children and young adults to explore math through virtual tools is amazing. They can discover new connections by engaging on new levels which will increase their academic success. Most of us grew up in classrooms with limited technology. Just a few decades ago there was typically only one computer lab for the whole school and we all took turns getting to experience the cutting edge technology of DOS, Windows and Macintosh.

This was the birth of technology in the classroom; integrating virtual lessons to enhance our comprehension while having fun. And those were the days that were the best. When it was your classroom’s turn to play Number Munchers, you were having a great day at school. You could sharpen your basic math skills while playing video games, which was living and learning in the future to our parents. Not to mention a ton of fun for us kids.

I think the most effective way to integrate technology in a math classroom in the 21st century is to use interactive apps that allow students to

Allowing children and young adults to explore math through virtual tools is amazing. They can discover new connections by engaging on new levels which will increase their academic success. Most of us grew up in classrooms with limited technology. Just a few decades ago there was typically only one computer lab for the whole school and we all took turns getting to experience the cutting edge technology of DOS, Windows and Macintosh.

This was the birth of technology in the classroom; integrating virtual lessons to enhance our comprehension while having fun. And those were the days that were the best. When it was your classroom’s turn to play Number Munchers, you were having a great day at school. You could sharpen your basic math skills while playing video games, which was living and learning in the future to our parents. Not to mention a ton of fun for us kids.

Adding excitement helps improve the ability to learn through increased engagement.

Our kids get the same thrill and benefit of having technology in their classrooms as well. Even though the technology has grown leaps and bounds since we were in grade school, the feeling is the same. We are instantly more enthusiastic when we are able to explore concepts through other platforms.

Our kids get the same thrill and benefit of having technology in their classrooms as well. Even though the technology has grown leaps and bounds since we were in grade school, the feeling is the same. We are instantly more enthusiastic when we are able to explore concepts through other platforms.

Students can drag the vertices of a triangle to see the various properties, or drag algebra tiles or fraction tiles for hands on learning. They can also see the more in depth concepts behind adding and subtracting integers with color coded counters. Plus, tech apps offer a great way for visualizing derivatives and integrals, by being able to see how they change as you adjust the function. There are so many wonderful apps and virtual tools for students to use. It can be hard to narrow down the ones that will benefit your children the best, but here are the ones that maximize the ability to investigate math concepts in hands-on ways. |

1. **GeoGebra** - It’s no secret that I love GeoGebra. Their application of dynamic mathematics software is great for all levels of education. It brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one easy-to-use package.

2.**Ooops App** - This “Order of Operations” activity is a fun way to practice the order of math equations and problem solving. This addictive game app sharpens math skills as well as cognitive development and critical thinking for any algebra students.

3**.** **Hands-On Equations** - Just like it sounds, this Hands-On app allows you to interact with algebra with games and lessons. Students will have fun with the experience of having success with sophisticated algebraic equations.

4.**Attributes by Math Doodles** - Discover the joy, wonder, and fun of mathematics through patterns and interactive puzzles while using different parts of your brain. The key to understanding math is understanding patterns. That’s why this app is perfect for exploring math. In a brain-based approach like my own favorite Math Doodle Notes, students get to creatively use both hemispheres of their brain while learning basic and complex concepts.

5.**Algebra Touch** - Students really get a hands on feel with this app. Using the touchscreen technology you can manipulate algebraic equations, such as isolating variable by dragging X’s and Y’s to either side of the equation. It’s a perfect all levels of algebra, from beginners to advanced.

6.**Geometry Pad** - For a more advanced geometry student, this app packs tons of benefits. Students and teachers can use it in class for a deeper understanding of geometric concepts. Easily create complex geometric sketches, measure everything you have in a document, and experiment with shapes and transformations.

7.**PhET Interactive Simulations: Math** - “Beautiful and responsive tools bring light to tricky math concepts.” This free app is great for students and teachers in beginner or advanced math subjects. It includes a draggable “Trig Tour,” an interactive “Function Builder,” curve fitting and graphing apps, and more!

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Technology will never fully replace performing math by hand, yet it is a fantastic way to explore mathematical concepts. I hope some of these help build your students understanding of higher-level math concepts! Share your own favorites in the comments area below to help us discover more great apps!