Are Geometry Proofs Still Needed?

My students wonder why they need to learn all those postulates, theorems, etc we normally teach in Geometry class.  Why write proofs using them if they can “see” why, then do they need to “prove” it. I ask them if they can explain why two triangles are congruent using the given information.  About 98 percent of the time, they can’t, so it proves they do need to learn it all.

I believe it is important because the theorems, etc, give us a much better grasp of geometry itself. I think half the problem is that many of us teach proofs the same way we learned them, in a very dry and straight forward manner.  We start with the given and go to the end, using the appropriate theorem, etc with the appropriate reason. We do not spend enough time helping them understand all the connections.

If you look on the internet or in app stores there is nothing that is set up to automatically do geometry proofs so students have to search each problem individually to find a proof for it.  In my search for this type of site, I discovered this site which has interactive proofs for a variety of problems.

By interactive, I mean it provides a problem with the given and prove part along with all the steps and reasons mixed up. You drag each step and reason into the correct location until all the spaces are filled. Once you are done, click the check button and it lets you know if your solution is correct. If it’s not, it simply says try again.

This makes it a great practice site because students do not have to figure out all the steps necessary. If a stun dent is incorrect the first time they try it, they can erase everything and start again.  The site allows access to certain proofs so you can assign them or you can create your own. In addition, students can hit a button to bring up a random proof.

As a teacher, you can copy and paste a link to a random problem that focuses on a proof you want them to practice.  If you do not find a proof you like, you have the ability to create your own interactive proof using the names of theorems you want. The great thing is that the site is free to signup for and use.

Tomorrow, I will share some great tips for teaching proofs so students end up with better notes and are easier to understand.  let me know what you think, I would love to hear.  Have a great day.

Accelerated Learning.

No this column is not about that math and reading system offered to schools.  I am referring to a topic I'm seeing in educational books.  Recently, I've seen books on Accelerated Learning or Brain friendly learning.

This type of learning refers to any activity that expedites the learning process.  The idea is to create a situation where people learn the material faster while retaining it for longer periods of time. 

This is not the same as cramming.  Something most people has done at some time in their life.  I did it only in so far as doing a review of all the material on the test but it was still trying to cram.  Accelerated learning has five stages to the process.

The first stage is to prepare both the mind and environment.  It is recommended a person find a nice quiet place where they need to sit upright.  Laying down is not good for studying because it makes people more tired while decreasing concentration.  Always look over the material and set a goal time wise to learn the material. 

One way to do this is to imagine yourself as the instructor of the class.  As you study the material decide what you want to teach in what order so you can picture how you'd teach to others.  By doing this, you are internalizing the material, making it easier for you to learn it.

The second stage involves using the whole brain.  The left side of the brain is the part of the brain that produces language, words, and numbers while the right side of the brain is the artistic center responsible for music, patterns, spacial relations, etc. The final part, the limbic system decides if the information is important enough to remember.

There is research indicating that music without words with a tempo of about one beat per second. The music helps relax the brain while stimulating the limbic system.  It is important to take notes while reading or watching something but it is just as important to stop and picture how the material fits together overall. 

In addition, stop and repeat material out loud to reinforce your memory.  Furthermore, always rewrite your notes to add that last bit of learning.  Be sure to include short 5 minute breaks every 30 minutes so your brain has a chance to process information.

Stage three is to understand the material and not just know it.  It is possible to know something such as a quote from Shakespeare but to understand the quote is a whole different level.  To effectively learn, we need both understanding and knowing and must do it by turning passive memory into active memory.

The most effective way of doing this is either by explaining it to another person, organizing it, or summarizing it.  For instance, if you take notes, always paraphrase them by putting them into your own words.  Paraphrasing material means you have to think more deeply about the material while developing a better understanding.  In addition, if you organize the material from most important to least, you are thinking deeply about the topic.  If you can make the material your own, you will remember it better.

One way to summarize the information is to create a mind map with the main topic in the center.  Reaching out from the center, place keywords which describe important ideas and connect to the center.  Continue this activity until all the material is covered. Keywords trigger information about the topic.  If the mind map is done with color, pictures, shapes, symbols, etc, it makes it easier to remember the material.A good way to remember the material is to try to recreate the mind map on a different sheet of paper.

Another way to summarize information is to write one sentence on a paper that summarizes a paragraph.  Do this until you've finished the chapter.  Then take and summarize each page or half page with one sentence so you have a compact summary.

Stage four requires reviewing the material because most people do not remember material unless they look at it several times.  Begin by summarizing the material at the end of the study period, then the next day for several days, then once a week, once a month and finally after about 6 months, you should know the material well.  To make this effective, try to remember the material from memory rather than just rereading it.  One way is to pull out an empty paper and write down everything you can remember or recreate a mind map.  The review should not take more than 2 or 3 minutes.

A good way to remember the material is to associate pictures or some form of visualization since we remember pictures well.  This helps us create an association between the abstract and concrete which our brains need. 

Stage five has you apply the material.  There is not reason to learn the material if you have no where to apply it. So it is necessary to figure out ways to apply the material.

Tomorrow, I'm going to focus on applying accelerated learning techniques to math.  Let me know what you think I'd love to hear.