Proof in Geometry

When I was a high school student I hated the sections in Geometry on proof. My teacher frequently said that "implied" too much. It was hard for me to wrap my brain around the idea of proof and how to prove something. After high school I went to college where I completed a bachelor program in Mathematics and Mathematics Education. My coursework in Mathematics has taught me that higher math is more about logic and proof than solving equations. So even though I hated proof as a student, I feel that it is really important.

Previous Status Quo
I am about to start my fourth year of teaching Geometry. In the past my students have always complained about proof (like I did when I was in high school). Because I feel that proof is important, I want to find a better way to introduce it and teach it. I have an idea, and I am interested to see what will happen this year.

The book we use (Prentice Hall (C) 2011) introduces proof in Chapter 2 and immediately has the students do Algebraic Proof and then proofs about angles. As if solving an equation on one side of a two-column proof and the reasons on the other is enough of an introduction to proof that students can jump into some complicated Geometric proofs in the next section. For example the very first proof in the book is The Vertical Angles Theorem, which is not simple. It requires students to think outside the box and use an angle other than the two vertical angles, notice that they are supplementary, and then do some pretty complicated Algebra.

It is during and after these first two sections of proof that my students started complaining about proof. I would then tell them that we have at least two more chapters of proof and they would grumble.

In Chapter 3 we would do proofs about proving lines to be parallel. Students got pretty good about recognizing the angle pairs and knowing which Theorem or Postulate needed to be used (ex: Corresponding Angles Postulate, or the Converse of the Alternate Interior Angles Theorem). But when it came time to put those ideas into a two-column proof, they would get so confused about where to put statements and reasons that they couldn't write the proof. The same thing happened in Chapter 4 when we would prove that triangles are congruent.

My Idea
My current thought is to not introduce two-column proofs until Chapter 4 when we prove triangles to be congruent. Before then, require that the students give a well thought out justification that includes Theorems when applicable. They could do this for triangles in Chapter 2 and parallel lines in Chapter 3. Then after students have been working on written justifications for two months, they will then be introduced to "formal proof" meaning, two-column proofs. We would only do two-column proofs for triangle congruence, CPCTC, and later in the second semester, proving that triangles are similar.

In essence, I am trying to encourage students to understand the concepts behind proof without getting confused by the formality of a two-column proof. I don't want the students to get lazy, but I do want students to think through why something is happening instead of guessing at statements and reasons.

Geometry In-Class Activities

Here is my brainstorm of activities that I might use next school year. I want to make every class period different so my students don't get bored.
 

“Variety’s the very spice of life 

That gives it all its flavour.”
-William Cowper (1731-1800)

Starter Activities
1. Summary and Question
Teacher Preparation:
· None

In Class:
· Students write a summary of what they learned the previous night and either ask a question or write something they found interesting
Pros:
· Quick summary/synthesizing activity.
· Writing activity
Cons:
· It will be difficult for some students.
2. Flash Cards
(This activity can be used as a short opener activity in sections that have lots of vocabulary.)
Teacher Preparation:
· Teacher creates flash cards that students can use to drill material. Name on one side, picture on the other.
In class:
· Done after the daily notes quiz.
· Students drill themselves going from name to picture and then from picture to name.
· Then students get in pairs and drill their partner.
Pros:
· Great for sections that have lots of vocabulary.
· Lots of quick practice.
· Solidifies important vocabulary.
Cons:
· Lots of teacher preparation, unless these flash cards can be found online.

Practice Activities
1. Textbook Problems
Teacher Preparation:
· None

In Class:
· Have students do the assignment that we assigned last year in their textbook.
· They should do it by themselves, but they are encouraged to ask their partner or other students for help before they ask their teacher.
· When they are finished they should check their answers in a different color pen or pencil, and then fix their mistakes in that same pen or pencil.
· Then students should take their “Mastery Check” by themselves to see if they can do about 3 problems without any help.

Pros:
· No preparation for the teacher.
· Students can work at their own pace through the problems.
· Students can help each other.

Cons:
· Student engagement might be low.
· Deeper understanding opportunities will be missed.
· Students who get too much help from friends will think they understand, but they don’t.

2. Peer Instruction http://blog.peerinstruction.net/
Teacher Preparation:
· Print off Multiple Choice Questions
· Teacher complete problems to know which of the problems are good “conceptual” questions. (Hard questions that get at the big idea of the mathematics).
· Cut the Standardized Test Prep paper into strips with 1 question and answers on each paper.
· The first time you do this, you will need to prepare (or have your first class prepare) Four cards for each student that say A, B, C, and D on them.
In Class:
· Present a problem and have the students complete it individually and commit to an answer.
· When all students have finished, have them all show their answer at the same time.
· The teacher counts how many “correct answers” there were but does not announce it to the class.
· The teacher then asks the students to find someone who has a different answer than you and discuss your thinking.
· After a few minutes the teacher asks the students to show their answer at the same time.
· Again the teacher counts how many “correct answers.” (This is used to see which questions are good questions. Good questions will have a big increase from first and second attempts).
· The teacher announces the correct answer. They can have a student present the answer, or complete it themselves, or if there are lots of students with the correct answer then they can move to the next question.
Pros:
· Students discuss their thinking and listen to each other’s thinking.
· Deeper understanding takes place.
Cons:
· A little bit of preparation for the teacher.
· Only a few problems will be attempted during a 45 or 60 minute time period. (Which is why the selection of quality problems is so important.)
· I fear about students changing their answer to the answer that their “smarter” friend’s answer.

3. Group Games
Teacher Preparation:
· (Same as Peer Instruction)
· Print off multiple choice questions
· Teacher complete problems to know which of the problems are good “conceptual” questions. (Hard questions that get at the big idea of the mathematic s).
· Cut the Standardized Test Prep paper into strips with 1 question and answers on each paper.
· White boards with markers and erasers – If available

In Class:
· Students get in teams of 3 or 4. – Pick a team name and decide on their ordering (1, 2, 3, 4)
· A question is presented.
· All the students complete the problem on their white board with work and an answer.
· They may help each other in their groups.
· At the end of the time limit. A number between 1 and 4 is randomly selected and that student in each group’s answers are compared and the teams with correct answers get points and teams with incorrect answers don’t get points.
· If multiple groups get the problem wrong, invite a student up to the board to complete the problem (for an additional point) or do the problem yourself. Answer any questions and probe for deeper questions.
Pros:
· Student engagement and enjoyment is high.
Cons:
· Weaker students rely on stronger group members for work and answers. Weaker students don’t get effective practice.
· Deeper understanding opportunities are missed.
4. Student Created Word Problems
(This will not take an entire class period. Should be paired with another practice activity. Maybe be used after the other activity)

Teacher Preparation: 
· None

In Class: 
· Students get in groups of 3 or 4. 
· Students write word problems that are difficult but clear. 
· Switch problems with another group.
Pros: 
· No Teacher Preparation 

Cons: 
· Quality of problems can very.

5. Student Transcriptions
Rubenstein, R. N., & Thompson, D. R. (2001). Learning mathematical symbolism: Challenges and instructional strategies. The Mathematics Teacher, 94(4), 265–271.
(Best used when notation in emphasized or when diagrams are emphasized)
Teacher Preparation: 
· Prepare cards of notation or diagrams.

In Class: 
· One partner reads a symbolic expression or sentence while the other writes what he or she hears using symbols. 
· In another variation of the transcription activity, students can practice using mathematical language by directing their partners to draw figures given to students on index cards.
Pros: 
· Prepare cards of notation or diagrams. Cons: 
· Drawing diagrams may not be the most effective way of practicing the mathematics of the day, but would be good for critical reading and thinking activities.
Content Review Activities
1. Student Created Problems
(I imagine this best being used as a review activity)
Teacher Preparation:
· None

In Class: 
· Students create (word) problems over the material that will be covered or assessed. They may use notes, textbook, etc. 
· The students either give them to the teacher and a group game is played.
Pros: 
· No teacher preparation Cons: 
· Quality of student created questions can very.
2. Student Section Summaries
Teacher Preparation:
· Prior to the review tell groups of students which section they will be assigned to review for the class the following class period. Highly recommend that they review that that section/topic.

In Class:
· Give the students time in class to finalize their presentations and pick three problems for the class to do.
· Give each group a chance to review their section and ask their questions.· The other groups complete the answers. Group points can be awarded.

Pros:
· No teacher preparation
· Students review one section in depth.

Cons:
· Students only review on section in depth.
· Fewer problems are presented.

Second English Lesson - Too Abstract

My confidence was boosted by my first lesson that went well because my students were much better than I had prepared. Prior to my lesson I decided on a new goal for teaching this course: "Try something new every lesson." I think I will stick with this goal, but probably not a drastic as I was for this second lesson.

The topic for my second lesson was on friendships and forgiveness. I decided on having them draw pictures on that topic and then discuss as a class. I had them sit in a circle around a few folding tables. I just wanted to do something very different from the week before.

Instead of having 5 students, I had about 13 and the abilities were very different. This week the majority of my students were definitely "beginning" students and I wasn't prepared to modify the task to fit lower ability students. My launch of the task was horrible. They all looked at me confused. One student commented that friendship and forgiveness are very abstract, so it is hard to draw a picture of them. I agreed with them. So I changed the task to draw a picture of one of their friends. We then went around the circle and talked about their friends. They were pretty embarrassed at their art abilities. The drawing task ended up being a hindrance instead of aid.

At the end of each student's explanation of their friend, I summarized and had the class repeat. I don't like repeat activities because I want to push them more to creating their own sentences. But  I also want them to talk more than listen.

I'll be less abstract and be more prepared to adapt my lessons for lower ability students.