Geometric Proof: Overcoming Low Self-Expectations

Task: Examine a topic that causes students to respond with low self-expectations.

Topic: Geometric Proof

Many of my students really struggle with the process of proof. In their mathematical experience, they have rarely (if ever) been given an answer and been asked to work “backwards” in order to figure out why it must be correct. This alternate way of thinking is overwhelming and challenging to many students, especially at first. Unless they are reproducing a proof that they have seen many times, they struggle to make those conceptual steps on their own. Lately I have been hearing of some ways to make proof more meaningful for students. Here is a collection of them, ranging to fun/game-like warm-ups to rigorous and relevant learning experiences.

1. Word game: start with a x letter word. Changing only one letter at a time, end with a specific x letter word. Each time you change a letter, you must have a real x letter word in order to proceed. (Ex: CAT - BAT - BAG - BOG - DOG). Many comparisons can be made between the idea of proof and this simple word game. I think it would be a fun warm-up before proofs are introduced. (Source: http://mathforum.org/sanders/exploringandwritinggeometry/proof.htm)
2. The Letter Game: I used this worksheet after Professor Golden shared it with our MTH 649 class this past fall. Some kids really went with it. It was a great way to see for which kids logic came naturally. Some kids really struggled with the worksheet, though, and I will spend more time going over examples in class next year. This is still a fun, logic puzzle kind of activity that really gets students familiar with the format  of two-column proofs. (Source: Mathematics Teacher)
3. Another great lead-in activity is one that I saw at the Adventures in Mathematics session at the Math-In-Action conference a few weeks ago. In this, students practice putting frames of a comic in order. They should also notice that sometimes there are frames that could be put in reverse order and the comic would still make sense - also true in proof writing. (Source: Adventures in Mathematics)
4. Battle Deck: In this method, students will make flashcards of theorems and definitions throughout the unit. This is their “weapon” when it comes to proofs. Though they wouldn’t be able to use it on a quiz or test, it allows students to familiarize themselves with the theorems/definitions as well as become comfortable with the process of proof. (Source: http://www.ehow.com/info_8072211_activities-geometry-proofs.html)
5. Similar to the comic book activity, cutting up the conclusions and justifications for a two-column proof and having students sort/order them is a great activity. It takes some of the stress away because all of the answers are right in front of them. They just have to make sense of it! I view this as a way to provide support when students need it, and gradually taking that support away. (Source: several)
6. The IXL Geometry website also has a neat way for students to practice two-column proofs. The questions are fill-in-the-blank, much like many assessment items that geometry teachers use. However, the blank has a drop down menu with a list of options. Again, this provides just enough support as students are getting used to this  format. It also gives some explanation if students make the wrong selection. Great way to give students feedback without creating extra work for the teacher! (Source: http://www.ixl.com/math/geometry)

When we as teachers are aware that a topic can cause students to have low self-expectations, we must really reflect on whether or not we are providing enough support. It all goes back to the “zone of proximal development.” Students can’t be given so much that they become bored, but they also can’t be given so little that they give up. Proof is one topic that I have always seen a need for ways to provide that extra support. If I can keep my students’ confidence level high throughout the learning process, they will be successful!