NCTM article 2

Cirillo, M. (2009). Ten things to consider when teaching proof. Mathematics Teaching. 103(4), 250.

The author of this article is really addressing her paper to new mathematics teachers. She gets this idea to write about how to teach proof when watching a new student teacher struggling at teaching proofs. She comes up with ten ways to think about proofs to make teaching proofs more effective. First, students should have sufficient understanding of the lower level mathematics needed for proof. Second, have the students think about proofs as a problem solving activity. This is more engaging. Third, make doing proofs a central part of the classroom and not a rare occasion. Fourth, be sure the students understand that proofs are used to explain why something works. Fifth, talk about the necessity of using assumptions and other rules of proofs. Sixth, make sure you as a teacher are fluent and ready to teach proofs. Seventh, she recommends using flow proofs rather then two column proofs because flow proofs are easier to understand and see. Eighth, provide enough wait time for students to think about proofs on their own before interjecting and solving the proof for them. Ninth, students should be making informed guesses before they ever try and prove anything. Tenth, and lastly, she suggests to teach proofs not theorems. By teaching proofs the teacher can show where theorems came from.

Teaching proofs can be a daunting task. It can seem very overwhelming, especially to a new teacher. This article is a great resource for beginning teachers to draw from. Just having guidelines is helpful for a student teacher who is overwhelmed with teaching. It is a source of comfort and suggestion. Just knowing that another teacher has struggled with teaching proof is nice to know. Its nice to know that it is common and normal to struggle with teaching proof. These are great guidelines written by someone with more then 20 years experience teaching. Teaching proof and having students understand is an accomplishable goal if done with great care and premeditation.

NCTM article

Givvin, Karen B. (2006). What does teaching look like around the world? ON-Math., 6(1).

My article was a summary of a study done where eighth grade mathematical classrooms were observed in seven different countries throughout the world. The team observed classrooms in Australia, Czech Republic, Hong Kong, Japan, The Netherlands, Switzerland, and The United States. It is interesting to see how math is taught very differently in each country. In Japan they devote class time to solving just two or three problems in great length and detail. In Australia half of the class is spent doing procedural instruction and the other half the class worked in groups on more conceptual problems. In Hong Kong 75% of class time was spent by solving problems as a whole class, with everyone working together. In switzerland most of the students time in the classroom is spent working in groups. On the flipside, the Netherlands places the responsibility of learning mathematics on the student. The students teach themselves through their textbooks only using the teacher to answer any questions that arise when they are studying. They concluded by using the United States stating that the U.S. used by far the most time reviewing previously taught subjects and taught in a very procedural manner. The observers commented that few if any connections between concepts were made. At the end of the paper the team concluded that not every type of teaching would work in every country because of cultural differences.

I agree with the research team concerning the fact that each type of teaching would not succeed in every country. I know the United States is falling behind other world leaders in our mathematical testing scores. I think it is interesting to compare some of the top countries and their teaching methods and see how lacking the United States is in teaching mathematics to our students. I think the United States needs to take notes from this study and make culturally appropriate adjustments to our teaching curriculum. I think for starters we could adjust how much of our time we spend reviewing in our classrooms. If we stop doing so much review students will be more apt to study and remember the things they've been taught because they know it is not going to be presented to them again. Plus with the time we just gained from not reviewing so much we can spend that time teaching conceptually rather then procedurally so the students will be more likely to remember the things they've learned. I think another way to help the students to remember the things they've been taught is to apply and connect things. Apply the things to the students lives and connect the concepts to previous and future concepts to help the students draw connections. I think with slow and steady changes we can slowly achieve a more affective mathematical program.