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.

Warrington

A huge advantage to teaching mathematics with constructivism is that students think for themselves. The students use their logical skills to solve problems on their own. Based on the ability to do similar problems the students can figure out how to do more complex problems. For example, in Warrington's paper her students had already done fraction addition, subtraction, and multipication. Warrington began her lesson with easy fraction division problems the students could easily do on their own and gradually increased their difficulty. Using the students' previous knowledge of fraction math she built on it and added more.

A disadvantage would be the time constraint. It takes so much longer for the students to learn constructionally. The teacher must give the students' adequate time to figure out the problems. It is also difficult time wise for the teacher to get the students used to thinking constructively. especially if the students has never had to think constructionally before.

Von Glasersfeld

Von Glasersfeld Uses the word 'construct' rather than 'acquire' because of his constructivism theory. to construct is a verb implying the person who is learning needs to be the active one and construct their understanding. Where acquire is a more passive verb that implies the person who is learning is just listening and retaining knowledge. they are simply acquiring it. The key difference here is the activity of the learner. A learner who is actively involved in the learning process is going to understand better and remember the information better and longer.

When I am teaching math i need to do activities where the student is not just watching and listening to me teach but rather are constructing their knowledge themselves. I need to have learning activities where my students can explore what they have learned so as to construct and heighten their knowledge.

Benny

Erlwanger's main point was to show that students can progress through the mathematical IPI program without a complete and adequate understanding of previous work. Benny, a 12 year old boy, who is involved in the IPI program was the subject of study for this paper. Benny's scores on homework and tests were quite high. Then during the interview, Benny couldn't compute simple mathematical equations correctly; such as converting between fractions and decimals. Benny thought he understood the concepts when in fact he did not.

After reading Erlwanger's paper, it got me thinking about when i become a teacher. If Benny was one of the highest scoring kids in the IPI program in his class and he had very little to no understanding of the concepts of mathematics, then when i am a teacher i need to be aware that even the students with the highest scores in my class may not have a complete understanding of the material.

In Skemp's paper titled 'Relational Understanding and Instrumental Understanding' Skemp addresses two types of mathematical teaching, those being relational and instrumental. Relational understanding is a way of teaching where students understand both how to do a math problem and why they did it that way; where instrumental understanding is a way of teaching just the how. Instrumental teaching would be much easier and faster, making it possible to cover more material than if teaching in a relational way. But if students are taught only in an instrumental fashion will they be able to apply what they have learned to other similar problems? Probably not. While both methods of teaching get the teaching done and both methods help the student's pass their tests; one method, relational, is better for the students in the long run.

Assignment #1

1. What is mathematics?

Mathematics is a study of numbers. It is a broad topic encompassing many aspects. Ranging from adding and subtracting numbers to solving multivariable equations. Mathematics is used in everyday lives from research to grocery shopping. Mathematics is used by everyone!

2. How do I learn mathematics best?

I best learn mathematics by doing examples. Doing them repeatedly, over and over again. I personally like to watch the teacher explain it then work on my homework right away while it is fresh in my head. I usually don't have enough time to get it all done right away so I then have to return to it later to test my memory. It works for me!

3. How will my students best learn mathematics?

I am thinking they will best learn similar to how I learn. I will show them how to do the subject making sure to have plenty of examples. Also making sure to use plenty of different ways of explaining the topic so as to best help everyone in the class. Then i will give them homework to practice what they have learned. after all... practice makes perfect.

4. What are some current practices in school mathematics classrooms that promote students' learning of mathematics?

Mathematics classrooms are equipped with several devices to help enhance the learning experience. The basics include marker boards and overheads, calculators, computers, books, etc... but now we have even more ways to help such as smartboards, overhead calculators, etc. I believe personally that if the teacher has a love for math and a great desire to teach it that students can tell and are more open to learning. so i personally believe the greatest tool a teacher can have is their attitude.

5. What are some current practices in school mathematics classrooms that are detrimental to students' learning of mathematics?

As i mentioned before, the teachers attitude. If the teacher is being negative and not willing to help, the student in turn is not going to want to be an active participant nor will they be willing to put the time necessary to learning mathematics into they schedule.

I think another practice would be to only teach in a way that you understand. It is important to remember everyone has their own unique learning style. It is important as a teacher to try and teach things several different ways so as to try and reach each of those learning types.