Orchestrating Classroom Discussions. NCTM article #2

Smith, M. S., Hughes, E. K., Engle, R. A., Stein, M. K. (May 2009). Orchestrating discussions. Mathematics Teaching in the Middle School, 14 (9), 548-556.

Orchestrating discussions in a classroom can be a difficult task for a teacher and requires much preparation on the teacher's part. These authors have developed a five practice model that helps teachers effectively use student responses during in-class discussions in a manageable way. The five points are; anticipating student responses, monitoring student' work and engagement on tasks, selecting particular students to present their work, choosing a specific order in which students present, and connecting student' responses to other students ideas and key concepts. The authors discuss in depth how to use each of these practices and how each can help "build on and honor student thinking" while ensuring that the key mathematical concepts are taught. An example of a task about fractions, ratios, and percentages using marbles is used to show different student responses and an efficient way to use those responses to teach statistics. The authors believe that by giving teachers a road map to follow in class discussions, teachers can better use class discussions in a effective way.

Overall, I think this was a very insightful and helpful article, and I believe that this model can be very helpful. As a student who has not yet had to do this type of classroom discussion yet, I may not fully understand the difficulty of orchestrating classroom discussions. Yet, I believe that by using this model, a teacher will be able to give their lesson plans more potency. In the example the article gave about a task concerning marbles, the authors show clearly how this model of teaching can be implemented, and how it can help class room discussions. I have also seen examples of teachers who have used other models to control class discussions ineffectively, and i think using this model would have been helpful for them. Although this model may take more time and better planning by a teacher, in the end it will save the teacher time and energy.

NCTM Article, Bridging the Math Gap

Switzer, J. M. (2010, March). Bridging the math gap. Mathematics Teaching in the Middle School, 15 (7), 400-405.

This article was dedicated to showing how knowing how math material is taught in elementary schools helps middle school teachers bridge the gap between elementary school, middle school, and even high school. The author uses an example of how in elementary schools teachers are now teaching an alternative algorithm for doing multiplication. She uses this to show that middle school teachers should be given the opportunity to work with elementary teachers to coordinate their work. The author, as a curriculum writer for a school district, saw this need to have better communication in school districts and encourages very adamantly awareness of this problem and better coordination among teachers.

I think that the article brings to light a very important point about the way in which middle school teachers go about bridging the gap from elementary schools, and I agree with the assessment the author makes. The article gives some important suggestions about how to effectively begin communication and I think the points would be very helpful as a teacher. The article uses a great example to illustrate the need for awareness, and the example helped me to realize that a gap does exist and as a teacher I would need to take some of the suggestions mentioned in the article to make myself more aware. The author effectively expresses his ideas and makes it obvious to the reader that his points are valid.

Entry #5: Reaching the Young Mind

It can be hard to see the advantages and disadvantages of teaching without procedures as we can’t line up the two side by side. Yet, the advantages do outweigh the disadvantages and they can have a profound effect on the future learning of students. One of the most important advantages is that this type of teaching provokes the inventive thinking of students as they use the skills they have to understand the world around them. This also causes them to think for themselves and gain real concrete understanding like the students in the paper did about fractions. I think that as teachers use this method, it also helps students learn to work, discover, and be productive in group settings. This can be seen in the paper as the students worked on 1 divided by 2/3. By the end of the paper, you could see that they had become confident in their ability to reason out things for themselves, a very useful skill in the world today.

There can be disadvantages to teaching math without teaching procedures and spoon feeding them. One that has been clear to me is certain students will not be successful in those types of situations due to different learning styles and even learning disabilities. Some students could be left behind. And sometimes, when a student thinks he or she is correct after doing their own thinking, and then is unable to understand the correct way of doing something, this can cause some to give up. Just like in all other types of teaching, it takes a lot of effort on the teacher’s part in order to make sure no one gets left behind. The other obvious disadvantage is that a teacher who is not skilled at this type teaching could spend way too much time on a given topic. It leaves a lot to be desired of in teachers. To impliment this teaching style, it would take a very collective effort of teachers from all grades.

Mastering Mathematics Needs More Than Just IPI

The main point of Erlwanger's article about IPI mathematics is to get across to the reader some of the inherent weaknesses in the IPI approach to mathematics. IPI mathematics is a scientific approach to teaching which uses assessment, own pace learning, and given rules to teach with very little influence by a teacher. Erlwanger uses the example of a young man names Benny who passed through the IPI program showing good results. Although he was able to make it very far in the IPI program, when assessed personally by Erlwanger, he was found to have a very inaccurate understanding of the material he covered in IPI. The main point that Erlwanger wanted to get across was that since one student was able to do "well" in the program without actually learning the material, a more thorough investigation should be done to determine the usefulness of IPI. Erlwanger's talk with Benny demonstrates that Benny has developed learning habits and views of the math that will slow down his learning progress in the future. He states that teacher should become more involved in the learning process than they are in the IPI approach. Through this investigation, Erlwanger concludes that Benny and other IPI students are taught with too much of an instrumental approach and that they are essentially sent on a "wild goose chase" after mathematical answers.

I think that Erlwanger's argument that teachers are taken out of the picture too much in IPI mathematics is valid and I agree that even today, different approaches to teaching math in a similar mode as IPI, puts teacher involvement to a minimum, which impedes student progress. I talk of teachers who rely too much on workbooks or even textbooks to do the teaching that they are supposed to be doing themselves. I have seen through my personal experience that many students are left behind when this approach is used too much. I really do not have a very good understanding of the concepts of high school geometry due to a teacher who thought that the textbook should do the teaching more than her. Although I received an A grade in the class, I came away with very little real understanding of geometry. Overall, I believe that a balance should be struck between the use of independent type teaching and real teacher involved teaching.

Relational and Instrumental Understanding in Math Education

In Richard R. Skemp’s article, “Relational Understanding and Instrumental Understanding” we find an excellent exposé about the advantages of relational understanding, one of two different types of teaching and learning in math education. Skemp defines relational understanding as learning both the “what” and “why” of mathematics and instrumental understanding as merely learning the rules without learning the reason behind those rules. They are similar in that they both teach the student the rules of math and many times how to apply those rules. Skemp leads his reader to believe that relational understanding includes in itself most or all of instrumental understanding. Both of these teaching types have been used over the years and each has its advantages and disadvantages. Skemp’s opinion is that relational understanding will give to a student a huge advantage in his or her math education by giving them solid bases on which he or she can develop new ideas. On the other hand, Skemp explains that in instrumental understanding, the student does not receive sufficient “why” to continue holding their interest and to help them actually learn and retain usable math, needful throughout their lives. The article explains in great detail this issue which is of most importance to math educators.

My Math Experience

To me, mathematics would be a study using symbols such as numbers to explain what occurs in such things as science and business. It is a way in which we can understand logically the world around us.

Because mathematics is a way to explain our world, I learn math best when I am able to understand where it applies logically in life. Math sinks in better when I am able to wrap my own mind around a concept to the point where I can explain it to others. It is at times detrimental when I am spoonfed a concept.

I think my future students will learn math best as they themselves make discoveries and put forth the effort to wrap their minds are each concept, and as I continually assist them in their endeavors. Then as they practice repeatedly each concept, they will adequately learn each given subject. I believe this will work because this method helps the students use their own mind to discover topics, and with practice, they will be able to remember the given section.

A few practices in school mathematics that really promote students' learning include basic traditions such as giving adequate homework to help the student practice what they have learned, giving reading assignments in which a student can make his or her own discoveries, and the practice of assessing a students learning through tests.

One practice that I believe is detrimental to students' learning in schools today is giving a test over a material and then thinking that the test signifies the end of the learning of the material on the test. Tests should be used to assist teachers in helping students learn each given concept. Another way teachers today are causing students to not learn math is through too much talk, and not enough participation. It is hard for a student to learn a concept if he or she is taking notes too quickly in a class, and is not given the opportunity to try out the concepts themselves in a structured and guided manner.