Want to Play Geometry?

Kaufmann, M. L., Bomer, M. A., Powel., N. N., (2009). Want to play geometry? Mathematics Teacher, 103(3), 190-195.

The main idea of the article is that sometimes alternative, fun, and interactive methods in the classroom can be used to explain mathematical concepts and ideas. The authors demonstrate this by describing an approach they took when teaching kids the structure and meaning of a proper proof. They put the class into groups and gave each group a die, 5 marbles, an egg carton, and 15 chips. They asked each group to use the pieces to develop their own game, for which they would write the rules for. They groups then exchanged games and critiqued the games made by other groups, looking for contradictions, incompleteness, and repetition. After this activity the teachers discussed with the class the similarities between the rules of their games and the rules and axioms involved in writing correct geometrical proofs. They argue that this is a quick, inexpensive, and motivating way to teach students about "analysis, synthesis, and evaluation".

I would definitely use something like this in my classroom. It is easy to see from the article that the students 1) enjoyed the activity 2) developed a conceptual, relational understanding of proofs 3) could relate these principles to the world outside the classroom. One student was even able to relate the principles they learned to the United States legislature, and the rules outlined in the Constitution. For each new big concept I wish to convey to my class, I want to develop an interactive activity to first allow them to explore the concept on their own and obtain a relational understanding.

Patterns Jumping out of a Simple Checker Puzzle

Staples, S. G., (2004). Patters jumping out of a simple checker puzzle. Mathematics Teacher, 98 (4), 224-227.

I believe that in her article, Susan G. Staples was trying to convey that unexpected mathematical patterns and problems can be found in simple games and activities, and that we can use these in our classrooms to convey to students the possibility of real-world applications of math. The article revolves around a simple checker game, where three black chips and three white chips lay on the edge of a seven space board, as shown below.

The object of the game is to get the black chips where the white are, and vise versa. The only possible moves are sliding one right or left, or jumping over another chip to the empty space. The motivation of the game is to accomplish this in the least amount of moves. While studying with her class the different possibilities of moves, they found that many patterns arose that could even be written as equations with variables. At the conclusion of the article she describes how her students found this activity "exhilarating" and were fascinated by the patterns they found.

I fully support the author's main idea, and believe that this article is a great demonstration of a good classroom activity for promoting mathematical applications outside of school. This is evidenced by the clear enjoyment conveyed by the students. Throughout the article it is clear to see their amusement and their internal motivation while working with the teacher on these patterns. It's clear also that they are mostly deriving the patterns and equations on their own without the teacher's help, so it seems they are gaining a relational type understanding. If mathematical patterns can be found in this simple game, I imagine there are many other activities we can use in future classroom to get similar effects.

Warrington

In Warrington's paper she describes her experience of creating fifth and sixth grade class environments in which the students develop understanding of division of fractions without first being given algorithms from the teacher. The results were fascinating and as far as the reader can tell from the article, very successful.

Some advantages of this teaching style as demonstrated in the article are the fact that the students' minds are free to explore and develop without being "shackled" by rules. Another clear advantage is demonstrated by the girl who disagreed with the rest of the group. Warrington describes her as having "intellectual autonomy", which is considered something positive out in the world. Overall the most important advantage is the extended understanding they develop as they work together and can't give up and ask the teacher. They build off of each others' ideas until what that come up with makes sense. After long discussions and heavy thinking, they develop deep and long-lasting understandings of fractions.

A disadvantage could be that some students might lead the discussions; ones who are further along or simply better at division of fractions might continue to grow and construct knowledge while the students not quite as proficient might just tag along and listen to the others without ever grasping how to do it on their own. I also question if the group could have come to a confident conclusion that was incorrect without knowing. If it weren't for the autonomic girl who pointed out that the last part was wrong, they might have stuck with their previous answer; so not being given the correct answer could be a problem.

Constructivism

In his "Learning as a Constructive Activity" von Glasserfeld discribes the concept of constructivism. I believe he uses the term construct because we gain knowledge so experience so it is constantly building upon what we already know, and it grows and grows and adds upon itself as we try to come to know truths. As we experience different situations, we adjust what we "know" to fit what we come to learn during these experiences. Certain conditions of course are required. We must be actively participating and working to come to solutions on our own. It is true however that we can never be sure as to whether what we know if absolutely correct because everything is relative and there is no way to know the absolute truth of anything. All we can do is try to come up with the best ideas we can from different problems we encounter.

In my classroom I want to begin lectures by presenting problems that I haven't yet taught them how to do. I want to give them an oportunity to try to figure it out own their own first or at least gain a greater concept of what the problem is asking for and what kind of information they would need to complete it. I hope that this will help my students construct their knowledge in such a way that in the future they will not always need a teacher to tell them how to do everything, but rather they can solve problems on their own.

Benny

I believe one of the main ideas of the article is that a teaching form which enforces purely instrumental understanding can permanently damage a students' attitude towards mathematics. The article suggests that because Benny wasn't given understanding of why the rules for fractions and decimal conversions worked, he developed his own meaning that made sense to him.

This is extremely applicable to us as math education majors because when we are teachers we'll need to make sure that our students know the reasons behind what we are teaching. If we don't, they may dislike math and believe that we are just making it up. They will work only towards "cracking the key" to our tests, and then fail to retain a deep understanding that will be lasting throughout the rest of their education.

Blog Entry #2

Richard R. Skemp presented an article in the Mathematics Teaching journal about two types of understanding that student can develop when learning mathematics. In the article, he describes one type of understanding as instrumental. From reading the article, my idea of what is meant by instrumental is an understanding of the rules and the steps to solving a mathematical problem, but having very little understanding of why those rules work. The other type of understanding presented is that of relational understanding. I believe that this is when a student can not only achieve the correct answer using the given rules and formulas, but also know where the rules and formulas came from and why they work.

These two types of understanding are not completely exclusive in that the students who have the relational understand also have the instrumental understanding. Additionally, I believe that students who have instrumental understanding may have an understanding of why things work, just not as extensively.

So if instrumental is included in relational, then why not just always teach relationally? For me personally, I learn better instrumentally because I am very good at memorizing, and storing what I memorize for a long time, rather than just forgetting it after a test. When a teacher spends a long time on how to get to the formula, equation, or rule that we will be using, the only thing I end up remembering is the end product. So for me, instrumental is preferable.

Since not everyone learns the way I do, both types of understanding have advantages and drawbacks. When learning relationally, the amount of material can be overwhelming and distracting. Students may learn how or why something works, but fail to know how to apply it to an actual problem. When learning instrumentally, students temporarily know the formulas and steps for say a quiz, test, or assignment, but that knowledge may not last, while if they had learned how to get to that formula relationally, they might otherwise recall.

Everyone learns differently, so it is hard to pick which way is ideal. Skemp clearly prefers relational, most likely because instrumental is included within it. I also agree because even though that's not how I learn, using relational can allow the student to decide for themselves what information they will work to recall and what is meaningless.

Blog Entry #1

1. Mathematics is the study of problem solving.

2. I learn math best through repetition. It helps me to memorize the steps to solving the problems, and puts it into my long term memory.

3. They will learn math the best by applying it to real life situations. This allows the students to experience math firsthand; once they realize that math can be used in places other than school, it will seem more fun and more practical.

4. Some current practices in math classes that I think help students are those that use technology and fun computer programs. Especially in geometry. I think these make math more exciting as well as hands on.

5. When teachers give tests and then move on to the next subject without individually helping the students who do poorly or fail, the students become permanently behind and never learn the material as well as they should. Tests should be a way of helping the teacher see where the students needs help.