Assignment #7

Daire, S. A. (2010). Celebrating mathematics all year 'round. Mathematics Teacher, 103(7), 509-513.

When students are given excitement and motivation to participate in mathematical thinking activites, they are brought to a greater knowledge and understanding of mathematical concepts after the activity is completed. Sandra Arguelles Daire read a magazine article in her early years of teaching that inspired her to really help students understand pi through activities such as puzzles, story problems, and contests. Throughout the years, this celebration of pi turned into a "Mathletes Boot Camp" which consisted of year round festivites promoting students to gain a love and understanding for mathematics. Competitions and prizes along with the enthusiam from the school faculty continue to draw more students in every year.

Mind games, contests, and other activities are great ways to get students involved with mathematical concepts. Daire did an excellent job at going above and beyond the average math teacher. The article made it clear that more students participate in Daire's activities and contests each year than the year before. More educators need to bring a bright side to mathematics to get students excited about the subject and erase the underlying hatred there has always been for math classes. Daire put an intense amount of work into planning activities, but sometimes that is necessary to get students motivated. Every math teacher will have their own way to get students involved, so as long as they put effort in, both the teacher and the students will benefit, which is the goal of teaching.

Assignment #6

D’Ambrosio, B. S., Kastberg, S. E., & dos Santos, J. R. (2010). Learning from student approaches to algebraic proofs. Mathematics Teacher, 103 ( 7), 489-495.

This article discusses the idea that students are not well trained in using algebraic methods to write or create a sufficient proof. Educators can discover if this is the case for their own students by asking them to write up their own ideas on a general statement or theorem. What each student writes will give the educator an idea of the level of understanding the class has reached on proving that particular concept. A study was conducted where the students were given the assignment to write a general statement regarding a pattern that results from squaring numbers with a 5 in the ones place. The students were struggling, so they were each giving a hint consisting of the binomial (10n + 5)^2. Most students thought the statement was obvious and found no point in proving it, so they just wrote specific examples of when it is true. Others set the binomial equal to zero and solved for n. Few students actually used the binomial correctly to algebraically prove the statement. This evidence shows that the students are not experienced in creating sufficient proofs through algebraic means.

If students are more familiar with algebraic proofs they will be more likely to have the ability to create their own sufficient proofs. Through the study conducted above, it was obvious that the students who were tested were uncertain how to connect algebra with written work. In order for students to become familiarized with these proofs, the educator must take the time to see how much knowledge they already have on the subject. As stated in the article, this information can be obtained through written proofs on statements they have never seen proven before. When the educator understands where they students are in the understanding process, they will know which proofs to expose them to, so the students will be better prepared to create their own proofs.

Assignment #5

In Warrington's article, she discusses the importance of allowing students to think for themselves and come up with thoughts on their own rather than teachers telling them procedures. First, when students speak their thoughts they give the educator an idea of what needs to be taught and how they should teach it. This clarifies understanding and helps the teacher give the right prompts to allow students to conctruct a more correct interpretation of a concept. Also, students do not accept anything until they understand it and they have the courage to ask questions since that is the atmosphere of the classroom. Warrington illustrates this through the girl who stood up to her class because she thought the answer was 13 and 1/5 rather that 13 and 1/15.
Disadvantages include the time constraint. It takes much longer to go through understanding that procedures. Also, if the educator cannot create a safe environment where students feel comfortable asking questions, then this method of teaching will never work.

When von Glasserfeld uses the term "constructing knowledge" because, as he points out, each person's life consists of different experiences and different views on life. As one experiences a specific moment of "knowing" they can add that to their former knowledge and either change or build upon their previous thoughts or experiences with that subject or concept. This is why von Glasserfeld calls it contructing rather than aquiring--sometimes previous experiences may have led one astray in their "knowledge" and so when a knew experience occurs, they can break down their former knowledge and make the necessary changes (reconstructions) so their new knowledge now coincides with every one of their previous experiences. Because experiences play such a crucial role in constructing knowlege, then, since no one has had every experience possible, knowledge is relative and more of a theory than a pool of facts.
Understanding that knowledge is constucted is important to teach mathematics. With this understanding, I, as an educator, can teach one concept in many different methods rather than just expecting every student to understand from one method. Different methods will fit into different experiences students have had with a concepts. Also, I can try to see where or why a student's knowledge may be scewed or needs "reconstructing" according to what concept I am teaching and the method I am using to teach it. If I apply this idea of "constructing" knowledge, I will be able to communicate more clearly and accurately to my students.

Assignment #3

In Erlwanger's article, he makes the point to his readers that using Individually Prescribed Instruction (IPI) is not successful when trying to help young students understand mathematics. The sixth grade subject, Benny, is a smart child. As he studied the rules that go along with fractions, he found patterns that were not necessarily correct. For example, since he knew 3 +5 is equivalent to 5+3, then he carried that communitive property over to say that 1/2 is equivalent to 2/1, which is not true. When there is no teacher there to explain why this does not work, Benny has no reason to believe he is doing anytyhing wrong. Right along with this reasoning, when Benny gets the correct answer, he doesn't know why it is correct. He tells Erlwanger that his answers are graded just by a key, and even if the answer is true it will get marked wrong just because of the key. This instills in Benny the idea that mathematics is just applying rules to try to get the same answer as the key, rather than explaining to him that the answers are right, and a teacher can show him why and how. A student-teacher relationship is necessary for students to progress in correct and accurate mathematics as well as to help the students learn to enjoy math.

It is imperative today to keep up this student-teacher relationship to prevent students from "plugging and chugging" with incorrect procedures. This goes right along with the instrumental verses relational understanding, and, in this case, the instrumental understanding does not even consist of the correct rules, let alone the correct understanding of how and why the rules can be applied. If young students today are not given the guidance they need at the early stages of mathematics, then when more concepts are tuaght to them they will not have any foundation to build upon and so their understanding will decrease rather than increase. Our responsibility as educators is to help students learn mathematics, and the only way we will help them is if we are personally teaching them.

Writing Assignment #2

Two types of understanding, as discussed in Richard Skemp's article, include relational understanding and instrumental understanding. Skemp pointed out the importance of recognizing the different meanings of understanding as educators. Relational understanding refers to the "what" and "why" of a concept whereas the instrumental includes memorizing rules and applying them without knowing exactly why or how they can actually be applied. Relational understanding is difficult to teach in the amount of time educators are given, but is definitely the prefered understanding for students to obtain. Intrstrumental understanding is quickly applied by students because they dont need deep explainations. Students often feel more successful with this type of understanding because they get the right answer quickly, without spending time on the "why". The problem, however, comes when the memorized problem comes in a question posed differently from what they memorized. Now instead of just plugging numbers in, they actually need to think about what is going on in the problem, which they never learned. By the definitions, instrumental is actually included in relational understanding, relational just goes deeper into the concept rather than just memorization of rules. Both are important in the learning of mathematics, but the deeper understanding (relational) will prove to be more useful all around application.

Follow up on post #1

Because I never discussed what mathematics is to me, I am posting this new post :) For the most part, mathematics includes numbers, calculations, and variables. These three components are used to understand structures, probablities, speeds, rates, reactions, and many more concepts. In other words, mathematics is a tad bit helpful in each human life.