NCTM Article and Evaluation #2

Daire S.A.(2010). Celebrating Mathematics All Year ’Round. Mathematics Teacher, vol 103(7), 509.

The main idea of this article was getting students involved and excited about mathematics. All it took was a simple article to spark an idea into a teacher who turned a high school into a math learning environment. In this article, Daire, took an idea from an article titled "Pi Day" (Waldner 1994), and started a celebration of the number Pi. It began within Daire's classroom celebrating pi day on the 14th of March. They made T-shirts with Pi designs on it. Something like, "wearing a “What’s your sine?” shirt with a graph that had p as an x-intercept intrigued my students," Daire observed that the shirt created opportunities to learn. As more participation came from students, a one day a year celebration turned into a weed long celebration. Then there came the whole school involvement, where there were questions posted outside the classroom for anyone to solve and win prizes like a graphing calculator, candy and T-shirts and stuff. This began "opening the door to a world of mathematics that [students] did not know existed."
As more and more participation came, the one week celebration turned into a year long celebration where pi was celebrated during major holidays like Halloween, Valentines, etc.
The principal allowed students to wear math themed t-shirts on March 14. "The ceramics teacher encouraged students to make pi art. The media center offered its display case to showcase an array of mathematics books, posters, student-made ceramics, and even songs that celebrate mathematics. And the custodians became accustomed to hallways covered with student-made digits of pi." Due to this involvement, the school's state test scores were significantly higher than other schools.

It's interesting to me that a simple article title with an imagination can spark an interest in so many students and create a math learning environment for everyone. I think that it is a great idea to tie in celebration to math learning because I feel that people learn best while having fun. And adding a little competition with actual prizes gives, considering that these students were from less financially stable families, gave them more incentive to participate and in doing so gave them a learning opportunity.

NCTM Article Summary and Evaluation

Otten S., Herbel-Eisenmann B.A. & Males L.M.(2010). Proof in Algebra: Reasoning Beyond Examples. Mathematics Teacher Vol 103(7),514.

The main point of the article is to "provide an image of what proof could look like in beginning algebra, a course that nearly every secondary student encounters." There are divisions of ways that students pick up on proofs which was studied by Stylianides. First there is the externally based schemes, which Stylianides defines as convincing the student from an outside source. Second, there is empirical proof schemes which are justifications based solely on observations. And third, there is analytic proof schemes which consist of valid mathematical arguments such as a treatment of the general case or formal deduction within a system. However, from another study by De Villiers (1995), he made the point that proof should be more explaining and understanding than convincing. The argument that the authors iare making is that proofs in secondary schools don't go beyond geometry which many of the secondary students don't get to. So the authors try to take proofs beyond geometry to see what it is like in beginning algebra which almost every secondary student takes. There are two cases presented. In the first case the teacher showed the students the pattern slowly by using examples. In this article the teacher showed that cross-multiplication of a number equal to itself gives the same product. for example 1/12=2/24. If we cross multiply we get 1 because 1*24=24 and 2*12=24, so we get 24/24=1. So after seeing several examples the students notice a pattern, that is, taking the cross-multiple of an equivalent fraction produces the same number. In the second case, the teacher used the examples to show the general case of using this concept of cross-multiplication which the students constructed. They represented the fraction as a/b and the number equivalent as n/n, so it was na/nb and they justified it by saying that it is the same as multiplying by 1, the n's cancel out. So from these two cases the authors point out that mathematics is a state of mind that is "characterized by inquiry and a thirst for justification." They conclude that it is a goal that can be reached by sharing, discussing and analyzing experiences in proofs in the classroom.

So this is an attempt to have students think for themselves and not be so dependent on authority. I believe that it is a critical study because of the many students who are very dependent on authority. I was one of those students. I was very dependent on the teacher, and became very instrumental in my learning. However, I am learning that it is a state of mind that requires a thirst for justification as the authors put it. From my own experience in secondary school math, just like in the article, the only proofs that I was exposed to were those in geometry and none after that. I am coming to see more that the thirst for justification that is talked about in the article is the questioning why? Students need to be able to question why? and not just accept the information given by the authority. They need to read through and construct ideas about the concepts and procedures for their own. This way they will be open minded when being taught by the teacher, they will have a wider vision of the concepts and procedures because they will be able to see more and question why? when being taught by an authority.

Warrington Paper

Some of the advantages of teaching the procedures or giving the correct answers gives the students the opportunity to think for themselves. In Warrington's paper she presented the students with problems without telling them what to do or how to solve it. She simply asked them what it meant to them and according to Warrington she states that children construct knowledge based on what they already know. So after she presented a fraction the students went on constructing knowledge about what they thought it meant based on what they already knew and in many cases they concluded correctly. Another advantage of this is that it encourages social interaction. When socially interacting with others it helps to bring more insights and helps to construct better knowledge because more minds are better than one. The students were able to put their ideas together and then conclude on an answer.

One disadvantage that I can see from this is that the children need to learn somewhat of the procedure and like in Benny's case, he was left to construct knowledge on his own and he came up with the wrong procedures and answers. That is something to consider, although the students are left to construct knowledge independently, they also need guidance to make sure they construct the right knowledge. Another thing, Warrington states that it is the responsibility of the teacher to provide learning environments, but how can the teacher know what a learning environment is that would suit each student? I think that that is a difficult thing to determine.

constructivism

von Glasersfled talks about "constructing" knowledge in the sense that it is more than the acquisition. He says that knowledge is that which human reason derives from experience. It fits observations. He also states that "the world we live in" is possible to be understood also as the world of our experience, as we see it, hear it and feel it. I think as far as constructivism is concerned, that "constructing" knowledge is gained through careful observation and then trial and error as opposed to the acquisition of knowledge which as I understand it is learning through what others have carefully observed and tired. Knowledge is such a profound concept when trying to think about it as something tangible and is called a theory because there is no real "proof" to it. There are so many different theories that remain only theories.

To me, if I were to apply constructivism to teaching mathematics, instead of teaching formulas and such I would have my students use objects like the blocks and shapes we use in class. I think that if they could carefully observe how and why concepts work using the blocks and shapes, it would create a better visual for them in their head. I think that it would help them to actually see why things such as division and multiplication work the way they do and not just to memorize them. I think this is compatible with a constructivist perspective because it seems to me to be more of constructing knowledge rather than me teaching and them acquiring.

S. H. Erlwanger

In Erlwanger's article, he conducts a study with a 6th grade student, Benny, on what he has learned from the IPI program. His main argument is that in the IPI program, (Individually Prescribed Instruction), which is suppose to help students learn more effectively independently, has some flaws that create disadvantages in teaching children incorrect forms of mathematics. He states in the beginning that it was expected that a student could not succeed in the program unless he or she had a sufficient understanding of the concepts. However, throughout the study Erlwanger argues that although students may be "successful," as Benny was, does not mean that they have sufficient understand the concepts.

Erlwanger supports this in his study with Benny. Benny is one of the more successful students in the IPI program, but when Erlwanger interviews him he finds misconceptions that Benny developed through the program. Some of the evidence Erlwanger presents is Benny's understanding of adding fractions and multiplying decimals. Benny has different logic when it came to this and to him he was correct, even though he wasn't. Benny merely developed his own methods for explaining his answers to his satisfaction.

Erlwanger also argues that the IPI program places heavy emphasis on finding the answers rather than on the process or the understanding behind it. As a result the programmed form of IPI was making it a hunt for answers the teacher wants just to get them correct. Thus, in Benny's case Erlwanger states that a "mastery of content and skill" does not imply understanding.

One of the arguments Erlwanger presents that I think is beneficial to today is that the aim to teaching mathematics should be to "free the pupil to think for himself." He goes on to say that students should be provided with opportunities to discover patterns in numerical relationships, they should realize the independency of their actions in reasoning, seeking relationships, making generalizations, and verifying discoveries. One other support Erlwanger provides is that mathematiccs should be a subject in which rules come from concepts and principles.

I think that this is applicable to today because I believe that most of mathematics taught in school are very instrumental in that instead of deriving rules from mathematical concepts, students are taught the rule without the concept. I think if the students are allowed to experiement independently and then share with others their ideas about their discoveries, it will create more of a relational understanding between them.

Instrumental vs. Relational Learning

Instrumental learning to me was more of a "robotic" type learning. It was like a formula, you plug in an "x" object and out comes a "y" result. I realized that most, if not all, of my High School learning was instrumental. I learned the formulas and then applied it to my homework and it helped me to finish my homework fairly quickly, but not so much on the tests.
Relational learning to me was more of a "curious" kind of learning where one would put question marks after everything until he or she received a background understanding of the concept. I like the ven diagram from class where Instrumental learning was encompassed within relational learning because although they are different, they really are similar and they are complimentary to each other. By this I mean that, sometimes it takes accepting the facts up front and then receive understanding, the "why", later. So in that sense one would receive the formula and apply it and find out later what the formula is used to find and how it applies to more than just the copmutation problems in the homework. So they compliment each other.
Another way I saw a difference between the two as I read the article was that the learning curve for Relational learning seemed to be a linear function where you can achieve a page full of correct answers, as Skemp said it, in a shorter amount of time and be done with the homework. On the other hand, Relational learning was more of an exponential function. By this I mean that it takes longer to learn the background and the concept, but once it is learned you can apply it to more than just the assignment and you can eventually achieve more than by instrumental learning. So it takes more preparation, if I may say, with relational learning which may result in quicker achievement and more understanding, whereas with instrumental learning one may be able to accomplish an assignment with the formulas given, but that would be it. He or she would not be able to apply the formulas to much else, if not anything else.

What is Mathematics?

What is mathematics?

I believe that mathematics is a system that we, humans, have developed over a long period of time to measure our lives. It is a system that we use to experiment, calculate and discover our everyday life essentials. We experiement with things such as vehicles, food, trees, animals, people, insects, elements, the human body, etc. This all includes mathematics. We calculate speeds, velocities, acelerations, amounts, how much of one substance we need in a certain type of dish, etc and then we discover what it is we need or could use to make it better.


How do I learn mathematics better?

I believe that the best way for me to learn mathematics is through hands-on experiementation. By that I mean, working on problems and discovering answers and different possible ways to solve a problem. In school I first read briefly through the chapter section, then pay close attention in lecture and take good notes and then jump straight into experimenting with the problems. Ive found that the best way for me is to use the formulas or the concepts repeatedly.
This works for me because it is when I am focused in study when I am better able to understand a concept. So while working on a problem, if I get stumped I refer back to the chapter explanations and examples and apply them repeatedly with different problems.


How will my students learn mathematics best?

I think that my studenst will learn best through math games and experimenting. By math games I mean, racing to see who gets the correct answer first. Its a little bit of competition but Ive learned that my mind is more focused when there is a little competition involved and it sticks better with me.
By experimenting I mean working with in private on a certain problem and then putting them on the spot to work it out in front of the class.


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

I am not sure what is meant by current practices, but from what I understand, I think that practices that promote students' success in the mathematics classroom are having TA's and Lab sesssions. Doing the homework and then going over it in Lab helps to ingrain and clarify the homework in the students' mind. The more times the students' see the material the more the understanding comes.


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

I think that everything in the school is aimed towards the students' success. From what I have seen the only reason why students' fail to learn is because they do not put in the time. I am aware that sometimes the material is very difficult, but there is plenty of help available.
I honestly believe that what you put in is what you get out. The more you put in the greater you will get out and then some.



















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