Blog Entry #7

Pendleton, K.L. (2010). Investigating the randomness of numbers. Mathematics Teacher, 103(5), 364-370.

In this article, Pendleton writes about the need to test the randomness of samples using statistical methods that are well known to those who study statistics. He explains that it is hard to test the randomness of a large sample, but fairly simple to test the randomness of a small sample. He goes through two tests to use in testing randomness. In the first you compare the sample mean with its population, and in the second you compare the sample's distribution with that of its population. He explains how these tests could be used in a classroom to teach finding the mean and standard deviation. Pendleton says testing samples for randomness will add to more confidence in the results found.

I agree that random samples should be tested for randomness. Nearly every time I read or hear the results of a study that included a random sample I can't help but wonder how random the sample was and how they can be sure of that it is. This article explained how such things can be tested and how they will add validity to results, and I believe this to be true. I also agree that it would be beneficial to do an activity such as this in a statistics classroom because students could pick random students from within their own classroom and then test the randomness of it. This would be teaching them to find the mean and standard deviation in a way that is more real to them.

Blog Entry #6

Gilbert, M.J., & Coomes, J. (2010). What mathematics do high school
teachers need to know. Mathematics Teacher, 103(6), 418-423.

Through research done with several different teachers and their classrooms, this conclusion was reached, "How teachers hold knowledge may matter more than how much knowledge they hold." Several points were made within this article to support that teachers deep understanding of a small amount of mathematical knowledge is more beneficial to their students than having large, but shallow, amounts of mathematical knowledge. It is vital that teachers have this deep understanding of the material that they are teaching to their students so that the teacher will be able to quickly recognize and even anticipate student misconceptions and errors. The sooner a teacher acknowledges a student misconception and corrects it, the better. If a students misconception is not properly realized and corrected, then they will take that incorrect information with them into their future mathematics learning. It is vital that teachers not only know how to answer the questions they ask their students, but that they will be able to understand several different solutions to any mathematical question they ask. If they do not have this understanding, then they may not be able to notice and fix students' errors. Having this understanding of why a student made a certain mistake will provide the teacher with the opportunity to guide the student to developing more careful thought processes that will prevent them from making that mistake in the future.

I agree with the main point made in "What Mathematics do High School Teachers Need to Know?" because of three reasons. First, I have had several experiences that support that teachers need to have a deeper understanding of what they are teaching. I cannot count the number of times that I, or another student in my class, has asked a question of our teacher wherein the teacher gives an answer, but they do so without addressing the question asked because they did not understand what the student was asking in the first place. They had simply assumed they knew and had continued from there. Second, I have had experiences wherein I was attempting to teach a mathematics principle to someone else and have been unsuccessful. I was unsuccessful because I did not have a deep enough understanding of the material that I was trying to teach, so I could not recognize their misconceptions and correct them. Third, the article gives excellent examples from the classrooms that were studied of numerous representations and misconceptions that students gave/had about one problem. If the teacher had not been able to first interpret the different solutions given, and then realize what precisely it was that the student had made an error on, they would not correct the student on the precise point that the student didn't understand and although the student might be shown how to get the right answer, they could still have that misconception. For these reasons I feel that it is more important for a teacher to have a deep understanding of a small amount of mathematical knowledge, rather than a shallow understanding of a large amount of mathematical knowledge.

Blog Entry #5

Two advantages that result from teaching mathematics without revealing specific procedures to students are that students will have a better understanding of why a problem is done a certain way, how to explain it, and more social interaction with/learning from peers. When a student must develop their own procedure for solving a problem, they will understand better why it is done that way. This is illustrated in the article by Warrington as the students explain that they know one divided by one-third equals three. They explain that 3 is the answer not because you invert and multiply the 1/3, but because 1/3 goes into 1 three times. Students taught in this method can explain a procedure and use mathematical jargon quite proficiently as well. This is exemplified by the children in Warringtons classroom as they were asked several times to explain what they had done, and they were able to explain their procedures in a way that the teacher and other students could understand. A third advantage for this method of teaching is that students interact more and share ideas with each other. In Warrington's classroom the students would converse with one another as they tried to form solutions to problems, they would build off of each others ideas and explain to one another how they got their answer. This would provide the opportunity for more students to understand as they talked together.
Some disadvantages for teaching mathematics without revealing specific procedures to students, and possibly not even revealing the right answers, include a possiblitiy for students to formulate incorrect procedures, and possible slower advancement through different mathematical subjects. When students are not given a procedure for solving a problem, but must solve their own, it is quite possible for them to formulate incorrect procedures in the process. An example of this with Warringtons students occured when they approached the problem four and two-fifths divided by one-third. Most all of the students came to the wrong answer because they did not completely understand how to solve the problem and the procedure they created to solve it was slightly incorrect. It also takes students longer to advance through mathematical subects if they must formulate their own mathematical procedures because they have to think more and at a deeper level about how to find a solution. Because of this, it is worrisome that students will not be able to learn all that they need to learn in the amount of time that they need to learn it.

Blog Entry #4

In "Learning as a Constructive Activity" von Glasserfeld describes how we come to know what we know through constructivism. von Glasserfield defines construct knowledge as a process of how we come to know things through experience. The term 'constructing' is used rather than 'acquire' because our knowledge is built through reasoning and experience, not simply gained or acquired through hearing it. As in the view of constructivism, we all gain knowledge through our own experience, von Glasserfeld explains that because of this our own personal knowledge is more of a theory because we do not know if what we know about a subject is the same as what someone else knows. This all can come into play with how teachers teach mathematics today.
As according to constructivism, if we gain knowledge through our experiences, then students must be actively involved in the learning process. If they will not learn simply through hearing, they must have more doing and practicing with mathematics so that they can gain more experience and reasoning with the material. As a teacher I would do this by having more in class assignments where the students can work together while practicing a concept and reasoning together about what is being taught. I would also make it possible for them to ask me questions about their homework assignments so that their reasoning and learning could be guided. Through these ways I would hope they could have more meaningful experiences with mathematics material.

Blog Entry #3

As Erlwanger examined some mathematical conceptions of 6th grader Benny, he concluded that Individually Prescribed Instruction had some serious weaknesses and flaws. IPI was a very ample effort to make mathematics teachable to the individual and was supposed to be beneficial to students in helping them learn math at their own pace. When Benny was asked by Erlwanger to explain some of the rules he had for different mathematical procedures, Erlwanger discovered that Benny had made up many rules of his own to try and make sense of math, and many of these rules were incorrect. Benny had been able to excell through the IPI course with these major misconceptions and this was not the intention of IPI. The teachers were also so uninvolved in IPI that none of Benny's teachers knew he had incorrect ways of doing mathematics, they all believed him to be a star pupil.
Currently in mathematics teaching, teachers can still be very distant from the students in their classes. I feel that in many math courses taken by students today, teachers have interest in the students learning collectively, but not individually. Many teachers ask students collectively if they understand a concept being taught, but they never bother to ask the individual. It seems as if many have an attitude of, "If at least this certain amount of students understand what is going on, then I have done my job, and too bad for the rest." Perhaps that is stated harshly, but it is an attitude that I have percieved. Teachers do not conference with their students about their learning and because of this, students feel less responsible for their learning and fall behind in their classes. Slight teacher interest for each individual student would make success for students more achievable.

Assignment #2

Relational and instrumental understanding are are two different definitions which are used for the word understanding in the field of mathematics. These two types of understanding each have advantages and disadvantages as well as areas where they overlap one another. Richard Skemp described relational understanding as knowing how to solve a problem and why the problem can be solved that way, while instrumental understanding is described by Skemp as only knowing how to solve a problem without understanding why it works. The benefits for relational understanding include being able to adapt what you have learned in one area and apply it to another, the ability to better remember what it is you have learned, becoming an effective goal, and producing a solid base from which to go and gain more mathematical knowledge while exploring the mathematical world. The main disadvantage of relational understanding is that it takes longer to learn the material. There are advantages and disadvantages for instrumental understanding as well. Instrumental understanding is generally easier to learn, faster to learn, and the only way to understand some mathematics because relational understanding would have to include science as well. Instrumental understanding is ill equipped to help a student apply material from one area in mathematics to another, remember what it is they have learned, and be able to solve mathematical problems in more than the one prescribed way. These two types of understanding do overlap one another. Included within relational understanding is instrumental understanding as both types of understanding include the ability to know how to complete the problem, but only relational understanding includes knowing why that works. Relational and Instrumental understanding can also both be successful in helping a student solve a mathematical problem successfully.

Assignment #1

To me, mathematics is a process and tool by which we are able to form relationships between quantities of different forms (such as measurements) and decipher the meaning of numbers including equations and geometry. Mathematics also helps to teach one how to think in processes.
I personally learn mathematics best through step by step explanation and example from someone who is knowledgeable about the subject, such as a teacher. It is really important that I am shown the process of solving a problem, not just given the equation for solving it. I also think that going through several examples is helpful in the learning process, and of course, doing several problems by myself helps me learn as well.
I believe that everyone learns mathematics best through slightly different processes. I think that a variety of examples with explanations and hands on experience is the best way for students to learn. Not all students may understand a concept right away, but when given several types of examples of how to apply a concept, it is likely then that different examples will help different students understand. I think it is then beneficial for students to try different problems that apply that concept with the availability of someone nearby to help if needed. It is also important for teachers to get constant feedback from the class to if they are explaining concepts in a way that they understand and clarifying as needed. It could also be quite helpful to the students learning if a student is occasionally asked to re-explain or discuss a concept to/with other students that was just taught. An environment needs to be present for the students that is comfortable where they feel they can at any time express whether they do or do not understand what is being taught.
In most all of my high school math classes I had teachers who clearly explained concepts and gave ample examples to help students understand what was being taught. They then would give us time in class to begin working homework problems and during this time classmates and the teacher would be available to help with any problem you might have trouble with. These practices in math classrooms are very beneficial to students and will really help them learn the material that is being taught.
I do not believe it to be beneficial at all when a teacher stands at the front of the classroom and simply gives a lecture on the material while writing on the board. In this situation they give a lecture of the material and explain the material in the way that THEY think students will understand, not in a way that the students actually will understand. In this type of scenario they rarely ask if students understand, and even if they do ask, in this type of a situation, students rarely feel comfortable stating that they do not understand.