Goodman, T. (2010) Shooting free throws, probability, and the golden ratio. Mathematics Teacher, 103(7), 482-487.
In this entry, Goodman is emphasizing the effectiveness of using real world application type of problems to help students in their math classes. His entry focuses more on a possible use of math outside the classroom, free throw shooting probability, than it does on using the situation to teach a concept. Goodman writes his article describing the process he and his students used to calculate the probability of a basketball player, Paige, making free throws in various situations: first in a "one and one" where Paige only shoots the second shot if she makes the first, second if she shoots both shots regardless of the outcome of the first, and third if there are three free-throws taken. They used the same process with t charts and graphs and writing equations using season percentage, p, as a basis. Once they determined Paige's probability, they applied the same methods and came up with a table for multiple athletes with different initial shooting percentages. Goodman concludes that this type of applicational situation allows students the opportunity to expand their understanding of mathematics and obtain a greater knowledge of the concept.
I agree with Goodman's thinking that contextual situations are beneficial for student learning and ultimate understanding. In all of my experience both attending math class and tutoring peers in them, the biggest road block for students' understanding is their doubt in themselves. They loose the motivation to try to understand a difficult concept because they don't see how it will benefit their life in the future. Goodman's example of free throw shooting proves the importance of having a basic knowledge of math and also the options available when more math knowledge is obtained. This topic is something all types of people can relate to and it is useful, especially in sports. I think more activities using this type of application can only benefit students.
Thursday, March 25, 2010
Monday, March 22, 2010
Blog Entry 6
Hunt, J. (2009) Prime or composite? Mathematics Teacher, On Math, 1-5.
This article focused on the importance of understanding prime and composite numbers and their factors. Hunt emphasizes that having a basic knowledge of these concepts is a necessity before students advance into middle school and higher math education. She describes her method of teaching these concepts to her class: using rectangle area, a factor game, and then a homework project. She first utilized a KWL chart and had the students list everything they remembered about factors and prime/composite numbers. They discussed student responses and together came up with a definition for the term factor. Hunt then gave her students colored pencils and graph paper and had her students use their new knowledge of factors to find different ways to form rectangle areas ranging from 1 to 30. After this activity, from student observation, the class established a meaning for prime and composite numbers. Hunt then reinforced this idea with a factor game played with partners on the computer. To end the lesson, she assigned a homework project, giving each student a different area of bathroom floor and each individually was supposed to figure out all possible floor plans to achieve that area.
I agree with Hunt's emphasis on understanding factors and her way of teaching them to her class. I really like how she balanced the amount of information she gave to the students and the amount they had to come up with on their own. She provided guidance after class discussion so that the students had a firm understanding of each concept, but the students had to think first and basically came upon the definitions themselves. Through the use of multiple activities to present the concept, Hunt reinforced the material multiple times in her students brains. This is a very useful practice from my personal experience, the more ways I see a concept, the more it sticks in my brain. I also thought, from her description, this method made very good use of time. The students were actively engaged for the whole class period and all the material planned on being covered was covered. I think this is a creative and effective way to teach this baseline information to students.
This article focused on the importance of understanding prime and composite numbers and their factors. Hunt emphasizes that having a basic knowledge of these concepts is a necessity before students advance into middle school and higher math education. She describes her method of teaching these concepts to her class: using rectangle area, a factor game, and then a homework project. She first utilized a KWL chart and had the students list everything they remembered about factors and prime/composite numbers. They discussed student responses and together came up with a definition for the term factor. Hunt then gave her students colored pencils and graph paper and had her students use their new knowledge of factors to find different ways to form rectangle areas ranging from 1 to 30. After this activity, from student observation, the class established a meaning for prime and composite numbers. Hunt then reinforced this idea with a factor game played with partners on the computer. To end the lesson, she assigned a homework project, giving each student a different area of bathroom floor and each individually was supposed to figure out all possible floor plans to achieve that area.
I agree with Hunt's emphasis on understanding factors and her way of teaching them to her class. I really like how she balanced the amount of information she gave to the students and the amount they had to come up with on their own. She provided guidance after class discussion so that the students had a firm understanding of each concept, but the students had to think first and basically came upon the definitions themselves. Through the use of multiple activities to present the concept, Hunt reinforced the material multiple times in her students brains. This is a very useful practice from my personal experience, the more ways I see a concept, the more it sticks in my brain. I also thought, from her description, this method made very good use of time. The students were actively engaged for the whole class period and all the material planned on being covered was covered. I think this is a creative and effective way to teach this baseline information to students.
Wednesday, February 17, 2010
Blog Entry #5
Warington describes the various advantages of how students learn more by finding the answers themselves, not being taught numerous different rules. Through her experiences, teaching a basic concept and then allowing the students to think through more difficult problems is a great way to determine what each student knows. Another advantage she talks about is that the students rely on themselves, not the teacher or a classmate, to find the answer. This encourages individual thinking as well as strengthening the confidence of each individual in their problem solving skills. Also, this method encouraged group work, talking with other students in order to find the answer. This also has the potential to prevent the struggling students from getting farther and farther behind as each individual is forced into discussion of a concept, not simply copying numbers on an assignment. Due to these numerous advantages, Warington presents the idea that this is the most effective way for students to learn a true understanding for concepts.
As with every situation, there is another side to the argument. There are disadvantages to this style of teaching to contrast the advantages. Without teaching the students the proper method to arrive at the solution, paired with not giving correct answers has the potential for students to get off track and stay off track. If the students are teaching themselves, they could be arriving at a correct answer through an incorrect method, which because they have found for themselves, is harder to remove from their brains. Also, this method takes a lot longer to accomplish and is nearly impossible to use to cover all the required curriculum today. This constructivist approach to teaching definitely had the potential to be effective, but it also has the potential for students to get behind: a debate for efficiency.
As with every situation, there is another side to the argument. There are disadvantages to this style of teaching to contrast the advantages. Without teaching the students the proper method to arrive at the solution, paired with not giving correct answers has the potential for students to get off track and stay off track. If the students are teaching themselves, they could be arriving at a correct answer through an incorrect method, which because they have found for themselves, is harder to remove from their brains. Also, this method takes a lot longer to accomplish and is nearly impossible to use to cover all the required curriculum today. This constructivist approach to teaching definitely had the potential to be effective, but it also has the potential for students to get behind: a debate for efficiency.
Monday, February 8, 2010
Blog Entry #4
In his article about constructivism, Van Glaserfeld talks about "construct knowledge." He uses this term to identify knowledge as understanding gained by both experience and experiments. His word choice is very strategic, by using the term construct he portrays the imagery of each student actually building, or constructing, their own knowledge. The students' building blocks are there own experiences, suggesting that in order for a student to accurately learn a new concept, he/she needs to be actively involved in problem solving. This can be done, but is not limited to, the use of hands-on classroom activities. These types of activities are helpful because they force the child to prove their individual understanding by applying a concept to a new situation. Van Glaserfeld emphasizes that we are always learning, through every experience we have we are constructing our own knowledge and theories of how the world works. Our knowledge is never correct though because we don't know what is correct in a world that is always changing.
If i was a math teacher, I would incorporate constructivism into my teaching by having student-teach days twice a month. On this day I, as the teacher, would be in the hall with a desk for myself and a desk for one student. I would have a movie showing in the classroom and call out students individually. When each student came out into the hall to meet with me, I would give them a problem and answer from the material we had been learning in class and have them teach me how to arrive at that answer. I would repeat this process with each student. This follows the constructivist idea because I would be able to better see each students individual understanding and what each is personally taking from class lectures, and how to benefit them in the future.
If i was a math teacher, I would incorporate constructivism into my teaching by having student-teach days twice a month. On this day I, as the teacher, would be in the hall with a desk for myself and a desk for one student. I would have a movie showing in the classroom and call out students individually. When each student came out into the hall to meet with me, I would give them a problem and answer from the material we had been learning in class and have them teach me how to arrive at that answer. I would repeat this process with each student. This follows the constructivist idea because I would be able to better see each students individual understanding and what each is personally taking from class lectures, and how to benefit them in the future.
Friday, January 22, 2010
Blog Entry #3
In the article about Benny's understanding of mathematics, Erlwanger is suggesting that IPI Mathematics is an ineffective and harmful way to teach math to students. Erlwanger explains the goal of IPI is that students can progress learning objectives in a chronological order. He then describes the process taken by the instructor and student: learning the objective, then being quizzed on it. Erlwanger criticizes the focus on test scores, with an 80% being required for the student to "pass" an objective. In his interviews with Benny, Benny tells him how he feels the key to the test is bad because there are more right answers than are on the key. Benny never thinks he is wrong because of this. As his conclusitory point, Erlwanger gives an interview in which he tried to correct Benny's knowledge and did not succeed because of Benny's previous years of incorrect learning. Erlwanger portrays that IPI was very damaging to Benny and implies it has the potential to be damaging to many other students, he advises against its use.
The overall idea of IPI teaching students individually could be useful today because it allows each student to progress at their own level. This type of individual progress is accomodating to many students because students of all different levels of math can progress individually and therefore better understand the material. Erlwanger strongly discourages the use of IPI, so an alternative style of individual assistance would need to be used to thus accomodate students. The use of many different examples can also be valid in mathematics today, but there needs to be more explanation behind each example. Move slower and allow the student to ask questions. Also, tests are a good examination of students' knowledge, but it is better to do different styles of tests to make sure the student understands. One idea could be having a student create a problem and then explain how he/she would solve it.
The overall idea of IPI teaching students individually could be useful today because it allows each student to progress at their own level. This type of individual progress is accomodating to many students because students of all different levels of math can progress individually and therefore better understand the material. Erlwanger strongly discourages the use of IPI, so an alternative style of individual assistance would need to be used to thus accomodate students. The use of many different examples can also be valid in mathematics today, but there needs to be more explanation behind each example. Move slower and allow the student to ask questions. Also, tests are a good examination of students' knowledge, but it is better to do different styles of tests to make sure the student understands. One idea could be having a student create a problem and then explain how he/she would solve it.
Thursday, January 14, 2010
Blog Entry #2
In Skemp's article he compares two types of mathematical understanding: relational and instrumental. While these two types have the same end result, the solution to the mathematical problem, they differ in the means of reaching the solution and therefore have different advantages and disadvantages. Skemp describes instrumental understanding as a list of equations, simply plugging numbers into the correct formulas. The advantage of this style of learning is how easily a teacher can teach it and how quickly the student can arrive at the answer. The main disadvantage of instrumental teaching is that the students typically don't understand the reasons behind their solutions, they understand "how" but not "why." Skemp describes relational understanding as having both the "how" and the "why" knowledge. This type of understanding takes a lot longer for the teacher to teach and is thus the reason that it is not used in schools today. This type of teaching has the huge advantage that students understand what they are learning. In this article, Skemp portrays relational as the better of the two understandings.
Tuesday, January 5, 2010
MthEd Blog Entry #1
Mathematics is the study of numbers and how they can be applied to everyday situations. It is a type of problem solving and even a way of thinking. I learn mathematics best by writing it down, figuring the problem out after I have been taught the correct process to complete the problem. Repetition is also critical for me, the more times i see a certain type of problem, the more i understand how to solve it and why it works. I think my students will learn mathematics best by numerous different methods. All students are different and different techniques work better for each individual student. I believe that teaching by whiteboard encouraging students to follow the steps by writing the problem on their own paper is effective for some. I also think visual aids can help many students. Simple things like hanging posters in the room with equations on them or showing how to do a problem using a picture. Applying the mathematical idea to real life situations helps lots of people too, because they feel it is a productive use of their time to be learning that particular material.
In schools today the beneficial methods of mathematics teaching in place are 10 minute quizzes at the start of class, the encouraging of group work both in and out of the classroom, and having students do certain problems on the board in front of the class. 10 minute quizzes at the start of class were very helpful to all students in my high school math classes in subjects such as the unit circle and derivatives and integrals. The students had to memorize these equations for the quizzes so when the equations were needed other than the quiz it was always fresh on the students mind. Group work in the class can sometimes be a waste of time but when monitored and forced to be productive by an assignment it is very helpful for students to talk with others to better understand the subject. Similarly, it is very helpful when a student does a problem on the board and walks the class through the steps he/she used to arrive at his/her solution. This allows students to hear different explanations of how to solve problems which might be more clear to them.
Some ineffective practices in school classrooms are teaching directly from the textbook and poor distribution of students in classes. Teaching directly from the textbook is boring and loses the attention of students. Use the textbook as an aid to a lesson that does not require students to have the textbook in front of them at all times. A big problem in most high schools is that there are not enough teachers to beneficially teach all the students. The result is that really smart kids end up in classes with those slower to learn. There is nothing wrong with the students that take longer to understand a subject, but their comments can be distracting to students who have already grasped the concept and end up second guessing themselves.
In schools today the beneficial methods of mathematics teaching in place are 10 minute quizzes at the start of class, the encouraging of group work both in and out of the classroom, and having students do certain problems on the board in front of the class. 10 minute quizzes at the start of class were very helpful to all students in my high school math classes in subjects such as the unit circle and derivatives and integrals. The students had to memorize these equations for the quizzes so when the equations were needed other than the quiz it was always fresh on the students mind. Group work in the class can sometimes be a waste of time but when monitored and forced to be productive by an assignment it is very helpful for students to talk with others to better understand the subject. Similarly, it is very helpful when a student does a problem on the board and walks the class through the steps he/she used to arrive at his/her solution. This allows students to hear different explanations of how to solve problems which might be more clear to them.
Some ineffective practices in school classrooms are teaching directly from the textbook and poor distribution of students in classes. Teaching directly from the textbook is boring and loses the attention of students. Use the textbook as an aid to a lesson that does not require students to have the textbook in front of them at all times. A big problem in most high schools is that there are not enough teachers to beneficially teach all the students. The result is that really smart kids end up in classes with those slower to learn. There is nothing wrong with the students that take longer to understand a subject, but their comments can be distracting to students who have already grasped the concept and end up second guessing themselves.
Subscribe to:
Posts (Atom)