Barger, R.H., & McCoy, A.C. (2009). Sacred cows make the best hamburgers. Mathematics Teacher, 102(6), 414-418.
The main idea of this article was that college mathematics courses should be more interactive and hands-on. Instead of the lecture and practice problems homework system that most courses consist of, Barger and McCoy did research and then tested out their own mathematics course, entitled Contemporary Mathematics, in which they implemented their creative and interactive material. The course is an equivalent of a mathematics general education course, in which most students are not majoring in anything related to the math or science areas. In the article, they highlighted the lessons on volume and surface area. Before starting this class, they noticed that a lot of students mix up the formulas for volume and surface area, proposing that this is because students do not understand the formulas. In this new course, the students built shapes and formulated their own hypotheses of what volume and surface area is and how to find them for different shapes. As was shown through test scores, the students understood mathematics so much more and grades improved. Additionally, Barger and McCoy point out that these interactive methods do not take that much more time than regular lectures and that understanding the formulas is worth much more than the extra time! Thus they concluded that mathematics can be taught in a different and more meaningful way to students in college.
From all my studies of education, I agree that mathematics courses can and should be more interactive. Firstly, the main reason they wanted mathematics courses to be more interactive and hands-on is so that students will understand more. I think this is supported by our study of relational learning in MathEd 117. By working it out and discovering "formulas" on their own, these students are doing relational learning, which will impact their learning and their tests scores. Secondly, this idea is supported from what I have studied in my Theatre Ed classes. "Dramatic learning" is when students are able to take control of their learning, by hands-on activities and the time develop their own ways to learn things. This article supported that type of learning because the students made their own formulas for volume and surface area. Thirdly, Barger and McCoy addressed my one concern - time. They acknowledged this but said the time was not so much more that it overpowered the understanding these students were gaining. Thus, I am sold on this new idea!
Wednesday, March 24, 2010
Thursday, March 18, 2010
Blog Entry #6
Switzer, J. M. (2010). Bridging the math gap. Mathematics Teaching in the Middle School, 15(7), 400-405.
In the article that I read, Switzer's main idea was that effective teaching requires building on previous knowledge. This idea came to him when teachers in the middle schools and high schools in his district could not teach their students because they did not understand the alternative algorithms that these students were doing for basic math. He quickly realized that understanding what students learned in elementary school can help teachers teach students better because they can connect that knowledge to what they will teach them in middle school and high school. The example Switzer used was the concept of partial products. Students in his district had learned the principles behind the distributive property of multiplication over addition (usually represented by the traditional algorithm of FOIL) in elementary school when learning multiplication of two digit numbers. If the algebra teachers realized the alternative algorithm that the students used for basic multiplication, then learning multiplication of polynomials would have been a lot easier. Overall, Switzer concluded to having meetings with all grade levels of teachers in his district so they could become in sync to how students are learning mathematics so they could integrate those alternative ways into their teaching of more advanced mathematics.
I think that Switzer was very perceptive when it came to the needs in his school district and I agree that teachers do need to build not only just on previous knowledge, but on the way that that knowledge was learned. I would say his secondary idea, which was that alternative algorithms are great and can easily be transferred to traditional algorithms if necessary, was also brilliant. When I was reading his article, I felt that it feel right in line with what we have been learning in class. Firstly, in class we have been exploring relational ways of understanding basic mathematics. In this article, he addressed an alternative way of understanding multiplication of double digit numbers, which seemed very relational to me. I had never thought about how the traditional algorithm of this process makes absolutely no sense! I mean, multiply the second number on the bottom with the first two numbers on the top, each separately, then drop a zero down and then multiply the first bottom number with the top two and then add it all up at the end! It has so many rules and steps to remember! The alternative way he talked about was a basis for algebra as well as teaching students the importance of place value with more than one digit numbers. I thought his explanation really illustrated what we have been learning in class about understanding why mathematics is the way it is. Secondly, his ideas would save a lot of time if the teachers did understand exactly how, not just what, the students learned before they got to their classrooms. It would make teaching effective and efficient. So many of the articles we have read don't necessarily have teaching ideas that are both of those things. Lastly, his idea is just very logical! You can't teach students certain principles or algorithms if you don't know how they learned what they know at this point. Having meetings like he did makes sense for the situation he described in his school district, but I am sure there are other ways of making sure upper division teachers are aware of how students learned in the lower division years. Overall, I am very supportive of Switzer's ideas.
In the article that I read, Switzer's main idea was that effective teaching requires building on previous knowledge. This idea came to him when teachers in the middle schools and high schools in his district could not teach their students because they did not understand the alternative algorithms that these students were doing for basic math. He quickly realized that understanding what students learned in elementary school can help teachers teach students better because they can connect that knowledge to what they will teach them in middle school and high school. The example Switzer used was the concept of partial products. Students in his district had learned the principles behind the distributive property of multiplication over addition (usually represented by the traditional algorithm of FOIL) in elementary school when learning multiplication of two digit numbers. If the algebra teachers realized the alternative algorithm that the students used for basic multiplication, then learning multiplication of polynomials would have been a lot easier. Overall, Switzer concluded to having meetings with all grade levels of teachers in his district so they could become in sync to how students are learning mathematics so they could integrate those alternative ways into their teaching of more advanced mathematics.
I think that Switzer was very perceptive when it came to the needs in his school district and I agree that teachers do need to build not only just on previous knowledge, but on the way that that knowledge was learned. I would say his secondary idea, which was that alternative algorithms are great and can easily be transferred to traditional algorithms if necessary, was also brilliant. When I was reading his article, I felt that it feel right in line with what we have been learning in class. Firstly, in class we have been exploring relational ways of understanding basic mathematics. In this article, he addressed an alternative way of understanding multiplication of double digit numbers, which seemed very relational to me. I had never thought about how the traditional algorithm of this process makes absolutely no sense! I mean, multiply the second number on the bottom with the first two numbers on the top, each separately, then drop a zero down and then multiply the first bottom number with the top two and then add it all up at the end! It has so many rules and steps to remember! The alternative way he talked about was a basis for algebra as well as teaching students the importance of place value with more than one digit numbers. I thought his explanation really illustrated what we have been learning in class about understanding why mathematics is the way it is. Secondly, his ideas would save a lot of time if the teachers did understand exactly how, not just what, the students learned before they got to their classrooms. It would make teaching effective and efficient. So many of the articles we have read don't necessarily have teaching ideas that are both of those things. Lastly, his idea is just very logical! You can't teach students certain principles or algorithms if you don't know how they learned what they know at this point. Having meetings like he did makes sense for the situation he described in his school district, but I am sure there are other ways of making sure upper division teachers are aware of how students learned in the lower division years. Overall, I am very supportive of Switzer's ideas.
Monday, February 15, 2010
Blog Entry #5
In the article, "How Children Think about Division with Fractions," Warrington advocates that a constructivist approach to mathematics can work in a classroom. She takes the reader through the process of division with fractions with her 5th and 6th grade class to illustrate her point. One of the advantages that she found was it gave the students intellectual autonomy. This meant that the students each had their own choice in discovering their intellect of mathematics. An example of intellectual autonomy in her classroom was when the one student did not agree with all the other students about the answer to 4 2/5 divided by 1/3. This stemmed to a huge discussion and debate over the right answer, in which each student had their own right and ability to edit their answer according to what they believed from the other students. This is a wonderful way in which Warrington's constructivist approach allows deeper understanding to take place in mathematics. Another advantage is that the students are able to increase their problem solving abilities because they are not relying on the teacher to teach them algorithms and the right answers, but instead, they are developing mathematics on their own and building upon their previous knowledge. In the first three problems that Warrington gave, she described the answers the students were giving her, and these answers came from the students' confidence in their mathematical abilities because they were building off of what they already knew. These abilities can be used in all subjects in school and in learning for the rest of their lives.
Although the article concluded with success in Warrington's classroom, I saw a few disadvantages as I was studying the situation. Firstly, I think the time that was spent teaching these students fractions was way too long! It said in the paper that she had been working with them on fractions for months, and when I learned them we did not spend that much time on fractions. This way is just too time consuming for everything that students need to learn. And frankly, I did not learn fractions this way and yet I am still able to do math at a university level, so therefore I must not be completely lacking in what I am learning, so the year that they spent learning this will put them behind in learning more mathematics, and more complicated mathematics, if learned in this way, will take even longer, and that time will not be found in schools. Another disadvantage I saw was that some students were figuring this out on their own. For all we know, there could have been many quiet students who never said anything, and never actually figured it out on their own, but instead just took their classmates' word for it, just as they would have taken the teacher's word for it if this had been a traditional classroom. So I am not sure if Warrington's approach really solved all the "problems" that she finds in traditional classrooms. In conclusion, I think her approach is a style that should be experimented with in more classrooms before it is truly accepted as greatness.
Although the article concluded with success in Warrington's classroom, I saw a few disadvantages as I was studying the situation. Firstly, I think the time that was spent teaching these students fractions was way too long! It said in the paper that she had been working with them on fractions for months, and when I learned them we did not spend that much time on fractions. This way is just too time consuming for everything that students need to learn. And frankly, I did not learn fractions this way and yet I am still able to do math at a university level, so therefore I must not be completely lacking in what I am learning, so the year that they spent learning this will put them behind in learning more mathematics, and more complicated mathematics, if learned in this way, will take even longer, and that time will not be found in schools. Another disadvantage I saw was that some students were figuring this out on their own. For all we know, there could have been many quiet students who never said anything, and never actually figured it out on their own, but instead just took their classmates' word for it, just as they would have taken the teacher's word for it if this had been a traditional classroom. So I am not sure if Warrington's approach really solved all the "problems" that she finds in traditional classrooms. In conclusion, I think her approach is a style that should be experimented with in more classrooms before it is truly accepted as greatness.
Monday, February 8, 2010
Blog Entry #4
Von Glasersfeld believes that all people construct the knowledge that they have. They do not gain or acquire knowledge, but construct and build their knowledge through a filter in their minds, shaped by the experiences that they have had in their lives. Therefore, knowledge is a theory because it is always changing because people's experiences are always happening that change and tweak the knowledge that they have previously constructed.
If I believed in constructivism, as a math teacher, I would emphasize the importance of understanding - or of seeing how it is that my students are filtering the knowledge that I am trying to teach. I would do this by having a few minutes at the end of each class period where they turn to their neighbor and each teach each other what they have learned that day. I would walk around and try to catch as many people as possible during their teaching descriptions. This would help the students, in a non-threatening way, share the knowledge they have constructed and give me a chance to catch any mistakes they have taken place in their understanding.
If I believed in constructivism, as a math teacher, I would emphasize the importance of understanding - or of seeing how it is that my students are filtering the knowledge that I am trying to teach. I would do this by having a few minutes at the end of each class period where they turn to their neighbor and each teach each other what they have learned that day. I would walk around and try to catch as many people as possible during their teaching descriptions. This would help the students, in a non-threatening way, share the knowledge they have constructed and give me a chance to catch any mistakes they have taken place in their understanding.
Saturday, January 23, 2010
Blog Entry #3
I think Erlwanger's most important point in this paper is that the individualism of the IPI program is the biggest downfall of its mathematical teaching system. By taking the reader on a journey with what Benny has learned about mathematics, Erlwanger exposes how individualistic the program is and how mathematics, if learned properly, demands a true teacher to teach students relational mathematics. After spending 4 years learning mathematics through IPI, Benny had learned on his own to distrust the teacher and the answer key by making up his own (false) rules for mathematical basics, especially when dealing the fractions and decimals. Thus, Erlwanger believes that learning individually is not how mathematics should be taught.
I believe that learning should not be an individual experience, which leads me to argue that this point of Erlwanger is applicable in teaching mathematics today because it shows how important a proper math teacher is in today's society. We, as math teachers, need to be trained well enough to teach students relational mathematics, stopping any kind of individual (false) learning as fast as possible in a child's learning lifetime. So not only is it valid to understand that the value of teachers becomes higher as individual learning becomes disintegrated, but we must use stories like Benny's to re-inspire us to continue our mathematical education to help prevent any other child suffering through an individualized program in mathematics.
I believe that learning should not be an individual experience, which leads me to argue that this point of Erlwanger is applicable in teaching mathematics today because it shows how important a proper math teacher is in today's society. We, as math teachers, need to be trained well enough to teach students relational mathematics, stopping any kind of individual (false) learning as fast as possible in a child's learning lifetime. So not only is it valid to understand that the value of teachers becomes higher as individual learning becomes disintegrated, but we must use stories like Benny's to re-inspire us to continue our mathematical education to help prevent any other child suffering through an individualized program in mathematics.
Wednesday, January 13, 2010
Blog Entry #2
Skemp revolutionized math educators' studies of mathematics with his discovery of rational understanding versus instrumental understanding, which has led the mathematics educational world to analyze and apply his findings. He found that many students and teachers clash with their views on what mathematics understanding means. Instrumental understanding is learning the how without the why. In mathematics, this is most commonly demonstrated though formulas with no meaning to the students. Rational understanding means that the students are learning how and why to every part of mathematics. According to Skemp, these two distinct types of understanding in mathematics are actually connected. Rational understanding is instrumental learning plus so much more. If students understand rationally, then they understand everything that instrumental understanders know plus the why to mathematics as well as applicable skills used in more advanced mathematics problems. Even though they are connected, both have advantages and disadvantages explored by Skemp. Instrumental understanding happens faster and simpler, while making it harder for students to remember mathematics longer and for them to apply their knowledge to different types of problems they did not see in class. On the other hand, rational understanding, although it is harder to grasp each concept at first, should carry a student on for a longer period of time, as well as teach them the skills they can apply to more advanced problems without learning more rules like instrumental understanding would require. Overall, Skemp deeply defined and explained both types of learning and left the reader able to make their own judgments and decisions about mathematical understanding.
Tuesday, January 5, 2010
Blog Entry #1
1. What is mathematics?
Mathematics is a subject in school, in which students learn about numbers and problem solving.
2. How do I learn mathematics best? Explain why you believe this.
I learn mathematics best by reading the book, paying attention especially to the sample problems. I study every step, and then I test myself by doing the practice problems. If I get stuck, I go back to the beginning of the chapter and re-read how to do that part before applying it again to the practice homework problems. This is how I learn mathematics best. I know it is the best way because it has produce successful results all through my mathematics past as well as it allows me to not be personally dependent on anyone else to learn mathematics.
3. How will my students learn mathematics best? Explain why you think this is true.
Even though I learn on my own from the book, I believe my students will learn mathematics the best from what I like to call "dramatic learning." Many students in today's society believe they are not good at math, and I do not think reading from a book will help them to improve. They will need someone who can explain the concepts clearly and in a fun, exciting (dramatic!) way. By developing hands-on methods (excluding having them write problems on the whiteboard!), I believe most students will learn mathematics the best because of the visual and more applicable ways of looking at mathematics.
4. What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? Justify your reasoning.
To be honest I do not know a lot of the current practices of mathematics in the classroom, but I know what my math classes did. Nightly homework promotes learning because the students need to try problems on their own in order to really learn mathematics; they cannot simply watch the teacher do a problem during lecture and then be able to reproduce a similar problem on the test without practicing other like problems. So I think daily practice problems are very helping in learning mathematics.
5. What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? Justify your reasoning.
I think one current practice that I noticed in my middle school and high school years in math classes that was detrimental to the learning of mathematics was too much daily homework. There is no need to do 20 problems that are all alike. 3 to 4 problems will suffice. I feel lots of students became overwhelmed at all the homework every night, and therefore decided that they hated math and gave up on trying. I think practice problems are good but within reason. Too much practice will cause students to not try at all and will hinder their learning of mathematics.
Mathematics is a subject in school, in which students learn about numbers and problem solving.
2. How do I learn mathematics best? Explain why you believe this.
I learn mathematics best by reading the book, paying attention especially to the sample problems. I study every step, and then I test myself by doing the practice problems. If I get stuck, I go back to the beginning of the chapter and re-read how to do that part before applying it again to the practice homework problems. This is how I learn mathematics best. I know it is the best way because it has produce successful results all through my mathematics past as well as it allows me to not be personally dependent on anyone else to learn mathematics.
3. How will my students learn mathematics best? Explain why you think this is true.
Even though I learn on my own from the book, I believe my students will learn mathematics the best from what I like to call "dramatic learning." Many students in today's society believe they are not good at math, and I do not think reading from a book will help them to improve. They will need someone who can explain the concepts clearly and in a fun, exciting (dramatic!) way. By developing hands-on methods (excluding having them write problems on the whiteboard!), I believe most students will learn mathematics the best because of the visual and more applicable ways of looking at mathematics.
4. What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? Justify your reasoning.
To be honest I do not know a lot of the current practices of mathematics in the classroom, but I know what my math classes did. Nightly homework promotes learning because the students need to try problems on their own in order to really learn mathematics; they cannot simply watch the teacher do a problem during lecture and then be able to reproduce a similar problem on the test without practicing other like problems. So I think daily practice problems are very helping in learning mathematics.
5. What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? Justify your reasoning.
I think one current practice that I noticed in my middle school and high school years in math classes that was detrimental to the learning of mathematics was too much daily homework. There is no need to do 20 problems that are all alike. 3 to 4 problems will suffice. I feel lots of students became overwhelmed at all the homework every night, and therefore decided that they hated math and gave up on trying. I think practice problems are good but within reason. Too much practice will cause students to not try at all and will hinder their learning of mathematics.
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