Shannon, M.G. (March 2010). The value of guess and check. Mathematics Teaching in the Middle School, 15(7), 392-398.
This article had a large main idea of keeping the method of guessing and checking for different problems, specifically story or word problems. By having students use this method in class, they will come up with a better understanding for the answer they find and will also find their "own way for applying this strategy" (as the author puts it). The author points out different steps the student needs to take in order to use guess and check properly. One needs to ask themselves "What information is given? What am I trying to find? and What do I need to know before I do anything else?" These questions help the student build up their own understanding before creating a table to guess and check the closest answer to the problem (or the answer). Besides the students using the steps and the guess and check method, they are also able to apply it to other places besides story problems, and moreover, create an equation for the specific problem. This equation helps the students fully understand what they were just guessing and checking in order to find the correct answer.
Through guess and check the students are able to find the answer through much work. And though it may help them understand the problem itself once they create an equation, I think another method would be more helpful for the students. I do believe the students should have a complete understanding of what the problem is asking for before finding the answer to the word problem (as described in the guess and check method). But by guessing and checking, the student is just trying different answers to see which is "closest." Then, by the end, the student hopefully has an understanding of the method applied to that specific problem. This is where the student now has an equation to use. But by understanding how or why exactly they are getting "closer" to the answer is a better choice than waiting for the students to understand what they are doing at the end. Many students may not even get that far.
Friday, March 26, 2010
Thursday, March 18, 2010
Assignment #6: NCTM Article #1
Switzer, J. M. (2010). Bridging the math gap. Mathematical Teaching in the Middle School, 15 (7), 400-405.
Through this article Switzer is describing the different algorithms used in elementary school in preparing children for middle school (junior high). Yet, the main idea throughout this paper is the importance of teaching the middle school and high school teachers these different algorithms used in the classrooms at earlier ages. Switzer says that in order for teachers to be effective, they need to be able to build on a child's prior knowledge. Therefore in knowing the algorithms used, they can see where the children are coming from. Also, the teacher will be able to see the connection the algorithms learned have to ones they will learn later on. This will help the teacher know how exactly to teach future algorithms. Therefore, through the different algorithms Switzer described in his journal, teachers will overall be able to help children make the connection to previous knowledge.
Teaching middle school and high school teachers different algorithms used in elementary school classrooms is a very good idea and will over all help school systems, mathematics programs and teachers. In showing teachers what is being taught in younger-aged classrooms, teachers will be able to make connections to the teaching of future algorithms. One example of this is when teachers were taught about the partial products algorithm taught in schools. Skewer then talks about the connection the teachers made to the distributive property. Another important reason for teaching teachers these algorithms used is that it will make it easier for children later in school. By teachers understanding what is being taught to the children before they come into their own classrooms, they will know what they need to build on in order to teach them other algorithms. This will help the children learn easier by building off their prior knowledge. Also, it is important to connect the different schools in their teaching because, like stated earlier, children will have their prior knowledge build upon. But this will then help the children build their confidence as they are able to make connections to ideas/algorithms they already know. From my own experience, I always found it easier to learn when a teacher in mathematics taught me knew ideas, procedures, algorithms, etc. by showing me how they connected to things I had already learned. Thus, the teacher knew what I had learned before coming into their classroom. Therefore, by showing the teachers the algorithms being used in elementary school, teachers will be able to build on prior knowledge, help the kids raise their self-esteem, and also be able to see the connection to future algorithms being taught.
Through this article Switzer is describing the different algorithms used in elementary school in preparing children for middle school (junior high). Yet, the main idea throughout this paper is the importance of teaching the middle school and high school teachers these different algorithms used in the classrooms at earlier ages. Switzer says that in order for teachers to be effective, they need to be able to build on a child's prior knowledge. Therefore in knowing the algorithms used, they can see where the children are coming from. Also, the teacher will be able to see the connection the algorithms learned have to ones they will learn later on. This will help the teacher know how exactly to teach future algorithms. Therefore, through the different algorithms Switzer described in his journal, teachers will overall be able to help children make the connection to previous knowledge.
Teaching middle school and high school teachers different algorithms used in elementary school classrooms is a very good idea and will over all help school systems, mathematics programs and teachers. In showing teachers what is being taught in younger-aged classrooms, teachers will be able to make connections to the teaching of future algorithms. One example of this is when teachers were taught about the partial products algorithm taught in schools. Skewer then talks about the connection the teachers made to the distributive property. Another important reason for teaching teachers these algorithms used is that it will make it easier for children later in school. By teachers understanding what is being taught to the children before they come into their own classrooms, they will know what they need to build on in order to teach them other algorithms. This will help the children learn easier by building off their prior knowledge. Also, it is important to connect the different schools in their teaching because, like stated earlier, children will have their prior knowledge build upon. But this will then help the children build their confidence as they are able to make connections to ideas/algorithms they already know. From my own experience, I always found it easier to learn when a teacher in mathematics taught me knew ideas, procedures, algorithms, etc. by showing me how they connected to things I had already learned. Thus, the teacher knew what I had learned before coming into their classroom. Therefore, by showing the teachers the algorithms being used in elementary school, teachers will be able to build on prior knowledge, help the kids raise their self-esteem, and also be able to see the connection to future algorithms being taught.
Wednesday, February 17, 2010
Assignment #5
Throughout Warrington's paper, she has many advantages and disadvantages to her idea of teaching children mathematics without procedures or algorithms. One of the advantages to this style of teaching is the idea that children, when they understand how to do problems, it will have been from their own thought process and would have created thinking in their part to get to an answer. An example of this is when she would ask fraction/division problems without teaching the children the "invert and multiply" rule. This then connects to the advantage of children having part of a relational understanding of the procedures they are creating in their head. Another advantage is Warrington's idea of constructivism of teaching the children new concepts by starting with a topic or concept they've already learned. As Warrington states, this helps them understand the new concept better because they are starting with something familiar.
Besides the advantages of Warrington's style of teaching, there are also many disadvantages. One of those is her idea of children not acquiring knowledge from other people. Yes people will think about things and come up with their own consensus on a concept or topic, but they need the influence of others in order to create their own ideas. Another disadvantage is how Warrington doesn't give the children answers. The procedure on how the child got to the answer is important, but informing them of the correct one is also important so they can move onto another problem and feel like they are understanding things correctly. The last disadvantage I believe Warrington has in teaching this way is that it takes the children longer to understand a procedure when they have to create it with their own knowledge. Many children think differently than others and may not have the prior knowledge enough to create that procedure on their own. Thus, if children are taught the correctly through relational understanding, they would learn the procedure as well as why they are using it for specific concepts or problems.
Besides the advantages of Warrington's style of teaching, there are also many disadvantages. One of those is her idea of children not acquiring knowledge from other people. Yes people will think about things and come up with their own consensus on a concept or topic, but they need the influence of others in order to create their own ideas. Another disadvantage is how Warrington doesn't give the children answers. The procedure on how the child got to the answer is important, but informing them of the correct one is also important so they can move onto another problem and feel like they are understanding things correctly. The last disadvantage I believe Warrington has in teaching this way is that it takes the children longer to understand a procedure when they have to create it with their own knowledge. Many children think differently than others and may not have the prior knowledge enough to create that procedure on their own. Thus, if children are taught the correctly through relational understanding, they would learn the procedure as well as why they are using it for specific concepts or problems.
Wednesday, February 10, 2010
Assignment #4
Throughout his paper, von Glasersfield gave different ideas and points to what constructivism is. The main idea that he followed and presented throughout the paper was the idea of construct knowledge. Construct knowledge is the idea that all knowledge we obtain is built upon previous knowledge that we have: it isn't created or acquired. He then describes how knowledge is constructed by the experiences we have in our life. Through these experiences we go through, there is then a truth to the knowledge that has been constructed. This truth is the reality of life around us. It can only be changed or our opinion of this knowledge being true only changes when we go through other experiences to counteract the truth that has been made by previous experiences.
In taking constructivism as a true idea and correct perspective of knowledge, an implication in teaching situations that could be used to teach mathematics would be the importance of creating experiences or life situations in the classroom These children then will use those experiences as truth for ideas and concepts in math. Besides creating experiences (such as different problems or examples) to acquire knowledge, one should also build on the experiences one is having in the outside world around them. By looking at mathematics in the world outside the classroom, children would then create a sense of truth to concepts and laws given them. It also would be important to build on knowledge and understanding the children already have constructed, such as in other mathematics classes. In doing all this, constructivism is being used by knowledge being built on knowledge that has already been created as true through prior experiences and then new experiences are being created or brought to one's attention by seeing them inside and outside the classroom.
In taking constructivism as a true idea and correct perspective of knowledge, an implication in teaching situations that could be used to teach mathematics would be the importance of creating experiences or life situations in the classroom These children then will use those experiences as truth for ideas and concepts in math. Besides creating experiences (such as different problems or examples) to acquire knowledge, one should also build on the experiences one is having in the outside world around them. By looking at mathematics in the world outside the classroom, children would then create a sense of truth to concepts and laws given them. It also would be important to build on knowledge and understanding the children already have constructed, such as in other mathematics classes. In doing all this, constructivism is being used by knowledge being built on knowledge that has already been created as true through prior experiences and then new experiences are being created or brought to one's attention by seeing them inside and outside the classroom.
Sunday, January 24, 2010
Assignment #3
S. H. Erlwanger's paper Benny's Conception of Rules and Answers in IPI Mathematics has many main points that are important to learning mathematics and that are pertinent to the way Benny himself learned math. Yet, the most important idea I feel Erlwanger keeps throughout his paper is the idea of teaching the "how's and why's" to concepts in mathematics. This is what relational understanding is. Through the IPI system, children are learning the processes of different concepts and are just learning how to do or accomplish specific problems given them. This is instrumental learning. This style of learning only teaches children the surface of mathematics. In Benny's experience, because he didn't understand why he was doing these certain processes, he created his own rules and reasons for accomplishing problems the way given him. In examples throughout Erlwanger's paper, he showed how detrimental this can be to a child's life. They begin creating ideas and rules on their own, and don't truly understand the reasons why they are going about mathematics and that specific way.
This main importance throughout Erlwanger's paper of learning relationally instead of instrumentally is still quite valid in today's society. It is still very important for children in the long-run to learn relationally the different concepts in mathematics. The children need to learn why they are using each process and how specifically it relates to the different problems, as well as problems OUTSIDE of what they are doing. This idea is very important in the world today. In learning more than just the process, the child can then connect what they are doing to problems outside the classroom. They will be able to use what they've learned in everyday problems and also be able to connect the different concepts they've learned to past and future education. They will understand how prior knowledge is pertinent to the concepts they are learning in the classroom today, and also remember the processes more in order to apply them to math they learn in the future. Learning relationally, as Erlwanger kept as a main idea throughout his paper, is still as important and valid today as it was when he was researching and studying Benny, a child who didn't have the best mathematics learning experience to help him throughout the rest of his life.
This main importance throughout Erlwanger's paper of learning relationally instead of instrumentally is still quite valid in today's society. It is still very important for children in the long-run to learn relationally the different concepts in mathematics. The children need to learn why they are using each process and how specifically it relates to the different problems, as well as problems OUTSIDE of what they are doing. This idea is very important in the world today. In learning more than just the process, the child can then connect what they are doing to problems outside the classroom. They will be able to use what they've learned in everyday problems and also be able to connect the different concepts they've learned to past and future education. They will understand how prior knowledge is pertinent to the concepts they are learning in the classroom today, and also remember the processes more in order to apply them to math they learn in the future. Learning relationally, as Erlwanger kept as a main idea throughout his paper, is still as important and valid today as it was when he was researching and studying Benny, a child who didn't have the best mathematics learning experience to help him throughout the rest of his life.
Friday, January 15, 2010
Assignment #2
In the article Relational Understanding and Instrumental Understanding by Richard R. Skemp, two different types of understanding in mathematics were described. Both of these are important and overlap in their description, yet they both have different characteristics that give them advantages and disadvantages. The first style of understanding is instrumental understanding. This is a type where people learn the basic laws of math and possibly the concept of accomplishing a task. They learn the basic steps and memorize things in order to accomplish what is necessary. The other type of understanding in math is called relational understanding. This style overlaps instrumental by people learning laws, concepts and steps on accomplishing a task. Yet, this style is different in the effect that people also learn how to accomplish it, what to do, and why they are using these ideas or this new understanding. Both of these different understandings are very important and have their pros and cons. A disadvantage of instrumental is that people are just learning the basics. They aren't learning why they are using the procedure described, or how someone came up with it in the beginning. This becomes an advantage to someone who learns by relational understanding. As stated earlier, this person can know how and why it is necessary to use a certain procedure for the learned concept. Yet, a disadvantage to relational understanding is that, for many people, it takes time. Learning why and how to apply the idea learned can take quite a while for someone to understand: time they could be using in practicing the task. The advantage to instrumental then takes place of this disadvantage by learning quickly and not taking the time to understand the application. Both of these understandings, though having advantages as well as disadvantages, are important for people and their understanding, yet are just different ways of learning.
Tuesday, January 5, 2010
Assignment #1
Mathematics. A word that, to many people, means something different. Elementary addition? Middle school algebra? High School calculus? Or how about long story problems? Or real-life measurements and areas? Math is something that is used in everyday life. So therefore each person needs to find their own style of "mathematics". Yet, to me? Math is in and around EVERYTHING.
Mathematics, as I said, can be the simple multiplication used in math classes. Yet, it can also be the dimensions of an object, a room, or a building. Math can also be the weight of a fruit and the mileage to my hometown. Or how about the simple numbers used in math? Counting the money one has, the number of songs someone has in their iTunes, or even the number of friends they have in Facebook. Mathematics is a HUGE concept. It is used everywhere. Whether people realize it or not.
How do I learn math best? This can be quite a difficult question. Yet, I feel the easiest way to learn math is by applying it to my life and the things I do. In explaining a concept, I've always found it easiest to understand when someone describes it by things I enjoy doing: sports, music, etc. This is how I feel my students will learn best. I feel a teacher should take the time to get to know their students and the things they enjoy participating in. They can then describe mathematics in a way their mind works, whether that be through football plays, music or dance beats, or even by using hands-on examples.
One of the things I find math teachers are getting better at is the different ways they are teaching. Many teachers are now understanding that students ALL learn in different ways and therefore use different styles to teach the same concept. My high school calculus teacher was amazing. He knew so many ways to describe the same concept. No one was stumped in the class. If you didn't understand it the first time, in coming to him for help, my teacher could describe it in just the perfect manner for your mind to wrap around the principle.
Some of the small and simple things teachers do or don't do, are actually extremely detrimental to students' learning. One of the things I feel math teachers don't do enough, is like I stated earlier, simply getting to know their students. One of the best teachers I ever had genuinely cared about his students. He wanted us to do well and cared for each of our well-beings. He therefore did all he could to help us use or skills, live up to our potential, and then receive the grade we deserved. If teachers simply got to know the characteristics of their students, the activities they find joy in, and the life they live, I feel students would feel more willing to learn and look to the teacher for help.
Mathematics, as I said, can be the simple multiplication used in math classes. Yet, it can also be the dimensions of an object, a room, or a building. Math can also be the weight of a fruit and the mileage to my hometown. Or how about the simple numbers used in math? Counting the money one has, the number of songs someone has in their iTunes, or even the number of friends they have in Facebook. Mathematics is a HUGE concept. It is used everywhere. Whether people realize it or not.
How do I learn math best? This can be quite a difficult question. Yet, I feel the easiest way to learn math is by applying it to my life and the things I do. In explaining a concept, I've always found it easiest to understand when someone describes it by things I enjoy doing: sports, music, etc. This is how I feel my students will learn best. I feel a teacher should take the time to get to know their students and the things they enjoy participating in. They can then describe mathematics in a way their mind works, whether that be through football plays, music or dance beats, or even by using hands-on examples.
One of the things I find math teachers are getting better at is the different ways they are teaching. Many teachers are now understanding that students ALL learn in different ways and therefore use different styles to teach the same concept. My high school calculus teacher was amazing. He knew so many ways to describe the same concept. No one was stumped in the class. If you didn't understand it the first time, in coming to him for help, my teacher could describe it in just the perfect manner for your mind to wrap around the principle.
Some of the small and simple things teachers do or don't do, are actually extremely detrimental to students' learning. One of the things I feel math teachers don't do enough, is like I stated earlier, simply getting to know their students. One of the best teachers I ever had genuinely cared about his students. He wanted us to do well and cared for each of our well-beings. He therefore did all he could to help us use or skills, live up to our potential, and then receive the grade we deserved. If teachers simply got to know the characteristics of their students, the activities they find joy in, and the life they live, I feel students would feel more willing to learn and look to the teacher for help.
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