So, how do we get students talking? Van de Walle and Lovin (2006) stresses that “classroom discussion based on students’ own ideas and solutions to problems is absolutely ‘foundational to children’s learning.’’ (p.5). Therefore, discussions must focus on students own thinking. What better place to start than to have students talk about themselves and what they did to solve the problem, or how they are thinking about what they are leaning. Hey, you must capitalize on children’s inflated egos. To begin this worthwhile mathematical journey, students must be given opportunities to talk in their math classrooms and teachers must learn how to listen. For teachers, this means stepping away from the whiteboard and stepping out of those teacher shoes to give students the control and power of classroom discussion, with helpful guidance and support, of course. Encouragement to talk about their own ideas, no matter how silly they may sound, will lead to students seeing that their thoughts and ideas are important. The ultimate goal is to have students talking about the math they are learning and how they are able to make sense of it. If students are really having a tough time trying to verbally communicate their thinking, have them use other mediums first, such as writing about their ideas, or give them something physical to talk about. For example, students may want to use manipulatives when explaining themselves as this redirects the focus from a shy student to the manipulative. The use of manipulatives also allows students to use the language of the manipulatives to make communication easier.
To get conversations going, Small (2008) suggests teachers encourage group discussions; allow more wait time after a question is posed and delay reaction to or evaluation of a student response. It may be worthwhile at this time to ask the class, or group, to evaluate the response having them justify their thinking. Teachers may want to be cognizant of the way they group students. Ensuring students are grouped with others they can talk to is important.
To help facilitate and guide discussions, prompting may be necessary. Asking students carefully thought out questions, such as ‘Can you show me how you did that?” “How do you know that is correct?” Why do you think that?”can help guide student thinking, especially when students have come to an inappropriate conclusion. Asking the right kinds of questions to guide student thinking may be all that is needed in order to bring them back on the right track. Most times, as students begin talking through or explaining their thinking, they realize their mistake and are able to come to correct conclusions. There will be more about questioning and how to create good questions later in the paper.
Mathematical conversations also become a valuable tool for the teacher. These discussions make it very helpful to assess student learning as it can lead to understanding how a student is thinking about something and how they were able to reach the appropriate/inappropriate solutions. As students explain or justify their work, teachers get to hear their explanations. As students explain their work to one another, they may be more able to help others understand the concept than the teacher. The student who is explaining gets to acquire a deeper understanding of the concept as he/she talks through his/her thinking and the others that are listening are given greater access to understanding. This deeper understanding of student learning makes instruction easier as the teacher does not have to guess or assume students know something. Teachers can use this vital information to plan their next lesson knowing exactly where student understanding on the concept is. This can not be achieved by teaching math in the traditional way, where students are only expected to produce the correct answers.
Tuesday, February 10, 2009
Conversations- Getting students talking
Perhaps one of the most important parts of having students construct mathematical understanding and truly gain a deep conceptual knowledge of the mathematical content being taught, students must be given opportunities to talk about what it is they are learning. It is through this talk and discussions with peers and teachers, new ideas are constructed and connections are made. To reinforce the importance of talking Boaler (2008) talks about a young Irish woman, Sarah Flannery, who won the European Young Scientist of the Year award who developed a wondrous mathematical algorithm. In her autobiography, Flannery recalls the events that led to her success. Boaler (2008) states “Flannery writes: “The first thing I realized about learning mathematics was that there is a hell of a difference between, on one hand, listening to math being talked about by someone else and thinking that you are understanding, and, on the other, thinking about math and understanding it yourself and talking about it to someone else.” (p.47) The National Council of Teachers of Mathematics also agree. NCTM (2007) suggest that one goal of mathematics education is for students to develop and expand their reasoning abilities. It is the mathematics classroom that is the main and central environment where students speak and write mathematics. Therefore, NCTM (2007) continues “it is essential for teachers to offer students opportunities to communicate mathematically by having them make, test, discuss, and refine conjectures, ultimately accepting or rejecting them” (p.1).
Student understanding of mathematical concepts is what teachers strive for, or at least, what teachers should strive for. Classrooms that have students interacting, communicating and helping each other solve problems is a classroom where students actually understand the mathematical concepts being taught. Marian Small also agrees. Small (2008) notes that “it is through interactions with other students as well as with the teacher, and with the opportunity to articulate their own thoughts that students are able to construct new mathematical knowledge” (p.4). Small (2008) goes on to note that it is in these classrooms, one in which she calls constructivists classrooms, “students are given the opportunity to develop richer and deeper cognitive structures related to mathematical ideas and students’ development of a level of mathematical autonomy”(p.4).
When students have mathematical conversations, they begin to see math as more than a collection of rules and procedures. They begin to see that math as a subject where they can have their own ideas, methods and perspectives. They will eventually see math as a connected subject with organized concepts and themes. Silence in the math classroom, where students are not asked about their own ideas and perspectives may feel disempowered and ultimately choose to leave mathematics even though they were good at it. Students that see their thought and ideas are valued are more apt to feeling responsible for the direction of their work, as they are being asked to use their intellect.
Student understanding of mathematical concepts is what teachers strive for, or at least, what teachers should strive for. Classrooms that have students interacting, communicating and helping each other solve problems is a classroom where students actually understand the mathematical concepts being taught. Marian Small also agrees. Small (2008) notes that “it is through interactions with other students as well as with the teacher, and with the opportunity to articulate their own thoughts that students are able to construct new mathematical knowledge” (p.4). Small (2008) goes on to note that it is in these classrooms, one in which she calls constructivists classrooms, “students are given the opportunity to develop richer and deeper cognitive structures related to mathematical ideas and students’ development of a level of mathematical autonomy”(p.4).
When students have mathematical conversations, they begin to see math as more than a collection of rules and procedures. They begin to see that math as a subject where they can have their own ideas, methods and perspectives. They will eventually see math as a connected subject with organized concepts and themes. Silence in the math classroom, where students are not asked about their own ideas and perspectives may feel disempowered and ultimately choose to leave mathematics even though they were good at it. Students that see their thought and ideas are valued are more apt to feeling responsible for the direction of their work, as they are being asked to use their intellect.
My Purpose of the paper
Today’s classrooms are filled with so many different learning styles, preferences, abilities and ethnicities. It is expected that teachers deliver a sound curriculum in a way that fosters each of these individual styles and abilities. This is becoming more and more difficult to achieve. Teachers and students can not afford to continue teaching and ‘learning’ in the way that is illustrated in Classroom One. It is here that teachers face the reality that everyone in their classrooms are not meeting their mathematical potential, or achieving the desired outcomes. Some may blame the curriculum for not allowing all students equal access to achievement, but the author of this paper will argue that it is the way the curriculum is delivered that causes students to become outsiders in the mathematical world.
It is not hard to see the difficulties students have with learning math. Current research, and even the very curriculum guides we use, suggests that math be taught through a problem based approach. It is here that tasks are open ended which allows students to learn the intended math outcomes through solving rich, contextual problems. The teacher’s role is merely a facilitator, whereby the students take on a more active role in constructing their own knowledge, as seen in Classroom Two. According to TIMMS (2007), Singapore is ranked the top country in the world when it comes to mathematical achievement. Problem solving is at the heart of Singapore’s math curriculum, where reasoning and communication are of utmost importance. Newfoundland and Labrador has just recently adapted the WNCP math curriculum from Western Canada. It is intended that this new curriculum will fix all of the problems created previously. However, the actual curriculum is not the single reason students in Newfoundland and Labrador have failed to meet basic mathematical understandings, we must also look at the way in which the math was taught. Simply adopting new curriculum does not mean students will understand math better. Students need different experiences in the classroom that enhances their opportunities for obtaining their mathematical potential! It is the experiences like that of the students in Classroom Two that will provide the opportunities all students need to become great mathematical problem solvers and thinkers.
With the introduction of this new curriculum, a curriculum that has been proven to work, the author of this paper feels it is the right time to begin a discussion about how to go about teaching mathematics. This discussion needs to focus on how math should be taught, a way where all students are given the opportunity to succeed, to become mathematical thinkers. The purpose of this paper is to discuss some strategies and suggest some possible ways to open up classrooms and ultimately, the curriculum that will provide all students with equal opportunities to learn mathematics in their classrooms, one where all students have access to the curriculum and the learning. This paper will discuss three major elements needed to make this type of learning situation possible, namely the importance of talking and discussions in math class, the important role of asking the right questions to facilitate these discussions and providing students with something worthwhile to talk about.
It is not hard to see the difficulties students have with learning math. Current research, and even the very curriculum guides we use, suggests that math be taught through a problem based approach. It is here that tasks are open ended which allows students to learn the intended math outcomes through solving rich, contextual problems. The teacher’s role is merely a facilitator, whereby the students take on a more active role in constructing their own knowledge, as seen in Classroom Two. According to TIMMS (2007), Singapore is ranked the top country in the world when it comes to mathematical achievement. Problem solving is at the heart of Singapore’s math curriculum, where reasoning and communication are of utmost importance. Newfoundland and Labrador has just recently adapted the WNCP math curriculum from Western Canada. It is intended that this new curriculum will fix all of the problems created previously. However, the actual curriculum is not the single reason students in Newfoundland and Labrador have failed to meet basic mathematical understandings, we must also look at the way in which the math was taught. Simply adopting new curriculum does not mean students will understand math better. Students need different experiences in the classroom that enhances their opportunities for obtaining their mathematical potential! It is the experiences like that of the students in Classroom Two that will provide the opportunities all students need to become great mathematical problem solvers and thinkers.
With the introduction of this new curriculum, a curriculum that has been proven to work, the author of this paper feels it is the right time to begin a discussion about how to go about teaching mathematics. This discussion needs to focus on how math should be taught, a way where all students are given the opportunity to succeed, to become mathematical thinkers. The purpose of this paper is to discuss some strategies and suggest some possible ways to open up classrooms and ultimately, the curriculum that will provide all students with equal opportunities to learn mathematics in their classrooms, one where all students have access to the curriculum and the learning. This paper will discuss three major elements needed to make this type of learning situation possible, namely the importance of talking and discussions in math class, the important role of asking the right questions to facilitate these discussions and providing students with something worthwhile to talk about.
Two Scenarios
Classroom One:
Teacher: (drawing a right triangle on the board) Who can tell me the name of this triangle?
Student 1: A right triangle.
Teacher: That is right, it is a right triangle because it has one right angle. We have been talking about finding the area of rectangles, and parallelograms and now I am going to show you how to find the area of a triangle. Any questions?
(No response)
Teacher: Great, so let’s move on and dig right in. Can anyone remember the formula to finding the area of a parallelogram?
Student 2: Length times width.
Teacher: Not quite, that is the area of a rectangle. What particular words am I looking for to describe the area of a parallelogram? Baaa times heee…
Student 3: Is it base times height?
Teacher: (feeling rather pleased) You are correct. Now let’s look at finding the area of a triangle. This is the base of a triangle (pointing to the particular line that represents the base) and this is the height, (indicating the imaginary line from the peak of the triangle to the base). To find the area you simply multiply the base and the height and then divide it by two. Any questions?
Student 1: Miss, so all you have to do is get the base and multiply it by the height and then half it?
Teacher: (again feeling rather pleased that the student ‘got it’) You are absolutely right. Let’s try an example and I’ll show you how it is done.
Student 3: Why do you divide it by two?
Teacher: Just watch and I will show you how to do it.
The teacher then continues to draw several examples of different triangles on the board and finds the area by applying the formula. Students are quietly copying examples in their note books. Once the teacher feels “all students” have got it, she assigns questions 1-17 from the next lesson entitled “Area of Triangles” in the student math book. She announces that any unfinished work is to be completed for homework.
The classroom described above, unfortunately, can be from any classroom or school around the country. This type of teaching and learning can be defined as ‘traditional’, where the teacher’s role is to transmit information to the student who in turn memorizes the procedure and applies the skill to repeating tasks of similar nature. It is no secret that this method of teaching and learning has proved disadvantageous and researchers of today are suggesting a different way, a more constructivist way, of teaching and learning math.
Let’s consider the next classroom.
Classroom Two:
Teacher: O.k, class, over the past few days we have been exploring how to find the area of different rectangles and parallelograms using various methods. Using what you have learned previously, I want you to create and find the area of any rectangle or parallelogram.
(Students begin working and can be seen discussing their creations with other students. During this time the teacher can be seen roaming around the room talking with individual and pairs of students about their shapes and asking them to explain how there were able to find the area, where any types of misconceptions were noted).
Teacher: (after about 15 minutes have passed) Who would like to come to the front of the class to show your shape and explain how you went about finding the area?
Student 1: (in front of the class) Well, the first thing I did was made a parallelogram on the geoboard (using the overhead goeboard, he reproduces the parallelogram Then I counted how many blocks the length was, or the base. It was five (again demonstrating how this is done). I then counted how many blocks high the parallelogram was (demonstrating how this was counted) and it is 6. I know a parallelogram is the same as a rectangle only squished a little, so I made this rectangle, a 5 x 6, out of snap cubes and then counted how many cubes I used. I used 30 snap cubes so I know the area of my parallelogram is going to be 30 units squared.
Teacher: Are there any questions or comments for Student 1? Would someone like to explain in their own words what Student 1 did to find the area?”
Student 3: Well, basically all he did was made a parallelogram on the geoboard, found the base and the height and then multiplied them together.
Student 4: Yeah, but he did not really use multiplication. He made a rectangle out of snap cubes and then counted how many blocks he used.
Student 3: Yes, but really he used multiplication to find the area because when you make a rectangle like he did out of cubes you are really making an array of 5 x 6. And 5 groups of 6 is 5, 10, 15, 20, 25, 30. See?
Student 4: Oh yeah, you are right. I can see that now. I never looked at doing this in that way before.
Student 1: Me either. I now see how you can use multiplication to find the area of the parallelogram. The parallelogram is the same as a rectangle, only it is slanted. So if you can find the area of a rectangle by multiplying length by width, than it would work the same for a parallelogram, only the names of length and width are different. Cool! Thanks Student 3.
Student 3: No problem!
Teacher: So now that we have seen a couple of different ways of getting the area of a parallelogram, I want you to check Student 1’s answer by using your method to find the area of a parallelogram, check to see if you get the same answer.
Students apply their own strategies to check the answer as the teacher walks around the room to check other people’s methods and understanding.
Teacher: So, what can we say about student 1’s answer? How can you be sure?
Student 7: Well, I drew the same parallelogram on graph paper and counted the complete units and got 30 units squared, just like student 1.
Teacher: So, we have all had a chance to come up with different ways to find the area of a rectangle and a parallelogram, all of which a perfectly right. There were a lot of great discussions happening in your groups and I am glad to see that you are helping each other. You are all becoming very great problem solvers. So, I now want you to put your thinking skills to the test. I want you to explore and find out as much as you can about the area of a triangle. I will give you about a half an hour and then we will come back as a whole class to discuss your findings.
Students began exploring the idea of trying to find the area of a triangle by using different manipulatives and through discussions with other students and the teacher. The teacher met with different students, asking questions to help them through their thinking. By the end of the half an hour, there were some pretty different discoveries made about triangles whereby all students were able to find out something about the area of a triangle and most had discovered strategies that worked for them to find the area.
Teacher: Now, that we have all had some time exploring the area of a triangle, let’s talk about what we have found out.
Student 7: Well, I found out that if I drew the triangle on a piece of grid paper, it was a little hard to count the number of squares because there are a lot of little pieces here and there.
Student 10: Yeah, I did the same thing but then I decided to make a triangle on the geoboard. I looked at it for a while and thought that maybe you would multiply the base and height but realized this would not work because that is how you get the area of a rectangle. Then I kept thinking about the triangle on my geoboard which was a right triangle and seen that if I made another triangle just like it, it would make a rectangle. Knowing how to find the area of the rectangle, I knew that if two triangles made one rectangle, then one half of the rectangle would be equal to the area of the triangle. So, I found the area of the rectangle and then cut it in half.
Student 9: Hey, that is almost the same way I did it, but I started off with the parallelogram I used from the last question that was still on my geoboard and cut it in half and realized I could get two equal triangles out of it. Knowing the area of the parallelogram I just divided it by 2 and got my answer.
Student 7: I can see that. I would have never of thought about doing it that way. I was too caught up in trying to figure out some other, more harder way. Hey let me try doing it that way.
Student 10: I’ll give you a hand, here let me show you.
Teacher: Why don’t you (Student 10) come up to the board and show the class.
Student 10: Sure.
(Student 10 continues to explain the process she went through to figure out how to find the area of the triangle on her geoboard.)
Teacher: That is wonderful. I want the rest of you to make a note of your strategy and when we come back tomorrow we will begin there. Before class is over I want you to think about this question and write a few things about it in your math journals – how is a parallelogram and a triangle alike? Different?
Students were given about five minutes to jot down a few ideas and then preceded on to the next class.
So, what is happening in Classroom Two? Well, the students are talking more, the teacher is talking and telling less, the students are constructing meaning through problem solving and coming up with their own personal strategies and methods to solve the problem. Students are learning from one another and they are not expected to simply memorize the formula or procedure. The focus of this classroom is on meaning and understanding, whereas in Classroom One, the focus was on getting the right answer and applying the correct formula. Classroom Two illustrates a reform approach to teaching and learning math. It is here that students learn the intended mathematical concepts through solving problems and discussions. The teacher’s role is to use appropriate questioning to guide student thinking to bring them to an appropriate understanding of concepts taught and to make meaningful connections among the various mathematical strands.
Teacher: (drawing a right triangle on the board) Who can tell me the name of this triangle?
Student 1: A right triangle.
Teacher: That is right, it is a right triangle because it has one right angle. We have been talking about finding the area of rectangles, and parallelograms and now I am going to show you how to find the area of a triangle. Any questions?
(No response)
Teacher: Great, so let’s move on and dig right in. Can anyone remember the formula to finding the area of a parallelogram?
Student 2: Length times width.
Teacher: Not quite, that is the area of a rectangle. What particular words am I looking for to describe the area of a parallelogram? Baaa times heee…
Student 3: Is it base times height?
Teacher: (feeling rather pleased) You are correct. Now let’s look at finding the area of a triangle. This is the base of a triangle (pointing to the particular line that represents the base) and this is the height, (indicating the imaginary line from the peak of the triangle to the base). To find the area you simply multiply the base and the height and then divide it by two. Any questions?
Student 1: Miss, so all you have to do is get the base and multiply it by the height and then half it?
Teacher: (again feeling rather pleased that the student ‘got it’) You are absolutely right. Let’s try an example and I’ll show you how it is done.
Student 3: Why do you divide it by two?
Teacher: Just watch and I will show you how to do it.
The teacher then continues to draw several examples of different triangles on the board and finds the area by applying the formula. Students are quietly copying examples in their note books. Once the teacher feels “all students” have got it, she assigns questions 1-17 from the next lesson entitled “Area of Triangles” in the student math book. She announces that any unfinished work is to be completed for homework.
The classroom described above, unfortunately, can be from any classroom or school around the country. This type of teaching and learning can be defined as ‘traditional’, where the teacher’s role is to transmit information to the student who in turn memorizes the procedure and applies the skill to repeating tasks of similar nature. It is no secret that this method of teaching and learning has proved disadvantageous and researchers of today are suggesting a different way, a more constructivist way, of teaching and learning math.
Let’s consider the next classroom.
Classroom Two:
Teacher: O.k, class, over the past few days we have been exploring how to find the area of different rectangles and parallelograms using various methods. Using what you have learned previously, I want you to create and find the area of any rectangle or parallelogram.
(Students begin working and can be seen discussing their creations with other students. During this time the teacher can be seen roaming around the room talking with individual and pairs of students about their shapes and asking them to explain how there were able to find the area, where any types of misconceptions were noted).
Teacher: (after about 15 minutes have passed) Who would like to come to the front of the class to show your shape and explain how you went about finding the area?
Student 1: (in front of the class) Well, the first thing I did was made a parallelogram on the geoboard (using the overhead goeboard, he reproduces the parallelogram Then I counted how many blocks the length was, or the base. It was five (again demonstrating how this is done). I then counted how many blocks high the parallelogram was (demonstrating how this was counted) and it is 6. I know a parallelogram is the same as a rectangle only squished a little, so I made this rectangle, a 5 x 6, out of snap cubes and then counted how many cubes I used. I used 30 snap cubes so I know the area of my parallelogram is going to be 30 units squared.
Teacher: Are there any questions or comments for Student 1? Would someone like to explain in their own words what Student 1 did to find the area?”
Student 3: Well, basically all he did was made a parallelogram on the geoboard, found the base and the height and then multiplied them together.
Student 4: Yeah, but he did not really use multiplication. He made a rectangle out of snap cubes and then counted how many blocks he used.
Student 3: Yes, but really he used multiplication to find the area because when you make a rectangle like he did out of cubes you are really making an array of 5 x 6. And 5 groups of 6 is 5, 10, 15, 20, 25, 30. See?
Student 4: Oh yeah, you are right. I can see that now. I never looked at doing this in that way before.
Student 1: Me either. I now see how you can use multiplication to find the area of the parallelogram. The parallelogram is the same as a rectangle, only it is slanted. So if you can find the area of a rectangle by multiplying length by width, than it would work the same for a parallelogram, only the names of length and width are different. Cool! Thanks Student 3.
Student 3: No problem!
Teacher: So now that we have seen a couple of different ways of getting the area of a parallelogram, I want you to check Student 1’s answer by using your method to find the area of a parallelogram, check to see if you get the same answer.
Students apply their own strategies to check the answer as the teacher walks around the room to check other people’s methods and understanding.
Teacher: So, what can we say about student 1’s answer? How can you be sure?
Student 7: Well, I drew the same parallelogram on graph paper and counted the complete units and got 30 units squared, just like student 1.
Teacher: So, we have all had a chance to come up with different ways to find the area of a rectangle and a parallelogram, all of which a perfectly right. There were a lot of great discussions happening in your groups and I am glad to see that you are helping each other. You are all becoming very great problem solvers. So, I now want you to put your thinking skills to the test. I want you to explore and find out as much as you can about the area of a triangle. I will give you about a half an hour and then we will come back as a whole class to discuss your findings.
Students began exploring the idea of trying to find the area of a triangle by using different manipulatives and through discussions with other students and the teacher. The teacher met with different students, asking questions to help them through their thinking. By the end of the half an hour, there were some pretty different discoveries made about triangles whereby all students were able to find out something about the area of a triangle and most had discovered strategies that worked for them to find the area.
Teacher: Now, that we have all had some time exploring the area of a triangle, let’s talk about what we have found out.
Student 7: Well, I found out that if I drew the triangle on a piece of grid paper, it was a little hard to count the number of squares because there are a lot of little pieces here and there.
Student 10: Yeah, I did the same thing but then I decided to make a triangle on the geoboard. I looked at it for a while and thought that maybe you would multiply the base and height but realized this would not work because that is how you get the area of a rectangle. Then I kept thinking about the triangle on my geoboard which was a right triangle and seen that if I made another triangle just like it, it would make a rectangle. Knowing how to find the area of the rectangle, I knew that if two triangles made one rectangle, then one half of the rectangle would be equal to the area of the triangle. So, I found the area of the rectangle and then cut it in half.
Student 9: Hey, that is almost the same way I did it, but I started off with the parallelogram I used from the last question that was still on my geoboard and cut it in half and realized I could get two equal triangles out of it. Knowing the area of the parallelogram I just divided it by 2 and got my answer.
Student 7: I can see that. I would have never of thought about doing it that way. I was too caught up in trying to figure out some other, more harder way. Hey let me try doing it that way.
Student 10: I’ll give you a hand, here let me show you.
Teacher: Why don’t you (Student 10) come up to the board and show the class.
Student 10: Sure.
(Student 10 continues to explain the process she went through to figure out how to find the area of the triangle on her geoboard.)
Teacher: That is wonderful. I want the rest of you to make a note of your strategy and when we come back tomorrow we will begin there. Before class is over I want you to think about this question and write a few things about it in your math journals – how is a parallelogram and a triangle alike? Different?
Students were given about five minutes to jot down a few ideas and then preceded on to the next class.
So, what is happening in Classroom Two? Well, the students are talking more, the teacher is talking and telling less, the students are constructing meaning through problem solving and coming up with their own personal strategies and methods to solve the problem. Students are learning from one another and they are not expected to simply memorize the formula or procedure. The focus of this classroom is on meaning and understanding, whereas in Classroom One, the focus was on getting the right answer and applying the correct formula. Classroom Two illustrates a reform approach to teaching and learning math. It is here that students learn the intended mathematical concepts through solving problems and discussions. The teacher’s role is to use appropriate questioning to guide student thinking to bring them to an appropriate understanding of concepts taught and to make meaningful connections among the various mathematical strands.
What Came Next?
Well it is now February and I am still learning, and thankfully, so are my students. After the requirements of my course was complete, I decided that blogging my trials and tribulations was a great way to get this out to my fellow collegues. However, there was a part of me that needed to bring all of my findings together into one place and really get my head wrapped around where I was professionally. So, I wrote a paper. It is way too big to post in one posting, so I will do it in several.
After I had my paper wrote and read it, that is when I came up with a title, and i think it kind of says it all:
Providing Access For All Through Problem Solving
So here is how it begins;
It is a fact that most American and Canadian math classrooms rid students of a natural tendency to be curious, to make sense of things and to understand them. Are you paying attention? According to Boaler (2008) before children enter school they are natural problem solvers. She goes on to state “many studies have shown that students are better at solving problems before they attend math class.” (p.43) Even at a young age, children reason through problems, using different methods in creative ways. Boaler (2008) suggests that after spending countless hours in a classroom where students are learning math in a passive way, their problem solving abilities are drained out of them. They think they need to memorize how to do math and forget about trying to make sense of it in order to follow these procedures.
After I had my paper wrote and read it, that is when I came up with a title, and i think it kind of says it all:
Providing Access For All Through Problem Solving
So here is how it begins;
It is a fact that most American and Canadian math classrooms rid students of a natural tendency to be curious, to make sense of things and to understand them. Are you paying attention? According to Boaler (2008) before children enter school they are natural problem solvers. She goes on to state “many studies have shown that students are better at solving problems before they attend math class.” (p.43) Even at a young age, children reason through problems, using different methods in creative ways. Boaler (2008) suggests that after spending countless hours in a classroom where students are learning math in a passive way, their problem solving abilities are drained out of them. They think they need to memorize how to do math and forget about trying to make sense of it in order to follow these procedures.
Wednesday, November 26, 2008
What's Next?
November 27th, 2008
Well, I have come to the last few days of my official blogging. (My course is over today!) I have enjoyed this experience as I find it very helpful to reflect, in writing, on my practices and the ups and downs I have experienced in this whirlwind journey. I am no where near where I want to be in terms of teaching through problem solving, but I feel I have come so far since I began. I am still working through some things such as how to ask the right questions, especially in class when you are on the spot teaching, and how to plan for this. Marian Small talks a lot about planning for questioning and has done some research on how to plan your math lessons to ask the right questions. This is something I will do more research on. I also feel I do not always know how to bring students where they need to be in terms of attainment and achievement of outcomes. I am resisting the urge of just telling them, but there are times when I do not know where to go from there.
The biggest change I want to see is for me to throw away the actual units of study that is now guiding me and helping me to choose investigations, and go toward a more holistic approach where I am not doing problems involving number patterns for two weeks, and then geometry for five and then on to something else. I want to be able to pose problems that does not necessarily achieve outcome 1, 2 and 3, in unit 4, for example, but pose problems in a way that at the end of the year students will have achieved all outcomes. In thinking about the logistics of this, I wonder then how would my assessment be different. How would I go about communicating student progress to home? Again, something to think about.
As I continue to learn how to teach math through problem solving, I know I will face more challenges, and more hiccups. I am confident I will work my way through these problems and be a better teacher for it. My students are doing very well thus far, and have come a very long way since September. They are becoming more independent and starting to pose their own questions more frequently. I want them to become better questioners and become more confident in themselves and their abilities to do math. I will continue to blog the happenings in my class, and please feel free to comment when you feel necessary. I would love to know what others think about what I am doing and how I am doing it. Please contact me with any questions or comments. I would love to hear from you.
Problem Solving at it's Finest
November 24th
After spending a few days talking about area and perimeter, I decided it was time to move on to getting students to understand the connections between length, width and area and how one influences the other. At the beginning of class I wrote two questions on the board, the first, a review of what was taught in the last few days, and the second, something a little different. Here are the two questions:
Mrs. John said that she had two different sized patios built onto her house, but they both had the same area. Can Mrs. John be correct? Prove it!
Mrs. John’s desk is the same width as Alex’s desk, but Alex’s desk is three times longer than Mrs. John’s. What can you say about the area of both desks? Use pictures, words and numbers to explain your thinking.
Question number one was just meant to be a means of assessment of the previously taught material and to get a sense of student reasoning and communication.
I would like to spend a few minutes talking about question two. Before today’s class, I did not do any work involving the influence of changing the dimensions on the area of the shape. We did, however, look at what happens to one of the dimensions when there is a fixed perimeter and the other dimension changes in the Changing Garden’s problem. I was hoping students would think back to this problem to help them solve the current one.
As usual, I spent a few moments discussing the context of the problem ensuring everyone knew what they had to do. I demonstrated, using my desk, that the widths of the two desks were the same, but the other desk, Alex’s desk was three times longer than mine. I entertained any questions and then let them go.
I was very pleased to see that all students were able to infer that three times the length required them to multiply the length of my desk by three to obtain the length of Alex’s desk. I seen all of the students using diagrams to sort out their thinking, which was a breakthrough for me as this is something, using illustrations to help solve the problem, I have been emphasizing all year. By using these diagrams students then decided to find the area of the two rectangles. Three students automatically seen the connection that the area was going to triple. They were so excited about this discovery that they went ahead and began questioning what would happen if they doubled both the length and the width. Of course, I let them go ahead and investigate that. The other students kept working on this and many, but not all, seen that the area tripled. The class ended with two students not being able to complete the question and it was left for next class.
I feel this illustrates learning math through problem solving at its finest. I did not tell students what was going to happen, nor did I lead them through conversing or probing or questioning. They discovered these connections on their own and when asked what would happen if the width, for example, tripled, they were quick to tell me that the area would also triple. I then knew they got it. But then I wanted to see how well they actually knew it, namely why the area was doubled when the length doubles. This will be the focus of my next lesson.
Discovery
November 21st, 2008
Today, I began a new unit on geometry. To begin, I posed questions to find out what they knew about length, perimeter and area of rectangles by using a problem called Changing Gardens. This problem can be found in Navigating through Measurement in Grades 3-5. (see reference list for details.) This required students to recognize that when given a fixed perimeter, there are different possibilities for the dimensions of the rectangle, leading them to recognize that the areas can change. At the end of the activity, there were several open ended questions that helped to lead students to make the necessary connections about length, width, perimeter and area of rectangles. Student’s responses to these questions enable me to plan the next activity for this unit. From these questions, I saw that there were still a few students whose ideas about area and perimeter were sketchy. Their concept of area and perimeter were not concrete, not where I thought they should be at grade six, and as a result, I knew I could not proceed in the way I had intended. Instead, I would focus my next lesson(s) on doing activities that would reinforce the concept of area and perimeter.
If I to give students a page or two, asking them to look at a figure, maybe a square or rectangle, and tell me what the area and perimeter was, I am confident they would be able to do this. This would lead me to believe, and themselves, that they understood what these concepts were. Hey, there is no real thinking involved, once you know how to do it. It wasn’t until I asked them to describe how they knew the area they had calculated was correct, or why, when the length of the rectangle decrease, the width had to increase with a fixed perimeter, that their ‘real’ understanding of these concepts came to light.
What would have happened if I did not give them these types of questions, or had them complete this activity? I would have continued with the lessons as planned and assumed that because they got the right answer; they knew what was meant by these terms. They would still be playing the game, and I would not have known the difference.
Communicationing and Refining
Nov. 17th, 2008
Student understanding of mathematical concepts is what teachers strive for, or at least, what teachers should strive for. Classrooms that see students interacting, communicating and helping each other solve problems, is what I would argue, a classroom where students actually understand the mathematical concepts being taught. Marian Small also agrees. Small (2008) notes that “it is through interactions with other students as well as with the teacher, and with the opportunity to articulate their own thoughts that students are able to construct new mathematical knowledge” (p.4). Small (2008) goes on to note that it is in these classrooms, one in which she calls constructivists classrooms, “that students are given the opportunity to develop richer and deeper cognitive structures related to mathematical ideas and students’ development of a level of mathematical autonomy”(p.4).
Often times, what you read in a book as theory, may seem really out to lunch. “yeah, it may be alright for them to say that, or do that in their classroom, but it would never work for me.” Well, I would like to demonstrate what this looks like as I take you, once again inside my classroom.
Please keep in mind we have only been engaged in learning mathematics through problem solving for a little less than two and a half months. The particular class I am about to describe was a concluding class after doing investigations with using patterns to explore division of 0.1, 0.01 and 0.001.
The question was posed at the beginning of class that went something like this : Frank divided 42.8 by 0.1 and got an answer between three and found hundred. How do you know his answer is incorrect? Show your thinking using pictures, numbers and words.
I talked a little about the problem, making sure everyone understood what the question was asking. I then let them go and begin solving the problem. What happened next, illustrates what Marian Small has just described.
Students began talking through the problem with their partners, they used things like Base Ten blocks, calculators (yes, I allow them to use calculators), graph paper and pictures to begin making sense of the problem. Most, if not all used Base Ten blocks to work out their thinking. I was hearing students correcting others thinking and explaining how they know in their own ways, talking about their own strategies in solving the problem. Once I seen that everyone had some type of answer written, I brought the class together as a whole and began discussing what they had found out.
I began by asking students to explain the problem, telling me what was required. I asked particular students to tell me how they thought about the problem. One student said she thought of it as money and you wanting to split $42.80 into dimes. Another student said that they thought of it by using Base Ten blocks and said that because they were dividing the number (42.8) into tenths, they would use a rod as a whole one and the unit block would represent a tenth. At this point I seen some students were a little lost, so I stopped. I asked another student, the one with a blank face, to explain what was just said in his own words. He began trying to make sense of what the other student was saying and as he was playing with the base ten blocks in his hands, things were beginning to make sense. Other students were helping this process of understanding evolve, understanding the concept of using a unit to represent one tenth and the rod would then equal one whole. With the support of the class and the use of the base ten blocks and conversations, the reluctant student understood why a unit was used as a tenth to represent this problem. He was then able to recognize that he would need 42 rods and 8 units to represent this number. I got him to come to the overhead (which is a part of every math class) and show the class what he had just discovered. But there was a problem. This student quickly realized there was not 42 overhead Base Ten blocks to put on the overhead. He looked at me so innocently and said, “but Miss, there’s not enough.” I laughed and said, you are right, so what are you going to do about it. He quickly recovered and said, well I know that a there are 10 rods in a flat, so I could use 4 flats. He then places 4 flats on the overhead. I asked him to look at the number he had on the overhead and say what it was. He said 400. Before I got a chance to respond, another student quickly corrected him and explained to him where he made the mistake. She pointed out that because he was using the unit as a tenth, the rod had to be a ‘one’ and the flat was now 10, so four flats would be 40. (Bingo, I now know that she understood the concept!) Then the reluctant student, who is now demonstrating a better understanding of this place value concept, placed two more rods and 8 units on the overhead and confidently said, “there you go miss, 42 and 8 tenths.” I looked at him as seriously as I could and said, prove it to me. After a little look, he went into detail about how he was able to come up with this arrangement. Once his explanation was complete, I looked at the class and asked if anyone agreed with him or if anyone would like to challenge him. I could see that everyone had the same base ten arrangement and they all agreed that this student’s explanation was correct.
So, from here, to ensure they really understood this, I posed the question: Why are we using the unit block as one tenth and why the rod would be equal to one? One of my weaker students, looked at me and simply told me because we are breaking the number into tenths miss, and a unit is one tenth of a rod, so that means all we have to do it count how many units or tenths there are and we have our answer. Well, I was a little blown away, to say the least. If my weakest student could think of it in that way, I knew something right was happening.
To keep the momentum going, I then asked the class, what would Frank’s answer be if he did divide 42.8 by 0.1. At once, they said 428. Eureka! One student stated,’ well miss we knew it was going to be that because dividing by tenths is just like multiplying by ten and 42.8 ‘times’ ten is 428. Horray!!
For the sake of time and space, I did not include all the conversations that was had during this class. More students were asked to come to the overhead in front of the class and explain how they went about solving the problem, or to prove that what was just said was indeed, true. The point I want to make is that through much interactions, conversations and student talk, students were given the opportunity to think about this problem, to talk about the problem and to talk through the problem. We used students’ lack of understanding as a springboard to help them, and the rest of the class, to have a deeper understanding of the problem and the concept being taught. I, as their teacher, did not intervene or disrupt this thinking, even if it was incorrect. They worked together to collaboratively ‘fix’ any misconceptions about this concept and used their own invented strategies and thinking to explain how they seen it to others. This type of classroom can be achieved, and this is proof that teaching through problem solving do work.
Reasoning and Communication
November 13th
Today I posed two different questions, an open and a closed. I wanted to once again see how these two questions compared in terms of what information it gave me about student understanding. I am noticing also, that open questions are giving me a lot of assessment information on student reasoning and communication skills, something that closed questions do not give me. Here are the two questions:
Mrs. J has a cell phone. She has a plan that allows her to talk for 1000 minutes a month. She gets her bill at the end of the month, and as she is scanning through, she spills coffee on it, blurring out some of the numbers. She notices that she talked for 345 minutes the first week, 210 minutes the second week, and 534 minutes the fourth week. The amount of minutes she was on her cell phone for the third week was missing. How many minutes was Mrs. J talking during week three? If she pays 5 cents a minute for her phone, how much was her bill?
#2. Mrs. J’s cell phone plan allows her to talk for 1200 minutes. As she is scanning her bill she notices that she talked the most in week four, the least in week two and about the same in weeks one and three. If Mrs. J has about 100 minutes left at the end of the month, what are some of the possible times Mrs. J could have talked in those four weeks? Mrs. J’s cell phone bill totaled 129.95. Knowing that she only pays 5 cents a minute, how did she know this was not correct? Explain.
I took about ten minutes to explain the two problems and entertained any questions students had. I gave students the entire class to work on the problems, where I collected their work and then compared their answers. As expected, the open question tasks allowed students greater access to the problem as every one of them was able to come up with appropriate answers, some even went as far to play with the numbers to see how many different combinations they could come up with. Students also did well with the closed question, but four out of ten students did not come up with an appropriate answer to the problem. Many, perhaps six of them had difficulty deciding what it was they had to do and they became frustrated, which may have contributed to the incorrect solutions. Using the open questioning technique, it required students to show me how they went about figuring out the problem. As a result of this, I was able to assess their level of thinking, reasoning and communication. I found that only three of my ten students were performing at a level four based on the rubric used in the CRT’s. All other students were scoring a level three with one student scoring a level two. This concerned me, but after doing this activity, I now see where I have to focus my teaching for this time being. I need to work with these 7 students to help them build their communication and reasoning skills. I intend to do this by using more open ended questions having the content scaled down a little so they will only need to focus on communicating their thinking and not on how to actually do the problem. This type of information would not have been gathered, nor would I have been able tole to realize I needed to work more closely with seven of my students to improve their reasoning and communication skills, if I had to only rely on the closed question.
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