Tuesday, February 10, 2009
READ THIS FIRST...
Teaching math for understanding has been a challenge for many years and is a challenge that still exists today. How do you teach mathematics in a way where all students understand concepts and not just merely memorizing a procedure or skill? Teaching Math 4 Understanding is a project that was undertaken by a grad student of Memorial University, Elaina Johnson, which investigates teaching math through a problem-based approach, which has been touted as a method of teaching mathematics for understanding. This blog is a product of this project and will be used as a means of communicating and highlighting the ups and downs, the observations, significant finding and events of using this approach in a Grade Six classroom at Bishop White School in Port Rexton, Newfoundland. Please note, the specific blog entries you will see here will not match the actual dates posted as the entries were entered after the dates indicated on the specific blog entry. This is due to the necessity of collecting data before any final product (the actual blog) could be created. It must be noted, however, the actual information within each posting is an accurate account of the processes, procedures, observations and findings that were completed at that time specified. It will be beneficial to the reader to begin reading the posts at the bottom of this blog, as the most recent entries are shown on top of the page. I have just recently included several entries that make up my paper that I wrote based on the findings from this experience. Hope you enjoy!
Concluding Remarks
Providing access to all in the mathematics classroom can be accomplished through problem solving. As students engage in worthwhile, rich, contextual math problems that have multiple entry points, and allow for various strategies to be used, all children can and will learn the intended math. Providing opportunities for children to talk through and about the math they are learning, facilitated by appropriate teacher questioning, are ingredients for a successful math classroom. Teachers are expected by law to teach all students in their classroom and give all students the opportunities to learn. This is becoming increasingly harder as today’s classrooms keep getting more diverse. Traditional methods of teaching, which emphasizes teachers showing and telling, or chalk and talk and then practice, have been proven to be ineffective in terms of allowing ALL students access to the curriculum and learning. It is time educators have a discussion on the important elements that make math more accessible to students, especially now with the introduction of the new curriculum. It is through these discussions that teachers will realize that students need to talk, teachers need to question without telling and most importantly, math needs to be taught through problem solving, where students are engaged in solving, personal, contextual and meaningful problems that involve the significant mathematical concepts, or the big ideas.
More problems
There are many different ways teachers can go about creating open ended tasks that engage and enrich the mathematical lives of students. One such suggestion, as outlined by Open Assess,,, (2008) is to get students to come up with or create a situation or an example that meets certain criteria. For example, if you want to assess student’s understanding on multiplication, as them to pose a problem that involves multiplying as a way to solve it. Tell students that hair grows 10 cm each year. Have them create a question that involves multiplication around this fact and have them solve it. Making students interested in what they are doing is over half the battle with teaching math. Think about the difference in these two questions and student’s interest level– ‘What is 45 x 6? How do you know?’ and ‘Choose two numbers to multiply. Talk about how you went about finding the product. Then tell three things about the product’. Students would probably be more willing to complete the second question just because they can choose the two numbers. Even if students choose two very simple numbers to multiply, teachers can use appropriate questioning techniques to upgrade this answer. Teachers will get more assessment data from this type of question than any type that is similar to the format of question one.
Open Assess… continues by suggesting another way open ended questions can be created is by asking students to explain why, or how they know something or someone is correct or incorrect. This could easily transform a traditional, procedural based question into an open one. Let look at the question – What is the perimeter of the room, given the dimensions? This could be turned around and look something like this – Mary said that when she found the area of the room, it was 62. Bill said that when he measured the room it was 36. Who is correct? How do you know? It is really fun when you make the teacher wrong and the student’s solution is the one that is correct!
Small (2008) suggests yet another way to create open ended tasks. Ask students to think about how things are similar and different, or the alike and different strategy. An example of this could be to ask students to tell you how a triangle is alike or different from a square, or how the number 35 is alike or different from 45.
Another effective strategy to creating open ended tasks, as Small (2008) suggests is to use the ‘use these digits’ strategy. Here you would ask students to create some type of situation of choice, addition, multiplication, where students would have to use specific numbers somewhere in their sentence to make it correct. For example, create a multiplication sentence where you would use the digits 2, 7, 1 and 15 somewhere in the question.
Asking students to solve the problem in more than one way is another way to create open ended tasks as suggested by open ass (2008). By doing this, you are requiring students to think of other ways to solve the problem. One thing to remember here, though, is to ensure there are more than one way to solve the problem and that students are able to come up with more than one possible way. Representing numbers, and asking student to show situations such a how to get a product and sum could be examples of how to use this particular strategy.
Small (2008) also suggests that by increasing or decreasing the value of the numbers in the various problems that are created can really open the problem up for a broad range of abilities. Making the value of the number greater than the initial problem would challenge students, whereas decreasing the value of a number may be just what someone else would need to access the situation.
Another engaging way to assess student learning through open ended tasks is to give students an answer, such as 45, and ask them what the question could be. This could turn boring fact questions, or outcomes, into interesting and engaging learning opportunities. Look at and think about the level of engagement, for example, between the following questions:
• What is 35 divided by 5? * The answer is 5. What is the question?
Students can also be engaged in open ended tasks when asked to tell or explain what it is they know about a certain number after being given certain criteria. For example, you may want to ask a student to tell you what they know about a number that is a three digit number whose digits add to 10. Then ask students to list all the possible things that must be true about the number.
Open Assess… continues by suggesting another way open ended questions can be created is by asking students to explain why, or how they know something or someone is correct or incorrect. This could easily transform a traditional, procedural based question into an open one. Let look at the question – What is the perimeter of the room, given the dimensions? This could be turned around and look something like this – Mary said that when she found the area of the room, it was 62. Bill said that when he measured the room it was 36. Who is correct? How do you know? It is really fun when you make the teacher wrong and the student’s solution is the one that is correct!
Small (2008) suggests yet another way to create open ended tasks. Ask students to think about how things are similar and different, or the alike and different strategy. An example of this could be to ask students to tell you how a triangle is alike or different from a square, or how the number 35 is alike or different from 45.
Another effective strategy to creating open ended tasks, as Small (2008) suggests is to use the ‘use these digits’ strategy. Here you would ask students to create some type of situation of choice, addition, multiplication, where students would have to use specific numbers somewhere in their sentence to make it correct. For example, create a multiplication sentence where you would use the digits 2, 7, 1 and 15 somewhere in the question.
Asking students to solve the problem in more than one way is another way to create open ended tasks as suggested by open ass (2008). By doing this, you are requiring students to think of other ways to solve the problem. One thing to remember here, though, is to ensure there are more than one way to solve the problem and that students are able to come up with more than one possible way. Representing numbers, and asking student to show situations such a how to get a product and sum could be examples of how to use this particular strategy.
Small (2008) also suggests that by increasing or decreasing the value of the numbers in the various problems that are created can really open the problem up for a broad range of abilities. Making the value of the number greater than the initial problem would challenge students, whereas decreasing the value of a number may be just what someone else would need to access the situation.
Another engaging way to assess student learning through open ended tasks is to give students an answer, such as 45, and ask them what the question could be. This could turn boring fact questions, or outcomes, into interesting and engaging learning opportunities. Look at and think about the level of engagement, for example, between the following questions:
• What is 35 divided by 5? * The answer is 5. What is the question?
Students can also be engaged in open ended tasks when asked to tell or explain what it is they know about a certain number after being given certain criteria. For example, you may want to ask a student to tell you what they know about a number that is a three digit number whose digits add to 10. Then ask students to list all the possible things that must be true about the number.
Some ideas
There are so many great math resources available, one being the actual resources that accommodates the curriculum. These resources, and those like it provide so many questions, but it is the job of the teacher to make these questions fit their class so every child can experience success and want to participate. Tasks must be engaging and provide interesting contexts to the students who will be involved in the working through the problem. Small (2008) suggests using the child’s personal world as a source of possible questions. Children’s literature could be a source whereby you create questions from student’s favorite stories or songs. Facts and everyday figures, such as the calendar, people facts, and world records could also be a place to start as you begin creating open ended, but engaging tasks. If we want kids to practice a skill it is not necessary to give them 120 practice questions. Ask a question that will peak their curiosity so they will want and need to solve the problem, while learning the intended outcomes.
Perhaps one of the most important things to remember when creating tasks that are accessible to all students, is to begin with the math and to think about the students involved. Have a clear idea in mind as to what students need to learn from the lesson. It is important to think about the big ideas, the concepts, not the procedures, or how to do’s. Also, it is important to know the students and what they already understand about the concept to be taught. The concept to be taught must be somewhat unfamiliar, but not out of reach. In order for learning to take place, students must be somewhat challenged. If these things are taken into consideration, it will be from here that a decision on a task can be made.
Open ended tasks that allow for differentiation in the classroom have a number of definite characteristics. Small (2008) identifies these characteristics in the following ways:
• The problem must allow students to connect new knowledge and ideas with old ones
• It can be solved in many different ways
• Has a variety of solutions
• It piques curiosity
• Is personal or has some relevance to the student
• Is creative
Perhaps one of the most important things to remember when creating tasks that are accessible to all students, is to begin with the math and to think about the students involved. Have a clear idea in mind as to what students need to learn from the lesson. It is important to think about the big ideas, the concepts, not the procedures, or how to do’s. Also, it is important to know the students and what they already understand about the concept to be taught. The concept to be taught must be somewhat unfamiliar, but not out of reach. In order for learning to take place, students must be somewhat challenged. If these things are taken into consideration, it will be from here that a decision on a task can be made.
Open ended tasks that allow for differentiation in the classroom have a number of definite characteristics. Small (2008) identifies these characteristics in the following ways:
• The problem must allow students to connect new knowledge and ideas with old ones
• It can be solved in many different ways
• Has a variety of solutions
• It piques curiosity
• Is personal or has some relevance to the student
• Is creative
Make math problematic
As we think about how to teach students mathematics, we should reflect upon the following questions: What is the purpose of teaching mathematics? Is it to have children memorize how to do something? Or is it to teach math so students can understand the concepts well enough to be able to apply them to other situations? Hopefully, you think the latter. Teaching math means we help students to understand the math they are learning, if there is no understanding going on, are they really learning? If a child can recite their multiplication facts, what does this tell us? Nothing, only they have a good memory. We need to teach for understanding and not merely for knowing how to do something. To teach for understanding, we need to create classrooms that promote this type of learning. Borgioli (2008) state that “in classrooms that promote teaching and learning mathematics with understanding (instead of memorizing abstract algorithms and mimicking teachers’ solution paths), students collaboratively engage in making sense of mathematical tasks in their own ways and communicate their thinking with one another.” (p.186) She then goes on to say that one of the three critical pieces to create this type of classroom is in the actual tasks that teachers pose or give to students. Borgiolo (2008) suggest that “the tasks be complex and problematic for the learner and involve meaningful problem solving as opposed to simple drill and practice exercises” (p.186).
Van de Walle and Lovin (2002) describes it best as they state “the single most important principle for improving the teaching of mathematics is to allow the subject of mathematics to be problematic for students. That is, students solve problems not to apply mathematics but to learn new mathematics. When students engage in well-chosen problem-based tasks and focus on the solution methods, what results is new a understanding of the mathematics embedded in the task. When students are actively looking for relationships, analyzing patterns, finding out which methods work and which don’t, justifying results, or evaluating and challenging the thoughts of others, they are necessarily and optimally engaging in reflective thought about the ideas involved”. (p.11)
Boaler (2002) states that “teachers who teach math through a problem based approach, believe that the open-ended approach they used was valuable for all students and that it was their job to make the work equitable accessible” (p.60). Students need to be able to have access to the problem and for them to enjoy success. Boaler (2002) states that students, who were taught in a problem based approach, had very different ideas about math, than those that learned math in the traditional way. Students who were taught through problem solving thought math was active, and “did not regard math to be a rule-bound subject involving set methods and procedures that they needed to learn” (p.77) All students need to think this way, so they can begin to apply what they know, and use this to construct new mathematical knowledge.
Engaging students in rich, contextual problems gives them something to talk about. These discussions can be prompted and guided by appropriate teacher questioning. One can argue that providing students with these worthwhile problems is the most important piece of this puzzle, and they are probably right. If students do not have something to talk about, they won’t. If there is no talking, then no type of questioning would spark their interest to have mathematical discussions. No talking, no learning, simple as that. So how is it done? How are these types of tasks created? What are the elements of these types of tasks that are open ended, having multiple entry points, so all students can experience success?
Van de Walle and Lovin (2002) describes it best as they state “the single most important principle for improving the teaching of mathematics is to allow the subject of mathematics to be problematic for students. That is, students solve problems not to apply mathematics but to learn new mathematics. When students engage in well-chosen problem-based tasks and focus on the solution methods, what results is new a understanding of the mathematics embedded in the task. When students are actively looking for relationships, analyzing patterns, finding out which methods work and which don’t, justifying results, or evaluating and challenging the thoughts of others, they are necessarily and optimally engaging in reflective thought about the ideas involved”. (p.11)
Boaler (2002) states that “teachers who teach math through a problem based approach, believe that the open-ended approach they used was valuable for all students and that it was their job to make the work equitable accessible” (p.60). Students need to be able to have access to the problem and for them to enjoy success. Boaler (2002) states that students, who were taught in a problem based approach, had very different ideas about math, than those that learned math in the traditional way. Students who were taught through problem solving thought math was active, and “did not regard math to be a rule-bound subject involving set methods and procedures that they needed to learn” (p.77) All students need to think this way, so they can begin to apply what they know, and use this to construct new mathematical knowledge.
Engaging students in rich, contextual problems gives them something to talk about. These discussions can be prompted and guided by appropriate teacher questioning. One can argue that providing students with these worthwhile problems is the most important piece of this puzzle, and they are probably right. If students do not have something to talk about, they won’t. If there is no talking, then no type of questioning would spark their interest to have mathematical discussions. No talking, no learning, simple as that. So how is it done? How are these types of tasks created? What are the elements of these types of tasks that are open ended, having multiple entry points, so all students can experience success?
Planning for end of class questions
Similarly, teachers also need to pre plan questions to actually end the lesson. If the last thing that is said or done in class is often what students remember the next day, then ending the class with a question for students to reflect on, find a solution to, or write about could be the most appropriate way to reinforce the newly acquired skills and concepts of the lesson. It is suggested that this type of question be posed in a way that requires students to produce some type of written answer. This ensures all students are engaged in the question, as many times when students are asked to answer orally, some students do not get a chance to become involved as they may tune out, or others answer before they get a chance to mentally understand what the question is asking. This could also take the form of self assessment, where students comment on their learning, explaining what it is they are learning, how they are making sense of it and most importantly, if and why they are finding it difficult. The hardest part of this is to manage time so this questioning can be done, as many times the bell rings even before the question is posed. Setting a timer, or having a student keep track of the time are possible suggestions to avoid this from happening.
Things to consider when questioning
As teachers find themselves in the middle of a lesson, there are other factors that need to be considered when engaging students in (verbal) open ended questions.
1. Do not lead students to the correct answer, only provide scaffolding when necessary. When told “Miss, I don’t know how to do this”, a teacher’s immediate response may be to tell the answer. Students can not learn by telling, they learn by doing. Asking students open questions such as finding out what they do know, followed by asking what it is that is making them stuck may help them to talk through their problems. Once students identify why they are stuck, the teacher can then decide the best action to take. If they are stuck because they do not know a particular skill, a quick mini-lesson may help. If students are stuck because of a lack of understanding of the concept being taught, a simpler version of the tasks can be provided. Students need to find success with solving the problems they work on; therefore it is essential to ensure the actual tasks fit the student.
2. Always answer a question with a question. Students ask questions when they either don’t know how to do something or they are unsure about what they are doing. They request an answer that will help them come to a solution. Students learn by doing, they learn by testing out their own conjectures and answering their own questions. Therefore, by answering student’s questions with appropriate teacher questions, students can learn how to eventually help themselves. Teachers may wish to begin asking students what they already know, and what is causing them trouble. As they talk through the problem and their thinking, they can be guided (and not told) how to proceed. This is hard for teaches to do as it is natural to ‘help’ students. This help traditionally have meant ‘telling’, therefore teachers must always be conscious of this strategy to ensure they are allowing all students a chance to become engaged in the math they are learning.
3. Never evaluate or confirm student answers. No matter how silly or outrageous a student response may be, always give the student a chance to explain his/her thinking. This is important to do as it is the teacher’s role to find out what it is that makes sense to the child and then build from there. If a student gives a wrong answer, they may feel upset and are likely to abandon the math they are working on. If they see that at least some of what they are saying or thinking is true, they will gain confidence, which is so important to math success.
4. Ask for others opinions before giving your own. As students offer solutions to a problem and explain it to the class, or the teacher, ask other students their opinions on the proposed answer. Have students use their own personal strategies to test out this solution to the problem. Then provide opportunities for these students to confirm or reject the solution, having them explain their reasoning. This not only creates a great conversation piece, it also allows students to see other ways of doing and thinking.
5. Be general with the questions, instead of being specific. For example, in a lesson where students are working on addition of two 2-digit numbers, a student has made the connection that the two numbers being added, the addends, need to be joined together. After representing these two separate addends using base ten blocks, the student decides to put all of the blocks together. As the student is trying to find the sum by counting all of the blocks, he/she comes to a point when they don’t know how to go from counting the rods to counting the units. The student asks the teacher how to continue counting. Using general questions, the teacher would ask the student to explain what s/he thinks s/he should do. Here the teacher can assess the students thinking and understanding of place value, an important concept in addition, and allow this response to guide further questioning. Other questions such as “Show me how you know”, “why do you think this?” “What would happen if…?” could also be used to help guide the student to solving the problem. Consider these questions the teacher could have asked “What should you do with the rods and the units? How many rods are here? How many units are here? If we are adding, what do we now have to do? If there are 6 rods, and each rod is 10, how much is here in rods? If there are 16 units, how many are there altogether? These types of questions would lead students to finding the right answer while applying the teacher’s method and not using their own. Therefore, these types of questions should be avoided.
1. Do not lead students to the correct answer, only provide scaffolding when necessary. When told “Miss, I don’t know how to do this”, a teacher’s immediate response may be to tell the answer. Students can not learn by telling, they learn by doing. Asking students open questions such as finding out what they do know, followed by asking what it is that is making them stuck may help them to talk through their problems. Once students identify why they are stuck, the teacher can then decide the best action to take. If they are stuck because they do not know a particular skill, a quick mini-lesson may help. If students are stuck because of a lack of understanding of the concept being taught, a simpler version of the tasks can be provided. Students need to find success with solving the problems they work on; therefore it is essential to ensure the actual tasks fit the student.
2. Always answer a question with a question. Students ask questions when they either don’t know how to do something or they are unsure about what they are doing. They request an answer that will help them come to a solution. Students learn by doing, they learn by testing out their own conjectures and answering their own questions. Therefore, by answering student’s questions with appropriate teacher questions, students can learn how to eventually help themselves. Teachers may wish to begin asking students what they already know, and what is causing them trouble. As they talk through the problem and their thinking, they can be guided (and not told) how to proceed. This is hard for teaches to do as it is natural to ‘help’ students. This help traditionally have meant ‘telling’, therefore teachers must always be conscious of this strategy to ensure they are allowing all students a chance to become engaged in the math they are learning.
3. Never evaluate or confirm student answers. No matter how silly or outrageous a student response may be, always give the student a chance to explain his/her thinking. This is important to do as it is the teacher’s role to find out what it is that makes sense to the child and then build from there. If a student gives a wrong answer, they may feel upset and are likely to abandon the math they are working on. If they see that at least some of what they are saying or thinking is true, they will gain confidence, which is so important to math success.
4. Ask for others opinions before giving your own. As students offer solutions to a problem and explain it to the class, or the teacher, ask other students their opinions on the proposed answer. Have students use their own personal strategies to test out this solution to the problem. Then provide opportunities for these students to confirm or reject the solution, having them explain their reasoning. This not only creates a great conversation piece, it also allows students to see other ways of doing and thinking.
5. Be general with the questions, instead of being specific. For example, in a lesson where students are working on addition of two 2-digit numbers, a student has made the connection that the two numbers being added, the addends, need to be joined together. After representing these two separate addends using base ten blocks, the student decides to put all of the blocks together. As the student is trying to find the sum by counting all of the blocks, he/she comes to a point when they don’t know how to go from counting the rods to counting the units. The student asks the teacher how to continue counting. Using general questions, the teacher would ask the student to explain what s/he thinks s/he should do. Here the teacher can assess the students thinking and understanding of place value, an important concept in addition, and allow this response to guide further questioning. Other questions such as “Show me how you know”, “why do you think this?” “What would happen if…?” could also be used to help guide the student to solving the problem. Consider these questions the teacher could have asked “What should you do with the rods and the units? How many rods are here? How many units are here? If we are adding, what do we now have to do? If there are 6 rods, and each rod is 10, how much is here in rods? If there are 16 units, how many are there altogether? These types of questions would lead students to finding the right answer while applying the teacher’s method and not using their own. Therefore, these types of questions should be avoided.
Planning questions
If using open questioning can give teachers a truer sense of student understanding and can provide differentiation for all students, why are teachers not doing it? Small (2008) notes that many times during the school day, as teachers are questioning students, they don’t pre-plan the questions they use and this often leads to over-scaffolding. When students answer a question incorrectly, teachers tend to ask simpler questions until students get the right answer. Teachers can not do this if the intent is to create students that are critical thinkers and good problem solvers. Questions are often used to check understanding rather than to start a conversation. Small (2008) suggests that because of this, teachers don’t always make sure the questions are broad enough to allow multiple entry points to allow all students the opportunity to access the math.
To make mathematics classroom more accessible to all students, and to make all students feel that it is o.k. to speak in class, Small (2008) suggests teaches pre-plan questions to ensure the questions asked are open, or broad enough to allow all students the opportunity to enter into the question and become engaged in the content and context of the problem. By pre-planning the questions asked in math class, teachers are more apt to ask the appropriate questions that will allow all students to engage. It is a good idea to think about and write specific open ended questions for the lesson to be taught. These questions need to focus on the main mathematical idea to be taught in the lesson, help guide student learning, and help refocus students thinking if inappropriate conclusions are made.
When planning questions, it is necessary to identify the important questions to ask in each phase of the lesson, namely how the lesson will begin, during, and end. To begin a class, teachers need to think about the purpose of the class. If introducing a new concept, maybe a question to assess student prior knowledge is needed. Another option is to pose the question to have students dive in to get them thinking about the new concept to be taught. Or it could be to assess previous learning. Questions such as “What do you know about…? How is _____ related to ______? How can we use _____ to help us find _____? Tell me everything you have learned about _____? can be used in these situations.
Just as questions can be planned for the beginning of a lesson, teachers should also think about questions they could ask during the lesson. These types of questions can be asked to keep the lesson going in rich directions. To do this, teachers may wish to think about how students will react to and answer the questions posed at the beginning of the lesson. Recording possible reactions and answers to each planned question, and subsequent open questions could be one way teachers could ensure his/her questions are open and accessible to all. Pre-planning questions and anticipating student response to questions decreases the chances of asking the typical ‘rapid fire’, closed questions that are found in a traditional classroom. Again, such questions could take the form of asking students what they think they should do, why they think that way and after offering a solution, asking them how they went about getting that answer. As students answer these types of questions, the teacher can help them move forward, at the same time help them understand math makes sense and that they can reason their way through any type of problem.
To make mathematics classroom more accessible to all students, and to make all students feel that it is o.k. to speak in class, Small (2008) suggests teaches pre-plan questions to ensure the questions asked are open, or broad enough to allow all students the opportunity to enter into the question and become engaged in the content and context of the problem. By pre-planning the questions asked in math class, teachers are more apt to ask the appropriate questions that will allow all students to engage. It is a good idea to think about and write specific open ended questions for the lesson to be taught. These questions need to focus on the main mathematical idea to be taught in the lesson, help guide student learning, and help refocus students thinking if inappropriate conclusions are made.
When planning questions, it is necessary to identify the important questions to ask in each phase of the lesson, namely how the lesson will begin, during, and end. To begin a class, teachers need to think about the purpose of the class. If introducing a new concept, maybe a question to assess student prior knowledge is needed. Another option is to pose the question to have students dive in to get them thinking about the new concept to be taught. Or it could be to assess previous learning. Questions such as “What do you know about…? How is _____ related to ______? How can we use _____ to help us find _____? Tell me everything you have learned about _____? can be used in these situations.
Just as questions can be planned for the beginning of a lesson, teachers should also think about questions they could ask during the lesson. These types of questions can be asked to keep the lesson going in rich directions. To do this, teachers may wish to think about how students will react to and answer the questions posed at the beginning of the lesson. Recording possible reactions and answers to each planned question, and subsequent open questions could be one way teachers could ensure his/her questions are open and accessible to all. Pre-planning questions and anticipating student response to questions decreases the chances of asking the typical ‘rapid fire’, closed questions that are found in a traditional classroom. Again, such questions could take the form of asking students what they think they should do, why they think that way and after offering a solution, asking them how they went about getting that answer. As students answer these types of questions, the teacher can help them move forward, at the same time help them understand math makes sense and that they can reason their way through any type of problem.
Questioning
Van de Walle and Lovin (2006) suggest that teachers “must transform classroom into a mathematical community of learners where student’s learning is enhanced by the reflective thought that social interactions promote.” (p.5) These social interactions, the conversations based on reflective thought, need to be focused on the intended mathematical concepts and it is through questioning that teachers can bring the conversation back on track. Good questions are those that give the teacher access to children’s mathematical thoughts, which will allow for supporting their development. As the teacher in the excerpt realized, the conversations about multiplication was not where s/he wanted it to go. S/he used guiding questions and comments to bring the conversation to a place where students were required to extend their thinking about what multiplication was, to how it could be used in different situations. Students need to be asked questions that lead to good conversations. Without thinking about it, questioning is something teachers do everyday. The reason for doing this may include to assess student thinking or to stimulate it. For this reason, it is important to ask the right types of questions that will allow each child in all classes the same opportunities to access the math and learning. There are particular questions that foster this communication and reasoning, namely open-ended questions, and it is these types of questions that we need to ask more often to open our classroom to allow all students to enter the world of mathematics.
Small (2008) defines an open ended question as a question “that can be approached very differently, but meaningfully, by students at different developmental levels” (p.642). She goes on to suggest that one of the key features of open questions that can allow for this differentiation is the fact that there are many different possible answers and many ways to get these answers. When students are asked open-ended questions, such as “What can you tell me about 13X52?, teachers are able to get insight into how students think about this question and what they know about the mathematics that surround it. When asking such a question, the desired answer is not for students to tell us the exact product, but to see how they go about thinking about what that product may be. Each and every student in this particular class would be able to offer some type of solution, even if it is something like the product will be greater than 100. The beauty of this question is that there is no one right answer or one right way of coming up with the solution, which is what makes it open ended. Teachers may wish to use scaffolding to make it even more accessible to all students as they could use different numbers, such as 1 digit numbers, then 1 and 2-digit number and so on, to help the student finally have some kind of success with the original question. More information about how to make questions more accessible to students will follow.
As teachers verbally ask more open ended questions, they get a chance to listen to students as they develop their own methods of doing. Students are asked to show their thinking and their way of knowing. In a more traditional classroom, where the main form of communication is in writing, students get really good at making teachers believe they understand something, and teachers get really good at believing them. If students were asked to calculate the product of 13 and 52 (a closed–ended question requiring only one correct answer), they may or may not get the answer correct. If they didn’t get it right, teachers would be lead to believe they don’t know how to multiply two two-digit numbers, when in fact they really did. Or, similarly, they may have gotten the product right, which would lead teachers into thinking they understood multiplication and can demonstrate this understanding by getting the product right, which again, could be the furthest from the truth. Through the use of open ended questioning student deception would be a lot more difficult. Therefore, teachers must ensure their questions do not lead to deception, as this is harmful to both teacher and student.
Small (2008) defines an open ended question as a question “that can be approached very differently, but meaningfully, by students at different developmental levels” (p.642). She goes on to suggest that one of the key features of open questions that can allow for this differentiation is the fact that there are many different possible answers and many ways to get these answers. When students are asked open-ended questions, such as “What can you tell me about 13X52?, teachers are able to get insight into how students think about this question and what they know about the mathematics that surround it. When asking such a question, the desired answer is not for students to tell us the exact product, but to see how they go about thinking about what that product may be. Each and every student in this particular class would be able to offer some type of solution, even if it is something like the product will be greater than 100. The beauty of this question is that there is no one right answer or one right way of coming up with the solution, which is what makes it open ended. Teachers may wish to use scaffolding to make it even more accessible to all students as they could use different numbers, such as 1 digit numbers, then 1 and 2-digit number and so on, to help the student finally have some kind of success with the original question. More information about how to make questions more accessible to students will follow.
As teachers verbally ask more open ended questions, they get a chance to listen to students as they develop their own methods of doing. Students are asked to show their thinking and their way of knowing. In a more traditional classroom, where the main form of communication is in writing, students get really good at making teachers believe they understand something, and teachers get really good at believing them. If students were asked to calculate the product of 13 and 52 (a closed–ended question requiring only one correct answer), they may or may not get the answer correct. If they didn’t get it right, teachers would be lead to believe they don’t know how to multiply two two-digit numbers, when in fact they really did. Or, similarly, they may have gotten the product right, which would lead teachers into thinking they understood multiplication and can demonstrate this understanding by getting the product right, which again, could be the furthest from the truth. Through the use of open ended questioning student deception would be a lot more difficult. Therefore, teachers must ensure their questions do not lead to deception, as this is harmful to both teacher and student.
The types of talk is important
It is clear that having students talk during math class is important. The types of conversations students have are just as, if not, the most important part. Students may be talking about math and about solving problems, but this talk may not be of any use to helping them construct the appropriate knowledge they need to make the necessary connections to acquire the conceptual understandings. This can be clearly illustrated in the excerpt below taken from a teacher’s blog, noting the ineffectiveness of student conversation about multiplication. In this example, students were talking about mathematical content but not in the way that allowed them to learn anything new.
October 21th, 2008
Today I wanted to get into multiplication strategies. I read a book entitled “Amanda’s Amazing Bean Dream” which is all about a girl, named Amanda, who loved to count everything and finally realizes that using multiplication can help her to count faster. A lot of my students have trouble with the whole concept of multiplication and I thought this would be a good place to start with getting their conceptual knowledge caught up with their procedures in multiplication. I read the book and we discussed different multiplication scenarios in the book. I used different open questions like asking particular students to look at the pictures and tell me a multiplication story and why it is multiplication. They, most, were getting good at that, and then I broadened it to telling me a multiplication story involving the classroom. Up until this point, I thought they all, finally, knew what multiplication meant. Until, they could not give me a multiplication scenario that involved them or the classroom. Oh, oh!
So what did I do? My mind went to Boaler’s book, specifically, to the exerpt by Sarah Flannery, about the importance of conversations in math. I was letting students talk about math, multiplication in this situation, but not in the right way. All they were doing really, was giving back to me what I had asked for, a simple multiplication sentence that involved a picture in a book which, on every page were similar. There were rows of books on so many shelves, how many books? There were rows of windows on so many floors, how many windows? There were 4 wheels on a bicycle, so many bicycles, how many wheels? There was no thinking needed. They could simply regurgitate the answer that I wanted.
So, I put the book down and we began talking. We talked about the book, but it wasn’t about the pictures of rows or columns, it was about the math that was going on there. We talked about how Amanda loved to count and how she did not want to use any other method of counting, only the way she already knew. We talked about counting and the different things that we counted each day, money, points in a game, step to the bus, minutes until the bell rings and so on. From there we talked about how as numbers started getting bigger Amanada was faced with the challenge of counting these larger numbers faster. We then talked about their experiences with wanting to count faster or using larger numbers. They came up with things like estimating how much money we spent on school supplies (this one made me happy as they were using one of their problem solving strategies – think of a similar problem), how many meals we have eaten in our life times and how many days we have been alive. (these last two made me happier as it gave me ideas for the next day) Then we talked about how Amanda finally realized that multiplying numbers together did make counting easier. This lead into a discussion about how in order to multiply you must have equal groups of something. I then asked the students to think of a multiplication question/situation that could be used in the classroom and asked them to write in on a piece of paper.
* Each and every one of those students was able to come up with an appropriate question that involved multiplication.
October 21th, 2008
Today I wanted to get into multiplication strategies. I read a book entitled “Amanda’s Amazing Bean Dream” which is all about a girl, named Amanda, who loved to count everything and finally realizes that using multiplication can help her to count faster. A lot of my students have trouble with the whole concept of multiplication and I thought this would be a good place to start with getting their conceptual knowledge caught up with their procedures in multiplication. I read the book and we discussed different multiplication scenarios in the book. I used different open questions like asking particular students to look at the pictures and tell me a multiplication story and why it is multiplication. They, most, were getting good at that, and then I broadened it to telling me a multiplication story involving the classroom. Up until this point, I thought they all, finally, knew what multiplication meant. Until, they could not give me a multiplication scenario that involved them or the classroom. Oh, oh!
So what did I do? My mind went to Boaler’s book, specifically, to the exerpt by Sarah Flannery, about the importance of conversations in math. I was letting students talk about math, multiplication in this situation, but not in the right way. All they were doing really, was giving back to me what I had asked for, a simple multiplication sentence that involved a picture in a book which, on every page were similar. There were rows of books on so many shelves, how many books? There were rows of windows on so many floors, how many windows? There were 4 wheels on a bicycle, so many bicycles, how many wheels? There was no thinking needed. They could simply regurgitate the answer that I wanted.
So, I put the book down and we began talking. We talked about the book, but it wasn’t about the pictures of rows or columns, it was about the math that was going on there. We talked about how Amanda loved to count and how she did not want to use any other method of counting, only the way she already knew. We talked about counting and the different things that we counted each day, money, points in a game, step to the bus, minutes until the bell rings and so on. From there we talked about how as numbers started getting bigger Amanada was faced with the challenge of counting these larger numbers faster. We then talked about their experiences with wanting to count faster or using larger numbers. They came up with things like estimating how much money we spent on school supplies (this one made me happy as they were using one of their problem solving strategies – think of a similar problem), how many meals we have eaten in our life times and how many days we have been alive. (these last two made me happier as it gave me ideas for the next day) Then we talked about how Amanda finally realized that multiplying numbers together did make counting easier. This lead into a discussion about how in order to multiply you must have equal groups of something. I then asked the students to think of a multiplication question/situation that could be used in the classroom and asked them to write in on a piece of paper.
* Each and every one of those students was able to come up with an appropriate question that involved multiplication.
How do we get students talking?
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.
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.
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.
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