Time - Match Clocks: http://nlvm.usu.edu/en/nav/frames_asid_317_g_1_t_4.html?from=grade_g_1.html
In this applet, students change the time on a digital clock to match the time on an analog clock, and change the time on an analog clock to match the digital clock. The objectives are that students will recognize that both representations show the same time and that students will read and create the time correctly on both types of clocks. It is easy to use, although I question that PreK students would be able to meet the objectives. The visual presentation is helpful, in that the clocks sit side-by-side where students can easily see the relationship between them. Once students have plenty of practice reading one type of clock, they are challenged to switch gears and read the next type. This could be difficult for some students. However, the activity really does not present sufficient challenges for first and second grade gifted students.
The information provided for the students is sufficient, since they are not at an age where they read proficiently and would not appreciate wordy directions. However, there is no information for teachers. Since the activity is straitforward, teachers should be able to use the applet to reinforce the relationships between the two representations of time, though. It promotes growth in understanding time, but does not require higher order thinking. Instead, he students practice one concept over and over. This repeated practice may reinforce what students have already learned, but does not push them to higher levels.
Monday, February 15, 2010
Pattern Blocks
Pattern Blocks: http://nlvm.usu.edu/en/nav/frames_asid_169_g_1_t_3.html?open=activities&from=grade_g_1.html
This applet designed for PK-2 grade students deals with making and describing patterns. Students drag pattern block shapes into a working area, creating a design of their choice. After playing with the shapes, the teacher instructs the students to make a wall with squares, a star with diamonds, and other simple shapes. While the names of the shapes are reinforced through the teacher's instructions, the teacher gradually guides students to create patterns based on variable names, like AAA, ABA, ABCCABCC. It is easy to drag the shapes, but PK-kindergarten students may have trouble rotating the shapes to fit into small areas. The visual display is attractive and the applet could keep students engaged. The challenges presented may vary depending on age level. For PK-K, simply learning the names of the shapes could be a challenge. For first and second graders, creating patterns in response to a teacher's use of variables may be a challenge.
The information presented to the students in this applet is focused strictly on shapes. Without teacher instruction, the applet might be a fun activity for children, but of little mathematical value. A simple lesson plan is provided for teachers to guide students to using mathematical discourse. The activity could potentially help students gain understanding of using different representations for the same concept (using letters instead of shapes to convey a pattern). It also provides early introduction to variables - using letters in place of a concept, albeit not a number in this case. It does not have much potential for improving problem solving skills, so would quickly bore a gifted student. The teacher could adapt the lesson, but the nature of the activity does not lend itself to problem solving, reasoning, and proof.
This applet designed for PK-2 grade students deals with making and describing patterns. Students drag pattern block shapes into a working area, creating a design of their choice. After playing with the shapes, the teacher instructs the students to make a wall with squares, a star with diamonds, and other simple shapes. While the names of the shapes are reinforced through the teacher's instructions, the teacher gradually guides students to create patterns based on variable names, like AAA, ABA, ABCCABCC. It is easy to drag the shapes, but PK-kindergarten students may have trouble rotating the shapes to fit into small areas. The visual display is attractive and the applet could keep students engaged. The challenges presented may vary depending on age level. For PK-K, simply learning the names of the shapes could be a challenge. For first and second graders, creating patterns in response to a teacher's use of variables may be a challenge.
The information presented to the students in this applet is focused strictly on shapes. Without teacher instruction, the applet might be a fun activity for children, but of little mathematical value. A simple lesson plan is provided for teachers to guide students to using mathematical discourse. The activity could potentially help students gain understanding of using different representations for the same concept (using letters instead of shapes to convey a pattern). It also provides early introduction to variables - using letters in place of a concept, albeit not a number in this case. It does not have much potential for improving problem solving skills, so would quickly bore a gifted student. The teacher could adapt the lesson, but the nature of the activity does not lend itself to problem solving, reasoning, and proof.
Wednesday, February 10, 2010
Mathematical Discourse
"Techniques for Small Group Discourse" from Teaching Children Mathematics
The subject of this article is using mathematical discourse in classrooms, including discourse between teachers and students and between students and other students. Two case studies highlighted two different teachers' responses to their students, who had just worked on problem solving in small groups. Some of the teachers' comments were helpful, in that they facilitated mathematical discourse, critical thinking and responsibility for learning; others of their comments hindered these qualities. The helpful responses from the teachers included requiring explanations from the students to help them reason through the math processes on their own instead of relying on the teacher. They also listened to the students' justifications, but did not evaluate them. Instead, one of the teachers in particular, turned to another group and encouraged their input. This promoted more mathematical discourse, while helping the students to take ideas from each other to note their own errors. One unhealthy response given by one of the teachers was that she did not allow her student to struggle and struggle with her own thought process. Instead, she jumped in and redirected her thinking so that she then figured out the answer using the teacher's preferred method. Not only did this hinder deep understanding contrived from the student's own reasoning, it halted mathematical discourse that could have stretched the student's understanding.
I definitely agree with the purpose of the article - that mathematical discourse improves student understanding and that teachers should be facilitators. It is my hope that I will allow time for my future students to engage in conversation and exchange ideas when working through math problems. Teaching math in this manner may not unveil immediate leaps in testing, but in the long run, I am sure that student understanding and achievement will increase.
The subject of this article is using mathematical discourse in classrooms, including discourse between teachers and students and between students and other students. Two case studies highlighted two different teachers' responses to their students, who had just worked on problem solving in small groups. Some of the teachers' comments were helpful, in that they facilitated mathematical discourse, critical thinking and responsibility for learning; others of their comments hindered these qualities. The helpful responses from the teachers included requiring explanations from the students to help them reason through the math processes on their own instead of relying on the teacher. They also listened to the students' justifications, but did not evaluate them. Instead, one of the teachers in particular, turned to another group and encouraged their input. This promoted more mathematical discourse, while helping the students to take ideas from each other to note their own errors. One unhealthy response given by one of the teachers was that she did not allow her student to struggle and struggle with her own thought process. Instead, she jumped in and redirected her thinking so that she then figured out the answer using the teacher's preferred method. Not only did this hinder deep understanding contrived from the student's own reasoning, it halted mathematical discourse that could have stretched the student's understanding.
I definitely agree with the purpose of the article - that mathematical discourse improves student understanding and that teachers should be facilitators. It is my hope that I will allow time for my future students to engage in conversation and exchange ideas when working through math problems. Teaching math in this manner may not unveil immediate leaps in testing, but in the long run, I am sure that student understanding and achievement will increase.
Dividing Fractions
Summary of "Dividing Fractions and Problem Solving" from Mathematics teaching in the Middle School
This article outlines how teachers helped a class of sixth graders comprehend the algorithm for dividing fractions (invert and multiply). In most textbooks, the article states, fraction division is explained with one or maybe two pictures showing why the problem turns out a larger answer than the whole (1/2 divided by 6 equals 12). Some students may gain a superficial understanding of dividing fractions from these pictures, but most of the time, the vague examples go over their heads as the teacher dives directly into teaching the algorithm. Unfortunately, many students are detached from any problems dealing with fractional division because it makes no sense to them - memorizing the algorithm becomes their crutch. In this lesson, however, students solve problems using their own pictorial representations. Not only do they solve one or two problems, they solve many problems and explain their answers verbally. After sharing together as a class, the students begin to see patterns in their math processes. They naturally find the common denominators between the numbers they are working with, and naturally note correlations between the written symbols and their pictorial representations.
At first, I wondered how students could really comprehend fractional division. It seems like such an abstract concept, and looking back on my own education, memorizing the algorithm worked great for me. I never struggled with dividing fractions. It was easy - just invert and multiply! But as I continued reading and computing the problems pictorially as the children in the article did, I began to notice patterns that I had never realized before. In addition, for the first time I felt like I understood why we invert and multiply. Working through the problems definitely assisted my own comprehension. I can only imagine how helpful it must be for a sixth grader to build understanding of fractional division through personally created representations. Indeed, I will keep this article and use it in my future classroom. Not only will I use the problems given to work on, I will use the method of teaching fractional division presented in this article. In so doing, I will be addressing many of the mathematical process standards: the students will create, use, and translate representations; they will communicate mathematically in both written and verbal contexts; they will problem solve by determining what the part and the whole is and what they are searching for; and they will reason and prove their answers with their representations and their words.
This article outlines how teachers helped a class of sixth graders comprehend the algorithm for dividing fractions (invert and multiply). In most textbooks, the article states, fraction division is explained with one or maybe two pictures showing why the problem turns out a larger answer than the whole (1/2 divided by 6 equals 12). Some students may gain a superficial understanding of dividing fractions from these pictures, but most of the time, the vague examples go over their heads as the teacher dives directly into teaching the algorithm. Unfortunately, many students are detached from any problems dealing with fractional division because it makes no sense to them - memorizing the algorithm becomes their crutch. In this lesson, however, students solve problems using their own pictorial representations. Not only do they solve one or two problems, they solve many problems and explain their answers verbally. After sharing together as a class, the students begin to see patterns in their math processes. They naturally find the common denominators between the numbers they are working with, and naturally note correlations between the written symbols and their pictorial representations.
At first, I wondered how students could really comprehend fractional division. It seems like such an abstract concept, and looking back on my own education, memorizing the algorithm worked great for me. I never struggled with dividing fractions. It was easy - just invert and multiply! But as I continued reading and computing the problems pictorially as the children in the article did, I began to notice patterns that I had never realized before. In addition, for the first time I felt like I understood why we invert and multiply. Working through the problems definitely assisted my own comprehension. I can only imagine how helpful it must be for a sixth grader to build understanding of fractional division through personally created representations. Indeed, I will keep this article and use it in my future classroom. Not only will I use the problems given to work on, I will use the method of teaching fractional division presented in this article. In so doing, I will be addressing many of the mathematical process standards: the students will create, use, and translate representations; they will communicate mathematically in both written and verbal contexts; they will problem solve by determining what the part and the whole is and what they are searching for; and they will reason and prove their answers with their representations and their words.
Wednesday, February 3, 2010
PBL Comparison
Lounging Around - Summary
In this PBL lesson, students will design a lounge for their school. This problem is quite broad, especially since the students must keep within their budget. The problem is open-ended, with no one right answer, and the students must first decide what "mini problems" are within the big problem. The lesson includes guiding questions that lead students to connect with prior knowledge. However, their prior knowledge is not sufficient to resolve the many details of the project. They must then inquire about their options, which leads to mini lessons. The mini lessons are student centered, in that they must research and collaborate to find information (like looking through the classified ads for a job). After connecting to prior knowledge, learning from a number of mini lessons, and collaborating with classmates, students do further research via technology and finally, synthesize their data and information to create their plan.
Lounging Around - Critique
I have learned that PBLs should be open, ill-structured problems with more than one possible solution. This problem reflects that characteristic, as well as the need to first define the problem and second, devise a plan of action. PBLs are also supposed to involve higher order thinking, which is encouraged through the provided teaching questions. One aspect of the lesson that I do not recall reading about in PBLs is having a large number of mini lessons. The mini lessons in this PBL are great lessons, but they really seem to guide the students down one certain path. In reality, I think it makes sense for the teacher's questions to guide the students to certain concepts. I just do not know if true PBLs allow the course to go in several directions.
Recycling At Our School - Summary
In this PBL lesson, 3rd, 4th, or 5th grade students will start a recycling program at their school. The students must decide what they will recycle and how much they can recycle. The teacher will ask guiding questions that lead them in the direction of finding volume for the storage containers. They also must learn how much they will earn from recycling and estimate how much can be recycled in a week's time. The students will compile all of their information into a report and present it to the class to finish their project.
Recycling At Our School - Critique
This lesson started out with a very open, broad and messy question - just like PBLs are meant to originate. However, the teachers provide more structure at the beginning than I remember reading about in articles about PBLs. For example, right from the beginning, the teacher dictated the process they should take, instead of asking them questions that lead them to come up with their own process. Also, the mini lessons were not as inquiry based as a PBL lesson would have. The volume lesson is teacher directed rather than student-derived. The lesson on fractions was better developed because the students helped discover the fractions instead of the teacher just telling them how to find the answer. They also experienced modeling, guided practice, and understanding checks. The independent practice did not seem to fit PBL because it was not relevant to the students' lives.
Comparison of the 2 PBLs
"Lounging Around" fit PBL criteria very closely. It was student centered, inductive, involved the students' prior knowledge, and collaborative. I would feel comfortable using this plan in my classroom. "Recycling At Our School" was not as student centered because the teacher instructed more and asked fewer guiding questions. The teacher was still "in charge" of the project, while the students took possession of "Lounging Around" and the teacher simply facilitated. Also, "Recycling At Our School" involved less of the students' prior knowledge during the mini lessons. The teacher told them the new information instead of guiding them to discovering that information through prior knowledge.
Changes to "Recycling At Our School"
I would have begun this PBL by making the topic more personal to the students. 3rd, 4th, and 5th graders only care about recycling because they know that their teachers want them to care. I would ask them questions like, "How much garbage do you think each of us makes each week? Each month? Each year?" "How much garbage would it take to fill this classroom? How long would it take?" When the students realize how quickly their classroom would fill up if all of the students' garbage was dumped in the room, they might start to get the picture of why recycling is so important. In the volume mini lesson, I would have led my students to access their prior knowledge - perhaps their knowledge of lengths or of area. I would have had tools available for the students to figure volume such as large, empty boxes, yard sticks, base ten blocks - anything that would help realize the concept of volume. I would not make them use the tools, but they would be available to the students.
Math as the focus
I feel that math is definitely the focus in "Lounging Around". Already on the first day, students are in groups thinking mathematically. The guiding questions direct their minds to graphing and money. On day two, they take their mathematical thinking a step deeper, applying prior knowledge and choosing the best representations for displaying their polling data. Each day, the students apply prior knowledge to seek solutions to their problems, and when they learn a new concept like creating excel charts, the students synthesize prior knowledge with new knowledge. The knowledge they access each day is always mathematical.
Math is not always the focu in "Recycling At Our School". The opening activity was a quiz testing students' prior knowledge about the recycling problem. They were indeed accessing prior knowledge, but that knowledge was not mathematical. Most of the daily activities involved math, however. On day 7, they have to use estimation to figure how much they can recycle each day.
Assessment
"Lounging Around" had a variety of assessment methods - checklists, teacher observations, and rubrics. The final rubric was thorough and clear, assessing mathematical concepts such as representation (excel chart and scale) and communication (expectation that students can express learning verbally).
"Recycling At Our School" assessed daily assignments for completion and accuracy, but not for mathematical reasoning. The final assessment focused largely on mathematical computations, like correct volumes and estimations. It did, however, assess representations, reflections, and reasoning. The assessment was clear and thorough.
In this PBL lesson, students will design a lounge for their school. This problem is quite broad, especially since the students must keep within their budget. The problem is open-ended, with no one right answer, and the students must first decide what "mini problems" are within the big problem. The lesson includes guiding questions that lead students to connect with prior knowledge. However, their prior knowledge is not sufficient to resolve the many details of the project. They must then inquire about their options, which leads to mini lessons. The mini lessons are student centered, in that they must research and collaborate to find information (like looking through the classified ads for a job). After connecting to prior knowledge, learning from a number of mini lessons, and collaborating with classmates, students do further research via technology and finally, synthesize their data and information to create their plan.
Lounging Around - Critique
I have learned that PBLs should be open, ill-structured problems with more than one possible solution. This problem reflects that characteristic, as well as the need to first define the problem and second, devise a plan of action. PBLs are also supposed to involve higher order thinking, which is encouraged through the provided teaching questions. One aspect of the lesson that I do not recall reading about in PBLs is having a large number of mini lessons. The mini lessons in this PBL are great lessons, but they really seem to guide the students down one certain path. In reality, I think it makes sense for the teacher's questions to guide the students to certain concepts. I just do not know if true PBLs allow the course to go in several directions.
Recycling At Our School - Summary
In this PBL lesson, 3rd, 4th, or 5th grade students will start a recycling program at their school. The students must decide what they will recycle and how much they can recycle. The teacher will ask guiding questions that lead them in the direction of finding volume for the storage containers. They also must learn how much they will earn from recycling and estimate how much can be recycled in a week's time. The students will compile all of their information into a report and present it to the class to finish their project.
Recycling At Our School - Critique
This lesson started out with a very open, broad and messy question - just like PBLs are meant to originate. However, the teachers provide more structure at the beginning than I remember reading about in articles about PBLs. For example, right from the beginning, the teacher dictated the process they should take, instead of asking them questions that lead them to come up with their own process. Also, the mini lessons were not as inquiry based as a PBL lesson would have. The volume lesson is teacher directed rather than student-derived. The lesson on fractions was better developed because the students helped discover the fractions instead of the teacher just telling them how to find the answer. They also experienced modeling, guided practice, and understanding checks. The independent practice did not seem to fit PBL because it was not relevant to the students' lives.
Comparison of the 2 PBLs
"Lounging Around" fit PBL criteria very closely. It was student centered, inductive, involved the students' prior knowledge, and collaborative. I would feel comfortable using this plan in my classroom. "Recycling At Our School" was not as student centered because the teacher instructed more and asked fewer guiding questions. The teacher was still "in charge" of the project, while the students took possession of "Lounging Around" and the teacher simply facilitated. Also, "Recycling At Our School" involved less of the students' prior knowledge during the mini lessons. The teacher told them the new information instead of guiding them to discovering that information through prior knowledge.
Changes to "Recycling At Our School"
I would have begun this PBL by making the topic more personal to the students. 3rd, 4th, and 5th graders only care about recycling because they know that their teachers want them to care. I would ask them questions like, "How much garbage do you think each of us makes each week? Each month? Each year?" "How much garbage would it take to fill this classroom? How long would it take?" When the students realize how quickly their classroom would fill up if all of the students' garbage was dumped in the room, they might start to get the picture of why recycling is so important. In the volume mini lesson, I would have led my students to access their prior knowledge - perhaps their knowledge of lengths or of area. I would have had tools available for the students to figure volume such as large, empty boxes, yard sticks, base ten blocks - anything that would help realize the concept of volume. I would not make them use the tools, but they would be available to the students.
Math as the focus
I feel that math is definitely the focus in "Lounging Around". Already on the first day, students are in groups thinking mathematically. The guiding questions direct their minds to graphing and money. On day two, they take their mathematical thinking a step deeper, applying prior knowledge and choosing the best representations for displaying their polling data. Each day, the students apply prior knowledge to seek solutions to their problems, and when they learn a new concept like creating excel charts, the students synthesize prior knowledge with new knowledge. The knowledge they access each day is always mathematical.
Math is not always the focu in "Recycling At Our School". The opening activity was a quiz testing students' prior knowledge about the recycling problem. They were indeed accessing prior knowledge, but that knowledge was not mathematical. Most of the daily activities involved math, however. On day 7, they have to use estimation to figure how much they can recycle each day.
Assessment
"Lounging Around" had a variety of assessment methods - checklists, teacher observations, and rubrics. The final rubric was thorough and clear, assessing mathematical concepts such as representation (excel chart and scale) and communication (expectation that students can express learning verbally).
"Recycling At Our School" assessed daily assignments for completion and accuracy, but not for mathematical reasoning. The final assessment focused largely on mathematical computations, like correct volumes and estimations. It did, however, assess representations, reflections, and reasoning. The assessment was clear and thorough.
Stoller PBL Article
In the article, "What Is Problem Based Learning?" Dr. De Gallow of The University of California outlines the outstanding points of PBLs, addresses critics' concerns, and explains how and why it is important to incorporate PBL in the classroom. PBL main points are similar to the Best Practices of Education. They include the following characteristics: student centered and experiential, inductive, builds on prior knowledge, context specific, complex and require metacognition, creates cognitive conflict, and collaborative. Although critics suggest that students do not know what is important for them to learn, and therefore may not reach their cognitive potential, Dr. De Gallow asserts that research shows learning occurring at a deeper level with PBL. He then provides specific tactics for teachers to use in their classes which will help them make PBL successful. For example, to create a student centered problem, teachers should create authentic problems that relate directly to the students' lives.
Reading the articles and documents about PBLs, I think that this method of "teaching as a facilitator" is the ideal way to teach. It is backed by research in the characteristics it promotes, especially the higher order thinking that must take place in order to solve the problem. Instruction could be differentiated easily so that all students can reach their personal potentials. It would be motivating to students as the problems relate more closely to their lives than traditional problems. Finally, it addresses the top complaints from employers about college graduates - poor verbal, problem solving, and collaborative skills. The only concern I would have is the practicality of implementing PBL in upper grades where the students are not used to that style of learning. I think it is important and should be used, but it would take a few years for students to adjust to the different approach. It would be much better if teachers would use PBL beginning in the earliest grades and throughout the upper grades.
De Gallow. (n.d.). What is problem based learning? Retrieved from http://www.pbl.uci.edu/whatispbl.html
Reading the articles and documents about PBLs, I think that this method of "teaching as a facilitator" is the ideal way to teach. It is backed by research in the characteristics it promotes, especially the higher order thinking that must take place in order to solve the problem. Instruction could be differentiated easily so that all students can reach their personal potentials. It would be motivating to students as the problems relate more closely to their lives than traditional problems. Finally, it addresses the top complaints from employers about college graduates - poor verbal, problem solving, and collaborative skills. The only concern I would have is the practicality of implementing PBL in upper grades where the students are not used to that style of learning. I think it is important and should be used, but it would take a few years for students to adjust to the different approach. It would be much better if teachers would use PBL beginning in the earliest grades and throughout the upper grades.
De Gallow. (n.d.). What is problem based learning? Retrieved from http://www.pbl.uci.edu/whatispbl.html
Stoller: PBL - What is it?
Problem Based Learning (PBL) is a method of teaching math that involves a problem relevant and age appropriate to the students. The problem is truly a problem, in that it features more than one answer. Emphasis is placed on the process of solving the problem, and students work together with the teacher serving as a facilitator. The teacher asks questions to guide learning and initiates an ill-structured problem which the students claim as their own, personal problems. The students are encouraged to seek multiple answers, the final assessment being student reflections of themselves, their group, and their product.
PBLs require a few important details. First, the problem must be ill-structured (messy) so that students first need to define the problem based on what they know already and what they need to know. It forces them to organize and gather information before they can generate solutions and decide on the best solution.
PBLs require a few important details. First, the problem must be ill-structured (messy) so that students first need to define the problem based on what they know already and what they need to know. It forces them to organize and gather information before they can generate solutions and decide on the best solution.
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