Do you think students in elementary school, middle school, and/or high school are too reliant on calculators? What prompted me to ask this question is the fact that a teller at a grocery store needed the assistance of a calculator to tally up how much change she needed to give me because she entered the wrong amount into the cash register. Also, it is obvious that Walgreens does not trust its employees because it has automatic change machines. Is this just a great advancement in technology, a move to get cheaper unskilled labor, or a sign that people are unable to do basic math in their heads? If people cannot do basic math in their heads then who is to blame?
Marc
Saturday, January 30, 2010
Converting Units Virtual Manipulative: A Virtual Manipulative That Every Middle School Math Teacher Should Consider
Title of Activity: Converting Units
Topic: Measurement
Grade Range: 6 - 8
Link: Converting Units Virtual Manipulative
Even though unit conversions is typically a middle school math topic, students generally struggle with it mightily in my eighth grade Earth Science and high school biology classes. This could be for several reasons. First, middle school math teachers may be teaching unit conversions without using any kind of a “hook” or attention grabber. The attention grabber could be a demonstration of how the unit conversions could be used in science classes. In fact, unit conversions is a perfect topic for an interdisciplinary project. However, the application of math concepts in other classes is unlikely. During my administrative internship, I conducted a survey with eighth graders as the subjects. The purpose was to find out what percentage of students applied what they learned in seventh grade math in a class other than math class. Would you believe that only about 30% of students had actually applied some aspect of the mathematics that they learned in another class?
Since the mathematics concepts are generally not being applied in classes outside of math class, it is extremely important that the time spent in the classroom is optimized. Students are so accustomed to using technology to do just about everything that it makes sense to incorporate technology into lessons whenever possible. Considering the important applications of unit conversions in science classes, it is particularly important that we, as future math teachers, implement technology into lessons regarding unit conversions. Maybe this strategy would help teachers attain greater success rates of getting students to retain the concept of converting from one unit to another for later use in science classes.
Several specific aspects of the Converting Units virtual manipulative program make it very appealing for middle school math and science teachers. First, it provides students with multiple options, which makes them think. Even when the students determine the correct conversion factor to use they must make the decision of how to orient it. For example, students must ask themselves which units should be in the numerator and which units should be in the denominator. Second, the activity includes multi-step problems. Thus, students can solve real world problems like converting 1 ¼ miles to meters.
Another nice feature of the Converting Units virtual manipulative program is that it tells students if they solved a problem correctly. If a student answered the question incorrectly then the program informs the student of this and allows him or her to try again. The nice thing about this is that it makes the program formative in nature. Students can keep trying to solve the problems until they learn the correct method of converting from one unit to another. Thus, students will get the immediate feedback that they crave, and theoretically, every student should be able to walk away from the program with an understanding of how to convert from one unit to another.
As a teacher, one must always consider when one can use a specific activity. This activity is perfect for individual student practice directly after direct instruction on how to perform unit conversions. After completing the Converting Units virtual manipulative activity, students would then be prepared to tackle problems with paper and pencil. A homework assignment on the same topic would then provide students with an opportunity to check their understanding of the topic after the school day has ended.
However, for students to retain the information learned in the Converting Units virtual manipulative activity, they need to apply the concepts in a real world setting. This is where this virtual manipulative could actually be tied into an interdisciplinary lab-based activity. Students could use the unit conversions that they learn in math class to convert units of length during microscope analysis in science class.
Topic: Measurement
Grade Range: 6 - 8
Link: Converting Units Virtual Manipulative
Even though unit conversions is typically a middle school math topic, students generally struggle with it mightily in my eighth grade Earth Science and high school biology classes. This could be for several reasons. First, middle school math teachers may be teaching unit conversions without using any kind of a “hook” or attention grabber. The attention grabber could be a demonstration of how the unit conversions could be used in science classes. In fact, unit conversions is a perfect topic for an interdisciplinary project. However, the application of math concepts in other classes is unlikely. During my administrative internship, I conducted a survey with eighth graders as the subjects. The purpose was to find out what percentage of students applied what they learned in seventh grade math in a class other than math class. Would you believe that only about 30% of students had actually applied some aspect of the mathematics that they learned in another class?
Since the mathematics concepts are generally not being applied in classes outside of math class, it is extremely important that the time spent in the classroom is optimized. Students are so accustomed to using technology to do just about everything that it makes sense to incorporate technology into lessons whenever possible. Considering the important applications of unit conversions in science classes, it is particularly important that we, as future math teachers, implement technology into lessons regarding unit conversions. Maybe this strategy would help teachers attain greater success rates of getting students to retain the concept of converting from one unit to another for later use in science classes.
Several specific aspects of the Converting Units virtual manipulative program make it very appealing for middle school math and science teachers. First, it provides students with multiple options, which makes them think. Even when the students determine the correct conversion factor to use they must make the decision of how to orient it. For example, students must ask themselves which units should be in the numerator and which units should be in the denominator. Second, the activity includes multi-step problems. Thus, students can solve real world problems like converting 1 ¼ miles to meters.
Another nice feature of the Converting Units virtual manipulative program is that it tells students if they solved a problem correctly. If a student answered the question incorrectly then the program informs the student of this and allows him or her to try again. The nice thing about this is that it makes the program formative in nature. Students can keep trying to solve the problems until they learn the correct method of converting from one unit to another. Thus, students will get the immediate feedback that they crave, and theoretically, every student should be able to walk away from the program with an understanding of how to convert from one unit to another.
As a teacher, one must always consider when one can use a specific activity. This activity is perfect for individual student practice directly after direct instruction on how to perform unit conversions. After completing the Converting Units virtual manipulative activity, students would then be prepared to tackle problems with paper and pencil. A homework assignment on the same topic would then provide students with an opportunity to check their understanding of the topic after the school day has ended.
However, for students to retain the information learned in the Converting Units virtual manipulative activity, they need to apply the concepts in a real world setting. This is where this virtual manipulative could actually be tied into an interdisciplinary lab-based activity. Students could use the unit conversions that they learn in math class to convert units of length during microscope analysis in science class.
Private Universe Project in Mathematics Workshop 1: Following Children's Ideas in Mathematics
The Following Children's Ideas in Mathematics video opens with the following questions: “Are these ideas [the mathematical principles taught in school] extended in subsequent grades? Will the mathematics my students have learned help them in their careers?” These are two very important questions. The first question is based on the fact that students need reinforcement, and thus, it is important that what they learn is used in subsequent grades levels. In addition, what the students use should be expanded upon as they get older. However, it is probably the second question that is more important. It seems to be a question very similar to what Grant Wiggins and Jay McTighe would ask with regards to their Understanding by Design unit planning strategy. Determining if the mathematics being used has a purpose in life is very important because teachers need to be able to identify what concepts students will need to remember five, ten, or even twenty years later.
When watching first half of the video entitled “The Youngest Mathematicians”, I realized that revisiting the same mathematical ideas in subsequent grades was vital to a student’s growth in terms of mathematical thinking. What was most surprising to me was the amount of growth that some students could show in just a few months. For example, the young boy in the video named Michael learned to apply a strategy in third grade that his classmate had applied in second grade. Learning from one’s classmates is something that is typically absent in a traditional math class. Also, the number of students who drew lines to form combinations increased dramatically from second to third grade. This indicates how the degree of mathematical thinking as it pertains to multiplication increases dramatically from second to third grade with the non-traditional style of teaching in the video. In addition, I noticed that the teachers employed a “learning by doing” strategy in the video, which is a lot better than the simple rote memorization strategies that the teachers used to use before the standards movement.
When watching the second part of the video entitled “From Towers to High School”, it made me think about how important it is that we challenge students. When I look in mathematics classes I almost always see the same thing. First, the teacher goes over the homework. Then, the teacher lectures about something new. After that, the students are given some problems to work on that relate to the new learning. Lastly, the students are assigned homework that requires them to practice what they learned in class. There is very little creativity in this style of teaching. Plus, it is too teacher-centered. Students need to learn through inquiry, and this type of teaching does not meet this need. However, the video shows students working with Unix Cubes, which is certainly a student-centered activity. It appears that the students in the video were challenged from a very early age to think critically about problems and develop their mathematical thinking skills. In essence, the video presents a feasible alternative to teaching math using rote memorization and formulas.
The film ends with two questions. “In what ways are the students in the study similar to your students? How are they different?” Personally, I think they are very similar to my students. Just like the students in the video, my students learn best when they are involved in the learning process. Thus, my students learn best by performing labs, building models of cells, DNA, proteins, and other molecules, and participating in their own learning experiences. The students referred to in the video performed better in the non-traditional math classes, but when they were put in a geometry class that was taught in a traditional way they did not learn much. My students are the same way. They do not retain as much from lectures as they do from labs or hands-on activities. The one major difference between the students in the video and my students is that the students in the video worked in an inquiry-based mathematics curriculum from grades two to eight. My students likely did not always have teachers that taught them in this fashion, and thus, I had to train them in how to learn in an inquiry-based setting at the beginning of the year.
When watching first half of the video entitled “The Youngest Mathematicians”, I realized that revisiting the same mathematical ideas in subsequent grades was vital to a student’s growth in terms of mathematical thinking. What was most surprising to me was the amount of growth that some students could show in just a few months. For example, the young boy in the video named Michael learned to apply a strategy in third grade that his classmate had applied in second grade. Learning from one’s classmates is something that is typically absent in a traditional math class. Also, the number of students who drew lines to form combinations increased dramatically from second to third grade. This indicates how the degree of mathematical thinking as it pertains to multiplication increases dramatically from second to third grade with the non-traditional style of teaching in the video. In addition, I noticed that the teachers employed a “learning by doing” strategy in the video, which is a lot better than the simple rote memorization strategies that the teachers used to use before the standards movement.
When watching the second part of the video entitled “From Towers to High School”, it made me think about how important it is that we challenge students. When I look in mathematics classes I almost always see the same thing. First, the teacher goes over the homework. Then, the teacher lectures about something new. After that, the students are given some problems to work on that relate to the new learning. Lastly, the students are assigned homework that requires them to practice what they learned in class. There is very little creativity in this style of teaching. Plus, it is too teacher-centered. Students need to learn through inquiry, and this type of teaching does not meet this need. However, the video shows students working with Unix Cubes, which is certainly a student-centered activity. It appears that the students in the video were challenged from a very early age to think critically about problems and develop their mathematical thinking skills. In essence, the video presents a feasible alternative to teaching math using rote memorization and formulas.
The film ends with two questions. “In what ways are the students in the study similar to your students? How are they different?” Personally, I think they are very similar to my students. Just like the students in the video, my students learn best when they are involved in the learning process. Thus, my students learn best by performing labs, building models of cells, DNA, proteins, and other molecules, and participating in their own learning experiences. The students referred to in the video performed better in the non-traditional math classes, but when they were put in a geometry class that was taught in a traditional way they did not learn much. My students are the same way. They do not retain as much from lectures as they do from labs or hands-on activities. The one major difference between the students in the video and my students is that the students in the video worked in an inquiry-based mathematics curriculum from grades two to eight. My students likely did not always have teachers that taught them in this fashion, and thus, I had to train them in how to learn in an inquiry-based setting at the beginning of the year.
In essence, this video points out how important it is for students to be challenged at a young age in mathematics. It is so important for students to begin thinking mathematically as soon as possible. It is also extremely important that students build on the skills that they learn in one grade level in the following grade level. In essence, this film gives me a lot of ideas on how math should be taught.
My Girlfriend and the Four-Block Tower Problem
I decided to have my girlfriend complete the Unifix Cube four-block tower problem. I started by verbally giving her the instructions. I told her to create as many towers as possible with only four blocks per tower and any combination of the orange and blue blocks. I also explained that the orientation of the towers must be like that of a chimney with the narrower portion pointing up. She immediately asked if the towers could be all one color, and I responded that they could be.
Then, my girlfriend got to work. She began by forming a pattern of opposites, but she quickly switched to creating a pattern with four orange blocks, followed by three orange blocks on top and one blue block on the bottom, followed by two orange blocks on top and two blue blocks on the bottom, followed by one orange block on top and three blue blocks on the bottom, followed by four blue blocks, followed by three blue blocks on top and one orange block on the bottom, followed by two blue blocks on top and two orange blocks on the bottom, followed by one blue block on top and three orange blocks on the bottom. She then said, “I think I have all of the combinations.” I asked her to prove it. This made her think for a few moments. I gave her a hint that the order of the cubes matters. Her response was very interesting and enlightening. She said, “Your directions were not clear.” While I felt that my instructions were clear, this proves that it is essential to ensure that every student understands a problem. What sounds good in the teacher’s head may not make any sense to the students.
Then, with a puzzled look on her face, my girlfriend began working with the Unifix Cubes again. However, this time she switched back to a pattern of opposites. When asked why she kept switching from one strategy to another, she simply replied that she kept seeing the problem differently. After a few more minutes, my girlfriend claimed that she thought she had found all of the combinations. However, she could not prove it. I told her to continue working and find more. Thus, she went back to using the Unifix Cubes, and she found two more possible towers for a total of sixteen towers. However, once again, she was unable to prove that this was the correct answer.
Then, I tried to get my girlfriend to come up with an algebraic method for solving the four-cube tower problem. She said the answer of sixteen is based on the algebraic expression 42 because there are four cubes, two colors, and four times four equals sixteen. I asked her how she could prove this, and she said that she could try three blocks. She did this and came up with eight combinations. She then decided to try two-block towers, and she found that there were four combinations. This led her to the solution of C^n, with C representing the number of colors and n representing the number of blocks.
Obviously, proving one’s answer to be correct is higher-level thinking, which means that many students will find it challenging. I also know that if an adult struggled with convincing me that her answer was correct then a middle school student or high school student likely will struggle as well. I feel that the best thing to do is use the Socratic Method and ask students questions that lead them to the right answer or the right explanation. However, it is important not to simply give the students the answer because this defeats the purpose, which is to get students to think mathematically. In addition, students are less likely to understand how to solve a problem and retain any concepts learned if the teacher simply tells them the answers.
Based on my experience in class and this observation, I can clearly see that there are many ways to solve a problem like the four-cube tower problem. However, it does appear that people will search for recognizable patterns rather than simply guessing. This obviously is a time saving measure, and it brings some degree of organization to the problem solving process. Interestingly, my girlfriend kept switching from one pattern to another. What this tells me is that depending on what one discovers while working on a problem can dramatically alter the strategy used.
In conclusion, using Unifix Cubes seems to be a great way to get students interested in a problem that requires mathematical thinking. From my experience as a science teacher, I know that students love working with their hands. Plus, if they learn by discovery it is much more likely for the concepts to remain in their long-term memories. Thus, lessons that utilize Unifix Cubes are highly likely to be effective because students get to learn by doing and learn by discovery.
Then, my girlfriend got to work. She began by forming a pattern of opposites, but she quickly switched to creating a pattern with four orange blocks, followed by three orange blocks on top and one blue block on the bottom, followed by two orange blocks on top and two blue blocks on the bottom, followed by one orange block on top and three blue blocks on the bottom, followed by four blue blocks, followed by three blue blocks on top and one orange block on the bottom, followed by two blue blocks on top and two orange blocks on the bottom, followed by one blue block on top and three orange blocks on the bottom. She then said, “I think I have all of the combinations.” I asked her to prove it. This made her think for a few moments. I gave her a hint that the order of the cubes matters. Her response was very interesting and enlightening. She said, “Your directions were not clear.” While I felt that my instructions were clear, this proves that it is essential to ensure that every student understands a problem. What sounds good in the teacher’s head may not make any sense to the students.
Then, with a puzzled look on her face, my girlfriend began working with the Unifix Cubes again. However, this time she switched back to a pattern of opposites. When asked why she kept switching from one strategy to another, she simply replied that she kept seeing the problem differently. After a few more minutes, my girlfriend claimed that she thought she had found all of the combinations. However, she could not prove it. I told her to continue working and find more. Thus, she went back to using the Unifix Cubes, and she found two more possible towers for a total of sixteen towers. However, once again, she was unable to prove that this was the correct answer.
Then, I tried to get my girlfriend to come up with an algebraic method for solving the four-cube tower problem. She said the answer of sixteen is based on the algebraic expression 42 because there are four cubes, two colors, and four times four equals sixteen. I asked her how she could prove this, and she said that she could try three blocks. She did this and came up with eight combinations. She then decided to try two-block towers, and she found that there were four combinations. This led her to the solution of C^n, with C representing the number of colors and n representing the number of blocks.
Obviously, proving one’s answer to be correct is higher-level thinking, which means that many students will find it challenging. I also know that if an adult struggled with convincing me that her answer was correct then a middle school student or high school student likely will struggle as well. I feel that the best thing to do is use the Socratic Method and ask students questions that lead them to the right answer or the right explanation. However, it is important not to simply give the students the answer because this defeats the purpose, which is to get students to think mathematically. In addition, students are less likely to understand how to solve a problem and retain any concepts learned if the teacher simply tells them the answers.
Based on my experience in class and this observation, I can clearly see that there are many ways to solve a problem like the four-cube tower problem. However, it does appear that people will search for recognizable patterns rather than simply guessing. This obviously is a time saving measure, and it brings some degree of organization to the problem solving process. Interestingly, my girlfriend kept switching from one pattern to another. What this tells me is that depending on what one discovers while working on a problem can dramatically alter the strategy used.
In conclusion, using Unifix Cubes seems to be a great way to get students interested in a problem that requires mathematical thinking. From my experience as a science teacher, I know that students love working with their hands. Plus, if they learn by discovery it is much more likely for the concepts to remain in their long-term memories. Thus, lessons that utilize Unifix Cubes are highly likely to be effective because students get to learn by doing and learn by discovery.
Tuesday, January 26, 2010
If You Are Looking to Learn About Math Manipulatives Then Look No Further ...
Hi Fellow Math Manipulatives I Students,
Welcome to my blog. As we progress through the semester, I am sure you will find things that I post to be very interesting. Please feel free to use my ideas in your classrooms. There is no need to reinvent the wheel! I look forward to working with all of you and learning from you. I think this course will be fun, and we will all get a lot out of it. The course certainly follows the mantra that we learn best by doing. From personal experience, I know this is true in science. Now, I will find out if it applies to mathematics.
I will see you online and on Monday nights!
Marc
Welcome to my blog. As we progress through the semester, I am sure you will find things that I post to be very interesting. Please feel free to use my ideas in your classrooms. There is no need to reinvent the wheel! I look forward to working with all of you and learning from you. I think this course will be fun, and we will all get a lot out of it. The course certainly follows the mantra that we learn best by doing. From personal experience, I know this is true in science. Now, I will find out if it applies to mathematics.
I will see you online and on Monday nights!
Marc
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