Well class is coming to a close and I am working hard to put the finishing touches on all of my final assignments and projects. It is hard to believe that the semester is already over. For this final post I am going to reflect back on some of the things that I have learned and some of the things that I am going to continue to work on before I get a job teaching math. As I have progressed through this class I have learned many teaching strategies for math, the different ways that students learn math, and some teaching aids.
One of the main concepts that I feel was emphasized during the class was to have the students explain their thinking. One reason for this is to see where students are getting things mixed up and then being able to help them figure out a way that will work for them. Another good reason for having students do this is to show that there may be more than one correct way to get to the same answer. So many students have this misunderstanding that there is only one way to do the problem, instead of them finding a way that works for them. Finally, students may know what they are doing, but are making simple error in their computation. Having them describe what they are doing will show if they understand the concept.
A couple of the teaching strategies that I thought would be very useful were the verbalizing and then algebrifying exercise and the complexifying exercise. The first one had students verbalize what they were finding in a problem and then having them write an algebraic formula based on what they were describing. Students struggle with the idea of variables and this exercise can help them find a way to clarify this concept. The other exercise has students take a very simple function and then add to it to make it more complex. Students then trade their work with someone else to see if they can then simplify it back down to the original function. By seeing how to make the problem more complex, it is easier to see how to work back wards to make it simpler.
During class we were able to work with a few different types of manipulatives. I had never had the chance to work with these while I was in High School or Middle School, but I have learned they have many benefits to students. For your students that learner more visually or with a more hands on approach, these will be great for them. Even for the students that are at the top of the class, they will see some benefits as well. I will be making sure I have some of these for my classroom and I will continue to look for resources on how to use them and what concepts can be taught using them.
Technology is a good resource to use today and the two programs that we worked with in class will be great tools. GeoGebra and Desmos are another great way for students to experiment with a new concept and to get more visual learning. I will continue to work with both of these programs to see what they are capable of and to see what concepts can benefit from being taught or introduced using one of these programs.
The final thing that I will need look at are the common core standards and the state standards. I want to have a better understanding of these standards. As I am learning about the various requirements for each grade level and class I want to start compiling resources that can be used with each one. I am thinking that I can organize all of the materials that I find into google docs. As I learn about a new standard or find something useful that I can use later I can easily add it to my list.
This class has been a great resource for me and I have learned so much about how to teach math. It has started my journey at looking at how I will teach math if and when I get a job. I am excited to continue to learn and to find new resources.
Saturday, December 6, 2014
Tuesday, November 18, 2014
Algebra Tiles...Who Knew!
A while back in class we did a lesson involving algebra tiles. This was the first time that I had ever experienced this type of a manipulative. I had never used them before and after being able to experiment with them in class and to see what you could do with them I am a big fan. You can use them for modeling, simplifying, solving, factoring and much more. I have been looking online for resources for how to use algebra tiles and it seems like the possibilities are endless. I started by looking up the prices and found that a classroom set would cost about $80 for a class of about 30. The cost seems a little high especially knowing that the teachers in my district normally only get about $80 a year for all of their classroom supplies. So I guess if you do not need anything else for your room you could go that way. I did find some other more cost effective alternatives. One would be to have the students make their own set using a template. You would want to use a little more of a heavier card stock paper and then you could get Ziploc bags for storage. One template I found was here http://mathbits.com/MathBits/AlgebraTiles/Algebra%20Tile%20Template.pdf .
I then found some great resources where you would not even need to have your own set of tiles, but instead you would use virtual tiles. If you had access to tablet, iPads, or laptops in your classroom this may be the best way to go. With these sites you can add tiles to a work area to create equations, to manipulate them, and even solve. The other thing that I liked about these sites was that a lot of them had built in lesson that you could use if you wanted and it would check the students work. You could also have problems on the board and the students could enter them into the site and then turn their screens around, where you could do a very easy visual check of their answers. Students could also work in groups very easily if you were not able to be one-to-one. Some of the ones I likes were:
http://nlvm.usu.edu/en/nav/topic_t_2.html
http://a4a.learnport.org/page/algebra-tiles
http://my.hrw.com/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html
http://www.classzone.com/cz/books/msmath_3_na/resources/applications/animations/chapter_3/swf/g8_3_3.swf
As I was searching I also came across some activities that you could use with either type of algebra tile, but some of them are a little more geared toward the ones students can physically manipulate. One is a starter activity and the other deals with the distributive property. I also found a couple of power points that had some different ways to use the tiles and some activities to do with your classes.
http://mathinscience.info/public/around_block/old/discover.pdf
http://westlakeeagles.weebly.com/uploads/1/3/0/3/13037123/distributivepropertywithalgebratilesv3.0.pdf
http://mathbits.com/MathBits/AlgebraTiles/AlgebraTiles/AlgebraTiles.html
http://www.trianglehighfive.org/pdf/005_algebra_tiles.pdf
I also found a site that you could use to test students knowledge and understanding of the algebra tiles.
http://www.ixl.com/math/grade-7/model-and-solve-equations-using-algebra-tiles
Overall I feel that this is a very valuable tool for all of your students. Students that learn better with a more hands-on approach will definitely get a lot out of the lessons that use these, but so will your top students. I have always been able to understand math very well and I think after working with these tiles I made a bit deeper connection with working with polynomials than I had before. This is a very valuable resource and I will be finding a way to use it in my math classroom.
I then found some great resources where you would not even need to have your own set of tiles, but instead you would use virtual tiles. If you had access to tablet, iPads, or laptops in your classroom this may be the best way to go. With these sites you can add tiles to a work area to create equations, to manipulate them, and even solve. The other thing that I liked about these sites was that a lot of them had built in lesson that you could use if you wanted and it would check the students work. You could also have problems on the board and the students could enter them into the site and then turn their screens around, where you could do a very easy visual check of their answers. Students could also work in groups very easily if you were not able to be one-to-one. Some of the ones I likes were:
http://nlvm.usu.edu/en/nav/topic_t_2.html
http://a4a.learnport.org/page/algebra-tiles
http://my.hrw.com/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html
http://www.classzone.com/cz/books/msmath_3_na/resources/applications/animations/chapter_3/swf/g8_3_3.swf
As I was searching I also came across some activities that you could use with either type of algebra tile, but some of them are a little more geared toward the ones students can physically manipulate. One is a starter activity and the other deals with the distributive property. I also found a couple of power points that had some different ways to use the tiles and some activities to do with your classes.
http://mathinscience.info/public/around_block/old/discover.pdf
http://westlakeeagles.weebly.com/uploads/1/3/0/3/13037123/distributivepropertywithalgebratilesv3.0.pdf
http://mathbits.com/MathBits/AlgebraTiles/AlgebraTiles/AlgebraTiles.html
http://www.trianglehighfive.org/pdf/005_algebra_tiles.pdf
I also found a site that you could use to test students knowledge and understanding of the algebra tiles.
http://www.ixl.com/math/grade-7/model-and-solve-equations-using-algebra-tiles
Overall I feel that this is a very valuable tool for all of your students. Students that learn better with a more hands-on approach will definitely get a lot out of the lessons that use these, but so will your top students. I have always been able to understand math very well and I think after working with these tiles I made a bit deeper connection with working with polynomials than I had before. This is a very valuable resource and I will be finding a way to use it in my math classroom.
Using T-Charts to Solve Equations?
A couple of years ago, when I first began to work as a paraprofessional in the middle school math classrooms, I was introduced to this new ways to solve equations. To use this method students were told to draw a line straight down from the equal sign and then draw a line underneath the equation. Students then designated one side of the chart for variables and the other for numbers. Then students moved the different terms around to get them on the correct side of the chart. The had a little saying that if it crosses the line you change the sign. Here is an example of how this works:
With this example I chose to make the left side numbers and the right side x values. If the term drops straight down it remains the same and if it needs to cross the line you change the sign. After that you add up the terms on each side and you have your answer. With this example we only needed to use adding and subtracting, but if you also needed to use multiplication and division you would just do that at the end after you take care of the adding and subtracting. Here is an example of one that will use dividing at the end.
So with this one you can see that it starts off the same way and when you get to the end you need to divide both sides by 3 to find the final answer.
I like the overall idea of this method, but I feel the way it is being taught is not useful for the students and ends up being a problem when they get to the high school. The main problem that I have with how it is being taught, is that students are not getting an explanation of what is really happening when they are moving terms across the line. I think that it would be beneficial for students to understand that they are undoing the problem to find the answer. In oder to undo something you need to go backwards or do the opposite. I like expelling to students that both sides of the equation need to remain balanced, so what ever you do to one side you need to do to the other. I think that also helps students understand that even though we are moving things around, the equation is remaining the same. I think that if the students have a solid understanding of keeping both sides of the equation balanced and how to undo the different operations, then the t-chart can be used to help students organize their work.
As I was looking for some examples of how other teachers were using t-charts, I actually did not find anything like what was used where I teach, but I did find a different use for them that I liked a lot better. This version was called a do/undo chart and on one side of the paper there were t-charts set-up that had "DO" on the left side and "UNDO" on the right side. The problems are presented as a mystery that needs to be solved, that the variable is the mystery. The first step is to look at what has been done to the variable, using the order of operations, and write it down in the "DO" column. The next step is to look at what they wrote in the "DO" column and use that information to fill in the "UNDO" column. They will need to know what the opposite operations are of those in the "DO" column. Now that they have there "UNDO" operations they go to the other side of the paper and solve the equation using the "UNDO" steps. So if the students had the equation 3x-6=3, they would dart by listing the operations in the "DO" column. They would use the order of operations and begin by writing x3 and then -6. Now in the "UNDO" column they would write +6 and they would go to the equation and add 6 to both sides of the equation. Then they would write /3 in the "UNDO" column and the divide both sides of the equation by 3. They would now have the solution of x=3. Here is a link to the worksheet and directions. http://www.teacherspayteachers.com/Product/Solving-Two-Step-Equations-with-a-DoUndo-chart-97427
I really like this idea a lot better then the t-charts from above. I feel like the students will have a better understanding of how to solve an equation with this method. The second method is also just how it is introduced and after students are comfortable with the process they do not need to use the chart anymore. The students that are learning the first method do not receive the necessary understanding that is needed to solve an equation and when the get to high school they have no idea what to do. The teachers have to start all over teaching them how to solve simple two step equations.
I like the overall idea of this method, but I feel the way it is being taught is not useful for the students and ends up being a problem when they get to the high school. The main problem that I have with how it is being taught, is that students are not getting an explanation of what is really happening when they are moving terms across the line. I think that it would be beneficial for students to understand that they are undoing the problem to find the answer. In oder to undo something you need to go backwards or do the opposite. I like expelling to students that both sides of the equation need to remain balanced, so what ever you do to one side you need to do to the other. I think that also helps students understand that even though we are moving things around, the equation is remaining the same. I think that if the students have a solid understanding of keeping both sides of the equation balanced and how to undo the different operations, then the t-chart can be used to help students organize their work.
As I was looking for some examples of how other teachers were using t-charts, I actually did not find anything like what was used where I teach, but I did find a different use for them that I liked a lot better. This version was called a do/undo chart and on one side of the paper there were t-charts set-up that had "DO" on the left side and "UNDO" on the right side. The problems are presented as a mystery that needs to be solved, that the variable is the mystery. The first step is to look at what has been done to the variable, using the order of operations, and write it down in the "DO" column. The next step is to look at what they wrote in the "DO" column and use that information to fill in the "UNDO" column. They will need to know what the opposite operations are of those in the "DO" column. Now that they have there "UNDO" operations they go to the other side of the paper and solve the equation using the "UNDO" steps. So if the students had the equation 3x-6=3, they would dart by listing the operations in the "DO" column. They would use the order of operations and begin by writing x3 and then -6. Now in the "UNDO" column they would write +6 and they would go to the equation and add 6 to both sides of the equation. Then they would write /3 in the "UNDO" column and the divide both sides of the equation by 3. They would now have the solution of x=3. Here is a link to the worksheet and directions. http://www.teacherspayteachers.com/Product/Solving-Two-Step-Equations-with-a-DoUndo-chart-97427
I really like this idea a lot better then the t-charts from above. I feel like the students will have a better understanding of how to solve an equation with this method. The second method is also just how it is introduced and after students are comfortable with the process they do not need to use the chart anymore. The students that are learning the first method do not receive the necessary understanding that is needed to solve an equation and when the get to high school they have no idea what to do. The teachers have to start all over teaching them how to solve simple two step equations.
Wednesday, November 12, 2014
Textbooks...To Use Them or Not To Use Them?
As I was sitting in the classroom doing my observations, a question came to my mind. Do we use textbooks any more? My experience last year when I was a parapro for the 8th grade math classes was that the textbook were not used very often. I really only remember them being used a handful of times. The teacher had handouts for taking notes on and then worksheets for practice and homework. So now that I have the opportunity to watch the high school classes, I am noticing the same thing. The teachers are giving handouts for taking notes and then worksheet for practice. After talking with the high school teachers I discovered that they do not really like their current textbook. They have the same series of textbooks for Algebra I, Geometry and Algebra II. They have a different book for Calculus, which the teacher said is actually not that bad, but she still does not use it very often.
So now I am wondering if other school are doing the same thing? And is this the best way to go? Are textbooks now obsolete? I know that we are stuck in this place between the way we have been programed to teach math, which many refer to as the drill and kill method, and the lass used discovery method. As I think about the textbook situation and this more unknown way of learning math for me, I am also wondering were I stand on these issues.
I begin with the method of teaching. I think that I am right in the middle right now, but maybe slightly drifting toward the discovery learning. I think what scares me the most about the discovery learning is that it is not the way that I was taught so it is uncomfortable for me. I do like the idea of the discovery learning. I imagine that students are working together cooperatively, they are asking themselves questions, they are using their computers to find more answers and then I snap back to the reality of where I teach. Most of the students that I teach are not really concerned with digging deeper into understanding anything. So then I steer back to the drill and kill method and I see students drowning in piles of papers that they will never finish and turn in for a grade. So many students just do not even care enough to do their homework. I also know that there are many students that are on the other side of this coin and they would want to know more and dig deeper and if you drown them in handouts they will finish them all because they do care about they grades. I am also aware that this is the situation where I teach and may not be the case everywhere, but if I start teaching a math class it will be where I am currently teaching so that is what will drive my decisions in the end. I feel that being in the end the middle of these two teaching styles may be the best place right now.
Moving on to the textbook question...I think that this is actually the same problem as the teaching styles. Most math textbooks are mainly drill and kill and some of the new ones have gone all of the way to discovery learning. There really is nothing right in the middle where you could have the students discover something and then reinforce their knowledge with more practice. One of the teachers that I am observing mentioned that she thinks that there is so many resources available online that can be tailored to exactly what she wants, that she is not sure the school should adopt a new textbook. She thinks that the resources she is finding and using in her classes is better than a textbook. After hearing her talk about this idea, I found that I was liking it more and more. It would be a lot more work than just following the book, but you could make it into exactly what you wanted and needed for your students. Are the students high level? Low level? Ready to do some self discovery leaning? do they need a little drill and kill? All of these questions can be answered if you can customize!
So I do not think that I have a definite answer on my teaching style, but I know that I will be able to use the many resources available online to help. Now my big job will be to go and start collecting those resources. Maybe that will make for a great future post?
So now I am wondering if other school are doing the same thing? And is this the best way to go? Are textbooks now obsolete? I know that we are stuck in this place between the way we have been programed to teach math, which many refer to as the drill and kill method, and the lass used discovery method. As I think about the textbook situation and this more unknown way of learning math for me, I am also wondering were I stand on these issues.
I begin with the method of teaching. I think that I am right in the middle right now, but maybe slightly drifting toward the discovery learning. I think what scares me the most about the discovery learning is that it is not the way that I was taught so it is uncomfortable for me. I do like the idea of the discovery learning. I imagine that students are working together cooperatively, they are asking themselves questions, they are using their computers to find more answers and then I snap back to the reality of where I teach. Most of the students that I teach are not really concerned with digging deeper into understanding anything. So then I steer back to the drill and kill method and I see students drowning in piles of papers that they will never finish and turn in for a grade. So many students just do not even care enough to do their homework. I also know that there are many students that are on the other side of this coin and they would want to know more and dig deeper and if you drown them in handouts they will finish them all because they do care about they grades. I am also aware that this is the situation where I teach and may not be the case everywhere, but if I start teaching a math class it will be where I am currently teaching so that is what will drive my decisions in the end. I feel that being in the end the middle of these two teaching styles may be the best place right now.
Moving on to the textbook question...I think that this is actually the same problem as the teaching styles. Most math textbooks are mainly drill and kill and some of the new ones have gone all of the way to discovery learning. There really is nothing right in the middle where you could have the students discover something and then reinforce their knowledge with more practice. One of the teachers that I am observing mentioned that she thinks that there is so many resources available online that can be tailored to exactly what she wants, that she is not sure the school should adopt a new textbook. She thinks that the resources she is finding and using in her classes is better than a textbook. After hearing her talk about this idea, I found that I was liking it more and more. It would be a lot more work than just following the book, but you could make it into exactly what you wanted and needed for your students. Are the students high level? Low level? Ready to do some self discovery leaning? do they need a little drill and kill? All of these questions can be answered if you can customize!
So I do not think that I have a definite answer on my teaching style, but I know that I will be able to use the many resources available online to help. Now my big job will be to go and start collecting those resources. Maybe that will make for a great future post?
Friday, October 10, 2014
Technology...
Today's society is so different from the one that I grew up in. I know that I will be aging myself, but when I was young remotes were still connected to the TV, phones were still attached to the wall, there was no internet or anything else we would label high tech. At school we used pencil and paper for all of our homework and we had to read textbooks to learn the materials. I know that does not sound too exciting, but I feel that I was very well educated. Technology has come along way, and I think I had an advantage by seeing the evolution unfold. I believe that my advantage is the disadvantage of students today. Students just do not appreciate what they have in front of them and how easy some things are now compared to 20 years ago, 10 years ago, even just a few years ago.
I think that in some ways we are too reliant on technology and all of its benefits. There really is not much around these days that does not have some sort of a technology component. Even when I go visit some of may family up north at the farm, where you would think there wouldn't be any technology, it is everywhere. There is no milking a cow by hand anymore, it is all done with high tech machines. The movies and shows on TV that deal with computers taking over the world and what would happen if we suddenly could not use computers always make me wonder if these scenarios were really to occur...would we survive. Is there such a things a too much technology? Or not enough technology?
I do believe there can be too much for some people. The people that spend every waking hour plugged in. I am related to some of these types of people. When I go to a family function we really do not socialize anymore. The new form of socialization is sitting in the living room with a phone or iPad in our hands surfing the web or playing online games. My favorite memory of this type of situation was all of us playing the popular game SongPop together in the same room. It still makes me laugh when I think about it. We did not even talk to each other! This is where I think the technology gets in the way. People not knowing how to interact with other people face-to-face.
I also believe that we can go the complete opposite direction and have too little technology. I would definitely not be a fan of going back to the dark ages. I like being able to cook a meal in the microwave, which I do recall getting our first one! I also love my DVR and being able to watch all of my favorite TV shows that I have missed. Drinking clean water is also a big one for me! These are just a few things, but I am sure I could go on and on.
So what does all of this have to do with how we use technology in our classrooms? I believe that we need to use technology and its many benefits in our classrooms, but we need to make sure we do not go overboard. There are also some very beneficial activities that do not require technology and also promote students interacting together to find an answer or complete a project. It is our job as an educator to make sure we are weighing the pros and cons of every activity, assignment, or project we give our students.
One other concern for technology is how many students do not always have access to the internet. My district is considered a low income/at-risk district and many families do not have the income to afford the conveniences of the internet. There are also districts that are not as equipped as others. Some districts are one-to-one with their technology and others do not even have a single computer lab. This is a caution for students that are just graduating from college and will be looking for a job soon. Your big ideas may not come to fruition if you end up in some of these lower income districts, so have some back up plans!
I think that in some ways we are too reliant on technology and all of its benefits. There really is not much around these days that does not have some sort of a technology component. Even when I go visit some of may family up north at the farm, where you would think there wouldn't be any technology, it is everywhere. There is no milking a cow by hand anymore, it is all done with high tech machines. The movies and shows on TV that deal with computers taking over the world and what would happen if we suddenly could not use computers always make me wonder if these scenarios were really to occur...would we survive. Is there such a things a too much technology? Or not enough technology?
I do believe there can be too much for some people. The people that spend every waking hour plugged in. I am related to some of these types of people. When I go to a family function we really do not socialize anymore. The new form of socialization is sitting in the living room with a phone or iPad in our hands surfing the web or playing online games. My favorite memory of this type of situation was all of us playing the popular game SongPop together in the same room. It still makes me laugh when I think about it. We did not even talk to each other! This is where I think the technology gets in the way. People not knowing how to interact with other people face-to-face.
I also believe that we can go the complete opposite direction and have too little technology. I would definitely not be a fan of going back to the dark ages. I like being able to cook a meal in the microwave, which I do recall getting our first one! I also love my DVR and being able to watch all of my favorite TV shows that I have missed. Drinking clean water is also a big one for me! These are just a few things, but I am sure I could go on and on.
So what does all of this have to do with how we use technology in our classrooms? I believe that we need to use technology and its many benefits in our classrooms, but we need to make sure we do not go overboard. There are also some very beneficial activities that do not require technology and also promote students interacting together to find an answer or complete a project. It is our job as an educator to make sure we are weighing the pros and cons of every activity, assignment, or project we give our students.
One other concern for technology is how many students do not always have access to the internet. My district is considered a low income/at-risk district and many families do not have the income to afford the conveniences of the internet. There are also districts that are not as equipped as others. Some districts are one-to-one with their technology and others do not even have a single computer lab. This is a caution for students that are just graduating from college and will be looking for a job soon. Your big ideas may not come to fruition if you end up in some of these lower income districts, so have some back up plans!
Sunday, September 7, 2014
Counting Circles
In class we have been doing these exercises called counting circles. Everyone in the class stands in a circle and the teacher gives a starting number and then the amount that it will change by as it is passed around the circle. The first time was a bit scary because I was afraid that I would get the answer wrong or that I would take way too long to figure out the answer. As the exercise progressed those anxieties eased up, and I realized that some people were fast and some were not. After we got all the way around the circle plus maybe a few more people going twice, the teacher picked a person down the line from where we stopped and asked the class to figure out what that person's response would be for the problem. Once we all had answers, shown by a thumbs up, we were asked to share the answer and how we came up with our answer. Multiple people shared out their answers and how they got to their answer. What I like a lot about this exercise is you are practicing mental math and at the end you see that there are many ways to get to the same answer. It is also a good way for you to realize that you made a mistake. When I shared my results and I explained what I did, I had made an error with my mental math, but I saw were I went wrong as I was talking through my process. I believe that this would be a great exercise for students to do at the beginning of class as a warm-up. I will be using this in my classroom and I my even try to figure out a way to tie it in to my band classes...Rhythm Circles! That could be fun!
Introduction
Well I am starting this blog for a class that I am taking. I have been teaching now for 14 years in the area of instrumental music. I am only part time so I have decided to finish my math minor so I can add the math endorsement to my teaching certificate. I am hoping that the district that I teach at will be able to bring up to full time by adding in some math classes. The class that I am taking is called Math For Secondary Teachers and I was a bit afraid that the class was going to be all about teaching basic, such as, classroom management, how to make a lesson plan, ect. We have only met three times and I am happy to be very wrong. This class has made me do a lot of thinking about how I will teach math an how I was taught math. I can't say that I thought teaching math was going to be easy, but I did not think that the transition would be too difficult. I am beginning to think that it is definitely going to be a little more involved than I thought. I am glad that I will be keeping this journal of what I have learned so when I do teach a math class I will be ready!
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