Throughout the semester we have used technology like Desmos and Geogebra to facilitate learning of different topics. Though different methods are used, it usually involves some correlation between equations and graphs, which I have found helpful in developing understanding. Specifically, sliders are something new to me; I have found them to be very helpful as I can manipulate different terms in an equation and immediately see how it affects the graphical representation. From there I can evaluate trends and build understating based on multiple sources, equation and graph.
I decided that it would fun and practical to learn how to make an activity; so, with students in mind, I built one in Desmos. One of my favorite activities up to this point has been the Marbleslide activities that we have done using a couple different types of equations, so I decided to make one for exponentials and logarithmics. I wanted to include both types with the idea that it would help students make connections between the two related forms.
When developing this activity I realized that a lot more planning has to go into it than I would have originally thought. The first thing I had to think about was, "what do I want students to get out of this?" I decide that I wanted to design an activity that would make students think about how the different terms in exponential and logarithmic equations relate to a graph, and how those graphs relate to each other. As such, I started with two activities that allowed the use of sliders to change the graph. The first challenge was uses an exponential equation and the second uses a logarithmic, though the ball drop and stars were in the same location. I wanted them to see that, even though the functions are inverses you can still make the graphs look very close to, or identical to each other. Moving forward I wanted to make sure that the game offered an opportunity to use a variety of different graphical shapes, and in a few cases I actually stipulated that the student must use one of each graph type (exponential and logarithmic) in collecting all of the stars. I also tried vary the difficulty of the tasks, creating some that were more simple, and one that I considered quite challenging.
This was a fun learning experience for me and something that I will definitely use in the future with my students.
Here is my finished product:
https://student.desmos.com/?prepopulateCode=cw6gs
Monday, October 23, 2017
Tuesday, October 3, 2017
Thinking About What Makes a Good Teacher
MTH 229, "Mathematical Activities for Secondary Teachers," has definitely got me thinking about activities, but perhaps more importantly how to prepare and teach the activities. The material is great, and every day I find myself saying "I have to remember that." I appreciate all the different ways we are learning to do things and the connections we are making between mathematical concepts, but what I really want to make sure I remember is the strategies and the ideas behind the lessons. We have not only been talking about math, but what strategies work well in teaching math, what keeps student's engaged, and the basic mentality behind planning a lesson. So what I want to do here is start gathering those ideas in one place so that I can think about how they all work together, reflect on what I was thinking when I underlined that note and put '!!!' next to it, and have a more permanent place to reference my thoughts.
Something we have discussed many times, and I want to be sure never to forget, is to always keep the goal of the lesson in mind when planning. And hopefully that is not just something like, 'learn polynomial division.' I want to make sure that they understand the material, but maybe more importantly how it can be useful to them. Not only in their daily lives or careers but how it relates to other mathematical topics we have discussed as well. We need to plan those connections in lessons, not just mention it and hope that a few kids pick it up. In other words, I want to make sure that I am planning ahead an appropriate lesson, to teach specific objectives, that promotes overall understanding. In addition, I think it is important to think about how we are going to measure the success of the lesson. Have the students gained what we had hoped they would from it?
I suppose this seems like a good point to transition to the idea of homework. I can see it as a way to assess where the students are and if they have understood a lesson. It can also serve as extra practice with the material. However, we must consider individual student needs and the fact that certain students have life situations that don't permit time outside of school for extra work. Something that stuck with me from our class discussion on homework is the question, 'is it supporting the students that need it most?' Given all that, if there is homework, the question of whether or not to grade it, and how to grade it comes to mind as well. My first inclination is to say that I would not want to attach a grade to homework.
Grades in general offer even further debate. It seems to be a growing trend, that I agree with, that grades should reflect an overall understanding of the material, not necessarily how well you do on a single test. It wasn't until late in my college career that I had a math professor who allowed us to re-assess on learning objectives that we had not yet shown understanding for. I will never forget how brilliant I thought that was. Not only does it reduce stress for students in so many ways, but it hits much closer to the point of assessments. If by the end of the class, you can show thorough understanding of the material, your grade should reflect that. A grade should not suffer because you don't test well, were sick, got buried in other classes for a couple weeks, or just life.
Another thing I have been thinking about as a result of this class is group work and whole class discussion. I think that group work can be greatly beneficial in different ways. Working on your own and then sharing with a group gives you time to think on your own before collaborating and seeing other ideas. Working with a group on a white board (VNPS) gives everyone the opportunity to participate in real time and see other thought processes immediately. I think that group work can create a safe space for students to share ideas and build relationships. Whole class work is a slightly different dynamic. This provides opportunities for small groups to present to each other, methods of standing with thumbs up (down, side, etc..), or clickers to poll the group for answers. I think that small groups presenting to the larger group helps create safe space and community, while thumbs and clickers provide anonymity and offer the instructor a good baseline of class understanding. I think we have to consider the right method of group work for each situation, and remember that perceived randomization of groups is important for students.
Finally, I want to talk a little bit about methods of discovery and ways to make things more fun. I think a discovery approach to mathematics can be really fun and engaging. By that I mean lessons where students are guided to find the answers for themselves but never specifically told how to do the problems. I experienced this a little bit in class just the other day with the Rational Function Puzzle. I slowly pieced this back together as we discussed it in class and other students brought things up. It is always really rewarding when you feel like you are figuring math out on your own without someone telling you how to do it, and I think this generates better understanding as well. Technological tools are great for this as well. I had worked with geogebra some in the past and it is cool, but Desmos is my new favorite. Not only is it easy to use and intuitive, but the sliders make understanding what happens when you change the value of a variable so much easier to see for me. Additionally, the games and programs people have made for both geogebra and Desmos are fun and functional learning tools ( Quadratics game, marbleslide, Rational Function Puzzle).
I realize that I basically brought up a bunch of ideas but didn't really settle on anything; at this point I am unable to say, 'this is how I am going to run my classroom.' I recognize that I have a lot more questions than answers, but I think that's okay. I don't know what the lesson plans will look like, or how I am going to assess the success or failure of a lesson plan. It is a little intimidating and sometimes overwhelming to think about having my own classroom and being somewhat responsible for at least the mathematical ideas of these kids. But on the other hand, I think it's good that I'm a little scared, and I know it is only because I care. I just want to make sure that I put as much effort into learning how to be a good teacher as I do into learning the math. Spending time thinking about the things listed here is a good start to being where I want to be. I am going to make mistakes, but I am going to keep trying to figure out the best way to solve the problem, it's what mathematicians do.
Something we have discussed many times, and I want to be sure never to forget, is to always keep the goal of the lesson in mind when planning. And hopefully that is not just something like, 'learn polynomial division.' I want to make sure that they understand the material, but maybe more importantly how it can be useful to them. Not only in their daily lives or careers but how it relates to other mathematical topics we have discussed as well. We need to plan those connections in lessons, not just mention it and hope that a few kids pick it up. In other words, I want to make sure that I am planning ahead an appropriate lesson, to teach specific objectives, that promotes overall understanding. In addition, I think it is important to think about how we are going to measure the success of the lesson. Have the students gained what we had hoped they would from it?
I suppose this seems like a good point to transition to the idea of homework. I can see it as a way to assess where the students are and if they have understood a lesson. It can also serve as extra practice with the material. However, we must consider individual student needs and the fact that certain students have life situations that don't permit time outside of school for extra work. Something that stuck with me from our class discussion on homework is the question, 'is it supporting the students that need it most?' Given all that, if there is homework, the question of whether or not to grade it, and how to grade it comes to mind as well. My first inclination is to say that I would not want to attach a grade to homework.
Grades in general offer even further debate. It seems to be a growing trend, that I agree with, that grades should reflect an overall understanding of the material, not necessarily how well you do on a single test. It wasn't until late in my college career that I had a math professor who allowed us to re-assess on learning objectives that we had not yet shown understanding for. I will never forget how brilliant I thought that was. Not only does it reduce stress for students in so many ways, but it hits much closer to the point of assessments. If by the end of the class, you can show thorough understanding of the material, your grade should reflect that. A grade should not suffer because you don't test well, were sick, got buried in other classes for a couple weeks, or just life.
Another thing I have been thinking about as a result of this class is group work and whole class discussion. I think that group work can be greatly beneficial in different ways. Working on your own and then sharing with a group gives you time to think on your own before collaborating and seeing other ideas. Working with a group on a white board (VNPS) gives everyone the opportunity to participate in real time and see other thought processes immediately. I think that group work can create a safe space for students to share ideas and build relationships. Whole class work is a slightly different dynamic. This provides opportunities for small groups to present to each other, methods of standing with thumbs up (down, side, etc..), or clickers to poll the group for answers. I think that small groups presenting to the larger group helps create safe space and community, while thumbs and clickers provide anonymity and offer the instructor a good baseline of class understanding. I think we have to consider the right method of group work for each situation, and remember that perceived randomization of groups is important for students.
Finally, I want to talk a little bit about methods of discovery and ways to make things more fun. I think a discovery approach to mathematics can be really fun and engaging. By that I mean lessons where students are guided to find the answers for themselves but never specifically told how to do the problems. I experienced this a little bit in class just the other day with the Rational Function Puzzle. I slowly pieced this back together as we discussed it in class and other students brought things up. It is always really rewarding when you feel like you are figuring math out on your own without someone telling you how to do it, and I think this generates better understanding as well. Technological tools are great for this as well. I had worked with geogebra some in the past and it is cool, but Desmos is my new favorite. Not only is it easy to use and intuitive, but the sliders make understanding what happens when you change the value of a variable so much easier to see for me. Additionally, the games and programs people have made for both geogebra and Desmos are fun and functional learning tools ( Quadratics game, marbleslide, Rational Function Puzzle).
I realize that I basically brought up a bunch of ideas but didn't really settle on anything; at this point I am unable to say, 'this is how I am going to run my classroom.' I recognize that I have a lot more questions than answers, but I think that's okay. I don't know what the lesson plans will look like, or how I am going to assess the success or failure of a lesson plan. It is a little intimidating and sometimes overwhelming to think about having my own classroom and being somewhat responsible for at least the mathematical ideas of these kids. But on the other hand, I think it's good that I'm a little scared, and I know it is only because I care. I just want to make sure that I put as much effort into learning how to be a good teacher as I do into learning the math. Spending time thinking about the things listed here is a good start to being where I want to be. I am going to make mistakes, but I am going to keep trying to figure out the best way to solve the problem, it's what mathematicians do.
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