Note: this is a simply designed lesson focused on the Thinking Classroom (TC) structure. I left out the commentary that would explain the teacher moves that go with the TC and how to run a "What do you Notice". The goal for students is specific, factor by grouping. Objectives:
~Preknowledge:
Lesson: Have the students multiply the two problems pictured below (using the area model). Then have the class do a Notice/Wonder with the results. (the problems were designed to be very similar with minimal changes to help focus on the difference in the factors that create a 4term polynomial instead of a trinomial). Send the students to the whiteboards in random groups to factor the progression of problems that can be factored with grouping (using an area model) Make sure to finish with a 4term polynomial that cannot be factored by grouping. Have students take notes on what they learned. Here are the google slides I used to teach during distance learning. The slides contain the problems ready to be copy/pasted into a jamboard.
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Note: this is a simply designed lesson focused on the Thinking Classroom (TC) structure. I left out the commentary that would explain the teacher moves that go with the TC and how to run a "What do you Notice". The goal for students is specific, factor sum and difference of cubes. Objectives:
~Preknowledge:
Lesson: What do you notice: factorizations of difference of squares (focus on the factors contain the square root of the original terms) What do you notice: factorizations of sum and difference of cubes (focus on the first factor and how it is the cube roots of the original terms, the contents of the 2nd factor are not important until they figure it out during the TC portion) Move students to the whiteboards in random groups. Give one problem at a time where students figure out the factoring. The last problem needs to have a monomial factored out first. Finish with students taking notes on what they learned.
Final Thoughts: I am not sure if giving the students a 2 by 3 box to start with is too much scaffolding. I will play around with this in the future. Here are the google slides I used for distance learning. The images are ready to be copy/pasted in the jamboard. The Prelude.I attended an Ethnomathemtics workshop about two years ago. I have always been fascinated by cultures, my own and others. The chance to tie mathematics into such an interesting topic was an opportunity I couldn't resist. One the presenters was Dr. Jim Barta from Bemidji State University. There were two things that he brought to the workshop that inspired me: the cultural interview and story cards. I saw both of these activities as tangible things that math teachers can do in our classrooms to get students to see mathematics in their cultures and to use their cultures in our mathematics classroom. (There is a lot more to this back story, but that will have to wait for another post). I have since experimented with both of these things, but now I want to get serious. I am doing my research and planning my lessons. I want to build a unit that other teachers can use in their classrooms. I want to build something that helps us build a sense of belonging for our students in our classrooms. I figure I have two types of students to consider here: 1) The students whose cultures are still beautifully immersed in their heritage. They can see many of their family's traditions as passed down from previous generations. I think of my good friend May who is Hmong and can tell me about so many customs that have been passed down through the generations. 2) The students who are a product of the melting pot that is the United States. Their family's traditions change with each generation as parents from different cultures create new traditions, or combine traditions, to work for their new family. I think of myself. My mother is German and my father is a combination of various Scandinavian ancestries. Neither share stories about why we do certain traditions during the holidays and other times. The closest comes to our Christian beliefs and traditions, but the actual tie to family culture and historical connection feels lost. (Of course, there will be students at various places between these two) Through this unit, we are going to focus at two levels:
The Outline of the Unit (so far).
My Final Thoughts
Today I was listening to the Making Math Moments Podcast, episode 98 with Peter Liljedahl and decided I wanted to write my reflection of what I learned from it. I fell in love with the Thinking Classroom over two years ago. Since then I have been studying articles, reading books, listening to podcasts, and exploring; basically doing anything I can to learn more and teach better. When Covid19 changed our classrooms to online environments, I refused to go back to lecturing. I used Jamboards and Zoom breakout rooms to keep my kids going. It worked pretty great and so I planned to keep doing that during the new school year. I started the year with 4 classes of students who were in my class last year, they were already familiar with working on whiteboards and learning through the Thinking Classroom. I also had 2 classes of student who were in my classroom for the first time and haven't been introduced to the Thinking Classroom yet. Also this year, I am teaching Hybrid. Half my students are in my physical classroom, half are at home attending through a live zoom link. I am super grateful for the technology that my school has made available for this as I feel that I can pretty seamlessly teach both groups at the same time. During the 8 weeks of school we have had so far, I have been putting my inperson students on the whiteboards and my athome students on Jamboards while working together in breakout rooms. It has not been going as well as it did in the spring. Over the last 4 weeks, I have been trying to figure out what to change while still honoring the 14 guidelines of the Thinking Classroom. Thanks to the podcast episode, I think I am ready to make some (informed) changes for Monday! My frustrations and possible solutions: My College Algebra class has not been doing well. They complain that when they are at home, the breakout rooms do not go well. They tell me that they are only learning during the consolidation time and the breakout room time is a waste. I actually quit doing the breakout rooms for two weeks, which broke my heart. But their tests have been backing up their claims of not learning. This week, I had a hearttoheart with the class about what needs to change to help them learn and they all chimed in that they want to be back at the boards. So my new mission is to get them back to the boards. So here are the issues I have been battling during Hybrid learning and what I am going to try differently. 1) Most of my seniors and juniors no longer have a touchscreen device. The amount work they have to do to write on the Jamboard has become enough of a nuisance that their boards are blank at the end of the hour. Besides that, the work space on a Jamboard is way too small. In my visit with my seniors, they all felt that having a physical whiteboard at home would help with groupwork in breakout rooms. So I called up the local lumberyard, ordered three sheets of shower board, had the shop class cut them into 2 foot by 2 foot pieces and handed them out to each student in College Algebra to keep at home. If this works, I will get more to hand out to the rest of my classes. 2) Random groupings of 3. In person, a random group of 3 is best, I have seen this over and over. But I have been questioning the "3 in a group" part this fall. I have heard from many students that being in a "bad group" means that they do all the work on their own and there is no interaction. I suppose this happens for many reasons, one of them being that I cannot easily monitor the groups and intervene when needed. (I wish there was a way to be able to see each breakout room simultaneously to make sure they are unmuted and interacting. I can monitor the Jamboard, but watching them write does not guarantee interaction and discourse). I have been playing around with the idea of having more students in a group. I was excited to hear that Peter Liljedahl also found the same problem and that 4 or 5 students in a virtual group worked better. (Listen to the podcast to hear the research behind the number in a group). So I plan to now randomly assign my students into groups of 4 or 5 during virtual breakout groups, still 3 when in person. 3) I don't like that I cannot listen in to the zoom breakout groups easily. During distance learning in the spring, I would hop between breakout rooms and check on each group. This worked relatively well. However, now that I am physically in a room of students who are physically distanced while working together and talking with masks on, my joining a group is almost impossible. They can't hear me, I can't hear them. I don't have a solution. But I did like Peter's suggestion of having a discussion board where virtual groups can list what they found difficult and how they worked through those difficulties. I am thinking I might assign some of that. Maybe have someone in the group keep a "group diary" summarizing their time?
4) I am horrible at leaving time for meaningful notes. This is even true for my nonCovid 19 teaching. The podcast was the first time I heard them referred to as "notes for your future, forgetful self". I love that. I am going to start assigning that for post class time each day. I have spent this first quarter modeling how to fill it in, I think my students are ready for it to be independently assigned for postclass reflection. 5) I still need to work on the "homework" part. But I am feeling overwhelmed and overworked, so I am going to be okay with that being left for future improvements. Although that is really hard for me. And that is what I learned today. I am looking forward to creating my lesson plans for the week. I hope to get back to feeling like I am living the dream of every teacher: engaged students, deep thinking, and great discussions. And I am confident that if I don't get to the great REM cycle dream, I can at least reach the daydream level. I keep seeing Sara Vanderwerf tell people on Twitter that I have some great thinking on this. So I decided I need to blog about my ideas so that I have something I can share with people. (I do not claim to be the best or know the best for all of this, I am just sharing my ideas). First off, my decisions on how I am teaching, whether in the classroom or from a distance are always founded in the Thinking Classroom. This is a set of elements that were researched and organized by Peter Liljedahl. Since I have built my classroom into a Thinking Classroom, my class has become a studentcentered, engaged room that gets kids thinking and motivated to figure things out for themselves while also helping each other. If you are not familiar with the Thinking Classroom, I am sharing my google slides and recording of a presentation I did with MDE and MCTM this summer. MDE is working on making the presentation accessible, but in the meantime, the raw recording is below. Link > Google Slides: The Thinking Classroom Distance Learning Last Spring My school is 1to1 chromebooks and we were able to get internet to every home. Most of what I did was Synchronous. Although what I did can be done using other platforms and technology, here is my list:
Here is a video of what I did in the Spring and what I plan to do during Virtual days (every Wednesday for my school, we are hybrid the other days) Link > Google Slides Hybrid Learning My school is starting with hybrid learning in the high school. I am intrigued to try it, but have a few worries about the execution. We will see. My goal is to still have a Thinking Classroom. I will be blending much of what I do in the classroom and what I did during distance learning. I will be using:
My school put together a little video showing what our hybrid model will look like. It also shows how the OWL works. You can see it here. Here is a video that explains more of my plans for hybrid. Asynchronous Learning Although this is not something that I have to do, I have some ideas. Thanks to my friend, May Vang Swanson, they are even better than what I was thinking on my own. My goal, as always, is to do as much as I can with the Thinking Classroom. Here is my list of what I would use:
As you can see above, one thing that I would add to what I use is Flipgrid. From what I have heard from May and others, it would very much help with making this studentfriendly and as mush student discourse as you can get asynchronously. Here is a video explaining what I would imagine Asynchronous would look like. Link > Google Slides I hope this is able to help spark some ideas for your classroom. I do not claim to be a knowitall for technology. So, if you have a great app, website, or resource that would help someone that cannot use what I have used, please share it in the comments.
Manipulatives I am a huge advocate of using manipulative at all levels of mathematics. During a pandemic, we have some challenges:
Good luck to everyone this year. If you would like to visit or as a question, feel free to add a comment, email me (jstrom@winemac.k12.mn.us) or find me on twitter (@strom_win). Have you heard of name tents? I hadn't until I had the opportunity to hear Sara Vanderwerf speak on them. Here is her blog on them. Sara taught in a large, urban school and used them to make a connection with her students in the first week of school, so important! Each night she writes a response on their paper and gives them back the next day. Make sure to read her blog to learn more about her experiences with name tents. I have two big differences in my experiences with name tents. Difference number 1: I teach in a small, rural school (k12 in one building). I know most of my students before I have them in class for the first time. I know their parents' names and where they live. So when I heard about name tents, I thought it was a great idea, but I also thought it was not necessary since I already knew my students. I was right and wrong. I didn't need the name tent part since I already knew their names. But I did need the inside response piece, even for the students I knew really well, even for the students I am related to. What I have found doing the name tents is that students use them as an avenue to tell you all kinds of things. The biggest thing I find out: how they feel about math. Many of them come to me feeling inadequate with their math abilities. Telling me this on the name tent opens the door to a conversation we need to have. It also cues me into knowing where this student stands and that I need to help nurture their selfimage with math. Difference number 2: I don't have paper name tents, I do them digitally using Google Docs and Google Classroom There are some pros and cons to doing them this way. Pros:
I designed my "feedback form" to be personal to me and my school, but very similar to what Sara uses (see below). If you would like to use it as a template, and make it personal to you and your school, please help yourself. Here is my form. I have also done shorter versions of this in the middle of the school year as a check in, per student request. I called it "Holiday Confessions". I ran it the same way using Google Classroom and doing one question a day.
Just like my Algebra 1 class, my Algebra 2 class is looking at the end of the quarter, year, distance learning, and chapter 10. A test just doesn't feel right. So I have decided to have the students complete a project based on sequences. There are so many great things that they could do, I have found it difficult to pick just a couple. I have settled on 5 projects, the students will pick one to do that interests them. The projects are: Dragon Fractals Patterns on a Grid Serpinski's Triangle Spirolaterals Visual Patterns I had a sixth project, Artwork Copycat, but decided to not give it to the students. In the end, I didn't feel like it analyzed the sequences enough. Dragon Fractals: I first came across this idea looking at the wonderful math art challenge website by Annie Perkins. I was intrigued by the paper folding and how you could develop the fractal by drawing on the previous iteration. I played with it for days, even doing it in spare moments during my zoom lessons. I hope that some of my students will find it just as interesting. If we were in class, I would love to explore the connection between the paper folding and drawing the iterations. But, since this is more of an independent project and I don't have them captive in my classroom to interact with, I left that part out. Patterns on a Grid: I first came across this idea at an art museum in Wurzburg, Germany where they had an amazing math art exhibit. The picture above is a number grid that starts in the middle and spirals out. The colored blocks are prime numbers. It did not occur to me to do this with other patterns until I was thinking about doing these projects. I had a start to a project with this idea when I stumbled upon Megan's Sprials in Annie Perkin's math art challenge. The rest, as they say, is history. It is quite interesting how the layout of the number grid and the coloring of the pattern can give a design. I highly recommend playing around with this. Serpinski's Triangle: Who doesn't love a good session with Serpinki's Triangle. It is such a fun fractal and lends itself so well to sequences. I love the idea of doing the 3D version of it and I really hope I have a student or two that will do it. I am also excited to sneak in a little bit of Pascal's Triangle. I did a different version of the 3D card with a class on Valentines day a couple years ago. Instead of straight lines, we did the top part of hearts. (I am secretly hoping to receive a card or two from my students). Spirolaterals: Have I mentioned Annie Perkins' website with math art challenges? Well, this one comes from there too! I actually saw it first when Annie tweeted about it. Of course, it screamed sequences and I had to include it. It was fun to play around with the idea. I am hoping to play around with it more after school is out. Visual Patterns: It wasn't that long ago that I was visiting with some teachers and they said that every math teacher should know who Fawn Nguyen is. Well, I didn't. So I figured I should find out, and now here I am, a fan girl. One of the things Fawn has brought to all of us is her visual patterns website. When I first looked at it, I didn't do anything with it because I didn't now what to do. After attending Fawn's workshop, I have become an avid user of visual patterns. To start the sequence chapter, I had my students do many visual patterns and then used them to introduce notation and concepts of arithmetic and geometric. It also helped us with being able to find the expressions. It seems fitting that one of the projects is playing with visual patterns. I am especially excited for this project because of the fun I had with my daughters creating a tiktok. Between the quarantine and being 13 years old, it can be tough to find things that my oldest daughter and I can enjoy together. We had so many laughs creating the video, I am hoping my students will enjoy doing something similar. Artwork Copycat:
And then there is the one that didn't make the final cut. Which I am somewhat sad about. In 2016, I went on an amazing trip to Germany. It was a weeklong school for math education researchers. My cup was filled from so many spouts: teaching math, traveling to Germany, exploring another culture, and visiting with math teachers from so many countries. One of our activities for the week was to visit a math art exhibit. Looking back on pictures, there were two art pieces that could be used to explore sequences. I am especially intrigued by the bright colored geometric design. Once you start exploring it, you see that it is a geometric sequence with a common ratio of 2. Trying to recreate it proved to be a fun, artistic challenge. I chose not to include it just because it wasn't quite as much sequence work as the rest and it didn't feel as rigorous as the others. I really want the projects to have the same amount of work and "value". That about covers it all. Please feel free to use any of these projects. I hope to save someone out there some work. Wishing everyone a good ending to the 20192020 school year. I am looking at the end of a quadratics chapter, the end of the year, the end of distance learning, and the end of 9th grade before summer. A normal assessment does not feel like the right thing to do. But I want to do something to culminate the end of the unit. Enter the end of chapter projects. I decided on 3 diverse projects that require knowledge about quadratics. Students get to pick which project they will do based on their interest. I am hoping I have given a diverse enough selection that all the students will be able to find interest in at least one of them. I am sharing them here for anyone that would like to use them with their students. String Art: I love the parabolic curve string art. The use of straight lines that create a curve is beautiful. But first I want them to understand what that design has to do with parabolas (it isn't completely obvious). In the 9th grade algebra class, I do not teach them about the focus and directrix of a parabola. So this project starts with a small lesson about that. I used a quick video from Khan Academy. I also used an idea from Sarah Carter's blog to have the students create a wax paper parabola. See here for her post. After that, it is a matter of teaching them the basics about creating a parabolic curve and then let them get creative. Below is my favorite recent creation and the link to the project I am giving my students. Catapult: When I have had time, I have had my students create a gummi bear catapult in class and then find the equation of the trajectory of the flight of the gummi bear. It has been a fun project in school, and I am hoping it is something students can do at home. The set up is pretty simple. First, they need to create a catapult with supplies they have or supplies I get to them through our delivery system during distance learning. In the google slides, I include a quick video to give them a couple ideas for creating their own catapult. Second, they need to collect data, with the help of another person or two. Third, they will use Desmos to create the quadratic regression. I include instructions for how to do this since they have done it for linear regressions, but not quadratic. Below is a link to the project I am giving my students. Desmos Words: Earlier this year, after our linear unit, I assigned an optional project where students did linear Desmos art. Some of my students really got into the project and gave me some great art! I wanted my third project to follow along those lines (pun intended). I struggled with what to have the students create in Desmos. I had a few students figure out that they could google Desmos art and get an already completed project to turn in as their own. Since they were working in class, I caught it pretty quickly. But now that I will not be able to observe them working on it, I am worried about the originality of their work. I decided to have the students create their name or a favorite phrase out of lines and parabolas. Hopefully this will be original enough that they won't be able to use someone else's work. We dabbled a little bit with transforming quadratics in the unit, but not enough to assess on it. So the beginning of the project is to have the students go back to the Desmos activity and reacquaint themselves with it. There are many activities out there, but I am particular to "Quadratics Graphing Lab" from Mrs. Turpin. I modeled the instructions after a Desmos Name Project I got from Dianna Hazelton. Each letter has to be constructed according to the alphabet chart. They also have to do a minimum of 8 letters. Below is a snap of the example I created for my students and the link of the project. I won't be giving this to my students for another week or two. I am hoping they will be a success and much more enjoyed than an assessment. Please feel free to use any of them in your classes.
My next task is to come up with projects for my juniors to do to finish their sequence unit. I am addicted to the thinking classroom. I have found so much value in the students thinking through their own work without me handing them everything. I have found so much joy in listening to their thinking and seeing them persevere through a problem. I have enjoyed the challenge of listening to their understanding and finding ways to push them farther. I have found pride in the daily occurrence of the type of class that many teachers dream of happening. But it all comes to a halt as we close the doors of our school due to COVID19. Or does it... I can't let it. I can't let the benefit go away because of distance learning. So, I believe I have found a solution! It only takes two different ingredients: Zoom (for random grouping) and Google Jamboards (for the vertical, nonpermanent surfaces) Random grouping Using zoom, I put students into Breakout rooms. I don't know if this is possible on other platforms, but I have found it to work beautifully in zoom. I thought it was only possible if you have a pro version, but it is possible in a basic version. You just have to go into settings and enable it. Breakout rooms allow you to randomly split the students into groups of any size. While in the groups, they can send a message to ask a question. You can drop into any group that you want to check on them. When you want them all back as a full class, you just close the rooms and they have a 60 second time frame to finish what they were doing and come back. The only drawback is that it takes a little bit of time to change between rooms (10 seconds). I have found very few stumbling blocks, which are pretty easily fixed. Vertical Nonpermanent surfaces I was originally going to use google slides and have the students use the "annotate" tool to write. But then I found out that Chromebooks cannot use the annotate tool. I was also going to try the explain everything app, but I found that difficult to assign to students in groups. I finally stumbled upon Google Jamboards. The easiest description is that they are a simple version of Google Slides. But the main, most important difference is that they are meant for writing on, just like a vertical nonpermanent surface. I create one Jamboard for a class period. Before class, I give each slide a number that matches the breakout room number and insert any needed information for the problem. I also make sure that I set the jamboard to "anyone with a link can edit". In Action:
During our class zoom, I let the students know we will do a breakout. I paste the link to the zoom in the chat window and give them time to open it before sending them to the breakout rooms. (Once they go to the rooms, they cannot access the chat window. At least we haven't figured it out, we are still new at this). I have also found that some have troubles opening from the link. We have found two workarounds. One, copy the link and past it into a new tab. Two, I email the link to them. Once everyone has the Jamboard open and ready, I give the typical verbal instructions that I would give in class. Then I click on breakout rooms. I pick the number of rooms that allow for 2 or 3 in a room. Since I have a para in the zoom also, I do a quick check on what room I want them in and move them there. I then "open the rooms". The students then click on the invitation to join the rooms. I wait a bit to let them settle in. I spend quite a bit of time watching the jamboard, switching between the different slides. This gives me a quick check on where the groups are and if they need me to step in. Then I start to drop into the groups based on what I see. This part works just like it does in class. They ask questions, I ask questions, and they sneak peaks at other slides. I have also found it helpful to sometimes just write on a slide instead of dropping into the room. It has been helpful because of the time it takes to drop in a room. It only takes 10 seconds, but it takes even less to drop a hint on their board. Plus it allows me to be with one group in the breakout room while also keeping an eye on and helping another group. When it is time for the wrap up, I close the rooms and the students join me back in the main room. I then share my screen and go back to the Jamboard. Just like in class, we go through the solutions and see each group's work. I have discovered something very important about myself during this time. I used to think that I loved teaching math for the challenge and enjoyment of teaching math. Teaching in a distance learning setting has revealed that, although the math is a great excuse to teach, I actually love teaching math because I enjoy the students. I am still teaching math all day, but it really sucks without the students in the room with me! Really, the only thing making this somewhat bearable is that I can still listen to and see their thinking. I can still push them to expand their thinking. And they can still work together to learn from and push each other. Annie Perkins, a math teacher in Minneapolis, does some great things with math and art. She is currently posting daily math art challenges. I plan to use some for the family math at home activities. Here is her challenge: Tons of triangles.
I would love to see the art you come up with. Share with me on Facebook, or share on Twitter with #mathartchallenge. 
AuthorI teach mathematics for grades 712. Teaching mathematics is my passion. Archives
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