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 (k-12 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 self-image 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.
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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 week-long 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 2019-2020 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 COVID-19. 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, non-permanent 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 Non-permanent 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 non-permanent 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 work-arounds. 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. We did a version of this game at Family Math Night this year. It was one of our most popular games. It is adapted from https://mathforlove.com/lesson/fill-the-stairs/. When we did it at the school we used:
To play this at home, you will need:
How to play the game: 1) On the ladder/stairs, label the bottom step 0 and the top step 70 with paper pieces. 2) Roll two dice (or one die twice). From those rolls, you have two possible numbers. For example, if you rolled a 4 and a 1, you could have the number 41 or 14. Decide which one you want and write it on a piece of paper. 3) Place that piece of paper on a step. Once you place a number, you cannot move it later. 4) If you are playing alone, roll again and follow the same steps. Your goal is to fill the steps/ladder with numbers, in order, from 0 to 60. If you roll and cannot place it on a step because it would be out of order, then you have lost. You win when all the steps are labeled in order.
5) If you are playing with more than one person, take turns rolling the dice and placing numbers on the steps. If a player rolls and cannot place a number, then they lose a turn. The winner is the last person to place a number on a step and successfully fill the steps/ladder. (Note, if you don't have any dice, get 12 pieces of paper and write the numbers 1 through 6, twice each, on the paper and put them in a hat. Then pull out two pieces of paper to figure out your number) This was a fun and busy game at math night. It proved to be a tough game to win. In 1.5 hours, there were only 5 winners. Have fun! This is fun to play on the video, but it would be even more fun if you got out painters tape and played in real life!
(This activity comes from natbanting.com) To begin, watch this video and play along: https://drive.google.com/file/d/1W01ERM18FtSsWJI9VEdUMevOJDyQRPyN/view If you want to play at home, get out painters tape, or use chalk on the garage floor or sidewalk. Make a 3x3 grid big enough for a person to stand in each box. It will help if you can color code the grid like a checkerboard and number each box. (The numbering is for your "David Copperfield" to tell you which box to remove without looking at what box you are standing in). David Copperfield gave you a choice of what box to start on, told you how many squares to move each time, and then removed a square from the grid. If we were at the school, the teacher in charge of the station would do that without watching you move around the grid (they would turn around so that their back is to you). Someone at home will have to take that role this time. For them to know how to do this, they will need to read the following document: https://drive.google.com/file/d/14SyKvIwliNTst7YBRlV-vy6X8KntQOrc/view Play the game several times. (Tip: when you are "removing" boxes, put a stuffed animal or paper plate in the square so that the person cannot stand in it anymore). Your Goal: Figure out how it works. When you think you know, switch roles and you be David Copperfield. If you have it figured out, try making a bigger grid and see if the rules stay the same or change. Have fun! Let me know how it goes! February is tough in the teaching world. At least I am assuming it is tough for all teaching, not just Minnesota. We (Minnesotans) sometimes blame the difficulties on the cold and snow. Our students don't get to go outside very much and we all get crabby from being cooped up indoors and missing out on the benefits of sunlight. But January/February is a really busy time of year. Sports, activities, snow days, and more activities, all get packed into two months. Plus, it is the middle of the school year. We are knee-deep into our standards and are starting to worry about the testing coming up in the spring. I recently had a tough month. Tough in the sense that I had too much going on and not enough hours in the day. Reflecting back, there isn't anything I would have removed from my to-do list if I were to do it over again. Everything was important and worthwhile, and I loved every moment of it! But it is the part of teaching that many do not see or appreciate. And in a few frustrating moments, I have done a lot of thinking about some things I would like to say to administrators, parents, the public and fellow teachers and staff (including myself). Part One: When my children were really young, I would sit in the back of the church so that I could take my noisy child out easily. On one particular Sunday, there was another mother with a few young children sitting a couple pews in front of me. I noticed, with irritation and judgement, that she would get up during the service and leave her YOUNG children alone for spans of time and then return. I am not proud of this, but I judged her for doing this. When she did this, her children were not well behaved, and there was nobody there to keep order. I kept thinking to myself, "I would never do that. Whatever could be so important that she would keep leaving and not taking her children with her?" It went on for some time like this, but then I learned the biggest lesson I have ever learned during a church service. A beautiful-souled woman from the front of the church got up and went to the mother-less pew of children. She smiled at them, loved them, and kept them contained. I didn't hear the sermon from the pastor that day because I was too busy learning a different lesson: it isn't my place to judge someone's shortcomings, but to ask how I can help them succeed. The mother kept leaving because she was sick. For whatever reason, she felt it was important to come to church, even though she was sick. She was doing the best she could with her situation. She was putting much more effort into being there than I was, sacrificing the comfort of her bed/couch to bring her family to church. I didn't need to judge her choices, but to find a way to help her. Part Two: I keep seeing this meme come across my Facebook newsfeed and it strikes my heartstrings every time. I am beyond that motherhood stage, my kids are older and much easier to take care of, but it brings me back to those early years and how it felt to be raising young children. In the chaos and exhaustion of raising babies and toddlers, it is hard to see that you are doing a good job. Especially if you fail to do everything you had on your list. There are loads of laundry to fold sitting around the house, dirty dishes in the sink, and you didn't read to them tonight because you were just too tired. And don't get me started on trying to keep up with the Pinterest moms!
Then you are visiting with someone and they tell you that you look tired or question your parenting choice. In reality, you have already questioned yourself about it and now you think that everyone is judging you for it. But what you really need to hear, especially since the two-year-old that you spend most of your time with isn't capable of doing it, is "You are a great mom". And once isn't going to cut it, you need to hear it often, and loudly, and maybe with some chocolate. Final Part: So let me bring it back to teaching in January and February. With all the activities, events, supervisory roles, planning of special lessons, and anticipation of testing, teachers can feel like these moms. For example, I recently spent the last month: planning a family math night, creating/building all the games for the math night, planning 3 different one-hour presentations, reading two books about teaching math, researching for three different day-long workshops that I will be presenting, driving to and from a day-long board meeting (10 hours round trip), designing/planning/hosting a kindergarten math night, creating 200 adorable ceramic magnets, and running concessions for the day-long elementary basketball tournament. This is all on top of the normal teaching duties and mom duties, which I won't list but keep me busy without all the extra stuff. Although this may seem extreme, I look at other teachers and see people coaching/supervising one, two and sometimes three extra-curricular activities at a time. What is easy to do, from the outside, is to judge these people for forgetting to do something or for putting something off until the last minute. They know they should make copies the day before, or order supplies a week ago, or get the master key ahead of time. But, for whatever reason, they didn't. They are probably beating themselves up for it. And then we are lounging in the background and judging them for their shortcomings. But, do you know what else is easy? To offer to help them. To tell them they are doing a good job. To bring them some chocolate. To recognize the stress and find a way to take a little of the weight off their shoulders. I am so lucky to be surrounded by co-workers that supported me through the last month. They built me up, shared some of the workload, and brought me chocolate. Now that my life has "calmed-down" (does it every really though?), I am working to recognize the same moments in their lives and find a way to support them. To administrators, parents, community, and fellow teachers. When faced with a judgemental moment, stop and ask yourself if that person needs help and support instead of criticism. A school environment can be completely changed with the way we view and treat each other. Side note: the family math night was worth all the work! Objectives:
~Preknowledge:
The problems I chose (in order):
To start, I had all the students join me at the front of the room to discuss our plans. I told them that I was going to give them a problem that I want them to factor using "the box". When they are all done, we are going to discuss their solutions. Then I would give them two more problems to do. I handed out the cards that randomly groups them on the boards and they got to work on the first problem. 2x^2+14x+24 Since they have played a little bit with have an a-value other than 1, they went right to work factoring with a 2x and x on the outside of the box. There were two answers that came out of the groups: (2x+6)(x+4) and (2x+8)(x+3) Since some groups finished quicker than others, I challenged them to find another solution (I told them there were 3 solutions). When the groups finished with at least one solution, I called the class to look at two different groups' work. They decided that both solutions were done correctly. I questioned how there could be two answers and challenged them to return to their boards to make sense of it. Most groups connected that dividing the factor with the 2x by 2 would give the other factor. (see image below) I took this opportunity to explain that we will FACTOR out a 2 instead of divide by 2, which means we will write the two in front of the factorization. I then also showed them what would happen if we factored out the 2 right away, and then factored the quadratic. We discussed the efficiency in doing that first. To wrap up, we agreed that 2(x+4)(x+3) was the complete factorization. (I also connected it to prime factorization from grade school: 20=2*2*5)
3x^2+18x+24 The students went back to their boards and I wrote this quadratic on the board. I chose this one so that it wasn't a 2 that got factored out, but I thought having a constant of 24 again would be interesting. (It turns out to not be interesting). This went quickly, the students factored out the 3 and then quickly factored the quadratic into two binomials. 2x^2+19x+24 I purposely gave them this one that does not have a monomial to factor out. I wanted to see what they would do and make sure they saw that not all quadratics with a non-one a-value would factor out a monomial. They started by trying to factor out a 2, but I heard discussions about being uncomfortable with the decimal coefficient that remained. They quickly scrapped that method and went right back to factoring using the area model. The groups all got (2x+3)(x+8). We wrapped up the board work by discussing why this factorization was different than the ones they originally got with the first problem (how they could factor out a 2 from their linear factor, but can't with this one). The remainder of the hour was spent practicing different problems. It went smoothly and any of their questions were discussed in their groups and figured out on their own. It was fun to listen to their discussions and reasoning. I have always been frustrated trying to teach students how to turn the slope-intercept form of a line into the standard form. I cannot count how many times I have pointed out that, to get rid of the denominator, students can multiply the equation by the denominator. Why wasn't this working? Because I was the one saying it. As the saying goes, "The person doing the talking is the person doing the thinking". I have heard this for years, but I didn't know how to utilize it until recently. I changed my lesson drastically, and this time, the students did the thinking! I started the lesson by showing them the following three equations. I asked them what they noticed and annotated what they said on the board with the equations. I then had them type all three equations on Desmos and tell me what they noticed about the graphs. They realized that the three equations graphed the same line. We went back to the equations and talked about slope-intercept form versus standard form of a line. The last two equations are in standard form and we concluded that standard form has no fractions and that x and y are on the same side of the equation, with the constant on the other side. Next, I randomly grouped the students into groups of 3 and sent them to the whiteboards. (One marker per group, the person doing the writing is not deciding what is being written). I challenged them to use algebra to change the slope-intercept form of the line from above into standard form. They worked for several minutes, trying different things. (I did not give them any ideas or tips at this point) It was time to move them forward and I decided to try something new at this point, workshops. (I read about workshops in Geoff Krall's book Necessary Conditions. I highly recommend the book! ) I paused the class and asked one person from each group to come over to me and the rest keep on working. I wasn't sure how this moment would go, but they did exactly what I wanted, one person came over to me and the rest turned back to their boards. It took a matter of 5 seconds. When the small group of students got to me, I wrote down the equation in slope-intercept form and asked the students what they have done so far. They gave me a couple of answers. One of their answers was that they moved the term with the x over to the other side by subtracting it on both sides. They also talked about how they wanted to get rid of the fraction, but weren't sure exactly how. I asked them what they had tried so far and I found out that they knew what to do, they were just insecure. I told them that they had great ideas and should try them. They went back to their groups to report what we did. Some groups went on with what they were doing and some went back with new ideas to try. When groups finished, they were able to check their answers against the standard forms on the board. I then gave them a new equation in slope-intercept form to change into standard form. As they finished, they wanted me to confirm if they were correct. I decided instead to have them check using Desmos. But, instead of telling them exactly what to do, I asked them what happened when we put our original equations into Desmos. This was enough to spark the idea of how to use Desmos to check their answers. I wanted the lesson to progress, so the next problem was a little bit different. I gave them the following problem: A line goes through the point (2,7) and is parallel to y=3/4x+5. Write the equation of the line in standard form. At first, the students were convinced that they couldn't do this problem. My first response was to tell them that I wouldn't give them something they couldn't do. About half of the groups went on to figure it out themselves. For the rest, I asked the following questions:
Students jumped back into the problem at various questions. This is the fun part as a teacher using the thinking classroom structure. Students are motivated to solve problems, they are invested in it and want to keep going! Once again, students used Desmos to make sure they got the correct answer. I visited each group to check how Desmos helped them. I wanted to make sure that they plotted the original parallel line, the point that went through the new line, the equation in slope-intercept form and their standard form. It turned out to be an important conversation with the groups. All the groups made it to the last problem (above), but not all of them made it to the next one. This is to be expected, given that the groups work at their own pace. I was happy as long as they made it to the last one. If they hadn't, I would have continued the next day. The next problem was intended to push students who were ready for a push. One of my favorite tasks for increasing Depth of Knowledge is from openmiddle.com. I finished this lesson with the following problem: It was fun to watch groups tackle this problem. None finished during class, but many came back the next day having worked on it outside of class. Of course, we used Desmos to check.
We followed the lesson up with practicing changing from slope-intercept form to standard form independently. Note: This was my first independent lesson in the thinking classroom. Seeing my students work independently and with self-motivation sold me on the teaching method. Now, I often stop and just soak in the math discourse and learning going on in my classroom. My students are now the ones doing the talking, and therefore doing the learning. I have recently realized that my classroom is now truly student-centered. Although my involvement with questioning and helping is important, when they leave and I reflect on the class, my thoughts are about how hard they worked and how well they did, not about how well I did presenting the lesson. |
AuthorI teach mathematics for grades 7-12. Teaching mathematics is my passion. Archives
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