I'm struggling to find time to write thoughtful posts. But I am taking pictures. Here's some fun shots of our Ball Drop experiment in Algebra 1.
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This past summer, I was honored by the Colorado Council of Teachers of Mathematics as the 2017 Outstanding Secondary Mathematics Teacher for Region 2. One for each region of the State. I didn't post this in September when it happened, because, honestly, it felt really weird to have people congratulating me. And then life showed up and it got lost as it moved farther down the 'to do' list.
I spent some time this past week reflecting on 2017. While going through my journal I was reminded of this wonderful little event. Part of my desire for this next year is to celebrate more. This is my first attempt.
My school district even gave me a shout out on Twitter:
I'm very thankful to all of the wonderful educators I've worked sidebyside with over the years and those who I've had the pleasure of interacting with online through the #MTBoS. Thank you for your kind words, your wisdom, your patience, and your friendship. I'm so very humbled that I get to be on this journey with such quality people. Without a doubt, I'm a better teacher because of your encouragement to try new things and to fail early and often. You all are the best! I truly believe we are better together. Always. I spent some time searching for a card matching exercise for exponential functions. I found a couple but none that did exactly what I wanted them to do. Rather than spending an hour searching for what I wanted, I spent an hour and made it. Although, to be real, I'm not sure what I made is what I wanted either. I plan to use it early next week. Students match percent growth and percent decay equations, graphs, descriptions, and tables. Not every function has a description or a table. I thought I'd cut down on the process of elimination solution method. I also included some function forms students would probably not be familiar with yet but I would like them to take a stab at looking at the form of the function and guessing what the matching graph might be like. I've included the files as both a pdf and word doc.
Update: I added an extra page with recursive definitions of most of the functions... just in case you want to add the extra step of complexity!
With systems of equations, I tried something new for review. Instead of doing the normal routine of making a review assignment and then having students solve it, and then having the students who don't do the assignment be the ones who need to do it the mostI had students make their own review assignment. Hey, why not? Here are the directions I gave to the class (link to Google Doc here):
Here are the sections we covered this chapter: 61: Graphing Systems of Equations 62: Substitution 63: Elimination using Addition and Subtraction 64: Elimination using multiplication 65: Applying systems of Linear Equations 66: Systems of Inequalities At first, there where the obligatory moans from the classroom when I introduced the assignment. But, when I told them they only had to create 6 problemone from each of the sections for the chapterthe mood changed. Don't get me wrong, they didn't cheer or anything, but I did have buyin from most of the class. When students asked me what a certain section was about, I directed them back to their textbook. I have both an inclass set and we have online pdf's so access should be an issue. Students checked their 'correct' solutions using Desmos. I appreciated how much this pushed students to understand the mechanics of solving systems of equations on a deeper level. It's one thing to use an algorithm. It's another to intentionally break the algorithm and see what you get when you break it. Sadly, this assignment didn't fix the problem of students who really really need to work on the assignment not doing the assignment. I guess if I had the magic fairy dust to fix that, I would be a very rich man. If I had to change one thing about this task next year, it would be that students didn't really know what I expected of them; there was some confusion as to what the end product might look like. So, I took pictures of this year's work to show to next year's students. I can usually go through the pictures without saying a word. Students start to get a pretty good intuition about what 'good work' versus 'bad work' feels like. This looks nice. This... doesn't look as nice. Here's some examples of studentcreated practice tests. I've included four different levels: the "Oh my goodness, can I frame this?" the "Good attempt but struggles with organization" the "I appreciate how much effort you put into this but it took me some time to figure it out", and the "Can you walk me through this, please?" Excellent Work: Good attempt but can use a little more organization to help the reader understand what's happening: Solid Effort: Can you walk me through this, please?: Thanks to Dan Greenberg for the idea!
I'm starting my second real book on formative assessment. I finished Embedding Formative Assessment by Wiliam and Leahy. Now I'm starting The Formative 5 by Fennel, Kobett, and Wray. Something I've noticed about both books: neither one suggests using warmups as a researchbacked strategy in the classroom.
I'm thinking about dropping them. Granted, I've only done them for like 3 months but I'm frustrated. My actual problem is that I can't fit everything into a lesson I want to fit in. I want to: i) have a discussion about a warmup ii) check and go over questions on the previous night's practice work/homework (quick stat: 8090% of my students do the homework on any given assignment) iii) teach an interesting lesson/investigation/experiment/whatever iv) and to top it all off, have students complete an exit ticket As it turns out, that's too much to pack into one day. I talked this out with one of our district middle school math coaches (Dan Greenberg) yesterday. He's like a ninjamatheducationguru. It's awesome. After turning my questions back on themselves several times, he helped me figure out that warmups are the least productive of the options I want to pursue. Rather than trying to spiral in old ideas, or bring up Visual Patterns, or Which Ones Don't Belong, or whatever other good 'extra' things I want to do in a day, maybe I could use problems from the previous night's homework as a launch for group discussions. It would help students see the value in doing the homework. It would be a natural tie in of the previous day's work to what we would be looking at today. And it would help me better understand what my students do and don't understand about what I'm attempting to teach. Those other things are totally cool, I simply need to plan time for them rather than trying to shove them into an already full plan for a day. So, I'm thinking about letting something good go. I spent days pulling together what I believe is an allstar cast of warmups to use in Algebra 1 this year. It's hard. This is my second attempt at a D.I.Y. iPad document camera stand. The height is adjustable via the screw at the back of the post. The leg towards the bottom of the picture is extended from the post to allow the iPad camera to center over the object better. I created a swivel to allow it to turn into an iPad tripod. I use it to take video in my classroom. Mostly, this ends up with me taking video of an experiment so my students can get better measurements. Here, my students are taking video of the experiment. Did Barbie hit the ground or not?! Instant replay. Here's a screenshot of the iPad video. Not too shabby. And, it's collapsible! I've added to a previous post to create some detailed instructions here, if you're interested.
This is the second in a series of posts about formative assessment using some of the tools outlined in the book Embedded Formative Assessment by William and Leahy. The first post is here.
No Hands Up, Except to Ask a Question AND Rough Draft Thinking Every teacher intimately knows the problem of only 4 or 5 kids raising their hands to answer questions. This is meant to address that problem. I want to know what every student is thinking not just 4 or 5 of them. Rather than asking the entire class a question and then calling on students that raise their hands, I've moved to calling on students based on a set of cards for the class. Before every class I shuffle the cards. When I ask a question, students put their hands up but I remind them that I'm only calling on students with the cards. By no means is this a new idea in education. I tried it in my first couple years of teaching and it went very poorly. Students felt uncomfortable with me putting them on the spot let alone without knowing how to do the problem. Add on top of that my inexperience as a young teacher who still struggled with classroom management, along with my inability to foster a productive classroom conversation, and you had a recipe for disaster. This time things went incredibly well! What went differently? Rough draft thinking. Last year, as a school, we worked on writing across subjects. When I sat down with the English and History teachers and heard them talk about the significant revision process in their classrooms I was struck by how different the math classroom tends to be. We want our math students to try new ideas and embrace failure as part of the learning process but we only take their first answer as either right or wrong. Based on that conversation I attempted to shift classroom culture to embrace the revision process by talking about rough draft thinking. When I ask students to do something, I don’t expect them to do it correctly the first time. Nowhere else in life do we expect that. We expect that our initial attempt will be riddled with problems but at least it’s a starting place. It’s now part of the mantra in our classroom. I expect students to be kind and to be brave. Because I’m going to call on students using the cards, I need them to be brave. I’m expecting them to present their rough draft thinking. It’s nerve racking to present information when you feel very unsure of yourself and you know it’s probably incorrect. That’s okay. That’s the expectation. I expect you to present something that will be a starting point for our classroom conversation not the ending of it. For the rest of the class, I need you to be kind. There can be no laughing, or snickering, or giggling. At all. It doesn’t really matter whether it’s directed at the presenter or not. The presenter will feel like it is. We will honor each other’s ideas and critique them with care. If you're the one who yells out “That’s stupid!” I will ask you to leave without any questions asked. Can we agree on this together? Everyone nods their heads. This has gone incredibly well! The level of discourse in my classroom has been phenomenal. In late August, I had students taking chances and buying into my narrative of what a math classroom could look like that normally doesn’t happen until sometime in November. There has been zero push back on me calling on students who don’t know the answers. None. I can’t believe it. When a student doesn’t know what’s going on, I ask them to give a starting point for the conversation. How might they approach the problem? What information do they see that might be relevant? IDK isn’t an option if you’re called on. You have to help the class move forward somehow. I don't know who I borrowed the rough draft thinking idea from, but if it's you, thank you! I do know that Amanda Jansen (@MandyMathEd) is doing some great work in this area. We've officially been in school for one month. That means I'm one month into my push into formative assessment. My current resource is Embedding Formative Assessment by William and Leahy. I really appreciate the tone as well as the content of the book. Sometimes when you read education books by professionals it feels condescending or guilt ridden. That's not my experience so far. On a side note, I've not written in a book this much since grad school. I'll post my notes on the book sometime later. Because I have to start somewhere, I'm pulling most of my initial strategies from this book. I have a note in the margin of page 95 that says, "This pageandahalf could be my entire year of implementation".
My goal is to implement a couple of formative assessment strategies every month. This month I've tried to implement 3 of them. Well, 4, if you count attempting to be consistent in posting learning targets. I plan to split this up into separate posts, otherwise, I’ll never end up hitting ‘submit’. 1) WarmUps This really isn't much of a formative assessment strategy by itself. When I first started teaching, I used it to keep the kids busy while I took attendance and got myself ready to rock and roll. I never did much with the information I got back from the students. A couple of years (more than a decade) later, things have changed. Now, instead of trying to give myself a chance to breathe or stave off discipline problems, this is about attempting to fill in gaps in my students knowledge, keep major ideas and skills fresh, and generally push students towards deeper thinking regarding topics they’ve already seen. I could probably keep the list going for another paragraph or so. Quick little aside on this being selfserving: I'm hoping this will end my issues with students not knowing how to create a line from two points come the end of May when we spend almost 3 months on linearity at the beginning of the year. If I’m honest with myself about warmups right nowI hate them. I love what they look like in class but the amount of time warmups take is unreal. I need to figure out a way to do this quicker. Do I have a classroom conversation about the material or don't I? What's the point of having a problem in class if we don't interact over it? How do I know how well my students understand the material without having a conversation about it? But warmups always seems to take 10 minutes! That's 1/5th of my class. It'd work better if I only did warmups a couple of times a week but I don't know how to make it into a rhythm if it doesn't happen normally. Maybe I can pull back on it after another month or so. Breathe in. Breathe out. I keep reminding myselfI need to go slow to go fast. I just don’t know that I believe it every day. If you’re interested in what I’m working with, I stole a bunch of stuff from the #MTBoS and decided to compile them in different Google presentations. Whatever the slide deck I’m using for the day, when I create it, I copy a slide from one of the warmup files and put it in the daily presentation. Here’s my current treasure trove. There comes a point in your teaching career where the problems you face are very different than the problems you faced in the first couple of years in the classroom. In my first years of teaching it was all about classroom management and handling the workload in healthy ways. Then it was about creating a safe, relationshipbased space for all my students. Then my attention shifted towards student engagement. Next, it was about incorporating more student voice into the classroom. Now, it's time for another shiftformative assessment. Sure, I've picked up pieces here and there but I've not taken the time to get good at it. You know what I mean right? There's a difference between knowing how to cook and knowing how to cook. My family doesn't complain when I cook dinner. I know how to cook. But neither do they celebrate when I cook like they do when Grandma cooks. There is passable and then there is... another level entirely. Here's the problem: though I want to be a better teacher, it's really, really hard. It feels like the amount of work necessary to keep upping my game increases with each next step. Kind of like this: If I’m realistic with myself, to keep pushing the boundaries I need a community to travel with me. I need collaboration to move past the safe and comfortable. And, if I'm also being real with myself, this is going to take years. Every other transition took time, why should this be any different? Here's the rough idea I'm currently working with: Throughout the year, I would like to implement various forms of formative assessment in my classroom. Though the plan is just starting to take shape, my current thinking is that I would implement/sample new assessment tools every few weeks to a month. Each month I would then reflect on the implementation and on any evidencebased outcomes I can see in my classroom. Then my support team would help me evaluate the toolhow I might change my implementation and whether to continue using the tool in my classroom. I would like to explore some of the areas underneath the rather large umbrella of formative assessment. Particularly, I’m thinking of standardsbased grading (SBG). Many of my online math colleagues have already made this jump. And they’ve written about the process. Jumping both feet into SBG seems like too much for next year. However, I think I can realistically pick one unit and implement it in that unit. After I have the experience of doing SBG for an entire unit, I would like to talk to the math department about aligning one or our assessments with the course objectives for a single unit. I don’t yet know the specific new skills or instructional practices I plan to demonstrate. What I do plan to do is immerse myself in formative assessment. I plan to make mistakes. I plan to be frustrated. I plan to fail forward. As with all new things, the more I do it the more I will know what works and what doesn’t in my classroom. As the process unfolds, I expect to develop a list of different assessment practices, how I’ve attempted to incorporate them into my classroom, and my reflections on whether the energyofimplementationtooutcome ratio is worthwhile. Action steps:
Ultimately, here's what I want: I want to understand where each of my classes and each of my students are with my course objectives. Based on that understanding, I want to be better be equipped to tailor interventions for my struggling students and extensions for my excelling students.
The month of May can be a really hard time to be a student; not only are you counting down the days but so are the adultsthough, in my experience, not in the same way as the students. At the end of the school year I start to get a bit panicky about "covering" all the standards for the course. This puts some strain on my beliefs that going slower is in fact going faster. My kneejerk reaction is to want to lecture every day for the last 4 weeks of school. I have to take deep breathes and remind myself that letting students explore and get out of their seats means much more in the long run to both student retention and understanding than just another day of talking at them. So, in Algebra 1 we took an entire day in May and reviewed proportions (based on this beautiful nugget from Jonathon Claydon). We measured objects around the school that were too tall to be measured otherwise: walls, flagpoles, ceilings, etc. We did this by comparing our 'real height' with our 'fake height'. Think of any proportions problem from middle school and you're halfway there. We start out by measuring our 'real heights'. Then we go out and measure our heights next to an object. This is our 'fake height'. For example, say a student is standing next to a wall, depending on where his or her parter is standing, say on the meter stick the student's 'fake height' is 3.5 cm and the wall's 'fake height' is 12 cm. But, since we know the student's 'real height', we can set up a proportion to find the wall's 'real height'. 'Real world' without being trite AND it gets students out of their seatsI can't ask for much more than that in a single lesson.
What to do in Geometry after finishing up our circles unit with one day left before Spring break? Algebra review in the form of art class!
Here's some of the work my students elected to turn in. I gave them one class period to work on it and asked them to turn in whatever they had. If students wanted to do more, they could but it was not required. Hence the varying degrees of completeness. I think they did brilliant! I keep up a couple separate web pages. One for my personal use. And one for my classroom use. Below are the analytics splash screens from Weebly. Personal: School: Whatever. I normally waltz on by and go into the site editor. Here's where things start getting interesting. I installed some Google Analytics trackers on my webpagesbecause I'm curious about how this whole internet deal works. Check this out. Personal: School: Wait. What?
Aside from the fact that Weebly makes me feel better about how many people visit my sites, this is stunning. Let's zoom in and narrow our focus for a moment. Look at unique users with me. Personal website Weebly: 1023 Google: 111 School website Weebly: 328 Google: 179 Not only have the numbers been reduced, they've switched! Weebly has 'personal' beating 'school' by 3:1. Google has 'school' beating 'personal' by 5:3. I didn't see that coming. I figure the numbers are reduced based on the differences in how Weebly and Google track users. I know I for one block and delete tracking cookies on my computer (thank you Ghostery and Better Privacy) and in doing so don't show up in Google Analytics tabulations. I do, however, show up in Weebly's calcuations because they count IP addresses. But I doubt most of the internet is doing the same thing. Even so, my expectation is that the numbers for both websites would be reduced by similar percentages. Not so my friends. Not so at all. There's an interesting discussion about these discrepancies on this thread. Let's be real for a secondalternative assessments are really hard in secondary math. I have like 87 objectives and you want my kids to make a poster? For real? I've tried doing them a couple of times over the years, but they were usually such bad initial experiences I've stuffed all memories of them deep into my mental closet. This project is the exception to the burnthelessonplantogetridofallevidence I've come to expect from my attempts at alternative assessments. This is my 3rd year doing this project. Apparently, the 3rd time is the charm. I posted about my experience last year here. You can smell the fear in that post. At this point in quadratics, we've covered: graphing, factoring, the quadratic equation, vertex form, and transformations. What I really want is for students to see the connections between multiple representations. I want students to see that you can find the roots from the graph in Desmos. Those roots are the same as the roots in the factored form of the equation. Those roots are also the solutions to the quadratic formula. Slowly, we're making progress. Here's the rough outline of the project: 1) Find a "realworld" example which you can fit a parabola to. If you want to fit a parabola to Payton Manning's forehead, or Donald Trump's smile, or some other hilarious thing, that's fine with meyou'll just lose the "realworld" point. 2) Fit a parabola to the image using vertex form. 3) Show all points of interest on the Desmos graph (roots, yintercept, vertex, axis of symmetry) 4) Demonstrate you know how to calculate the avalue for the vertex form of the equation using the vertex and a point. 5) Show me you know how to find the factored form of the equation from your Desmos graph. 6) From either vertex or factored form, calculate standard form of the equation. 7) From the standard form, show me you know how to calculate the vertex of the parabola. It should be the same vertex as your graph. 8) From the standard form, use the quadratic formula to calculate the roots of the equation. These should be the same roots as your graph. 9) Put your name on the back of the poster so I can take pictures of your work and put them online without worrying about student identities (that's one of the reasons at least). Let's talk pacing for a moment. I give two work days in class for this project. On the first day, we intro the project, work on finding a super cool picture, and fit a curve to it (#13 above). On the second day, we work on doing the math and I hand out poster board to students. I have the project due several days after our second work day. There is no time to work on the project on the due date. Students waltz into class, I collect the posters, and we delve into radical functions. Next time, I will strongly suggest to students that it's worth it to do the work before starting on the poster board. I made a template for next year (below this paragraph). I'll probably import it into a Google Doc to encourage students to type out their work. Typed posters are usually much nicer to look at than the scrawl of most of my students.
If you're interested, here's my files for the project. The directions on the left are what I hand out to students. The checklist, which is included in the directions, gets paper clipped to the front of the poster so I can evaluate student understanding without writing all over their beautiful projects.
And now for the gallery. I'm going to include the good, the bad, and the ugly so you get a realistic picture of what to expect. I make it very clear to students that it is possible to get full credit without making these display worthy. I'm most interested in the math. However, I do offer bonus points for making things pretty. First, a couple of pretty ones. The math isn't perfect, but I'm not going to highlight the mistakes for you. Here are some more student examples to give you an idea of the range of quality to expect. Not everything is super pretty all of the time. A few years ago I started introducing trigonometry using circles. It started in Algebra 2 and moved down into Geometry. By basing all trig off the unit circle, we make an easy transition from Geometry trig (finding missing sides and angles of triangles) to Algebra 2 trig (sine and cosine functions and transformations). After a couple of years, I'm never going back.
Q: If you were the person taking this video, what would the graph of your height look like? Sketch a graph. I get all sorts of mountain looking graphs or semicircles. It's what I want. We are now going to get some intellectual dissonance. After students committed to a graph, we break out the hula hoops. I borrow them for a couple of days from our PE teacher. As an aside, I borrow stuff from the PE teacher and the science teachers all the time. Believe it or not, they are excited that your doing fun stuff in math class. They want you to borrow their stuff. Not kidding. We give a twominute talk through regarding how we are going to collect data. Something like: Put some tape on the hula hoops and put a dot on the side of the hula hoop. This dot is you. Throughout the experiment, you will always measure the height of the dot. You will roll the hula hoop 10 cm and then measure the height of the dot. In cm. When you get to the end of your meter stick, keep your fingers on the bottom point and slide it back to the beginning of the meter stick. Do this for one complete revolution of the hula hoop/ferris wheel. Most groups finish with the data gathering in 30 minutes or less. As another aside, I have students start at the bottom, like they are loading onto a Ferris wheel. Technically, this will look like a cosine function with an avalue of 1. However, we're measuring height, which is a sine function. Whatever. In four years of doing this, I've never had it throw students off the scent of the math we're doing.
Here's some of the resulting graphs. I take pictures of some of the student work and we have a conversation about it. Q: What do you like or dislike about this graph? Students tend to not like the "lumpy" portions of the graph. I put the following graph on the screen in response. Students like this one much better. As if almost on cue, some student always says we should connect the points. Then we get to have a good discussion about whether the graph should be continuous or discrete. Students generally settle on the following as their "ideal" graph. We then talk about rate of change in table groups. How do you talk about rate of change with this graph? After flailing about wildly, we usually are able to give a decent explanation about whey when the point we are measuring is at the bottom or the top of the circle it doesn't have much vertical change in distance as opposed to when it is on the sides. Day 1 usually ends here and students finish their 7 questions on the handout. This year, Day 1 was half eaten by questions over a previous assignment and we gave up half of Day 2 to finish gathering data, graphing, and having this discussion. Day 2 begins with a discussion about the unit circle. The following is the definition of "sine" from Wolphram Alpha. Notice the first part of the definition. Forget the "common schoolbook definition". As a class, we define 'sine' as the vertical distance of a point on the circle to the xaxis and the 'cosine' as the horizontal distance of a point on the circle to the yaxis. It's *expletive* amazing! We talk about what this distance looks like when we move the point around the circle. Here's some of the webbased visualizations I use: Sine and Cosine as vertical and horizontal distances in the unit circle http://www.analyzemath.com/unitcircle/unit_circle_applet.html https://www.mathsisfun.com/algebra/triginteractiveunitcircle.html https://www.desmos.com/calculator/v7x6br3w6a A visualization of Sine and Cosine in the Unit Circle: (If you want to moving one, click on the link below) http://i.stack.imgur.com/p8O4P.gif This is THE move that sets up everything! It sets up talking about sine and cosine as functions. It sets up a point on the unit circle being (cos, sin). It sets up the Law of Cosines not being some weird abnormality that's never really understood by students. It sets up that what we commonly refer to as trig ratios are lengths of lines in the unit circle. It also sets up any triangle with a hypotenuse other than 1 being a dilation of a triangle on the unit circle, with a scale factor of the length of the hypotenuse. Of course we end up deriving the shortcut methods for finding sine, cosine and tangent without having to go back to the unit circle every time. However, the fact that these are real lengths and not just abstractions goes a long way with my concrete learners. Sine is a vertical distance. Cosine is a horizontal distance. Tangent is the distance from the tangent to the point on the circle to the xaxis. Which is why tangent of 90 degrees and 270 degrees is undefined. This also helps discuss Cotangent, Secant, and Cosecant when they come up later in the year. They are all actual lengths in the unit circle!
If you're teaching Algebra 2, after this throw in a little Cosine Ferris Wheel from MARS and you've got yourself a decent start to a trig unit.

Andrew Busch
I teach Math and Programming at Summit Middle School in Boulder, CO. Categories
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