So, I’ve been using my ipad pretty much every day. I mainly use it as a sort of mobile document camera, snapping pictures as I walk around. I usually then project the work on the board and ask students to explain it for the class. This gives me a chance to ask some good questions like: “Why did you multiply by 99 instead of 100?” “Is there really a force pushing on he ball?” “If your rule is +2 every step, how many will there be in the 73rd step?” Sometimes we spot mistakes, sometimes I purposefully get two pics to compare, sometimes we are surprised by different methods, always good learning happens! I’ve got hundreds of these now:
At Twitter Math Camp this summer, I got the opportunity to work with a small group of amazing Algebra I teachers on planning a linear functions unit. While the crazy of school has thwarted me from following through on all the plans, I still got in a thrown together version of one today.
So, the idea was to compare pizza companies and see who had the best deal. But each company provides us with different info about their prices. Check it:
To get things started, we all did Pizza Hut as a class. We decided on the function c(t) because cost depends on toppings. We wrote the equation c(t) = 10 + .85t. We used the function to generate a table using the domain [0,5] because some people might want cheese pizza and 5 is a good max number of toppings. Then we graphed y = c(t) using the same domain and we were done! They did most of the work, I was just there to prod them along if they didn’t know what to do next.
Next, I told one half of the class they were to be my “Papa John’s Experts” and the other half they were my “Domino’s Experts”. If someone had an issue figuring out one of the representations, they were to see an “expert” from the other side of the aisle. It worked great most of the time; they found help on their own when they needed it.
I thought the best part was finding the cost per topping for Domino’s. Since they were only given two points, it was really like asking them to find the slope of the line between (2 , 10.7) and (5 , 14). But in context, it was not a scary math problem, just “well, there’s three more toppings there and it costs $3.30 more, so it’s $1.10 per topping”(their words). Later I showed them (out of context) that they had done the good ol’ (y2-y1)/(x2-x1). One kid even seemed mad that I had “tricked” him into calculating slope, as if he was proud of not being able to do it and I ruined all that for him. Weird.
Note: It is fun being able to refer to the y-int as a cheese pizza all day! (Also, it was a great follow up to the 100×100 burger problem from Kaplinsky in which the y-int is a bun and dressings.)
Next class, we’ll choose a company and make whiteboards to defend the choice. Is Domino’s really always the cheapest? Is cost all we care about? Tune in next time to find out. :)
Oh yeah, and I started class with a pizza themed counting circle! We started at 10.5 and counted up by 1.25 at a time. It was a good one, it went pretty fast. And we did the “toss the ball and make a prediction” game, where I call “stop” and whoever has the ball tosses it across and we have to predict what that person will say. I <3 Counting Circle!
It has been way too long since I’ve blogged, but from what I’ve been reading lately I am not alone on this. Like many others, I have been feeling a little overwhelmed this year. It is tough trying to keep up with new curricula in Algebra and Physics, and I’m basically constructing my textbook as I go too. </whine> So far, I’m happy with the way things are going in both classes. In Algebra, I’m using the learning targets that I wrote over the summer (modifying as I go of course) and I’ve been using a bunch of the handouts from Connecticut’s model Alg 1 curriculum (click “CSDE” then “Mathematics” then “CT Common Core Alg 1″, thanks to Jen Silverman for sharing that resource with me!). If you teach Algebra I or MS Math, I’d give them a look at least. (Edit: guest password is CSDE.)
I’ve also been writing a class blog to keep in touch with students and parents. That blog also serves as a model for my students when they finally start their blogs (we have them set up but haven’t started posting yet). I’m thinking maybe we should do an Explore-MTBoS-style series of missions! Gotta get to work on that with all my free planning time.
A highlight for me this year has been doing a lot of problem based lessons. Some I got from the CT curriculum, but mostly from online sources like MARS, 101qs.com, or the twitterblogosphere in general. I’ve really been enjoying this type of lesson and I think most of my kids have too. Many parents told me at back to school night that their kids were enjoying the class. However, I’ve found that many kids are struggling to make the connection between the lesson and the assessment (I have the same problem with modeling instruction in physics). My worry is that if I don’t come up with some fixes for this problem, I’ll revert to old ways and abandon the problem based course design (which I think is going well otherwise). Anyway, here’s an example of my assessment scheme:
First, the lesson handout:
So, we did this handout. As a class, we noticed that the cost would depend on whether or not the dog needed a bath each visit. We decided that it was a dirty dog, so yes it did need a bath every time. Then, we wondered which would be the better deal and decided that it would depend on how many times you visit the place. With some guiding questions, groups of kids modeled the scenario, made some nice TI and Desmos graphs and most decided that they should go with Super Dog Delight because it would be cheaper in the long run.
Ok, so one of the skills I am looking to assess after this activity is solving equations with variables on both sides. Here’s how I did that:
1. Formatively assess during the activity. Notice the different methods for writing and solving equations, point them out to the whole class. Have students demonstrate their methods. Celebrate mistakes!
2. Multiple choice summative quiz. 5 questions, randomly generated by Socratic Brain (according to my specifications). Students must get 5/5 twice (on different days) for full credit. They can take the quiz as many times as they need to. Questions look like this:
3. Screencast! Our school got an iPad cart (20), so I checked it out and had students make screencasts explaining their solutions to a particular equation. Here’s an example of one. He made some nice mistakes in there. I gave feedback, but I’d love to work out a system where they view and critique each other’s screencasts (working on it!). We also used Infuse Learning to practice literal equations:
4. The Exam. So, to assess this skill one more time, I have a scenario similar to the Dog Spa problem on their quarter exam. My strategy on the Algebra exam is inspired by a History teacher I had in high school who would give us essays on every test, but let us know the topics beforehand so that we could prepare. I decided to try something similar by giving the students the scenarios, but not the questions. Here’s what I showed them:
And here’s the questions I asked:
And I gave them some class time to prepare for them. I told them to think of questions they have and how they might go about answering those questions. I also told them to think about what questions I might ask and how they would answer those. Feelings about the exam were mixed. Some students were excited that I was giving them a peek at the exam, but others were confused as to how to proceed because of the lack of questions. I think the latter group of kids is the group I referred to earlier. They’re the ones who are not making the connections between the activities we do in class and the skills required for the assessments. Like maybe they participate in the group activity by making a graph, but then never learn about writing an equation (“my part’s done here”). Now that I’m thinking about it though, it seems like the solution might be more problem solving activities, not less. That would mean more chances for that kid to write some equations, right?
So why the title “Function notation can wait”? Well, I started the year with my Unit 1 called “Modeling with Functions”. But, I found that my students were thrown off by the notation and it was hurting the learning process. For instance, we did the checkerboard border problem 3-act-style (thanks to Dan Meyer for the fancy graphics). Here’s a few examples of what they came up with:
They explained how they saw the pattern very well and then also used function notation in their equations. But when I asked them to explain why they wrote the equation that way, I got “Because that’s how you did it”, and that’s not what I want! They should use function notation because it’s useful as a labeling tool, not because I said so. Other groups wrote the equation as (# of blue tiles) = 4*(width of smaller square) + 8, which I like much better anyway. I don’t want to confuse anyone when they clearly get the idea conceptually.
So, new title for Unit 1: Patterns and Modeling. Function notation can wait.
Here are just a few of the ways I’ve used Desmos in my class along with the CCSS Math standards that apply.
1.) Recognizing that graphing the functions f(x) = 3x – 4 and g(x) = 6 gives you the solution to 3x – 4 = 6. I used to think that this was a waste of time, since solving this algebraically seems easier. But having the resource of Desmos there (that kids like using and see the value in), makes it simple for them to see the connection. Then when we go to solve Ix-3I = 7, they can find the solution graphically first and work up to an algebraic solution. Same goes for exponential, quadratic, anything! I never saw the value in it until we starting using Desmos regularly. But it’s even a CCSS! Check it:
A-REI.11 states: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using DESMOS to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
(I may have made a small edit there)
2.) Finding a fit line and then adjusting a fit line (Are you doing Barbie Bungee this year? Me too!). This one is so easy to do with Desmos. The one I did last year was ok; I had the kids measure their arm length (from elbow to fingertip) and their foot length (no shoes, heel to big toe). Once each group was done measuring, they came up and entered their data in Desmos on my computer which was projected on the board. It looked like this. Then we discussed the best model to use here (linear was easy to agree on) and used that. Then it looked like this. So then I just shared that graph with every group (I used email to send it out, but Google drive may be an easier way to share), and had them write the equation of the fit line as they saw it. The great part here is that each group comes up with a different equation. Then, I had each group make a prediction about my arm length given my foot length and we could see who was closest. It’s a nice way to add a little friendly competition and get to discuss why the best prediction was the best. This is also a good place to discuss the meaning of slope and y-intercept. Should the y-intercept be zero? Does the slope have units? There’s really a bunch of CCSS that go with this one, but mainly I was thinking:
S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
- Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
- Informally assess the fit of a function by plotting and analyzing residuals.
- Fit a linear function for a scatter plot that suggests a linear association.
3.) Exploring function graphs. Check out Building Functions 3:
F-BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
OMG! That’s so complicated sounding! But if you have students build on a parent function, they’ll start to be able to tell the difference between f(x) + k and f(x+k) even if that’s not how they’d explain it. Here’s the handout I used to do vertex form quadratics last year (needs a little work, but you’ll get the idea I think). Once they’d built up some rules for how to shift the graph around, I could show them the general vertex form graph and they could make some predictions about what the a, h, and k would do. Bam! Function built!! (of course that’s not the end of our quadratics explorations, it’s the beginning!)
More specifically, Twitter Math Camp 2013! Here’s what I got:
(note: these are just some highlights, I learned way too much to capture in one post)
First and foremost, I learned about how to teach math better. That was definitely the focus of the conference. And it wasn’t just obvious to us classroom teachers, see Eli’s take on it (from Team Desmos):
So, the ”big picture” takeaway for me was definitely Max Ray’s ”I notice, I wonder”. I just love the wording of that! It’s so inviting; everyone can notice something, and everyone wonders about things. The basic idea is that when you present the students with the scenario you want them to analyze, first ask them what they notice, then what they wonder (and give credit to all ideas). Simple, brilliant! And when I really thought about it, I noticed that I had been trying to ask these questions of my students all along (in both my physics and algebra classes), but I had been fumbling the wording. I would say things like “so how does this work?” or “what’s going on here?”. Duh, Mr. Owen, if they knew what was going on already, then they wouldn’t need the lesson. In Max’s session, I noticed that after we had done a few minutes of noticing, some people started throwing out things that they wondered too. I wonder if it’ll be that easy with students, I hope so.
I also learned about what makes for a good group work task and how to implement one. Thanks to Anna (@Borschtwithanna) and Jessica (@algebrainiac1) for that one. Main takeaway from their session was that to be a good group task, there need to be multiple obvious ways to solve it. What a great way to get kids talking about math, have them solve the same problem and then debate who did it the “best” way! That may not have been the main point of the session, but that’s what I got out of it.
While that session was on the topic of group dynamics, I also attended a great session on building a class community. It was Sadie’s (@wahedahbug) session on Counting Circles. A counting circle is a classroom routine that builds number sense. It’s pretty much what you would think a counting circle is. Sadie suggests chapter 4 of this book to get started (I have not had a chance to check it out yet, but will soon). The book is for K-3, but the concept is totally scale-able. It’s not just about counting! For instance, let’s say we’re going around the circle counting up by 1/2s, then I stop and ask “what number will Suzie be?” (Suzie is 7 kids from where we stopped). Some of my kids will go straight to a linear function to solve that, some won’t. But that’s just it, the methods will all be on the table so we can share and learn. But I think the biggest reason I’m gonna use it is for the confidence building. As we started the counting circle with a group of math teachers, several of us commented that we were nervous. Imagine how the kids will feel! The thing is, we’re all in it together. I felt like a part of a mini community in our session and that was after doing it once. If I make this part of my class routine, my hope is that the kids will feel that same sense of community. This should make them more confident and less afraid to share out, and that’s in addition to the numeracy skills we’re building.
It was also great having Team Mathalicious there. I had already planned on utilizing some of their lessons, so having them explain their thought process really helped me out. I feel that after attending their sessions, I’ll be not only better at implemented their lessons, but also better at writing my own real world lessons, based on my own and my students specific interests. It didn’t hurt that Steve Leinwand and Ann Lawrence were both in my lesson writing session, throwing out ideas and asking good questions.
I also learned way too much to type by just talking to people. The importance of these face to face encounters cannot be overstated. I think that was the best thing about the camp. Every conversation I had was amazing! Some were not math related (which is a good thing!), but a lot of them were. And it was super obvious that everyone there has a passion for what they do. Specifically though, I saw a passion for math (or maths for our British friends) in many of the attendees. What I mean is that they were in one way or another involved in doing mathematics, not just teaching it. In his Mathematicians Lament, Lockhart talks about how we don’t get music teachers that don’t play music. The implication is that we do have math teachers that don’t “play math”. And that has definitely been true of me. It’s not that I don’t like math (especially to play with), but I haven’t been doing it, really.
So I had a few conversations with Edmund Harriss (@Gelada) and he totally convinced me that I need to. Not because I’m a terrible teacher if I don’t play with math and constantly work at new problems, but rather because it is fun. For example, he asked me if I had heard of the 3x+1 problem. It goes: take a number, if even —> take half, if odd —> 3x+1 it, then repeat till you get to a loop. Try it with 3: 3 – 10 – 5 – 16 – 8 – 4 – 2 - 5. And bam! there’s the loop! Now try 23. Have fun! But where it really gets interesting is in the mathematical art. Check out some of Edmund’s stuff, amazing! And after seeing the things he did with Desmos after only a few days of playing around with it, I’m convinced that I need to play around with math more. Of course, kids like to play too, I just need to give them some toys!
So, there it is. Teach math better, play with math more.
I have enjoyed “lurking” around the conversation about what the mathtwitterblogosphere is, where it could go, and how it can be more welcoming to newcomers. I really liked Sam’s post because he talks about how it’s different for everyone and you get out of it what you want or need. He’s great at welcoming newcomers and making them feel comfortable contributing. I joined his new blogger initiation and that’s what moved me from lurker to contributor. I also really liked Kate’s post on the culture of the MTBoS. Her comparisons are spot on and very interesting, you should definitely check that one out if you haven’t already.
Like most things I write here, I feel like someone (or lots of someones) already thought of this and wrote their own similar posts. But then I write it anyways. Because this is my blog and I write what helps me. If I stopped every time I felt like my idea was unoriginal, I wouldn’t have written ANY posts this year. So here goes:
I think the #MTBoS is like a game.
Ok, so I just got back from ISTE13 and one of the keynotes was by Jane McGonigal. And I just started reading her book Reality is Broken. Fifty pages in and it already blew my mind. So that’s where the idea comes from obviously (or not obviously if you’ve never heard of her), but I think it’s a good comparison. Here’s why:
1.) You tend to get out of it what you put in. You can lurk around (like everyone does at first), but then you’re gonna stay at level one. If you want to “level up” you’ve got to participate. Joined a conversation: achievement unlocked! You earn two follows and one sweet lesson idea. Wrote a good blog post: achievement unlocked! You earn two more follows and a worksheet on factoring.
2.) Participation is voluntary. Why do we put in the extra time here? To become better teachers? I would think that would be the most common answer if I gave a survey, but does it really answer the question? Why are we trying to become better teachers in this specific way? Especially since…
3.) It can be really hard work. I put in a lot of time (as I’m sure you do too if you’re reading my little blog) on Twitter and Feedly, trying to get everything I can out of them. I am constantly checking my feeds on my phone, my iPad, and my computer, at home and at school. Why so obsessive? Well, I think it has to do with this:
“Strangers tell me at conferences how much the blogosphere has meant to their practice and their happiness.” Dan Meyer said that earlier today in a comment on his blog. That’s it! Their practice AND their happiness. Just like gamers wouldn’t keep challenging themselves if it didn’t make them happy, I think members of the MTBoS are putting in the extra hours because it makes them happy too! Hence…
4.) Intrinsic rewards. From Reality is Broken: “The scientific term for this kind of self-motivated, self-rewarding activity is autotelic (from the Greek words for “self” auto and “goal” telos“)”.
First, that quote made me think of this clip (a student talking about why he likes to play Minecraft) (Side note: that clip is from this video by Douglas Kiang which is a wonderful breakdown on what gamer mindsets can teach us about education.)
Second, it made me think about what keeps me putting in the extra hours to keep up with the MTBoS. Jane McGonigal describes the four intrinsic rewards that are most “essential to our happiness”. I’ll just direct quote those bad boys here (emphasis hers):
“First and foremost, we crave satisfying work, every single day. The exact nature of this “satisfying work” is different from person to person, but for everyone it means being immersed in clearly defined, demanding activities that allow us to see the direct impact of our efforts.”
“Second, we crave the experience, or at least the hope, of being successful. We want to feel powerful in our own lives and show off to others what we’re good at. We want to be optimistic about our own chances for success, to aspire to something, and to feel like like we’re getting better over time.”
“Third, we crave social connection. Humans are extremely social creatures, and even the most introverted among us derive a large percentage of our happiness from spending time with the people we care about. We want to share experiences and build bonds, and we most often accomplish that by doing things that matter together.”
“Fourth, and finally, we crave meaning, or the chance to be a part of something larger than ourselves. We want to feel curiosity, awe, and wonder about things that unfold on epic scales. And most importantly, we want to belong to and contribute to something that has lasting significance beyond our own individual lives.”
So, there she was describing what makes us happy, not exactly what about games makes us happy, just in general. You might read that and think “Yes! And that’s why I am a teacher!!” So where is the connection between games and the MTBoS? I think it’s here, where she makes the connection between games and happiness:
“[Games] actively engage us in satisfying work that we have the chance to be successful at. They give us a highly structured way to spend time and build bonds with people we like. And if we play a game long enough, with a big enough network of players, we feel a part of something bigger than ourselves – part of an epic story, an important project, or a global community. Good games help us experience the four things we crave most – and they do it safely, cheaply, and reliably.”
There it is! Just replace gaming with “math nerd internet time” and she makes a great case for how the MTBoS makes me happy and keeps me coming back for more.
So, here’s some stuff I’m working on this summer:
1.) Re-working Algebra I units and learning targets to make them more coherent. After using SBG this year, I found that some units had way too many targets and some had too few. Also, some targets were really the same as others, and they needed to be combined; whereas some targets were too complex and needed to be split into two separate ones. I also need to split up my learning targets into the categories Fundamental (procedural), Core (conceptual), and Advanced (synthesis). I think I’ll find that after doing this, I am woefully lacking in the Advanced category. Must fix!
2.) Writing algorithm generated questions for practice and assessments. For any target that I can legitimately assess with a multiple choice or numerical response question, I want to have a bank of algorithm generated questions so that I don’t have to write new ones every time a student wants to re-assess. They just go to SocraticBrain, log in, then enter the quiz password to re-assess. I want to get these written and out of the way so I can spend more time writing task and project based (Advanced category) assessments.
3.) Getting to know SocraticBrain, the SBAR system I will be using for Algebra I and Physics next year. There are some new features and I’ll need to figure out how they work and how I’m going to implement them. For instance, it is now set up to automatically generate practice for students based on their scores on any recent work they’ve done. Awesome! But it means that I’ll have to be very careful when deciding what practice problems or tasks the system will spit out based on the areas the student is struggling in. When a student keeps missing a fundamental question, maybe they just need to see an example worked out and then practice some more. In this case, the system should spit out both resources (a video with a worked out example, and some practice problems). When a student fails on an Advanced task, more skills practice won’t necessarily help them. What they probably need is more specific feedback. I think in this case the system should spit out a self reflection form based on the task. Questions like “What parts of the task did you get stuck on?” and “What skills do you think are required for this task?” could go on there. Then when it comes time for me to enter the picture, they will already have thought about these things and my feedback will be more useful to them.
4.) Re-working physics units and learning targets to match the NGSS. This one I’ll be working on with the whole science dept at my school. We’ve got a full two weeks blocked off for this. It’s awesome that we’re doing it together; we will hopefully end up with a nice vertically aligned science curriculum. Once the physics learning targets are made, I can start to align the Algebra targets to them.
I plan to post my aligned Algebra I/Physics units and learning targets by the end of the summer for review by the mathtwitterblogosphere.