tag:blogger.com,1999:blog-89387994324057505252018-03-05T17:34:13.399-05:00Encouraging Problems in the ClassroomChrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-8938799432405750525.post-33737921123436099392011-02-24T08:16:00.000-05:002011-02-24T08:16:11.408-05:00Why have I Insisted on Driving?I'm not happy with the way I've taught graphs of Trig. Functions this year (the last few years, honestly). My students inevitably get confused by the details (adding fractions of pi is extremely difficult for them, and therefore extremely distracting from what is actually going on), and I haven't really provided any real world motivation... how hard would it have been to throw some simple harmonic motion at them and ask for a model? Why didn't I start that way and let them play around with the functions to really <i>feel</i> what's going on?<br /><br />Ok, now that I've vented: What this really shows me is that way to often the "natural flow" of material in a textbook is completely bass akwards. A whole bunch of math, neatly divided into six or seven sections, followed by a section dedicated to "applications." By the time they get around to the part where we (speaking as a typical math student) finally find out why we're supposed to care, it's too late: we already don't.<br /><br />I know that is far from a groundbreaking or original statement, and if you had said to me yesterday, or even last year, I would have given you a smug smile and congratulated you for seeing things the way I always have... but now I'm really starting to become aware of the <i>allure </i>of the textbook approach: the math builds so nicely upon itself... initial problems are easy to understand, and we can apply the concepts to more difficult problems when we finally encounter them. Its so perfect... what could go wrong?<br /><br />Here's what goes wrong: no 17 year old in their right mind<i> cares</i> about anything you want to tell them unless its the answer to a question they've asked (go find a teenager on the street and regale them with some interesting facts, I dare you). If you don't get them to ask the question first, they'll never care about (or remember) the answer.<br /><br />Here's the interesting part: I'm exactly the same way. I've been re-inspired as a teacher this year by the likes of <a href="http://blog.mrmeyer.com/">Dan Meyer</a> and <a href="http://101studiostreet.com/wordpress/">Shawn Cornally</a> (two of many, but two that were the doorway to the many), who <i>scream </i>the last paragraph I wrote at me in every post or tweet. I read, nod, agree, give an "Amen" to myself, and then<b> </b>start a class talking about how the graphs for the sine and cosine functions are a natural extension of their relationship to the unit circle. Wow... feeling inspired, kids?<br /><br />I've been told the solution to this over and over again: make them want to know <i>before</i> you give them answers. Then you won't need to <i>give</i> them anything, they'll just go get it. Just like them, however, I hadn't run into the wall yet. I didn't <i>need</i> that solution to satisfy me. I don't know why it took so long (probably the same reason it takes <i>them</i> so long), but now I <i>need</i> them to ask the questions. I <i>need </i>them to <i>need</i> the answers.<br /><br />Why have I been driving the buggy this whole time? Time to drop the spurs, hang a few carrots out there, and enjoy the ride.Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com1tag:blogger.com,1999:blog-8938799432405750525.post-50531003679959903092011-02-09T10:56:00.001-05:002011-02-09T11:03:28.295-05:00Writing a Calculus Paper I had my Calculus students each write a paper on single function. I was pretty happy with the results, not because they were so great, but because they really exposed some weak spots in the group's grasp of some ideas. When the whole class is working on the same problems, its pretty rare that mistakes go uncaught and uncorrected. I think that this is often a good thing, but it was becoming apparent that the same students were correcting the same mistakes that kept being made by the same students (that sentence can't possibly make sense, can it?)<br /><br /> When forced to work independently, each student was faced pretty directly with their weak points. Some of the randomly assigned functions were more difficult to work with algebraically, but in today's <a href="http://www.wolframalpha.com/">Wolfram Alpha</a> and <a href="http://goo.gl/g7LAg">Microsoft Math</a> world, finding and simplifying derivatives wasn't the issue... interpreting them was. I'm now starting to get some second drafts, and its pleasing to see them wrap their heads around the relationship between a function and its derivatives.<br /><br /><a href="https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B4mpjRlur1yMN2IyNTFjNDItMmM2My00MzVlLTlmYWEtNjE5NGQ5ZmYxMTk4&hl=en">Here is a copy of my work</a>, which I gave to show my expectations.Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com0tag:blogger.com,1999:blog-8938799432405750525.post-51716753396615703312010-12-07T09:34:00.001-05:002010-12-14T08:06:36.788-05:00Yo-Yo's and Calculus I like problems that make my students write their own equations, and we've been working with position, velocity, and acceleration recently. Yesterday, I had them write an equation to model the height of a yo-yo from the ground and then differentiate to find the maximum velocity. The situation lends itself nicely to writing a sinusoidal function, which was what I wanted them to do, really. We got to practice relating position, velocity and acceleration, differentiating trig functions, and using the chain rule all at once.<br /> The problem is, I don't think a yo-yo behaves sinusoidally. Neither did my students, so we tried to write an equation that worked better. Short on time, we wound up with the absolute value of a natural log (it moves quickly out of the hand, changes direction abruptly, and slows down on its way back up). For an impromptu activity, its going to segue nicely into finding the derivative of <i>y</i> = ln <i>x</i>... but I still want to find a good model for a yo-yo. I may head down to the Physics lab and see if there's an easy way to take some data, but I'm wondering if this has been done before (my googling reveals much more thorough models than we're ready for).<br /><br /><br /><iframe frameborder="0" height="342" src="https://docs.google.com/present/embed?id=dgwh7mz2_2314hdrn4mr6" width="410"></iframe>Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com0tag:blogger.com,1999:blog-8938799432405750525.post-84784916407612954202010-12-06T21:33:00.000-05:002010-12-06T21:33:49.013-05:00My Grading System I asked my students what they thought of my grading system (thank you, Google Forms). As I mentioned in my last post, I'm using SBG for the first time this year, and its exciting and nerve-wracking. Most of the <a href="https://spreadsheets.google.com/pub?key=0AompjRlur1yMdGVHWllZTzNoc3JOT1ZldGNURXNRelE&output=html">feedback</a> I've gotten so far has been positive, but naturally, I want to focus on the negative.<br /><br /> Most student think that my system penalizes low scores on skills to heavily. They might be right. <a href="https://docs.google.com/document/pub?id=1y1aXogGVVKJDUV8-y44oq8SrsqdLWHEn1D2GiFtzTyo">Here is the whole system</a>, but the dirty of it is that if their score on any skill is a 2, they highest grade they can get is a B, even if every other score is a 4. If they have a 1, the highest possible grade is a C-.<br /><br /> My reasoning is that if there is a skill that you have absolutely no understanding of (score of 1), then it should be your first priority to strengthen that skill. To raise a skill from a 1 to a 2 doesn't take much... I just need to see that you've got <i>some</i> clue what's going on and what you're trying to do. You don't even need to be able to successfully solve a single problem.<br /><br /> If you're sitting on a whole bunch of 2's, then I'm pretty confident you're a C student. You can take part in the conversation and not make a fool of yourself, but you're not blazing the trail (usually). The real issue comes when a student has a whole bunch of 4's, and a 2 on one skill. They argue that the one 2 shouldn't bring their grade down to a B. I argue that they should turn that 2 into a 3. Is that too simplistic?<br /><br /> I know there are so many other systems out there for converting a list of skills into a letter grade... Anyone have a favorite?Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com0tag:blogger.com,1999:blog-8938799432405750525.post-27194153735906770042010-12-01T21:36:00.002-05:002010-12-01T21:57:44.815-05:00Tell 'em What They Know... So a colleague (who I will pretend is not the only one reading this) asked me about how I share my students progress with them. Apparently, he'd heard good things from students, which is nice, because I only hear about the bugs in the system. He also made fun of the fact that I took the crappy worksheets I wrote for when I was out of town and shared them with the world, saying that it wasn't very "21st Century" of me, so I need to dig myself out of that hole...<br /><br /> I use Google Spreadsheet to keep my gradebook, which makes it "easy" to share my gradebook with my students. As you probably realize, high school students love comparing themselves to one another, so giving them access to everyone else's grades isn't a great idea (probably not great for job security, either). I'll try to explain in this post how I show each student <i>only</i> their own grades, and keep it updated automatically when I update... buckle up.<br /><br />Step One:<br /> I published the gradebook as a webpage. This sounds terrifying, but no one gets the address. This is really the only way I could find to export the data to another spreadsheet (<i>not</i> just another sheet in the same file), which is what I needed to do next...<br /><br />Step Two:<br /> When you publish a Google Spreadsheet, you have the option of copying a link to specific portions of it. I chose the column that represents each student's scores, and imported that into a separate spreadsheet for each student (be sure to "republish whenever changes are made". Now, every student has their own spreadsheet that has a list of the skills we've covered so far and their current score on each (I use <a href="http://101studiostreet.com/wordpress/?page_id=114">Standards Based Grading,</a> which <a href="http://larkolicio.us/blog/?s=sbg">many others</a> have <a href="http://alwaysformative.blogspot.com/p/standards-based-grading-implementation.html">covered </a>much better than <a href="http://blog.mrmeyer.com/?p=346">I ever will</a>). This is the tedious step, since I had to create a spreadsheet for each student... probably only took a couple of hours total, but its a lot of annoying cutting and pasting. Not fun.<br /><br />(<a href="http://docs.google.com/support/bin/answer.py?hl=en&answer=82233">Here</a> is where I learned to export (publish) and import into another spreadsheet. Much clearer than my explanation)<br /><br />Step Three:<br /> This I learned the hard way: I hid the row that contained the original link to the published data. It turned out my students could click the link and see my whole gradebook. Not good, but luckily they told me about it early on, and I could fix it. Luckily, I had imported only the first row, then auto-filled the rest. Only the first row contained the link to the published data, so I could hide it and not worry.<br /><br />Step Four:<br /> I gave each student <i>read-only</i> access to their spreadsheet. Vwah-la (French for "Holla!").<br /><br /><br />***Of course I would never be able to do this sort of thing without the incredible tech department at my school. They're a great resource that I know most teachers don't have! Thanks, guys!Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com3tag:blogger.com,1999:blog-8938799432405750525.post-58014916661867642922010-11-30T20:34:00.000-05:002010-11-30T20:34:47.411-05:00Teaching in absentia...So, I missed three days of class to attend a wedding in Belize... feel bad for me? Three days is a lot, and I really wanted them to be somewhat productive. For my Precalculus classes, I designed some <a href="https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B4mpjRlur1yMM2NhOTFhNmMtYmUwNC00MzI2LWIyN2QtOTM3NzdjYjdiMzcz&hl=en&authkey=CNSJ-t4O">worksheets </a>that just barely pushed the boundaries of what we'd been doing. We'd been translating graphs, and I inched them into some dilation (or "scale-change" as our text book calls it) on top of that. I think they were productive, mostly because I didn't try to go too far.<br /><br />In Calculus, I was ready to push the class a little harder, so I recorded a <a href="http://www.vimeo.com/17350372">video</a> introducing them to related rates. This is the first time I've ever done this, and it was (eventually) pretty easy. I used <a href="http://camstudio.org/">CamStudio </a>(free), and didn't even bother with any editing. A single take... eat your heart out, Scorcese.<br /><br />I think the video got pretty boring, but I like the idea, and might try it some more. I know there are tons of this type of thing out there, but I like making my own and I'm interested in tips for making them better. For instance, if anyone has any recommendations for editing software?<br /><br />As for the effectiveness of the video, I think it worked well as an introduction. I didn't push things too far (no substitution of variables from another equation, for example) and I tried to hammer on the idea that this is just the chain rule. This is just the chain rule. This is just the chain rule. Now I've been back for a couple of days, and we can start to explore those realms. But the video helped me at least feel like I had a head start in making up for the three days I fell behind.Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com0tag:blogger.com,1999:blog-8938799432405750525.post-82723324837791350422010-11-15T12:22:00.000-05:002010-11-15T12:22:12.470-05:00Graph Translation Reflection So I did my <a href="http://encouragingproblems.blogspot.com/2010/11/brainstorming-graph-translation.html">graph translation lesson</a> with two of my three Precalculus classes (the conversation in the third class went a different direction, and it seemed to me at the time that the students in that class were more comfortable seeing the correct direction in a horizontal translation). The lesson ended up being much smaller in scale than I initially imagined, with 5-6 students in the front of the room, each with an assigned <i>x-</i>value. I then gave them each a different shape to hold (I borrowed some extremely large pipe cleaner type things from a colleague). I explained carefully that the shape represented the "value" of the function. Finally, I said, "Instead of your value, please take the value from <i>x</i> minus 3." After some fumbling, the function moved three spaces to the right. After a few repeats, they had it cold.<br /><br /> So did it help them understand? They seemed to shrug it off as if I had gone to a lot of trouble to demonstrate the most obvious thing in the world. However, in the days since, I've not had a single question about why the graph moves in the "opposite" direction... <i>except</i> from the class that didn't do the demo. Feels like positive feedback to me!<br /><br />Note: I didn't end up video taping the lesson... it felt like that would make the students think it was more of an "event" than it actually was. Plus, I started worrying about putting my students out there on the internet without notarized consent from their parents, their attorneys, and their parents' attorneys.Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com0tag:blogger.com,1999:blog-8938799432405750525.post-77492794143264747192010-11-09T20:54:00.001-05:002010-11-09T21:04:33.053-05:00Brainstorming: Graph Translation I'm working an idea out in my head... I'm about to start discussing graph translations with my Precalculus students, which is a topic that I've never quite felt like I've really been able to guide my students to <i>understanding</i>. Instead, they memorize rules (this happens in all sorts of topics, of course, but I'll leave those for another day). In particular, I'm not satisfied with their understanding that when <i>x</i> is replaced by (<i>x</i> -2) in a function, the resulting graph has shifted by 2 in the <i>positive</i> direction, while (<i>x</i> + 2) would shift the graph in the <i>negative</i> direction. They seem to balk at first, and then just accept it as one of those weird math rules that I invented to make their lives a bit more complicated. They don't seem to reach the level of understanding that I want to see...<br /><br /> So here's my idea, though I'm changing details as I write: I want to line the students up (outside?) so that each of them represents a number on a number line (the <i>x</i>-axis). They will literally hold a function (colored ribbon at certain heights? a rigid model of a function?) in their hands. This way each student (<i>x</i>), will have a value for the function (<i>y</i>) in front of them. What I want them to see is that when I ask them to take the <i>y</i> from the person<i> </i>(<i>x</i>) who is two to their <b><i>left </i></b>as their own, the function moves to the <b><i>right</i></b>.<br /><br />Still TBD:<br />--will they have a solid object as the function? numbers to hold up? a ribbon? a lane line from the pool? does it matter?<br />--will all of the students participate, or will some watch? is watching or participating more effective?<br />--shall we record some video to look back on and share with the world?<br /><br />--<b>is this going to help them understand the concept?</b>Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com5tag:blogger.com,1999:blog-8938799432405750525.post-37611246738535473742010-11-08T19:54:00.000-05:002010-11-08T19:54:08.344-05:00Compelling Problems So I've stumbled into the math teacher blogosphere, and I like what I see. I like it so much that I steal lessons, videos, pictures, and even entire grading systems. So far, though, I've been engaging with the Web 2.0 world in a very Web 1.0 way, so I figure it's time to give back.<br /><br /> After having my world opened to the<a href="http://coxmath.blogspot.com/2010/10/wcydwt-toast.html"> kinds </a>of <a href="http://blog.mrmeyer.com/?p=4778">compelling </a>problems that <a href="http://blog.mrmeyer.com/?p=7649">excite</a> and pull my students into the process, I'm hungry for more. I figure the more voices in the conversation the more great ideas that will come out of it, even if one those voices is just my own. To do my part, I'm trying desperately to train myself to spot mathematical applications in the wild. Maybe blogging about it will keep my eyes open...Chrishttp://www.blogger.com/profile/09506027195685580723noreply@blogger.com0