tag:blogger.com,1999:blog-44775183483471612182024-03-13T21:02:25.771-07:00Michael ManganelloMy thoughts on issues that interest and concern me.
Follow me on Twitter: @m_manganelloMichael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.comBlogger27125tag:blogger.com,1999:blog-4477518348347161218.post-289212688218230862016-11-10T17:02:00.001-08:002016-11-10T17:02:05.266-08:00The 2016 Presidential Elections<span style="font-size: large;">President Donald Trump.</span><br />
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<span style="font-size: large;">Words I never thought I would have to speak aloud, now my reality.</span><br />
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<span style="font-size: large;">I admit that I spent the evening of November 8 getting more and more upset as the results came in. And I spent the day of November 9 wandering around my house, crying, doing chores, and pretending to work from home. But now it is time to get down to business and figure out what we can and should learn from this election.</span><br />
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<span style="font-size: large;">There are a lot of things that worry me about the impending Trump administration. I worry that my right to marry my same-sex partner could disappear. I worry that law makers will make it legal to discriminate against me, my boyfriend, and my other LGBTQ friends. I am concerned that we elected a vice president who is anti-LGBT, anti-science, anti-intellectual, and anti-woman. I am nervous about the kinds of policies that a Trump administration and Republican Congress might put in place, with little opposition. But mostly I find myself deeply disturbed about how normalized hate-speech has become. I feel much more unsafe today than I did a few days ago. I fear that I will get called a faggot while walking down the street. I am sincerely afraid of being gay-bashed, for the first time in my adult life. And I dread that I will hear chants of "Trump! Trump! Trump!" as I get hit, as though his election justifies violence against my person. And yet, we must learn from this election.</span><br />
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<span style="font-size: large;">In the past year and half, Trump called Mexicans rapists, referred to women as pigs, implied that all Muslims are jihadists, said that the inner cities (read: black folk) are terrible, mocked individuals with disabilities, and promised to strip away rights from LGBTQ people. If you are in one of those marginalized groups, like I am, you might have rejected his words and ignored his message. I know I did. His message to white, working class America was clear: you have been pushed out of the mainstream for too long and the America you love is gone; it is time to take our country back.</span><br />
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<span style="font-size: large;">Trump's is a powerful message, if you can set aside the hate-speech. That this message resonated with voters should not be a surprise. After all, it was very similar to the message Bernie Sanders used to propel himself through the primaries. Of course, Bernie's message was that we should get government to work better for us and that we can fix it together. The core of Trump's message was government is broken and corrupt and he alone is the solution. Both men put together a populist platform. Bernie's was built on hope and optimism and collaboration. Trump's was constructed out of fear and despair and authoritative leadership. Maybe we would be having a very different conversation if it were Sanders vs Trump. Would Sanders' positive message have bested Trumps' hate-filled one? There is no way to know. But we must learn that people across America believe that politicians at the federal level are out of touch with the real day-to-day lives of good, hard-working folks.</span><br />
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<span style="font-size: large;">Obviously Hillary Clinton did not prevail. Maybe her message was not strong enough. Maybe she was too disliked. Maybe she represented too much of the establishment. Those of us who supported her have to reflect critically on this election and wonder if maybe we were so excited to make the kind of her-story we wanted to see that we ignored the warning signs. However, we must remember that Hillary Clinton has gone farther along the path to the presidency than any woman ever. She didn't get the golden ring, but she did win the popular vote. There will be a first woman president of the United States. It just was not meant to be this time around.</span><br />
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<span style="font-size: large;">Ignorance is its own kind of hate-speech. White, working-class, rural people have become a marginalized group in America. They feel ignored and dismissed by the ruling elites. I have to believe that not every vote for Trump was a vote for hatred. I know it is difficult to see that today. And maybe it will be difficult to recognize that for a long time. People vote for lots of different reasons. I know military and law-enforcement personnel who voted for Trump because he represented safety and security in a way the Clinton did not. There are small business owners who experience regulations which prevent them from running a thriving enterprise. They did not think about the LGBTQ implications of their vote because that issue is not as important to them. We have to learn about our differences and why what matters so much to you does not matter as much to me and vice versa.</span><br />
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<span style="font-size: large;">This one is going to be a difficult pill to swallow for those of us with progressive mindsets. Many Republicans and those on the conservative right have spent nearly 40 years pounding away at the same set of messages: Government is broken, Politicians are corrupt, Liberals want to take away your guns, Democrats want to kill your unborn babies, Gay people are immoral, Immigrants will destroy this country, Free trade erodes jobs... The list goes on. All the other side has been able to say is something along the lines of "No, that's not true." Decades of being bludgeoned with mis-truths and outright lies cannot be countered with a simple refutation. If we want to move forward with a progressive agenda, we need to learn how to change the narrative.</span><br />
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<span style="font-size: large;">I hope we learn, in time, that Trump's election was a watershed moment for American history. One that threatened to send our country back decades but instead thrust forward the next generation of progressive, liberal, intellectual leaders who fought for equality, decency, and prosperity.</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-35212802005533107942016-07-13T14:02:00.000-07:002016-07-13T14:02:35.231-07:00The end of another year - Algebra 1 edition<span style="font-size: large;">As the school year has ended and vacation has been underway, it is important to reflect on the year that past and look forward to next year. By nature, I spend a lot of time dwelling on both the past and the future while not considering the present too often so this should be a breeze for me!</span><br />
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<span style="font-size: large;">In truth, I have had an extremely difficult school year. This, without doubt, the most challenging group of students I have dealt with in 17 years of teaching mathematics. There have been ups and downs and while it is easy to focus on the bad stuff without acknowledging the good stuff I want to try to give equal treatment to both.</span><br />
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<span style="font-size: large;">My 9th grade Algebra 1 students were tough... tough to manage, tough to motivate, tough to encourage, tough to engage, tough to have them think... Many of them came from special education classes where the teacher never taught a full year's worth of math. They had low expectations from the adults in their lives and they had low perceptions of themselves. Despite the deck being stacked against them in pretty much every way possible, we managed to work our way through the entire Algebra 1 curriculum with two weeks to spare before the state test. I am incredibly impressed with what they accomplished this year. Maybe we went a little too fast but I am reasonably confident that most of the students can solve an equation, graph a line, calculate/identify/interpret slope, add/subtract/multiply/factor polynomials, and perform some data analysis. What more can I ask? Well, I suppose I could ask for my students to comply with requests the first time I ask. In fact that is probably the single biggest reason why this year has been so challenging in Algebra 1. I ask nicely for a student to do something and he/she refuses because he/she simply does not want to do it. These are just a few samples of the exchanges I've had:</span><br />
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<span style="font-size: large;"><u>Vignette 1</u></span><br />
<span style="font-size: large;">M: Please put your phone away. </span><br />
<span style="font-size: large;">S: No, man. Why? </span><br />
<span style="font-size: large;">M: Because I asked you to and because we are going to take notes and you won't need it right now. </span><br />
<span style="font-size: large;">S: I pay attention better when I'm listening to music.</span><br />
<span style="font-size: large;">M: You probably don't. Please put your phone away.</span><br />
<span style="font-size: large;">S: I'm not even using it.</span><br />
<span style="font-size: large;">M: That doesn't matter. Please put your phone away.</span><br />
<span style="font-size: large;">S: Fuck you, man. I don't have to listen to you.</span><br />
<span style="font-size: large;">M: I've asked you nicely several times. I'm going to ask you one more time. Please put your phone away.</span><br />
<span style="font-size: large;">S: No.</span><br />
<span style="font-size: large;">M: OK, please go to the office.</span><br />
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<span style="font-size: large;"><u>Vignette 2</u></span><br />
<span style="font-size: large;">M: Please get away from the window.</span><br />
<span style="font-size: large;">S1: (ignoring me, speaking to another student) Yo man, you won't drop your phone out of the window.</span><br />
<span style="font-size: large;">M: Please do not drop your phone out of the window. Please get away from the window.</span><br />
<span style="font-size: large;">S2: How much will you pay me?</span><br />
<span style="font-size: large;">M: Please get away from the window. Please do not drop your phone out of the window.</span><br />
<span style="font-size: large;">(S2's phone gets dropped out of the window and students cheer. S2 runs out of the room to retrieve his phone)</span><br />
<span style="font-size: large;">M: S1, please get away from the window. (no response) </span><br />
<span style="font-size: large;">M: <b>S1</b>, please get away from the window. (no response)</span><br />
<span style="font-size: large;">M: </span><b><span style="font-size: x-large;">S1</span></b><span style="font-size: large;">, please get away from the window.</span><br />
<span style="font-size: large;">S1: Chill, bro. You don't have to yell.</span><br />
<span style="font-size: large;">M: I was trying to get your attention. I asked you several times to get away from the window.</span><br />
<span style="font-size: large;">S1: You don't have to be a fucking dick about it.</span><br />
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<span style="font-size: large;"><u>Vignette 3</u></span><br />
<span style="font-size: large;">M: S3, please stop talking and write down what's on the board.</span><br />
<span style="font-size: large;">S3: Why should I stop talking, S4 is talking too.</span><br />
<span style="font-size: large;">S4: You never shut up. Just do what he asks.</span><br />
<span style="font-size: large;">M: Both of you please stop talking. I will worry about who is working and who is not working.</span><br />
<span style="font-size: large;">S4: Yeah. You should do your work. You're so annoying.</span><br />
<span style="font-size: large;">S3: Shut up. You're annoying.</span><br />
<span style="font-size: large;">M: Both of you! Stop talking and copy down the notes. </span><br />
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<span style="font-size: large;">An exchange like one of these happened at least once per period. It has been demoralizing and depressing and debilitating. Every two steps forward was met with one step backwards. I know that's a bit of a cliche, but it feels so true. Setbacks are a necessary component of learning and a natural part of school but to be in a near constant state of rehabilitation and recovery has been mentally and physically exhausting. If I had to measure the good and the bad and weight them against each other I think Algebra 1 would be pretty balanced. The significant academic growth my students made was systematically offset by considerable behavioral issues.</span><br />
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<span style="font-size: large;">I know that I am teaching Algebra 1 again next year. I am not sure how I feel about it. I recognize that I get this class because very few teachers in my department would be willing to teach this class (or "these students") and I think that the administration believes that I can handle it. I know there are things that I have to do better this year, starting with a new classroom management plan. Luckily, I get to spend the summer contemplating what I need to do to make this coming school year run more smoothly. </span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-66170192428359666442016-04-15T06:53:00.002-07:002016-04-15T12:28:23.239-07:00NCTM 2016 - Making Sense of Logarithms<span style="font-size: large;"><a href="https://drive.google.com/open?id=0B3qYfYNB-KxMY2xGX0JMQmpUV3c">Powerpoint and Notes</a></span><br />
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<span style="font-size: large;">Thank you for the attendees who noticed that I have an error in the slides. I have a decimal place in the wrong position: </span><br />
<span style="font-size: large;">10^3 * 10^0.6435 should be 1000 * 4.40, not 1000 * 0.440</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-84508038338346607252016-02-21T15:51:00.000-08:002016-02-21T15:51:59.419-08:00My Review of "Building a Better Teacher"<span style="font-size: large;">I've been struggling with my classes this year, so I've been revisiting some books about teaching. When none of them gave me any insight into my dilemma this year, I decided to try some new material. Aimlessly browsing through the neighborhood Barnes and Noble, I came across Elizabeth Green's book with the incredulous title "<a href="http://www.amazon.com/Building-Better-Teacher-Teaching-Everyone/dp/0393351084/ref=sr_1_1?ie=UTF8&qid=1455669464&sr=8-1&keywords=elizabeth+green">Building a Better Teacher: How Teaching works (and How to Teach It to Everyone)</a>". After skimming a few chapters and recognizing some names from the world of education (Lee Shulman, <a href="http://www-personal.umich.edu/~dball/">Deborah Ball</a>*, Magdalene Lampert, Doug Lemov) I decided to give it a try.</span><br />
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<span style="font-size: large;">I'm not sure what I was expecting when I brought it home, but what I read wasn't it. That said, I liked the book. It was an enjoyable read and I would recommend it to anyone with an interest in teacher education. It probably would have been more accurately titled, "A layman's guide to the history of the research on teaching from 1980 through 2015", but who would buy that book? I thought it would be helpful to write a short review of the book to help me process what I read.</span><br />
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<span style="font-size: large;">Truthfully, I'm not sure that I read a lot that was very new to me. I knew that it had been long argued that teaching is difficult intellectual work. I was pleased to see that Green was not at all dismissive of the challenges of the classroom, as are some writers on educational issues. What was helpful for me was Green's agnostic and critical stance on schools of education and educational entrepreneurs. She argued that each camp was engaged in the same goal: to describe and classify effective pedagogy so that it can be taught to aspiring teachers. However, she was careful to acknowledge that each camp was differently equipped to attack the problem. Schools of education are often fighting a culture and value war about what teaching should look like, what Dan Lortie called the <i>apprenticeship of observation</i> in his seminal work, <a href="http://www.amazon.com/Schoolteacher-Sociological-Dan-C-Lortie/dp/0226493539/ref=sr_1_1?ie=UTF8&qid=1456005561&sr=8-1&keywords=schoolteacher">Schoolteacher</a>. The education start-ups entered the fray explicitly rejecting what schools <b>were</b> in the hopes of creating what schools <b>could be</b>. Many schools of education are research institutions which means (hopefully) that they are thoughtful about what teaching methods to try and how to try them responsibly. And they should be well-positioned to collect and analyze the data in both a qualitative and quantitative way. The charter schools were more or less haphazard in collecting information about student achievement and trying to match it to effective teachers. </span><br />
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<span style="font-size: large;">What both camps seem to have in common is the belief that <u>good teaching can be taught</u>! That is a pretty revolutionary idea. I've heard a lot of people say that good teachers are born not made. I've had administrators tell me that I'm just a "naturally good" teacher. I believe that the notion that some people have a gift for teaching and others don't is dangerous for (at least) two reasons. For teachers who work very hard at improving their craft, being told that they are "naturally good" demeans and discredits the effort they have put forth. For teachers who think that teaching is a talent, it releases them from any responsibility to get better; it absolves them from all pedagogical sins. How can we improve as a profession if we don't acknowledge that some instructional practices are more likely to result in student learning and others are less likely to bring about the desired results?</span><br />
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<span style="font-size: large;">In the conclusion to Green's book, both camps constructed lists of effective pedagogies. Deborah Ball and colleagues put together what they call "High Leverage Practices" which are a set of "fundamental capabilities" that they believe all teachers should master. All of the practices and more are available on their <a href="http://www.teachingworks.org/">website</a> dedicated to improving teaching. Meanwhile, Doug Lemov published a book. "<a href="http://teachlikeachampion.com/">Teach Like a Champion</a>" that documents his taxonomy of effective teaching "techniques". I know that some people have been dismissive of Lemov's approach because it feels like behavior modification. But, Ball's cognitive stance could be critiqued for emphasizing thoughts over actions. I don't think it's fair to compare them. I think that a thoughtful and critical teacher should be reading and using both of them to inform and improve his or her practice. That's what I'll be doing. </span><br />
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*Deborah Ball is a widely recognized authority on mathematics education and I had read much of her research. Also, my graduate school advisor was a student of Deborah Ball so I like to think of her as my academic grandmother.**<br />
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**You know, a grandmother who lives halfway across the country, whom you've never met, and who most likely has no idea what you exist. So, we're not close.<br />
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<br />Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-62870177863437206482016-02-14T13:56:00.000-08:002016-02-15T04:29:55.836-08:00Monmouth University - February 15, 2016<span style="font-size: large;">Thank you for allowing me to speak about my research. Here are links to PDFs of some relevant additional information:</span><br />
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<a href="https://drive.google.com/file/d/0B3qYfYNB-KxMLUxtZi01TGxsVGM/view?usp=sharing"><span style="font-size: large;">Power Point Slides</span></a><br />
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<a href="https://drive.google.com/file/d/0B3qYfYNB-KxManZlS3VTM3U4djQ/view?usp=sharing"><span style="font-size: large;">Task Interview</span></a><br />
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<a href="https://drive.google.com/file/d/0B3qYfYNB-KxMNk9yZE9CRjRzbFk/view?usp=sharing"><span style="font-size: large;">Framework</span></a><br />
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<a href="https://drive.google.com/file/d/0B3qYfYNB-KxMU2t6ZzV4WjhzWlE/view?usp=sharing"><span style="font-size: large;">Classroom Vignettes</span></a><br />
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Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-7620921081309966452015-05-02T19:23:00.003-07:002015-05-02T19:23:43.836-07:00Nuts & Bolts of Online Teaching<span style="font-size: large;">After reading a chapter from Kristin Kipp's book on online and blended learning, I couldn't help but think, "why didn't we read this weeks ago?". The chapter provided some very grounded suggestions in creating a syllabus, managing student information, and how to maintain teacher presence. All of which were exciting to read, but I felt much too late in the sequence of readings. This chapter, and maybe more chapters from her book, could have been used to ground some of the other more theoretical readings. Don't get me wrong, I love reading about learning theory. I really do find it an interesting subject! I just wish we had tried to create more of a bridge between theory and practice. </span><br />
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<span style="font-size: large;">That said, what I mostly got out of Kipp's chapter was that online instructors must put a lot of thought into their course design. But, face-to-face teachers should put a lot of effort into their course designs also. In an online or blended format, teachers also need to consider how the user interface could affect how students engage with the course content. Well, face-to-face teachers should probably consider those things too. How does the classroom environment influence what and how students will engage with the content. Teachers should think about the environments in which their students will learn whether online or blended or face-to-face? Yup.</span><br />
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<span style="font-size: large;">Actually, that there isn't really that much difference between high quality face-to-face teaching and online teaching is pretty much the same take-away I had from DiPietro et al's piece on best practices in online teaching. (As an aside, I think it was on Twitter this week that someone criticized the term "best practices" because it encourages copying the technique without analyzing its affordances and constraints and thus effectively shuts down innovation.) Their extremely detailed and extensively descriptive table of findings was interesting. While reading through it, I just kept thinking, "shouldn't all teachers be doing these things?". The medium is obviously different in online teaching, but the actions and intentions of the teacher aren't that different. Good teaching looks like good teaching whether in an online or blended or face-to-face format? Yup.</span><br />
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<span style="font-size: large;">This entire semester hasn't really convinced me that digital technologies can supplant traditional ones. Supplement? Yes! Replace? Not sure. There's some research that suggests that writing notes by hand is better than typing on a computer. However students who need help visualizing some of the more esoteric and/or abstract mathematical concepts can reap huge rewards because of what digital technologies can do. Form should follow function. First we need to figure out what students are supposed to learn then we can choose the appropriate technologies.</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com5tag:blogger.com,1999:blog-4477518348347161218.post-55002415170839767202015-04-27T18:36:00.003-07:002015-05-02T18:11:57.707-07:00Technological Tools for Teaching<div>
<span style="font-size: large;">Mishra and Koehler have a really interesting take on technology and teaching. I think I read somewhere that if you want to get ahead, you have to get a theory. It seems like they've done just that. Rather, they've expanded an existing theory to explicitly include educational technologies.</span></div>
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<span style="font-size: large;">I'm pretty familiar with Shulman's construct of Pedagogical Content Knowledge. (In the math ed world, Deborah Ball and a few others have proposed a special domain of teacher knowledge called Mathematical Knowledge for Teaching (MKT) which is located partly within PCK and partly with Content knowledge.) I really liked Misha & Koehler's Venn diagram for PK and CK and how they included a separate but overlapping domain for Technology Knowledge. I couldn't help but wonder if there are aspects of MKT that exist within the sphere of TK or any of the other categories they created. I'm sure there is. </span></div>
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<span style="font-size: large;">I also couldn't help but wonder if TK is more correctly placed within one or both of the other domains. Is it really a distinct domain of teacher knowledge? Maybe it is. I could go either way on it. Maybe the tipping point for me is that TK is not just about knowing about and how to use both traditional (black/white boards) and advanced technologies (computers/mobile devices) but is also about the ability to learn about existing and emerging technologies. When we started to introduce Interactive White Boards at my school, there were more than a few teachers who resisted them and complained loudly that the IWBs were placed in the center of the existing white boards essentially removing at least 1/3 of the board space. I liked the idea of the IWB but was also wary about its location in the room. Now, I am so glad that my principal had the foresight to put it front and center in the classrooms. I cannot imagine teaching without an IWB now. Many of the same teachers never really explored what an IWB can really do for your instruction. Students used to marvel at things that I did on the board; things that seemed routine to me but that other teachers hadn't tried to do.* I always figured that I couldn't do any permanent damage and that I should just try some stuff. We used to characterize that kind of attitude towards technology as being a "digital native". Mishra and Koehler conjecture that it's part of a special domain of teacher knowledge. Perhaps it is.</span></div>
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<span style="font-size: large;">Perhaps TK is becoming so much a part of what we do that it really isn't separate domain of teacher knowledge. Last semester I had the opportunity to teach a course on technology in mathematics classes. I had anticipated that the students in the class would want to know about how to use hand-held graphing calculators to enhance instruction. I was very wrong. We did a few activities but they students seemed mostly disinterested. I have a few ideas why: 1) graphing calculators have permeated mathematics instruction so thoroughly that the students in the class, who were both preservice and inservice teachers, had already developed a familiarity with what the graphing calculators could do ; 2) the way I was using the calculators had not really occurred to them before and what I was doing was maybe a little too weird and different to really understand ; 3) the graphing calculators have been around for a while and the graphing capabilities are subpar when compared with newer technologies ; 4) the graphing calculator has a pretty steep learning curve and it often produces static results while newer technologies are considerably more intuitive and are much more adept at producing dynamic results which makes the graphing calculator a difficult tool to use. These are just a few of the reasons why I think the students were uninterested in graphing calculator but did take to other technologies.</span><br />
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One anecdote can't make a theory, but it might provide a little evidence that maybe TK is just part of what we do as teachers. Teachers with better / more developed TK or TPCK can use all sorts of technologies better or more flexibly. They can also adapt to new technologies faster because they have that kind of disposition. And maybe they even seek out or invent new technologies when existing ones can't do what he/she wants. It's certainly an interesting construct and I'd like to learn some more about it.<br />
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*Kids used to marvel at the IWBs. Nowadays there's a lot less awe. I think its because the crop of students I have now have had IWBs for pretty much their entire school career. It's hard to get excited about something that's been part of your routine for 10 years.</div>
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Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-15367989184805080202015-04-26T15:05:00.001-07:002015-04-26T15:05:17.592-07:00Gaming<span style="font-size: large;">I've been playing video games for almost as long as I can remember. We had a Magnavox Odessey 4000, then an Atari 2600, then a Nintendo Entertainment System. For as many fond memories I have of playing football outside with the kids in my neighborhood, I have even fonder ones of my brother and I sitting in our basement trying to beat Contra. My roommates sophomore and junior years in college had a Sega Genesis, which they used to play an NHL game incessantly. I've never liked playing sports games, so I sat out those sessions. A college professor introduced me to the Sony PlayStation and the revolutionary games that were being created for that system, like the Resident Evil and Final Fantasy series. I bought myself one when I was a senior and spent many Saturdays during my first year of teaching in front of the TV. I'm pretty sure that I should not try to accurately chronicle how much of my life I've poured into Final Fantasy VII, VIII, and IX. I used to visit friends and we would play Grand Theft Auto III or Final Fantasy X or Guitar Hero pretty much non-stop for an entire weekend. Even though a lot of the games we played were single-player, we used to watch and give suggestions to each other or cheer each other on. More eyes on the screen meant that few items went unnoticed. It also meant that we progressed through the games a little faster because if the person holding the controller couldn't figure out what do to, someone else in the room had an idea. After an extended break from gaming, and missing it quite a bit, my incredible partner bought me a PS3. I only play on weekend mornings now, because I know I could lose myself in the games.</span><br />
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<span style="font-size: large;">There was a time when you almost had to read the instruction manual for a game because the controls for each game would be slightly different and often the designers would put tips and hints into the manual. Do games even have instruction manuals anymore? I haven't read one in years. Most of the games that I like to play build in a tutorial and then set you loose. You figure stuff out along the way by accomplishing the tasks that move the story forward. I guess I never really thought of it as learning, but I suppose that you do have to learn how to interact with the environment of the game world.</span><br />
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<span style="font-size: large;">I can absolutely see the appeal of using video games and virtual environments in education. (I did love to play "Where in the World is Carmen Sandiego?") Games are designed to give you instant feedback on what you've done correctly or where you've made a mistake. You don't get a bad grade if you mess up. Even if you're playing a game where there is a score element, chances are you can replay that section of the game again to improve your score. (Some games are even designed in such a way that you can't maximize your score until you revisit early sections with the better abilities you acquire later in the game.) I wonder if a game that has an overt educational message or tone can be designed to do similar things. Maybe it can. I'm wary though. I wonder if what a player learns through a video game or through a virtual environment transfers to other domains. Perhaps it depends on what the player learns. </span><br />
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<span style="font-size: large;">But, assessing what students know through a virtual environment seems really promising. The idea that a virtual world can give educators a better sense of what students know and can do in novel situations is very intriguing. I would use something like that in my classroom, if it were available. We have some computer based assessments, but they are basically just a paper-and-pencil test on the computer. Science assessments seems to be a near-perfect fit. Sometimes math and science get lumped together, but I think there is a bigger divide between them than people realize. Science is pretty much based on empirical experiments while math relies heavily on thought experiments. Im sure someone can come up with a good math assessment that isn't just algorithm recall and implementation - which is often what we end up assessing in math class anyway. It's really hard to figure out if your students have developed any mathematical habits of mind. Hard, but not impossible. Maybe I should try my hand at that for a while...</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-48786365233091580092015-04-25T19:51:00.000-07:002015-04-25T19:51:00.789-07:00New Literacies<span style="font-size: large;">Thompson's book chapter on the "New Literacies" was enlightening. I did feel like it was really multiple chapters that had been unceremoniously lumped together. Each of the new literacies - data, video, photo - felt like it could have been expanded to fill a chapter a piece and form a section of his book. Maybe in the expanded second edition.</span><div>
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<span style="font-size: large;">I am going to restrict my comments to data literacy. Not because I have nothing to say about the other ones but because I've been thinking about quantitative literacy for a long time. Data (quantitative) literacy is a vital skill. Since the economic collapse in 2008, I've been wondering about whom many people got themselves into trouble purchase houses or cars or whatever that they couldn't really afford but were lulled into thinking that they could. I know it's not really a mortgage broker's job to dissuade a potential homeowner from getting an inappropriate loan but maybe they should consult more. How many Wall Street stockbrokers and other financial retailers didn't really understand the mathematics behind the derivatives they were selling? Quantitative literacy is important because data is becoming more prevalent in society. (You can't even be a 1st grade teacher without being confronted with data on your students almost daily.) Because of the vaulted status that mathematics holds in our society and because so many people have anxiety linked to numbers, knowledgable people can use data and mathematics intimidate the less literate (or numerate, if you prefer). To become responsible citizens, people must learn that just because there is a number attached to a statement that does not automatically imply that the statement should be believed. The mathematics classroom seems like a natural place to expose students to data analysis. Some people might argue that practical applications of data should be the only thing we teach in school. I think that is misguided. We can't limit math class to data analysis. It would be like only teaching kids how to fill out job applications in English class. Sure, job applications have reading and writing but its just one kind. Mathematics has existed since the beginning of humanity because it can be used to answer questions about the world and it can be used to explore questions with no physical manifestation. </span></div>
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<span style="font-size: large;">Puntambekar and Hubscher present an informative argument about how the term "scaffolding", now so prominent in classrooms, had taken on a very different meaning than originally intended. Scaffolding was designed to be one-on-one, tutor and student. As the idea of scaffolds were applied to the classroom, the notions behind what constitutes a scaffold have been generalized. Puntambekar and Hubscher argue that maybe we've overgeneralized too much. I couldn't help but wonder if the term scaffold was really appropriate. Regardless, they present a few suggestions for how to implement scaffolds in classrooms to keep with the spirit, if not the letter, of the original idea of a scaffold. I found it refreshing that the researchers took into account that classrooms are (often) diverse and complex learning environments. I anticipated their suggestion that scaffolding tools be removed from environment over time as students need them less. That seemed perfectly reasonable to me.</span></div>
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<span style="font-size: large;">I hadn't encountered Zygotsky's work until I started graduate school. I knew about Thorndike and Skinner and behaviorism, and Piaget and constructivism, but not Zygotsky and Brunner, and cognitive psychology. I wonder if other teacher preparation programs introduce the concept of Zone of Proximal Development. From the Puntambekar and Hubscher article, I got the sense that some teachers don't really understand ZPD. Without some basic understanding of ZPD, teachers would naturally struggle with how to structure supports for students and the importance of removing them when students no longer need them. I see this subtle tension whenever we use manipulatives in math classrooms. Some teachers and parents will argue against using base-10 blocks to model addition and subtraction because students need to know how to add and subtract without them. But that's not the right argument. Some students will need to use the models because the abstraction of adding and subtracting is beyond their *current* cognitive development. The base-10 blocks are one way for an adult to provide guidance to a student. Maybe addition and subtraction without a model are outside of the student's current ability but by providing the base-10 blocks, those concepts move into the child's ZPD. Maybe. I feel like I might have to read Mind in Society again over the summer. </span></div>
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Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-77647681818178987752015-04-25T04:32:00.003-07:002015-04-25T04:32:39.357-07:00Is there a grand unified learning theory?<span style="font-size: large;">As I was reading Yasmin's Kafai's chapter on Constructionism from the Cambridge Handbook of the Learning Sciences, I couldn't help but think about James Greeno's chapter from the same text. They present two different, but compatible, theories of learning. (To be clear, I mean "theory" scientifically, as a way to explain data collected by experimentation or observation, not "theory" colloquially, as an idea or thought one has about something.) It felt to me that Greeno was writing about learning as participation in activity and Kafai was writing about learning as acquiring knowledge through construction. I remembered another piece I read a while ago by Anna Sfard where discussed learning through the participation and acquisition metaphors and cautioned against privileging one over the other. I don't think that Greeno or Kafai (at least in these pieces) was trying to say that their individual learning theory was superior to any other one. However, it is tempting to try to explain all learning through one lens... even though I think it's pointless.</span><br />
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<span style="font-size: large;">I know I shouldn't pick on this, but Kafai's chapter seemed less like a survey of constructionism as it did a homage to the work of Seymour Papert. That said, it was an interesting read. I don't think I ever really considered the differences between constructivism and constructionism. It's easy to say that constructivism is a theory of knowledge while constructionism is a theory of learning, but that doesn't really explain the difference. At the core, constructivism is descriptive (what are the stages of knowledge development) where constructionism aims to be prescriptive (what teachers can do to induce learning). You can't teach a little kid object permanence. Either he has or he doesn't. But you can give a little kid Logo and help them figure out how to draw a square. </span><br />
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<span style="font-size: large;">Chapter 6 from How People Learn was also an insightful read. At the beginning, I was fearful that it was going to advocate for a Learner-Centered environment over a Knowledge-Centered environment. Then the authors introduce Assessment-Centered and Community-Centered environments and I got all confused. I found it satisfactory to learn that the authors were saying that all four of these centers should be present in any kind of learning environment. I've always found that a good theory should helpfully explain what I already know. These four things seem so obvious now that they've been named. A good teacher will try to figure out what students bring to the classroom and leverage that information into the design of the environment (learner-centered). She will also make sure that norms and ways of knowing and core concepts of the discipline are a major component of the course work (knowledge-centered). Of course, the teacher needs to figure out what students have learned and must give students feedback about what they still need to learn (assessment-centered). And the classroom is a community located within and alongside other communities. The teacher has to take that into account when designing activities (community-centered). I think I try to do these things but I have to be more intentional about them. </span><br />
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<span style="font-size: large;">Personal aside: Since I've read a few different pieces about computer programming (Kafai's Constructionism chapter and Resnick et al's short piece on Scratch) I've become even more convinced that including some programming in my math classes could enhance student learning. I just have to figure out how to incorporate it. I was the kid who learned to program in Basic in elementary school and I tinkered with Logo in an after school program. I taught programming early in my career and I really wish I had kept up with it. My skills are rusty, but I think I should pick it up again. </span><br />
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<br />Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-14846505343247472622015-04-23T19:33:00.002-07:002015-07-06T07:25:35.980-07:00Situativity<span style="font-size: large;">I like to think that I have a pretty good working knowledge of the main learning theories - behaviorism, cognitivism, situativity - but it's always good to get a little refresher on them. James Greeno's excellent synthesis of the situative research perspective in the Cambridge Handbook of Learning Sciences made me think about my classroom environment and how students might be interacting within it.</span><br />
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<span style="font-size: large;">My big take away from Greeno's piece was that "[s]ituativity is a general scientific perspective and as such does not say what educational practices should be adopted". I thought that Greeno was implying throughout the piece that you cannot create a situative classroom or situative learning environment. Rather, situativity is an analytical framework that researchers in the learning sciences can use to try to understand how people interact and learn within some kind of system. That system could be a classroom or a school, but it does not have to be either of those.* </span><br />
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<span style="font-size: large;">However, since I work in a school and teach classes in that school, I started to wonder about how my classroom environment, or any classroom environment, could be analyzed or examined through a situative lens. It seems to me that the classroom is a kind of activity system and that what students learn about content of the class is intricately linked to the learning environment. How students interact with mathematical concepts, with each other, with the teacher in my classroom <strike>might be</strike> is almost certainly different than how students are expected to to behave in other classrooms. Can differences in the classroom system explain or account for differences in student achievement? I think Greeno might say yes. </span><br />
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<span style="font-size: large;">Furthermore, classroom activity systems exist within the larger school activity systems. What happens when a single classroom system rejects the prevailing norms of the larger school system? Is that a sustainable situation for the teacher and the students? What will students learn about the content of the class? Will students be able to participate within the systems authentically? I guess I'm thinking about how many teachers design their instruction in such a way that students are positioned as passive recipients of knowledge where other teachers expect students to actively make sense of the subject matter. Do students find it difficult to switch between these environments?</span><br />
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<span style="font-size: large;">A situative analysis could also helpful in framing the short article on the Scratch programming language. The activity system has been constructed in such a way that participants can interact with the language individually or they can make their creations public to garner feedback from a wider audience. The threefold goals of tinkerability, meaningfulness, and social interaction can be viewed as providing learners opportunities to engage in authentic problem solving and inquiry in much the same way that professionals do. Even if programming is not a career goal, making sense of problems and persevering in solving them is a goal most educators can support.</span><br />
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<span style="font-size: large;">As an aside, I believe strongly that we teach much too much computation in high school mathematics classes. We expect students to be able to do algebra that a computer can do much better. As such, we never get to the analysis and problem solving that (it seems) everyone values more than mathematical facts. Programming has felt to me (for a long time) like a nearly perfect way to incorporate critical thinking and problem solving into math class. The real mathematics is in developing the algorithm to solve a family of problem not from executing the algorithm flawlessly on a contrived assessment. I'm not sure that I can get enough student buy-in but I want to try. </span><br />
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<span style="font-size: x-small;">*Greeno passingly referred to Lave and Wenger's work. They studied some apprentice-like situations (butchering, midwifery) and posited the existence of a construct they called "legitimate peripheral participation". This means that novices to a system perform roles that are outside the core practices of the system but are necessary for the system to run effectively. The novices gradually take on more responsibility as they become full members of the activity. </span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-25595989350614548892015-04-22T20:16:00.001-07:002015-04-22T20:16:15.670-07:00Can schools meet the challenges of the digital revolution?<span style="font-size: large;">After reading a few things about emerging technologies and what the future world might look like, I'm not sure. As a public school teacher for over 15 years, I know that schools in their current state are not the answer. They often do not serve their populations well. But, I'm not ready to abandon schools just yet.</span><br />
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<span style="font-size: large;">In an interesting piece by Collins and Halverson, they lay out an argument for why schools reject technology and the digital revolution. There were a few parts of the argument that struck me as very true and some that struck me as not particularly current. The piece is relatively old by digital standards, being published in 2009. I'm sure that in 2009 many facets of the piece were more true than they are now. (In 2009, I didn't have a SmartBoard in my classroom. Now, I can't imagine teaching without one.) Perhaps Collins & Halverson need to write a follow-up documenting what schools have and have not done to incorporate digital tools into classrooms.</span><br />
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<span style="font-size: large;">Here are my major disagreements with Collins & Halverson's argument: (1) They put a lot of faith in the ability of computers to respond to the interests of students. I'm not sure that their faith has borne itself out yet. Computers are very good at some things. Interacting authentically with a human is not one of them. No computers have yet been able to pass the Turing test, which is widely accepted as the bar for human-like behavior. Simulations are realistic, but always programmed not authentic. Games can have open-world features but are governed by algorithms; they are not truly interactive. Computers can deliver content but they are not very good at providing personalized feedback. (2) One part of their argument entails giving learners, primarily children, more significant control over what they learn. Obviously children should be encouraged to follow their passions. However, over a century ago, John Dewey warned against the dangers of a curriculum that was too child-centered. Children do not know enough of the world to make the most informed choices. Schools, teachers, parents have a responsibility to expose children to different facets of how we make sense of the world. I know that anecdotes should be taken lightly, but I know that I would never have pursued mathematics without the guidance or the excellent high school math teachers who inspired me with their passion for the subject. (3) It was only a small portion, but the idea that in adult life one rarely needs to have readily available "knowledge in the head" rang hollow for me. When you go to buy a car, do you want the salesperson to know the facts about different models or should they read them to you from the brochure? I can read the specs online myself! I came to the dealership to interact with a person. Would you feel good about an English teacher who needed to check the text of Romeo & Juliet to figure out which one was a Montague and which was a Capulet? Certainly not. I want my physician to know some things about what might be causing my fever and cough even if he/she has to check his/her computer for a backup diagnosis. Internalize knowledge is still important, but so is knowing how to use resources to support your work.</span><br />
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<span style="font-size: large;">I agree with a lot of what they said. Here goes: (1) Schooling needs to change from knowledge acquisition to learning to learn. This is absolutely true. However, I think they failed to take into account that this kind of schooling is different from what parents experienced. That means that parents will be expected to support their children in a type of education with which they are unfamiliar. The issues with learning to learn is probably most prevalent in mathematics. Many new elementary school mathematics programs focus on ways of thinking instead of fast fact recall. You only have to scroll through Facebook to see the objections to the "new math". Yes, parents object to teachers who want children to think mathematically and construct viable arguments and use multiple representations instead of memorizing facts. Crazy, but true. (2) Math teachers need to stop teaching computation skills and need to teach thinking skills. There are some substantial objections to the Common Core State Standards, but they did put out the Standards for Mathematical Practice. These are ways of thinking and acting that are prevalent in the mathematical sciences but also cut across disciplines. It is far more valuable to have students analyze the conclusions drawn from quantitative data than it is to have them graph a context-free parabola. Mathematics can be co-opted by people who understand it, or who think they understand it, to intimidate people who don't. That's a dangerous situation. </span><br />
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<span style="font-size: large;">Can schools meet the challenges of the digital revolution? I don't know. But I hope that schools can make some changes to content and format and pedagogy so that they remain relevant places of learning far into the future. </span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-62546364439008979022015-03-26T12:31:00.000-07:002015-03-26T12:31:50.097-07:00A Silver Bullet for Math Teaching?<span style="font-size: large;">This school year, I had the opportunity to step away from the classroom and move into a district level teacher leadership position. I teach one class in the morning, and then I visit K-12 math teachers around the district for the rest of the day. Or I attend meetings and provide input about the needs and desires of the math department. Or I plan professional development sessions for the middle school and high school math teachers. And I'm supposed to do some teacher coaching too.</span><br />
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<span style="font-size: large;">It sounds kind of great, doesn't it?</span><br />
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<span style="font-size: large;">I do not like it. There are a lot of reasons why but I'm not writing about that right now. Maybe another time. What I do want to reflect on now is the challenges of how to respond when teachers ask me questions about teaching. </span><br />
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<span style="font-size: large;">A few days ago, I was at a school and stopped in to see a teacher I had visited before. We have a pretty good rapport and she said hello when I walked in. Mind, I hadn't been to the school since November. I saw that all of her students were on computers working on various math tutorials. I've been in the room for about 30 seconds and then this conversation takes place:</span><br />
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<span style="font-size: large;">Teacher: So, do you have any advice?</span><br />
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<span style="font-size: large;">Teacher: About what we can do to teach math better?</span><br />
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<span style="font-size: large;">At this point, I think I mumbled something about not knowing what they were doing or how the year was going or that I couldn't possibly give any feedback since I had only just stepped foot in the classroom. The weird thing is, she asked me the same question during lunch, "Do you have any suggestions for what we can do to teach better?" Again, I had to mumble something and try to avoid the conversation. What could I possibly say?</span><br />
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<span style="font-size: large;">It's taken a few days of unpacking and reflecting, but I think I know why the question has stuck with me. I should also put out there that I think the question was an earnest attempt to engage me in a conversation about pedagogy but nonetheless, two things about it has bothered me a bit. First, I think the question betrayed her belief about her own teaching. She believes that her mathematics pedagogy is not good and is in need of repair. Perhaps she was being self-deprecating; perhaps she was telling me what she thought I wanted to hear; perhaps she was trying to learn something. It is entirely possible that she was really asking for help to improve her practice but didn't really know how to ask. I'd be happy to do some coaching with her, if that's what she really wanted but I suspect that is not what she wanted.</span><br />
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<span style="font-size: large;">The second thing about the question is more important to me. It felt to me like the teacher expected some kind of silver bullet. She was hoping for a quick fix. She expected me to have some new technique that she had never heard before that would turn her instruction on its head, be easy to implement, and raise standardized test scores in weeks. I have no such tricks. I doubt those tricks exist. My suggestions for how to improve your math teaching isn't popular or easy. Here goes:</span><br />
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<span style="font-size: large;"><b>1) Make sure you know the math</b> - Seems obvious, right? It's not. Read the textbook. The student edition AND the teacher edition. Work the problems. Work all of the problems. Don't skim the notes to the teacher. Most modern textbooks for elementary teachers (and middle and high school too) have pretty substantial notes to the teacher that alert the teacher to student misconceptions and tips for teaching. The best books also explain why a particular topic is important and how that topic fits into the overall curriculum plan. It's also worthwhile to plan with your colleagues and collaborate on the problems and explore the mathematics. Learn different approaches and algorithms and models. Be sure that you can attack a problem from multiple angles. Be able to respond to students' questions about "why" with something more than, "that's the rule". Make sure you know the math.*</span><br />
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<span style="font-size: large;"><b>2) Make connections with kids</b> - Seems obvious, right? It isn't. I suspect that we all knew or had a teacher that seemed to hate kids. You know the ones: the teacher who snaps at kids when they have a question, the teacher who talks down to children who are just trying to learn, the teacher who blames a child for his inability to keep up with the rest of the class, the teacher who thinks that some kids just aren't capable of learning math. The list could go on and on. I have a lot of colleagues who spend their day talking at kids, a.k.a. lecturing, and them complain that kids don't pay attention and they haven't learned any math. I say, of course kids haven't learned. You need to engage kids where they are and nudge them towards understanding mathematical concepts. If you leave work bothered because your students have not complied with your requests, you need to ask yourself why the students didn't comply. Maybe they didn't understand the instructions. </span><span style="font-size: large;">On the other hand, if you leave work every day wondering what you could do differently to help them learn more and better math, that's a sign that you're trying to improve. Keep at it. Talk to colleagues about it. But not the people who hate kids. </span><br />
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<span style="font-size: large;"><b>3) Only scratch when you have an itch</b> - Every single mathematical concept or algorithm was developed as the answer to a question. A lot of times, we teach a concept or procedure before kids consider why anyone would want to know that. Think about these different ways to start a unit on fractions: "Today we're going to start talking about fractions. This can get tricky so make sure you pay attention." versus "I was wondering, are there numbers between numbers?" Which one creates an itch? One of these openers will almost certainly deviate from the district sanctioned pacing guide but might result in deep and meaningful understanding of fractions. When we don't ask any questions, we essentially tell kids what to think. </span><span style="font-size: large;">I sometimes like to call this, "opening everyone's presents for them". Even worse, </span><span style="font-size: large;">we give students the false impression that mathematics has no human aspect, that the rules of mathematics are irrefutable because they have been gifted to us from the Creator, that mathematics is done by memorizing and acting with certainty and not by understanding and exploring. I'm not saying that we need to give kids a practical, real-world example of how mathematics is useful. Sometimes there are physical situations that lend themselves to mathematics. Sometimes we try too hard to manufacture a scenario where impose mathematics that no one would actually do. A question about an interesting phenomenon can trump a physical situation. Think about which one will create a better itch to scratch.</span><br />
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<span style="font-size: large;"><b>4) Be patient</b> - This one seems especially obvious but it is probably the least apparent suggestion. You have to understand learning theories. You have to remember that kids learn at different paces and in different ways. Kids will learn math. But, it might not be on the timetable that the textbook or the school district give you. It will be stressful for you and for them. But you have to embrace the stress and not let it consume you. Most of all, you have to rid yourself of the noxious belief that some kids can do math and other kids cannot. I firmly believe that all kids can learn math. (Excepting children with severe developmental disabilities.) If a strategy you find appealing isn't working then you need to try another one. Use a different kind of representation. Get a concrete model. Draw a picture. Use base-10 blocks. Don't resort to giving the child an algorithm before he or she is ready for it. You're not helping when you prematurely move him to the finish. She needs to get there on her own with your support and guidance. He will never get there if you don't give him adequate time to play, experiment, and figure things out.</span><br />
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<span style="font-size: large;">These are not glamorous suggestions, but I think they are true. More than my own belief about teaching, I think we can find pretty solid research for each of these suggestions. It's not an easy path to take, but it can work. There is no silver bullet for teaching math. Teaching math well entails knowing the subject and how to represent it in a variety of ways, connecting with students and their individual learning progressions, and asking questions instead of giving all the answers. It's hard work... and I love working at it. </span><br />
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<span style="font-size: x-small;">*By the way, it's still pretty much an open question about how much math you need to know to teach it well. There is a very robust area of research devoted to this very topic. Many scholars have worked for many years investigating a construct called "Mathematical Knowledge for Teaching". Still, we don't know exactly what comprises MKT. </span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com2tag:blogger.com,1999:blog-4477518348347161218.post-29735255761296291422015-03-09T05:15:00.002-07:002015-03-09T05:15:09.079-07:00Why all the buzz around "flipping" the classroom?<span style="font-size: large;">I have a problem with the flipped classroom movement.</span><br />
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<span style="font-size: large;">OK, maybe not the movement itself, but I do have a problem with the usual justifications used by the flipped classroom movement. Every single thing I have ever seen about the flipped classroom movement has stated that the main reason to flip is so that students can watch videos of the content at their own pace and come to class prepared to work problems or investigate further. This approach leaves the teacher free to work individually with students.</span><br />
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<span style="font-size: large;">I don't even know where to begin, but I'll try.</span><br />
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<span style="font-size: large;">First, the entire movement seems to be predicated on the notion that lecture is the primary instructional method. It's 2015, people! If you're lecturing more than you're engaging kids in conversation about mathematical ideas or discovering interesting properties of mathematical structures or undertaking problem solving, then you're probably not a very good math teacher. There, I said it.</span><br />
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<span style="font-size: large;">Second, is the ridiculous idea that lecturing is the reason why you can't work individually with students. If you can't find time during class to address a particular student's need, then you're probably not a very good math teacher. There, I said it again.</span><br />
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<span style="font-size: large;">Third, the videos that are often peddled to math teachers are nothing more than how-to's. These videos tell kids how to get answers. They do not help kids learn how to think mathematically. If all you're doing in math class is telling kids how to perform procedures, not getting them involved in the solutions and not motivating the algorithms with the interesting and vital historical questions that accompany the problems that the algorithms were invented to solve, </span><span style="font-size: large;">then you're probably not a very good math teacher. </span><span style="font-size: large;">Yup, I said it a third time.</span><br />
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<span style="font-size: large;">OK, I'll soften a bit. I can see where a flipping aspects of the classroom might be beneficial. You might want to show students a short video to get them interested in a topic. It could be worth having students view the content ahead of time and then take all of class time to explore. That I get. But, is the flipped classroom really all that different from what we expect in other disciplines? In literature, students read a section of a novel and then come to class prepared to discuss it. That's not flipping the classroom, that's just good teaching. In history, the same thing should happen. Maybe it doesn't, but it should.</span><br />
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<span style="font-size: large;">I don't have any problem with students preparing in advance; that's a very good thing. I don't have any problem with not talking at kids; that's a good thing too. You have to get kids involved in learning! It's not about flipping the classroom; it's about good teaching.</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com2tag:blogger.com,1999:blog-4477518348347161218.post-26994450322596313312015-03-01T19:02:00.003-08:002015-03-01T19:02:31.893-08:00The future classroom<span style="font-size: large;">I've been trying to envision what my classroom might look like in 10 years. It occurred to me today that I have absolutely no idea what might be on the horizon for me. But there are some things that I'd like to happen. </span><br />
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<span style="font-size: large;">First, I think there's a temptation to project physical attributes into the classroom of the future. My ideal classroom has looked pretty much the same in my mind for many years. I would rather have tables and chairs instead of desks and incandescent lights to replace the fluorescent ones. Maybe a wall color that isn't institutional beige and a floor that doesn't look like in belongs in a hospital. Books, games, puzzles, video game consoles, computers, outlets for charging hand-held devices, and some kind of interactive projection technology. Basically, I want a Muggle version of the Gryffindor common room (or, a hipster coffee shop).</span><br />
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<span style="font-size: large;">The physicality of the space is a pipe-dream; it's never going to happen and I know that. The changes to the environment are simply the window dressing for the kind of education with which I'd like to be involved. I want students to collaborate more. They should work on projects that are compelling and authentic. Not necessarily practical or real-world, but interesting.* Students can consult outside sources, in real time or on video, to work on their questions. They can make a record of what they've accomplished, either digitally or manually, so that they know where they've come from and where they might go next. The engage with different kinds of tools to help them make progress in their task. Maybe the tool is a dynamic geometry suite or maybe it is hand-held graphing technology or maybe it is pencil and paper. The tool chosen assists in the task completion. They present their work to their peers, have it constructively and good-naturedly critiqued and then they revise it to make it better. </span><br />
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<span style="font-size: large;">In the best kind of environment my students will near the completion of one task and realize that they have developed more questions to pursue. Wouldn't that be amazing?</span><br />
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<span style="font-size: x-small;">*An awful lot of mathematics was done for purposes of curiosity and only later took on some usefulness. The computer was a thought experiment before someone built one and realized its utility.</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com7tag:blogger.com,1999:blog-4477518348347161218.post-59298174299585347112015-02-22T15:02:00.003-08:002015-02-22T15:02:23.238-08:00Mathematical Knowledge Space<span style="font-size: large;">I've been thinking a lot about Linear Knowledge Space and Random Knowledge Space. (Incidentally, the term "Random" seems to connote haphazard and unconnected, and I don't think that's the point. If I come up with a better term, I'll put it out there.)</span><br />
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<span style="font-size: large;">Linear Knowledge Space seems pretty self-explanatory. Start at the beginning and proceed to the end. This is certainly the most common kind of presentation of mathematics seen in schools. Orderly and hierarchical. Step 1, Step 2, etc. And don't dare miss any steps!</span><br />
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<span style="font-size: large;">This view of mathematical knowledge is the one that permeates the culture. It is popularly believed that "math people"</span><span style="font-size: large;">*</span><span style="font-size: large;"> are rigid and linear and excessively rational in their thoughts and actions. School mathematics often gets framed, or positioned, in these linear knowledge spaces. The conventional wisdom says that in order to learn section 12.5 of the textbook, you must have first mastered chapter 1-11 and sections 12.1-12.4. But, where is the evidence that this is necessary? Check any two textbooks for the same course (at any level) and you will see that there are differences. These differences could be minor, like the order of the topics, or they could be major, such as the philosophical stance on mathematics that the authors take. Regardless, differences exist. If mathematics was linear, there would be only one line!</span><br />
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<span style="font-size: large;">History could be our guide here. Mathematics, like all other human endeavors (and mathematics is as susceptible to humans foibles as any other discipline) is written by the victor. If history of mathematics can teach us anything, it should be that mathematics was developed in fits and starts, through argument, experimentation, and collaboration. A question was posed and people tried to answer it. Newton and Leibniz knew an awful lot about Calculus and they didn't have any idea what a "limit" was. Weierstrass comes along (about 100 years later) and defines "limit" to everyone's satisfaction, despite a century of doing calculus without that definition. What is the first chapter of pretty much every major calculus textbook? Limits. There is absolutely no historical reason for it. We force students to learn linearly that which does not need to be linear. What we need is better questions to answer.</span><br />
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<span style="font-size: large;">"But", you say, "But, certainly there are topics that have prerequisite knowledge." Yes, of course there are. It's pretty hard to learn how to multiply if you are sketchy on what it means to add. However, the analogy that learning mathematics is like building a tower is fundamentally flawed. Learning mathematics is more like building a city. Yes, there are towers. But there are expansions to the towers and there are other buildings, and some of those buildings have additions. And there's an infrastructure of roads, and maybe even tunnels, that connect the various parts of the city. Put another way, mathematics is less like a novel and more like short stories that take place in the same universe. Do you have to have seen episodes 1-99 of "Friends" before watching the 100th one? Certainly not. Could it help you understand something different about the characters if you have seen episodes 1-99? Quite possibly.</span><br />
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<span style="font-size: large;"><a href="http://www.amazon.com/Out-Labyrinth-Setting-Mathematics-Free/dp/1608198707/ref=sr_1_1?s=books&ie=UTF8&qid=1424646070&sr=1-1&keywords=out+of+the+labyrinth">Bob and Ellen Kaplan</a>, whom I consider mentors, have said that when teaching math, if you want the children to learn topic A then you should ask them about topic B where A is necessary. I think this is the "sweet spot" that instruction needs to hit. There are prerequisites, but many fewer than we think there are. Mathematical Knowledge Space isn't necessarily Linear, but it's not completely Random either. It doesn't matter if the instruction is face-to-face or online. What matters is that there is a compelling question, some strategies to answer it, and collaboration. What turns it from the answer to a specific problem into mathematical knowledge is connecting the specific answer to answers of other problems.</span><br />
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*Frankly, that there are "math people" and "non-math people" is a ridiculous notion.Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com1tag:blogger.com,1999:blog-4477518348347161218.post-18848086498123811722015-02-19T18:12:00.000-08:002015-04-07T09:43:50.836-07:00I do like what John Dewey had to say<div class="separator" style="clear: both; text-align: center;">
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<br />Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-73544567078720668872015-02-10T08:57:00.000-08:002015-02-16T13:40:31.966-08:00The most important reason to understand online teaching and learning is...<span style="font-size: large;">... because they exist.</span><br />
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<span style="font-size: large;">As our lives become more and more technologically integrated it becomes vital to understand how people consume, use, and generate information. That is, what do we learn in this increasingly digital age, and how do we learn it.</span><br />
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<i><span style="font-size: large;">How many times has this happened: you are out with friends and you will be involved in a conversation where some details are missing. For example, when a particular event started, or what years a TV show aired, or the name of an actor in a movie, or the title of a sing or book, or how to convert from cups to gallons, or whatever. After a few minutes of banter and deciding that you do not know the answer to the question, someone says, "if only there was a way that we could figure that out". At which point several people take out their phones, retreat into the internet for a few minutes, and emerge with some facts that address the question. </span></i><br />
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<span style="font-size: large;">This is the most obvious (and maybe most prevalent) kind of online learning. The sort where facts can be found relatively quickly. I'm not sure how much teaching is happening in the above scenario. The question I wonder about is whether this fact acquisition is the only type of learning that can happen in online environment. A search engine is much better than a human at coming up with facts. But, a computer is not very good at synthesizing information or asking good questions. It is not even very good at so-called adaptive instruction. (Technically, the program/website/app is not reacting at all. It is only following an algorithm designed by a human.)</span><br />
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<span style="font-size: large;">My questions are: How does a human teacher use technology to enhance learning? How does a teacher use digital technologies to move beyond fact acquisition and procedural knowledge?</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com4tag:blogger.com,1999:blog-4477518348347161218.post-10676832656758552802013-10-29T19:12:00.003-07:002014-11-18T08:42:14.800-08:00Blogging my Guerrilla PLC<span style="font-size: large;">My district has mandatory PLCs but I hate being told what I have to do to grow as a professional. Anyhow, I am rejecting the official PLC and going guerrilla (which is my understanding about how PLCs work best anyway).</span><br />
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<span style="font-size: large;">So, I'm teaching Algebra 2 for the first time in a long time. (Not to get too far off topic here, but I hate it when teachers say, "I haven't taught that in 10 years. I don't know any of it." Really?! You don't remember the big ideas of Algebra 2? I haven't taught it in 10 years and I'm pretty sure that it hasn't changed that much. I get that you might forget a formula here and there, but the entire curriculum?You obviously just memorized it and didn't really know it in the first place.) I remember that Quadratics always give kids a hard time. And, I know that other teachers feel the same way. Fortunately, Twitter to the rescue! James Tanton (who I've reference elsewhere) has a free, online <a href="http://gdaymath.com/courses/quadratics/">Quadratics course</a>. I asked a few colleagues to join me in watching some videos and trying to create a good, cohesive, conceptually based unit on Quadratics. </span><br />
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<span style="font-size: large;">So, we made some goals: (1) We need to create a good unit on Quadratics. (2) Students should understand something about the history of Quadratics, it helps to ground the unit. (3) We're going to watch/do everything ourselves but will probably adapt for our students.</span><br />
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<span style="font-size: large;">This might not be the best goals so far, but I think they'll get adjusted through the process. </span><br />
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<span style="font-size: large;">Then we got down to work. Watched a few (pretty awesome) videos and learned some (pretty awesome) math. Some of it I had seen before, some of it was new to me. I'm excited about the possibility of making something fun and interesting for class. We didn't make it through all that we intended because the practice problems were pretty challenging, both computationally and conceptually. Then we gave ourselves homework. We have to watch three more videos each and do the practice. When we get together next week, we can go over stuff together, share what we've learned and then go on to the next section. </span><br />
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<span style="font-size: large;">I'm pretty pumped. I think this is going to be awesome. And I think to myself, isn't this what teachers in other countries do all the time? Why did I have to carve out personal time to learn something for my job? Whatever... my new motto: when it comes to professional development, go guerrilla!</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-16346958059753332302013-10-24T11:07:00.002-07:002013-10-24T11:07:59.834-07:00Low Expectations for Algebra 2<span style="font-size: large;">First off, I really don't believe in natural ability. When I hear teachers saying things like, "These kids just aren't math people", I want to scream. I understand that there are gifted people out there in the world, but hard-work trumps talent any day. Many kids are mistakenly led to believe that they aren't "math people" by elementary teachers who think that there are "math people". Instead of encouraging productive struggle, we let kids with learning challenges off the hook and make them think that they're stupid. By the time they get to high school, teachers are fighting a battle whose outcome had mostly been decided by a teacher who got to the kind long before you did. It's an unfortunate state of affairs. </span><br />
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<span style="font-size: large;">That said, my Algebra 2 class this year is chock-full of students who consider themselves weak at math ... and, truthfully, they are. I haven't taught Algebra 2 in a few years but I've been in conversation about it with other teachers in my school. In my grand plan for what my Algebra 2 course would look like, I figured that I could run through a quick review of linear functions and systems of linear equations and then plunge headfirst into quadratics before Halloween. Well, Halloween is next week and I just gave a unit assessment on Linear Functions. Now we are going to delve into systems of equations ... but first I thought I'd do something investigative.</span><br />
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<span style="font-size: large;">I feel like I have low expectations for the class. A large percentage of the class are classified Learning Support, which really doesn't bother me very much. And I also don't buy into the idea that I have to go slower for them. Well, maybe slower, but not less rigorous. All students should have access to high quality mathematics. (Do I sound like an ad for the CCSS-M?) I found a good activity in a recent Mathematics Teacher which compared miles per gallon and gallons per hundred miles. I prepared for a large group instruction/discussion and expected the worst. I was sure that I was going to get a barrage of, "this is stupid" or "this is too hard" or "this doesn't make any sense" or "are you going to the football game on friday". Pretty much anything except a high quality discussion about these ratios/measurements. </span><br />
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<span style="font-size: large;">What happened did surprise me. The class was quiet and paying attention. Admittedly, there were only about 4 students actively answering questions, but the remaining ones were either working independently or following along and making sense as we went. There was a lot of head nodding and button pushing and I'm pretty sure that by the end they had a good idea of where we had been and picked up a few skills along the way. We still had to end the discussion 5 minutes before the end of the class because they can't concentrate for 45 straight minutes... but really, who can in 2013? </span><br />
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<span style="font-size: large;">The rest of the discussion, and the real meat of the activity, is tomorrow. If it goes half as well as today, I'll be pretty ecstatic. </span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-4222833110010213132013-10-16T15:12:00.003-07:002013-10-16T15:13:17.552-07:00Why do I need to learn this stuff?<span style="font-family: inherit;"><span style="font-size: large;">Just so that everyone knows that I'm not a total complainer:</span></span><br />
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<span style="font-family: inherit;"><span style="font-size: large;">A few weeks ago, one of my 9th grade students asked me the question I hate to answer: "why do I need to learn this stuff?" First, I gave the terrible reason that he has to know it to pass the state-mandated test at the end of the year. But, because I believe that there's a lot of superfluous content in high school math, I also told him that he might never need to know it ever. I went on to say that because we cannot predict the future, it is impossible for me or for him or for anyone else to know whether or not he will ever need to know it. And, often, when adults are asked, they say that they wish that had paid more attention in school especially in math class. By now I was on a roll so I stepped up on my soapbox and continued. Mathematics is not just about getting right answers. Mathematics is a way to experience the world and a way to think. We, as humans, do two things that are different from all the other animals on the planet: we explore the human condition through history, literature, art, music, etc and we attempt to understand the natural world through chemistry, biology, physics, mathematics… Mathematics helps us think logically and creatively. It gives us a way to explain and explore the patterns present in the universe. Without mathematics, our 21st century way of life does not exist. Without mathematics, beauty does not exist. </span></span><br />
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<span style="font-family: inherit; font-size: large;">At the end of my ranting and raving, another young man raised his hand and said, "Most of the time, when you ask a teacher that question you get a lame answer. But that one was pretty awesome." I hope it sticks with a few of them.</span></div>
Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-53361907797827015392013-10-15T11:18:00.000-07:002013-10-15T11:18:03.987-07:00Formative Assessment<span style="font-size: large;">Each year, my district decides on a focus for the professional development. This year it's formative assessment. I like the idea of formative assessment. I like doing un-graded exit problems, or asking a question and then choosing a student's name from a shuffled deck of index cards, or Poll Everywhere or Bell-ringer/Warm-up questions or even something as mundane as a practice quiz. I don't use it nearly as much as I should, but I like it.</span><br />
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<span style="font-size: large;">Here's the thing that confuses me: Isn't formative assessment just part of good teaching? If you're just standing up there talking at kids and only taking the time to figure out what they know/don't know, what misconceptions there are, what the pre-existing knowledge base is, when you've finished talking, then you're not really doing your job. The advantage of having a real, living, breathing classroom teacher is the ability to collect data and make near-instantaneous changes to improve the learning experience. This is something that a virtual presence (read: online video tutorials) can't do right now.</span><br />
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<span style="font-size: large;">But, I'm over the fact that someone who probably doesn't know as much as I do is going to tell me how to start incorporating formative assessment. The issue I have is that my district is also really big into conformity right now. What I mean is, everyone has to use the same formative assessment. What is the purpose for this? No one can give me an answer beyond, "Everyone should do the same thing" or "Collaboration is good". If we really want everyone to do the same thing, then we should plop our students in front of on-line videos and hope for the best. I agree the collaboration is good, but in my experience, a group of more than 4 math teachers who have been assigned to work together have too many different philosophies of education and mathematics to create a usable product. </span><br />
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<span style="font-size: large;">Moreover, I believe that formative assessment is supposed to give the teacher information about how things are going in your classroom. It should be short and only on 1 topic (2 at most). You need to read the answers yourself to get any information. And the kids should get them back the next day (at the latest) so that they can use your feedback to progress.</span><br />
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<span style="font-size: large;">These are the things I overheard today in the teacher's room:</span><br />
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<span style="font-size: large;">"I had the students grade each others' papers and then I wrote down the score."</span><br />
<span style="font-size: large;">WHAT?! How are you (the teacher) supposed to get any information if you didn't read and evaluate the answers yourself? You've missed the whole point.</span><br />
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<span style="font-size: large;">"I don't understand why it can't be a minor grade for the student."</span><br />
<span style="font-size: large;">WHAT?! The whole purpose of formative assessment is for you (the teacher) and them (the students) to get some information about who knows what and how well. If it's a grade, then you are potentially punishing students for not being proficient when you should help them learn it first. And don't get me started on how <a href="http://mathteacherlament.blogspot.com/2013/10/why-i-love-idea-of-sbg.html">ridiculous grades are anyway</a>. </span><br />
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<span style="font-size: large;">I'm going to keep doing formative assessment because it's helpful and it's important and it's revealing. But I won't give the crappy assessments designed in committee because they're crappy and because they're not going to tell me anything worth knowing. </span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-55332992488987275652013-10-12T08:13:00.002-07:002013-10-14T06:04:51.487-07:00The germ of an idea...<span style="font-size: large;">Full disclosure: I read a lot. Sometimes novels, sometimes magazines, sometimes YA, sometimes SciFi, sometimes philosophy, sometimes books about teaching/education, sometimes books about math, sometimes blogs, sometimes Twitter... pretty much anything (except historical fiction... gotta draw the line somewhere, right?) The point is that when I wonder about something, my default position is to do some reading and then reflect. This post is mostly reflecting about teaching math as it pertains to a specific pedagogy.</span><br />
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<span style="font-size: large;">I've been assessing my own teaching lately and been pretty down about it. I lament the fact that I can't seem to get my 9th grade students to engage with Algebra 1 beyond a "how-to" level. I know it's a process and I know that it'll get better if I plug away at it... but it seems so far away.</span><br />
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<span style="font-size: large;">There has been some success: I just have to remind myself about a student from last year who told me that the only reason that she passed the state test was because when she got to a question that she didn't know how to do, she thought to herself, "What would Mr Mango do?". It was very nice of her to say, but of course, I don't think I deserve that much credit.</span><br />
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<span style="font-size: large;">I digress: I've decided that I need to re-root myself. I'm going to re-read Bob & Ellen Kaplan's book, "<a href="http://www.amazon.com/Out-Labyrinth-Setting-Mathematics-Free/dp/0195368525/ref=sr_1_1?ie=UTF8&qid=1381590139&sr=8-1&keywords=Out+of+the+Labyrinth">Out of the Labyrinth</a>". (Christopher Rice, I love you, but you're going to have wait a few weeks. I'll get to you.) I've been to a few of their training sessions and I've totally drunk the Kool-Aid on Math Circle techniques. But. What Am I Doing Wrong?</span><br />
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<span style="font-size: large;">I think it's that I'm not starting the lesson (planning or instruction) with a question. Bob and Ellen always start their Math Circle sessions with a question; a place from which inquiry can venture forth. I've known for some time, but maybe haven't verbalized the fact that all of the <strike>crap</strike> stuff that we teach kids in HS math is the answer to some question. The teacher's job should be to ask the questions and help kids see where the answers come from... not to tell them the answers and imply that the questions are unimportant. No wonder math is boring for most kids/people. We don't ask enough questions.</span><br />
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<span style="font-size: large;">Why do kids always ask, "Why do I need this?" or "Who made this up?" or "What is this good for?"? Well, not entirely absent from a lot of teaching is the inquiry that precedes the mathematical technique. I'm not asserting that no one will ever ask again "When am I ever going to use this?" but starting with a question might help alleviate some of that. Resolution: I will incite as many lessons as possible with a question.</span><br />
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<span style="font-size: large;">So, I'm embarking on "<a href="http://www.amazon.com/Out-Labyrinth-Setting-Mathematics-Free/dp/0195368525/ref=sr_1_1?ie=UTF8&qid=1381590139&sr=8-1&keywords=Out+of+the+Labyrinth">Out of the Labyrinth</a>" again... for the 4th or 5th time. New insights this time? Pretty sure that a given.</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com0tag:blogger.com,1999:blog-4477518348347161218.post-80618765386935496082013-10-11T15:04:00.001-07:002013-10-11T15:04:17.691-07:00Motivating Taylor Series<span style="font-size: large;">I'm teaching AP Calculus BC for the first time this year. I was sort of excited about it. I taught AB calculus last year and in the school where I teach, BC students have already taken AB. Which means that I get a bunch of (really awesome) kids for second year. Then there was the realization that there actually isn't that much new content that is BC only and the stuff that is new mostly fits into the story you already unfolded last year. At least, that's how I feel right now.</span><br />
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<span style="font-size: large;">Of course, then there are Taylor Series. I only vaguely remembered these from my college courses, and somehow I managed to avoid them in grad school. My recollection is that I had no idea what they were or why they were used. And then there was the fear that I'm going to be forced to, the horror!, tell my students how to do something instead of getting them to explore the idea. So, I took an AP summer institute and it assuaged many of my fears. The instructor was good and did develop the idea of Taylor series and motivated their use. I was pretty excited about it. </span><br />
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<span style="font-size: large;">Then I remembered that I pretty extensively used <a href="http://www.jamestanton.com/">James Tanton's</a> <a href="http://www.lulu.com/spotlight/jtanton">Thinking Mathematics (Vol 6)</a> in developing the AB course that I really liked to teach. Fortunately for me he also has a Volume 7 for BC calculus! He has an excellent chapter (actually, he has lots of excellent chapters, but for the purposes of this lesson, there's only one) on motivating series. It goes something like this:</span><br />
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<span style="font-size: large;">A lot of functions are difficult to evaluate, but polynomials are not one of them. Maybe, I wonder if, we could develop a polynomial that could be used to approximate the values of a not-nice function. He leads the reader through the development of the polynomial, which is really just a series). But what makes Tanton's books awesome is that you feel like he's trying to engage you in a conversation. So my job is taking that conversation apart and making it an activity for students to explore. Right now I'm thinking that I have small groups of 3 or 4. Each group gets a different function (sin, cos, e, ...) and get them thinking: </span><br />
<span style="font-size: large;">What would this mystery approximating polynomial look like if the function and the polynomial only agreed at 1 point? (Who knows? Couldn't it just be a horizontal line?) </span><br />
<span style="font-size: large;">What about the slopes? Should the slopes agree at that one point? (That makes sense... isn't that a derivative?) </span><br />
<span style="font-size: large;">How good is a tangent line at approximating the values of the function? (Good in a small range around the tangent point; pretty bad beyond that.) </span><br />
<span style="font-size: large;">What if you tried to make the function and the polynomial have the same concavity too? (Really? Second derivative? I guess I can try...)</span><br />
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<span style="font-size: large;">Look at the graph of the function and the approximating polynomial. How close are they? Does the polynomial do a good job of estimating the points? By how much? When does the polynomial start to fail? What do you think you could do to get a better polynomial</span><span style="font-size: large;">?</span><br />
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<span style="font-size: large;">Then jigsaw the groups and get some similarities and differences going. </span><br />
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<span style="font-size: large;">That's my idea right now. It'll go through many mental iterations before it goes through a few print alterations. But I'm looking forward to it.</span>Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com3tag:blogger.com,1999:blog-4477518348347161218.post-79824051389459104682013-10-10T14:39:00.000-07:002013-10-11T15:04:55.747-07:00"Well, you're a numbers guy."<span style="font-size: large;">Actually, no, I'm not. I'm not very good at arithmetic and the very idea of statistics (beyond high school level) makes me queasy. I find the theory of numbers fascinating, but that's not the same thing as being a "numbers guy".</span><br />
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<span style="font-size: large;">The assumption made by many of my students and most of the people I meet and a lot of the folks that I interact with daily is that I love everything to do with numbers because I teach math. The numbers part of math is what I find the most boring and mundane. It's the ideas that count. The ideas, the concepts are amazing. The numbers? Hate them.</span><br />
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<span style="font-size: large;">And maybe it's not really so much the numbers as it is the assumption that I both love and excel at arithmetic. I don't. When the check comes when I'm out to dinner with friends, someone always glances my way like I'm supposed to jump at the chance to perform some feat of computation. I routinely joke that I don't do arithmetic on the weekend. Mostly I'm trying to save myself the embarrassment of getting the answer wrong. Then I'll have to endure the taunts of, "You're a math teacher and you can't divide 102 by 7?" You know what? I can't. And I'm OK with that. Somehow, I've managed to graduate from college, get a series of teaching jobs at both secondary and post-secondary levels, acquire a master's degree (in math) and get so close to the completion of a doctorate (in education) that I can almost taste it and I can't do a lot of mental math. But I can help kids learn the ideas that really make up math and let them know that it's OK to suck at arithmetic if you understand the underlying concepts. We have machines that are much better at computations than humans will ever be. But those machines are pretty bad at identifying similarities and creating analogies. That's where a person comes in.</span><br />
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<br />Michael Manganellohttp://www.blogger.com/profile/04011970934231802046noreply@blogger.com1